Topics in Galois Theory

Jean-Pierre Serre

Course at Harvard University, Fall 1988

Notes written by Henri Darmon

Henri Darmon, Mathematics Department, Fine Hall, Princeton University, Princeton NJ 08544, USA.

Jean-Pierre Serre, Collège de France, 3 rue d’Ulm, 75005 Paris, France.

iv Contents

Contents

Foreword ix

Notation xi

Introduction xiii

1 Examples in low degree 1 1.1 The groups Z/2Z, Z/3Z, and S_{3} . . . . . . . . . . . . . . . . . 1 1.2 The group C_{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Application of tori to abelian

Galois groups of exponent 2, 3, 4, 6 . . . . . . . . . . . . . . . 6

2 Nilpotent and solvable groups as Galois groups over Q 9 2.1 A theorem of Scholz-Reichardt . . . . . . . . . . . . . . . . . . 9 2.2 The Frattini subgroup of a finite group . . . . . . . . . . . . . 16

3 Hilbert’s irreducibility theorem 19 3.1 The Hilbert property . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Properties of thin sets . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Extension of scalars . . . . . . . . . . . . . . . . . . . . 21 3.2.2 Intersections with linear subvarieties . . . . . . . . . . 22

3.3 Irreducibility theorem and thin sets . . . . . . . . . . . . . . . 23 3.4 Hilbert’s irreducibility theorem . . . . . . . . . . . . . . . . . 25 3.5 Hilbert property and weak approximation . . . . . . . . . . . 27 3.6 Proofs of prop. 3.5.1 and 3.5.2 . . . . . . . . . . . . . . . . . . 30

4 Galois extensions of Q(T): first examples 35 4.1 The property Gal_{T} . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Example: the quaternion group Q_{8} . . . . . . . . . . . . . . . 38 4.4 Symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5 The alternating group A_{n} . . . . . . . . . . . . . . . . . . . . 43 4.6 Finding good specializations of T . . . . . . . . . . . . . . . . 44 5 Galois extensions of Q(T) given by torsion on elliptic curves 47

5.1 Statement of Shih’s theorem . . . . . . . . . . . . . . . . . . . 47 5.2 An auxiliary construction . . . . . . . . . . . . . . . . . . . . 48 5.3 Proof of Shih’s theorem . . . . . . . . . . . . . . . . . . . . . . 49 5.4 A complement . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.5 Further results on PSL_{2}(F_{q}) and SL_{2}(F_{q}) as Galois groups . . 53

6 Galois extensions of C(T) 55 6.1 The GAGA principle . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Coverings of Riemann surfaces . . . . . . . . . . . . . . . . . . 57 6.3 From C to Q̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.4 Appendix: universal ramified coverings of Riemann surfaces

with signature . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Rigidity and rationality on finite groups 65 7.1 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Counting solutions of equations in finite groups . . . . . . . . 67 7.3 Rigidity of a family of conjugacy classes . . . . . . . . . . . . 70 7.4 Examples of rigidity . . . . . . . . . . . . . . . . . . . . . . . 72

7.4.1 The symmetric group S_{n} . . . . . . . . . . . . . . . . . 72 7.4.2 The alternating group A_{5} . . . . . . . . . . . . . . . . 73 7.4.3 The groups PSL_{2}(F_{p}) . . . . . . . . . . . . . . . . . . 74 7.4.4 The group SL_{2}(F_{8}) . . . . . . . . . . . . . . . . . . . . 75 7.4.5 The Janko group J_{1} . . . . . . . . . . . . . . . . . . . . 77 7.4.6 The Hall-Janko group J_{2} . . . . . . . . . . . . . . . . 78 7.4.7 The Fischer-Griess Monster M . . . . . . . . . . . . . 78

8 Construction of Galois extensions of Q(T)

by the rigidity method 81 8.1 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 Two variants . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.2.1 First variant . . . . . . . . . . . . . . . . . . . . . . . . 83 8.2.2 Second variant . . . . . . . . . . . . . . . . . . . . . . 84

8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.3.1 The symmetric group S_{n} . . . . . . . . . . . . . . . . . 85 8.3.2 The alternating group A_{5} . . . . . . . . . . . . . . . . 86 8.3.3 The group PSL_{2}(F_{p}) . . . . . . . . . . . . . . . . . . . 87 8.3.4 The Gal_{T} property for the smallest simple groups … 88

8.4 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 88 8.4.2 A problem on good reduction . . . . . . . . . . . . . . 89 8.4.3 The real case . . . . . . . . . . . . . . . . . . . . . . . 90 8.4.4 The p-adic case: a theorem of Harbater . . . . . . . . . 93 9 The form Tr(x^{2}) and its applications 95

9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.1.1 Galois cohomology (mod 2) . . . . . . . . . . . . . . . 95 9.1.2 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . 95 9.1.3 Cohomology of S_{n} . . . . . . . . . . . . . . . . . . . . 97

9.2 The quadratic form Tr (x^{2}) . . . . . . . . . . . . . . . . . . . . 98 9.3 Application to extensions with Galois

group Ã_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10 Appendix: the large sieve inequality 103 10.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . 103 10.2 A lemma on finite groups . . . . . . . . . . . . . . . . . . . . . 105 10.3 The Davenport-Halberstam theorem . . . . . . . . . . . . . . 105 10.4 Combining the information . . . . . . . . . . . . . . . . . . . . 107

Bibliography 109

Index 117

viii Contents

Foreword

These notes are based on “Topics in Galois Theory,” a course given by J-P. Serre at Harvard University in the Fall semester of 1988 and written down by H. Darmon. The course focused on the inverse problem of Galois theory: the construction of field extensions having a given finite group G as Galois group, typically over Q but also over fields such as Q(T).

Chapter 1 discusses examples for certain groups G of small order. The method of Scholz and Reichardt, which works over Q when G is a p-group of odd order, is given in chapter 2. Chapter 3 is devoted to the Hilbert irreducibility theorem and its connection with weak approximation and the large sieve inequality. Chapters 4 and 5 describe methods for showing that G is the Galois group of a regular extension of Q(T) (one then says that G has property Gal_{T}). Elementary constructions (e.g. when G is a symmetric or alternating group) are given in chapter 4, while the method of Shih, which works for G = PSL_{2}(p) in some cases, is outlined in chapter 5. Chapter 6 describes the GAGA principle and the relation between the topological and algebraic fundamental groups of complex curves. Chapters 7 and 8 are devoted to the rationality and rigidity criterions and their application to proving the property Gal_{T} for certain groups (notably, many of the sporadic simple groups, including the Fischer-Griess Monster). The relation between the Hasse-Witt invariant of the quadratic form Tr (x^{2}) and certain embedding problems is the topic of chapter 9, and an application to showing that Ã_{n} has property Gal_{T} is given. An appendix (chapter 10) gives a proof of the large sieve inequality used in chapter 3.

The reader should be warned that most proofs only give the main ideas; details have been left out. Moreover, a number of relevant topics have been omitted, for lack of time (and understanding), namely:

a) The theory of generic extensions, cf. [Sa1].

b) Shafarevich’s theorem on the existence of extensions of Q with a given solvable Galois group, cf. [ILK].

c) The Hurwitz schemes which parametrize extensions with a given Galois group and a given ramification structure, cf. [Fr1], [Fr2], [Ma3].

d) The computation of explicit equations for extensions with Galois group PSL_{2}(F_{7}), SL_{2}(F_{8}), M_{11}, …, cf. [LM], [Ma3], [Ma4], [Ml1], …

e) Mestre’s results (not yet published) on extensions of Q(T) with Galois group 6 · A_{6}, 6 · A_{7}, and SL_{2}(F_{7}).

We wish to thank Larry Washington for his helpful comments on an earlier version of these notes.

Paris, August 1991.

H. Darmon  J-P. Serre

Notation

If V is an algebraic variety over the field K, and L is an extension of K, we denote by V (L) the set of L-points of V and by V_{/L} the L-variety obtained from V by base change from K to L. All the varieties are supposed reduced and quasi-projective.

A^{n} is the affine n-space; A^{n}(L) = L^{n}.

P_{n} is the projective n-space; P_{n}(L) = (L^{(n+1)} − {0})/L^{∗}; the group of automorphisms of P_{n} is PGL_{n} = GL_{n}/G_{m}.

If X is a finite set, |X| denotes the cardinality of X.

xii Notation

Introduction

The question of whether all finite groups can occur as Galois groups of an extension of the rationals (known as the inverse problem of Galois theory) is still unsolved, in spite of substantial progress in recent years.

In the 1930’s, Emmy Noether proposed the following strategy to attack the inverse problem [Noe]: by embedding G in S_{n}, the permutation group on n letters, one defines a G-action on the field Q(X_{1}, …, X_{n}) = Q(\underline{X}). Let E be the fixed field under this action. Then Q(\underline{X}) is a Galois extension of E with Galois group G.

In geometric terms, the extension Q(\underline{X}) of E corresponds to the projection of varieties: π : A^{n} ⟶ A^{n}/G, where A^{n} is affine n-space over Q. Let P be a Q-rational point of A^{n}/G for which π is unramified, and lift it to Q ∈ A^{n}(Q̄). The conjugates of Q under the action of Gal(Q̄/Q) are the sQ where s ∈ H_{Q} ⊂ G, and H_{Q} is the decomposition group at Q. If H_{Q} = G, then Q generates a field extension of Q with Galois group G.

A variety is said to be rational over Q (or Q- rational) if it is birationally isomorphic over Q to the affine space A^{n} for some n, or equivalently, if its function field is isomorphic to Q(T_{1}, …, T_{n}), where the T_{i} are indeterminates.

Theorem 1 (Hilbert, [Hi]) If A^{n}/G is Q-rational, then there are infinitely many points P, Q as above such that H_{Q} = G.

This follows from Hilbert’s irreducibility theorem, cf. §3.4.

Example: Let G = S_{n}, acting on Q(X_{1}, …, X_{n}). The field E of S_{n}-invariants is Q(T_{1}, …, T_{n}), where T_{i} is the ith symmetric polynomial, and Q(X_{1}, …, X_{n}) has Galois group S_{n} over E: it is the splitting field of the polynomial

X^{n} − T_{1}X^{n−1} + T_{2}X^{n−2} + ⋯ + (−1)^{n}T_{n}.

Hilbert’s irreducibility theorem says that the T_{i} can be specialized to infinitely many values t_{i} ∈ Q (or even t_{i} ∈ Z) such that the equation

X^{n} − t_{1}X^{n−1} + t_{2}X^{n−2} + ⋯ + (−1)^{n}t_{n} = 0

has Galois group S_{n} over Q. In fact, “most” t_{i} work: of the N^{n} n-tuples (t_{i}) with t_{i} ∈ Z, 1 ≤ t_{i} ≤ N, only O(N^{n− \frac{1}{2}} log N) may fail to give S_{n}, cf. [Ga], [Coh], [Se9].

In addition to the symmetric groups, the method works for the alternating groups A_{n} with n ≤ 5, cf. [Mae] (For n ≥ 6, it is not known whether the field of A_{n}-invariants is rational.) Somewhat surprisingly, there are groups for which the method fails (i.e. A^{n}/G is not Q-rational):

• Swan [Sw1] has shown that the field of G-invariants is not rational when G is a cyclic group of order 47. The obstruction is related to the automorphism group of G which is a cyclic group of order 46 = 2 × 23, and to the fact that Q(ζ_{23}) does not have class number 1 (since h(−23) = 3).

• In [Le] H. Lenstra gives a general criterion for the field of G-invariants to be rational when G is an abelian group: in particular, he shows that this criterion is not satisfied when G is cyclic of order 8.

(The above counter-examples are over Q. Counter-examples over C (involving a non-abelian group G) are given by the following result of Saltman [Sa2]: if there is a non-zero α ∈ H^{2}(G, Q/Z) such that Res_{H}^{G}(α) = 0 for all abelian subgroups H generated by two elements, then A^{n}/G is not C-rational. It is not hard to construct groups G satisfying the hypothesis of Saltman’s theorem: for example, one may take a suitable extension of abelian groups of type (p, …, p).)

It is easy to see (e.g., using the normal basis theorem) that the covering map

π : A^{n} ⟶ A^{n}/G

is generic (or versal) in the sense that every extension of Q (or of any field of characteristic zero) with Galois group G can be obtained by taking the π-fibre of a rational point of A^{n}/G over which π is unramified. Hence, if A^{n}/G is Qrational, then the set of all G-extensions of Q can be described by a system of n rational parameters. Such a parametrization implies the following property of extensions with Galois group G [Sa1]:

Theorem 2 Assume A^{n}/G is Q-rational. Let {p_{i}} be a finite set of primes, L_{i} extensions of Q_{p_{i}} with Galois group G. Then there is an extension L of Q with Gal(L/Q) = G such that L ⊗ Q_{p_{i}} = L_{i}.

Remark: There is a more general statement, where the L_{i} are allowed to be Galois algebras, and Q is replaced by a field endowed with finitely many independent absolute values.

Proof (sketch): Each L_{i} is parametrized by (\underline{X}^{(i)}) ∈ A^{n}(Q_{p_{i}}). A “global” parameter (\underline{X}) ∈ A^{n}(Q) which is sufficiently close to each of the (\underline{X}^{(i)}) in the Q_{p_{i}}-topology gives an extension of Q with group G having the desired local behaviour (Krasner’s lemma). QED.

The cyclic group of order 8 does not satisfy the property of th. 2. Indeed, if L_{2}/Q_{2} is the unique unramified extension of Q_{2} of degree 8, there is no cyclic extension L of degree 8 over Q such that L_{2} ≃ L ⊗ Q_{2} (an easy exercise on characters, see [Wa]).

One could perhaps extend Hilbert’s theorem to a more general class of varieties. There is an interesting suggestion of Ekedahl and Colliot-Thélène in this direction [Ek], [CT] (see §3.5).

Since A^{n}/G is not always Q-rational, one has to settle for less:

Question: If G is a finite group, can it be realized as a Galois group of some regular extension F of Q(T)? (Recall that “F is regular” means that F ∩ Q̄ = Q.)

Remarks:

1. If F is a function field of a variety V defined over Q, then F is regular if and only if V is absolutely irreducible. The regularity assumption is included to rule out uninteresting examples such as the extension E(T) of Q(T) where E is a Galois extension of Q.

2. If such an F exists, then there are infinitely many linearly disjoint extensions of Q with Galois group G.

The existence of regular extensions of Q(T) with Galois group G is known when G is:

• Abelian;

• One of the 26 sporadic simple groups, (with the possible exception of the Mathieu group M_{23}); •PSL_{2}(F_{p}), where at least one of (\frac{2}{p}), (\frac{3}{p}), (\frac{7}{p}) is −1 [Shih1], [Shih2];

• A_{n}, or S_{n}, cf. [Hi];

• Ã_{n}, cf. N. Vila [Vi] and J-F. Mestre [Me2];

• G_{2}(F_{p}) [Th2], where G_{2} is the automorphism group of the octonions, and the list is not exhaustive.

The method for finding F proceeds as follows:

1. Construction (by analytic and topological methods) of an extension F_{C} of C(T) with Galois group G.

2. A descent from C to Q. This is the hardest part, and requires that G satisfies a so-called rigidity criterion.

The outline of the course will be:

xvi Introduction

1. Elementary examples, and the Scholz-Reichardt theorem.

2. Hilbert’s irreducibility theorem and applications.

3. The “rigidity method” used to obtain extensions of Q(T) with given Galois groups.

4. The quadratic form x ↦ Tr (x^{2}), and its applications to embedding problems, e.g., construction of extensions with Galois group Ã_{n}.

Chapter 1

Examples in low degree

1.1 The groups Z/2Z, Z/3Z, and S_{3}

• G = Z/2Z: all quadratic extensions can be obtained by taking square roots: the map P_{1} ⟶ P_{1} given by X ↦ X^{2} is generic (in characteristic different from 2 - for a characteristic-free equation, one should use X^{2} − T X + 1 = 0 instead).

• G = Z/3Z: A “generic equation” for G is:

X^{3} − T X^{2} + (T − 3)X + 1 = 0,

with discriminant Δ = (T^{2} − 3T + 9)^{2}. (In characteristic 3, this reduces to the Artin-Schreier equation Y^{3} − Y = −1/T by putting Y = 1/(X + 1).) The group G acts on P_{1} by

σX = \frac{1}{1 − X} ,

where σ is a generator of G. The function

T = X + σX + σ^{2}X = \frac{X^{3} − 3X + 1}{X^{2} − X}

is G-invariant and gives a map Y = P_{1} ⟶ P_{1}/G. To check genericity, observe that any extension L/K with cyclic Galois group of order 3 defines a homomorphism ϕ : G_{K} ⟶ G ⟶ Aut Y which can be viewed as a 1-cocycle with values in Aut Y. The extension L/K is given by a rational point on P_{1}/G if and only if the twist of Y by this cocycle has a rational point not invariant by σ. This is a general property of Galois twists. But this twist has a rational point over a cubic extension of K, and every curve of genus 0 which has a point over an odd-degree extension is a projective line, and hence has at least one rational point distinct from the ones fixed by σ.

• G = S_{3}: The map

S_{3} ↪ GL_{2} ⟶ PGL_{2} = Aut (P_{1})

gives a projection

P_{1} ⟶ P_{1}/S_{3} = P_{1}

which is generic, although the reasoning for C_{3} cannot be applied, as the order of S_{3} is even. But S_{3} can be lifted from PGL_{2} to GL_{2}, and the vanishing of H^{1}(G_{K}, GL_{2}) can be used to show that

P_{1} ⟶ P_{1}/S_{3}

is generic.

Exercise: Using the above construction (or a direct argument), show that every separable cubic extension of K is given by an equation of the form

X^{3} + T X + T = 0, with T ≠ 0, −27/4.

1.2 The group C_{4}

Let K_{4}/K be Galois and cyclic of degree 4, and suppose that Char K ≠ 2. The extension K_{4} is obtained from a unique tower of quadratic extensions:

K ⊂ K_{2} ⊂ K_{4},

where K_{2} = K(\sqrt{ϵ}), and K_{4} = K_{2}(\sqrt{a + b\sqrt{ϵ}}).

Conversely, let K_{2} = K(\sqrt{ϵ}) be a quadratic extension of K, where ϵ ∈ K^{∗} is not a square. If a, b ∈ K and K_{4} = K_{2}(\sqrt{a + b\sqrt{ϵ}}), then K_{4} may not be Galois over K (its Galois closure could have Galois group isomorphic to D_{4}, the dihedral group of order 8).

Theorem 1.2.1 The field K_{4} is cyclic of degree 4 if and only if a^{2} − ϵb^{2} = ϵc^{2} for some c ∈ K^{∗}.

Proof: Let G be a group, ϵ a non-trivial homomorphism from G to Z/2Z, and χ a homomorphism from H = Ker ϵ to Z/2Z. Let H_{χ} denote the kernel of χ.

Lemma 1.2.2 The following are equivalent :

(a) H_{χ} is normal in G, and G/H_{χ} is cyclic of order 4.

1.2. The group C_{4} 3

(b) Cor_{H}^{G}χ = ϵ, where Cor_{H}^{G} is the corestriction map.

(We abbreviate H^{1}(G, Z/2Z) = Hom(G, Z/2Z) to H^{1}(G). The corestriction map H^{1}(H) ⟶ H^{1}(G) can be defined by

(Cor_{H}^{G}χ)(g) = χ(Ver_{H}^{G}g),

where Ver_{H}^{G} : G/(G, G) ⟶ H/(H, H) is the transfer.)

The proof that (a) ⇒ (b) is immediate: replacing G by G/H_{χ}, it suffices to check that the transfer C_{4} ⟶ C_{2} is onto: but this map is given by s ↦ s^{2}.

Now, assume (b). Select s ∈ G − H. The transfer is given by:

Ver_{H}^{G}(h) = h · shs^{−1} mod (H, H).

Hence for all h ∈ H:

χ(Ver_{H}^{G}h) = χ(h) + χ(shs^{−1}) = ϵ(h) ≡ 0 (mod 2).

But if h ∈ H_{χ}, then χ(h) = 0. It follows that χ(shs^{−1}) = 0, so that H_{χ} is normal in G. Now, applying the hypothesis to s shows

χ(s^{2}) = Cor_{H}^{G}χ(s) = ϵ(s) ≡ 1 (mod 2),

so s^{2} ≠ 1 (mod H_{χ}). It follows that G/H_{χ} is cyclic of order 4, and this completes the proof of lemma 1.2.2.

Now, let G = G_{K} = Gal(K̄/K). The extensions K_{2} and K_{4} define homomorphisms ϵ and χ as in the lemma. Via the identification of H^{1}(G_{K}) with K^{∗}/K^{∗2}, the corestriction map Cor : H^{1}(G_{K_{2}}) ⟶ H^{1}(G_{K}) is equal to the norm, and the criterion Cor_{H}^{G}χ = ϵ becomes:

N(a + b\sqrt{ϵ}) = ϵc^{2},

where c ∈ K^{∗}. This completes the proof of th. 1.2.1.

Remark: In characteristic 2, Artin-Schreier theory gives an isomorphism H^{1}(G_{K}) ≃ K/℘K, where ℘x = x^{2} + x, and the corestriction map corresponds to the trace. Hence the analogue of th. 1.2.1 in characteristic 2 is:

Theorem 1.2.3 Suppose Char K = 2, and let K_{2} = K(x), K_{4} = K_{2}(y), where ℘x = ϵ, ℘y = a + bx. Then K_{4} is Galois over K and cyclic of degree 4 if and only if Tr (a + bx)(= b) is of the form ϵ + z^{2} + z, with z ∈ K.

Observe that the variables ϵ, a, z of th. 1.2.3 parametrize C_{4}-extensions of K. In particular, it is possible in characteristic 2 to embed any quadratic extension in a cyclic extension of degree 4. This is a special case of a general result: the embedding problem for p-groups always has a solution in characteristic p (as can be seen from the triviality of H^{2}(G, P) when G is the absolute Galois group of a field of characteristic p and P is an abelian p-group with G-action. See for example [Se1].)

The situation is different in characteristic ≠ 2: the criterion a^{2} − b^{2}ϵ = ϵc^{2} implies that ϵ must be a sum of 2 squares in K: if b^{2} + c^{2} ≠ 0, then:

ϵ = (\frac{ab}{b^{2} + c^{2}} )^{2} + (\frac{ac}{b^{2} + c^{2}} )^{2} .

Otherwise \sqrt{−1} ∈ K, and any element of K can be expressed as a sum of 2 squares. Conversely, if ϵ is the sum of two squares, ϵ = λ^{2} + μ^{2}, then setting

a = λ^{2} + μ^{2}, b = λ, c = μ,

solves the equation a^{2} − b^{2}ϵ = c^{2}ϵ. Hence we have shown:

Theorem 1.2.4 A quadratic extension K(\sqrt{ϵ}) can be embedded in a cyclic extension of degree 4 if and only if ϵ is a sum of two squares in K.

Here is an alternate proof of th. 1.2.4: the quadratic extension K_{2} can be embedded in a cyclic extension K_{4} of degree 4 if and only if the homomorphism ϵ : G_{K} ⟶ Z/2Z given by K_{2} factors through a homomorphism G_{K} ⟶ Z/4Z. This suggests that one apply Galois cohomology to the sequence:

0 ⟶ Z/2Z ⟶ Z/4Z ⟶ Z/2Z ⟶ 0,

obtaining:

H^{1}(G_{K}, Z/4Z) ⟶ H^{1}(G_{K}, Z/2Z) ⟶^{δ} H^{2}(G_{K}, Z/2Z).

The obstruction to lifting ϵ ∈ H^{1}(G_{K}, Z/2Z) to H^{1}(G_{K}, Z/4Z) is given by δϵ ∈ H^{2}(G_{K}, Z/2Z) = Br_{2}(K), where Br_{2}(K) denotes the 2-torsion in the Brauer group of K. It is well-known that the connecting homomorphism δ : H^{1} ⟶ H^{2}, also known as the Bockstein map, is given by δx = x · x (cup-product). This can be proved by computing on the “universal example” P_{∞}(R) = K(Z/2Z, 1) which is the classifying space for Z/2Z. The cup product can be computed by the formula:

α · β = (α, β),

where H^{1}(G, Z/2Z) is identified with K^{∗}/K^{∗2} and (α, β) denotes the class of the quaternion algebra given by

i^{2} = α, j^{2} = β, ij = −ji.

But (ϵ, −ϵ) = 0 (in additive notation), so (ϵ, ϵ) = (−1, ϵ). Hence, δϵ is 0 if and only if (−1, ϵ) = 0, i.e., ϵ is a sum of two squares in K.

Similarly, one could ask when the extension K_{4} can be embedded in a cyclic extension of degree 8. The obstruction is again given by an element of Br_{2}(K). One can prove (e.g., by using [Se6]):

Theorem 1.2.5 The obstruction to embedding the cyclic extension K_{4} in a cyclic extension of degree 8 is given by the class of (2, ϵ) + (−1, a) in Br_{2}(K), if a ≠ 0, and by the class of (2, ϵ) if a = 0.

Hence, when a ≠ 0, the C_{8}-embedding problem is possible if and only if the quaternion algebra (2, ϵ) is isomorphic to (−1, a).

There is also a direct proof of th. 1.2.1. Let K_{2} and K_{4} be as before, with K_{2} = K(\sqrt{ϵ}), K_{4} = K_{2}(\sqrt{a + b\sqrt{ϵ}}). Let x = \sqrt{a + b\sqrt{ϵ}} and y = \sqrt{a − b\sqrt{ϵ}}. If Gal(K_{4}/K) = C_{4}, then we may choose a generator σ of Gal(K_{4}/K) taking x to y, and hence y to −x. Setting

c = xy/\sqrt{ϵ},

we have σc = y(−x)/(−\sqrt{ϵ}) = c, so c ∈ K^{∗}. Also

ϵc^{2} = x^{2}y^{2} = a^{2} − b^{2}ϵ,

and one obtains the same criterion as before. Conversely, if a, b, c, ϵ satisfy the equation a^{2} − b^{2}ϵ = c^{2}ϵ, one verifies that K_{4} is a cyclic extension with Galois group C_{4}.

Remarks:

1. The minimal polynomial for x over K is

X^{4} + AX^{2} + B = 0,

where A = 2a, B = a^{2} − ϵb^{2}. The condition for a general polynomial of this form to have Galois group C_{4} is that A^{2} − 4B is not a square and that

(\frac{A^{2} − 4B}{B} ) ∈ K^{∗2}.

2. The C_{4}-extensions are parametrized by the solutions (ϵ, a, b, u) of the equation

a^{2} − ϵb^{2} = ϵu^{2},

with u ≠ 0 and ϵ not a square. This represents a rational variety: one can solve for ϵ in terms of a, b, and u. Hence the class of C_{4}-extensions of Q satisfies the conclusion of th. 2: there are C_{4}-extensions of Q with arbitrarily prescribed local behaviour at finitely many places; recall that this is not true for the cyclic group of order 8.

Exercises:

1. The group C_{4} acts faithfully on P_{1} via the map C_{4} ⟶ PGL_{2} which sends a

generator σ of C_{4} to ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} ). The corresponding map P_{1} ⟶ P_{1}/C_{4} is given

by z ↦ (z^{4} + 6z^{2} + 1)/(z(z^{2} − 1)). This gives rise to the equation:

Z^{4} − T Z^{3} + 6Z^{2} + T Z + 1 = 0

with Galois group C_{4} over Q(T). 1.1 If i ∈ K, show that this equation is generic: in fact, it is equivalent to the Kummer equation. 1.2 If i ∉ K, show that there does not exist any one-dimensional generic family for C_{4}-extensions. 1.3 If a C_{4}-extension is described as before by parameters ϵ, a, b and c, show that it comes from the equation above if and only if (−1, a) = 0 or a = 0.

2. Assume K contains a primitive 2^{n}-th root of unity z. Let L = K( ^{2^{n}} \sqrt{a}) be a cyclic extension of K of degree 2^{n}. Show that the obstruction to the embedding of L in a cyclic extension of degree 2^{n+1} is (a, z) in Br_{2}(K).

1.3 Application of tori to abelian Galois groups of exponent 2, 3, 4, 6

A K-torus is an algebraic group over K which becomes isomorphic to a product of multiplicative groups G_{m} × … × G_{m} over the algebraic closure K̄ of K. If this isomorphism is defined over K, then the torus is said to be split.

Let T be a K-torus and denote by X(T) its character group,

X(T) = HomK̄(T, G_{m}).

It is well known that X(T) is a free Z-module of rank n = dim T endowed with the natural action of G_{K}. The functor T ↦ X(T) defines an anti-equivalence between the category of finite dimensional tori over K and the category of free Z-modules of finite rank with G_{K} action.

A split K-torus is clearly a K-rational variety; the same holds for tori which split over a quadratic extension K′ of K. This follows from the classification of tori which split over a quadratic extension (whose proof we shall omit - see [CR]):

Lemma 1.3.1 A free Z-module of finite rank with an action of Z/2Z is a direct sum of indecomposable modules of the form :

1.3. Application of tori to abelian Galois groups of exponent 2, 3, 4, 6 7

1. Z with trivial action.

2. Z with the non-trivial action.

3. Z × Z with the “regular representation” of Z/2Z which interchanges the two factors.

The corresponding tori are :

1. G_{m}

2. A “twisted form ” of G_{m}, which corresponds to elements of norm 1 in (3) below.

3. The algebraic group R_{K′/K}G_{m} obtained from G_{m}_{/K′} by “restriction of scalars” to K ( cf. §3.2.1).

It is not difficult to show that the three cases give rise to K-rational varieties, and the result follows.

If G is a finite group, the group G of invertible elements of the group algebra Λ = K[G] defines an algebraic group over K. In characteristic 0, we have

G ≃ ∏ GL_{n_{i}} over K̄,

where the product is taken over all irreducible representations of G and the n_{i} denote the dimensions of these representations.

In particular, if G is commutative, then G is a torus with character group Z[Ĝ], where Ĝ = HomK̄(G, G_{m}). Therefore, G splits over the field generated by the values of the characters of G. There is an exact sequence of algebraic groups:

1 ⟶ G ⟶ G ⟶ G/G ⟶ 1,

and the covering map G ⟶ G/G is generic for extensions of K with Galois group G. If G is of exponent 2,3, 4 or 6, then G splits over a quadratic extension, since the characters values lie in Q, Q(\sqrt{3}), or Q(i). By the previous result, G - and hence a fortiori G/G - is Q-rational. So the abelian groups of exponent 2,3,4 or 6 yield to Noether’s method (but not those of exponent 8).

Exercise: Show that all tori decomposed by a cyclic extension of degree 4 are rational varieties, by making a list of indecomposable integer representations of the cyclic group of order 4 (there are nine of these, of degrees 1, 1, 2, 2, 3, 3, 4, 4, 4). See [Vo2].

8 Chapter 1. Examples in low degree

Nilpotent and solvable groups as Galois groups over Q

2.1 A theorem of Scholz-Reichardt

Our goal will be to prove the following theorem which is due to Scholz and Reichardt [Re]:

Theorem 2.1.1 Every l-group, l ≠ 2, can be realized as a Galois group over Q. (Equivalently, every finite nilpotent group of odd order is a Galois group over Q.)

Remarks:

1. This is a special case of a theorem of Shafarevich: every solvable group can be realized as a Galois group over Q. [The proofs of that theorem given in [Sha1] and [Is] are known to contain a mistake relative to the prime 2 (see [Sha3]). In the notes appended to his Collected Papers, p.752, Shafarevich sketches a method to correct this. See also [ILK], ch. 5.]

2. The proof yields somewhat more than the statement of the theorem. For example, if |G| = l^{N}, then the extension of Q with Galois group G can be chosen to be ramified at at most N primes. It also follows from the proof that any separable pro-l-group of finite exponent is a Galois group over Q.

3. The proof does not work for l = 2. It would be interesting to see if there is a way of adapting it to this case.

4. It is not known whether there is a regular Galois extension of Q(T) with Galois group G for an arbitrary l-group G.

An l-group can be built up from a series of central extensions by groups of order l. The natural approach to the problem of realizing an l-group G as a Galois group over Q is to construct a tower of extensions of degree l which ultimately give the desired G-extension. When carried out naively, this approach does not work, because the embedding problem cannot always be solved. The idea of Scholz and Reichardt is to introduce more stringent conditions on the extensions which are made at each stage, ensuring that the embedding problem has a positive answer.

Let K/Q be an extension with Galois group G, where G is an l-group. Choose N ≥ 1 such that l^{N} is a multiple of the exponent of G, i.e., s^{l^{N}} = 1 for all s ∈ G. The property introduced by Scholz is the following:

Definition 2.1.2 The extension L/Q is said to have property (S_{N}) if every prime p which is ramified in L/Q satisfies :

1. p ≡ 1 (mod l^{N}).

