Universal deformations, rigidity,

and Ihara’s cocycle

Colin Stewart Department of Mathematics and Statistics McGill University, Montréal January, 2001

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements of the degree of Master of Science.

\begin{matrix}c \\ ◯ \end{matrix}Colin Stewart, 2000

Abstract

In [Iha86b], Ihara constructs a universal cocycle

Gal (\overline{ℚ}/ℚ) ⟶ ℤ_{p}[[t_{0}, t_{1}, t_{∞}]]/ ((t_{0} + 1)(t_{1} + 1)(t_{∞} + 1) − 1)

arising from the action of Gal (\overline{ℚ}/ℚ) on certain quotients of the Jacobians of the Fermat curves

x^{p^{n}} + y^{p^{n}} = 1

for each n ≥ 1. This thesis gives a different construction of part of Ihara’s cocycle by considering the universal deformation of certain two-dimensional representations of Π\overline{ℚ}, where Π\overline{ℚ} is the algebraic fundamental group of ℙ^{1}(\overline{ℚ})\\{0, 1, ∞\}. More precisely, we determine, with and without certain deformation conditions, the universal deformation ring arising from a residual representation

ρ̄: Π\overline{ℚ} ⟶ GL_{2}(𝔽_{p}).

Belyĭ’s Rigidity Theorem is used to extend each determinant one universal deformation to a representation of Π_{K}, where K is a finite cyclotomic extension of ℚ(μ_{p^{∞}}). For a particular ρ̄, we give a geometric construction of one such extended universal deformation ρ, and show that part of Ihara’s cocycle can be recovered by specializing ρ at infinity.

Résumé

Dans [Iha86b], Ihara construit un cocycle universel

Gal (\overline{ℚ}/ℚ) ⟶ ℤ_{p}[[t_{0}, t_{1}, t_{∞}]]/ ((t_{0} + 1)(t_{1} + 1)(t_{∞} + 1) − 1)

provenant de l’action de Gal (\overline{ℚ}/ℚ) sur certains quotients des jacobiennes des courbes de Fermat

x^{p^{n}} + y^{p^{n}} = 1

pour chaque n ≥ 1. Cette thèse présente une construction différente d’un cas particulier du cocycle d’Ihara en considérant la déformation universelle de certaines représentations de dimension deux de Π\overline{ℚ}, où Π\overline{ℚ} est le groupe fondamental de ℙ^{1}(\overline{ℚ}) \ \{0, 1, ∞\}. Plus précisement, nous décrivons, avec et sans certaines conditions de déformation, l’anneau de déformation universelle provenant d’une représentation residuelle

ρ̄: Π\overline{ℚ} ⟶ GL_{2}(𝔽_{p}).

Le théorème de rigidité de Belyĭ est utilisé pour étendre chaque déformation universelle de déterminant un à une représentation du groupe Π_{K}, où K est une extension cyclotomique de degré fini de ℚ(μ_{p^{∞}}). Pour un ρ̄ particulier, une construction géométrique d’une de ces déformations universelles étendues ρ est fournie. Ceci permet de récupérer un cas particulier du cocycle d’Ihara par spécialisation de ρ à l’infini.

Acknowledgments

First and foremost, I would like to thank my supervisor, Henri Darmon, for his boundless patience and optimism, and for always finding the time to give me help when I needed it. I am also grateful to him for suggesting this project, which has allowed me to encounter a wide range of beautiful mathematics. My brother, Ian Stewart, has also earned my gratitude for his many helpful editorial suggestions, and for lending his expertise with L^{A}TEX. In addition, I would like to thank Eyal Goren for a number of stimulating discussions, Marc-Hubert Nicole for his help with translating the abstract, and, not least of all, Laura Mark for her encouragement and distractions. Finally, I would like to thank NSERC for their generous financial support of this research.

Contents

1 Introduction 7

2 Deformation Theory of Π\overline{ℚ} 12 2.1 Profinite Groups and Infinite Galois Theory . . . . . . . . . . 12 2.2 The Algebraic Fundamental Group . . . . . . . . . . . . . . . 16 2.3 The 𝔪-adic Topology . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 The Universal Deformation . . . . . . . . . . . . . . . . . . . . 26 2.6 Conditions on Deformations . . . . . . . . . . . . . . . . . . . 36

3 Lowering the Field of Definition 41 3.1 The Cyclotomic Character . . . . . . . . . . . . . . . . . . . . 41 3.2 The Rigidity Theorem . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Rigidity in GL_{2}(R^{univ}) . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Extending the Universal Deformation . . . . . . . . . . . . . . 53

4 Geometric Construction of Universal Deformations 57 4.1 Jacobians of Curves . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Tate Modules and ℓ-adic Representations . . . . . . . . . . . . 61 4.3 Reduction of Curves . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Mumford Curves . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.5 Hypergeometric Families of Curves . . . . . . . . . . . . . . . 69 4.6 The Reduction Type of C_{n}, C_{n}^{−} and the Associated Galois Rep-

resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.7 A Theorem of Katz . . . . . . . . . . . . . . . . . . . . . . . . 88

4.8 The Universal Deformation . . . . . . . . . . . . . . . . . . . . 97 5 Relation to Ihara’s Cocycle 102

5.1 Ihara’s Construction . . . . . . . . . . . . . . . . . . . . . . . 102 5.2 The Inertia Group at Infinity . . . . . . . . . . . . . . . . . . 107

6 Conclusion 115

1 Introduction

One approach to studying the absolute Galois group G_{ℚ} = Gal (\overline{ℚ}/ℚ) has been via its canonical representation in the outer automorphism group of the algebraic fundamental group Π\overline{ℚ} of ℙ^{1}(\overline{ℚ}) \ \{0, 1, ∞\}. Let M denote the maximal algebraic extension of \overline{ℚ}(t) unramified outside t = 0, 1, ∞. Conjugating in Gal (M/ℚ(t)) by a lift of γ ∈ G_{ℚ} gives rise to an automorphism of Π\overline{ℚ} whose class modulo the group of inner automorphisms depends only on γ. Thus G_{ℚ} acts on Π\overline{ℚ} = Gal (M/\overline{ℚ}(t)) as a group of outer automorphisms, and we obtain a representation

ϕ : G_{ℚ} ⟶ Out (Π\overline{ℚ}) .

By a theorem of Belyĭ, ϕ is injective; as a result, studying the full representation ϕ seems to be too difficult. However, as a first step in this direction, Ihara considered, for each prime p, the representation

ψ : G_{ℚ} ⟶ Out (F/F′′) ,

where F denotes the maximal pro-p quotient of Π\overline{ℚ}, and F′′ = [[F, F], [F, F]] denotes the double commutator subgroup of F. We define a ℤ_{p}-algebra A by

A = ℤ_{p}[[t_{0}, t_{1}, t_{∞}]]/ ((t_{0} + 1)(t_{1} + 1)(t_{∞} + 1) − 1) .

Letting χ_{p} : G_{ℚ} ⟶ ℤ_{p}^{×} denote the p-cyclotomic character, G_{ℚ} acts as ℤ_{p}-algebra automorphisms on A by

γ · (1 + t_{i}) = (1 + t_{i})^{χ_{p}(γ)}

for each γ ∈ G_{ℚ}, and each i = 0, 1, ∞. In [Iha86b], Ihara shows that ψ is encoded by a cocycle

F : G_{ℚ} ⟶ A^{×}.

For each n, F describes in a precise way the action of G_{ℚ(μ_{p^{n}})} on the p-adic Tate module of the primitive quotients of the Jacobian of the Fermat curve F_{n} : x^{p^{n}} + y^{p^{n}} = 1 (see Theorem 5.4).

Let r : A ⟶ ℤ_{p}[[T]] be the ℤ_{p}-algebra homomorphism which maps t_{0} and t_{1} to T. In this paper, we describe a new construction of r ∘ F for each odd p, obtained via deformation theory of two-dimensional representations of Π\overline{ℚ} and the rigidity method of Belyĭ, Matzat, and Thompson.

We begin in Chapter 2 by considering deformations of arbitrary absolutely irreducible residual representations

ρ̄: Π\overline{ℚ} ⟶ GL_{2}(𝔽_{p}).

First we consider general deformations, then deformations subject to certain conditions; namely, the condition of having determinant equal to one, as well as certain “ordinariness” conditions combined with this determinant condition (see §2.6 for precise definitions). In each case, we determine the universal deformation ring, which is a power series ring with coefficients in ℤ_{p}, where the number of parameters depends only on the deformation conditions (see Theorems 2.27 to 2.31). In particular, let σ_{0}, σ_{1}, σ_{∞} ∈ Π\overline{ℚ} be topological generators of inertia groups above t = 0, 1, ∞ respectively, satisfying σ_{0}σ_{1}σ_{∞} = 1; then if ρ̄ has determinant one and is {σ_{0}, σ_{1}}-ordinary, the {σ_{0}, σ_{1}}-ordinary determinant one universal deformation ring of ρ̄ is the power series ring ℤ_{p}[[T]].

The arithmetic content of the various determinant one universal deformations (R^{univ}, ρ^{univ}) of Chapter 2 arises in Chapter 3 by means of rigidity. In order to use Belyĭ’s Rigidity Theorem (Theorem 3.5) to extend these universal deformations, we study rigidity in GL_{2}(R), where R is a local unique factorization domain, proving in particular that (ρ^{univ}(σ_{0}), ρ^{univ}(σ_{1}), ρ^{univ}(σ_{∞})) is rigid in GL_{2}(R^{univ}) (see Theorem 3.10). This result allows us to extend each representative of ρ^{univ} to a representation of Π_{K(t)} := Gal (M/K(t)), where K is a cyclotomic extension of ℚ(μ_{p^{∞}}) of degree at most p^{2} − 1 which depends on ρ̄ (see Theorem 3.12).

In Chapter 4, we fix the residual representation ρ̄ to be the representation describing the action of Π\overline{ℚ} on the p-torsion points of the Legendre family E_{L} of elliptic curves given by

E_{L} : y^{2} = x(x − 1)(x − t).

In this case, ρ̄ is {σ_{0}, σ_{1}}-ordinary, and the extension theorem of Chapter 3 shows that any representative of the {σ_{0}, σ_{1}}-ordinary universal deformation of ρ̄ can be extended to a representation

ρ : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T]]) .

Let μ_{p^{n}} be the group of p^{n}th roots of unity in \overline{ℚ}, and let ℤ_{p}[μ_{p^{n}}] be the corresponding group ring. We construct ρ as the inverse limit of the representations

ρ_{n} : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2}(ℤ_{p}[μ_{p^{n}}])

associated to the curves C_{n}/ℚ(t) given by

C_{n} : y^{2} = x (x^{2p^{n}} + (4t − 2)x^{p^{n}} + 1) ,

where the action of μ_{p^{n}} on C_{n} is given by ζ_{n} · (x, y) = (ζ_{n}x, ζn^{\frac{p^{n}+1}{2}} y) for any primitive p^{n}th root of unity ζ_{n}. In order to obtain a detailed understanding of each ρ_{n}, we make use of Mumford’s uniformization (Theorem 4.11) of Jacobians of curves C/L having a specific reduction type, where L is a field which is complete with respect to a non-archimedean valuation. We also use a general theorem of Katz (Theorem 4.31) which gives a geometric construction of any representation

κ : Π\overline{ℚ} ⟶ GL_{2} (ℚ_{p}(ζ_{n}))

for which (κ(σ_{0}), κ(σ_{1}), κ(σ_{∞})) is rigid.

Finally, we show in Chapter 5 how to specialize ρ at ∞ so as to obtain the representation r ∘ F (see Theorem 5.7). To prove that these representations are equal, we use the geometric construction of Chapter 4 to show that the given specialization ρ_{∞} of ρ describes the action of G_{ℚ(μ_{p^{∞}})} on certain quotients of the Jacobian J_{n} of the Fermat curve F_{n}. This property together with the corresponding property of r∘F implies that r∘F is a direct summand of ρ_{∞}.

This thesis is comprised of a combination of known and original results. Whenever possible, I have listed a source for known results. The main results of Chapter 2, namely Theorem 2.27 and the results contained in §2.6, may be known to some people, but, to my knowledge, have not previously been written down. The theorems of §§3.3 and 3.4 are original, as are all results appearing after Proposition 4.25 except those that are clearly marked otherwise.

2 Deformation Theory of Π\overline{ℚ}

2.1 Profinite Groups and Infinite Galois Theory

Throughout the sequel, we will be working extensively with Galois groups of infinite Galois extensions. In this section, we present the basic theory of such extensions, and show how profinite groups arise naturally as Galois groups in this context.