2. If v is a place of L dividing p, the inertia group I_{v} at v is equal to the decomposition group D_{v}.

Condition 2 is equivalent to saying that the local extension L_{v}/Q_{p} is totally ramified, or that its residue field is F_{p}.

Now, let

1 ⟶ C_{l} ⟶ G̃ ⟶ G ⟶ 1

be an exact sequence of l-groups with C_{l} central, cyclic of order l. The “embedding problem” for G̃ is to find a Galois extension L̃ of K containing L, with isomorphisms Gal(L̃/L) ≃ C_{l} and Gal(L̃/K) ≃ G̃ such that the diagram

1 ⟶  C_{l}  ⟶  G̃  ⟶  G  ⟶ 1
||  ||  ||
1 ⟶ Gal(L̃/L) ⟶ Gal(L̃/K) ⟶ Gal(L/K) ⟶ 1

is commutative.

The Scholz-Reichardt theorem is a consequence of the following (applied inductively):

Theorem 2.1.3 Let L/Q be Galois with Galois group G, and assume that L has property (S_{N}). Assume further that l^{N} is a multiple of the exponent of G̃. Then the embedding problem for L and G̃ has a solution L̃, which satisfies (S_{N}) and is ramified at at most one more prime than L. (Furthermore, one can require that this prime be taken from any set of prime numbers of density one.)

The proof of th. 2.1.3 will be divided into two parts: first, for split extensions, then for non-split ones.

First part: the case G̃ ≃ G × C_{l}

Let (p_{1}, …, p_{m}) be the prime numbers ramified in L. Select a prime number q with the following properties:

1. q ≡ 1 (mod l^{N}), 2. q splits completely in the extension L/Q,

3. Every prime p_{i}, (1 ≤ i ≤ m) is an l-th power in F_{q}.

Taken together, these conditions mean that the prime q splits completely in the field L( ^{l^{N}} \sqrt{1}, ^{l} \sqrt{p_{1}}, …, ^{l} \sqrt{p_{m}}). The following well-known lemma guarantees the existence of such a q:

Lemma 2.1.4 If E/Q is a finite extension of Q, then there are infinitely many primes which split completely in E. In fact, every set of density one contains such a prime.

Proof: The second statement in the lemma is a consequence of Chebotarev’s density theorem; the first part can be proved by a direct argument, without invoking Chebotarev. Assume E is Galois, and let f be a minimal polynomial with integral coefficients of a primitive element of E. Suppose there are only finitely many primes p_{i} which split completely in E or are ramified. Then f(x) is of the form ±p_{1}^{m_{1}} … p_{k}^{m_{k}}, for x ∈ Z. When x is between 1 and X, the number of distinct values taken by f(x) is at least \frac{1}{n}X. But the number of values of f(x) which can be written in the form ±p_{1}^{m_{1}} … p_{k}^{m_{k}} is bounded by a power of log X. This yields a contradiction.

Having chosen a q which satisfies the conditions above, fix a surjective homomorphism

λ : (Z/qZ)^{∗} ⟶ C_{l}.

(Such a λ exists because q ≡ 1 (mod l).) We view λ as a Galois character. This defines a C_{l}-extension M_{λ} of Q which is ramified only at q, and is linearly disjoint from L. The compositum LM_{λ} therefore has Galois group G̃ = G×C_{l}. Let us check that LM_{λ} satisfies property (S_{N}). By our choice of q, we have q ≡ 1 (mod l^{N}). It remains to show that I_{v} = D_{v} at all ramified primes. If p is ramified in L/Q, it splits completely in M_{λ}, and hence D_{v} = I_{v} for all primes v|p. The only prime ramified in M_{λ} is q, and q splits completely in L by assumption. Hence, for all primes v which are ramified in LM_{λ}, we have D_{v} = I_{v} as desired.

Second part: the case where G̃ is a non-split extension

The proof will be carried out in three stages:

(i) Existence of an extension L̃ giving a solution to the embedding problem.

(ii) Modifying L̃ so that it is ramified at the same places as L.

(iii) Modifying L̃ further so that it has property (S_{N}), with at most one additional ramified prime.

(i) Solvability of the embedding problem

The field extension L determines a surjective homomorphism ϕ : G_{Q} → G. The problem is to lift ϕ to a homomorphism ϕ̃ : G_{Q} → G̃. (Such a ϕ̃ is automatically surjective because of our assumption that G̃ does not split.) Let ξ ∈ H^{2}(G, C_{l}) be the class of the extension G̃, and let

ϕ^{∗} : H^{2}(G, C_{l}) ⟶ H^{2}(G_{Q}, C_{l})

be the homomorphism defined by ϕ. The existence of the lifting ϕ̃ is equivalent to the vanishing of ϕ^{∗}(ξ) in H^{2}(G_{Q}, C_{l}). As usual in Galois cohomology, we write H^{2}(G_{Q}, −) as H^{2}(Q, −), and similarly for other fields. The following well-known lemma reduces the statement ϕ^{∗}ξ = 0 to a purely local question:

Lemma 2.1.5 The restriction map

H^{2}(Q, C_{l}) ⟶ ∏_{p} H^{2}(Q_{p}, C_{l})

is injective.

(A similar result holds for any number field.)

Sketch of Proof: Let K = Q(μ_{l}). Since [K : Q] is prime to l, the map H^{2}(Q, C_{l}) ⟶ H^{2}(K, C_{l}) is injective. Hence, it is enough to prove the lemma with Q replaced by K. In that case, H^{2}(K, C_{l}) is isomorphic to Br_{l}(K), the l-torsion of the Brauer group of K. The lemma then follows from the Brauer-Hasse-Noether theorem: an element of Br(K) which is 0 locally is 0. (Note that, since l ≠ 2, the archimedean places can be ignored.)

By the above lemma, it suffices to show that ϕ^{∗}ξ = 0 locally at all primes. In other words, we must lift the map ϕ_{p} : G_{Q_{p}} ⟶ D_{p} ⊂ G to ϕ̃_{p} : G_{Q_{p}} ⟶ G̃. There are two cases:

1. p is unramified in L, i.e., ϕ_{p} is trivial on the inertia group I_{p} of G_{Q_{p}}. Then ϕ_{p} factors through the quotient G_{Q_{p}}/I_{p} = Ẑ. But one can always lift a map Ẑ ⟶ G to a map Ẑ ⟶ G̃: just lift the generator of Ẑ.

2. p is ramified in L. By construction, p ≡ 1 (mod l^{N}), hence p ≠ l and L_{v}/Q_{p} is tamely ramified (as in 2.1.2, v denotes a place of L above p); since its Galois group D_{v} is equal to its inertia group I_{v}, it is cyclic. The homomorphism G_{Q_{p}} ⟶ D_{v} ⊂ G factors through the map G_{Q_{p}} ⟶ Gal(E/Q_{p}), where E is the maximal abelian tame extension of Q_{p} with exponent dividing l^{N}. The extension E can be described explicitly: it is composed of the unique unramified extension of Q_{p} of degree l^{N}, (obtained by taking the fraction field of the ring of Witt vectors over F_{p^{l^{N}}}) and the totally ramified extension Q_{p}( ^{l^{N}} \sqrt{p}) (which is a Kummer extension since p ≡ 1 (mod l^{N}) ). It follows that Gal(E/Q_{p}) is an abelian group of type (l^{N}, l^{N}); it is projective in the category of abelian groups of exponent dividing l^{N}. The inverse image of D_{v} in G̃ belongs to that category (a central extension of a cyclic group is abelian). This shows that the local lifting is possible.

(ii) Modifying the extension L̃ so that it becomes unramified outside the set ram(L/Q) of primes ramified in L/Q

Lemma 2.1.6 For every prime p, let ϵ_{p} be a continuous homomorphism from Gal(Q̄_{p}/Q_{p}) to a finite abelian group C. Suppose that almost all ϵ_{p} are unramified. Then there is a unique ϵ : Gal(Q̄/Q) ⟶ C, such that for all p, the maps ϵ and ϵ_{p} agree on the inertia groups I_{p}.

(The decomposition and inertia groups D_{p}, I_{p} are only defined up to conjugacy inside G_{Q}. We shall implicitly assume throughout that a fixed place v has been chosen above each p, so that D_{p} and I_{p} are well-defined subgroups of G_{Q}. )

Proof of lemma: By local class field theory, the ϵ_{p} can be canonically identified with maps Q_{p}^{∗} ⟶ C. The restrictions of ϵ_{p} to Z_{p}^{∗} are trivial on a closed subgroup 1 + p^{n_{p}}Z_{p}, where n_{p} is the conductor of ϵ_{p}. Since almost all n_{p} are zero, there is a homomorphism ϵ : (Z/MZ)^{∗} ⟶ C, with M = ∏ p^{n_{p}}, and ϵ(k) = ∏ ϵ_{p}(k^{−1}). If we view ϵ as a Galois character, class field theory shows that it has the required properties. (Equivalently, one may use the direct product decomposition of the idèle group I_{Q} of Q, as:

I_{Q} = (∏_{p} Z_{p}^{∗} × R_{+}^{∗}) × Q^{∗}.)

Proposition 2.1.7 Let 1 → C → Φ̃ → Φ → 1 be a central extension of a group Φ, and ϕ be a continuous homomorphism from G_{Q} to Φ which has a lifting ψ : G_{Q} ⟶ Φ̃. Let ϕ̃_{p} : G_{Q_{p}} ⟶ Φ̃ be liftings of ϕ_{p} = ϕ|_{D_{p}}, such that the ϕ̃_{p} are unramified for almost all p. Then there is a lifting ϕ̃ : G_{Q} ⟶ Φ̃ such that, for every p, ϕ̃ is equal to ϕ̃_{p} on the inertia group at p. Such a lifting is unique.

This proposition is also useful for relating Galois representations in GL_{n} and PGL_{n} (Tate, see [Se7, §6]).)

Proof of prop. 2.1.7: For every p, there is a unique homomorphism

ϵ_{p} : G_{Q_{p}} ⟶ C

such that

ψ(s) = ϵ_{p}(s)ϕ̃_{p}(s)

for all s ∈ G_{Q_{p}}. By the previous lemma, there exists a unique ϵ : G_{Q} ⟶ C which agrees with ϵ_{p} on I_{p}. The homomorphism ϕ = ψϵ^{−1} has the required property. This proves the existence assertion. The uniqueness is proved similarly.

Corollary 2.1.8 Assuming the hypotheses of prop. 2.1.7, a lifting of ϕ can be chosen unramified at every prime where ϕ is unramified.

Proof: Choose local liftings ϕ̃_{p} of ϕ which are unramified where ϕ is; this is possible since there is no obstruction to lifting a homomorphism defined on Ẑ. Then, apply prop. 2.1.7.

The corollary completes the proof of part (ii): L̃ can be modified so that it is ramified at the same places as L.

(iii) Modifying L̃ so that it satisfies property (S_{N})

We have obtained an extension L̃ which is ramified at the same places as L and which solves the extension problem for G̃. Let p be in ram(L/Q) = ram(L̃/Q). Denote by D_{p}, I_{p} (resp D̃_{p}, Ĩ_{p}) the decomposition and inertia groups for L (resp L̃) at p. We have I_{p} = D_{p} ⊂ G; this is a cyclic group of order l^{α}, say. Let Ip^{′} be the inverse image of I_{p} in G̃. We have Ĩ_{p} ⊂ D̃_{p} ⊂ Ip^{′}. If Ip^{′} is a non-split extension of I_{p} (i.e., is cyclic of order l^{α+1}) we even have Ĩ_{p} = D̃_{p} = Ip^{′}, and the Scholz condition is satisfied at p. Let S be the set of p ∈ ram(L/Q) for which Ip^{′} is a split extension of I_{p}; since Ĩ_{p} is cyclic, we have Ip^{′} = Ĩ_{p} × C_{l}. The Frobenius element Frob_{p} ∈ D̃_{p}/Ĩ_{p} ⊂ Ip^{′}/Ĩ_{p} may be identified with an element c_{p} of C_{l}; the Scholz condition is satisfied at p if and only if c_{p} = 1. If all c_{p}’s are equal to 1, L̃ satisfies (S_{N}). If not, we need to correct ϕ̃ : G_{Q} ⟶ G̃ by a Galois character χ : (Z/qZ)^{∗} ⟶ C_{l} which satisfies the following properties:

1. q ≡ 1 (mod l^{N}).

2. For every p in S, χ(p) = c_{p}.

3. The prime q splits completely in L/Q.

Conditions 1, 2, and 3 impose conditions on the behaviour of q in the fields Q( ^{l^{N}} \sqrt{1}), Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S), and L respectively. Write Q( ^{l^{N}} \sqrt{1}) = Q( ^{l} \sqrt{1}) · F, where F is cyclic of order l^{N−1} and totally ramified at l.

Lemma 2.1.9 The fields L, F, and Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S) are linearly disjoint over Q.

Proof: Since L and F have distinct ramification, L and F are linearly disjoint: L · F has Galois group G × C_{l^{N−1}}. The extension Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S) has Galois group V = C_{l} × C_{l} … × C_{l} (|S| times) over Q( ^{l} \sqrt{1}). The action of Gal(Q( ^{l} \sqrt{1})/Q) = F_{l}^{∗} on V by conjugation is the natural action of multiplication by scalars. The Galois group of Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S) over Q is a semi-direct product of F_{l}^{∗} with V. Since l ≠ 2, this group has no quotient of order l: there is no Galois subfield of Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S) of degree l over Q. This implies that L · F and Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S) are linearly disjoint. QED.

If S = {p_{1}, …, p_{k}}, define integers ν_{i}, 2 ≤ i ≤ k, by c_{p_{i}} = c_{p_{1}}^{ν_{i}}. (This is possible if c_{1} ≠ 1, which we may assume.) In order to satisfy conditions 1, 2 and 3, the prime q must have the following behaviour in the extension L · F · Q( ^{l} \sqrt{1}, ^{l} \sqrt{p}, p ∈ S):

Frob_{q} = 1 in L · F and Q( ^{l^{N}} \sqrt{1});
Frob_{q} ≠ 1 in Q( ^{l} \sqrt{1}, ^{l} \sqrt{p_{1}});
Frob_{q} = 1 in Q( ^{l} \sqrt{1}, ^{l} \sqrt{p_{1}/p_{i}^{ν_{i}}}), i = 2, …, k.

By the Chebotarev density theorem and lemma 2.1.9, such a q exists. One can then define the character χ so that χ(p_{i}) = c_{p_{i}}. This completes part (ii): the homomorphism ϕ̃χ^{−1} defines a new G̃-extension L̃ with property (S_{N}), and with one additional ramified prime, namely q.

The proof allows us to generalize the theorem somewhat. Let us make the following definition:

Definition 2.1.10 If G is a profinite group, the following are equivalent :

1. The topology of G is metrizable.

2. G can be written as a denumerable projective limit

G = lim_{⟵}(⋯ → G_{n} → G_{n−1} → ⋯),

where the G_{n}’s are finite (and the connecting homomorphisms are surjective).

3. The set of open subgroups of G is denumerable.

A group G which satisfies these equivalent properties is said to be separable.

If G = Gal(L/K), these properties are equivalent to [L : K] ≤ ℵ_{0}; if G is a pro-l-group, they are equivalent to dim H^{1}(G, Z/lZ) ≤ ℵ_{0}.

The proof that the four properties in the definition are equivalent is elementary.

Theorem 2.1.11 If G is a separable pro-l-group of finite exponent, then there is a Galois extension of Q with Galois group G.

Proof: If l^{N} is the exponent of G, write G as proj.lim(G_{n}) where each G_{n} is a finite l-group, the connecting homomorphism being surjective, with kernel of order l. By th. 2.1.3, one can construct inductively an increasing family of Galois extensions L_{n}/Q with Galois group G_{n} which have the (S_{N}) property; the union of the L_{n}’s has Galois group G.

Remark: The finiteness condition on the exponent cannot be dropped: for example, Z_{l} × Z_{l} is not a Galois group over Q.

A more general result has been proved by Neukirch [Ne] for pro-solvable groups of odd order and finite exponent.

2.2 The Frattini subgroup of a finite group

Let G be a finite group.

Definition 2.2.1 The Frattini subgroup Φ of G is the intersection of the maximal subgroups of G.

The Frattini subgroup is normal. If G ≠ 1, then Φ ≠ G. If G_{1} ⊂ G satisfies Φ · G_{1} = G, then G_{1} = G. (Otherwise, choose a maximal subgroup M such that G_{1} ⊂ M ⊂ G. Since Φ ⊂ M, it follows that ΦG_{1} ⊂ M, which is a contradiction.) In other words, a subset of G generates G if and only if it generates G/Φ: elements of Φ are sometimes referred to as “non-generators”.

Examples:

1. If G is a simple group, then Φ = 1.

2. If G is a p-group, the maximal subgroups are the kernels of the surjective homomorphisms G ⟶ C_{p}. Hence Φ is generated by (G, G) and G^{p}, where (G, G) denotes the commutator subgroup of G; more precisely, we have

Φ = (G, G) · G^{p}.

The group G/Φ is the maximal abelian quotient of G of type (p, p, …, p).

Proposition 2.2.2 ([Hu], p. 168) Let G be a finite group, Φ its Frattini subgroup, N a normal subgroup of G with Φ ⊂ N ⊂ G. Assume N/Φ is nilpotent. Then N is nilpotent.

Corollary 2.2.3 The group Φ is nilpotent.

This follows by applying prop. 2.2.2 to N = Φ.

Let us prove prop. 2.2.2. Recall that a finite group is nilpotent if and only if it has only one Sylow p-subgroup for every p. Choose a Sylow p-subgroup P of N, and let Q = ΦP. The image of Q by the quotient map N ⟶ N/Φ is a Sylow p-subgroup of N/Φ which is unique by assumption. Hence this image is a characteristic subgroup of N/Φ; in particular it is preserved by inner conjugation by elements of G, i.e., Q is normal in G. Let

N_{G}(P) = {g|g ∈ G, gP g^{−1} = P }

be the normalizer of P in G. If g ∈ G, then gP g^{−1} is a Sylow p-subgroup of Q. Applying the Sylow theorems in Q, there is a q ∈ Q such that

qgP g^{−1}q^{−1} = P.

Hence qg ∈ N_{G}(P). It follows that G = QN_{G}(P) = ΦN_{G}(P). Therefore G = N_{G}(P), and P is normal in G, hence in N; this implies that P is the only Sylow p-subgroup of N.

Application to solvable groups

Proposition 2.2.4 Let G be a finite solvable group ≠ 1. Then G is isomorphic to a quotient of a group H which is a semi-direct product U · S, where U is a nilpotent normal subgroup of H, and S is solvable with |S| < |G|.

Proof: Let Φ be the Frattini subgroup of G; since G/Φ is solvable and ≠ 1, it contains a non-trivial abelian normal subgroup, e.g., the last non-trivial term of the descending derived series of G/Φ. Denote by U its inverse image in G. Since Φ ⊂ U ⊂ G, with U/Φ abelian, U is nilpotent by prop. 2.2.2. Choose a maximal subgroup S of G which does not contain U: this is possible since U ≠ Φ. Since U ·S ≠ S and S is maximal, G = U ·S. Hence, writing H = U ·S (with S acting by conjugation on the normal subgroup U), there is a surjective map H ⟶ G.

The relevance of prop. 2.2.4 to Galois theory lies in the following result which asserts that the embedding problem for split extensions with nilpotent kernel has always a solution.

Claim 2.2.5 ([Sha2], [Is]) Let L/K be an extension of number fields with Galois group S, let U be a nilpotent group with S-action, and let G be the semi-direct product U · S. Then the embedding problem for L/K and for

1 → U → G → S → 1

has a solution.

Theorem 2.2.6 Claim 2.2.5 implies the existence of Galois extensions of Q with given solvable Galois group.

Proof: Let G be a solvable group. We proceed by induction on the order of G. We may asume G ≠ 1. By prop. 2.2.4, write G as a quotient of U · S with U nilpotent and S solvable, |S| < |G|. The induction hypothesis gives a Galois extension L/Q with Galois group S. By the claim above, U · S can be realized as a Galois group; hence, so can its quotient G.

Let us give a proof of claim 2.2.5 in the elementary case where U is abelian of exponent n. Observe that:

1. If claim 2.2.5 is true for an extension L′ of L, it is true for L: for, if S′ = Gal(L/K), there is a natural quotient map US′ → US. Hence we may assume μ_{n} ⊂ L, where μ_{n} denotes the nth roots of unity.

2. We may also assume

U ≃ direct sum of copies of Z/nZ[S],

because any abelian group of exponent n on which S acts is a quotient of such an S-module.

Suppose that h is the number of copies in the decomposition of U as a direct sum of S-modules Z/nZ[S]. Choose places v_{1}, …, v_{h} of K which split completely in L, w_{1}, …, w_{h} places of L which extend them; any place of L extending one of the v_{i} can be written uniquely as sw_{i}, for some s ∈ S. Choose elements ϕ_{j} ∈ L^{∗} such that

(sw_{i})(ϕ_{j}) = \{ \begin{matrix}1 & if s = 1 and i = j. \\ 0 & otherwise. \end{matrix}

Let M be the field generated over L by the ^{n} \sqrt{sϕ_{j}}, for s ∈ S and j = 1, …, h. This is a Galois extension of K, with Gal(M/L) ≃ U. Its Galois group over K is an extension of S by U; since U is a free Z/nZ[S]-module, it is known that such an extension splits (see, e.g., [Se2, ch. IX]). Hence Gal(M/K) is isomorphic to the semi-direct product of S by U.

Hilbert’s irreducibility theorem

3.1 The Hilbert property

Fix a ground field K with Char K = 0, and let V be an irreducible algebraic variety over K. (In what follows, algebraic varieties will be tacitly assumed to be integral and quasi-projective.) Denote by V (K) the set of K-rational points of V.

A subset A of V (K) is said to be of type (C_{1}) if there is a closed subset W ⊂ V, W ≠ V, with A ⊂ W (K), i.e., if A is not Zariski-dense in V.

A subset A of V (K) is said to be of type (C_{2}) if there is an irreducible variety V′, with dim V = dim V′, and a generically surjective morphism π : V′ ⟶ V of degree ≥ 2, with A ⊂ π(V′(K)).

Definition 3.1.1 A subset A of V (K) is called thin (“mince” in French) if it is contained in a finite union of sets of type (C_{1}) or (C_{2}).

Alternately, a set A is thin if there is a morphism

π : W → V with dim W ≤ dim V

having no rational cross-section, and such that A ⊂ π(W (K)).

Example: If V = P_{1}, V (K) = K ∪ {∞}. The set of squares (resp. cubes,…) in K is thin.

Definition 3.1.2 (cf. [CTS1], p. 189) A variety V over K satisfies the Hilbert property if V (K) is not thin.

This is a birational property of V.

Definition 3.1.3 A field K is Hilbertian if there exists an irreducible variety V over K, with dim V ≥ 1, which has the Hilbert property.

It is easy to show (cf. exerc. 1) that if K is Hilbertian, then the projective line P_{1} over K has the Hilbert property (hence, our definition is equivalent to the standard one, see e.g. [L]).

The fields R, Q_{p} are not Hilbertian. A number field is Hilbertian (see §3.4).

Remark on irreducible varieties: The variety V is said to be absolutely irreducible if the algebraic closure K′ of K in the field K(V) of rational functions on V is equal to K. Equivalently, V must remain irreducible upon extension of scalars to the algebraic closure K̄ of K. If V is not absolutely irreducible, then V (K) ⊂ W (K), where W is a subvariety of V, W ≠ V. Indeed, if V is a normal variety, then V (K) = ∅. For, the residue field of the local ring at P ∈ V contains K′, and hence no point of V is K-rational. The general case follows from this by normalization. In particular, an irreducible variety which has the Hilbert property is absolutely irreducible.

Therefore, in our definition of C_{2}-type subsets, we could have asked that V′ be absolutely irreducible.

Remark: Let π : V′ ⟶ V be a finite morphism (we also say that π is a “covering”, even though it can be ramified). Assume that V and V′ are absolutely irreducible, and let K(V′)/K(V) be the corresponding field extension. Let K(V′)^{gal} be the Galois closure of K(V′) over K(V), and let W be the normalisation of V′ in K(V′)^{gal}. The variety W with its projection W ⟶ V may be called the Galois closure of V′ ⟶ V. Note that K is not always algebraically closed in K(V′)^{gal}, i.e., W need not be absolutely irreducible. For example, take V = V′ = P_{1}, π(x) = x^{3}; the Galois closure of V′ is P_{1/K(μ_{3})}, and hence is not absolutely irreducible over K if K does not contain μ_{3}.

Exercises:

1. Let V be an affine irreducible variety over K, with dim V ≥ 1. Let W_{1}, …, W_{r} be absolutely irreducible coverings of the projective line P_{1}. Show that there exists a morphism f : V ⟶ P_{1} such that the pullback coverings f^{∗}W_{i} of V are absolutely irreducible. Use this to show that if V has the Hilbert property, then so has P_{1}.

2. Let V and T be absolutely irreducible varieties, and A ⊂ V (K) a thin subset. Show that A × T (K) is thin in V × T. More generally, let f : W ⟶ V be a generically surjective morphism whose generic fiber is absolutely irreducible (i.e. the function field extension K(W)/K(V) is regular). If B is a subset of W (K) such that f(B) is thin in V (K), show that B is thin in W (K).

3. Let K be a Hilbertian field, and let A be the set of elements of K which are sums of two squares. Show that A is not thin. (Use exerc. 2.)

4. Let K be a number field and V an abelian variety over K with dim V ≥ 1. Show that V (K) is thin (i.e., V does not have the Hilbert property). Hint: use the Mordell-Weil theorem.

Problem: If V and V′ are irreducible varieties with the Hilbert property, is it true that V × V′ has the same property?

3.2 Properties of thin sets

3.2.1 Extension of scalars

Let L/K be a finite extension, V an absolutely irreducible variety over K. Extension of scalars to L yields a variety over L, denoted V_{/L}.

Proposition 3.2.1 If A ⊂ V (L) is thin with respect to L, then A ∩ V (K) is thin with respect to K.

The proof uses the restriction of scalars functor R_{L/K} : (Var_{L}) ⟶ (Var_{K}) from L-varieties to K-varieties, cf. [We], [Oe]. Here are two equivalent definitions of R_{L/K}:

1. It is the right adjoint to the extension of scalars (Var_{K}) ⟶ (Var_{L}), i.e., for every K-variety T and L-variety W, one has:

Mor_{K}(T, R_{L/K}W) = Mor_{L}(T_{/L}, W);

In particular, taking T to be a point which is a rational over K, the above formula yields

(R_{L/K}W)(K) = W (L).

2. Let Σ_{L} be the set of embeddings of L in some fixed algebraic closure K̄; for each σ ∈ Σ_{L}, let W^{σ} be the variety deduced from the given L-variety W by extension of scalars via σ. Then the product X = ∏_{σ} W^{σ} is a K̄-variety. Moreover, one has natural isomorphisms from X to X^{s} for every s ∈ G_{K}. By Weil’s descent theory, these isomorphisms give rise to a K-variety from which X comes by extension of scalars; this variety is R_{L/K}W.

If A is of type (C_{1}), then A ∩ V (K) is clearly of type (C_{1}). Hence we may assume that A ⊂ π(W (L)), that W is absolutely irreducible over L with dim W = dim V, and π is a covering W ⟶ V with deg π > 1. By restricting suitably V, we may assume that π is finite étale. The functor R_{L/K} then gives an étale covering R_{L/K}W ⟶ R_{L/K}V_{/L}. Using the diagonal embedding Δ : V ⟶ R_{L/K}V_{L}, we obtain an étale covering π′ : V′ ⟶ V, and a Cartesian diagram:

V′ ⟶ R_{L/K}W
↓^{π′}  ↓
V ⟶^{Δ} R_{L/K}V_{/L}.

The set A∩V (K) is contained in π′(V′(K)), and it is easy to check that all the components of V′ have degree over V at least equal to deg π. Hence π′(V′(K)) is thin, and the same is true for A ∩ V (K).

Corollary 3.2.2 If K is Hilbertian, then so is L.

Suppose L were not. Then A = P_{1}(L) is thin in P_{1} with respect to L. This implies that A∩P_{1}(K) = P_{1}(K) is thin in P_{1} with respect to K; contradiction. Remark: The converse to cor. 3.2.2 is not true; see e.g. [Ku].

3.2.2 Intersections with linear subvarieties

Let V be the projective space P_{n} of dimension n, and let A ⊂ V (K) be a thin set. We denote by Grass_{n}^{d} the Grassmann variety of d-linear subspaces of P_{n}, where 1 ≤ d ≤ n.

Proposition 3.2.3 There is a non-empty Zariski-open subset U ⊂ Grass_{n}^{d} such that if W belongs to U(K), then A ∩ W is thin in W.

It is enough to prove this when A is either of type (C_{1}) or of type (C_{2}). The first case is easy. In the second, there is a map π : V′ ⟶ P_{n}, with V′ absolutely irreducible, deg π ≥ 2, and A ⊂ π(V′(K)). By Bertini’s theorem (see e.g., [Jou, ch. I, §6], [Ha, p. 179], [De2], [Z]), there exists a non-empty open set U in Grass_{n}^{d} such that π^{−1}(W) is absolutely irreducible for all W ∈ U. Hence, if W ∈ U(K), then W ∩ A is of type (C_{2}), and hence is thin.

An interesting case occurs when d = 1. Let π : V′ ⟶ P_{n} be a generically surjective map of degree > 1, and Φ the hypersurface of ramification of π. Consider the set U of lines which intersect Φ transversally at smooth points. Then for L ∈ U, the covering π^{−1}(L) ⟶ L is irreducible: one proves this over C by deforming the line into a generic one, and the general case follows.

Example: Consider the “double plane” VF^{′} with equation t^{2} = F (x, y), where F is the equation for a smooth quartic curve Φ in P_{2}. The natural projection of VF^{′} onto P_{2} is quadratic and ramified along the curve Φ. A line in P_{2} which intersects Φ transversally in 4 points lifts to an irreducible curve of genus 1 in VF^{′}; a line which is tangent at one point and at no other lifts to an irreducible curve of genus zero; finally, if the line is one of the 28 bitangents to Φ, then its inverse image is two curves of genus zero. In that case, one can take for U the complement of 28 points in P_{2} = Grass_{2}^{1}.

Corollary 3.2.4 If P_{n} has the Hilbert property over K for some n ≥ 1, then all projective spaces P_{m} over K have the Hilbert property.

Proof: P_{1} has the Hilbert property over K: if not, P_{1}(K) = A is thin, and A × P_{1}(K) × … × P_{1}(K) is thin in P_{1} × … × P_{1}. This cannot be the case, since P_{n} has the Hilbert property (and hence also P_{1} × ⋯ × P_{1} which is birationally isomorphic to P_{n}). This implies the same for P_{m}, with m ≥ 1.

For, if A = P_{m}(K) is thin in P_{m}, then by prop. 3.2.3, there is a line L such that L ∩ A = L(K) is thin in L = P_{1}. But this contradicts the fact that P_{1}(K) has the Hilbert property.

3.3 Irreducibility theorem and thin sets

Let π : W ⟶ V be a Galois covering with Galois group G, where V, W denote K-irreducible varieties, V = W/G, and G acts faithfully on W. Let us say that P ∈ V (K) has property Irr(P) if P is “inert”, i.e., the inverse image of P (in the scheme sense) is one point, i.e., the affine ring of the fiber is a field K_{P} (or, equivalently, G_{K} acts freely and transitively on the K̄-points of W above P). In this case, π is étale above P and the field K_{P} is a Galois extension of K with Galois group G.

Proposition 3.3.1 There is a thin set A ⊂ V (K) such that for all P ∉ A, the irreducibility property Irr(P) is satisfied.

Proof: By removing the ramification locus, we may assume that W ⟶ V is étale, i.e., G acts freely on each fiber. Let Σ be the set of proper subgroups H of G. We denote by W/H the quotient of W by H, and by π_{H} the natural projection onto V. Let

A = ⋃_{H∈Σ} π_{H}(W/H)(K).

The set A is thin, since the degrees of the π_{H} are equal to [G : H] > 1. If P ∉ A, then Irr(P) is satisfied: for, lift P to P̄ in W (K̄), and let H be the subgroup of G consisting of elements g ∈ G such that gP̄ = γP̄ for some γ ∈ G_{K} = Gal(K̄/K). Then H = G. Otherwise, H would belong to Σ, and since the image of P̄ in W/H is rational over K, the point P would be in A.