Let (I, ≤) be a directed set, that is, ≤ is a partial order on I such that for each i, j ∈ I, there is some k ∈ I such that i ≤ k and j ≤ k.

Definition 2.1 A directed system of groups (G_{i}, (ϕ_{ji})) is a collection of groups {G_{i}}_{i∈I} indexed by I, together with homomorphisms ϕ_{ji} : G_{j} ⟶ G_{i} for each i ≤ j such that ϕ_{ii} = Id_{G_{i}} and ϕ_{ki} = ϕ_{ji} ∘ ϕ_{kj} for i ≤ j ≤ k.

Given a directed system of groups (G_{i}, (ϕ_{ji})), a group G together with homomorphisms g_{i} : G ⟶ G_{i} for each i ∈ I will be called a commuting system above (G_{i}, (ϕ_{ji})) if the diagrams

\xymatrix{ & G \ar[dl]_{g_{j}} \ar[dr]^{g_{i}} \\ G_{j} \ar[rr]_{ϕ_{ji}} &  & G_{i}}

commute for all i ≤ j.

Proposition 2.2 Given a directed system of groups (G_{i}, (ϕ_{ji})), there is a commuting system (G, (g_{i})) above (G_{i}, (ϕ_{ji})) satisfying the following universal property: given any commuting system (H, (f_{i})) above (G_{i}, (ϕ_{ji})), there exists a unique homomorphism f : H ⟶ G such that the diagram

\xymatrix{ & H \ar[ddl]_{f_{j}} \ar[d]^{f} \ar[ddr]^{f_{i}} \\  & G \ar[dl]_{g_{j}} \ar[dr]^{g_{i}} \\ G_{j} \ar[rr]_{ϕ_{ji}} &  & G_{i}}

commutes for all i ≤ j.

Proof: Take G to be the set of all sequences of elements of {G_{i}}_{i∈I} compatible under the maps ϕ_{ji}; that is

G = {(σ_{i})_{i∈I} ∈ ∏_{i∈I}G_{i} : σ_{i} ∈ G_{i}, ϕ_{ji}(σ_{j}) = σ_{i} for all i ≤ j} .

Then G is a subgroup of ∏_{i∈I}G_{i} and satisfies the given universal property (see [Mor96], App. C, Proposition 4.2 for details). □ Remark: By the usual argument for universal objects, (G, (g_{i})) is unique up to unique isomorphism (see, e.g., [Lan93], p.57). We write G = \underleftarrow{lim}_{i∈I} G_{i}, and call (G, (g_{i})) the inverse limit of (G_{i}, (ϕ_{ji})).

By the same construction, inverse limits exist in the categories of rings, modules, and topological groups, among others.

Definition 2.3 A profinite group is a group which can be expressed as the inverse limit of a directed system of finite groups. A profinite group is said to be procyclic if it can be expressed as the inverse limit of a directed system of finite cyclic groups.

Given a profinite group G = \underleftarrow{lim} G_{i} (where each G_{i} is finite), we may view G as a subgroup of the direct product ∏_{i∈I}G_{i} as in the proof of Proposition 2.2 above. Giving each G_{i} the discrete topology, we may define a topology on G by taking the topology induced from the product topology on ∏_{i∈I}G_{i}. This definition gives G the structure of a topological group, and plays an essential role in the theory of infinite Galois extensions. For more on profinite groups, see [Sha72].

Consider an infinite Galois extension L/K. Let G = Gal (L/K). Given any finite Galois extension M/K contained in L, the group G_{M} := Gal (L/M) is a normal subgroup of G of finite index [M : K], and G/G_{M} is isomorphic to Gal (M/K), as in the case of finite extensions. Let M denote the set of all such intermediate fields M. Then M forms a directed set by inclusion, and {G/G_{M}}_{M∈M} together with the canonical maps ϕ_{M′M} : G/G_{M′} ⟶ G/G_{M} whenever G_{M′} ⊂ G_{M} (i.e. whenever M′ ⊃ M) forms a directed system of finite groups. The canonical maps G ⟶ G/G_{M} define a commuting system above (G/G_{M}, (ϕ_{M′M})), so by the universal property of the inverse limit, we obtain a homomorphism ϕ : G ⟶ \underleftarrow{lim}_{M∈M}G/G_{M}. In fact, ϕ is an isomorphism (see [Lan93], Ch. VI, Theorem 14.1). Thus G is naturally a profinite group. The topology on G is called the Krull topology. The Krull topology may also be defined without realizing G as a profinite group by taking as a base for open sets {σG_{M} : σ ∈ G, M ∈ M}.

As with finite Galois extensions, one defines the Galois correspondence between the set of intermediate fields M between K and L, and the set of subgroups H of G = Gal (L/K). This correspondence takes the intermediate field M to the subgroup Gal (L/M), and the subgroup H to the intermediate field L^{H} consisting of those elements of L fixed pointwise by H. In the case of infinite Galois extensions, not every subgroup of G arises as Gal (L/M) for some intermediate field M. However, the Krull topology on G allows us to identify which subgroups correspond to intermediate fields, in a way which is made precise by the Fundamental Theorem of Infinite Galois Theory:

Theorem 2.4 The Galois correspondence defines an inclusion-reversing bijection between the set of closed subgroups of G and the set of intermediate fields between K and L. Moreover, a closed subgroup H ⊂ G is normal if and only if the corresponding extension L^{H}/K is Galois, in which case Gal (L^{H}/K) ≅ G/H, the isomorphism being one of topological groups if we give G/H the quotient topology.

Outline of Proof: The main observation is that given any subgroup H ⊂ G, Gal (L/L^{H}) = \overline{H}, where \overline{H} denotes the closure of H in G with respect to the Krull topology. This observation together with the usual fundamental theorem of Galois theory reduces the proof to verifying certain details, which may be found in [Mor96], Ch. IV, §17. □

Given any group G, let N denote the set of all normal subgroups of G of finite index. Then N is naturally a directed set with respect to inclusion, and {G/N}_{N∈N} together with the canonical homomorphisms forms a directed system of finite groups.

Definition 2.5 For any group G, the profinite group

Ĝ := \underleftarrow{lim}_{N∈N}G/N

is called the profinite completion of G.

The profinite completion of a group is indeed a topological completion in the usual sense; it is possible to define Cauchy sequences in G with respect to a directed set of normal subgroups, in which case Ĝ is the completion of G with respect to these sequences. See [Lan93], Ch. I, §10 for details.

It is often useful to consider the subset N_{p} of N consisting of all normal subgroups of G of p-power index, where p is a fixed prime. In this case, \underleftarrow{lim}_{N∈N_{p}}G/N is called the pro-p completion of G. A collection of elements \{γ_{i}\}_{i∈I} of a profinite group G is said to topologically generate G if the subgroup of G generated by {γ_{i}}_{i∈I} is dense in G. Thus, for example, if Ĝ is the profinite completion of a group G, and {γ_{i}}_{i∈I} generates G, then viewing each γ_{i} as an element of Ĝ via the natural map G ⟶ Ĝ, the system {γ_{i}}_{i∈I} topologically generates Ĝ.

2.2 The Algebraic Fundamental Group

In this section, we give an explicit description of the group structure of Gal (K/\overline{ℚ}(t)), where K is the maximal algebraic extension of the function field \overline{ℚ}(t) ramified only at a fixed finite set of places.

Given fields K and F, and a place ϕ : K ⟶ F ∪ {∞}, the set ϕ^{−1}(F) of finite elements under ϕ is a local subring R of K with maximal ideal 𝔭 = ϕ^{−1}(0). We call R the valuation ring corresponding to ϕ, and 𝔭 its valuation ideal. If V/K is a variety with function field K(V), one may define a place ϕ_{P} : K(V) ⟶ K ∪ {∞} for each point P ∈ V by ϕ_{P} (f) = f(P), where we let f(P) = ∞ if f is not defined at P. In this case, the valuation ideal of ϕ_{P} is also called the valuation ideal corresponding to P.

Let L/K be a (possibly infinite) Galois extension. Let 𝔭 be a valuation ideal of K, and suppose that 𝔭̂ is a valuation ideal of L lying above 𝔭, with corresponding valuation ring A ⊂ L.

Definition 2.6 The group

D (𝔭̂/𝔭) := {σ ∈ Gal (L/K) : σ(𝔭̂) = 𝔭̂}

is called the decomposition group of 𝔭̂/𝔭. The inertia group I (𝔭̂/𝔭) of 𝔭̂/𝔭 is the subgroup of D (𝔭̂/𝔭) given by

I (𝔭̂/𝔭) := {σ ∈ Gal (L/K) : σ(a) ≡ a mod 𝔭̂ for all a ∈ A}.

We say that 𝔭̂ is unramified over 𝔭 if I (𝔭̂/𝔭) = 1.

If every valuation ideal of L lying above 𝔭 is unramified over 𝔭, then we say that 𝔭 is unramified in L. We will also say that a place ϕ of K is unramified in L if the valuation ideal corresponding to ϕ is unramified in L.

Let k be an algebraically closed subfield of ℂ. Let P_{1}, …, P_{r} be distinct points in ℙ^{1}(k), and 𝔭_{1}, …, 𝔭_{r} their corresponding valuation ideals in k(t). Let k(t)_{S} denote the maximal algebraic extension of k(t) unramified outside S = {𝔭_{1}, …, 𝔭_{r}}. Give ℙ^{1}(ℂ) the topology of the Riemann sphere, and choose a point P ∈ ℙ^{1}(ℂ)\{P_{1}, …, P_{r}}. Let Π be the topological fundamental group π_{1}(ℙ^{1}(ℂ) \ {P_{1}, …, P_{r}}, P) ≅ ⟨γ_{1}, …, γ_{r} | γ_{1} ⋯ γ_{r} = 1⟩.

Theorem 2.7 The extension k(t)_{S}/k(t) is Galois and Gal (k(t)_{S}/k(t)) is isomorphic to the profinite completion Π̂ of Π. Moreover, there are generators γ_{1}, …, γ_{r} of Π such that for each i = 1, …, r, the image of γ_{i} in Gal (k(t)_{S}/k(t)) topologically generates the (procyclic) inertia group I (𝔭̂_{i}/𝔭_{i}) of some valuation ideal 𝔭̂_{i} above 𝔭_{i}.

Outline of Proof: First assume k = ℂ. Then there exists a universal covering u : U ⟶ ℙ^{1}(ℂ) \ {P_{1}, …, P_{r}}. Using the Riemann Existence Theorem, one may show that finite Galois extensions N/k(t) unramified outside S are in bijective correspondence with finite coverings p : Y ⟶ ℙ^{1}(ℂ) of compact Riemann surfaces unramified outside {P_{1}, …, P_{r}} in such a way that the surface Y corresponds to its function field N/k(t) (see [Vol96], Theorem 5.14). Moreover, Gal (N/k(t)) is isomorphic to the group Deck(p) of deck transformations of the covering p. Now p̃:= p|_{Y \p^{−1}({P_{1},... ,P_{r}})} is a covering of ℙ^{1}(ℂ) \ {P_{1}, …, P_{r}}, and Deck(p) ≅ Deck(p̃). Using the universal covering u above, one sees that such coverings p̃ are in bijective correspondence with normal subgroups H of Π of finite index in such a way that Deck(p̃) ≅ Π/H. Thus, letting N_{k,S} = {N ⊂ k(t)_{S} : N/k(t) is finite, Galois}, we have

Gal (k(t)_{S}/k(t)) ≅ \underleftarrow{lim}_{N∈N_{k,S}}Gal (N/k(t))
≅ \underleftarrow{lim}_{H◁Π ; finite index}Π/H = Π̂.

This proves the first statement when k = ℂ.

To prove the second statement when k = ℂ, let Y and N be as above, fix a point P̂ ∈ u^{−1}(P), and let P̃ ∈ Y be the image of P̂. It is possible to choose lifts P̃_{i} ∈ Y of each P_{i}, and d_{i} ∈ Deck(p) so that d_{i}(P̃_{i}) = P̃_{i} and d_{1} ∘ ⋯ ∘ d_{r} = Id. Let 𝔭̄_{i} be the valuation ideal in N corresponding to P̃_{i}. Let σ_{i} ∈ Gal (N/k(t)) be the automorphism satisfying σ_{i} (f(P̃)) = d_{i}(P̃) for f ∈ N. Then σ_{i} generates I (𝔭̄_{i}/𝔭_{i}), and the various σ_{i} obtained in this way are compatible as N varies over finite extensions of k(t). Viewing Gal (k(t)_{S}/k(t)) as the inverse limit \underleftarrow{lim}_{N∈N_{k,S}}Gal (N/k(t)) and taking γ_{i} = (σ_{i})_{N∈N_{k,S}} gives generators of Gal (k(t)_{S}/k(t)) satisfying the assertions of the theorem with 𝔭̂_{i} = ⋃_{N∈N_{k,S}} 𝔭̄_{i}. This proves the theorem when k = ℂ.

For any algebraically closed subfield k of ℂ, let S′ denote the set of valuation ideals in ℂ(t) corresponding to the points P_{1}, …, P_{r} ∈ ℙ^{1}(k) ⊂ ℙ^{1}(ℂ). One may show that the assignment N ⟼ N ⊗_{k} ℂ defines a bijection N_{k,S} ⟶ N_{ℂ,S′}. This bijection gives rise to an isomorphism

Gal (k(t)_{S}/k(t)) ≅ \underleftarrow{lim}_{N∈N_{k,S}}Gal (N/k(t))
≅ \underleftarrow{lim}_{N⊗_{k}ℂ ∈ N_{ℂ,S′}}Gal (N ⊗_{k} ℂ/ℂ(t))
≅ Gal (ℂ(t)_{S}/ℂ(t)) ,

as desired. See [MM99], Ch. I, Theorems 1.3, 1.4, and 2.2 for full details. □ Remark: The above theorem is true for any algebraically closed field k of characteristic 0. We will only need the result when k = \overline{ℚ}.

Theorem 2.7 is part of a much more general connection between Galois groups over function fields and topological fundamental groups. Let k be as above, and X/k a smooth projective curve of genus g. Given distinct points P_{1}, …, P_{r} ∈ X(k), the maximal algebraic extension k(X)_{S} of the function field k(X) of X unramified outside the set S of valuation ideals of P_{1}, …, P_{r} is Galois. The group Gal (k(X)_{S}/k(X)) is called the algebraic fundamental group of X \ {P_{1}, …, P_{r}}, and is denoted by π_{1}^{alg} (X \ {P_{1}, …, P_{r}}). There is, up to homeomorphism, a unique compact connected oriented surface X_{g} of genus g (see, e.g. [Arm97], §7.4, 7.5). Theorem 2.7 may be generalized to this context as follows: let Q_{1}, …, Q_{r} ∈ X_{g} be distinct points, and choose any point Q ∈ X_{g} \ {Q_{1}, …, Q_{r}}; then π_{1}^{alg} (X \ {P_{1}, …, P_{r}}) is isomorphic to the profinite completion of π_{1} (X_{g} \ {Q_{1}, …, Q_{r}}, Q). See [Ser92], §6.3 for more details.

2.3 The 𝔪-adic Topology

This section collects some results concerning rings with which we will be working below. All rings will be assumed to be commutative.

Definition 2.8 A topological ring R is a ring together with a topology on its underlying set such that R forms a topological group under its addition, and the multiplication law R × R ⟶ R is continuous.

Let (R, 𝔪) be a local noetherian ring. There is a natural topology on R, called the 𝔪-adic topology, obtained by taking {𝔪^{n}}_{n∈ℕ} to be a fundamental system of neighbourhoods of 0 (and thus defining a fundamental system of neighbourhoods of each point by translation). This topology gives (R, 𝔪) the structure of a topological ring. Since R is noetherian, ⋂_{n∈ℕ}𝔪^{n} = {0} (see [Lan93], Ch. X, Corollary 5.7); thus the 𝔪-adic topology is Hausdorff. This topology is precisely that obtained from the metric d on (R, 𝔪) given by

d(r, s) = \begin{cases}0 & if r = s \\ e^{−v(r−s)} & otherwise \end{cases}  (2.9)

where v : R \ {0} ⟶ ℕ, called the 𝔪-adic valuation on R, is given by v(r) = max{n ∈ ℕ : r ∈ 𝔪^{n}}. Thus we may consider Cauchy sequences and convergence in R. We say that (R, 𝔪) is complete if R is complete with respect to the metric d.

Proposition 2.10 Given (R, 𝔪) as above, there exists a complete local ring (R̂, 𝔪̂) together with a continuous injective homomorphism ϕ : R ⟶ R̂ satisfying the following universal property: given any complete local ring (A, 𝔫) together with a continuous homomorphism ψ : R ⟶ A, there is a unique continuous homomorphism ψ̂ : R̂ ⟶ A such that

\xymatrix{R \ar[r]^{ϕ} \ar[dr]_{ψ} & R̂ \ar[d]^{ψ̂} \\  & A}

commutes.

Proof: See [GS71], §2. □

The completion R̂ of R may be identified with \underleftarrow{lim}_{n∈ℕ}R/𝔪^{n}, where the inverse limit is taken with respect to the canonical maps. In this case, ϕ is the natural injection R ⟶ R̂. Moreover, 𝔪̂ is the ideal generated by ϕ(𝔪) and R is itself complete if and only if ϕ is an isomorphism. For details, see [GS71], §2.

Example 2.11 Let R be the localization of ℤ at a prime ideal (p), so that (R, pR) is a local ring. The completion of (R, pR), denoted ℤ_{p}, is called the ring of p-adic integers. By the above remark, ℤ_{p} is isomorphic to \underleftarrow{lim}_{n∈ℕ}ℤ/p^{n}ℤ. The quotient field ℚ_{p} of ℤ_{p} is called the field of p-adic numbers.

Example 2.12 Let k = 𝔽_{p^{n}} denote the finite field of order p^{n}. The ring W (k) of Witt vectors over k is the integral closure of ℤ_{p} in the splitting field of x^{p^{n}} − x over ℚ_{p}. The ring W (k) is a complete local ring with residue field k. In particular, W (𝔽_{p}) is equal to ℤ_{p}, and if p is odd, W (𝔽_{p^{2}}) is equal to ℤ_{p}[\sqrt{α}], where α ∈ ℤ_{p}^{×} is not a square in ℤ_{p}. See [Ser68], Ch. II, §6 for details.

Example 2.13 Let (A, 𝔪_{A}) be a complete noetherian local ring, and let R be the localization of A[t_{1}, …, t_{n}] at the maximal ideal 𝔪 := (𝔪_{A}, t_{1}, …, t_{n}). The completion of (R, 𝔪R) is isomorphic to the ring A[[t_{1}, …, t_{n}]] of formal power series in n variables with coefficients in A.

Proposition 2.14 Let R be a noetherian ring. Then the ring R[[t_{1}, …, t_{n}]] is also noetherian.

Proof: See [Lan93], Ch. IV, Theorem 9.5 and its Corollary. □

Theorem 2.15 (Hensel’s Lemma) Let (R, 𝔪) be a complete local noetherian ring with residue field k, and let f(x) ∈ R[x] be a monic polynomial. Suppose that a ∈ k is a nonrepeated root of the reduction of f(x) mod 𝔪. Then f(x) has a unique root α ∈ R such that α reduces to a mod 𝔪.

Proof: See [Lan93], Ch. XII, Corollary 7.4. □

2.4 Deformation Theory

Fix a prime p. Let Π be a group having the property that its pro-p completion is topologically finitely generated. Let k be a finite field of characteristic p, and fix an absolutely irreducible continuous representation ρ̄: Π ⟶ GL_{n}(k), which will be called the residual representation. Let (R, 𝔪) be a complete local noetherian W (k)-algebra with residue field k, where W (k) is the ring of Witt vectors of k. A lift of ρ̄ to R is a continuous homomorphism ρ : Π ⟶ GL_{n}(R) such that the diagram

\xymatrix{Π \ar[r]^{ρ} \ar[dr]_{ρ̄} & GL_{n}(R) \ar[d]^{r} \\  & GL_{n}(k)}

commutes, where r is the map which takes a matrix to its entrywise reduction mod 𝔪. We define an equivalence relation ∼, called strict equivalence, on the set of lifts of ρ̄ to R by ρ_{1} ∼ ρ_{2} if there exists an M ∈ GL_{n}^{∘}(R) := ker(r) satisfying ρ_{1} = Mρ_{2}M^{−1} (that is, ρ_{1}(γ) = Mρ_{2}(γ)M^{−1} for all γ ∈ Π). A deformation of ρ̄ to R is a strict equivalence class [ρ] of lifts of ρ̄ to R. Note that [ρ̄] = {ρ̄} and whenever M ∈ GL_{n}^{∘}(R), conjugating a lift ρ of ρ̄ by M gives another lift of ρ̄. We will often write ρ in place of [ρ] when there is no risk of confusion.

Define a category DEF(ρ̄) whose objects are pairs (R, [ρ]), where R is a complete local noetherian W (k)-algebra with residue field k, and [ρ] is a deformation of ρ̄ to R. A morphism from (R_{1}, [ρ_{1}]) to (R_{2}, [ρ_{2}]) in DEF(ρ̄) is a continuous homomorphism ϕ : R_{1} ⟶ R_{2} reducing to the identity on k, such that for some ρ′_{2} ∈ [ρ_{2}], the diagram

\xymatrix{Π \ar[r]^{ρ_{1}} \ar[dr]_{ρ′_{2}} & GL_{n}(R_{1}) \ar[d]^{ϕ̃} \\  & GL_{n}(R_{2})}

commutes, where ϕ̃ denotes the map obtained by applying ϕ entrywise to a given matrix. Using a result of Schlessinger which guarantees the representability of functors satisfying certain criteria, Mazur proved the following theorem:

Theorem 2.16 (Mazur, 1989) There exists a universal element in the category DEF(ρ̄); that is, there exists a pair (R^{univ}, ρ^{univ}) ∈ DEF(ρ̄) such that for each (R, ρ) ∈ DEF(ρ̄), there is a unique ϕ : R^{univ} ⟶ R such that ϕ ∈ Mor (DEF(ρ̄)).

Proof: See [Maz89], §1.2. □

As usual, (R^{univ}, ρ^{univ}) is well-defined up to unique isomorphism in the category DEF(ρ̄). We call ρ^{univ} the universal deformation of ρ̄. In [dL97], Lenstra and de Smit give an explicit construction of R^{univ} in terms of generators and relations; however, their construction requires many more generators than are usually necessary, and is not very practical when considering specific examples. In what follows, we will consider only the cases k = 𝔽_{p} or 𝔽_{p^{2}} and n = 2.

Proposition 2.17 Let ρ̄ be a residual representation, and (R^{univ}, ρ^{univ}) its universal deformation. Then the entries of the elements of Imρ^{univ} topologically generate R^{univ}.

Proof: Let S denote the complete subring of R^{univ} (topologically) generated by the entries of the elements of Imρ^{univ}. Then ρ^{univ} maps to S, so the universal property of R^{univ} gives a morphism ι : R^{univ} ⟶ S in DEF(ρ̄). By the definition of S, ι is surjective. Given (A, ρ) ∈ DEF(ρ̄), the universal property of R^{univ} gives a morphism τ : R^{univ} ⟶ A in DEF(ρ̄), which restricts a morphism on S. On the other hand, if τ_{1}, τ_{2} : S ⟶ A are two such morphisms, then τ_{1} ∘ ι, τ_{2} ∘ ι : R^{univ} ⟶ A are two such morphisms, and hence are equal. Since ι is surjective, it follows that τ_{1} is equal to τ_{2}, and therefore (S, ρ^{univ}) is universal in DEF(ρ̄), and ι is an isomorphism. □

Let (R, 𝔪_{R}) be a local noetherian W (k)-algebra.

Definition 2.18 The (Zariski) cotangent space of R is the k-vector space t_{R}^{∗} := 𝔪_{R}/(p, 𝔪_{R}^{2}). The (Zariski) tangent space t_{R} of R is the dual space Hom_{k}(t_{R}^{∗}, k) of the cotangent space of R.

Note that since R is noetherian, t_{R}^{∗} and t_{R} are finite-dimensional vector spaces, and hence are abstractly isomorphic.

Proposition 2.19 Let R and S be local noetherian W (k)-algebras, and let f : R ⟶ S be a W (k)-algebra homomorphism reducing to the identity on k. Then f induces a k-linear map f_{∗} : t_{R}^{∗} ⟶ t_{S}^{∗} which is surjective if and only if f is surjective.