Corollary 3.3.2 If V has the Hilbert property over K, the existence of a Gcovering W ⟶ V as above implies that there is a Galois extension of K with Galois group G.

Assume furthermore that W, and hence V, are absolutely irreducible, and that V has the Hilbert property. Let us say that an extension L/K is of type W if it comes from lifting a K-rational point on V to W.

Proposition 3.3.3 Under these assumptions, for every finite extension L of K, there is a Galois extension E/K of type W with Galois group G which is linearly disjoint from L.

Corollary 3.3.4 There exist infinitely many linearly disjoint extensions with Galois group G, of type W.

Proof of prop. 3.3.3: Extend scalars to L; applying prop. 3.3.1 there is a thin set A_{L} ⊂ V_{/L}(L) such that for all P ∉ A, the property Irr(P) is satisfied over L. Set A = A_{L} ∩ V (K). This is a thin set by prop. 3.2.1. Choose P ∈ V (K) with P ∉ A. Then Irr(P) is true over L, and hence a fortiori over K. This P gives the desired extension E.

Exercise: Show that the set of rational points P satisfying property Irr(P) and giving a fixed Galois extension of K is thin (if G ≠ 1).

Polynomial interpretation Let V be as above, let K(V) be its function field, and let

f(X) = X^{n} + a_{1}X^{n−1} + ⋯ + a_{n}, a_{i} ∈ K(V)

be an irreducible polynomial over K(V). Let G ⊂ S_{n} be the Galois group of f viewed as a group of permutations on the roots of f. (This group can be identified with a subgroup of S_{n}, up to conjugacy in S_{n}, i.e., one needs to fix a labelling of the roots.) If t ∈ V (K) and t is not a pole of any of the a_{i}, then a_{i}(t) ∈ K, and one can define the specialization of f at t:

f_{t}(X) = X^{n} + a_{1}(t)X^{n−1} + ⋯ + a_{n}(t).

Proposition 3.3.5 There exists a thin set A ⊂ V (K) such that, if t ∉ A, then :

1. t is not a pole of any of the a_{i},

2. f_{t}(X) is irreducible over K,

3. the Galois group of f_{t} is G.

By replacing V by a dense open subset, we may assume that the a_{i} have no poles, that V is smooth and that the discriminant Δ of f is invertible. The subvariety V_{f} of V × A^{1} defined by

(t, x) ∈ V_{f} ⟺ f_{t}(x) = 0

is an étale covering of degree n. Its Galois closure W = V_{f}^{gal} has Galois group G. The proposition follows by applying prop. 3.3.1 to W.

Examples:

• G = S_{3}. Let f(X) = X^{3} +a_{1}X^{2} +a_{2}X +a_{3} be irreducible, with Galois group S_{3} over K(V). The specialization at t has Galois group S_{3} if the following properties are satisfied:

1. t ∈ V (K) is not a pole for any of the a_{i}.

2. Δ(t) ≠ 0 (where Δ = a_{1}^{2}a_{2}^{2} + 18a_{1}a_{2}a_{3} − 4a_{1}^{3}a_{3} − 4a_{2}^{3} − 27a_{3}^{2} is the discriminant of f).

3. X^{3} + a_{1}(t)X^{2} + a_{2}(t)X + a_{3}(t) has no root in K.

4. Δ(t) is not a square in K. Conditions 3 and 4 guarantee that G is not contained in either of the maximal subgroups of S_{3}, namely S_{2} and A_{3}. It is clear that the set A of t ∈ V (K) which fail to satisfy these conditions is thin.

• G = S_{4}. The maximal subgroups of G are S_{3}, A_{4}, and D_{4}, the dihedral group of order 8. The case of G ⊂ S_{3} or A_{4} can be disposed of by imposing the same conditions as in the case G = S_{3}; to handle the case of D_{4}, one requires the cubic resolvent

\begin{matrix}(X − (x_{1}x_{2} + x_{3}x_{4})) & (X − (x_{1}x_{3} + x_{2}x_{4})) & (X − (x_{1}x_{4} + x_{2}x_{3})) \\ = X^{3} − a_{2}X^{2} + (a_{1}a_{3} − 4a_{4})X + (4a_{2}a_{4} − a_{1}^{2}a_{4} − a_{3}^{2}) \end{matrix}

to have no root in K, where x_{1}, …, x_{4} are the roots of the polynomial

f = X^{4} + a_{1}X^{3} + … + a_{4}.

• G = S_{5}. The maximal subgroups of G are S_{4}, S_{2} × S_{3}, and A_{5}, which give conditions similar to the above, and the Frobenius group F_{20} of order 20, which is a semi-direct product C_{5}C_{4} and can be viewed as the group of affine linear transformations of the form x ↦ ax + b on Z/5Z. If the Galois group of the specialized polynomial is contained in F_{20}, then the sextic resolvent, which is the minimal polynomial over K(V) for

\frac{(x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + x_{4}x_{5} + x_{5}x_{1}) − (x_{1}x_{3} + x_{2}x_{4} + x_{3}x_{5} + x_{4}x_{1} + x_{5}x_{2})}{∏_{i<j}(x_{i} − x_{j})},

has a root in K when specialized.

3.4 Hilbert’s irreducibility theorem

Theorem 3.4.1 (Hilbert [Hi]) If K is a number field, then for every n, the affine space A^{n} (or equivalently the projective space P_{n}) has the Hilbert property over K. (In other words, K is Hilbertian.)

By cor. 3.2.2 and cor. 3.2.4, it is enough to show that the projective line P_{1} over Q has the Hilbert property. There are several ways of proving this:

1. Hilbert’s original method ([Hi]): The proof uses Puiseux expansions (cf. [L]). It shows that if A ⊂ A^{1}(Q) is thin, then the number of integers n ∈ Z∩A with n < N is O(N^{δ}) for some δ < 1, when N → ∞. (But it does not give a good estimate for δ.)

2. Proof by counting points of height smaller than N on P_{1}(Q): write ξ ∈ P_{1}(Q) as (x, y) with x, y ∈ Z and x, y relatively prime. This can be done in a unique way, up to a choice of sign. The height of ξ is defined to be

height(ξ) = sup(|x|, |y|).

The number of points in P_{1}(Q) with height less than N is asymptotically \frac{12}{π^{2}} N^{2}. On the other hand:

Proposition 3.4.2 If A ⊂ P_{1}(Q) is thin, then the number of points of A with height ≤ N is ≪ N, when N ⟶ ∞.

Sketch of proof: We may assume A is of type (C_{2}), A ⊂ π(X(Q)), where X is is an absolutely irreducible curve and π : X ⟶ P_{1} has degree ≥ 2. Let Rat_{X}(N) be the number of points x in X(Q) with height(π(x)) ≤ N, and let g be the genus of X. Case 1: g ≥ 2. Then X(Q) is finite by Faltings’s theorem; hence Rat_{X}(N) is O(1). One could also invoke an earlier result of Mumford (cf. [L],[Se9]) which gives Rat_{X}(N) = O(log log N). Case 2: g = 1: By the Mordell-Weil theorem, and the Néron-Tate theory of normalized heights, (see [L]), one has Rat_{X}(N) = O((log N)^{γ/2}), where γ is the rank of X(Q). Case 3: g = 0: let H_{X}, H denote the heights on X and P_{1} respectively. It is known that

H_{X} ⌢^{⌣} (H ∘ π)^{m},

where m = deg π. Hence, the number of points on X with H ∘ π ≤ N is at most O(N^{1/m}).

3. Proof by counting integral points: A variant of the second proof shows that the number of integral points in A with height less than N is O(N^{\frac{1}{2}} ), which is an optimal bound.

4. Proof by counting S-units: let S = {p_{1}, …, p_{k}} be a finite non-empty set of primes, E_{S} = {±p_{1}^{m_{1}} ⋯ p_{k}^{m_{k}}}, m_{i} ∈ Z. Let α ∈ Q, and let α + E_{S} be the set of α + e, where e ∈ E_{S}.

Proposition 3.4.3 If A is a thin set in P_{1}(Q), then A ∩ (α + E_{S}) is finite for all but a finite number of α.

Recall that, if V is an affine variety over Q, a subset B of V (Q) is called quasi-S-integral if for every regular function f on V, the set f(B) has bounded denominators in the ring of S-integers, i.e., there is a non-zero integer θ (depending on f and B) such that θf(b) is an S-integer for all b ∈ B.

Let now π_{j} : X_{j} ⟶ P_{1} be a finite number of coverings, and choose α outside the finite sets ram(π_{j}) of X at which π_{j} is ramified. If B_{j} ⊂ X_{j}(Q) is the set of elements of X_{j}(Q) with π_{j}(b) ∈ α+E_{S}, then B_{j} is a quasi-S-integral set in the affine curve X_{j} − π_{j}^{−1}(∞) − π_{j}^{−1}(α). Over Q̄, π_{j}^{−1}(∞) ∪ π_{j}^{−1}(α) has at least 3 elements, since π_{j}^{−1}(α) ∉ ram(π_{j}). A theorem of Siegel, Mahler and Lang then shows that B_{j} is finite ([L], [Se9]). Prop. 3.4.3 follows.

Remark: The bound given in proof no. 3 can be extended to affine n-space A^{n}. More precisely, let A be a thin set in A^{n}(Q), and let Int_{A}(N) be the number of integral points (x_{1}, …, x_{n}) ∈ A with |x_{i}| ≤ N. Then:

Theorem 3.4.4 (S.D. Cohen) Int_{A}(N) = O(N^{n− \frac{1}{2}} log N).

One can replace the log N term in this inequality by (log N)^{γ}, where γ < 1 is a constant depending on A. The proof is based on the large sieve inequality: one combines th. 3.6.2 with cor. 10.1.2 of the appendix, cf. [Coh] and [Se9].

Problem: Let X ⊂ P_{n} be an absolutely irreducible variety of dimension r. As above, denote by Rat_{X}(N) the number of points of X(Q) with height ≤ N. If X is linear, deg X = 1, then

Rat_{X}(N) ⌢^{⌣} N^{r+1}.

If deg X ≥ 2, one can show, using th. 3.4.4 (cf. [Se9]), that

Rat_{X}(N) = O(N^{r+ \frac{1}{2}} log N).

A better result follows from results of Schmidt [Schm], namely,

Rat_{X}(N) = O(N^{r+ \frac{4}{9}} ).

Can this estimate be improved to:

Rat_{X}(N) = O(N^{r+ϵ}), for every ϵ > 0?

3.5 Hilbert property and weak approximation

Let K be a number field, Σ_{K} the set of places in K (including the archimedean ones). For v ∈ Σ_{K}, let K_{v} denote the completion of K at v, and let Nv be the cardinality of the residue field of K_{v} in case v is non-archimedean. If V is an absolutely irreducible integral variety over K, V (K_{v}) is naturally endowed with a K_{v}-topology which gives it the structure of a K_{v}-analytic space (resp. manifold, if V is smooth).

Proposition 3.5.1 If W ⊂ V, W ≠ V, then W (K_{v}) ≠ V (K_{v}) for all but a finite number of v ∈ Σ_{K}.

Proposition 3.5.2 Let W be absolutely irreducible, of same dimension as V, and π : W ⟶ V be a generically surjective morphism, deg π > 1. Let K_{π} be the algebraic closure of K in the extension K(W)^{gal}/K(V). If v ∈ Σ_{K} splits completely in K_{π} and Nv is large enough, then π(W (K_{v})) ≠ V (K_{v}).

The proofs of prop. 3.5.1 and 3.5.2 will be given in § 3.6.

Example: Take K = Q, V = P_{1}, W the curve defined by

y^{3} = (x^{2} + 3)/(x^{2} + 12),

and define π : W ⟶ V by π(x, y) = x. Then K_{π} = Q(\sqrt{−3}). A prime p splits in K_{π} when p ≡ 1 (mod 3). If p ≡ −1(mod 3), p ≥ 5, then one checks that π : W (Q_{p}) ⟶ V (Q_{p}) is an isomorphism of analytic Q_{p}-manifolds. This shows that the condition “v splits completely in K_{π}” cannot be omitted.

Theorem 3.5.3 Let A be a thin subset of V (K), and let S_{0} be a finite subset of Σ_{K}; then there is a finite set S of places of K satisfying :

a) S ∩ S_{0} = ∅.

b) The image of A in ∏_{v∈S} V (K_{v}) is not dense.

Observe first that if the theorem holds for A_{1} and A_{2}, it holds also for A_{1} ∪A_{2}: for, choose S_{1} satisfying the conclusion of the theorem for A_{1}; then choose S_{2} satisfying the conclusion of the theorem for A_{2}, but with S_{0} replaced by S_{0} ∪ S_{1}. Then, taking S = S_{1} ∪ S_{2}, one has

A_{1} ∪ A_{2} ⊂ ∏_{v∈S} V (K_{v})

is not dense: the point (x_{1}, x_{2}), where x_{i} ∈ ∏_{v∈S_{i}} V (K_{v}) − closure(A_{i}), is not in the closure of A_{1} ∪ A_{2} in ∏_{v∈S} V (K_{v}).

Hence it is enough to prove th. 3.5.3 for sets of type (C_{1}) and (C_{2}).

1. If W ⊂ V, W ≠ V, A ⊂ W (K), then choose S = {v} with v large enough, so that v ∉ S_{0}, W (K_{v}) ≠ V (K_{v}) (prop. 3.5.1). Since Ā ⊂ W (K_{v}), A is not dense in W (K_{v}).

2. Assume A ⊂ π(W (K)), where π : W → V is generically surjective, dim W = dim V, W is absolutely irreducible, and deg π ≥ 2. We may also assume that π is a finite morphism (replace V by a suitable open subvariety). By prop. 3.5.2, there exists a v ∉ S_{0} such that π(W (K_{v})) ≠ V (K_{v}). Since π is finite, it transforms closed subsets into closed subsets. This shows that π(W (K_{v})) is closed in V (K_{v}); hence A is not dense in V (K_{v}).

Corollary 3.5.4 Assume V is a projective variety, and let Ā denote the closure of A in the compact space

∏_{s∉S_{0}} V (K_{v}).

Then the interior of Ā is empty, i.e., A is nowhere dense in the product.

One says that V has the weak approximation property for a finite set S of places if V (K) is dense in ∏_{v∈S} V (K_{v}).

Lemma 3.5.5 If V, V′ are smooth, birationally equivalent, then V has the weak approximation property for S if and only if V′ has the weak approximation property for S. (In other words, the weak approximation property is a birational property for smooth varieties).

It is enough to prove the lemma when V′ = V −W, with W a closed subvariety, W ≠ V. Clearly, if V has the weak approximation property for S, so does V − W. Conversely, if V − W has the weak approximation property for S, one uses smoothness to prove that V (K_{v}) − W (K_{v}) is dense in V (K_{v}). Hence, any point in W (K_{v}) can be approximated by points in V (K) − W (K).

As a special case, any smooth K-rational variety has the weak approximation property for any finite set S of places.

Remark: The smoothness assumption is necessary: for example, consider the Q-rational curve

y^{2} = (x^{2} − 5)^{2}(2 − x^{2}).

Its rational points are not dense in the set of its real points (the points (−\sqrt{5}, 0) and (\sqrt{5}, 0) are isolated).

Definition 3.5.6 A variety V is said to have property (WA) if it satisfies the weak approximation property with respect to S for all finite S ⊂ Σ_{K}. It is said to have property (WWA) (“weak weak approximation property”) if there exists a finite set S_{0} of places of K such that V has the weak approximation property with respect to S ⊂ Σ_{K}, for all S with S ∩ S_{0} = ∅.

Examples:

1. A K-rational variety has property (WA).

2. A K-torus has property (WWA), but not necessarily (WA). More precisely, if it is split by a finite Galois extension L/K, one can take for exceptional set S_{0} the places of K whose decomposition group in Gal(L/K) is not cyclic. (See, e.g., [Vo1],[CTS1].)

Theorem 3.5.7 ([Ek],[CT]) A variety which has the WWA property satisfies the Hilbert property.

Proof: If not, A = V (K) would be thin. By th. 3.5.3 there would exist S disjoint from S_{0} such that V (K) would not be dense in ∏_{v∈S} V (K_{v}); this contradicts WWA.

The following conjecture is due to Colliot-Thélène [CT]; it is closely related to the questions discussed in [CTS2]:

Conjecture 3.5.8 A K-unirational smooth variety has the WWA property.

Recall that a variety is K-unirational if there exists a generically surjective map P_{n} ⟶ V defined over K, for some n - one may always take n equal to dim V.

Theorem 3.5.9 Conjecture 3.5.8 implies that every finite group is a Galois group over Q.

Proof: Make G act faithfully on W = A^{n} for some n, and let V = W/G. Then V is K-unirational. By conjecture 3.5.8, V^{smooth} has the WWA property, and hence satisfies the Hilbert property. By cor. 3.3.2, G can be realized as a Galois group over Q.

3.6 Proofs of prop. 3.5.1 and 3.5.2

Let O denote the ring of integers of K, and choose a scheme \underline{V} of finite type over O having V as its generic fiber; any two choices for \underline{V} coincide outside a finite set of primes, i.e., they become isomorphic as schemes over SpecO[\frac{1}{d}] for some non-zero d. If v is a non-archimedean place, P_{v} the corresponding prime of O, denote by κ(v) = O/P_{v} the residue field at v, which is a finite field with Nv elements. Let \underline{V} (κ(v)) be the set of κ(v)-rational points of \underline{V} (or, equivalently, of the fiber of V at v). We shall use the following known result:

Theorem 3.6.1 (Lang-Weil) If V is absolutely irreducible over K, then

|\underline{V} (κ(v))| = Nv^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ).

The original proof of Lang-Weil is by reduction to the case of curves for which one can use the bound proved by Weil. A different method, which gives a more precise error term, is to use Deligne’s estimates for the eigenvalues of the Frobenius endomorphism, together with Bombieri’s bounds for the number of zeros and poles of the zeta function, see [Bo].

Proof of prop. 3.5.1 We may assume that V is smooth and W = ∅ (by replacing V by V − W). If Nv is large enough, th. 3.6.1 implies that \underline{V} (κ(v)) is not empty. Choose x ∈ \underline{V} (κ(v)). Since \underline{V} is smooth at v (for Nv large enough), x lifts to a point x̃ ∈ \underline{V} (O_{v}), which is contained in V (K_{v}). Hence V (K_{v}) is not empty.

Proof of prop. 3.5.2 Recall that we are given a generically surjective map π : W ⟶ V, with deg π > 1, dim W = dim V. By replacing W and V by open subsets if necessary, we may assume that π is finite étale and V is smooth. We may also choose a scheme of finite type \underline{W} for W over some O[\frac{1}{d}], such that π comes from a map (also denoted π) \underline{W} → \underline{V}. By changing d, we may further assume that π : \underline{W} ⟶ \underline{V} is étale and finite and \underline{V} is smooth (see, e.g. EGA IV §8). For v prime to d, we have a diagram

\underline{W} (κ(v)) ⟵ \underline{W} (O_{v}) ↪ W (K_{v})
↓ ↓ ↓
\underline{V} (κ(v)) ⟵ \underline{V} (O_{v}) ↪ V (K_{v}).

The fact that π : \underline{W} ⟶ \underline{V} is finite implies that the right square is Cartesian, i.e. a point z ∈ W (K_{v}) is an O_{v}-point if and only if π(z) ∈ V (K_{v}) has the same property. Moreover, the reduction map \underline{V} (O_{v}) ⟶ \underline{V} ((v)) is surjective, since \underline{V} is smooth.

To show that W (K_{v}) ≠ V (K_{v}) it is thus enough to prove that

\underline{W} (κ(v)) ≠ \underline{V} (κ(v)).

This is a consequence of the following more precise result:

Theorem 3.6.2 Let m = deg π (m ≥ 2). Then, for v splitting completely in K_{π}, one has

|π(\underline{W} (κ(v)))| ≤ c(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ),

where c = 1 − \frac{1}{m!}. (This implies:

|π(\underline{W} (κ(v)))| ≤ c′(Nv)^{dim V}

with c′ < 1, for Nv large enough, v splitting completely in K_{π}.)

Let \underline{W}^{gal} be the Galois closure of \underline{W}; its Galois group G injects into S_{m}, and hence |G| ≤ m!. Now, divide the points in \underline{W} (κ(v)) into two sets:

\underline{W} (κ(v)) = A ∪ B,

where A is the set of points which can be lifted to \underline{W}^{gal}(κ(v)), and B is the set of the remaining points. By th. 3.6.1 applied to \underline{W}, we have:

|A| + |B| = (Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ).

If v splits completely in K_{π}, then all the connected components of the fiber at v are absolutely irreducible, and hence, letting e be the number of these components, we have

|\underline{W}^{gal}(κ(v))| = e(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ).

We have A = \underline{W}^{gal}(κ(v))/H where H = Gal(\underline{W}^{gal}/\underline{W}). Since H acts freely, this gives

|A| = \frac{e}{|H|}(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ).

The same argument applied to the action of G on \underline{W}^{gal}(κ(v)) gives

|π(A)| = \frac{e}{|G|}(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} )

= \frac{1}{m}|A| + O((Nv)^{dim V − \frac{1}{2}} ).

Hence:

|π(A)| + |π(B)| ≤ \frac{1}{m}|A| + |B| + O((Nv)^{dim V − \frac{1}{2}} )

≤ |A| + |B| − (1 − \frac{1}{m})|A| + O((Nv)^{dim V − \frac{1}{2}} )

≤ (Nv)^{dim V} − (1 − \frac{1}{m})|A| + O((Nv)^{dim V − \frac{1}{2}} ),

and therefore

π(\underline{W} (κ(v))) ≤ (Nv)^{dim V} − (1 − \frac{1}{m})\frac{e}{|H|}(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ).

Finally, since (1 − \frac{1}{m})\frac{e}{|H|} ≥ \frac{1}{m|H|} ≥ \frac{1}{|G|} ≥ \frac{1}{m!}, we get:

π(\underline{W} (κ(v))) ≤ (1 − \frac{1}{m!})(Nv)^{dim V} + O((Nv)^{dim V − \frac{1}{2}} ),

and this completes the proof of th. 3.6.2, and hence of prop. 3.5.2.

Application to the distribution of Frobenius elements Let E be an elliptic curve over Q without complex multiplication over Q̄, and let a_{p}, for p prime, be the “trace of Frobenius”:

a_{p} = 1 + p − N_{p}

where N_{p} is the number of points of \underline{E} over F_{p}. The following is well-known:

Theorem 3.6.3 If f ≠ 0 is any polynomial in two variables over Q, then the set of primes p such that f(p, a_{p}) = 0 has density 0.

(The proof uses the l-adic representation ρ : G_{Q} ⟶ GL_{2}(Z_{l}) attached to E: since det ρ(Frob_{p}) = p, Tr ρ(Frob_{p}) = a_{p}, one is lead to consider the set of x ∈ ρ(G_{Q}) such that f(Tr x, det x) = 0. Since ρ(G_{Q}) is known to be open in GL_{2}(Z_{l}), this set has Haar measure zero; the theorem follows by applying Chebotarev’s density theorem.)

More generally,

Theorem 3.6.4 Let A be a thin subset of Z × Z. The set of p’s such that (p, a_{p}) ∈ A has density 0.

The theorem is already proved for A of type (C_{1}): so assume A is of type (C_{2}). Let A_{l} be the image of A in Z/lZ × Z/lZ and let S_{l} be the set of x ∈ GL_{2}(Z/lZ) such that (Tr (x), det(x)) belongs to A_{l}. One checks that

|S_{l}| ≤ |A_{l}| · l^{2}(1 + 1/l).

By th. 3.6.2, we have |A_{l}| ≤ cl^{2} for c < 1 and l sufficiently large, splitting completely in some fixed extension K of Q; hence

|S_{l}| ≤ cl^{4}(1 + 1/l).

This shows that the density of S_{l} in GL_{2}(Z/lZ) is

≤ c(1 + 1/l)(1 − 1/l)^{−1}(1 − 1/l^{2})^{−1} = c(1 − 1/l)^{−2}

if l is large enough. Recall now that, if l_{1}, ⋯, l_{m} are large enough distinct prime numbers, the Galois group of the l_{1} ⋯ l_{m}-division points of E is ∏ GL_{2}(Z/l_{i}Z). If each l_{i} splits completely in K the Chebotarev density theorem, applied to the field of l_{1} ⋯ l_{m}-division points of E, shows that the upper density of the set of primes p with (p, a_{p}) ∈ A is ≤ c^{m} ∏_{i=1}^{m}(1 − 1/l_{i})^{−2}. Since this can be made arbitrarily small by taking m large enough, the theorem follows.

Remarks:

1. The above implies, for example, that the set of the primes p for which |\underline{E}(F_{p})| − 3 is a square has density 0.

2. One can prove more than density zero in 3.6.4; in fact, one has

|\{p|p < N, (p, a_{p}) ∈ A\}| = O (\frac{N}{(log N)^{1+δ}} ) , for some δ > 0.

This implies that

∑_{(p,a_{p})∈A} \frac{1}{p} < ∞.

The proof (unpublished) uses the Selberg “Λ^{2}” sieve.

3. There are similar results for the Ramanujan τ-function.

Galois extensions of Q(T): first examples

4.1 The property Gal_{T}

Let E be a finite Galois extension of Q(T) with group G which is regular, i.e., Q̄ ∩ E = Q. Geometrically, E can be viewed as the function field of a smooth projective curve C which is absolutely irreducible over Q; the inclusion Q(T) ↪ E corresponds to a (ramified) Galois covering C ⟶ P_{1} defined over Q with group G.

Conjecture 4.1.1 Every finite group G occurs as the Galois group of such a covering.

Let us say that G has property Gal_{T} if there is a regular G-covering C ⟶ P_{1} as above. In that case, there are infinitely many linearly disjoint extensions of Q, with Galois group G (cf. th. 3.3.3). Remark: If a regular G-covering exists over P_{n}, n ≥ 1, then such a covering also exists over P_{1}, by Bertini’s theorem (cf. e.g. [Jou]).

Examples: The property Gal_{T} is satisfied for:

1. Abelian groups.

2. A_{n} and S_{n} (Hilbert); Ã_{n} (Vila, Mestre).

3. Some non-abelian simple groups, such as the sporadic ones (with the possible exception of M_{23}), most PSL_{2}(F_{p}), p prime, and a few others.

4. If G has property Gal_{T}, then so does every quotient of G.

Proposition 4.1.2 If G_{1}, G_{2} have property Gal_{T}, then so does their product G_{1} × G_{2}.

Proof: Let C_{1}, C_{2} be regular coverings of P_{1} with groups G_{1}, G_{2}, and let Σ_{1}, Σ_{2} be their ramification loci.

36 Chapter 4. Galois extensions of Q(T)

1. If Σ_{1} ∩ Σ_{2} = ∅, then the extensions corresponding to C_{1}, C_{2} are linearly disjoint, because P_{1} is algebraically simply connected (see §4.4 below). One can take for covering C the fibered product C_{1} ×_{P_{1}} C_{2}, which has function field Q(C_{1}) ⊗_{Q(T)} Q(C_{2}), and hence has G_{1} × G_{2} as Galois group.

2. If Σ_{1} ∩ Σ_{2} ≠ ∅, one modifies the covering C_{1} ⟶ P_{1} by composing it with an automorphism of P_{1} so that the new ramification locus is disjoint from Σ_{2}. One is thus reduced to case 1.

Remark: If G_{1} and G_{2} have property Gal_{T}, one can show (cf. e.g. [Ma3], p. 229, Zusatz 1) that the wreath product G_{1} Wr G_{2} also has property Gal_{T}. This can be used to give an alternate proof of prop. 4.2.2 below.

Exercises:

1. Show that the profinite group Z_{p} is not the Galois group of any regular extension of Q(T). (Hence conjecture 4.1.1 does not extend to profinite groups, not even when they are p-adic Lie groups.)

2. Let G be a finite group having property Gal_{T}. Show that there exists a regular Galois extension L of Q(T), with Galois group G, such that:

(a) Every subextension of L distinct from Q(T) has genus ≥ 2.

(b) Every Q-rational point P of P_{1} has property Irr(P) with respect to L. (Use a suitable base change P_{1} ⟶ P_{1}, combined with Faltings’s theorem.)

4.2 Abelian groups

A torus defined over Q is said to be a “permutation torus” if its character group has a Z-basis which is stable under the action of Gal(Q̄/Q), or equivalently, if it can be expressed as a product of tori of the form R_{K_{i}/Q}G_{m}, where the K_{i} are finite extensions of Q. A permutation torus is clearly rational over Q.

Now, let A be a finite abelian group. The following proposition implies that A has property Gal_{T}:

Proposition 4.2.1 There exists a torus S over Q, and an embedding of A in S(Q), such that the quotient S′ = S/A is a permutation torus. (In particular, S′ is a Q-rational variety.)

The proof uses the functor Y which to a torus associates the Z-dual of its character group. An exact sequence of the form 1 ⟶ A ⟶ S ⟶ S′ ⟶ 1 gives rise to the exact sequence

1 ⟶ Y (S) ⟶ Y (S′) ⟶ Ã ⟶ 1,

where

Ã = Ext^{1}(Â, Z) = Hom(μ_{n}, A) = Hom(Â, Q/Z),

and Â denotes as usual the Cartier dual Hom(A, G_{m}). Choose K finite Galois over Q such that the action of G_{Q} on Ã factors through Gal(K/Q), e.g. K = Q(μ_{n}), where n is the exponent of A. Now express Ã as a quotient of a free Z[Gal(K/Q)]-module F, and let S′ be a torus such that Y (S′) = F; it follows that S′ is a permutation torus, and there is an A-isogeny S ⟶ S′.

Proposition 4.2.2 Let G be a finite group having property Gal_{T}, and let M be a finite abelian group with G-action. Then the semi-direct product G̃ = M · G also has property Gal_{T}.

We may assume without loss of generality that M is an induced G-module,

M = ⨁_{g∈G} gA, so that

G̃ = \underbrace{(A × ⋯ × A)}_{|G| times} ·G

is the wreath product of A and G. By prop. 4.2.1, there is an isogeny S ⟶ S′ defined over Q, with S′ a permutation torus and with kernel A ⊂ S(Q). By hypothesis, there is a regular étale G-covering C ⟶ U where U is a Q-rational variety (e.g. an open subvariety of P_{1}). The actions of A on S, and of G on C and on S × … × S give rise to a natural G̃-action on X = S × … × S × C. This action is free. Let Y = X/G̃. Prop. 4.2.2 then follows from the following lemma:

Lemma 4.2.3 The variety Y = X/G̃ is Q-rational.

Define X′ = X/(A × … × A) = S′ × … × S′ × C. We have Y = X′/G. This shows that Y is the fiber space over U with fiber the torus S′ × … × S′, which is associated to the principal G-bundle C ⟶ U. We may thus view Y as a torus over U. In particular, the generic fiber Y_{U} of Y ⟶ U is a torus over the function field Q(U) (this torus is obtained from S′ × … × S′ by a Gtwisting, using the Galois extension Q(C)/Q(U)). Since S′ is a permutation torus over Q, Y_{U} is a permutation torus over Q(U). Hence the function field Q(Y) of Y_{U} is a pure transcendental extension of Q(U), which itself is a pure transcendental extension of Q. The lemma follows. (One can also deduce lemma 4.2.3 from lemma 4.3.1 below.)

Exercise: Let H be a finite group generated by an abelian normal subgroup M and a subgroup G having property Gal_{T}. Show using prop. 4.2.2 that H has property Gal_{T}.

4.3 Example: the quaternion group Q_{8}

We need first:

Lemma 4.3.1 Let Y ⟶ X be an étale Galois covering with Galois group G, and G ⟶ GL(W) be a linear representation of G, where W is a finitedimensional vector space. Let E be the associated fiber bundle with base X and fiber W. Then E is birationally isomorphic to X × W.

This follows from Hilbert’s theorem 90: H^{1}(K(X), GL_{n}) = 0, where n is dim W, and K(X) is the function field of X over the ground field (which we assume to be Q).

[Alternate proof: Use descent theory to show that E is a vector bundle over X, hence is locally trivial.]

This lemma implies that Q(E) = Q(X)(T_{1}, …, T_{n}), hence that Q(E) and Q(X) are stably isomorphic. (Recall that two extensions k_{1} and k_{2} of a field k are stably isomorphic if there exist integers n_{1}, n_{2} ≥ 0 such that the extensions k_{1}(T_{1}, …, T_{n_{1}}) and k_{2}(T_{1}, …, T_{n_{2}}) are k-isomorphic.)

Application: Let G act linearly on vector spaces W_{1} and W_{2}, the action on W_{2} being faithful. Letting n = dim W_{1}, we have:

4.3.2

Q(W_{1} × W_{2})/G ≃ Q(W_{2}/G)(T_{1}, …, T_{n}).