Proof: For each m ∈ 𝔪_{R}, f(m) ∈ 𝔪_{S}, so f restricts to an additive homomorphism f̄: 𝔪_{R} ⟶ 𝔪_{S}/(p, 𝔪_{S}^{2}). Checking that f̄(p, 𝔪_{R}^{2}) = 0, we obtain a k-linear map f_{∗} : t_{R}^{∗} ⟶ t_{S}^{∗}.

Suppose now that f is surjective. Then f̄: R ⟶ S/(p, 𝔪_{S}^{2}) is surjective, and hence f̄(𝔪_{R}) = 𝔪_{S}/(p, 𝔪_{S}^{2}). Thus f_{∗} is surjective.

Conversely, suppose that f_{∗} is surjective. The reduction of f mod p makes 𝔪_{S}/p𝔪_{S} into an R/pR-module; thus f gives rise to an R/pR-module homomorphism f^{+} : 𝔪_{R}/p𝔪_{R} ⟶ 𝔪_{S}/p𝔪_{S}, which reduces to a homomorphism f̄^{+} : 𝔪_{R}/(p, 𝔪_{R}^{2}) ⟶ 𝔪_{S}/(p, 𝔪_{R}𝔪_{S}). Given α ∈ 𝔪_{S}^{2}, write α = m_{1}m_{2} with m_{1}, m_{2} ∈ 𝔪_{S}, m_{1} ∉ 𝔪_{S}^{2}. Since f_{∗} is surjective, there is some m′_{1} ∈ 𝔪_{R} such that m_{1} = m′_{1} + m̃, where m̃ ∈ (p, 𝔪_{S}^{2}), and hence α = (m′_{1} + m̃)m_{2}. Thus we have shown that

𝔪_{S}^{2}/p𝔪_{S}^{2} ⊂ (𝔪_{R}/p𝔪_{R})𝔪_{S}/p𝔪_{S} + 𝔪_{S}^{3}/p𝔪_{S}^{3}.

By induction,

𝔪_{S}^{2}/p𝔪_{S}^{2} ⊂ (𝔪_{R}/p𝔪_{R})𝔪_{S}/p𝔪_{S} + 𝔪_{S}^{n}/p𝔪_{S}^{n}

for all n, which implies that 𝔪_{S}^{2}/p𝔪_{S}^{2} = (𝔪_{R}/p𝔪_{R})𝔪_{S}/p𝔪_{S} since S is noetherian. Thus f̄^{+} is surjective, and by a corollary of Nakayama’s lemma, f^{+} is itself surjective (see [Lan93], Ch. X, Proposition 4.5). Viewing 𝔪_{R} and 𝔪_{S} as W (k)-modules and applying Nakayama’s lemma shows that f(𝔪_{R}) = 𝔪_{S}. Every element of S can be expressed as λ + m with λ ∈ W (k) and m ∈ 𝔪_{S}, so this proves that f is surjective since f(W (k)) = W (k). □

2.5 The Universal Deformation

Let K be an algebraic extension of ℚ. Throughout the sequel, let

Π := Gal ({ \atop \overline{K}(t)}/\overline{K}(t)) ,

where { \atop \overline{K}(t)} denotes the maximal algebraic extension of \overline{K}(t) unramified outside 0,1, and ∞. Fix a prime p, and let k be a finite field of characteristic p. It follows from Theorem 2.7 that the pro-p completion of Π is topologically finitely generated. Let ρ̄: Π ⟶ GL_{2}(k) be a residual representation.

Proposition 2.20 The universal deformation ring R^{univ} of ρ̄ is isomorphic to a power series ring with coefficients in W (k).

Proof: By Proposition 2.14 and Examples 2.12 and 2.13 of §2.3, any power series ring W (k)[[t_{1}, …, t_{d}]] is a complete noetherian local ring. Writing R for R^{univ}, let d denote the k-dimension of t_{R}^{∗}, and let x̄_{1}, …, x̄_{d} ∈ t_{R}^{∗} be a collection of elements which forms a basis for t_{R}^{∗}. Choose lifts x_{1}, …, x_{d} ∈ 𝔪_{R} of x̄_{1}, …, x̄_{d} respectively. Defining ϕ(t_{i}) = x_{i} for each i = 1, …, d gives rise to a continuous W (k)-algebra homomorphism ϕ : W (k)[[t_{1}, …, t_{d}]] ⟶ R which reduces to the identity on k. Since the reductions t̄_{1}, …, t̄_{d} of t_{1}, …, t_{d} mod (p, 𝔪_{W (k)[[t_{1},... ,t_{d}]]}) form a basis for t_{W (k)[[t_{1},... ,t_{d}]]}^{∗}, and ϕ_{∗}(t̄_{i}) = x̄_{i} for each i, ϕ_{∗} is a k-vector space isomorphism. In particular, ϕ_{∗} is surjective, and therefore, by Lemma 2.19, ϕ is itself surjective.

Fix elements σ_{0}, σ_{1} of Π which generate Π topologically, and choose for each i = 0, 1 a lift M_{i} ∈ ϕ̃^{−1} (ρ^{univ}(σ_{i})) of ρ^{univ}(σ_{i}), where ϕ̃ denotes the map induced from ϕ. We obtain a deformation ρ : Π ⟶ GL_{2}(W (k)[[t_{1}, …, t_{d}]]) such that ρ(σ_{i}) = M_{i} for i = 0, 1, and ϕ̃∘ρ = ρ^{univ}. By the universal property of R, there is a map ψ : R ⟶ W (k)[[t_{1}, …, t_{d}]] such that ρ = ψ̃ ∘ ρ^{univ}. We claim that ψ splits ϕ, that is, ϕ ∘ ψ = Id_{R}. Given M ∈ Imρ^{univ}, let σ ∈ Π be a preimage of M; then

ϕ̃ ∘ ψ̃(M) = ϕ̃ ∘ ψ̃(ρ^{univ}(σ)) = ϕ̃(ρ(σ)) = ρ^{univ}(σ),

so if r ∈ R is an entry of some M ∈ Imρ^{univ}, then ϕ ∘ ψ(r) = r. Applying Proposition 2.17 proves the claim. Now ϕ∘ψ = Id_{R} implies that ϕ_{∗}∘ψ_{∗} = Id_{t_{R}^{∗}}, so ψ_{∗} is an isomorphism. In particular, ψ is surjective. Therefore, ψ is an isomorphism, as desired. □

If Π were to be replaced with some other profinite group in Proposition 2.20, it would not necessarily be possible to lift ρ^{univ} to W (k)[[t_{1}, …, t_{d}]]. However, the proof that we have given works with only minor changes provided that the cohomology group H^{2}(Π, ad(ρ̄)) is trivial, where ad(ρ̄) denotes the matrix ring M_{2}(k) together with the action of Π given by

σ · M = ρ̄(σ)Mρ̄(σ)^{−1}

for each σ ∈ Π, M ∈ M_{2}(k). Mazur showed moreover that the Krull dimension of R^{univ}/pR^{univ} is at least d_{1} − d_{2}, where d_{i} = dim_{k}H^{i}(Π, ad(ρ̄)), with equality when d_{2} = 0 (see [Maz89], §1.6 and [Gou], p.50 for details). As we shall see, H^{1}(Π, ad(ρ̄)) is naturally isomorphic to t_{R^{univ}} (as a k-vector space), so Mazur’s result agrees with the choice of d in the proof of Proposition 2.20.

Fix a residual representation

ρ̄: Π ⟶ GL_{2}(𝔽_{p}).

In order to determine R^{univ}(ρ̄) explicitly, it may be convenient to extend scalars to 𝔽_{p^{2}}, and thus replace ρ̄ with ρ̄′, where ρ̄′ is obtained by composing ρ̄ with the inclusion 𝔽_{p} ↪ 𝔽_{p^{2}}. Let R′ be the universal deformation ring corresponding to ρ̄′; by Proposition 2.20, R′ = W (𝔽_{p^{2}})[[t_{1}, …, t_{d′}]] for some d′. We will show that R^{univ} = ℤ_{p}[[t_{1}, …, t_{d′}]], so that R^{univ} may be recovered from R′. By Proposition 2.20, R^{univ} = ℤ_{p}[[t_{1}, …, t_{d}]] for some d, so it suffices to show that d = d′. If we show that for any residual representation ϱ̄: G ⟶ GL_{2}(k), there is a k-vector space isomorphism

t_{R^{univ}(ϱ̄)} ≅ H^{1}(G, ad(ϱ̄)),

then we have

d′ = dim_{𝔽_{p^{2}}}H^{1}(Π, ad(ρ̄′))
= dim_{𝔽_{p^{2}}}H^{1}(Π, ad(ρ̄) ⊗_{𝔽_{p}} 𝔽_{p^{2}})
= dim_{𝔽_{p^{2}}}H^{1}(Π, ad(ρ̄)) ⊗_{𝔽_{p}} 𝔽_{p^{2}}
= dim_{𝔽_{p}}H^{1}(Π, ad(ρ̄)) = d,

as desired.

Let (R^{univ}, ϱ^{univ}) be the universal deformation of a residual representation ϱ̄ : G ⟶ GL_{2}(k). The isomorphism t_{R^{univ}} ≅ H^{1}(G, ad(ϱ̄)) arises naturally through deformations of ϱ̄ to the ring of dual numbers k[ϵ], where ϵ^{2} = 0. First, there is a k-vector space isomorphism t_{R^{univ}} ≅ Hom_{W (k)}(R^{univ}, k[ϵ]), where Hom_{W (k)}(R^{univ}, k[ϵ]) consists of continuous W (k)-algebra homomorphisms reducing to the identity on k. Given ϕ ∈ Hom_{W (k)}(R^{univ}, k[ϵ]), and r ∈ R^{univ}, let r̄ ∈ k denote the reduction of r mod 𝔪_{R^{univ}}; since ϕ reduces to the identity on k, there is some ϕ′(r) ∈ k for which ϕ(r) = r̄ + ϕ′(r)ϵ. Restricting ϕ′ to 𝔪_{R^{univ}} gives an additive homomorphism whose kernel contains (p, 𝔪_{R^{univ}}^{2}), and thus ϕ′|_{𝔪_{R^{univ}}} factors through a map ϕ^{′∗} : t_{R^{univ}}^{∗} ⟶ k which is k-linear since ϕ is W (k)-linear. Furthermore, since ϕ is a W (k)-algebra homomorphism, it is completely determined by ϕ′|_{𝔪_{R^{univ}}}, so the correspondence ϕ ⟷ ϕ^{′∗} defines a bijection Hom_{W (k)}(R^{univ}, k[ϵ]) ⟷ t_{R^{univ}} which is k-linear.

On the other hand, there is a natural k-vector space isomorphism

Hom_{W (k)}(R^{univ}, k[ϵ]) ≅ H^{1}(G, ad(ϱ̄)).  (2.21)

First, there is a bijective correspondence between Hom_{W (k)}(R^{univ}, k[ϵ]) and the set of deformations of ρ̄ to k[ϵ], given by ϕ ⟷ ϕ̃ ∘ ρ^{univ}. For any lift ϱ : G ⟶ GL_{2}(k[ϵ]) of ϱ̄, let ϱ′ : G ⟶ M_{2}(k) denote the set-theoretic map satisfying

ϱ(g) = ϱ̄(g)(1 + ϱ′(g)ϵ)

for all g ∈ G. Then ϱ′ is a 1-cocycle with values in ad(ϱ̄), and a lift ϱ_{1} of ϱ̄ is strictly equivalent to ϱ if and only if ϱ′_{1} differs from ϱ′ by a coboundary. Thus deformations of ϱ̄ to k[ϵ] correspond to elements of H^{1}(G, ad(ϱ̄)); in fact, this correspondence defines the desired k-vector space isomorphism Hom_{W (k)}(R^{univ}, k[ϵ]) ≅ H^{1}(G, ad(ϱ̄)), and therefore gives rise to the isomorphism t_{R^{univ}} ≅ H^{1}(G, ad(ρ̄)). In particular, when G = Π and ρ̄ = ϱ̄, we may conclude that if R′ = W (𝔽_{p^{2}})[[t_{1}, …, t_{d}]], then R^{univ} = ℤ_{p}[[t_{1}, …, t_{d}]].

To determine the value of d, we will single out a distinguished representative for each deformation [ρ] of ρ̄. We will need the following lemma:

Lemma 2.22 Suppose that p > 3. Then there exist elements σ_{0}, σ_{1} ∈ Π such that σ_{0}, σ_{1} topologically generate Π, and ρ̄(σ_{0}), ρ̄(σ_{1}) each have distinct eigenvalues in 𝔽_{p^{2}}.

Proof: Let γ_{0}, γ_{1} ∈ Π be any two elements which (topologically) generate Π. Extending scalars to 𝔽_{p^{2}}, the matrices ρ̄(γ_{0}), ρ̄(γ_{1}) have eigenvectors v_{0}, v_{1} respectively. Since ρ̄ is absolutely irreducible, v_{0}, v_{1} form a basis for 𝔽_{p^{2}}^{2}, and writing ρ̄(γ_{0}), ρ̄(γ_{1}) with respect to this basis gives ρ̄(γ_{0}) = ( \begin{matrix}a & b \\ 0 & c \end{matrix}) and ρ̄(γ_{1}) = ( \begin{matrix}d & 0 \\ f & g \end{matrix} ), for some a, b, c, d, f, g ∈ 𝔽_{p^{2}}. Since ρ̄ is absolutely irreducible, b and f are both nonzero. Rescaling v_{0}, v_{1} (equivalently, conjugating by an appropriate diagonal matrix), we may assume that b = 1. Suppose first that only one of ρ̄(γ_{0}) or ρ̄(γ_{1}) has distinct eigenvalues. Then without loss of generality, we have d ≠ g. Now ρ̄(γ_{0}γ_{1}) has characteristic polynomial

f(X) = X^{2} − (ad + f + ag)X + a^{2}dg,

which has a repeated root if and only if (\frac{ad+f+ag}{2})^{2} = a^{2}dg (since p ≠ 2). Similarly, ρ̄(γ_{0}γ_{1}^{−1}) has a repeated eigenvalue if and only if (\frac{ad+ag−f}{2} )^{2} = a^{2}dg. In particular, if both ρ̄(γ_{0}γ_{1}) and ρ̄(γ_{0}γ_{1}^{−1}) have repeated eigenvalues, then (ad + ag − f)^{2} = (ad + f + ag)^{2}; expanding gives d = −g since a ≠ 0 and f ≠ 0. Also, the equalities (\frac{ad+f+ag}{2})^{2} = a^{2}dg and d = −g imply that f^{2} = −4a^{2}d^{2}. If ρ̄(γ_{0}γ_{1}) and ρ̄(γ_{0}γ_{1}^{−1}) both have repeated eigenvalues, then a similar calculation shows that ρ̄(γ_{0}^{2}γ_{1}) has a repeated eigenvalue if and only if f^{2} = −a^{2}d^{2}, which is impossible when p ≠ 3 since f^{2} = −4a^{2}d^{2}, a ≠ 0 and d ≠ 0. Similarly, ρ̄(γ_{0}^{3}γ_{1}) has a repeated eigenvalue if and only if 9f^{2} = −4a^{2}d^{2}, which is impossible when p ≠ 2 since f^{2} = −4a^{2}d^{2}. Therefore, at least one of the pairs (γ_{0}γ_{1}, γ_{1}), (γ_{0}γ_{1}^{−1}, γ_{1}), or (γ_{0}^{2}γ_{1}, γ_{0}^{3}γ_{1}) gives the desired (σ_{0}, σ_{1}).

Suppose now that ρ̄(γ_{0}), ρ̄(γ_{1}) both have repeated eigenvalues. Without loss of generality, we may assume that ρ̄(γ_{0}) = ( \begin{matrix}a & 1 \\ 0 & a \end{matrix}) and ρ̄(γ_{1}) = ( \begin{matrix}b & 0 \\ c & b \end{matrix}), for some a, b, c ∈ 𝔽_{p^{2}}^{×}. A simple calculation shows that ρ̄(γ_{0}γ_{1}) has a repeated eigenvalue if and only if 4ab + c = 0. Similarly, ρ̄(γ_{1}γ_{0}γ_{1}) has a repeated eigenvalue if and only if 2ab + c = 0, which cannot be the case when p ≠ 2 if ρ̄(γ_{0}γ_{1}) has a repeated eigenvalue. Therefore at least one of the pairs (γ_{0}γ_{1}, γ_{1}) or (γ_{1}γ_{0}γ_{1}, γ_{1}) generates Π and has the property that the image of its first component has distinct eigenvalues. This reduces the problem to the case considered above, thus proving the lemma. □

Let F be a free module over a ring R, and M an endomorphism of F.

Definition 2.23 An element v ∈ F is said to be an eigenvector of M (with eigenvalue λ) if there exists some λ ∈ R satisfying Mv = λv, and v may be completed to a basis of F.

Remark: If R is a local ring, and F is finitely generated over R, then by Nakayama’s lemma, v ∈ F may be completed to a basis of F if and only if the reduction of v mod 𝔪_{R} is nontrivial.

Proposition 2.24 Let (R, 𝔪) be a local ring with residue field k. Suppose that M ∈ GL_{2}(R) does not reduce to a scalar matrix mod 𝔪. Then M has an eigenvector in R^{2} with eigenvalue λ ∈ R if and only if λ is a root of the characteristic polynomial ch(M) of M.

Proof: Let M̄ denote the reduction of M mod 𝔪. Since M̄ is not a scalar matrix, there is a basis {b̄_{1}, b̄_{2}} of k^{2} with respect to which M̄ has at least three nonzero entries. Let b_{1}, b_{2} ∈ R^{2} be elements reducing to b̄_{1}, b̄_{2} mod 𝔪. By Nakayama’s lemma, \{b_{1}, b_{2}\} forms a basis for R^{2}. Let M = ( \begin{matrix}a & b \\ c & d \end{matrix}) with respect to {b_{1}, b_{2}}. Assume that a, b, d ∈ R^{×} (if not, one may apply a similar argument using the three entries of M which are units). Suppose that λ ∈ R is a root of ch(M). Then we claim that v = b_{1} + (\frac{λ−a}{b})b_{2} is an eigenvector of M having eigenvalue λ. Clearly v reduces to a nontrivial vector mod 𝔪. Expanding gives Mv = λb_{1} + (c + d(\frac{λ−a}{b})) b_{2}. Since λ is a root of ch(M), we have (a − λ)(d − λ) − bc = 0, and hence λ(\frac{λ−a}{b}) = c + d(\frac{λ−a}{b}). Substituting into the above expression for Mv proves the claim.

Conversely, suppose that v ∈ R^{2} is an eigenvector of M with eigenvalue λ ∈ R. By Nakayama’s lemma, there is a vector v′ ∈ R^{2} such that {v, v′} forms a basis for R^{2}. With respect to this basis, M = ( \begin{matrix}λ & b \\ 0 & d \end{matrix}) for some b, d ∈ R. Thus ch(M) = (X − λ)(X − d), so λ is indeed a root of ch(M). □

Conjugating ρ̄ only affects ρ^{univ} by conjugation, for if M ∈ GL_{2}(𝔽_{p}), then choosing any lift M̃ ∈ GL_{2}(R^{univ}) of M, the deformation (R^{univ}, M̃ρ^{univ}M̃^{−1}) is the universal deformation of Mρ̄M^{−1}. Thus in order to determine R^{univ}, we are free to alter ρ̄ by changing to any basis of 𝔽_{p}^{2}. Let σ_{0}, σ_{1} be as in Lemma 2.22; since σ_{0}, σ_{1} topologically generate Π, the residual representation ρ̄ is completely determined by ρ̄(σ_{0}), ρ̄(σ_{1}). Extending scalars to 𝔽_{p^{2}}, we may assume (as in the proof of Lemma 2.22) that ρ̄(σ_{0}) = ( \begin{matrix}a_{0} & 1 \\ 0 & d_{0} \end{matrix} ) and

ρ̄(σ_{1}) = ( \begin{matrix}a_{1} & 0 \\ c_{1} & d_{1} \end{matrix} ) for some a_{0}, d_{0}, a_{1}, c_{1}, d_{1} ∈ 𝔽_{p^{2}}^{×} satisfying a_{0} ≠ d_{0} and a_{1} ≠ d_{1}.

Fix lifts α_{0}, δ_{0}, α_{1}, η_{1}, δ_{1} of a_{0}, d_{0}, a_{1}, c_{1}, d_{1} respectively to W (𝔽_{p^{2}}). The following lemma will suggest a candidate for ρ^{univ}:

Lemma 2.25 Let (A, [ρ]) be a deformation of ρ̄⊗𝔽_{p^{2}}. Then there is a unique representative ρ_{g} ∈ [ρ] for which there exist m_{0}, m_{1}, n_{0}, n_{1}, n_{2} ∈ 𝔪_{A} such that

ρ_{g}(σ_{0}) = (\begin{matrix}α_{0}(1 + m_{0}) & 1 \\ 0 & δ_{0}(1 + m_{1}) \end{matrix})
and  ρ_{g}(σ_{1}) = (\begin{matrix}α_{1}(1 + n_{0}) & 0 \\ η_{1}(1 + n_{1}) & δ_{1}(1 + n_{2}) \end{matrix}) .

Proof: Let f(x) be the characteristic polynomial of ρ(σ_{0}). Since the roots a_{0}, d_{0} of the reduction f̄(x) of f(x) mod 𝔪_{A} are distinct, f(x) satisfies the hypotheses of Hensel’s lemma, and therefore has a root λ_{0} ∈ A reducing to a_{0} mod 𝔪_{A}. By Proposition 2.24, ρ(σ_{0}) has an eigenvector x_{0} ∈ A^{2} with eigenvalue λ_{0}. Similarly, ρ(σ_{1}) has an eigenvector x_{1} ∈ A^{2} with eigenvalue λ_{1} ∈ A such that λ_{1} reduces to d_{1} mod 𝔪_{A}. Since ρ̄ is absolutely irreducible, the reductions x̄_{0}, x̄_{1} of x_{0}, x_{1} mod 𝔪_{A} are linearly independent; hence by Nakayama’s lemma, {x_{0}, x_{1}} forms a basis for M. Let ρ_{g} : Π ⟶ GL_{2}(A) denote the homomorphism obtained by writing ρ with respect to this basis, so that ρ_{g}(σ_{0}) is upper-triangular and ρ_{g}(σ_{1}) is lower-triangular. Rescaling \{x_{0}, x_{1}\} if necessary, we may assume that ρ_{g}(σ_{0}) = ( \begin{matrix}∗ & 1 \\ 0 & ∗ \end{matrix}). Since λ_{0} reduces to a_{0} and λ_{1} reduces to d_{1}, and since ρ_{g} is conjugate to ρ, the reduction of ρ_{g} mod 𝔪_{A} is equal to ρ̄. Let B ∈ GL_{2}(A) be such that ρ_{g} = BρB^{−1}. Since ρ̄ is absolutely irreducible, Schur’s lemma together with the fact that ρ̄_{g} = ρ̄ imply that B must reduce to a scalar matrix mod 𝔪. Multiplying B by an appropriate scalar thus gives B ∈ GL_{2}^{∘}(A), so ρ_{g} ∈ [ρ]. This proves the existence of ρ_{g}.

To prove uniqueness, suppose ρ′ ∈ [ρ_{g}] is also of the given form. Let

b_{0}, b_{1}, b_{2}, b_{3} ∈ 𝔪 be such that B = ( \begin{matrix}1+b_{0} & b_{1} \\ b_{2} & 1+b_{3} \end{matrix} ) satisfies ρ′ = Bρ_{g}B^{−1}. In

particular, we have

ρ′(σ_{0}) = Bρ_{g}(σ_{0})B^{−1}  (2.26)
= \frac{1}{detB} ( \begin{matrix}∗ & (1+b_{0})((1+b_{0})−α_{0}(1+m_{0})b_{1}+b_{1}δ_{0}(1+m_{1})) \\ b_{2}((α_{0}−δ_{0})(1+b_{3})−b_{2}) & ∗ \end{matrix} ) .

By assumption, the lower-left entry of ρ′(σ_{0}) is zero, that is,

b_{2}((α_{0} − δ_{0})(1 + b_{3}) − b_{2}) = 0.