(In particular, Q(W_{1} × W_{2})/G is stably isomorphic to Q(W_{2}/G).) This is a consequence of the lemma applied to E = W_{1} × W_{2} and X = W_{2}.

Corollary 4.3.3 If G acts faithfully on W_{1} and W_{2}, then Q(W_{1}/G) and Q(W_{2}/G) are stably isomorphic.

Let R = ⊕W_{i} be a decomposition of the regular representation of G as a sum of Q-irreducible ones. By the corollary, if one of the W_{i} is a faithful G-module, then Q(R/G) is stably isomorphic to Q(W_{i}/G). If Q(W_{i}/G) is a rational field, then so is Q(R/G), by 4.3.2.

Application to the quaternion group Q_{8}. Let Q_{8} be the quaternion group of order 8. The group algebra Q[Q_{8}] decomposes as

Q[Q_{8}] = Q × Q × Q × Q × H,

where H denotes the standard field of quaternions (over Q), and Q_{8} acts on H by left multiplication. By the previous remark, the Q-rationality of Q[Q_{8}]/Q_{8} is equivalent to the Q-rationality of H^{∗}/Q_{8}, where H^{∗} is the multiplicative group of H viewed as a 4-dimensional Q-algebraic group. The group Q_{8} has a center {±1} of order 2, and D = Q_{8}/{±1} is a group of type (2, 2). On the other hand,

H^{∗}/{±1} ≃ G_{m} × SO_{3},

by the map

(N, ϕ) : H^{∗} ⟶ G_{m} × SO_{3},

where N is the reduced norm, and ϕ : H^{∗} ⟶ SO_{3} maps a quaternion x to the rotation y ↦ xyx^{−1} (on the 3-dimensional vector space of quaternions of trace 0). Therefore,

H^{∗}/Q_{8} = G_{m} × SO_{3}/D,

and it suffices to show that SO_{3}/D is a rational variety over Q. But the group D is the stabilizer in SO_{3} of a flag in A^{3}. Hence SO_{3}/D is isomorphic to an open subvariety of the flag variety of A^{3}, which is rational over Q. Noether’s method therefore applies to Q_{8}; in particular Q_{8} has property Gal_{T}. For explicit formulae, see [JY].

Exercise (L. Schneps) Show that every p-group of order p^{3} has property Gal_{T} (use the exercise at the end of §4.2).

4.4 Symmetric groups

The symmetric group S_{n} acts on the affine space A^{n} with quotient A^{n} (“symmetric functions theorem”). This shows that S_{n} has property Gal_{T}. Let us give some explicit constructions of polynomials with S_{n} as Galois group. For example, consider a polynomial

f(X) = X^{n} + a_{1}X^{n−1} + ⋯ + a_{n}, with a_{i} ∈ Q,

and put

f(X, T) = X^{n} + a_{1}X^{n−1} + ⋯ + a_{n} − T.

Theorem 4.4.1 (Hilbert [Hi]) If f is a Morse function, then the splitting field of f(X, T) over Q(T) is a regular extension with Galois group S_{n}.

(The polynomial f is called a Morse function if:

1. The zeros β_{1}, …, β_{n−1} of the derivative f′ of f are simple.

2. f(β_{i}) ≠ f(β_{j}) for i ≠ j.)

We will need the following simple facts about the symmetric group S_{n}:

Lemma 4.4.2 S_{n} is generated by transpositions.

This is well-known: indeed S_{n} is generated by the transpositions

(12), (23), …, (n − 1, n).

Lemma 4.4.3 Let G be a transitive subgroup of S_{n} which contains a transposition. Then the following are equivalent :

1. G contains an (n − 1)–cycle.

2. G is doubly transitive.

3. G = S_{n}.

If G contains an (n − 1)–cycle, then the stabilizer of a point is transitive on the complement of the point, hence G is doubly transitive. If G is doubly transitive, then G contains all the transpositions in S_{n}, hence G = S_{n} by lemma 4.4.2. That 3 ⇒ 1 is obvious.

Lemma 4.4.4 (cf. [Hu], p.171) Let G be a transitive subgroup of S_{n} which is generated by cycles of prime orders. Then :

1. G is primitive.

2. If G contains a transposition, then G = S_{n}.

3. If G contains a 3-cycle, then G = A_{n} or S_{n}.

Let {Y_{1}, …, Y_{k}}, with k > 1, be a partition of {1, …, n} which is stable under G. Our assumptions imply that there is a cycle s of G, of prime order p, such that Y_{1} ≠ sY_{1}. Since no element of Y_{1} ∪ sY_{1} ∪ … ∪ s^{p−1}Y_{1} is fixed by s, we have:

|Y_{1}| + |sY_{1}| + … + |s^{p−1}Y_{1}| ≤ p,

and hence |Y_{1}| = 1. This shows that {Y_{1}, …, Y_{k}} is the trivial partition of {1, …, n}. Hence G is primitive. To show 2, let G′ be the subgroup of G generated by the transpositions belonging to G. Since G′ ≠ 1, it is transitive (a non-trivial normal subgroup of a primitive group is transitive). For Ω ⊂ {1, …, n}, let us denote by S_{Ω} (resp. A_{Ω}) the symmetric (resp, alternating) group on Ω. Let Ω ⊂ {1, …, n} be maximal with the property S_{Ω} ⊂ G′, and suppose that Ω ≠ {1, …, n}. By the transitivity of G′, there exists (xy) ∈ G′ with x ∈ Ω, y ∉ Ω. Hence S_{Ω∪{y}} ⊂ G′, contradicting the maximality of Ω. It follows that Ω = {1, …, n} and hence G = G′ = S_{n}, proving 2. The proof of 3 is similar, taking G′ this time to be the subgroup of G generated by the 3-cycles belonging to G. The hypothesis implies that G′ is non-trivial, and hence is transitive. Choose Ω ⊂ {1, …, n} maximal with the property A_{Ω} ⊂ G′. As before, if Ω ≠ {1, …, n}, there is a 3-cycle (xyz) which does not stabilize Ω. There are two cases: Case 1: {x, y, z}∩Ω has two elements, say y and z. Then clearly A_{Ω∪{x}} ⊂ G′, contradicting the maximality assumption for Ω.

Case 2: {x, y, z} ∩ Ω has 1 element, say x. Choose two elements y′, z′ ∈ Ω distinct from x; it is easy to see that (xyz) and (xy′z′) generate the alternating group A_{5} on {x, y, z, y′, z′}. In particular, the cycle (xy′z) is in G; since this 3-cycle meets Ω in two elements, we are reduced to case 1, QED.

More generally, Jordan has shown that a primitive subgroup of S_{n} which contains a cycle of prime order ≤ n − 3 is equal to A_{n} or S_{n} (see [Wi], p.39).

Th. 4.4.1 will be proved in the following more general form:

Theorem 4.4.5 If K is any field of characteristic 0, or of characteristic p not dividing n, and f(X) ∈ K[X] is Morse, then Gal(f(X) − T) = S_{n} over K(T).

Proof: We may assume K to be algebraically closed. The polynomial f can be viewed as a ramified covering of degree n

\begin{aligned}f : & P_{1} & ⟶ & P_{1} \\ & x & ↦ & t = f(x). \end{aligned}

The corresponding field extension is K(X) ⊃ K(T); it is separable because p |̸ n. Let G ⊂ S_{n} be the Galois group of the Galois closure of K(X) over K(T), i.e., the Galois group of the equation f(X) − T = 0.

The ramification points of the covering f are

X = ∞, f(β_{1}), …, f(β_{n−1}).

At X = ∞, the ramification is tame, and the inertia group is generated by an n-cycle. At the f(β_{i}), the hypothesis on f implies that the inertia group is tame for p ≠ 2 and wild for p = 2, and that (in both cases) it is generated by a transposition.

Hence the theorem is a consequence of the following proposition, combined with lemma 4.4.4.

Proposition 4.4.6 Let C ⟶ P_{1} be a regular Galois covering with group G, tamely ramified at ∞. Then G is generated by the inertia subgroups of points outside ∞, and their conjugates.

Proof: Let H be the normal subgroup generated by the inertia subgroups outside ∞. Then C/H is a G/H-covering of P_{1} which is tame at ∞, and unramified outside. The Riemann-Hurwitz formula implies that the genus of C/H is ≤ \frac{1}{2}(1 − |G/H|); hence G = H.

Corollary 4.4.7 a) P_{1} is algebraically simply connected.

b) In characteristic 0, the affine line is algebraically simply connected.

Remark: In characteristic p > 0, the affine line A^{1} is not simply connected (as shown e.g. by the Artin-Schreier equation X^{p} −X = T). If G is the Galois group of an unramified covering of A^{1}, then prop. 4.4.6 implies that G is generated by its Sylow p-subgroups. There is a conjecture by Abhyankar [Ab1] that, conversely, every group G having this property occurs. This is known to be true when G is solvable, and in many other cases (Nori, Abhyankar, Harbater, Raynaud, see e.g. [Ab2], [Se11]).

Example of an S_{n}-extension of ramification type (n, n − 1, 2)

The above method for constructing polynomials over Q(T) with Galois group S_{n} gives us polynomials with ramification type (n, 2, …, 2). For a different example, consider the polynomial

f(X) = X^{n} − X^{n−1},

so that

f(X, T) = X^{n} − X^{n−1} − T.

Then

f′ = nX^{n−2}(X − α), α = 1 − \frac{1}{n},

and hence the ramification is given (in char. 0) by

at ∞ : cycle of order n;
at 0 : cycle of order n − 1;
at α : a transposition.

Hence G = S_{n} by lemma 4.4.3. The polynomial f has ramification type (n, n − 1, 2).

Remarks:

1. Consider f(X, T) = X^{p+1} − X^{p} − T. This polynomial has S_{p+1} as Galois group in characteristic different from p (the proof is similar to the one above). In characteristic p, one can show that it has Galois group PGL_{2}(F_{p}).

2. One might ask for an explicit polynomial f_{n} over Q such that f_{n} has Galois group S_{n}. Here is an example: f_{n}(X) = X^{n} − X − 1. Indeed, Selmer [Sel] has shown that f_{n} is irreducible over Q. Assuming this, let us prove that f_{n} has Galois group S_{n}. We look at the primes p dividing the discriminant of f_{n}, i.e., those modulo which f_{n}(X) has a multiple root. This happens if f_{n}(X) and fn^{′}(X) = nX^{n−1} − 1 have a common root mod p. Substituting X^{n−1} ≡ 1/n in the equation f(X) ≡ 0, one gets X ≡ n/(1 − n). Hence there can be at most one double root mod p for each ramified prime p. This shows that the inertia subgroup at p is either trivial, or is of order two, generated by a transposition. But G = Gal(f) is generated by its inertia subgroups, because Q has no nontrivial unramified extension. By Selmer’s result, G is transitive; we have just shown that G is generated by transpositions; hence G = S_{n} by lemma 4.4.4.

4.5. The alternating group A_{n} 43

Many more examples can be found in the literature. For instance, the “truncated exponential”

1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + … + \frac{x^{n}}{n!}

has Galois group S_{n} when n ≢ 0 (mod 4), and Galois group A_{n} otherwise (I. Schur).

Exercises:

1. Let Y_{n} be the subvariety of P_{n−1} defined by the homogeneous equations

X_{1}^{i} + ⋯ + X_{n}^{i} = 0 for i = 1, 2, …, n − 2.

1.1. Y_{n} is an absolutely irreducible smooth curve, whose genus g_{n} is given by:

g_{n} = 1 + (n − 2)!\frac{n^{2} − 5n + 2}{4}.

(e.g. g_{3} = g_{4} = 0, g_{5} = 4, g_{6} = 49, g_{7} = 481, …)

1.2. The quotient of Y_{n} by S_{n} (acting by permutation of coordinates) is isomorphic to P_{1}.

1.3. The Galois covering Y_{n} ⟶ P_{1} is the Galois closure of the degree n covering given by the polynomial f(X) = X^{n} − X^{n−1}.

2. In characteristic 11, show that the equation

X^{11} + 2X^{9} + 3X^{8} − T^{8} = 0

is an unramified extension of A^{1} whose Galois closure has for Galois group the Mathieu group M_{11}. (Hint: reduce mod 11 the equation of [Ma4], after dividing the X-variable by 11^{1/4}.)

4.5 The alternating group A_{n}

One exhibits the alternating group A_{n} as a Galois group over Q(T) by using the following lemma (“double group trick”).

Lemma 4.5.1 Let G be the Galois group of a regular extension K/k(T), ramified at most at three places which are rational over k, and let H be a subgroup of G of index 2. Then the fixed field K_{1} of H is rational. (In particular, if k = Q, then H has property Gal_{T}.)

Because of the conditions on the ramification, the curve corresponding to K_{1} has genus zero, and has a k-rational point. The lemma follows.

For example, the polynomials with ramification type (n, n − 1, 2) discussed in the previous section can be used to construct A_{n}-extensions of Q(T). More precisely, let us change variables, and put

h(X, T) = (n − 1)X^{n} − nX^{n−1} + T.

Then the discriminant of h (with respect to X) is

Δ(h) = (−1)^{\frac{n(n−1)}{2}} n^{n}(n − 1)^{n−1}T^{n−2}(T − 1).

Up to square factors, we have

Δ(h) ∼ \{ \begin{matrix}(−1)^{\frac{n}{2}} (n − 1)(T − 1) & if n is even; \\ (−1)^{\frac{n−1}{2}} nT (T − 1) & if n is odd. \end{matrix}

Hence the equation D^{2} = Δ defines a rational curve. For example, if n is even, by replacing T by 1 + (−1)^{\frac{n}{2}} (n − 1)T^{2}, we get the equation

(n − 1)X^{n} − nX^{n−1} + 1 + (−1)^{\frac{n}{2}} (n − 1)T^{2} = 0,

which gives rise to a regular Galois extension of Q(T) with Galois group A_{n}.

Hilbert’s original construction [Hi] was somewhat different. For the sake of simplicity, we reproduce it here only in the case where n = 2m is even. Choose a polynomial

g(X) = nX ∏_{i=1}^{m−1}(X − β_{i})^{2},

with the β_{i} distinct and non-zero. Then, take f(X) so that df/dX = g. Assume that the f(β_{i}) are all distinct, and distinct from f(0). Then f has ramification type (n, 2, 3, 3, …, 3). Hence its Galois group is S_{n} by lemma 4.4.4. But then the quadratic subfield fixed by A_{n} is only ramified at the two places ∞ and 0; hence it is a rational field.

Exercise: Show that the condition that the f(β_{i}) are ≠ f(0) can be suppressed.

4.6 Finding good specializations of T

Let f be a polynomial over Q(T) with splitting field a regular G-extension of Q(T). Although Hilbert’s irreducibility theorem guarantees that for “most” values of t, the specialized polynomial f(X, t) will have Galois group G over Q, it does not give a constructive method for finding , say, an infinite number of such t’s.

4.6. Finding good specializations of T 45

For p ∉ S, where S denotes a suitable finite set of primes, the equation f(X, T) = 0 can be reduced mod p. If p is large enough (see exercise below), then all conjugacy classes in G occur as Frobenius elements at t for some t ∈ F_{p}. Letting C_{1}, …, C_{h} be the conjugacy classes of G, one can thus find distinct primes p_{1}, …, p_{h} and points t_{1}, …, t_{h}, with t_{i} ∈ F_{p_{i}}, such that Frob_{p_{i}}(f(X, t_{i})) = C_{i}. Specializing T to any t ∈ Q such that t ≡ t_{i} (mod p_{i}) for all i gives a polynomial f(X) = f(X, t) whose Galois group over Q intersects each of the conjugacy classes C_{i}, and hence is equal to G, by the following elementary result:

Lemma 4.6.1 (Jordan, [J2]) Let G be a finite group, and H a subgroup of G which meets every conjugacy class of G. Then H = G.

Indeed, if H is a subgroup of G, then the union of the conjugates of H in G has at most

1 + (G : H)(|H| − 1) = |G| − ((G : H) − 1)

elements.

Exercises:

1. Use lemma 4.6.1 to show that every finite division algebra is commutative (cf. Bourbaki, A.VIII, §11, no.1).

2. Show that lemma 4.6.1 can be reformulated as: “Every transitive subgroup of S_{n}, n ≥ 2, contains a permutation without fixed point”. (This is how the lemma is stated in Jordan, [J2].)

3. Let π : X ⟶ Y be a G-covering of absolutely irreducible projective smooth curves over F_{p}. Let N be the number of geometric points of X where π is ramified, and let g be the genus of X. Assume that 1 + p − 2g\sqrt{p} > N. Show that, for every conjugacy class c in G, there is a point t ∈ Y (F_{p}) over which π is unramified, and whose Frobenius class in G is c. (Apply Weil’s bound to the curve X, twisted by an element of c.)

46 Chapter 4. Galois extensions of Q(T)

Galois extensions of Q(T) given by torsion on elliptic curves

5.1 Statement of Shih’s theorem

Consider an elliptic curve E over Q(T) with j-invariant equal to T, e.g. the curve defined by the equation

y^{2} + xy = x^{3} − \frac{36}{T − 1728}x − \frac{1}{T − 1728}.

(Any other choice of E differs from this one by a quadratic twist only.) By adjoining to Q(T) the coordinates of the n-division points of E, one obtains a Galois extension K_{n} of Q(T) with Gal(K_{n}/Q(T)) = GL_{2}(Z/nZ). More precisely, the Galois group of C · K_{n} over C(T) is SL_{2}(Z/nZ), and the homomorphism G_{Q(T)} ⟶ GL_{2}(Z/nZ) ⟶^{det} (Z/nZ)^{∗} is the cyclotomic character. Hence the extension K_{n} is not regular when n > 2: the algebraic closure of Q in K_{n} is Q(μ_{n}). So the method does not give regular extensions of Q(T) with Galois group PGL_{2}(F_{p}), nor PSL_{2}(F_{p}). Nevertheless, K-y. Shih was able to obtain the following result [Shih1], [Shih2]:

Theorem 5.1.1 There exists a regular extension of Q(T) with Galois group PSL_{2}(F_{p}) if (\frac{2}{p}) = −1, (\frac{3}{p}) = −1, or (\frac{7}{p}) = −1.

Shih’s theorem will be proved in §5.3.

Remark: It is also known that PSL_{2}(F_{p}) has property Gal_{T} when (\frac{5}{p}) = −1; this follows from [Ml2] combined with th. 5.1.1.

5.2 An auxiliary construction

Let E be an elliptic curve defined over a field k of characteristic 0. The ptorsion of E, denoted E[p], is a two-dimensional F_{p}-vector space; let us call P E[p] the associated projective line P E[p] ≃ P_{1}(F_{p}). The actions of the Galois group G_{k} on E[p] and P E[p] give rise to representations

ρ : G_{k} ⟶ GL(E[p]) ≃ GL_{2}(F_{p})

and

ρ̄: G_{k} ⟶ PGL(P E[p]) ≃ PGL_{2}(F_{p}).

The determinant gives a well-defined homomorphism PGL_{2}(F_{p}) ⟶ F_{p}^{∗}/F_{p}^{∗2}. The group F_{p}^{∗}/F_{p}^{∗2} is of order 2 when p is odd (which we assume from now on). Hence the homomorphism ϵ_{p} = det ∘ρ̄ is a quadratic Galois character,

ϵ_{p} : G_{K} ⟶ {±1}.

The Weil pairing gives a canonical identification of ∧^{2}(E[p]) with μ_{p}, so that det ρ = χ, where χ is the pth cyclotomic character giving the action of G_{k} on the pth roots of unity. Therefore ϵ_{p} is the quadratic character associated to k(\sqrt{p^{∗}}), where p^{∗} = (−1)^{\frac{p−1}{2}} p.

Let K be a quadratic extension of k, and let σ be the involution in Gal(K/k). Let N be an integer ≥ 1 and E an elliptic curve defined over K such that E and E^{σ} are N-isogenous (i.e., there is a homomorphism ϕ : E ⟶ E^{σ} with cyclic kernel of order N). Assume for simplicity that E has no complex multiplication, so that there are only two N-isogenies, ϕ and −ϕ, from E to E^{σ}.

If p |̸ N, the maps ϕ, −ϕ induce isomorphisms (also denoted ϕ, −ϕ) from E[p] to E^{σ}[p]. By passing to the projective lines, one gets an isomorphism ϕ : P E[p] ⟶ P E^{σ}[p] which is independent of the choice of the N-isogeny E ⟶ E^{σ}, and commutes with the action of G_{K}. We define a map ρ_{E,N} : G_{k} ⟶ PGL(P E[p]) as follows:

1. If s belongs to the subgroup G_{K} of G_{k}, then ρ_{E,N}(s) is defined via the natural action of G_{K} on P E[p].

2. If s ∈ G_{k} − G_{K}, then the image of s in G_{K/k} is σ. Hence s gives an isomorphism P E[p] ⟶ P E^{σ}[p], and ρ(s) : P E[p] ⟶ P E[p] is defined by composing this isomorphism with ϕ^{−1} : P E^{σ}[p] ⟶ P E[p].

(In other words, ρ_{E,N} describes the action of G_{k} on the pairs of points (x, y) in P E[p] × P E^{σ}[p] which correspond to each other under the isogeny ϕ.) One checks easily that the map

ρ_{E,N} : G_{k} ⟶ PGL(P E[p])

so defined is a homomorphism.

As before, the projective representation ρ_{E,N} of G_{k} gives a character

ϵ_{E,N} : G_{k} ⟶ ±1.

Let ϵ_{K} denote the character G_{k} ⟶ ±1 corresponding to the quadratic extension K/k. Recall that ϵ_{p} : G_{k} ⟶ ±1 corresponds to k(\sqrt{p^{∗}})/k.

Proposition 5.2.1 We have :

ϵ_{E,N} = \{ \begin{matrix}ϵ_{p} & if (\frac{N}{p} ) = 1 \\ ϵ_{p}ϵ_{K} & if (\frac{N}{p} ) = −1 \end{matrix}

Corollary 5.2.2 If K = k(\sqrt{p^{∗}}), and (\frac{N}{p} ) = −1, then ϵ_{E,N} = 1, i.e., the image of ρ_{E,N} is contained in PSL_{2}(F_{p}).

Proof of 5.2.1: If s ∈ G_{K}, then ρ_{E,N}(s) acts on ∧^{2}E[p] ≃ μ_{p} by χ(s). Hence ϵ_{E,N}(s) = (\frac{χ(s)}{p}) = ϵ_{p}(s). If s ∉ G_{K}, then using the fact that the homomorphism ϕ^{−1} : E^{σ}[p] ⟶ E[p] induces multiplication by N^{−1} on

μ_{p} ≃ ∧^{2}(E^{σ}[p]) ≃ ∧^{2}(E[p])

(where the group operation on μ_{p} is written additively), one finds that

det ρ_{E,N}(s) = N^{−1}χ(s).

Hence ϵ_{E,N}(s) = (\frac{N}{p} ) ϵ_{p}(s). This completes the proof.

5.3 Proof of Shih’s theorem

Let K_{N} denote the function field of the modular curve X_{0}(N) of level N over Q,

K_{N} = Q[j_{1}, j_{2}]/F_{N}(j_{1}, j_{2}),

where F_{N} is the (normalized) polynomial relating the j-invariants of N-iso-genous elliptic curves (cf. [F]). The Fricke involution W_{N} acts on K_{N} by interchanging j_{1} and j_{2}. Let X_{0}(N)^{p} be the twist of X_{0}(N) by the quadratic character ϵ_{p}, using the involution W_{N}. The function field of X_{0}(N)^{p} can be described as follows: let k_{N} be the field of invariants of W_{N} in K_{N}, i.e., the function field of X_{0}(N)^{+}. Then K_{N} and k_{N}(\sqrt{p^{∗}}) are disjoint quadratic extensions of k_{N}. The third quadratic extension, k, of k_{N} contained in K = K_{N}(\sqrt{p^{∗}}) is the function field of X_{0}(N)^{p}

K
↗ ↑ ↖
k_{N}(\sqrt{p^{∗}})  k  K_{N}.
↖ ↑ ↗
k_{N}

Now, let E denote an elliptic curve over K with j-invariant j_{1}. Then E^{σ} has j-invariant j_{2}, and hence E and E^{σ} are N-isogenous. Assuming (\frac{N}{p} ) = −1, cor. 5.2.2 gives a projective representation G_{k} ⟶ PSL_{2}(F_{p}). It is not hard to see that this representation is surjective, and gives a regular extension of k.

Remark: Instead of taking E over K one can take it over K_{N}. One thus gets a projective representation G_{k_{N}} ⟶ PGL_{2}(F_{p}) and prop. 5.2.1 shows that it is surjective, and gives a regular extension if (\frac{N}{p}) = −1. Since k_{N} is known to be isomorphic to Q(T) if N belongs to the set

S = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71},

this shows that PGL_{2}(F_{p}) has property Gal_{T} provided there exists N ∈ S with (\frac{N}{p}) = −1 (a computer search shows that this is true for all p < 5329271).

We are interested in values of N for which k is isomorphic to Q(T), i.e., such that:

1. X_{0}(N) has genus 0.

2. X_{0}(N)^{p} has a Q-rational point.

Assuming that N is not a square, condition 1 implies that N = 2, 3, 5, 6, 7, 8, 10, 12, 13, 18. Condition 2 is satisfied when N = 2, 3, 7 (and also for N = 6, (\frac{2}{p}) = 1 and N = 10, (\frac{5}{p}) = 1, cf. [Shih1]). More precisely:

Proposition 5.3.1 For N = 2, 3, or 7, the two fixed points of W_{N} in X_{0}(N) are rational over Q.

(Hence, these points stay rational on X_{0}(N)^{p}, which is thus isomorphic to P_{1} over Q; this concludes the proof of th. 5.1.1.)

Let us give two proofs of prop. 5.3.1.

First proof: Let O be an order of class number 1 in a quadratic imaginary field, and E_{O} an elliptic curve with endomorphism ring O. Assume that the unique prime ramified in the field is N, and that O contains an element π with ππ̄ = N. This is indeed possible for N = 2, 3, 7:

N O  π
2 Z[i]  1 + i
Z[\sqrt{−2}] \sqrt{−2}
3 Z [\frac{1+\sqrt{−3}}{2}] \sqrt{−3}
Z[\sqrt{−3}] \sqrt{−3}
7 Z [\frac{1+\sqrt{−7}}{2}] \sqrt{−7}
Z[\sqrt{−7}] \sqrt{−7}

By the theory of complex multiplication, the pair (E_{O}, π) then defines a rational point of X_{0}(N) which is fixed under W_{N}.

Second proof: Let Δ be the discriminant modular form

Δ(q) = q ∏(1 − q^{n})^{24}, q = e^{2πiz},

and let f be the power series defined by

\begin{aligned}f & = & (Δ(z)/Δ(Nz))^{\frac{1}{N−1}} \\ & = & q^{−1} ∏_{{n≢0(N) \atop n≥1}}(1 − q^{n})^{2m}, where m = \frac{12}{N − 1}. \end{aligned}

One can prove that f generates K_{N} over Q, and that f^{W_{N}} = N^{m}/f. For N = 2, 3, and 7 we have m = 12, 6, 2, and N^{m} is a square. This shows that the fixed points of W_{N} are rational.

One can also show that the quaternion algebra corresponding to X_{0}(N)^{p} is given by (N^{m}, p^{∗}). More generally:

Exercise: Let F be a field, and let X be the variety obtained by twisting P_{1} by the quadratic character attached to the extension F (\sqrt{b}) and the involution of P_{1} x ↦ a/x. Then the quaternion invariant of this curve is the element (a, b) of Br_{2}(F).

The ramification in the Shih covering constructed above is as follows:

N=2: The ramification is of type (2, p, p). One ramification point is rational over Q, with inertia group of order 2 generated by the element ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). The two others are defined over Q(\sqrt{p^{∗}}) and are conjugate to each other. Their inertia groups, of order p, correspond to the two different conjugacy classes of unipotent elements in PSL_{2}(Z), ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & α \\ 0 & 1 \end{matrix} ), where α is a non-quadratic residue mod p.

N=3: The situation is similar; we have ramification of type (3, p, p), the inertia group of order 3 being generated by the element ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ). N=7: The ramification is of type (3, 3, p, p); with a suitable choice of coordinates, the ramification points are located at ±\sqrt{−27p^{∗}} and ±\sqrt{p^{∗}}.

Exercise: Show that the statement of th. 5.1.1 remains true when PSL_{2}(F_{p}) is replaced by PSL_{2}(Z/p^{n}Z).

5.4 A complement

Assume as before that (\frac{N}{p} ) = −1. Then there is a regular Galois covering C ⟶ X_{0}(N)^{p} with Galois group G = PSL_{2}(F_{p}) by the above. In Shih’s theorem, one takes N such that X_{0}(N) has genus 0. One might try to exploit the case where X_{0}(N) is of genus 1; for if the twisted curve X_{0}(N)^{p} has a Qrational point, and is of rank > 0 (where X_{0}(N)^{p} is viewed as an elliptic curve over Q by fixing this rational point as origin) then the following variant of Hilbert’s irreducibility theorem allows us to deduce the existence of infinitely many extensions of Q with Galois group G:

Proposition 5.4.1 Let C ⟶ E be a regular Galois covering with group G, where E is an elliptic curve over a number field K. Assume that for every proper subgroup H of G containing the commutator subgroup (G, G), the corresponding G/H-covering of E is ramified at least at one point. Then all P ∈ E(K) except a finite number have property Irr(P) of §3.3.

The hypothesis in the proposition implies that for all subgroups H ≠ G, the covering C/H ⟶ E is ramified somewhere, since the only unramified coverings of E are abelian. Hence the genus of C_{H} = C/H is at least 2; by Faltings’ theorem, C_{H} has finitely many rational points. Let S_{K} ⊂ E(K) be the union of the ramification points and the images of the C_{H}(K). This is a finite set, and if P ∉ S_{K}, then property Irr(P) is satisfied. This proves the proposition.

The above result was first obtained by Néron (Proc. Int. Cong. 1954) in a weaker form, since Mordell’s conjecture was still unproved at that time.

Corollary 5.4.2 Assume E(K) is infinite. Then, there are infinitely many linearly disjoint Galois extensions of K with Galois group G.

Proof: If L is any finite extension of K, we can find P ∈ E(K), P ∉ S_{L} (where S_{L} is defined as above).The property Irr(P) is then satisfied both over K and over L. The corresponding G-extension K_{P} is then linearly disjoint from L. The corollary follows.

Example: For N = 11, p = 47, the twisted elliptic curve X_{0}(N)^{p} has rank 2 over Q. Hence there are infinitely many extensions of Q with Galois group PSL_{2}(F_{47}). (An explicit example has been written down by N. Elkies.)

Problem: Is it possible to generalize prop. 5.4.1 to abelian varieties? More precisely, let π : C ⟶ A be a finite covering of an abelian variety A. Assume that a) A(K) is Zariski dense in A; b) π is ramified (i.e., C is not an abelian variety). Is it true that π(C(K)) is “much smaller” than A(K), i.e., that the number of points of logarithmic height ≤ N in π(C(K)) is o(N^{ρ/2}), where ρ = rank A(K)? There is a partial result in this direction in the paper of A. Néron referred to above.

5.5 Further results on PSL_{2}(F_{q}) and SL_{2}(F_{q}) as Galois groups

Concerning the groups PSL_{2} and SL_{2} over finite fields, there are the following results:

1. PSL_{2}(F_{q}), for q not a prime, is known to have property Gal_{T} when:

(a) q = 4 and q = 9, because PSL_{2}(F_{q}) is isomorphic to A_{5} and A_{6} respectively in these cases.

(b) q = 8, by a result of Matzat ([Ma3]).

(c) q = 25, by a result of Pryzwara ([Pr]).

(d) q = p^{2}, for p prime, p ≡ ±2 (mod 5), cf. [Me1].

2. SL_{2}(F_{q}) is known to have property Gal_{T} for:

(a) q = 2, 4, 8, for then SL_{2}(F_{q}) ≈ PSL_{2}(F_{q}).

(b) q = 3, 5 or 9, for then SL_{2}(F_{q}) is isomorphic to Ã_{4}, Ã_{5}, or Ã_{6}, which are known to have property Gal_{T}, cf. [Me2], or §9.3.

(c) q = 7, Mestre (unpublished).

It seems that the other values of q have not been treated (not even the case q = 11).

There are a few examples of Galois extensions of Q with Galois group SL_{2}(F_{2^{m}}) for m = 1, 2, …, 16. Their construction is due to Mestre (unpublished), who uses the representations of G_{Q} given by modular forms mod 2 of prime level ≤ 600.