Since α_{0} − δ_{0} is a unit, so is (α_{0} − δ_{0})(1 + b_{3}) − b_{2}, and therefore b_{2} = 0. Applying the same argument to the upper-right entry of ρ′(σ_{1}) gives b_{1} = 0. Putting b_{2} = b_{1} = 0 in (2.26) gives ρ′(σ_{0}) = \frac{1}{(1+b_{0})(1+b_{3})} ( \begin{matrix}∗ & (1+b_{0})^{2} \\ 0 & ∗ \end{matrix} ), and hence \frac{(1+b_{0})^{2}}{(1+b_{0})(1+b_{3})} = 1, which implies that b_{0} = b_{3}, and therefore ρ_{g} = ρ′. □

Theorem 2.27 Suppose that p > 3 and let ρ̄, σ_{0}, σ_{1}, α_{0}, δ_{0}, α_{1}, η_{1}, and δ_{1} be as in Lemma 2.25. Then R^{univ}(ρ̄ ⊗ 𝔽_{p^{2}}) = W (𝔽_{p^{2}})[[t_{0}, t_{1}, u_{0}, u_{1}, u_{2}]], and the corresponding universal deformation ρ^{univ} of ρ̄ ⊗ 𝔽_{p^{2}} is conjugate to the deformation ρ given by

ρ(σ_{0}) = (\begin{matrix}α_{0}(1 + t_{0}) & 1 \\ 0 & δ_{0}(1 + t_{1}) \end{matrix}) , ρ(σ_{1}) = (\begin{matrix}α_{1}(1 + u_{0}) & 0 \\ η_{1}(1 + u_{1}) & δ_{1}(1 + u_{2}) \end{matrix}) .

Moreover, R^{univ}(ρ̄) = ℤ_{p}[[t_{1}, …, t_{5}]].

Proof: Given any deformation [ρ] of ρ̄ ⊗ 𝔽_{p^{2}} to A, choose ρ_{g} ∈ [ρ] as in Lemma 2.25. Define a W (𝔽_{p^{2}})-algebra homomorphism

ϕ : W (𝔽_{p^{2}})[t_{0}, t_{1}, u_{0}, u_{1}, u_{2}] ⟶ A

by ϕ(t_{i}) = m_{i} and ϕ(u_{i}) = n_{i} for each i, extended by W (𝔽_{p^{2}})-linearity. By Proposition 2.10, we may extend ϕ to a continuous homomorphism

ϕ : W (𝔽_{p^{2}})[[t_{0}, t_{1}, u_{0}, u_{1}, u_{2}]] ⟶ A.

In fact, ϕ is a morphism in DEF(ρ̄⊗ 𝔽_{p^{2}}). To show that ϕ is unique, suppose that ϕ′ : W (𝔽_{p^{2}})[[t_{0}, t_{1}, u_{0}, u_{1}, u_{2}]] ⟶ A is another such morphism. Letting ϕ′ also denote the induced map on the general linear groups, ϕ′(ρ^{univ}(σ_{0})) and ϕ′(ρ^{univ}(σ_{1})) are of the form given in Lemma 2.25; hence by the uniqueness statement of Lemma 2.25, we have ϕ′(ρ^{univ}(σ_{i})) = ρ_{g}(σ_{i}) for i = 0, 1. This implies that ϕ(t_{i}) = ϕ′(t_{i}) and ϕ(u_{i}) = ϕ′(u_{i}) for each i, and therefore ϕ = ϕ′, as desired. The final statement now follows from the discussion preceding Lemma 2.22. □

2.6 Conditions on Deformations

If the determinant of a given residual representation ρ̄ is 1 (that is, if the image of ρ̄ is contained in SL_{2}(k)), then it is natural to insist that deformations of ρ̄ also have determinant 1. Accordingly, let DEF^{1}(ρ̄) denote the subcategory of DEF(ρ̄) consisting of only those objects (A, [ρ]) such that ρ has determinant one. Mazur’s proof of the existence of the universal deformation carries over to show that there is a universal object (R_{1}^{univ}, ρ_{1}^{univ}) in DEF^{1}(ρ̄) (see [Gou], p.68). In fact, imposing a fixed determinant on deformations of ρ̄ is perhaps the simplest example of a “deformation condition”, that is, a property of deformations which defines a subcategory of DEF(ρ̄) in which a universal object is guaranteed to exist. See [Gou], Lecture 6 for a detailed discussion of such conditions.

Theorem 2.28 Let notation be as in Theorem 2.27, and suppose that ρ̄ has determinant one. Then R_{1}^{univ}(ρ̄ ⊗ 𝔽_{p^{2}}) = W (𝔽_{p^{2}})[[t_{0}, u_{0}, u_{1}]], and the corresponding universal deformation ρ_{1}^{univ} of ρ̄⊗𝔽_{p^{2}} is conjugate to the deformation ρ_{1} given by

ρ_{1}(σ_{0}) = (\begin{matrix}α_{0}(1 + t_{0}) & 1 \\ 0 & (α_{0}(1 + t_{0}))^{−1} \end{matrix})
and  ρ_{1}(σ_{1}) = (\begin{matrix}α_{1}(1 + u_{0}) & 0 \\ η_{1}(1 + u_{1}) & (α_{1}(1 + u_{0}))^{−1} \end{matrix}) .

Moreover, R_{1}^{univ}(ρ̄) = ℤ_{p}[[t_{1}, t_{2}, t_{3}]].

Proof: If (A, [ρ]) ∈ DEF^{1}(ρ̄ ⊗ 𝔽_{p^{2}}), then the representative ρ_{g} ∈ [ρ] of Lemma 2.25 has determinant one, and hence δ_{0}(1 + m_{1}) = (α_{0}(1 + m_{0}))^{−1}, and δ_{1}(1 + n_{2}) = (α_{1}(1 + n_{0}))^{−1}. Thus defining ϕ : ℤ_{p}[[t_{0}, u_{0}, u_{1}]] ⟶ A by ϕ(t_{0}) = m_{0}, ϕ(u_{0}) = n_{0}, and ϕ(u_{1}) = n_{1} gives a morphism in DEF^{1}(ρ̄). Uniqueness again follows from the uniqueness of ρ_{g}.

To obtain R_{1}^{univ}(ρ̄) from R_{1}^{univ}(ρ̄ ⊗ 𝔽_{p^{2}}), one applies the same argument that was used above for the usual universal deformation, with two minor changes. First, one must choose the lift of ρ^{univ} to W (k)[[t_{1}, …, t_{d}]] in the proof of Proposition 2.20 to have determinant one. Such a choice is possible since the homomorphism ϕ : W (k)[[t_{1}, …, t_{d}]] ⟶ R^{univ} reduces to the identity on k. Also, one must check that deformations of determinant one correspond to cocycles of trace zero under the isomorphism (2.21). In other words, t_{R_{1}^{univ}} ≅ H^{1}(Π, ad^{0}(ρ̄)), where ad^{0}(ρ̄) is the subgroup of ad(ρ̄) consisting of the trace zero matrices. Thus replacing H^{1}(Π, ad(ρ̄)) with H^{1}(Π, ad^{0}(ρ̄)), one may apply the above argument to R_{1}^{univ}, which gives the desired result. □

Given a residual representation ρ̄: Π ⟶ GL_{2}(k), suppose that for some closed subgroup I ⊂ Π, the subspace of k^{2} consisting of all fixed points of ρ̄(I) has dimension one. A deformation ρ of ρ̄ to a ring R is said to be I-ordinary if the submodule of R^{2} of fixed points of ρ(I) is a direct summand of R^{2} of rank one. Note that the condition of being I-ordinary is preserved by strict equivalence, and is therefore a well-defined property of deformations. If I = ⟨δ⟩ for some δ ∈ Π, we will say that ρ is δ-ordinary. By essentially the same argument that he used to prove the existence of the universal deformation, Mazur showed in his original paper [Maz89] that there is a universal I-ordinary deformation whenever ρ̄ is itself I-ordinary. If ρ̄ has determinant one, then there is a universal I-ordinary deformation of determinant one. Throughout the following, all deformations will be assumed to have determinant one.

Theorem 2.29 Let σ_{0}, σ_{1} be topological generators of Π. For p ≠ 2, let ρ̄ : Π ⟶ SL_{2}(𝔽_{p}) be a residual representation which is σ_{i}-ordinary for i = 0 or i = 1. Let R_{ord}^{univ} denote the (determinant one) σ_{i}-ordinary universal deformation ring. Then R_{ord}^{univ} = ℤ_{p}[[t_{1}, t_{2}]].

Proof: By conjugating ρ̄ and interchanging σ_{0} and σ_{1} if necessary, we may assume that i = 0, and that ρ̄(σ_{0}) = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). Moreover, every σ_{0}-ordinary deformation ρ of ρ̄ has a representative satisfying ρ(σ_{0}) = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). Once again, we may apply the same argument as for R^{univ} above to show that R_{ord}^{univ} is a power series ring with coefficients in W (k), except that we must lift ρ^{univ}(σ_{0}) to a matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) and ρ^{univ}(σ_{1}) to a matrix of determinant one in the proof of Proposition 2.20. Moreover, if R_{ord}^{univ}(ρ̄ ⊗ 𝔽_{p^{2}}) = W (𝔽_{p^{2}})[[t_{1}, …, t_{d}]], then R_{ord}^{univ} = ℤ_{p}[[t_{1}, …, t_{d}]]. This again follows from the above argument, except that H^{1}(Π, ad(ρ̄)) must be replaced by H^{1}(Π, ad_{0}^{σ_{0}}(ρ̄)), where ad_{0}^{σ_{0}}(ρ̄) denotes the subgroup of ad_{0}(ρ̄) consisting of those matrices whose kernel contains the subspace of k^{2} fixed by ρ̄(σ_{0}).

The last paragraph of the proof of Lemma 2.22 shows that we may assume that σ_{1} has distinct eigenvalues (in 𝔽_{p^{2}}) by replacing σ_{1} with σ_{1}σ_{0} or σ_{0}σ_{1}σ_{0} if necessary. Thus without loss of generality, any σ_{0}-ordinary deformation ρ of ρ̄ ⊗ 𝔽_{p^{2}} to a ring R has a unique representative of the form ρ(σ_{0}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

and ρ(σ_{1}) = ( \begin{matrix}α+m_{1} & 0 \\ β+m_{2} & (α+m_{1})^{−1} \end{matrix} ) for some m_{1}, m_{2} ∈ 𝔪_{R}, where α, β ∈ W (𝔽_{p^{2}})

are fixed. An argument similar to that in the proof of Theorem 2.27 shows that R_{ord}^{univ}(ρ̄ ⊗ 𝔽_{p^{2}}) = W (𝔽_{p^{2}})[[t_{1}, t_{2}]], where the corresponding universal deformation ρ_{ord}^{univ} is given by ρ_{ord}^{univ}(σ_{0}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ρ_{ord}^{univ}(σ_{1}) = ( \begin{matrix}α+t_{1} & 0 \\ β+t_{2} & (α+t_{1})^{−1} \end{matrix} ). Therefore, by the above remarks, R_{ord}^{univ} = ℤ_{p}[[t_{1}, t_{2}]]. □

If ρ̄(σ_{i}) ∼ ( \begin{matrix}−1 & ∗ \\ 0 & −1 \end{matrix} ), then by abuse of language we will also say that a deformation ρ of ρ̄ is σ_{i}-ordinary if ρ(σ_{i}) ∼ ( \begin{matrix}−1 & ∗ \\ 0 & −1 \end{matrix} ).

Corollary 2.30 Let σ_{0} and σ_{1} be as in Theorem 2.29. Suppose that for i = 0 or 1, ρ̄(σ_{i}) ∼ ( \begin{matrix}−1 & ∗ \\ 0 & −1 \end{matrix} ). Then there is a universal σ_{i}-ordinary deformation (R_{ord}^{univ}, ρ_{ord}^{univ}), and R_{ord}^{univ} = ℤ_{p}[[t_{1}, t_{2}]].

Proof: Without loss of generality, we may assume that i = 0. Given any deformation ρ of any residual representation ρ̄, let ρ_{−} denote the deformation given by ρ_{−}(σ_{0}) = −ρ(σ_{0}) and ρ_{−}(σ_{1}) = ρ(σ_{1}). Since ρ̄(σ_{0}) ∼ ( \begin{matrix}−1 & ∗ \\ 0 & −1 \end{matrix} ), we have ρ̄_{−}(σ_{0}) ∼ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). Since ρ̄ is absolutely irreducible, so is ρ̄_{−}; hence by Theorem 2.29, there is a universal σ_{0}-ordinary deformation ρ_{−ord}^{univ} corresponding to ρ̄_{−}, with R_{−ord}^{univ} = ℤ_{p}[[t_{1}, t_{2}]]. The universal σ_{0}-ordinary deformation of ρ̄ is given by (ρ_{−ord}^{univ})_{−}. □

Let S ⊂ Π be a finite set. We say that a deformation ρ of ρ̄: Π ⟶ GL_{2}(k) is S-ordinary if ρ is σ-ordinary for every σ ∈ S. Assuming that ρ̄ is itself S-ordinary, we once again obtain a universal deformation (R_{S−ord}^{univ}, ρ_{S−ord}^{univ}).

Theorem 2.31 Let Π be as above, σ_{0}, σ_{1} topological generators of Π. Let S = {σ_{0}, σ_{1}}, and suppose that ρ̄ : Π ⟶ SL_{2}(𝔽_{p}) is an S-ordinary residual representation. Then R_{S−ord}^{univ} = ℤ_{p}[[t]], and ρ_{S−ord}^{univ} is conjugate to the deformation ρ given by

ρ(σ_{0}) = ± (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and ρ(σ_{1}) = ± (\begin{matrix}1 & 0 \\ α + t & 1 \end{matrix})

for some α ∈ ℤ_{p}, where the sign of each ρ(σ_{i}) corresponds to the sign of the eigenvalue ±1 of ρ̄(σ_{i}).

Proof: Conjugating ρ̄ if necessary and applying a similar argument to that in the proof of Corollary 2.30, we may assume that ρ̄(σ_{0}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ρ̄(σ_{1}) = ( \begin{matrix}1 & 0 \\ a & 1 \end{matrix} ). Fix a lift α ∈ ℤ_{p} of a. Any S-ordinary deformation ρ of ρ̄ to R has a unique representative of the form ρ(σ_{0}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ρ(σ_{1}) = ( \begin{matrix}1 & 0 \\ α+m & 1 \end{matrix} ) for some m ∈ 𝔪_{R}. The same argument as for the universal deformations above now gives the result. □

3 Lowering the Field of Definition

3.1 The Cyclotomic Character

Let ˆ \overline{ℚ}(t) denote the maximal algebraic extension of \overline{ℚ}(t) unramified outside three places, each of which is fixed by Gal (\overline{ℚ}(t)/ℚ(t)), and let 𝔭_{0}, 𝔭_{1}, 𝔭_{2} denote the valuation ideals corresponding to these places. By Theorem 2.7, there exist topological generators γ_{0}, γ_{1}, γ_{2} of inertia groups above 𝔭_{0}, 𝔭_{1}, 𝔭_{2} respectively, which topologically generate Π := Gal ( ˆ\overline{ℚ}(t)/\overline{ℚ}(t)), and satisfy γ_{0}γ_{1}γ_{2} = 1. By the fundamental theorem of infinite Galois theory, Π is a normal subgroup of Γ_{ℚ} := Gal ( ˆ\overline{ℚ}(t)/ℚ(t)); thus Γ_{ℚ} acts on Π by conjugation.

The action of σ ∈ Γ_{ℚ} on Π is determined up to conjugation in Π by the restriction σ̄ of σ to \overline{ℚ}(t). Viewing σ̄ as an element of G_{ℚ} via the natural isomorphism Gal (\overline{ℚ}(t)/ℚ(t)) ≅ G_{ℚ}, the action of σ on each γ_{i} is determined up to conjugation in Π by the action of σ̄ on the roots of unity in \overline{ℚ}. To make this explicit, we define the cyclotomic character χ as follows: let ℤ̂ := \underleftarrow{lim}_{n∈ℕ} ℤ/nℤ, and fix a compatible system (ζ_{n})_{n∈ℕ} of primitive nth roots of unity ζ_{n}. Given σ̄ ∈ G_{ℚ}, for each n ∈ ℕ we have

σ̄(ζ_{n}) = ζ_{n}^{χ_{n}(σ̄)}

for some χ_{n}(σ̄) ∈ (ℤ/nℤ)^{×} which is independent of the choice of ζ_{n}. Moreover, this action is compatible in the sense that whenever m|n, the natural map ℤ/nℤ ⟶ ℤ/mℤ takes χ_{n}(σ̄) to χ_{m}(σ̄). Thus (χ_{n}(σ̄))_{n∈ℕ} ∈ ℤ̂^{×}, and we define the cyclotomic character χ : G_{ℚ} ⟶ ℤ̂^{×} by χ(σ̄) = (χ_{n}(σ̄))_{n∈ℕ}. For σ ∈ Γ_{ℚ}, we will often write χ(σ) to mean χ(σ̄).

Given any profinite group G = \underleftarrow{lim}_{i∈I} G_{i}, where each G_{i} is finite, there is a natural way to define exponentiation in G by elements of ℤ̂. Given (g_{i})_{i∈I} ∈ G, α = (α_{n})_{n∈ℕ} ∈ ℤ̂, define (g_{i})^{α} := (g_{i}^{α_{n(i)}}) where n(i) = |G_{i}|. The compatibility conditions on (g_{i}) and on (α_{n}) ensure that (g_{i}^{α_{n(i)}}) is indeed an element of G.

Theorem 3.1 For each σ ∈ Γ_{ℚ} and each i = 0, 1, 2,

γ_{i}^{σ} ∼ γ_{i}^{χ(σ)},

where ∼ denotes conjugacy in Π, and χ is the cyclotomic character.

Proof: The proof given here follows that of [MM99], Ch. I, Theorem 2.3. For each i = 0, 1, 2, let 𝔭̂_{i} be the valuation ideal of ˆ \overline{ℚ}(t) such that γ_{i} generates the inertia group I_{i} := I (𝔭̂_{i}/𝔭_{i}). Since σ(𝔭_{i}) = 𝔭_{i} for each i = 0, 1, 2, we have I_{i}^{σ} = I (σ(𝔭̂_{i})/𝔭_{i}), and in particular γ_{i}^{σ} ∈ I (σ(𝔭̂_{i})/𝔭_{i}). Since Π acts transitively on the primes above 𝔭_{i}, there is some δ ∈ Π such that δ (σ(𝔭̂_{i})) = 𝔭̂_{i}, and thus (γ_{i}^{σ})^{δ} ∈ I_{i}. The group I_{i} is generated by γ_{i} as a procyclic group, so there is some α ∈ ℤ̂ such that (γ_{i}^{σ})^{δ} = γ_{i}^{α}; in particular, we have γ_{i}^{σ} ∼ γ_{i}^{α}.

It remains to show that α = χ(σ). For each i = 0, 1, 2, let f_{i} ∈ ℚ(t) be an element which generates 𝔭_{i} in its corresponding valuation ring. For each i = 0, 1, 2, \overline{ℚ}(t)(f_{i}^{1/n})_{n∈ℕ} is an abelian extension of \overline{ℚ}(t) contained in ˆ \overline{ℚ}(t), where we choose each f_{i}^{1/n} so that they are compatible in the sense that (f_{i}^{1/kn})^{k} = f_{i}^{1/n} for all k, n ∈ ℕ. We now fix some i = 0, 1, or 2. Since

ℚ(t)(f_{i}^{1/n})_{n∈ℕ} ⋂ \overline{ℚ}(t) = ℚ(t),

there is some σ̃ ∈ Γ_{ℚ} whose restriction to \overline{ℚ}(t) is σ̄, and which fixes f_{i}^{1/n} for all n. Now γ_{i}(f_{i}) = f_{i}, so γ_{i}(f_{i}^{1/n}) is an nth root of f_{i} in ˆ \overline{ℚ}(t), and is therefore of the form ζ_{n}f_{i}^{1/n} for some nth root of unity ζ_{n}. Moreover,

I ((f_{i}^{1/n})/(f_{i})) = Gal (\overline{ℚ}(t)(f_{i}^{1/n})/\overline{ℚ}(t))

is generated by the restriction of γ_{i} to \overline{ℚ}(t)(f_{i}^{1/n}), so ζ_{n} is a primitive nth root of unity, and the various ζ_{n} obtained in this way are compatible under the canonical maps. Since σ̃ restricts to σ̄, there is some δ ∈ Π such that σ̃ = δσ, and hence γσ̃ ; i ∼ γ_{i}^{σ} ∼ γ_{i}^{α}. Therefore, the restrictions of γσ̃ ; i and γ_{i}^{α} to the maximal abelian extension ˆ \overline{ℚ}(t)^{ab} of \overline{ℚ}(t) in ˆ \overline{ℚ}(t) are equal; in particular, γ_{i}^{α}(f_{i}^{1/n}) = γσ̃ ; i(f_{i}^{1/n}) for all n. Thus we have

ζ_{n}^{α}f_{i}^{1/n} = γ_{i}^{α}(f_{i}^{1/n}) = γσ̃ ; i(f_{i}^{1/n}) = σ̃γ_{i}σ̃^{−1}(f_{i}^{1/n})

= σ̃γ_{i}(f_{i}^{1/n}) = σ̃(ζ_{n}f_{i}^{1/n}) = σ̄(ζ_{n})f_{i}^{1/n},

and therefore σ̄(ζ_{n}) = ζ_{n}^{α} for all n, which proves that α = χ(σ̄). □

3.2 The Rigidity Theorem

In this section, we introduce the notion of rigidity, which will be used to extend the universal deformation of a given residual representation

ρ̄: Π ⟶ SL_{2}(𝔽_{p})

to a representation of Π_{K(μ)} := Gal ({ \atop \overline{K}(t)}/K(μ, t)), where K is an algebraic extension of ℚ and μ is a collection of roots of unity in \overline{K} which depends on ρ̄.

Let G be a group, and let C_{0}, …, C_{n} be conjugacy classes in G (not necessarily distinct). We denote by Σ̄(C_{0}, …, C_{n}) the set of all n + 1-tuples (g_{0}, …, g_{n}) ∈ C_{0} × ⋯ × C_{n} which satisfy g_{0} ⋯ g_{n} = 1. An n + 1-tuple (h_{0}, …, h_{n}) ∈ G^{n+1} is said to be locally conjugate to an element (g_{0}, …, g_{n}) of Σ̄(C_{0}, …, C_{n}) if (h_{0}, …, h_{n}) belongs to Σ̄(C_{0}, …, C_{n}) and the subgroups ⟨g_{0}, …, g_{n}⟩ and ⟨h_{0}, …, h_{n}⟩ of G are isomorphic. Note that G acts on Σ̄(C_{0}, …, C_{n}) by componentwise conjugation; thus for g ∈ G, we will write (g_{0}, …, g_{n})^{g} to mean (gg_{0}g^{−1}, …, gg_{n}g^{−1}). Two elements of Σ̄(C_{0}, …, C_{n}) are said to be globally conjugate if they lie in the same G-orbit under this action.

Definition 3.2 The n + 1-tuple (g_{0}, …, g_{n}) ∈ Σ̄(C_{0}, …, C_{n}) is said to be rigid if every element of G^{n+1} which is locally conjugate to (g_{0}, …, g_{n}) is globally conjugate to (g_{0}, …, g_{n}).

For any algebraic extension K of ℚ, let G_{K} := Gal(\overline{K}/K), and let Π_{K} := Gal ({ \atop \overline{K}(t)}/K(t)), where { \atop \overline{K}(t)} denotes the maximal algebraic extension of \overline{K}(t) unramified outside 0, 1, ∞. Let γ_{0}, γ_{1}, γ_{∞} ∈ Π\overline{K} be topological generators of inertia groups I_{0}, I_{1}, I_{∞} above 0, 1, ∞ respectively such that γ_{0}γ_{1}γ_{∞} = 1.

Lemma 3.3 (Belyĭ) For each i = 0, 1, ∞, the natural surjection Π_{K} ↠ G_{K} has a splitting ϕ_{i} : G_{K} ↪ Π_{K} whose image is contained in N_{Π_{K}}(I_{i}).

Outline of Proof: Without loss of generality, suppose that i = 0. Let

Γ := {γ ∈ Π_{K} : γγ_{0}γ^{−1} = γ_{0}^{χ(γ)}, γγ_{1}γ^{−1} ≈ γ_{1}^{χ(γ)}} ,

where ≈ denotes conjugacy by an element of the commutator subgroup [Π\overline{K}, Π\overline{K}] of Π\overline{K}, and χ denotes the cyclotomic character. One may show that restricting the natural map Π_{K} ↠ G_{K} to Γ defines an isomorphism Γ ≅ G_{K}. Letting ϕ_{i} be the inverse of this isomorphism gives the result. See [Bel80], §1 for details. □

Corollary 3.4 The group Π_{K} is isomorphic to Π\overline{K} ⋊ G_{K}.

The following theorem, which we will use to extend the universal deformations of §2.6, is a variant of the rigidity theorem of Belyĭ, Fried, Matzat, Shih, and Thompson. For other variants, see [Ser92], [Vol96], and [MM99].