54 Chapter 5. Elliptic curves

Galois extensions of C(T)

6.1 The GAGA principle

Our goal in this chapter is to construct Galois extensions of C(T), using the tools of topology and analytic geometry. To carry out this program, we need a bridge between analysis and algebra.

Theorem 6.1.1 (GAGA principle) Let X, Y be projective algebraic varieties over C, and let X^{an}, Y^{an} be the corresponding complex analytic spaces. Then 1. Every analytic map X^{an} ⟶ Y^{an} is algebraic.

2. Every coherent analytic sheaf over X^{an} is algebraic, and its algebraic cohomology coincides with its analytic one.

For a proof, see [Se4] or [SGA1], exposé XII. In what follows, we will allow ourselves to write X instead of X^{an}.

Remarks:

1. The functor X ↦ X^{an} is the “forgetful” functor which embeds the category of complex projective varieties into the category of complex analytic spaces. Th. 6.1.1 implies that it is fully faithful.

2. By the above, there is at most one algebraic structure on a compact analytic space which is compatible with it.

3. Th. 6.1.1 implies Chow’s theorem: every closed analytic subspace of a projective algebraic variety is algebraic.

4. The analytic map exp : G_{a} ⟶ G_{m}, where

G_{a} = P_{1} − {∞}, and G_{m} = P_{1} − {0, ∞},

is not algebraic; hence the hypothesis that X is projective is essential.

Exercise: If X and Y are reduced varieties of dimension 1, prove that any analytic isomorphism of X on Y is algebraic; disprove this for non-reduced varieties.

56 Chapter 6. Galois extensions of C(T)

Theorem 6.1.2 (Riemann) Any compact complex analytic manifold of dimension 1 is algebraic.

The proof is easy, once one knows the finiteness of the (analytic) cohomology groups of coherent sheaves, see e.g. [Fo], chap. 2. A generalization for a broader class of varieties is given by a theorem of Kodaira:

Theorem 6.1.3 (see e.g. [GH], ch. I, §4) Every compact Kähler manifold X whose Kähler class is integral (as an element of H^{2}(X, R)) is a projective algebraic variety.

In the above theorems, the compactness assumption is essential. For coverings, no such assumption is necessary:

Theorem 6.1.4 Let X be an algebraic variety over C, and let π : Y ⟶ X be a finite unramified analytic covering of X. Then there is a unique algebraic structure on Y compatible with its analytic structure and with π.

The proof is given in [SGA1], exposé XII. Let us explain the idea in the case X = X̄ − Z, where X̄ is an irreducible projective normal variety, and Z a closed subspace. One first uses a theorem of Grauert and Remmert [GR] to extend Y ⟶ X to a ramified analytic covering Ȳ ⟶ X̄. Such a covering corresponds to a coherent analytic sheaf of algebras over X̄. Since X̄ is projective, one can apply th. 6.1.1, and one then finds that this sheaf is algebraic, hence so are Ȳ and Y.

(The extension theorem of Grauert and Remmert used in the proof is a rather delicate one - however, it is quite easy to prove when dim X = 1, which is the only case we shall need.)

The GAGA principle applies to real projective algebraic varieties, in the following way: we may associate to any such variety X the pair (X^{an}, s), where X^{an} = X(C) is the complex analytic space underlying X, and

s : X^{an} ⟶ X^{an}

is the anti-holomorphic involution given by complex conjugation on X. The real variety X can be recovered from the data (X^{an}, s) up to a unique isomorphism. Furthermore, any complex projective variety X together with an anti-holomorphic involution s determines a projective variety over R: for by GAGA, s is an algebraic isomorphism from X to X̄, the conjugate variety of X, and hence gives descent data which determines X as a variety over R. Similar remarks apply to coherent sheaves, and coverings: for example, giving a finite covering of X over R is equivalent to giving a finite covering of the complex analytic space X(C), together with an anti-holomorphic involution which is compatible with the complex conjugation on X(C).

6.2 Coverings of Riemann surfaces

Recall that for all g ≥ 0, there exists, up to homeomorphism, a unique compact, connected, oriented surface X_{g} of genus g (i.e., with first Betti number 2g). This can be proved either by “cutting and pasting” or by Morse theory (in the differentiable category). The surface X_{g} has a standard description as a polygon with 4g edges labelled a_{1}, b_{1}, a_{1}^{−1}, b_{1}^{−1}, …, a_{g}, b_{g}, a_{g}^{−1}, b_{g}^{−1}, and identified in the appropriate manner. From this description one can see that the fundamental group π_{1}(X_{g}) relative to any base point has a presentation given by 2g generators a_{1}, b_{1}, …, a_{g}, b_{g}, and a single relation

a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1} = 1.

(To show this, one may express X_{g} as the union of two open sets U and V, where U is the polygon punctured at one point, which is homotopic to a wedge of 2g circles and has fundamental group the free group F on 2g generators, and V is a disk in the center of the polygon. Applying the van Kampen theorem, one finds that

π_{1}(X_{g}) = 1 ∗_{R} F = F/⟨R⟩,

where R is a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} … a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}. )

Let now P_{1}, …, P_{k} be distinct points of X_{g}, and let π_{1} be the fundamental group of X_{g} − {P_{1}, …, P_{k}}, relative to a base point x. Each P_{i} defines in π_{1} a conjugacy class C_{i} corresponding to “turning around P_{i} in the positive direction” (choose a disk D_{i} containing P_{i} and no other P_{j}, and use the fact that π_{1}(D_{i} −P_{i}) can be identified with Z). A similar argument as above proves that the group π_{1} has a presentation given by 2g + k generators a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{k} and the single relation:

a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}c_{1} ⋯ c_{k} = 1,

where the c_{i} belong to C_{i} for all i.

We denote this group by π_{1}(g, k). Observe that, if k ≥ 1, then π_{1}(g, k) is the free group on 2g +k −1 generators. In the applications, we shall be mainly interested in the case g = 0, k ≥ 3.

6.3 From C to Q̄

Let K be an algebraically closed field of characteristic 0, X a projective smooth curve of genus g over K, and let P_{1}, …, P_{k} be distinct points in X(K). The algebraic fundamental group of X − {P_{1}, …, P_{k}} may be defined as follows. Let \overline{K(X)} be an algebraic closure of the function field K(X) of X over K, and let Ω ⊂ \overline{K(X)} be the maximal extension of K(X) unramified outside the points P_{1}, …, P_{k}. The algebraic fundamental group is the Galois group of Ω over K(X), namely,

π_{1}^{alg}(X − {P_{1}, …, P_{k}}) = Gal(Ω/K(X)).

It is a profinite group. Let us denote it by π.

Theorem 6.3.1 The algebraic fundamental group π is isomorphic to the profinite completion of the topological fundamental group :

π ≃ π̂_{1}(g, k).

(Recall that the profinite completion Γ̂ of a discrete group Γ is the inverse limit of the finite quotients of Γ.)

Proof: Let first E be a finite extension of K(X) with Galois group G, let Q_{i} be a place of E above P_{i}, and let I(Q_{i}) denote the inertia subgroup of G at Q_{i}. Since the ramification is tame, we have a canonical isomorphism:

I(Q_{i}) ≃ μ_{e_{i}},

where e_{i} is the ramification index at Q_{i}; this isomorphism sends an element s of I(Q_{i}) to sπ/π (mod π), where π is any uniformizing element at Q_{i}.

From now on, let Q_{i} be a place of Ω above P_{i}. The isomorphism above can be generalized to such infinite extensions, by passing to the limit: write Ω as a union of finite extensions E_{j} with ramification indices e_{j} at Q_{i}. Then one has:

I(Q_{i}) ≃ lim_{⟵} μ_{e_{j}}.

Choose a coherent system of roots of unity in K, i.e., a fixed generator z_{n} for each μ_{n}, such that z_{nm}^{m} = z_{n} for all n, m. This provides I(Q_{i}) with a canonical generator c(Q_{i}). Changing Q_{i} above P_{i} merely affects c(Q_{i}) by inner conjugation. Hence, the conjugacy class C_{i} of c(Q_{i}) depends only on P_{i}. (Changing the coherent system {z_{n}} – i.e., replacing it by {z_{n}^{α}}, where α ∈ Ẑ^{∗} = Aut Ẑ– replaces c(Q_{i}) by c(Q_{i})^{α}.)

Th. 6.3.1 is a consequence of the following more precise statement:

Theorem 6.3.2 There exists a choice of elements x_{j}, y_{j}, c_{i} in π, with c_{i} ∈ C_{i} for all i, such that:

1. x_{1}y_{1}x_{1}^{−1}y_{1}^{−1} ⋯ x_{g}y_{g}x_{g}^{−1}y_{g}^{−1}c_{1} ⋯ c_{k} = 1.

2. The map π_{1}(g, k) ⟶ π mapping the generators x_{j}, y_{j}, c_{i} of π_{1}(g, k) to the elements x_{j}, y_{i}, c_{i} extends to an isomorphism

π̂_{1}(g, k) ⟶^{∼} π.

In other words, π is presented as a profinite group by the generators x_{j}, y_{j}, c_{i} with the relations above. The proof will be done in two stages: first, for K = C, then, for arbitrary algebraically closed fields K of characteristic 0.

Step 1: Proof when K = C, with the standard choice of roots of unity, {z_{n}} = {e^{2πi/n}}. In this case, the result follows from the GAGA dictionary between algebraic and topological coverings (see th. 6.1.3 and [SGA1]).

Step 2: K arbitrary. One has the following general result

Theorem 6.3.3 Let V be an algebraic variety over an algebraically closed field K of characteristic 0, and let K′ be an algebraically closed extension of K. Then any covering of V over K′ comes uniquely from a covering defined over K.

(Note that this is not true in characteristic p, unless the coverings are tame. For example, the Artin-Schreier covering Y^{p} − Y = αT of the affine line with α ∈ K′, α ∉ K, does not come from a covering defined over K.)

Two coverings of V defined over K and K′-isomorphic are K-isomorphic; this is clear, e.g. by using a specialization argument. Next, one has to show that any covering which is defined over K′ can be defined over K. If K′ is of finite transcendence degree over K, then by using induction on tr.deg(K′/K), it is enough to show this for tr.deg(K′/K) = 1. In this case, a covering W ⟶ V over K′ corresponds to a covering W × C ⟶ V × C over K, where C is the curve over K corresponding to the extension K′/K. This can be viewed as a family of coverings W ⟶ V over K parameterized by C. But it is a general fact that in characteristic 0, such families of coverings are constant. (One can deduce this from the similar geometrical statement over C, where it follows from the fact that π_{1}(X × Y) = π_{1}(X) × π_{1}(Y).) By choosing a K-rational point c of C, on then gets a covering W_{c} ⟶ V defined over K which is K′-isomorphic to W.

Step 3: Using th. 6.3.3, we may replace (K, {z_{n}}) by (K_{1}, {z_{n}}), where tr.degK_{1}/Q < ∞. One can then find an embedding K_{1} ⟶ C which transforms the {z_{n}} into the {e^{2πi/n})} (irreducibility of the cyclotomic polynomials). Another application of th. 6.3.3 concludes the proof.

Remark: Let Γ be a discrete group, Γ̂ its profinite completion

Γ̂ = lim_{⟵} Γ/N

where N runs over the normal subgroups of Γ of finite index. The canonical homomorphism Γ ⟶ Γ̂ is universal for maps of Γ into profinite groups. It is not always true that this map is injective. There are examples of finitely presented Γ with Γ infinite but Γ̂ = {1}, e.g. the group defined by four generators x_{i} (i ∈ Z/4Z) with the four relations x_{i}x_{i+1}x_{i}^{−1} = x_{i+1}^{2} (G. Higman).

Problem: Can such a Γ be the fundamental group of an algebraic variety over C?

In the case of curves, which is the case we are mostly interested in, the map π_{1}(g, k) ⟶ π̂_{1}(g, k) is injective. This follows from the fact that π_{1}(g, k) is isomorphic to a subgroup of a linear group (e.g., SL_{3}(R) or even better, SL_{2}(R)); one then uses a well-known result of Minkowski and Selberg (see e.g. [Bor], p. 119).

6.4 Appendix: universal ramified coverings of Riemann surfaces with signature

Let X be a Riemann surface, S ⊂ X a finite set of points in X. Let us assign to each point P ∈ S an integer n_{P}, with 2 ≤ n_{P} < ∞. Such a set S along with a set of integers n_{P} is called a signature on X.

One defines a ramified covering subordinate to the signature (S; n_{P}) to be a holomorphic map f : Y ⟶ X, with the following properties:

1. If S_{Y} = f^{−1}(S), the map f : Y − S_{Y} ⟶ X − S is a topological covering.

2. Let P ∈ S and choose a disk D in X with D ∩ S = {P }. Then f^{−1}(D) splits into connected components D_{α}, and the restriction f_{α} of f to D_{α} is isomorphic to the map z ↦ z^{n_{α}}, for some n_{α} dividing n_{P}.

(When Y ⟶ X is a finite map, condition 2 just means that the ramification above P divides n_{P}.)

For every P ∈ S, let C_{P} be the conjugacy class “turning around P ” in the fundamental group π_{1}(X − S). If Y ⟶ X is as above, Y − f^{−1}(S) ⟶ X − S is a covering, and the action of C_{P}^{n_{P}} is trivial. Conversely, given a covering of X − S with that property, one shows (by an easy local argument) that it extends uniquely to a map Y ⟶ X of the type above.

Theorem 6.4.1 Let X, S, and n_{P} be given. Then there is a universal covering Y subordinate to the signature (S; n_{P}) with faithful action of

Γ = π_{1}(X − S)/N,

where N is the normal subgroup generated by the s^{n_{P}}, s ∈ C_{P}, P ∈ S. This Y is simply connected, and one has Y/Γ = X.

Sketch of proof: Let Y_{N} be the covering of X − S associated to N. This is a Galois covering, with Galois group π_{1}(X − S)/N. As above, one can complete it to a covering Y ⟶ X. One checks that Y is universal, simply connected, and that Γ acts properly on Y, with X = Y/Γ.

We apply this to the case X = X̄ − T, where X̄ is a compact Riemann surface of genus g, and T is a finite subset of X̄. Letting S = {P_{1}, …, P_{s}} and T = {P_{s+1}, …, P_{s+t}}, one associates to X a “generalized signature” on X̄, where the indices are allowed to take to value ∞, by setting

\{ \begin{matrix}n_{i} = n_{P_{i}} & if i ≤ s, \\ n_{i} = ∞ & otherwise. \end{matrix}

The corresponding group Γ is then defined by generators a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, … c_{s+t} and relations:

\{ \begin{matrix}a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}c_{1} ⋯ c_{s+t} = 1 \\ c_{1}^{n_{1}} = 1, …, c_{s}^{n_{s}} = 1. \end{matrix}

Example: In the case g = 0 and s+t = 3, the group Γ and the corresponding universal covering X̄ ⟶ P_{1}(C) can be constructed geometrically as follows. Set λ = 1/n_{1} + 1/n_{2} + 1/n_{3} (with the convention that 1/∞ = 0). If λ > 1, let X̄ be the Riemann sphere S_{2} equipped with its metric of curvature 1; if λ = 1, let X̄ be C with the euclidean metric; and if λ < 1, let X̄ be the Poincaré upper half plane, equipped with the metric of curvature −1. In each case, by the Gauss-Bonnet theorem there is a geodesic triangle on X̄ with angles π/n_{1}, π/n_{2}, and π/n_{3}. (The case n_{i} = ∞ gives an angle equal to 0, and the corresponding vertex is a “cusp” at infinity.) Denoting by s_{i} the reflection about the ith side (i = 1, 2, 3), the elements C_{1} = s_{2}s_{3}, C_{2} = s_{3}s_{1}, and C_{3} = s_{1}s_{2} satisfy the relation

C_{1}C_{2}C_{3} = 1

and the C_{i} are of order n_{i}. Hence the group Γ′ generated by the C_{i} is a quotient of Γ, by the map sending c_{i} to C_{i} (i = 1, 2, 3). In fact, one can show that the projection Γ ⟶ Γ′ is an isomorphism, and that X̄ with this action of Γ is the universal covering of th. 6.4.1. A fundamental domain for the action of Γ is given by the union of the geodesic triangle with one of its reflections about a side.

Theorem 6.4.2 The element c_{i} of Γ has order n_{i}, except in the following two cases :

1. g = 0, s + t = 1;

2. g = 0, s + t = 2, n_{1} ≠ n_{2}.

Consider first the case g = 0. This case can itself be divided into three subcases:

1. s + t = 2, and n_{1} = n_{2}. In this case Γ is isomorphic to a cyclic group of order n_{1} (if n_{1} = ∞, then Γ = Z), and the statement follows.

2. s + t = 3. The explicit construction of Γ above shows that the c_{i} are of order n_{i}.

3. s + t ≥ 4. One reduces this to case 2 by adding the relations c_{i} = 1 for a set I of indices with |I| = s + t − 3. One thus gets a surjective homomorphism onto a group of the type treated in case 2, and the c_{i} with i ∉ I are therefore of order n_{i}. By varying I, one obtains the result.

To treat the case g ≥ 1, it suffices to exhibit a group containing elements A_{1}, …, A_{g}, B_{1}, …, B_{g}, C_{1}, …, C_{s+t} satisfying the given relation for Γ, with the C_{i} of order n_{i}, (i = 1, … s + t). For this, one may choose elements C_{i} of order n_{i} in the special orthogonal group SO_{2}. Every element x of SO_{2} can be written as a commutator (y, z) in the orthogonal group O_{2}; for, choosing y ∈ SO_{2} such that y^{2} = x, and z ∈ O_{2} − SO_{2}, one has yzy^{−1}z^{−1} = y · y = x. Hence, it suffices to take A_{j}, B_{j} = 1 if j > 1, and A_{1}, B_{1} ∈ O_{2} such that

(A_{1}, B_{1}) = (C_{1}C_{2} ⋯ C_{s+t})^{−1}.

Assume that the signature (n_{1}, …, n_{s+t}) is not one of the exceptional cases in the theorem above. We then have a “universal” ramified covering Y which is simply connected. By the uniformization theorem (cf. [Fo]), such a Y is isomorphic to either P_{1}(C), C, or the upper half plane H. To tell which of these three cases occur, let us introduce the “Euler-Poincaré characteristic” E of the signature:

E = 2 − 2g − ∑_{i=1}^{s+t}(1 − \frac{1}{n_{i}}),

with the convention that \frac{1}{∞} = 0.

Theorem 6.4.3 1. If E > 0, then Y is isomorphic to P_{1}(C).

2. If E = 0, then Y is isomorphic to C.

3. If E < 0, then Y is isomorphic to the upper half plane H.

This can be proved using the Gauss-Bonnet formula.

1. Case 1 occurs only if g = 0 and the signature is:

• empty, corresponding to the identity covering P_{1}(C) ⟶ P_{1}(C).

• (n, n), n < ∞, corresponding to the cyclic covering z ↦ z^{n}.

• (2, 2, n), corresponding to Γ = D_{n}, the dihedral group of order 2n.

• (2, 3, 3), corresponding to Γ = A_{4} (tetrahedral group).

• (2, 3, 4), corresponding to Γ = S_{4} (octahedral group).

• (2, 3, 5), corresponding to Γ = A_{5} (icosahedral group).

2. Case 2 occurs for

• g = 1 and empty signature, corresponding to the covering of a complex torus by the complex plane, Γ = a lattice.

• g = 0 and signature (∞, ∞), corresponding to the map

exp : C ⟶ C, Γ = Z.

• signature (2, 4, 4), corresponding to Γ = group of affine motions preserving the lattice Z + Zi.

• signature (2, 3, 6), corresponding to Γ = group of affine motions preserving the lattice Z + (\frac{1+\sqrt{−3}}{2})Z.

• signature (3, 3, 3), corresponding to a subgroup of index 2 of the previous one.

• signature (2, 2, 2, 2), corresponding to coverings

C ⟶ T ⟶ P_{1},

where T is an elliptic curve and C ⟶ T is the universal unramified covering map.

3. Case 3 corresponds to the case where Γ is a discrete subgroup of PSL_{2}(R) which is a “Fuchsian group of the first kind”, cf. e.g. Shimura [Shim], Chap. 1; one then has X = H/Γ and X̄ is obtained by adding to X the set of cusps (mod Γ). (By a theorem of Siegel [Sie], such groups Γ can be characterized as the discrete subgroups of PSL_{2}(R) with finite covolume.) The generators c_{i} with n_{i} < ∞ are elliptic elements of the Fuchsian group Γ, the c_{i} with n_{i} = ∞ correspond to parabolic elements, and the a_{j} and b_{j} to hyperbolic elements.

64 Chapter 6. Galois extensions of C(T)

Rigidity and rationality on finite groups

7.1 Rationality

Let G be a finite group, Cl(G) the set of its conjugacy classes. Choose N such that every element of G has order dividing N; the group Γ_{N} = (Z/NZ)^{∗} acts on G by sending s to s^{α}, for α ∈ (Z/NZ)^{∗}, and acts similarly on Cl(G).

Let X(G) be the set of irreducible characters of G. They take values in Q(μ_{N}), and hence there is a natural action of Gal(Q(μ_{N})) ≃ Γ_{N} on X(G). The actions of Γ_{N} on Cl(G) and on X(G) are related by the formula

σ_{α}(χ)(s) = χ(s^{α}),

where σ_{α} ∈ Gal(Q(μ_{N})/Q) is the element sending the Nth roots of unity to their αth powers.

Definition 7.1.1 A class c in Cl(G) is called Q-rational if the following equivalent properties hold :

1. c is fixed under Γ_{N}.

2. Every χ ∈ X(G) takes values in Q (and hence in Z) on c. (The equivalence of these two conditions follows from the formula above.)

The rationality condition means that, if s ∈ c, then all of the generators of the cyclic group < s > generated by s are in c, i.e., are conjugate to s.

For instance, in the symmetric group S_{n}, every conjugacy class is rational. More generally, let K be any field of characteristic zero. The class c of

an element s ∈ G is called K-rational if χ(s) ∈ K for all χ ∈ X(G), or equivalently, if c^{α} = c for all α such that σ_{α} ∈ Gal(K(μ_{N})/K).

For example, the alternating group A_{5} has five conjugacy classes of order 1, 2, 3, 5, and 5 respectively. Let us call 5A and 5B the two conjugacy classes of exponent 5. (They can be distinguished as follows: given a five-cycle s, let α(s) be the permutation of {1, 2, 3, 4, 5} sending i to s^{i}(1). Then s is in the conjugacy class 5A if and only if α(s) is even.) If s ∈ 5A, then s^{−1} ∈ 5A and s^{2}, s^{3} ∈ 5B, so the classes 5A and 5B are not rational over Q. However, they are rational over the quadratic field Q(\sqrt{5}).

Remark: The formula σ_{α}(χ)(s) = χ(s^{α}) has an analogue for supercuspidal admissible representations of semi-simple p-adic groups. This can be deduced from the fact, proved by Deligne (cf. [De1]), that the character of such a representation is supported by the elliptic elements (note that these are the only elements s for which the notation s^{α} makes sense).

Rationality of inertia Let K be a local field with residue field k of characteristic zero, and let M be a Galois extension of K with group G. There is a unique maximal unramified extension L of K in M; it is Galois over K. Let I be the inertia subgroup of G, i.e., I = Gal(M/L). Since the ramification is tame, I is cyclic. In fact, there is a canonical isomorphism

I ⟶^{∼} μ_{e}(l),

where e denotes the ramification index of L/K, and l is the residue field of L. (This isomorphism sends σ ∈ I to σπ/π (mod π), where π is a uniformizing element of M.)

Proposition 7.1.2 The class in G of an element of I is rational over k.

Indeed, the group H = Gal(L/K) = Gal(l/k) acts on μ_{e}(l) in a natural way, and on I by conjugation. These actions are compatible with the isomorphism above. If now σ_{α} is an element of Gal(k(μ_{e})/k), there exists t ∈ H such that t acts on μ_{e}(l) by z ↦ z^{α}. If s ∈ I, then s^{α} and tst^{−1} have the same image in μ_{e}(l). Hence s^{α} = tst^{−1}. This shows that the class c of s is such that c^{α} = c, q.e.d.

Corollary 7.1.3 If k = Q, then the classes in I are rational in G.

Remark on the action of Γ_{N} on Cl(G) and X(G)

The Γ_{N}-sets Cl(G) and X(G) are the character sets of the étale Q-algebras Q ⊗ R(G) and ZQ[G], where R(G) is the representation ring of G (over Q̄), and ZQ[G] is the center of the group algebra Q[G]. These Γ_{N}-sets are easily proved to be “weakly isomorphic” in the sense of exercise 1 below. However, they are not always isomorphic (see [Th1]).

Exercises:

1. Let X and Y be finite sets on which acts a finite group Γ.

(a) Show the equivalence of the following properties:

i. |X^{C}| = |Y^{C}| for every cyclic subgroup C of Γ;

ii. |X/C| = |Y/C| for every cyclic subgroup C of Γ;

iii. |X/H| = |Y/H| for every subgroup H of Γ;

iv. The Q-linear representations of Γ defined by X and Y are isomorphic. If those properties are true, the Γ-sets X and Y are said to be weakly isomorphic.

(b) Show that weak isomorphism is equivalent to isomorphism when Γ is cyclic (“Brauer’s Lemma”). Give an example where Γ is a non-cyclic group of order 4, |X| = |Y | = 6, and X and Y are weakly isomorphic but not isomorphic.

2. Use exerc. 1.(b) to prove that, if G is a p-group, p ≠ 2, then Q ⊗ R(G) is isomorphic to ZQ[G].

3. Extend prop. 7.1.2 to the case where k has characteristic p > 0, assuming that M/K is tame.

7.2 Counting solutions of equations in finite groups

Let G be a compact group, equipped with its Haar measure of total mass 1. Let χ be an irreducible character of G, with ρ : G ⟶ GL(E) the corresponding linear representation. By Schur’s lemma, ∫_{G} ρ(txt^{−1}) dt is a scalar multiple λ · 1_{E} of the identity in GL(E). Taking traces gives χ(x) = λχ(1), hence

∫_{G} ρ(txt^{−1})dt = \frac{χ(x)}{χ(1)}1_{E}.

Multiplying on the right by ρ(y), we get:

∫_{G} ρ(txt^{−1}y)dt = \frac{χ(x)}{χ(1)}ρ(y).

Taking traces gives:

∫_{G} χ(txt^{−1}y)dt = \frac{χ(x)χ(y)}{χ(1)}.

This formula extends by induction to k elements x_{1}, …, x_{k}:

∫_{G} ⋯ ∫_{G} χ(t_{1}x_{1}t_{1}^{−1} ⋯ t_{k}x_{k}t_{k}^{−1}y) dt_{1} dt_{2} ⋯ dt_{k} = \frac{χ(x_{1}) ⋯ χ(x_{k})χ(y)}{χ(1)^{k}} (7.1)

More generally, let ϕ be a class function on G,

ϕ = ∑_{χ} c_{χ}χ,

where c_{χ} is the inner product of ϕ with the irreducible character χ,

c_{χ} = ∫_{G} ϕ(x)χ̄(x) dx,

and assume that ∑ |c_{χ}|χ(1) < ∞ so that the sum is normally convergent. By summing over all characters in formula 7.1, we see that the integral

I(ϕ) = ∫_{G} ⋯ ∫_{G} ϕ(t_{1}x_{1}t_{1}^{−1} ⋯ t_{k}x_{k}t_{k}^{−1}y)dt_{1} dt_{2} ⋯ dt_{k}

is equal to

∑_{χ} c_{χ} \frac{χ(x_{1}) ⋯ χ(x_{k})χ(y)}{χ(1)^{k}}.

Let us compute I(ϕ) in the case where G is a finite group and ϕ is the Dirac function which is 1 at the identity element, and 0 elsewhere. One has

ϕ = \frac{1}{|G|} ∑_{χ} χ(1)χ,

and hence c_{χ} = χ(1)/|G|. Given elements x_{1}, …, x_{k}, and y in G, the value of I(ϕ) is N/|G|^{k}, where N is the number of solutions (t_{1}, …, t_{k}) of the equation

t_{1}x_{1}t_{1}^{−1} ⋯ t_{k}x_{k}t_{k}^{−1}y = 1.

Hence

N = |G|^{k} ∑_{χ} \frac{χ(1)}{|G|} \frac{χ(x_{1}) ⋯ χ(x_{k})χ(y)}{χ(1)^{k}}

= |G|^{k−1} ∑_{χ} \frac{χ(x_{1}) ⋯ χ(x_{k})χ(y)}{χ(1)^{k−1}}.  (7.2)

Let C_{1}, …, C_{k} denote the conjugacy classes of the elements x_{1}, …, x_{k}, and define n(C_{1}, …, C_{k}) to be the number of solutions (g_{1}, …, g_{k}) of the equation

g_{1}g_{2} ⋯ g_{k} = 1, g_{i} ∈ C_{i}.

Letting Z_{i} be the order of the centralizer of an element of C_{i}, one has

n(C_{1}, …, C_{k}) = \frac{N}{Z_{1} ⋯ Z_{k}} .

By applying formula 7.2 and observing that Z_{i} = |G|/|C_{i}|, one therefore gets:

Theorem 7.2.1 The number n = n(C_{1}, …, C_{k}) of solutions of the equation g_{1} ⋯ g_{k} = 1, g_{i} ∈ C_{i}, is given by

n = \frac{1}{|G|}|C_{1}| ⋯ |C_{k}| ∑_{χ} \frac{χ(x_{1}) ⋯ χ(x_{k})}{χ(1)^{k−2}},

where x_{i} is a representative of the conjugacy class C_{i}, and χ runs through all the irreducible characters of G.

Exercises:

1. Let G be a compact group and let ρ be an irreducible representation of G with character χ. Show that

∫ ∫ ρ(txt^{−1}x^{−1})dx dt = 1/χ(1)^{2}.

(Hint: Show that the left hand side is an intertwining operator for ρ, and hence is a scalar by Schur’s lemma. Then compute its trace.) Conclude that

∫ ∫ ρ(txt^{−1}x^{−1} · y)dx dt = ρ(y)/χ(1)^{2}, for all y ∈ G,

and that

∫ ∫ χ(txt^{−1}x^{−1} · y)dx dt = χ(y)/χ(1)^{2}.

Hence show by induction that:

∫ ⋯ ∫ χ(t_{1}x_{1}t_{1}^{−1}x_{1}^{−1} ⋯ t_{g}x_{g}t_{g}^{−1}x_{g}^{−1} · y)dx_{1} dt_{1} ⋯ dx_{g} dt_{g} = χ(y)/χ(1)^{2g}.

2. Suppose now that G is finite. We denote the commutator uvu^{−1}v^{−1} of u and v by (u, v). Let g ≥ 0, and let z_{j} be fixed elements of G, for j = 1, …, k. For y ∈ G, let N = N(g, z_{j}, y) denote the number of tuples of elements of G,

(u_{1}, v_{1}, …, u_{g}, v_{g}, t_{1}, …, t_{k}),

such that

(u_{1}, v_{1}) ⋯ (u_{g}, v_{g})t_{1}z_{1}t_{1}^{−1} ⋯ t_{k}z_{k}t_{k}^{−1} = y^{−1}.

Using exerc. 1, show that:

N = |G|^{2g+k−1} ∑_{χ} χ(z_{1}) ⋯ χ(z_{k})χ(y)/χ(1)^{2g+k−1}.

3. Show that an element y of G is a product of g commutators if and only if the sum

∑_{χ} χ(y)/χ(1)^{2g−1}

is non-zero. In particular, y is a commutator if and only if ∑ χ(y)/χ(1) ≠ 0. It is a well-known conjecture of Ore that every element in a finite non-abelian simple group is a commutator. The reader may wish to verify this (for one of the sporadic groups, say) by using the formula above and the character tables in [ATLAS].

Remark: Th. 7.2.1 can be used to compute the number of subgroups in a finite group which are isomorphic to the alternating group A_{5}. Indeed, A_{5} has a presentation given by three generators x, y, z, and relations

x^{2} = y^{3} = z^{5} = xyz = 1.

The problem therefore amounts to finding the number of solutions of the equation xyz = 1, where x, y z belong to conjugacy classes of exponent 2, 3, and 5 respectively. The same remark applies to S_{4}, A_{4}, and the dihedral groups, which have similar presentations, cf. th. 6.4.3, case 1.

7.3 Rigidity of a family of conjugacy classes

Let G be a finite group, and fix conjugacy classes C_{1}, …, C_{k} in G. Let Σ̄ = Σ̄(C_{1}, …, C_{k}) be the set of all (g_{1}, …, g_{k}), with g_{i} ∈ C_{i}, such that g_{1} ⋯ g_{k} = 1. (Hence, by the previous section, n(C_{1}, …, C_{k}) = |Σ̄|.) Let Σ = Σ(C_{1}, …, C_{k}) be the set of (g_{1}, …, g_{k}) in Σ̄ such that g_{1}, …, g_{k} generate the group G. The group G acts by conjugation on Σ and Σ̄.

Assume that the center of G is trivial (as is the case, for example, when G = S_{n}, n ≥ 3, or when G is a non-abelian simple group). Then the action of G on Σ is free: for, if g ∈ G fixes (g_{1}, …, g_{k}), it commutes with the g_{i}’s, and hence with all of G, since the g_{i}’s generate G; but then g = 1, because G is assumed to have trivial center.

One says that a k-tuple of conjugacy classes (C_{1}, …, C_{k}) is rigid if Σ ≠ ∅ and G acts transitively on Σ, i.e.,

|Σ| = |G|.

One says that (C_{1}, …, C_{k}) is strictly rigid if it is rigid, and Σ = Σ̄. The order of Σ̄ can be computed using the formula of the previous section:

|Σ̄| = \frac{1}{|G|}|C_{1}| ⋯ |C_{k}| ∑_{χ} χ(c_{1})χ(c_{2}) ⋯ χ(c_{k})/χ(1)^{k−2},

where c_{i} ∈ C_{i} for i = 1, …, k, and χ runs through the irreducible characters of G. If z_{i} is the order of the centralizer Z(c_{i}), this can also be written as:

|Σ̄| = \frac{|G|^{k−1}}{z_{1} … z_{k}} ∑_{χ} χ(c_{1})χ(c_{2}) ⋯ χ(c_{k})/χ(1)^{k−2}.  (7.3)

Rigidity is often proved in two steps:

1. Compute the order of Σ̄, by using formula 7.3 and the character table of G.

2. Compute the order of Σ̄ − Σ, by finding k-tuples in Σ̄ which do not generate all of G: for this, one uses a knowledge of the maximal subgroups of G (whenever possible).

Remarks:

1. Let σ be an outer automorphism of G. Suppose (C_{1}, …, C_{k}) is a rigid k-tuple of conjugacy classes. Then σ(C_{i}) ≠ C_{i} for some i. Indeed, suppose that σ preserves each C_{i}. Then, letting (g_{1}, …, g_{k}) be an element of Σ, it follows that (σg_{1}, …, σg_{k}) belongs to Σ. Since G acts transitively on Σ by inner conjugation, there exists g ∈ G such that

σg_{i} = gg_{i}g^{−1} for all i.

But the g_{i} generate G and hence σ is an inner automorphism, which contradicts the assumption.

2. In many cases, the term ∑_{χ} χ(C_{1}) ⋯ χ(C_{k})/χ(1)^{k−2} in formula 7.3 is not very large, the main contribution to the sum being given by the unit character χ = 1. For a rigid family of conjugacy classes, one might therefore expect that the order of magnitude of |C_{1}| ⋯ |C_{k}| is close to |G|.

As in §7.1, let α ∈ (Z/NZ)^{∗}, where N is a multiple of the exponent of G, so that (Z/NZ)^{∗} acts on Cl(G). Then one has:

Proposition 7.3.1

1. |Σ̄(C_{1}^{α}, …, C_{k}^{α})| = |Σ̄(C_{1}, …, C_{k})|;

2. |Σ(C_{1}^{α}, …, C_{k}^{α})| = |Σ(C_{1}, …, C_{k})|.

The first identity follows from formula 7.3 above, combined with the formula χ(c^{α}) = σ_{α}(χ)(c) of §7.1. The second is proved by induction on the order of G, as follows: if H is a subgroup of G, let Σ^{(H)}(C_{1} ∩ H, …, C_{k} ∩ H) denote the set of (g_{1}, …, g_{k}), with g_{i} ∈ C_{i} ∩ H for all i, such that g_{1} ⋯ g_{k} = 1 and the g_{i} generate H. (In general, the C_{i} ∩ H are not conjugacy classes in H, but unions of conjugacy classes.) The formula

Σ̄ − Σ(C_{1}, …, C_{k}) = ⋃_{H⊂G, H≠G} Σ^{(H)}(C_{1} ∩ H, …, C_{k} ∩ H)

supplies the induction step.

Corollary 7.3.2 If (C_{1}, …, C_{k}) is rigid (resp. strictly rigid), so is the family (C_{1}^{α}, …, C_{k}^{α}).

Remark: Here is another way to prove prop. 7.3.1. Let F be the group with presentation given by generators x_{1}, …, x_{k} and relation x_{1} ⋯ x_{k} = 1, and let F̂ be its profinite completion. Then one has:

Proposition 7.3.3 For each α ∈ Ẑ^{∗}, there is an automorphism θ of F̂ such that θ(x_{i}) belongs to the conjugacy class of x_{i}^{α}. (Equivalently, there exist elements y_{1}, …, y_{k} with y_{i} conjugate to x_{i}^{α}, satisfying y_{1} ⋯ y_{k} = 1, and generating F̂.)

Let us use the interpretation of F̂ as the algebraic fundamental group π of the projective line P_{1} with k points removed. By choosing a coherent system {z_{n}} of roots of unity in Q̄, one has an identification (cf. §6.3)

F̂ ⟶^{∼} π.

Replacing the system {z_{n}} by {z_{n}^{α}} gives a different isomorphism, and composing the two yields the desired automorphism θ of F̂.

Prop. 7.3.3 implies that |Σ(C_{1}^{α}, …, C_{k}^{α})| = |Σ(C_{1}, …, C_{k})|. For, the elements of Σ(C_{1}, …, C_{k}) are in one-one correspondence with the surjective homomorphisms F̂ ⟶ G sending each x_{i} to an element of C_{i}, while the elements of Σ(C_{1}^{α}, …, C_{k}^{α}) correspond to the surjective homomorphisms sending each x_{i} to an element in C_{i}^{α}. The automorphism θ : F̂ ⟶ F̂ induces a map Hom(F̂, G) ⟶ Hom(F̂, G) which gives a bijection between the two sets Σ(C_{1}, …, C_{k}) and Σ(C_{1}^{α}, …, C_{k}^{α}).

Exercise: Show that, if (C_{1}, …, C_{k}) is rigid (resp. strictly rigid), then so is the family (C_{σ(1)}, …, C_{σ(k)}) for any permutation σ of {1, …, k}.

7.4 Examples of rigidity

We give only a few such examples. For more, the reader should consult [Ma3] and the papers quoted there.

7.4.1 The symmetric group S_{n}

The symmetric group S_{n} (n ≥ 3) has conjugacy classes nA, 2A, and C^{(1)} corresponding to cycles of order n, 2, and n − 1 respectively. The triple (nA, 2A, C^{(1)}) is strictly rigid. For, giving an n-cycle x ∈ nA determines a cyclic arrangement of {1, …, n} (i.e., an oriented n-gon). Composing this permutation of order n with a transposition gives an (n − 1)-cycle if and only if the two vertices which are permuted are consecutive. Hence, the solutions x, y, z of the equation xyz = 1, with x, y, z cycles of order n, 2 and n − 1 respectively, are in one to one correspondence with the oriented n-gons with a distinguished edge. Any two such configurations can be tranformed into one another by a unique permutation in S_{n}; hence |Σ̄| = |G|. But Σ̄ = Σ, since (12) and (12 … n) are known to generate S_{n}; this shows that (nA, 2A, C^{(1)}) is strictly rigid.

More generally, consider the conjugacy classes nA, 2A, and C^{(k)}, where C^{(k)} is the class of the permutation

(1 … k)(k + 1 … n).

As before, an element (x, y, z) ∈ Σ̄ corresponds to an oriented n-gon with two distinguished vertices separated by k edges. If k ≠ n/2, any two such configurations can be transformed into one another by a unique permutation in S_{n}, and hence |Σ̄| = |G|. However, (nA, 2A, C^{(k)}) is not rigid in general; to get rigidity, one must assume that (k, n) = 1. In that case, any triple (x, y, z) in Σ̄ generates S_{n}: by relabelling if necessary, we may write x = (1 … n), and y = (1, k + 1). Since (k, n) = 1, the permutation x^{k} is still an n-cycle. By relabelling again, the group generated by x^{k} and y is isomorphic to the group generated by the permutations (1 … n) and (12). This in turn contains the group generated by (12), (23), …, (n − 1, n), which is equal to S_{n} by a well-known result (cf. §4.4).

Exercise: Check that |Σ̄| = n! for the conjugacy classes (2A, nA, C^{(1)}) by applying formula 7.3 of §7.3 (prove that the only non-zero terms come from the two characters of degree 1 of S_{n}).

7.4.2 The alternating group A_{5}

The alternating group A_{5} has unique conjugacy classes of order 2 and 3, denoted by 2A and 3A respectively. It has two conjugacy classes of order 5 which are rational over Q(\sqrt{5}) and conjugate to each other, denoted by 5A and 5B.

Proposition 7.4.1 The following triples of conjugacy classes are strictly rigid : (2A, 3A, 5A), (2A, 5A, 5B), and (3A, 5A, 5B).

To prove this, one can compute the order of Σ̄ in each case from the character table of A_{5}. In ATLAS style, it is:

characters 60 4 3 5 5 ← orders of centralizers
↓  1A 2A 3A 5A 5B ← classes
χ_{1}  1 1 1 1 1
χ_{2}  3 −1 0 z′ z z = \frac{1+\sqrt{5}}{2}
χ_{3}  3 −1 0 z z′ z′ = \frac{1−\sqrt{5}}{2}
χ_{4}  4 0 1 −1 −1
χ_{5}  5 1 −1 0 0

One then gets:

|Σ̄(2A, 3A, 5A)| = \frac{60^{2}}{4 · 3 · 5}(1 + 0 + 0 + 0 + 0) = 60

|Σ̄(2A, 5A, 5B)| = \frac{60^{2}}{4 · 5 · 5}(1 + \frac{1}{3} + \frac{1}{3} + 0 + 0) = 60

|Σ̄(3A, 5A, 5B)| = \frac{60^{2}}{3 · 5 · 5}(1 + 0 + 0 + \frac{1}{4} + 0) = 60.

One checks easily that any triple in any of these Σ̄ generates A_{5}, and prop. 7.4.1 follows.

Exercises:

1. Show that the triples (3A, 3A, 5A), (3A, 5A, 5A), and (5A, 5A, 5A) are strictly rigid.

2. Show that (2A, 2A, 5A) is not rigid, even though |Σ̄| = 60 in that case (the triples in Σ̄ generate dihedral subgroups of order 10).

7.4.3 The groups PSL_{2}(F_{p})

The group PSL_{2}(F_{p}), with p > 2, contains unique conjugacy classes of elements of order 2 and 3, denoted by 2A and 3A respectively. There are two classes pA and pB of elements of order p, which are represented by unipotent matrices, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), and ( \begin{matrix}1 & α \\ 0 & 1 \end{matrix} ), where (\frac{α}{p}) = −1.

Proposition 7.4.2 The triple (2A, 3A, pA) is strictly rigid.

One checks that (x_{0}, y_{0}, z_{0}) ∈ Σ, where x_{0}, y_{0}, and z_{0} are represented by the matrices:

x_{0} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,  y_{0} = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) ,  z_{0} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

Conversely, let (x, y, z) be in Σ̄. We lift x, y, z to x̃, ỹ, z̃ in SL_{2}(F_{p}), with x̃ of order 4, ỹ of order 3 and z̃ of order p, so that we have x̃ỹz̃ = ±1. We view these elements as automorphisms of the vector space V = F_{p} ⊕ F_{p}. Let D be the line of V fixed by z̃ and let D′ = x̃D be its transform by x̃. One has D′ ≠ D (otherwise, ±x̃z̃ would not be of order 3). After conjugating by an element of SL_{2}(F_{p}), we may assume that D (resp. D′) is the first (resp. the second) axis of coordinates in V. This means that we have

x̃ = ( \begin{matrix}0 & −λ \\ λ^{−1} & 0 \end{matrix} ) , and z̃ = ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) ,

for some λ, μ in F_{p}^{∗}. By assumption, z belongs to the class pA of ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ); this

implies that μ is a square. If we write μ = ν^{2}, and conjugate by ( \begin{matrix}ν & 0 \\ 0 & ν^{−1} \end{matrix} ), we see that we can further assume that μ = 1, i.e., that z̃ = z_{0}. Moreover, since z̃x̃ is of order 3 or 6, we have Tr (z̃x̃) = ±1. This gives λ = ±1, hence x = x_{0}, y = y_{0}, and z = z_{0}, which proves the result.

Proposition 7.4.3 The triple (2A, pA, pB) is strictly rigid if (\frac{2}{p}) = −1.

One checks that (x_{0}, y_{0}, z_{0}) is in Σ, where

x_{0} = ( \begin{matrix}1 & −1 \\ 2 & −1 \end{matrix} ) ,  y_{0} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,  z_{0} = ( \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ) .

The element z_{0} is conjugate to ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ); since (\frac{2}{p}) = −1, it belongs to the class pB. Conversely, let (x, y, z) ∈ Σ̄. We lift (x, y, z) as above to (x̃, ỹ, z̃) with x̃ of order 4 and ỹ, z̃ of order p. Let D be the line fixed by ỹ and D′ be the line fixed by z̃. We may again assume that these lines are the standard coordinate lines, and that

ỹ = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,  z̃ = ( \begin{matrix}1 & 0 \\ λ & 1 \end{matrix} ) .

Writing that ỹz̃ has order 4, one gets Tr (ỹz̃) = 0, i.e. λ = −2, q.e.d.

Proposition 7.4.4 The triple (3A, pA, pB) is strictly rigid if (\frac{3}{p}) = −1.

One checks that (x_{0}, y_{0}, z_{0}) is in Σ, where

x_{0} = ( \begin{matrix}1 & −1 \\ 3 & −2 \end{matrix} ) ,  y_{0} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,  z_{0} = ( \begin{matrix}1 & 0 \\ −3 & 1 \end{matrix} ) .

The assumption that 3 is not a quadratic residue mod p ensures that z_{0} is in the class pB.

Conversely, let (x, y, z) be in Σ̄. Using liftings (x̃, ỹ, z̃) as above, one may assume that

ỹ = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ,  z̃ = ( \begin{matrix}1 & 0 \\ λ & 1 \end{matrix} ) .

Writing that ỹz̃ has order 3 or 6, one gets Tr (ỹz̃) = ±1, i.e., λ = −1 or λ = −3. However, λ = −1 is impossible (it would imply that z̃ belongs to the class pA); hence λ = −3, q.e.d.

7.4.4 The group SL_{2}(F_{8})

The simple group G = SL_{2}(F_{8}), of order 504, has three distinct conjugacy classes of order 9, denoted 9A, 9B, and 9C which are rational over the cubic field Q(cos \frac{2π}{9}) and conjugate to each other (cf. [ATLAS], p.6).

Proposition 7.4.5 The triple (9A, 9B, 9C) is strictly rigid.

The character table for G is ([ATLAS], loc. cit.):

504 8 9 7 7 7 9 9 9 ← orders of centralizers
1A 2A 3A 7A 7B 7C 9A 9B 9C ← classes
χ_{1} 1  1 1 1 1 1 1 1 1
χ_{2} 7 −1 −2 0 0 0 1 1 1
χ_{3} 7 −1 1 0 0 0 x x′ x′′
χ_{4} 7 −1 1 0 0 0 x′′ x x′
χ_{5} 7 −1 1 0 0 0 x′ x′′ x
χ_{6} 8  0 −1 1 1 1 −1 −1 −1
χ_{7} 9  1 0 y y′ y′′ 0 0 0
χ_{8} 9  1 0 y′′ y y′ 0 0 0
χ_{9} 9  1 0 y′ y′′ y  0 0 0

x = −2 cos \frac{2π}{9},  x′ = −2 cos \frac{4π}{9},  x′′ = −2 cos \frac{8π}{9},  xx′x′′ = 1;
y = 2 cos \frac{2π}{7},  y′ = 2 cos \frac{4π}{7},  y′′ = 2 cos \frac{8π}{7},  yy′y′′ = 1.

Using formula (7.3) of §7.3, one gets:

|Σ̄(9A, 9B, 9C)| = \frac{504^{2}}{9^{3}} (1 + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} − \frac{1}{8} + 0 + 0 + 0) = 504 = |G|.

Hence it suffices to show that any (x, y, z) ∈ Σ̄ generates G to prove rigidity.

The only maximal subgroups of G containing an element of order 9 are the normalizers of the non-split Cartan subgroups, which are isomorphic to a semi-direct product C_{2}(F_{64}^{∗})_{1}, where (F_{64}^{∗})_{1} denotes the multiplicative group of elements of F_{64} of norm 1 over F_{8}, and the non-trivial element of C_{2} acts on (F_{64}^{∗})_{1} by x ↦ x^{−1}; they are dihedral groups of order 18.

If (x, y, z) ∈ Σ̄ does not generate G, then x, y, z are contained in such a normalizer. It follows that (by interchanging y and z if necessary):

y = x^{±2}, z = x^{±4}.

But then xyz = x^{1±2±4} is not equal to 1. This contradiction completes the proof.

Exercises:

1. Show that the triples (7A, 7A, 7A), (2A, 3A, 7A), (2A, 3A, 9A) are strictly rigid, and that (7A, 7B, 7C) is not rigid.

2. Let G · 3 be the automorphism group of SL_{2}(F_{8}), cf. [ATLAS], p. 6. Show that the triple (9A, 3B, 3C) is strictly rigid; the class 9A is rational (as a class of G · 3); the classes 3B and 3C are rational over Q(\sqrt{−3}), and conjugate to each other.

7.4.5 The Janko group J_{1}

The sporadic simple group J_{1} discovered by Janko is of order

175560 = 2^{3} · 3 · 5 · 7 · 11 · 19.

It contains conjugacy classes 2A, 5A, and 5B of orders 2, 5, and 5; the classes 5A and 5B are rational over Q(\sqrt{5}) and conjugate to each other. If x ∈ 5A, then x^{−1} ∈ 5A, but x^{2}, x^{3} ∈ 5B; these conjugacy classes behave like the ones of the same order in A_{5}.

Proposition 7.4.6 (cf. [Ho]) The triple (2A, 5A, 5B) is rigid but not strictly rigid.

The relevant part of the character table of G = J_{1} is (cf. [ATLAS], p.36):

175560 120 30 30 ← orders of centralizers
1A  2A 5A 5B ← classes
χ_{1}  1  1  1  1
χ_{4}  76  4  1  1
χ_{5}  76  −4 1  1
χ_{6}  77  5  2  2 z = (1 + \sqrt{5})/2
χ_{7}  77  −3 −z′ −z z′ = (1 − \sqrt{5})/2
χ_{8}  77  −3 −z −z′
χ_{12} 133  5 −2 −2
χ_{13} 133  −3 z′  z
χ_{14} 133  −3 z  z′
χ_{15} 209  1 −1 −1

Using formula (7.3) of §7.3, one obtains:

\begin{aligned}|Σ̄| & = & \frac{175560^{2}}{120·30^{2}} (1 + \frac{4}{76} − \frac{4}{76} + \frac{20}{77} + \frac{3}{77} + \frac{3}{77} + \frac{20}{133} + \frac{3}{133} + \frac{3}{133} + \frac{1}{209}) \\ & = & 438900 = \frac{5}{2}|G|. \end{aligned}

Hence (2A, 5A, 5B) is not strictly rigid. One can check that the triples in Σ̄ − Σ generate subroups of J_{1} isomorphic to A_{5}. It is known that J_{1} contains 2 conjugacy classes of such subgroups:

1. There are |J_{1}|/(2|A_{5}|) conjugate subgroups isomorphic to A_{5} which are contained in the centralizer of an involution in J_{1}. (Indeed, J_{1} was first defined abstractly by Janko as a simple group having the property that it contains an involution whose centralizer is isomorphic to {±1} × A_{5}.)

2. There is a conjugacy class of A_{5}-subgroups which are self-normalizing: there are |J_{1}|/|A_{5}| such subgroups.

In all, one has \frac{3}{2}|J_{1}|/|A_{5}| subgroups of J_{1} which are isomorphic to A_{5}. Since the conjugacy classes (2A, 5A, 5B) are rigid in A_{5}, each subgroup gives |A_{5}| solutions in Σ̄. This shows that

|Σ̄ − Σ| = \frac{3}{2}|J_{1}|,

and hence |Σ| = |J_{1}|, i.e., (2A, 5A, 5B) is rigid.

7.4.6 The Hall-Janko group J_{2}

This sporadic simple group has order 604800 = 2^{7}3^{3}5^{2}7. It has a rational class 7A of order 7, and two conjugate classes 5A and 5B of order 5, rational over Q(\sqrt{5}), see e.g. [ATLAS], pp.42-43.

Proposition 7.4.7 (cf. [Ho]) The triple (5A, 5B, 7A) is strictly rigid.

No proper subgroup of J_{2} has order divisible by 35, hence |Σ̄| = |Σ|. On the other hand, formula (7.3) of §7.3 and the character table of J_{2} ([ATLAS], loc.cit.) give:

|Σ̄| = \frac{|J_{2}|^{2}}{300^{2} · 7}(1 + \frac{16}{36} − \frac{25}{90} − \frac{25}{160} + \frac{9}{288}) = |J_{2}|.

Hence the result.

7.4.7 The Fischer-Griess Monster M

The Fischer-Griess group M, known as the “Monster”, is the largest of the sporadic simple groups. Its order is

2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71,

yet it has only 194 conjugacy classes. Its character table is therefore of manageable size (in fact, it was computed before M had been shown to exist). The group M contains rational conjugacy classes 2A, 3B, and 29A of exponent 2, 3, and 29 (ATLAS notation).

Proposition 7.4.8 (Thompson, cf. [Hunt], [Ma3], [Th2]) The triple (2A, 3B, 29A) is strictly rigid.

It can be verified by computer that |Σ̄| = |M|. To prove rigidity, one must show that Σ = Σ̄, i.e., no (x, y, z) ∈ Σ̄ generate a proper subgroup of M. Unfortunately, the maximal subgroups of M are not completely known at present. Hence, one must take the following indirect approach: suppose there is a proper subgroup G in M which is generated by (x, y, z) ∈ Σ̄. Let S be a simple quotient of G. Clearly the elements x, y, z have non-trivial image in S, and hence the order of S is divisible by 2 · 3 · 29. Hence, it suffices to check that there are no simple groups S with 2 · 3 · 29 dividing |S| and |S| dividing |M|, such that S is generated by elements x, y, z coming from the conjugacy classes 2A, 3B, and 29A in the Monster. This is done by checking that no group in the list of finite simple groups satisfies these properties. One is thus forced to invoke the classification theorem for the finite simple groups to prove rigidity in this case.

[Although the proof of the classification theorem has been announced, described, and advertized since 1980, it is not yet clear whether it is complete or not: the part on “quasi-thin” groups has never been published.] 80 Chapter 7. Rigidity and rationality on finite groups

Construction of Galois extensions of Q(T) by the rigidity method

8.1 The main theorem

Let K be a field of characteristic zero, let P_{1}, …, P_{k} be distinct K-rational points of P_{1}, and let C_{1}, …, C_{k} be a family of conjugacy classes of a finite group G with trivial center. The following result is due to Belyi, Fried, Matzat, Shih, and Thompson (see [Ma3] and [Se8] for references).

Theorem 8.1.1 Assume that the family (C_{1}, …, C_{k}) is rigid and that each C_{i} is rational. Then there is a regular G-covering C ⟶ P_{1} defined over K which is unramified outside {P_{1}, …, P_{k}} and such that the inertia group over each P_{i} is generated by an element of C_{i}. Furthermore, such a covering is unique, up to a unique G-isomorphism.

By taking K = Q, one has:

Corollary 8.1.2 G has property Gal_{T} (and hence is a Galois group over Q).

Proof of th. 8.1.1 Let L be the maximal extension of K̄(T) unramified outside {P_{1}, …, P_{k}}, and let π denote the Galois group of L over K̄(T). This is the algebraic fundamental group of P_{1}−{P_{1}, …, P_{k}} over K̄. (It is also called the geometric fundamental group because the ground field is algebraically closed.) Since L is a Galois extension of K(T), one has an exact sequence:

1 ⟶ π ⟶ π_{K} ⟶ Γ ⟶ 1,  (8.1)

where π_{K} is the Galois group of L over K(T), and Γ = Gal(K̄/K).

Let I_{i} be the inertia group of π at P_{i}. As a profinite group, π has a presentation given by k generators x_{1}, …, x_{k} and a single relation x_{1} ⋯ x_{k} = 1, cf. th. 6.3.1. More precisely, choose a coherent system {z_{α}} of roots of unity in Q̄. This choice determines an element x_{i} in each I_{i} up to conjugacy in π. One can then choose the x_{i} so that they satisfy the relation x_{1} ⋯ x_{k} = 1 (cf. §6.3).

The set Hom(π, G) of continuous homomorphisms π ⟶ G is equipped with natural G- and π_{K}-actions. The G-action is defined (on the left) by

(g ∗ f)(x) = gf(x)g^{−1}, g ∈ G, f ∈ Hom(π, G),

and the π_{K}-action is defined (on the right) by

(f ∗ α)(x) = f(αxα^{−1}), α ∈ π_{K}, f ∈ Hom(π, G).

The two actions commute, i.e.,

(g ∗ f) ∗ α = g ∗ (f ∗ α).

Consider the set H ⊂ Hom(π, G) defined by:

H = {ϕ|ϕ is surjective and ϕ(x_{i}) ∈ C_{i} for all i}.

This set is stable under both the G and π_{K}-actions:

• The action of G on itself by inner automorphisms stabilizes the C_{i}, and hence G preserves H, which is isomorphic to Σ(C_{1}, …, C_{k}) as a G-set (cf. §7.3). By the rigidity assumption, G acts freely and transitively on H.

• Conjugation by an element σ ∈ π_{K} sends an inertia group I_{i} at P_{i} to an inertia group at P_{i}^{σ}. Since the P_{i} are K-rational, π_{K} permutes the inertia groups above P_{i}. Hence, σ sends each of the x_{i} ∈ I_{i} to a conjugate of x_{i}^{α}, for some α ∈ Ẑ (namely, α = χ(σ), where χ is the cyclotomic character). By the rationality of the C_{i}, it follows that f ∗ σ maps each x_{i} to an element of C_{i}, and hence f ∗ σ ∈ H.

Any ϕ ∈ H defines a Galois extension E of K̄(T) with Galois group G. To descend from K̄(T) to K(T), it suffices to show that ϕ can be extended to π_{K}. This is an immediate consequence of the following:

Lemma 8.1.3 Let 1 ⟶ A ⟶ B ⟶ C ⟶ 1 be an exact sequence of groups, let G be a finite group, and let G and B act on Hom(A, G) as above (i.e., (g ∗ f)(x) = gf(x)g^{−1} if g ∈ G, x ∈ A, and (f ∗ b)(x) = f(bxb^{−1}) if b ∈ B, x ∈ A). If H is a non-empty subset of Hom(A, G) on which G acts freely and transitively, then the following are equivalent :

1. Any ϕ ∈ H extends uniquely to a homomorphism B ⟶ G.

2. H is stable under the action of B.

Proof : Suppose 1 holds, i.e., any ϕ ∈ H extends uniquely to a homomorphism ψ : B ⟶ G. If b ∈ B, then (ϕ ∗ b)(x) = ϕ(bxb^{−1}) = ψ(bxb^{−1}) = (ψ(b) ∗ ϕ)(x). Hence ϕ ∗ b ∈ H, since by hypothesis H is preserved by the action of G. Conversely, if property 2 is satisfied, then given ϕ : A ⟶ G, one may define ψ : B ⟶ G by:

ϕ ∗ b = ψ(b) ∗ ϕ.

Such a ψ exists (G acts transitively on H) and is unique (G acts freely on H). One verifies that ψ defines a homomorphism B ⟶ G which extends ϕ; this follows from the compatibility of the G- and B-actions.

This completes the proof of the lemma, and hence of th. 8.1.1.

Alternate method of proof for th. 8.1.1:

(a) Prove that the required G-covering C ⟶ P_{1} exists over K̄ and is unique up to a unique isomorphism.

(b) Use Weil’s descent criterion ([Se3], chap. V, no. 20) to prove that C, together with the action of G, can be defined over K. (General principle: every “problem” over K which has a unique solution - up to a unique isomorphism - over K̄ has a solution over K.)

Remark: When k = 3, one can suppose without loss of generality that (P_{1}, P_{2}, P_{3}) = (0, 1, ∞). In this way, a rigid triple (C_{1}, C_{2}, C_{3}) of rational conjugacy classes of G determines a canonical extension of K(T). Several natural questions arise in this context:

1. Can one describe what happens when K is a local field? We will do this (in a special case) for K = R in §8.4.

2. Can one describe the decomposition group above P_{i}? For example, if G is the Monster, and (C_{1}, C_{2}, C_{3}) = (2A, 3B, 29A) as in §7.4.7, then the decomposition group D_{1} above P_{1} must be contained in the normalizer of an element of the class 2A. This normalizer is 2 × B, where B is the “Baby Monster” sporadic group. Aside from this, nothing seems to be known about D_{1}.

Exercise: Show that the alternating group A_{n} (n = 4, 5, 6, 7, 8) does not contain any rigid family (C_{1}, …, C_{k}) of rational conjugacy classes (use remark 1 of §7.3).

8.2 Two variants

8.2.1 First variant

Th. 8.1.1 can be generalized to the case where the classes are only K-rational. More precisely, let us fix a choice of primitive Nth roots of unity over K, i.e., an orbit under Gal(K̄/K) of primitive Nth roots of unity. This amounts to choosing a K-irreducible factor of the nth cyclotomic polynomial ϕ_{n}. (For example, if K = Q(\sqrt{5}), N = 5, the cyclotomic polynomial ϕ_{5} factors as

ϕ_{5}(X) = X^{4} + X^{3} + X^{2} + X + 1 = (X^{2} + \frac{1 + \sqrt{5}}{2}X + 1)(X^{2} + \frac{1 − \sqrt{5}}{2}X + 1).

A choice of 5th roots of unity is just the same as choosing a square root of 5 in Q(\sqrt{5}).) Such a choice determines a generator x_{i} for each inertia group I_{i} at P_{i} (which is well-defined up to conjugation in π_{K}). After such a choice has been made, one has:

Theorem 8.2.1 If C_{1}, …, C_{k} is a rigid family of K-rational classes of G, and P_{1}, …, P_{k} are K-rational points of P_{1}, then there is a regular G-covering C ⟶ P_{1} defined over K which is unramified outside {P_{1}, …, P_{k}} and such that the x_{i}-generator of the inertia group above P_{i} belongs to the class C_{i}. This covering is uniquely defined up to a unique G-isomorphism.

The proof is essentially the same as that of th. 8.1.1.

Since conjugacy classes are always rational over the maximal cyclotomic extension Q^{cycl} of Q, one only needs the rigidity condition to ensure that a group can be realized as a Galois group over Q^{cycl}(T). This property is known for:

• Most of the classical Chevalley groups over finite fields (Belyi [Be2]);

• All the sporadic groups;

• Most of the exceptional groups G_{2}, F_{4}, E_{6}, E_{7} and E_{8} (and also the twisted forms ^{2}G_{2}, ^{3}D_{4}, and ^{2}E_{6}) over finite fields (Malle [Ml1]).

8.2.2 Second variant

The assumption that the conjugacy classes are rational is often too restrictive for applications. The following variant of th. 8.1.1 is useful in practice:

Theorem 8.2.2 Let (C_{1}, C_{2}, C_{3}) be a rigid triple of conjugacy classes of G, with C_{1} rational and C_{2} and C_{3} conjugate to each other over a quadratic field Q(\sqrt{D}). Let P_{1} ∈ P_{1}(Q), P_{2}, P_{3} ∈ P_{1}(Q(\sqrt{D})), with P_{2} and P_{3} conjugate to each other. Then there is a regular G-extension of Q(T) which is ramified only at P_{1}, P_{2} and P_{3}, and such that the canonical generator of the inertia group at P_{i} (which is well-defined after a choice of roots of unity over Q(\sqrt{D})) is in C_{i}.

Corollary 8.2.3 The group G has property Gal_{T}.

The proof of th. 8.2.2 is similar to that of th. 8.1.1. The set H ⊂ Hom(π, G) is defined in the same way. The key point is to prove that H is still preserved under the action of π_{Q}.

• If σ ∈ π_{Q} is trivial on Q(\sqrt{D}), then σ fixes P_{1}, P_{2} and P_{3}, and also fixes the choice of roots of unity over Q(\sqrt{D}). Hence σ preserves H, as before.

• If σ is not trivial on Q(\sqrt{D}), then σ interchanges P_{2} and P_{3}, and hence I_{2} and I_{3}. But σ also changes the choice of roots of unity, and these effects compensate each other.

Remarks:

1. The assumptions on the number of classes and the field of rationality are only put to simplify the proof, and because this is the principal case which occurs in practice. In fact, the same conclusion holds in greater generality, e.g., if {C_{1}, …, C_{k}} is stable under the action of Gal(Q̄/Q), and the map {P_{1}, …, P_{k}} ⟶ {C_{1}, …, C_{k}} defined by P_{i} ↦ C_{i} is an anti-isomorphism of Gal(Q̄/Q)-sets.

2. For other variants of th. 8.1.1 using the braid group, see [Fr1] and [Ma3].

8.3 Examples

Here also, we only give a few examples. For more, see [Ma3].

8.3.1 The symmetric group S_{n}

Recall from §7.4.1 that the symmetric group S_{n} has a rigid triple of conjugacy classes (nA, 2A, C^{(k)}), when (k, n) = 1. The covering P_{1} ⟶ P_{1} given by X ↦ X^{k}(X − 1)^{n−k} has ramification of this type, namely:

\{ \begin{matrix}t = ∞ & nA \\ t = 0 & C^{(k)} \\ t = k^{k}(k − n)^{n−k}n^{−n} & 2A \end{matrix}

Hence, by th. 8.1.1, the polynomial

X^{k}(X − 1)^{n−k} − T = 0

has Galois group S_{n} over Q(T) when (k, n) = 1. Note that when (k, n) = l ≠ 1, the splitting field of the equation

X^{k}(X − 1)^{n−k} − T = 0

contains Q(T^{1/l}, μ_{l}). Its Galois group is strictly smaller than S_{n} (and the extension is not regular when l > 2).

8.3.2 The alternating group A_{5}

Recall from §7.4.2 that the triple (2A, 3A, 5A) of conjugacy classes in A_{5} is rigid. The conjugacy class 5A is rational over Q(\sqrt{5}) (but not over Q). By th. 8.2.1, there is a regular extension of Q(\sqrt{5})(T) with Galois group A_{5}, and ramification of type (2A, 3A, 5A). The corresponding curve C has genus 0 (but is not isomorphic to P_{1} over Q(\sqrt{5}), cf. [Se5]).

The action of A_{5} on C can be realized geometrically as follows (loc. cit.). Consider the variety in P_{4} defined by the equations:

\{ \begin{matrix}X_{1} + ⋯ + X_{5} = 0 \\ X_{1}^{2} + ⋯ + X_{5}^{2} = 0. \end{matrix}

Since the first equation is linear, this variety can be viewed as a quadric hypersurface in P_{3}. The variety of lines on this quadric is a curve over Q which becomes isomorphic over Q(\sqrt{5}) to the disjoint union of two curves of genus 0 which are conjugate over Q(\sqrt{5}). The obvious action of S_{5} on V (permuting coordinates) induces an action of S_{5} on this curve. The extension of Q(T) corresponding to this curve is a non-regular extension with Galois group S_{5}, which contains Q(\sqrt{5}); it can also be viewed as a regular A_{5}-extension of Q(\sqrt{5})(T).

An A_{5}-covering of P_{1} with ramification (2A, 5A, 5B) can be realized by taking an S_{5}-covering with ramification of type (2, 4, 5) (e.g., the covering X ↦ X^{5} − X^{4}) and using the double group trick (cf. §4.5). This defines a regular A_{5}-covering of P_{1} over Q, with two ramification points conjugate over Q(\sqrt{5}); this is the situation of th. 8.2.2. This covering can be shown to be isomorphic to the Bring curve, defined in P_{4} by the homogeneous equations

\{ \begin{matrix}X_{1} + ⋯ + X_{5} = 0, \\ X_{1}^{2} + ⋯ + X_{5}^{2} = 0, \\ X_{1}^{3} + ⋯ + X_{5}^{3} = 0, \end{matrix}

cf. §4.4, exercise.

Exercise: Let C be the Bring curve in P_{4} (see above).

a) Let E be the quotient of C by the group of order 2 generated by the transposition (12) in S_{5}. Show that E is isomorphic to the elliptic curve defined in P_{2} by the homogeneous equation:

(x^{3} + y^{3} + z^{3}) + (x^{2}y + x^{2}z + y^{2}z + y^{2}x + z^{2}x + z^{2}y) + xyz = 0,

(put x = x_{3}, y = x_{4} = z = x_{5}). Show that this curve is Q-isomorphic to the curve 50E of [ANVERS], p.86, with minimal equation

Y^{2} + XY + Y = X^{3} − X − 2

and j-invariant −5^{2}/2.

b) Use the action of S_{5} on C to show that the Jacobian of C is Q-isogenous to the product of 4 copies of E.

c) Show that the quotient of C by A_{4} is Q-isomorphic to the elliptic curve 50H of [ANVERS], loc.cit., with j-invariant 2^{−15} · 5 · 211^{3}; this curve is 15-isogenous to E.

d) Show that C has good reduction (mod p) for p ≠ 2, 5. If N_{p}(C) (resp N_{p}(E)) denotes the number of points on C (resp. E) modulo p, deduce from b) that

N_{p}(C) = 4N_{p}(E) − 3 − 3p.

Use [ANVERS], p.117, to construct a table giving N_{p}(C) for p < 100:

\begin{matrix}p & 7 & 11 & 13 & … & 83 & 89 & 97 \\ N_{p}(C) & 0 & 24 & 30 & … & 120 & 30 & 90 \end{matrix}

Check these values by determining the polynomials X^{5} + aX + b over F_{p}, with (a, b) ≠ (0, 0), which have all their roots rational over F_{p}. For instance, if p = 83, there is only one such polynomial (up to replacing a by at^{4} and b by bt^{5}, with t ∈ F_{p}^{∗}), namely:

X^{5} + 11X + 11 ≡ (X + 33)(X + 13)(X − 4)(X − 20)(X − 22).

This fits with N_{p}(C) = 120.

e) Show that C has semi-stable reduction at 2, the reduced curve being isomorphic (over F_{4}) with the union of two copies of P_{1} intersecting each other at the five points of P_{1}(F_{4}); describe the action of S_{5} on this curve, using the fact that A_{5} is isomorphic to SL_{2}(F_{4}).

8.3.3 The group PSL_{2}(F_{p})

Th. 8.2.2 applied to the rigid triples of conjugacy classes (2A, pA, pB) (when (\frac{2}{p}) = −1) and (3A, pA, pB) (when (\frac{3}{p}) = −1) shows that there are regular PSL_{2}(F_{p})-extensions of Q(T) with ramification of this shape. These correspond to the Shih coverings with N = 2 and N = 3 (cf. §5.1). The rigidity method does not predict the existence of the Shih coverings related to N = 7, which are ramified at four points.

A covering of P_{1} having ramification type (2A, 3A, pA) is given by the covering of modular curves defined over Q(\sqrt{p^{∗}}), X(p) ⟶ X(1). This covering has Galois group PSL_{2}(F_{p}) over Q(\sqrt{p^{∗}}); the rigidity property shows that it is the only PSL_{2}(F_{p})-covering with this ramification type. This was first pointed out by Hecke [He].

In particular, the rigid triple (2A, 3A, 7A) in PSL_{2}(F_{7}) gives rise to the Klein covering of P_{1} defined over Q(\sqrt{−7}) and having Galois group PSL_{2}(F_{7}). Its function field E is a Galois extension of Q(T) with Galois group PGL_{2}(F_{7}), which is not regular.

8.3.4 The Gal_{T} property for the smallest simple groups

The following table lists the smallest ten non-abelian simple groups. All of them, except the last one SL_{2}(F_{16}), are known to have the Gal_{T} property. The last column of the table indicates why.

group  order  Gal_{T} property
A_{5} = SL_{2}(F_{4}) 60 = 2^{2} · 3 · 5  Hilbert (§§4.5, 8.3.2)
= PSL_{2}(F_{5})  Shih with N = 2, 3
(§§5.1, 8.3.3);
SL_{3}(F_{2}) 168 = 2^{3} · 3 · 7 Shih with N = 3
= PSL_{2}(F_{7})  (§§5.1, 8.3.3);
A_{6} = PSL_{2}(F_{9}) 360 = 2^{3} · 3^{2} · 5  Hilbert (§4.5);
SL_{2}(F_{8})  504 = 2^{3} · 3^{2} · 7  Th. 8.2.2 and §7.4.4;
PSL_{2}(F_{11})  660 = 2^{2} · 3 · 5 · 11  Shih with N = 2
(§§5.1, 8.3.3);
PSL_{2}(F_{13})  1092 = 2^{2} · 3 · 7 · 13 Shih with N = 2
(§§5.1, 8.3.3);
PSL_{2}(F_{17}) 2448 = 2^{4} · 3^{2} · 17 Shih with N = 3
(§5.1, 8.3.3);
A_{7}  2520 = 2^{3} · 3^{2} · 5 · 7 Hilbert (§4.5);
PSL_{2}(F_{19})  3420 = 2^{2} · 3^{2} · 5 · 19 Shih with N = 2, 3
(§§5.1, 8.3.3);
SL_{2}(F_{16})  4080 = 2^{4} · 3 · 5 · 17 ?

In addition, it has been shown that all the sporadic simple groups satisfy the Gal_{T} property, with the possible exception of the Mathieu group M_{23}, cf. [Hunt], [Ma3], [Pa].

8.4 Local properties

8.4.1 Preliminaries

Let π : C ⟶ P_{1} be a G-covering defined over a field K which is complete with respect to a real valuation. Let x be a point of P_{1}(K) − S, where S is the ramification locus of the covering. The fiber Λ_{x} at x is an étale K-algebra with action of G; its structure is defined by a continuous homomorphism

ϕ_{x} : Gal(K̄/K) ⟶ G,

which is well-defined up to inner conjugation in G. Let us denote by H the quotient of the set of such homomorphisms by the action of G by conjugation; we endow H with the discrete topology.

Proposition 8.4.1 The map from P_{1}(K) to H defined by x ↦ ϕ_{x} is continuous with respect to the topology of P_{1}(K) induced by the valuation on K. (In other words, for every x ∈ P_{1}(K), there is a neigbourhood U of x such that the Galois algebras Λ_{x} and Λ_{y} are isomorphic, for all y ∈ U.)

This can be deduced (with some care) from Krasner’s lemma: if

P (X) = ∏(X − α_{i}) and Q(X) = ∏(X − α_{i})

are irreducible polynomials over K, with α_{i} and β_{i} sufficiently close for each i, then K(α_{1}) = K(β_{1}). See [Sa1].

Another way to prove prop. 8.4.1 is as follows: the map C ⟶ P_{1} gives an extension O_{x} ⊂ Õ_{x}, where O_{x} is the local ring at x, and Õ_{x} is the semi-local ring above x. Let M_{x} be the maximal ideal of O_{x}. The quotient Õ_{x}/M_{x}Õ_{x} is Λ_{x}. One has a natural inclusion O_{x} ⊂ O_{x}^{hol}, where O_{x}^{hol} denotes the ring of power series with coefficients in K which converge in a neighbourhood of x. Since O_{x}^{hol} is Henselian (cf. [Ra], p. 79), and the extension is étale, one has:

Õ_{x}^{hol} ≃ Λ_{x} ⊗_{K} O_{x}^{hol}.

This means that, for its analytic structure, the covering C ⟶ P_{1} is locally a product. The proposition follows.

By the discussion above, we may write P_{1}(K) − S as a finite disjoint union of open sets:

P_{1}(K) − S = ⋃_{ϕ∈H} U_{ϕ},

where U_{ϕ} = {t ∈ P_{1}(K) − S|ϕ_{t} = ϕ}. For example, if K = Q_{p}, p ≠ 2, the universal Z/2Z-covering defined by x ↦ x^{2} gives a decomposition of P_{1}(Q_{p}) − {0, ∞} into four open and closed pieces, corresponding to the four non-isomorphic Galois algebras of rank 2 over Q_{p}.

Remark: A propos of the U_{ϕ}, let us mention an unpublished result of Raynaud: if the order of G is prime to the residue characteristic, every unramified G-covering of a rigid polydisk becomes trivial after a finite extension of the local field K.

8.4.2 A problem on good reduction

Assume that π : C ⟶ P_{1} is a G-covering obtained by the rigidity method (th. 8.1.1) from rational points P_{1}, …, P_{k} and conjugacy class C_{1}, …, C_{k}.

Let p be a prime number. Assume the reductions of the P_{i} are distinct in P_{1}(F_{p}).

Problem: Assume that p does not divide the orders of the elements of C_{i} for i = 1, …, k. Is it true that the curve C has good reduction at p?

This is true at least when p does not divide the order of G, cf. [Bc].

Exercise: Check that the above problem has a positive answer for the examples given in 8.3.1, 8.3.2, 8.3.3.

8.4.3 The real case

We now restrict our attention to the case when π : Y ⟶ P_{1} comes from a rigid triple of conjugacy classes (C_{1}, C_{2}, C_{3}), and where K = R. By prop. 8.4.1, to each connected component of P_{1}(R) − S there is attached a unique conjugacy class of involutions in G, corresponding to complex conjugation. There are two cases, depending on the number of such connected components:

Case 1: We suppose that the three conjugacy classes are rational over R. The corresponding covering is ramified at exactly three real points P_{1}, P_{2}, P_{3}, which divide the circle P_{1}(R) into three connected components.

Let us choose x_{i} ∈ C_{i}, with x_{1}x_{2}x_{3} = 1 and G = ⟨x_{1}, x_{2}, x_{3}⟩. We assume G ≠ {1}, hence x_{i} ≠ 1 for i = 1, 2, 3. Since the C_{i} are rational over R, we have

x_{1}^{−1} ∈ C_{1}, x_{2}^{−1} ∈ C_{2}.

Put x3^{′} = x_{2}x_{1} = x_{2}x_{3}x_{2}^{−1}. Then (x_{1}^{−1}, x_{2}^{−1}, x3^{′}) is in Σ. By rigidity, there exists a unique s_{3} ∈ G such that:

\{ \begin{matrix}s_{3}x_{1}s_{3}^{−1} & = & x_{1}^{−1} \\ s_{3}x_{2}s_{3}^{−1} & = & x_{2}^{−1} \end{matrix}

The element s_{3} thus defined is such that s_{3}^{2} = 1.

Theorem 8.4.2 The complex conjugation attached to a point x in the connected component (P_{1}, P_{2}) of P_{1}(R) between P_{1} and P_{2} is in the conjugacy class of s_{3}.

Remark: An analogous statement holds for the involutions s_{1} and s_{2}, which are defined in the same way as s_{3}, and correspond to the complex conjugations attached to the components (P_{2}, P_{3}) and (P_{1}, P_{3}) respectively. This gives the relations:

\{ \begin{matrix}s_{1} & = & s_{3}x_{2}, \\ s_{2} & = & s_{1}x_{3}, \\ s_{3} & = & s_{2}x_{1}. \end{matrix}

Hence, if s_{1} and s_{3} are non-trivial, the group generated by s_{1}, s_{3} and x_{2} is a dihedral group of order 2n, where n is the order of x_{2}. There is a single conjugacy class of involutions in a dihedral group of order 2n with n odd. Hence, one has:

Corollary 8.4.3 If the order of x_{2} is odd, then s_{1} and s_{3} are conjugate in G.

Corollary 8.4.4 If two of the three classes C_{1}, C_{2} and C_{3} have odd exponent, then the s_{i}’s are conjugate.

Example: The rigid triple of conjugacy classes (2A, 3B, 29A) in the Monster M satisfies the hypotheses of cor. 8.4.4. Hence, complex conjugation corresponds to a unique class of involutions in M (the class 2B, in ATLAS notation).

Remark: The case where s_{3} = 1 occurs only in the case G = D_{n} and (C_{1}, C_{2}, C_{3}) = (2, 2, n), n odd. For, if s_{3} = 1, then x_{1}, x_{2} are of order 2, equal to s_{2} and s_{1} respectively. The group generated by x_{1}, x_{2}, x_{3} is thus a dihedral group of order 2n, where n is odd (since the group has no center). The only dihedral group of this type satisfying the rationality condition over Q is G = S_{3}. This means that the rigidity method of th. 8.1.1, when applied with three classes to a group G ≠ S_{3}, never gives totally real extensions of Q.

Case 2: C_{1} is rational over R, and C_{2} and C_{3} are complex conjugate to each other.

Then, P_{1}(R)−{P_{1}, P_{2}, P_{3}} is connected, since P_{2} and P_{3} do not lie on P_{1}(R). Hence there is a single conjugacy class of involutions in G corresponding to complex conjugation.

Let (x_{1}, x_{2}, x_{3}) ∈ Σ(C_{1}, C_{2}, C_{3}). By rigidity, there is a unique involution s in G such that:

sx_{1}s^{−1} = x_{2}^{−1},  sx_{2}s^{−1} = x_{1}^{−1},  sx_{3}s^{−1} = x_{3}^{−1}.

Theorem 8.4.5 Complex conjugation belongs to the conjugacy class of s.

Observe that in this case s ≠ 1. (Otherwise G would be cyclic.)

Proof of th. 8.4.2 and th. 8.4.5 Choose a base point x on P_{1}(R), (lying on the connected component (P_{1}, P_{2}) in case 1) and let π denote as before the geometric fundamental group of X = P_{1} −{P_{1}, P_{2}, P_{3}}, with base point x. Let α_{1}, α_{2}, α_{3} denote the generators of π corresponding to paths around P_{1}, P_{2}, and P_{3} respectively. The complex conjugation σ_{X} on X acts as a symmetry around the equator on the Riemann sphere. Hence σ_{X} acts on the generators α_{1}, α_{2}, α_{3} of π by: Case 1: α_{1} ↦ α_{1}^{−1}, α_{2} ↦ α_{2}^{−1} Case 2: α_{2} ↦ α_{3}^{−1}, α_{3} ↦ α_{2}^{−1}

Let σ_{Y} denote the complex conjugation acting on Y. Since the G-covering π : Y ⟶ X is defined over R, the following diagram commutes:

Y ⟶^{σ_{Y}} Y
π ↓  ↓ π
X ⟶^{σ_{X}} X

Since σ_{X}x = x, the fiber Y_{x} of Y over x is preserved by σ_{Y}. The action of σ_{Y} on Y_{x} commutes with the action of G.

The group G acts freely and transitively on Y_{x}, so a choice of y ∈ Y_{x} determines a surjective map ϕ_{y} : π_{1}(X; x) ⟶ G. Because the action of σ commutes with that of G, the following diagram is commutative:

π_{1}(X; x) ⟶^{ϕ_{y}} G
σ_{X} ↓  ||
π_{1}(X; x) ⟶^{ϕ_{σy}} G

On the other hand, we also have:

ϕ_{σy} = σ_{Y} ϕ_{y}σ_{Y}^{−1}.

Hence the diagram below commutes:

π_{1}(X; x) ⟶^{ϕ_{y}} G
σ_{X} ↓  ↓ Inn(σ_{Y,t})
π_{1}(X; x) ⟶^{ϕ_{σy}} G.

This proves the theorem.

Exercise: Let G be a finite group, generated by elements g_{1}, …, g_{k}. Choose points z_{1}, …, z_{k} with (say) Re(z_{i}) = 0 for i = 1, … k, and

Im(z_{1}) > Im(z_{2}) > … > Im(z_{k}) > 0.

Let z1^{′}, …, zk^{′} be the complex conjugates of z_{1}, …, z_{k} and choose a real base point x < 0; let π denote the fundamental group π_{1}(P_{1}(C) − S; x), where S =

{z_{1}, …, z_{k}, z1^{′}, …, zk^{′}}. It is generated by elements x_{1}, …, x_{k}, x1^{′}, …, xk^{′}, where x_{i}

(resp. xi^{′}) denotes the homotopy class of paths going in a straight line from x to z_{i} (resp. zi^{′}) and going around this point in the positive direction. These generators, together with the relation x_{1} ⋯ x_{k}xk^{′} ⋯ x1^{′} = 1, give a presentation for π. Let ϕ be the homomorphism π ⟶ G sending x_{i} to g_{i} and xi^{′} to g_{i}^{−1}. Show that ϕ defines a regular G-extension of R(T). (Hence every finite group is the Galois group of a regular extension of R(T). This result can also be deduced from [KN]. The analogous statement for Q_{p}(T) is also true; see next section.)

8.4.4 The p-adic case: a theorem of Harbater

Let p be a prime number.

Theorem 8.4.6 ([Harb]) Every finite group is the Galois group of a regular extension of Q_{p}(T).

The proof shows more, namely that for every finite group G, there exists an absolutely irreducible G-covering X ⟶ P_{1} over Q_{p} having a “base point”, i.e., an unramified point P ∈ P_{1}(Q_{p}) which is the image of a Q_{p}-point of X. Call R_{p} this property of G. The theorem clearly follows from the following two assertions:

(i) Every cyclic group has property R_{p}.

(ii) If G is generated by two subgroups G_{1} and G_{2} having property R_{p}, then G has property R_{p}.

Assertion (i) is easy (and true over Q, as the construction of §4.2 shows). Assertion (ii) is proved by a gluing process which uses rigid analytic geometry. Namely, let X_{i} ⟶ P_{1} (i = 1, 2) be a G_{i}-covering as above with a base point P_{i}. By removing a small neighborhood of P_{i} one gets a rigid analytic G_{i}- covering V_{i} ⟶ U_{i} where U_{i} is a p-adic disk; this covering has the further property that it is trivial on U_{i} −Ui^{′} where Ui^{′} is a smaller disk contained in U_{i}. Let W_{i} ⟶ U_{i} be the (non-connected) G-covering of U_{i} defined by induction from G_{i} to G. (It is a disjoint union of |G/G_{i}| copies of V_{i}.) One then embeds U_{1} and U_{2} as disjoint disks in P_{1}, and defines a rigid analytic G-covering of P_{1} by gluing together W_{1} on U_{1}, W_{2} on U_{2}, and the trivial G-covering on P_{1} −U1^{′} −U2^{′}. If this is done properly, the resulting G-covering W is absolutely irreducible. By the “GAGA” theorem in the rigid analytic setting (cf. [Ki], [Kö]), this covering is algebraic, and (ii) follows.

Remark: Harbater’s original proof uses “formal GAGA” instead of “rigid GAGA”; the idea is the same.

94 Chapter 8. The rigidity method

The form Tr(x^{2}) and its applications

9.1 Preliminaries

9.1.1 Galois cohomology (mod 2)

Let K be a field of characteristic ≠ 2, and let G_{K} be Gal(K_{s}/K), where K_{s} is a separable closure of K. We will be interested in the Galois cohomology of K modulo 2; for brevity, let us denote the cohomology groups H^{i}(G_{K}, Z/2Z) by H^{i}(G_{K}). In the case where i = 1, 2, Kummer theory provides the following interpretation of H^{i}(G_{K}):

H^{1}(G_{K}) = K^{∗}/K^{∗2},
H^{2}(G_{K}) = Br_{2}(K),

where Br_{2}(K) is the 2-torsion in the Brauer group of K. If a ∈ K^{∗}, we denote by (a) the corresponding element of H^{1}(G_{K}). The cup-product

(a)(b) ∈ H^{2}(G_{K})

corresponds to the class of the quaternion algebra (a, b) in Br_{2}(K) defined by

i^{2} = a, j^{2} = b, ij = −ji.

9.1.2 Quadratic forms

Let f be a non-degenerate quadratic form over K of rank n ≥ 1. By choosing an appropriate basis, we may write

f = ∑_{i=1}^{n} a_{i}X_{i}^{2}, with a_{i} ∈ K^{∗}.

96 Chapter 9. The form Tr (x^{2}) and its applications

The element (1 + (a_{1}))(1 + (a_{2})) ⋯ (1 + (a_{n})) in the cohomology ring H^{∗}(G_{K}) depends only on f. One defines the i-th Stiefel-Whitney class w_{i} of f by:

(1 + (a_{1})) ⋯ (1 + (a_{n})) = 1 + w_{1} + w_{2} + ⋯ + w_{n}, w_{i} ∈ H^{i}(G_{K}).

In particular, we have:

w_{1} = ∑(a_{i}) = (∏ a_{i}) = (Disc(f))

w_{2} = ∑_{i<j} (a_{i})(a_{j}).

The cohomology class w_{2} is known as the Hasse (or Witt) invariant of the quadratic form f. If K is a number field, then f is completely characterized by its rank, signature, and the invariants w_{1} and w_{2}. (The same is true for arbitrary K, when n ≤ 3.) The following results can be found in [Sch], pp. 211-216.

Theorem 9.1.1 (Springer) If two quadratic forms over K become equivalent over an odd-degree extension K′ of K, then they are already equivalent over K.

(For a generalization to hermitian forms, see [BL].)

Let us now consider K(T), where T is an indeterminate. If v is a place of K(T) which is trivial on K and ≠ ∞, there is a unique uniformizing parameter π_{v} at v which is monic and irreducible in K[T]. Let K(v) = K[T]/(π_{v}) denote the residue field at v, and let a ↦ \overline{a} be the reduction map K[T] ⟶ K(v). If f = ∑_{i=1}^{n} a_{i}X_{i}^{2} is a quadratic form over K(T), we may assume (since the a_{i} can be modified by squares) that v(a_{i}) = 0 or 1. The K(v)-quadratic forms:

∂_{1}(f) = ∑_{v(a_{i})=0} \overline{a_{i}} X_{i}^{2}, ∂_{2}(f) = ∑_{v(a_{i})=1} \overline{(a_{i}/π_{v})} X_{i}^{2}

are called the first and second residues of f. One shows (cf. e.g. [Sch], p. 209) that their images in the Witt group W (K(v)) do not depend on the chosen representation of f as ∑ a_{i}X_{i}^{2}. (Recall that the Witt group W (L) of a field L is the Grothendieck group of the set of quadratic forms over L, with the hyperbolic forms identified to 0.)

Theorem 9.1.2 (Milnor) If a quadratic form f over K(T) has second residue 0 at all places of K(T) except ∞, then f is equivalent to a quadratic form over K.

(More precisely, one has an exact sequence:

0 ⟶ W (K) ⟶ W (K(T)) ⟶ ∐_{v≠∞} W (K(v)) ⟶ 0.)

A quadratic form over a ring R is said to be strictly non-degenerate if its discriminant is invertible in R.

Theorem 9.1.3 (Harder) A strictly non-degenerate quadratic form over the ring K[T] comes from K.

This theorem can be formulated more suggestively: a quadratic vector bundle over A^{1} is constant. (For a generalization to other types of bundles, see [RR].)

9.1.3 Cohomology of S_{n}

We need only H^{i}(S_{n}) for i = 1, 2. These groups are well-known:

H^{1}(S_{n}) = Z/2Z, n ≥ 2,
H^{2}(S_{n}) = Z/2 ⊕ Z/2Z, n ≥ 4 (Schur).

The non-trivial element in H^{1}(S_{n}) is the signature homomorphism

ϵ_{n} : S_{n} ⟶ {±1}.

The cohomology group H^{2}(S_{n}), n ≥ 4, has a Z/2Z-basis given by ϵ_{n} · ϵ_{n} (cup-product) and an element s_{n} corresponding to the central extension

1 ⟶ C_{2} ⟶ S̃_{n} ⟶ S_{n} ⟶ 1

which is characterized by the properties:

1. A transposition in S_{n} lifts to an element of order 2 in S̃_{n}.

2. A product of two disjoint transpositions lifts to an element of order 4 in S̃_{n}.

(The element ϵ_{n}·ϵ_{n} corresponds to the extension Sn^{′} of S_{n} obtained by taking the pullback:

Sn^{′} ⟶ C_{4}
↓  ↓
S_{n} ⟶ {±1}.

This extension is characterized by the property that a transposition lifts to an element of order 4, while a product of two disjoint transpositions lifts to an element of order 2.)

The image of ϵ_{n} by the restriction map H^{1}(S_{n}) ⟶ H^{1}(A_{n}) is zero. The cohomology group H^{2}(A_{n}) is isomorphic to Z/2Z for n ≥ 4 and is generated by a_{n} = Res(s_{n}). The corresponding central extension of A_{n} is denoted by Ã_{n} (or by 2 · A_{n} in ATLAS’ style); it is a subgroup of index 2 of S̃_{n}.

The cohomology classes ϵ_{n} and s_{n} can be given the following topological interpretations: the map S_{n} ⟶ O_{n}(R) gives a map of classifying spaces

BS_{n} ⟶ BO_{n}(R),

and corresponding maps on the cohomology rings. But

H^{∗}(BO_{n}(R)) = F_{2}[w_{1}, …, w_{n}],

where w_{i} is the ith Stiefel-Whitney class. The class w_{1} corresponds to the element ϵ_{n} in H^{1}(BS_{n}) = H^{1}(S_{n}), and w_{2} gives s_{n}.

Exercises:

1. Show that Ã_{4} ≃ SL_{2}(F_{3}), Ã_{5} ≃ SL_{2}(F_{5}), and Ã_{6} ≃ SL_{2}(F_{9}).

2. Let Ŝ_{n} be the central extension of S_{n} by {±1} corresponding to the element s_{n} + ϵ_{n} · ϵ_{n} of H^{2}(S_{n}).

(a) Show that Ŝ_{4} ≃ GL_{2}(F_{3}).

(b) Show that any outer automorphism of S_{6} lifts to an isomorphism of Ŝ_{6} onto S̃_{6}.

(c) Show that the groups Ŝ_{n} and S̃_{n} are not isomorphic if n ≥ 4, n ≠ 6.

9.2 The quadratic form Tr (x^{2})

Let E be an étale K-algebra of finite rank n over K; it is a product of separable field extensions of K. There is a dictionary between such algebras and conjugacy classes of homomorphisms e : G_{K} ⟶ S_{n}, which works as follows: given E, let ϕ(E) be the set of K-algebra homomorphisms E ⟶ K_{s}. The set ϕ(E) is of cardinality n, and the natural action of G_{K} on ϕ(E) gives the desired homomorphism e : G_{K} ⟶ S_{n}, after identifying ϕ(E) with {1, …, n}. Conversely, E can be constructed as the twist of the split algebra K × ⋯ × K by the 1-cocycle e : G_{K} ⟶ S_{n} = Aut (K^{n}). The image of e is the Galois group of the smallest extension of K over which the algebra E splits; if E is a field, it is Gal(E^{gal}/K).

The function x ↦ Tr (x^{2}) defines a non-degenerate quadratic form Q_{E} of rank n over K. This invariant was studied extensively by 19th century mathematicians such as Jacobi and Hermite.

Theorem 9.2.1 Let E/K(T) be a finite separable extension of degree n. Let G ⊂ S_{n} be the corresponding Galois group. Assume that, for all places v of K(T) not equal to ∞, the order of the inertia group at v is odd. Then Q_{E} is constant, i.e., comes from a form over K.

9.3. Application to extensions with Galois group Ã_{n} 99

Let Λ be the integral closure of K[T] in E, and denote by D the different. Using the fact that the inertia groups are of odd order, one can show that D^{−1} is the square of a fractional ideal,

D^{−1} = A^{2}.

One checks that Q_{E} induces a strictly non-degenerate quadratic form over K[T] on the K[T]-module A. By th. 9.1.3 the result follows. (One may also show that the second residues of Q_{E} are 0, and apply Milnor’s theorem 9.1.2.)

Example: Let E be a regular Galois extension of Q(T) with Galois group the Monster, obtained from the rigid family (2A, 3B, 29A), cf. §7.4.7. Since two of the conjugacy classes in this family have odd exponent, the form Q_{E} comes from Q, and does not depend on T. One can prove that it is hyperbolic.

The following theorem is proved in [Se6].

Theorem 9.2.2 Let E be an étale K-algebra of rank n and discriminant d associated to a homomorphism e : G_{K} ⟶ S_{n}, and let Q_{E} denote the trace form of E. Then :

1. w_{1}(Q_{E}) = e^{∗}ϵ_{n},

2. w_{2}(Q_{E}) = e^{∗}s_{n} + (2)(d)

Suppose that G = e(G_{K}) is contained in A_{n}, i.e., that d is a square. By th. 9.2.2, we have w_{1}(Q_{E}) = 0 and w_{2}(Q_{E}) = e^{∗}a_{n}. The element e^{∗}a_{n} is the obstruction to lifting the homomorphism G_{K} ⟶ A_{n} to a homomorphism G_{K} ⟶ Ã_{n}. Hence, we have

Corollary 9.2.3 The homomorphism e : G_{K} ⟶ A_{n} lifts to a homomorphism G_{K} ⟶ Ã_{n} if and only if the Witt invariant w_{2}(Q_{E}) is 0.

9.3 Application to extensions with Galois group Ã_{n}

The previous corollary applies when e is surjective: an extension E of K of degree n with Galois group A_{n} can be embedded in an Ã_{n}-extension if and only if w_{2}(Q_{E}) = 0.

This will be used to show the following result:

Theorem 9.3.1 (Mestre, [Me2]) The group Ã_{n} has the Gal_{T} -property for all n.

(This was already shown for n ≡ 0, 1 (mod 8), and for some other n, by Vila [Vi].)

The proof constructs a covering C ⟶ P_{1} of degree n whose Galois closure has Galois group A_{n}, and which has the following additional properties:

1. There is a point of P_{1} whose inverse image is a set {a_{1}, …, a_{n}}, where the a_{i} are rational and distinct.

2. The non-trivial inertia groups are all generated by cycles of order 3.

Let E/K(T) be the degree n extension corresponding to this covering. Condition 2 implies that the quadratic form Q_{E} comes from Q, by th. 9.2.1. But by condition 1, there is one rational point where this quadratic form is equivalent to the standard one, ∑ X_{i}^{2}. It is easy to see that this implies that Q_{E} is equivalent to ∑ X_{i}^{2} over Q(T), and hence has trivial Witt invariant; by cor. 9.2.3, we can thus solve the embedding problem for this extension.

Let us say that a property Σ of a polynomial

P = X^{n} + s_{1}X^{n−1} + ⋯ + s_{n}

is generally true if there exists a Zariski open dense subset U of A^{n} such that P has the property Σ for all (s_{1}, …, s_{n}) in U. Mestre’s construction (for n odd) relies on the following:

Proposition 9.3.2 It is generally true that there exist polynomials Q and R in Q[T] of degree n − 1 with the following properties :

a) Q′P − P′Q = R^{2} (i.e., (Q/P)′ = (R/P)^{2}).

b) P, Q, R are pairwise relatively prime.

c) The zeros b_{1}, …, b_{n−1} of R in Q̄ are distinct.

d) The values t_{i} of Q/P at the b_{i} are distinct.

Sketch of proof: The matrix M with ij-entry 1/(a_{i} − a_{j}) for i ≠ j and 0 for i = j is a skew symmetric matrix of odd dimension. Hence it has zero determinant and there exists a non-zero (c_{1}, …, c_{n}) in the kernel of M. We have:

∑_{j≠i^{j=1}}^{n} \frac{c_{j}}{a_{i} − a_{j}} = 0 for all i.

Now, let Q, R be the polynomials such that:

Q/P = ∑_{i=1}^{n} \frac{−c_{i}^{2}}{X − a_{i}} .

R/P = ∑_{i=1}^{n} \frac{c_{i}}{X − a_{i}} .

One has:

(R/P)^{2} = ∑_{i,j=1}^{n} \frac{c_{i}c_{j}}{(X − a_{i})(X − a_{j})}
= ∑_{i=1}^{n} \frac{c_{i}^{2}}{(X − a_{i})^{2}} + ∑_{i≠j} \frac{c_{i}c_{j}}{a_{i} − a_{j}} (\frac{1}{X − a_{i}} − \frac{1}{X − a_{j}})
= (Q/P)′

Moreover, one can check (by looking at a well-chosen example) that it is generally true that M has rank n − 1 (so that the c_{i} are essentially unique) and that b), c), d) hold.

If (P, Q, R) are chosen as in prop. 9.3.2, with P = ∏(X−a_{i}) with a_{i} ∈ Q, the a_{i} being distinct, then the map P_{1} ⟶ P_{1} given by X ↦ T = P (X)/Q(X) has degree n, and is ramified at the zeros b_{i} of R, the ramification groups being generated by 3-cycles. Let G ⊂ S_{n} (resp. Ḡ ⊂ S_{n}) be the Galois group of the corresponding Galois extension of Q(T) (resp. of Q̄(T)). The group Ḡ is transitive and generated by 3-cycles (cf. prop. 4.4.6). By lemma 4.4.4, it is equal to A_{n}. Hence G is equal to A_{n} or S_{n}. However, we have seen that the Tr (x^{2}) form attached to the extension is the standard form ∑ X_{i}^{2}. In particular, its discriminant is a square. This shows that G is contained in A_{n}, hence G = Ḡ = A_{n}, QED.

There is a similar, but more complicated, construction when n is even. Exercise 2 below proves that Ã_{n} has the Gal_{T} property for n even by reducing to the case of odd n.

Remarks:

1. One may also prove Mestre’s theorem by showing that the Witt invariant w_{2} ∈ Br_{2}(Q(T)) of the trace form has “no poles” (because all the ramification is odd), and hence is constant, i.e., belongs to Br_{2}(Q). Since it is 0 at the base point, it is zero. Another possibility is to prove by a direct construction that the trace form of E/K(T) is constant, cf. exerc. 1.

2. For explicit formulas related to the above constructions, see [Cr], [Schn].

3. For n = 6, 7, the Schur multiplier of A_{n} is cyclic of order 6. The corresponding groups 6 · A_{6} and 6 · A_{7} also have property Gal_{T}. This has been proved by J-F. Mestre (unpublished).

Exercises:

1. Assume P, Q, R are as in prop. 9.3.2 (n odd). Put

E = K(X) and T = P (X)/Q(X),

so that [E : K(T)] = n.

a) Let Λ be the integral closure of K[T] in E. Show that Λ = K[X, 1/Q(X)] and that the different of Λ over K[T] is the principal ideal generated by R(X)^{2}. b) Let

Tr : E ⟶ K(T)

be the trace map. If f ∈ K[X], show that Tr (f(X)/R(X)^{2}) belongs to K[T] and that:

deg_{T} Tr (f(X)/R(X)^{2}) ≤ sup(0, deg(f) − 2n + 2).

In particular, Tr (f(X)/R(X)^{2}) belongs to K if deg(f) ≤ 2n − 2.

c) Let V be the n-dimensional K-subspace of E spanned by

1/R(X), X/R(X), …, X^{n−1}/R(X).

One has E = V ⊗_{K} K(X). Show, using b), that if v_{1}, v_{2} ∈ V, then Tr (v_{1}v_{2}) ∈ K. Conclude that the trace form of E/K(T) comes from K.

2. Let n be even, let f : P_{1} ⟶ P_{1} be an n + 1-covering given by Mestre’s construction above, and let C_{f} be it Galois closure. One has maps

C_{f} ⟶^{g} P_{1} ⟶^{f} P_{1},

and g : C_{f} ⟶ P_{1} is a regular A_{n}-covering. Show that this covering lifts to a regular Ã_{n}-covering of P_{1} (hence Ã_{n} also has property Gal_{T}).

Appendix: the large sieve inequality

10.1 Statement of the theorem

Let N be an integer ≥ 1, and, for each prime p, let ν_{p} be a real number with 0 < ν_{p} ≤ 1. Let A be a subset of Λ = Z^{n}, such that for all primes p,

|A_{p}| ≤ ν_{p}p^{n},

where A_{p} ⊂ Λ/pΛ denotes the reduction of A mod p. Given a vector x = (x_{1}, …, x_{n}) ∈ R^{n}, and N ∈ R, we denote by A(x, N) the set of points in A which are contained in the cube of side length N centered at x, i.e.,

A(x, N) = {(a_{1}, …, a_{n}) ∈ A | |x_{i} − a_{i}| ≤ N/2}.

Then:

Theorem 10.1.1 (Large sieve inequality) For every D ≥ 1, we have

|A(x, N)| ≤ 2^{n} sup(N, D^{2})^{n}/L(D),

where

L(D) = ∑_{d square−free^{1≤d≤D}} ∏_{p|d} (\frac{1 − ν_{p}}{ν_{p}} ) .

Taking D = N^{\frac{1}{2}} :

Corollary 10.1.2 |A(x, N)| ≤ (2N)^{n}/L(N^{\frac{1}{2}} ).

Examples:

1. If ν_{p} = \frac{1}{2} for every p, then

L(D) = ( ∑_{d square−free^{1≤d≤D}} 1) ∼ \frac{6}{π^{2}} D.

Hence |A(x, N)| << N^{n− \frac{1}{2}} . This is a typical “large sieve” situation.

2. Assume there is a set S of primes of density > 0, such that ν_{p} = C for p ∈ S, with 0 < C < 1. One may estimate L(D) from below by summing over primes ≤ D:

L(D) ≥ 1 + ∑_{p prime^{p≤D}} \frac{1 − ν_{p}}{ν_{p}} ≫ \frac{D}{log D}.

Hence |A(x, N)| ≪ N^{n− \frac{1}{2}} log N. A more careful estimate of L(D) by summing over all square-free d ≤ D allows one to replace the factor log N by (log N)^{γ}, with γ < 1, under a mild extra condition on S, cf. [Se9], chap. 13.

3. Suppose n = 1, and ν_{p} = 1− \frac{1}{p}. Then one can show that L(D) ⌢^{⌣} log D, and hence |A(x, N)| ≪ \frac{N}{log N}, a weak form of the prime number theorem: however, the method also allows one to conclude that in any interval of length N, there are at most O(\frac{N}{log N}) primes. (More precisely, their number is ≤ 2N/ log N, cf. [MoV].)

Historically, a weaker form of the sieve inequality was discovered first, where the sum giving L(D) was taken over the primes ≤ D; this only gave interesting results in large sieve situations (hence the name “large sieve inequality”). The possibility of using square-free d’s was pointed out by Montgomery, [Mo1].

Exercise: Use th. 10.1.1 to show that the number of “twin primes” (primes p such that p + 2 is also prime) ≤ N is asymptotically ≪ \frac{N}{(log N)^{2}} . Conclude that

∑_{twin prime^{p}} \frac{1}{p} < ∞.

Proof of th. 10.1.1: preliminaries Let us assume without loss of generality that A = A(x, N). Given vectors a = (a_{1}, …, a_{n}), t = (t_{1}, …, t_{n}) belonging to R^{n}, put

χ_{a}(t) = exp (2πi ∑_{j=1}^{n} a_{j}t_{j}) .

We identify Λ = Z^{N} with the character group of the torus T = R^{n}/Z^{n} by a ↦ χ_{a}, and associate to A = A(x, N) the function ϕ whose Fourier expansion is:

ϕ = ∑_{a∈A} χ_{a}.

The condition on the reduction of A mod p and the fact that A is contained in a cube of side length N give rise to inequalities satisfied by ϕ; combining these will give the sieve inequality.

10.2 A lemma on finite groups

Let C_{i} for 1 ≤ i ≤ h be finite abelian groups (written additively), Ĉ_{i} their character groups, ϕ a function on C = ∏ C_{i}. Suppose there are subsets Ω_{i} of Ĉ_{i} with |Ω_{i}| ≤ ν_{i}|C_{i}|, such that the Fourier coefficient of ϕ relative to the character χ = (χ_{i}) ∈ Ĉ = ∏ Ĉ_{i} is 0 outside ∏ Ω_{i}. Let us call x ∈ C primitive if its image in each C_{i} is ≠ 0. Then:

Lemma 10.2.1 We have :

∑_{x primitive^{x∈C}} |ϕ(x)|^{2} ≥ |ϕ(0)|^{2} ∏_{i} (\frac{1 − ν_{i}}{ν_{i}} ) .

We give the proof in the case of a single group C: the general case follows by induction on the number of factors. Write ϕ = ∑ c_{χ}χ, the sum being taken over all characters χ ∈ Ω. Then:

∑ |c_{χ}|^{2} = \frac{1}{|C|} ∑_{x∈C} |ϕ(x)|^{2}.

Applying the Cauchy-Schwarz inequality, we get

|ϕ(0)|^{2} = | ∑ c_{χ} · 1|^{2} ≤ ∑ |c_{χ}|^{2} ∑_{χ∈Ω} 1,

and hence

|ϕ(0)|^{2} ≤ ν_{1}(∑_{x≠0} |ϕ(x)|^{2} + |ϕ(0)|^{2}).

The lemma follows by rearranging terms in this inequality.

10.3 The Davenport-Halberstam theorem

Define a distance on R^{n} by |x| = sup |x_{i}|; this defines a distance on the torus T = R^{n}/Z^{n}, which we also denote by | |. Let δ > 0; a set of points {t_{i}} in T is called δ-spaced if |t_{i} − t_{j}| ≥ δ for all i ≠ j.

Theorem 10.3.1 (Davenport-Halberstam) Let ϕ = ∑ c_{λ}χ_{λ} be a continuous function on T whose Fourier coefficients c_{λ} vanish when λ is outside some cube Σ of size N. Let t_{i} ∈ T be δ-spaced points for some δ > 0. Then

∑_{i} |ϕ(t_{i})|^{2} ≤ 2^{n} sup(N, \frac{1}{δ})^{n}||ϕ||_{2}^{2},

where ||ϕ||_{2} is the L^{2}-norm of ϕ.

If δ > 1/2, there is at most one t_{i} and the inequality follows from the Cauchy-Schwarz inequality applied to the Fourier expansion of ϕ. Let us now suppose that δ ≤ \frac{1}{2}. One constructs an auxiliary function θ on R^{n}, such that

1. θ is continuous and vanishes outside the cube |x| < \frac{1}{2}δ. This allows us to view θ as a function on T.

2. The Fourier transform of θ has absolute value ≥ 1 on the cube Σ.

3. ||θ||_{2}^{2} ≤ 2^{n}M^{n}, where M = sup(N, \frac{1}{δ}).

Let λ ∈ R^{n} be the center of the cube Σ. Then one checks, by an elementary computation, that the function θ defined by

θ(x) = \{ \begin{matrix}χ_{λ}(x)M^{n} ∏ 2 cos πMx_{i} & if |x| ≤ \frac{1}{2M} \\ 0 & elsewhere. \end{matrix}

has the required properties. For each λ ∈ Λ, let c_{λ}(ϕ) be the λ-th Fourier coefficient of ϕ; define similarly c_{λ}(θ). We have:

c_{λ}(ϕ) = 0 if λ ∉ Σ and |c_{λ}(θ)| ≥ 1 if λ ∈ Σ.

We may thus define a continuous function g on T whose Fourier coefficients are:

c_{λ}(g) = \{ \begin{matrix}c_{λ}(ϕ)/c_{λ}(θ) & if λ ∈ Σ \\ 0 & if λ ∉ Σ \end{matrix}

Since c_{λ}(ϕ) = c_{λ}(θ)c_{λ}(g) for every λ ∈ Λ, ϕ is equal to the convolution product θ ∗ g of θ and g. Therefore:

ϕ(t_{i}) = ∫_{T} θ(t_{i} − t)g(t)dt = ∫_{B_{i}} θ(t_{i} − t)g(t)dt,

where B_{i} is the set of t such that |t−t_{i}| < \frac{δ}{2}. By the Cauchy-Schwarz inequality:

|ϕ(t_{i})|^{2} ≤ ||θ||_{2}^{2} ∫_{B_{i}} |g(t)|^{2}dt ≤ 2^{n}M^{n} ∫_{B_{i}} |g(t)|^{2}dt.

Since the t_{i} are δ-spaced, the B_{i} are disjoint. Summing over i then gives

∑_{i} |ϕ(t_{i})|^{2} ≤ 2^{n}M^{n}||g||_{2}^{2} ≤ 2^{n}M^{n}||ϕ||_{2}^{2},

because

||g||_{2}^{2} = ∑ |\frac{c_{λ}(ϕ)}{c_{λ}(θ)}|^{2} ≤ ||ϕ||_{2}^{2}.

This completes the proof.

Remark: In the case n = 1, the factor 2 sup(N, \frac{1}{δ}) can be improved to N + \frac{1}{δ} (Selberg, see e.g. [Mo2]); it is likely that a similar improvement holds for any n.

10.4 Combining the information

Let D be given; the set {t_{i}} of all d-division points of T, where d ranges over positive square-free integers ≤ D, is δ-spaced, for δ = 1/D^{2}. Applying th. 10.3.1 to ϕ = ∑_{a∈A} χ_{a}, we have

∑_{i} |ϕ(t_{i})|^{2} ≤ 2^{n} sup(N, D^{2})^{n}|A|.  (10.1)

On the other hand, for each d ≤ D square-free, the kernel T [d] of d : T ⟶ T splits as

T [d] = ∏_{p|d} T [p]

and its character group is Λ/dΛ = ∏_{p|d} Λ/pΛ. Hypothesis (2) on A allows us to apply lemma 10.2.1 to the restriction of ϕ to T [d]. We thus obtain

∑_{t of order d^{t∈T [d]}} |ϕ(t)|^{2} ≥ |A|^{2} ∏_{p|d} \frac{1 − ν_{p}}{ν_{p}} .

Hence, by summing over all square-free d ≤ D, we obtain:

∑_{i} |ϕ(t_{i})|^{2} ≥ |A|^{2}L(D).  (10.2)

Combining equations 10.1 and 10.2 and cancelling a factor of |A| on both sides gives the large sieve inequality. (The case |A| = 0 does not pose any problem.) QED.

Remark: A similar statement holds for a number field K; Λ is replaced by O_{K} ×⋯×O_{K}, where O_{K} denotes the ring of integers of K; the corresponding torus T is then equipped with a natural action of O_{K}. The technique of the proof is essentially the same as in the case K = Q, see [Se9], ch. 12.

Exercises:

1. Let p_{i} (i ∈ I) be integers ≥ 1 such that (p_{i}, p_{j}) = 1 if i ≠ j. Let A be a subset of Z^{n} contained in a cube of side length N. Let ν_{i} be such that the reduction of A mod p_{i} has at most ν_{i}p_{i}^{n} elements. Show (by the same method as for th. 10.1.1) that

|A| ≤ 2^{n}sup(N, D^{2})^{n}/L(D),

with

L(D) = ∑_{J} ∏_{i∈J}(1 − ν_{i})/ν_{i},

where the sum runs through all subsets J of I such that ∏_{i∈J} p_{i} ≤ D. (This applies for instance when the p_{i}’s are the squares or the cubes of the prime numbers.)

2. Let H be the set of pairs (x, y) of integers ≠ 0 such that the Hilbert symbol (x, y) is trivial (i.e., the conic Z^{2} − xX^{2} − yY^{2} = 0 has a rational point). Show (by using exerc. 1.) that the number of points of H in a cube of side length N is << N^{2}/ log N. (Whether or not this bound is sharp is not known.)

For generalizations, see [Se10].

Bibliography

[Ab1] S. Abhyankar. Coverings of algebraic curves, Amer. J. of Math. 79 (1957), 825-856.

[Ab2] S. Abhyankar. Galois theory on the line in nonzero characteristic, Bull. A.M.S., 27 (1992), 68-133.

[ANVERS] B.J. Birch and W. Kuyk (edit.). Modular Functions of One Variable IV, Lect. Notes in Math. 476, Springer-Verlag, 1975.

[ATLAS] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson. Atlas of finite groups: maximal subgroups and ordinary characters for simple groups. New York: Clarendon press, 1985.

[Bc] S. Beckmann. Ramified primes in the field of moduli of branched coverings of curves, J. of Algebra 125 (1989), 236-255.

[Be1] G.V. Belyi. On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR 43 (1979), 267-276 (= Math. USSR Izv. 14 (1980), 247-256).

[Be2] G.V. Belyi. On extensions of the maximal cyclotomic field having a given classical Galois group, J. Crelle 341 (1983), 147-156.

[BL] E. Bayer-Fluckiger and H.W. Lenstra. Forms in odd degree extensions and self-dual normal bases, Amer. J. of Math. 112 (1990), 359-373.

[Bo] E. Bombieri. On exponential sums in finite fields II, Invent. Math. 47 (1978), 29-39.

[Bor] A. Borel. Introduction aux groupes arithmétiques, Paris, Hermann, 1969.

[Coh] S.D. Cohen. The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. London Math. Soc.(3) 43 (1981), 227-250.

[CT] J-L. Colliot-Thélène. Letter to T. Ekedahl, 9/21/1988, unpublished.

[CTS1] J-L. Colliot-Thélène and J-J. Sansuc. Principal homogeneous spaces and flasque tori: applications, J. of Algebra 106 (1987), 148-205.

[CTS2] J-L. Colliot-Thélène and J-J. Sansuc. La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375-492.

[Cr] T. Crespo. Explicit construction of Ã_{n}-type fields, J. of Algebra 127 (1989), 452-461.

[CR] C.W. Curtis and I. Reiner. Representation Theory of Finite Groups and Associative Algebras, Inters. Publ., 1962.

[Da] H. Davenport. Multiplicative Number Theory (revised by H.L. Montgomery), Springer-Verlag, 1980.

[DD] R. and A.Douady. Algèbre et théories galoisiennes; 2/ théories galoisiennes, CEDIC/F.Nathan, Paris, 1979.

[De1] P. Deligne. Le support du caractère d’une représentation supercuspidale, C.R. Acad. Sci. Paris 283 (1976), 155-157.

[De2] P. Deligne. Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien, Sém. Bourbaki 1979/80, no. 543.

[Ek] T. Ekedahl. An effective version of Hilbert’s irreducibility theorem, preprint, Stockholm, 1987. (cf. sém. Théorie des Nombres, Paris 1988-1989, Birkhäuser, 1990, 241-248).

[F] R.Fricke. Lehrbuch der Algebra. III. Algebraische Zahlen, Braunschweig, 1928.

[Fo] O. Forster. Lectures on Riemann surfaces, Springer-Verlag, 1981.

[Fr1] M.D. Fried. Fields of definition of function fields and Hurwitz families - Groups as Galois groups, Comm. Alg. 5 (1977), 17-82.

[Fr2] M.D. Fried. Rigidity and applications of the classification of simple groups to monodromy, preprint.

[Ga] P.X. Gallagher. The large sieve and probabilistic Galois theory, Proc. Symp. Pure Math. 24, 91-101, A.M.S., 1973.

[GH] P. Griffiths and J. Harris. Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978.

[GR] H. Grauert and R. Remmert. Komplexe Räume, Math. Ann. 136 (1958), 245-318.

[Harb] D. Harbater. Galois coverings of the arithmetic line, Lect. Notes in Math. 1240, Springer-Verlag, 1987, 165-195.

[Ha] R. Hartshorne. Algebraic Geometry, Springer-Verlag, 1977.

[He] E. Hecke. Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften, Math. Ann. 111 (1935), 293-301 (= Math. Werke, 568-576).

[Hi] D. Hilbert. Ueber die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten, J. Crelle 110 (1892), 104-129 (= Ges. Abh. II, 264-286).

[Ho] G. Hoyden-Siedersleben. Realisierung der Jankogruppen J_{1} und J_{2} als Galoisgruppen über Q, J. of Algebra 97 (1985), 14-22.

[Hu] B. Huppert. Endliche Gruppen I, Springer-Verlag, 1967.

[Hunt] D.C. Hunt. Rational rigidity and the sporadic groups, J. of Algebra 99 (1986), 577-592.

[ILK] V.V. Ishanov, B.B. Lur’e, D.K. Faddeev. Embedding problems in Galois theory, Moscow, 1990.

[Is] V.V. Ishanov. On the semi-direct embedding problem with nilpotent kernel, Izv. Akad. Nauk SSSR 40 (1976), 3-25, (=Math. USSR Izv. 10 (1976),1-23).

[J1] C. Jordan. Traité des substitutions et des équations algébriques, Gauthiers-Villars, Paris, 1870; 2ème édition, Paris, 1957.

[J2] C. Jordan. Recherches sur les substitutions, J. Liouville 17 (1872), 351-367 (= Oeuvres, I, no. 52).

[J3] C. Jordan. Sur la limite de transitivité des groupes non alternés, Bull. Soc. Math. France 1 (1873), 40-71 (= Oeuvres, I, no. 57)

[Jou] J-P. Jouanolou. Théorèmes de Bertini et applications, Birkhäuser, [JY] C.U. Jensen and N. Yui. Quaternion extensions, Algebraic Geom-

etry and Commutative Algebra, in honor of M. Nagata (1987), 155-182.

[Ki] R. Kiehl. Der Endlichkeitsatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie, Inv. Math. 2 (1967), 191-214.

[Kö] U. Köpf. Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schriftenreihe Univ. Münster, 2. serie Heft 7, 1974.

[KN] W. Krull and J. Neukirch. Die Struktur der absoluten Galoisgruppe über dem Körper R(t), Math. Ann. 193 (1971), 197-209.

[Ku] W. Kuyk. Extensions de corps hilbertiens, J. of Algebra 14 (1970), 112-124.

[L] S. Lang. Fundamentals of Diophantine Geometry, Springer-Ver-lag, New York, 1983.

[Le] H.W. Lenstra. Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299-325.

[LM] M.E. LaMacchia. Polynomials with Galois group PSL(2, 7), Commun. Alg. 8 (1980), 983-992.

[Mae] T. Maeda. Noether’s problem for A_{5}, J. of Algebra 125 (1989), 418-430.

[Ma1] B.H. Matzat. Zwei Aspekte konstruktiver Galoistheorie, J. of Algebra 96 (1985), 499-531.

[Ma2] B.H. Matzat. Topologische Automorphismen in der konstruktiver Galoistheorie, J. Crelle 371 (1986), 16-45.

[Ma3] B.H. Matzat. Konstruktive Galoistheorie, Lect. Notes in Math. 1284, Springer-Verlag, 1987.

[Ma4] B.H. Matzat. Konstruktion von Zahlkörpern mit der Galoisgruppe M_{11} über Q(\sqrt{−11}), Man. Math. 27 (1979), 103-111.

[Me1] J-F. Mestre. Courbes hyperelliptiques à multiplications réelles, C.R.Acad.Sci.Paris 307 (1988), 721-724.

[Me2] J-F. Mestre. Extensions régulières de Q(T) de groupe de Galois Ã_{n}, J. of Algebra 131 (1990), 483-495.

[Ml1] G. Malle. Exceptional groups of Lie type as Galois groups, J. Crelle 392 (1988), 70-109.

[Ml2] G. Malle. Genus zero translates of three point ramified Galois extensions, Man. Math. 71 (1991), 97-111.

[Mo1] H.L. Montgomery. A note on the large sieve, J. London Math. Soc. 43 (1968), 93-98.

[Mo2] H.L. Montgomery. The analytic principle of the large sieve, Bull. A.M.S. 84 (1978), 547-567.

[MoV] H.L. Mongomery and R.C. Vaughan. The large sieve, Mathematica 20 (1973), 119-134.

[Ne] J. Neukirch. On solvable number fields, Invent. Math. 53 (1979), 135-164.

[Noe] E. Noether. Gleichungen mit vorgeschriebener Gruppe, Math. Ann. 78 (1918), 221-229 (=Ges. Abh. no. 42).

[Oe] J. Oesterlé. Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math 78 (1984), 13-88.

[Os] H. Osada. The Galois groups of the polynomials x^{n} + ax^{l} + b, J. Number Theory 25 (1987), 230-238.

[Pa] H. Pahlings. Some sporadic groups as Galois groups, Rend. Sem. Mat. Univ. Padova 79 (1988), 97-107; II, ibid. 82 (1989), 163-171; correction, to appear.

[Pr] B. Pryzwara. Die Operation der Hurwitzschen Zopfgruppe auf den Erzeugendensystemklassen endlicher Gruppen, Diplomarbeit, Karlsruhe, 1988.

[Ra] M. Raynaud. Anneaux locaux henséliens, Lect.Notes in Math. 169, Springer-Verlag, 1970.

[Re] H. Reichardt. Konstruktion von Zahlkörpern mit gegebener Galoisgruppe von Primzahlpotenzordnung, J. Crelle 177 (1937), 1-5.

[RR] M.S. Raghunathan and A. Ramanathan. Principal bundles on the affine line, Proc. Indian Acad. Sci. (Math. Sci.) 93 (1984), 137-145.

[Sa1] D. Saltman. Generic Galois extensions and problems in field theory, Adv. Math. 43 (1982), 250-283.

[Sa2] D. Saltman. Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), 71-84.

[Sch] W. Scharlau. Quadratic and Hermitian Forms, Springer-Verlag, 1985.

[Schm] W.M. Schmidt. Integer points on hypersurfaces, Monat. für Math. 102 (1986), pp. 27-58.

[Schn] L. Schneps. Explicit construction of extensions of K(t) of Galois group Ã_{n} for n odd, J. of Algebra, 146 (1992), 117-123.

[Se1] J-P. Serre. Cohomologie Galoisienne, Lect. Notes in Math. 5, Springer-Verlag, 1973 (4th edition).

[Se2] J-P. Serre. Local Fields, GTM 67, Springer-Verlag, 1979.

[Se3] J-P. Serre. Algebraic Groups and Class Fields, GTM 117, Springer-Verlag, 1988.

[Se4] J-P. Serre. Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1-42 (=C.P. no. 32).

[Se5] J-P. Serre. Extensions icosaédriques, Sém. Th. des Nombres, Bordeaux, 1979-80, exposé 19 (=C.P. no. 123).

[Se6] J-P. Serre. L’invariant de Witt de la forme Tr (x^{2}), Comm. Math. Helv. 59 (1984), 651-679. (=C.P. no. 131).

[Se7] J-P. Serre. Modular forms of weight one and Galois representations, Algebraic Number Fields (A. Fröhlich edit.) Acad. Press, 1977, 193-268. (=C.P. no. 110).

[Se8] J-P. Serre. Groupes de Galois sur Q, Sém. Bourbaki 1987-1988, no. 689.

[Se9] J-P. Serre. Lectures on the Mordell-Weil theorem, translated and edited by M. Brown from notes by M. Waldschmidt, Vieweg-Verlag, 1989.

[Se10] J-P. Serre. Spécialisation des éléments de Br_{2}(Q(T_{1}, …, T_{n})), C.R. Acad. Sci. Paris 311 (1990), 397-402.

[Se11] J-P. Serre. Revêtements de courbes algébriques, Sém. Bourbaki, 1991-1992, no. 749.

[Sel] E.S. Selmer. On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287-302.

[SGA1] A. Grothendieck. Revêtement étales et groupe fondamental, Lect. Notes in Math. 224, Springer-Verlag, 1971.

[Sha1] I.R. Shafarevich. Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR 18 (1954), 525-578 (=Amer. Math. Transl. 4 (1956), 185-237 = C.P., 139-191).

[Sha2] I.R. Shafarevich. The embedding problem for split extensions, Dokl. Akad. Nauk SSSR 120 (1958), 1217-1219 (=C.P., 205-207).

[Sha3] I.R. Shafarevich. On the factors of a descending central series (in Russian), Mat.Zam. 45 (1989), 114-117.

[Shih1] K-y. Shih. On the construction of Galois extensions of function fields and number fields, Math. Ann. 207 (1974), 99-120.

[Shih2] K-y. Shih. p-division points on certain elliptic curves, Comp. Math. 36 (1978), 113-129.

[Shim] G. Shimura. Introduction to the arithmetic theory of automorphic functions, Publ. Math.Soc.Japan 11, Iwanami Shoten and Princeton University Press, 1971.

[Sie] C.L. Siegel. Topics in complex function theory, vol. 2, Wiley-Interscience, New York, 1971.

[Sw1] R. Swan. Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148-158.

[Sw2] R. Swan. Noether’s problem in Galois theory, Emmy Noether in Bryn Mawr (J.D. Sally and B. Srinivasan edit.), 21-40, Springer-Verlag, 1983.

[Th1] J.G. Thompson. A non-duality theorem for finite groups, J. of Algebra 14 (1970), 1-4.

[Th2] J.G. Thompson. Some finite groups which appear as Gal(L/K), where K ⊆ Q(μ_{n}), J. of Algebra 89 (1984), 437-499.

[Th3] J.G. Thompson. PSL_{3} and Galois groups over Q, Proc. Rutgers group theory year 1983-1984, Cambridge Univ. Press, 1984, 309-319.

[Vi] N. Vila. On central extensions of A_{n} as Galois group over Q, Arch. Math. 44 (1985), 424-437.

[Vo1] V.E. Voskresenskii. Birational properties of linear algebraic groups, Izv. Akad. Nauk SSSR 34 (1970), 3-19 (= Math. USSR Izv. 4 (1970), 1-17).

[Vo2] V.E. Voskresenskii. On the question of the structure of the subfields of invariants of a cyclic group of automorphisms of the field Q(x_{1}, …, x_{n}), Izv. Akad. Nauk SSSR 34 (1970), 366-375 (=Math. USSR Izv. 4 (1970), 371-380).

[Wa] S. Wang. A counterexample to Grunwald’s theorem, Ann. Math. 49 (1948), 1008-1009.

[We] A. Weil. Adeles and algebraic groups. Notes by M. Demazure and T. Ono, Birkhäuser-Boston, 1982.

[Wi] H. Wielandt. Finite Permutation Groups, Acad. Press, New York, 1964.

[Z] O. Zariski. Pencils on an algebraic variety and a new proof of a theorem of Bertini, Trans. A.M.S. 50 (1941), 48-70 (=C.P., I, 154-176).

Index

117