Let G be a profinite group, and r the natural map G ⟶ G/Z(G). Given any homomorphism ρ^{geom} : Π\overline{K} ⟶ G, let μ denote the set of all nth roots of unity in \overline{K} for which ρ^{geom}(γ_{i}) has exact order n in some finite quotient of G for some i = 0, 1, ∞.

Theorem 3.5 Suppose that (ρ^{geom}(γ_{0}), ρ^{geom}(γ_{1}), ρ^{geom}(γ_{∞})) forms a rigid triple in G. Suppose moreover that Z_{G}(Im(ρ^{geom})) = Z(G).

(1) The composed map ρ̂^{geom} := r ∘ ρ^{geom} : Π\overline{K} ⟶ G/Z(G) extends uniquely to a homomorphism ρ̂: Π_{K(μ)} ⟶ G/Z(G).

(2) Let ϕ_{i} be as in Lemma 3.3, and suppose that for some i the inclusion Z(G) ↪ r^{−1} (ρ̂ ∘ ϕ_{i}(G_{K(μ)})) splits. Then ρ^{geom} extends to a homomorphism ρ : Π_{K(μ)} ⟶ G which is unique up to multiplication by a homomorphism ψ : G_{K} ⟶ Z(G).

Proof: Let γ ∈ Π_{K(μ)}. By Theorem 3.1, γγ_{i}γ^{−1} ∼ γ_{i}^{χ(γ)} in Π\overline{K} for each i = 0, 1, ∞, and hence ρ^{geom}(γγ_{i}γ^{−1}) ∼ ρ^{geom}(γ_{i})^{χ(γ)} in G. Let H be any finite quotient of G, and let n be the order of the image of ρ^{geom}(γ_{i}) in H. Since γ fixes K(μ) pointwise and K(μ) contains all of the nth roots of unity in \overline{K}, we have χ(γ) ≡ 1 mod n. Therefore ρ^{geom}(γ_{i})^{χ(γ)} = ρ^{geom}(γ_{i}), and hence ρ^{geom}(γ_{i}) ∼ ρ^{geom}(γγ_{i}γ^{−1}) in G. For each i = 0, 1, ∞, let δ_{i} = ρ^{geom}(γ_{i}), and δ_{i}^{γ} = ρ^{geom}(γγ_{i}γ^{−1}). Since ⟨δ_{0}, δ_{1}, δ_{∞}⟩ = ⟨δ_{0}^{γ}, δ_{1}^{γ}, δ_{∞}^{γ}⟩ = Imρ^{geom}, (δ_{0}^{γ}, δ_{1}^{γ}, δ_{∞}^{γ}) is locally conjugate to (δ_{0}, δ_{1}, δ_{∞}) in G; thus by the rigidity of (δ_{0}, δ_{1}, δ_{∞}), there is some g_{γ} ∈ G such that g_{γ}δ_{i}g_{γ}^{−1} = δ_{i}^{γ} for each i = 0, 1, ∞. Define a set-theoretic map ρ̂: Π_{K(μ)} ⟶ G/Z(G) by

ρ̂(γ) = r(g_{γ}) ∈ G/Z(G)

for all γ ∈ Π_{K(μ)}. We claim that ρ̂ is a homomorphism extending ρ̂^{geom}. Note that r(g_{γ}) is uniquely determined since Z_{G}(Imρ^{geom}) = Z(G); thus to show that ρ̂ is a homomorphism, it suffices to show that given γ, γ′ ∈ Π_{K(μ)}, we have

g_{γ}g_{γ′}δ_{i}g_{γ′}^{−1}g_{γ}^{−1} = ρ^{geom}(γγ′γ_{i}γ^{′−1}γ^{−1})

for each i = 0, 1, ∞. In fact, since γ_{0}, γ_{1}, γ_{∞} generate Π\overline{K}, we have ρ^{geom}(γσγ^{−1}) = g_{γ}ρ^{geom}(σ)g_{γ}^{−1} for all σ ∈ Π\overline{K}, so ρ̂ is indeed a homomorphism. The uniqueness of ρ̂ follows from the uniqueness of r(g_{γ}) and the fact that for any homomorphism ρ̂ : Π_{K(μ)} ⟶ G/Z(G) extending ρ̂^{geom}, ρ̂(γγ_{i}γ^{−1}) = ρ̂(γ)ρ̂^{geom}(γ_{i})ρ̂(γ)^{−1} for each i = 0, 1, ∞.

To prove (2), choose i so that the inclusion Z(G) ↪ r^{−1} (ρ̂ ∘ ϕ_{i}(G_{K(μ)})) splits. Let N = r^{−1} (ρ̂ ∘ ϕ_{i}(G_{K(μ)})) ⊂ G. Since the inclusion Z(G) ↪ N splits, the surjection r|_{N} : N ⟶ N/Z(G) is split by some homomorphism ψ : N/Z(G) ⟶ N. Thus we obtain a homomorphism

ψ ∘ ρ̂ ∘ ϕ_{i} : G_{K(μ)} ⟶ G.

By Corollary 3.4, Π_{K(μ)} ≅ Π\overline{K} ⋊G_{K(μ)}, so writing γ ∈ Π_{K(μ)} as γ = αβ where α ∈ Π\overline{K} and β ∈ ϕ_{i}(G_{K(μ)}), we may define a homomorphism ρ : Π_{K(μ)} ⟶ G extending ρ^{geom} by ρ(γ) = ρ^{geom}(α)ψ∘ρ̂(β). The uniqueness statement follows immediately from that of (1). □

3.3 Rigidity in GL_{2}(R^{univ})

The universal deformation rings of Theorems 2.28, 2.29, and 2.31 are power series rings over ℤ_{p}; in particular, they are local unique factorization domains (UFDs). Thus in order to extend the corresponding universal deformations using Theorem 3.5, it is necessary to study rigidity in GL_{2}(R), where (R, 𝔪) is a local UFD with residue field k. We will show that if ρ is a determinant one deformation of a residual representation ρ̄: Π ⟶ GL_{2}(k) to such a ring R, then (ρ(γ_{0}), ρ(γ_{1}), ρ(γ_{∞})) is rigid in GL_{2}(R).

Definition 3.6 For any domain R, a subgroup G of GL_{n}(R) is said to be irreducible if there is no eigenvector common to all elements of G in any domain containing R. The subgroup G is said to be acentral in GL_{n}(R) if the centralizer Z_{M_{n}(R)}(G) of G in the matrix ring M_{n}(R) consists only of the scalar matrices.

If R = k is a field, then G is irreducible if and only if the identity map of G is an absolutely irreducible representation; moreover, by Schur’s lemma, every irreducible subgroup is acentral (see [Isa94], p.145).

Proposition 3.7 Let (R, 𝔪) be a local domain with residue field k, and suppose that M_{0}, M_{1} ∈ GL_{2}(R) have the property that the reductions M̄_{0}, M̄_{1} of M_{0}, M_{1} mod 𝔪 generate an irreducible subgroup of GL_{2}(k). Then for any domain R′ ⊃ R, the subgroup of GL_{2}(R′) generated by M_{0} and M_{1} is both irreducible and acentral.

Proof: Let λ_{0}, λ_{1} be eigenvalues of M_{0}, M_{1} respectively in some domain containing R. Since λ_{0} and λ_{1} are integral over R, there is a maximal ideal 𝔭 of R[λ_{0}, λ_{1}] lying above 𝔪 (see [Lan93], Ch. VII, Propositions 1.10, 1.11). Let v_{0}, v_{1} ∈ R[λ_{0}, λ_{1}]_{𝔭}^{2} be eigenvectors corresponding to λ_{0}, λ_{1} respectively. The reductions of v_{0} and v_{1} mod 𝔭 must be distinct, for otherwise R[λ_{0}, λ_{1}]_{𝔭}/𝔭 is an extension of k in which M̄_{0}, M̄_{1} have a common eigenvector; in particular, v_{0} and v_{1} are distinct. Therefore M_{0} and M_{1} generate an irreducible subgroup G of GL_{2}(R), and hence also of GL_{2}(R′) where R′ is any domain containing R. Furthermore, G is an irreducible and thus acentral subgroup of GL_{2} (Qu(R′)). Since

Z_{GL_{2}(R′)}(G) = Z_{GL}_{2}_{(Qu(R′))}(G) ⋂ GL_{2}(R′),

G is also an acentral subgroup of GL_{2}(R′). □

In proving the rigidity of certain triples (M_{0}, M_{1}, M_{2}) of matrices in GL_{2}(R), the easiest case occurs when M_{0} and M_{1} both have eigenvalues in R. The following lemmas will allow us to extend R to a domain in which M_{0} and M_{1} have eigenvalues, then descend to obtain conjugacy in GL_{2}(R).

Lemma 3.8 Let L be a quadratic extension of a field K. Suppose that the pair M_{0}, M_{1} ∈ GL_{2}(K) generates an irreducible subgroup of GL_{2}(K), and that (M′_{0}, M′_{1}) ∈ GL_{2}(K)^{2} is conjugate to (M_{0}, M_{1}) by an element of GL_{2}(L). Then (M′_{0}, M′_{1}) is conjugate to (M_{0}, M_{1}) by an element of GL_{2}(K).

Proof: Let σ denote the nontrivial element of Gal(L/K), and G the subgroup of GL_{2}(K) generated by M_{0} and M_{1}. Let M ∈ GL_{2}(L) be such that MM_{i}M^{−1} = M′_{i} for i = 0, 1. Since M_{i}, M′_{i} ∈ GL_{2}(K), applying σ gives

σ(M)M_{i}σ(M)^{−1} = M′_{i} = MM_{i}M^{−1},

and therefore M^{−1}σ(M) ∈ Z_{M_{2}(L)}(G). By Proposition 3.7, since G is an irreducible subgroup of GL_{2}(K), it is an acentral subgroup of GL_{2}(L). Thus M^{−1}σ(M) = ζId for some ζ ∈ L. Applying σ to the equation σ(M) = ζM gives M = σ(ζ)σ(M) = σ(ζ)ζM, and therefore σ(ζ)ζ = 1. By Hilbert’s Theorem 90, there exists some α ∈ L^{×} such that ζ = \frac{α}{σ(α)}. Hence we have σ(α)σ(M) = αM, so αM is invariant under Gal(L/K), and therefore αM ∈ GL_{2}(K). Conjugating each M_{i} by αM gives (αM)M_{i}(αM)^{−1} = M′_{i}, as desired. □

Lemma 3.9 Let (R, 𝔪) be a local UFD with residue field k and quotient field K. Suppose that the reductions mod 𝔪 of M_{0}, M_{1} ∈ GL_{2}(R) together generate an irreducible subgroup of GL_{2}(k), and that (M′_{0}, M′_{1}) ∈ GL_{2}(R)^{2} is conjugate to (M_{0}, M_{1}) by an element of GL_{2}(K). Then (M′_{0}, M′_{1}) is conjugate to (M_{0}, M_{1}) by an element of GL_{2}(R).

Proof: Let M ∈ GL_{2}(K) be such that MM_{i}M^{−1} = M′_{i} for i = 0, 1. Multiplying M by a suitable scalar, we may assume that M ∈ M_{2}(R), det(M) ≠ 0, and det(M) has minimal 𝔪-adic valuation among all such multiples of M.

Let M^{∗} = det(M)M^{−1} ∈ M_{2}(R). If det(M) ∈ R^{×} then we are done. Otherwise, there is an irreducible element α ∈ R which divides det(M). We will show that α divides each entry of M, and therefore \frac{1}{α}M ∈ M_{2}(R) is such that det(\frac{1}{α}M) has lesser 𝔪-adic valuation than det(M), contradicting the assumption on M.

Since R is a UFD and α is irreducible, 𝔭 = (α) is a prime ideal. Let G be the subgroup of GL_{2}(R) generated by M_{0} and M_{1}, let k̃ = R_{𝔭}/𝔭, and let G̃ denote the subgroup of GL_{2}(k̃) obtained by taking the mod 𝔭 reduction of G viewed as a subgroup of GL_{2}(R_{𝔭}). The diagram

\xymatrix{R \ar@{^{(}->}[r] \ar[d] & R_{𝔭} \ar[d] \\ R/𝔭 \ar@{^{(}->}[r]_{Qu} \ar[d] & k̃ \\ k}

commutes, where the injection R/𝔭 ↪ k̃ is obtained by viewing k̃ as the quotient field of R/𝔭. Since k is the residue field of R/𝔭 and the reduction Ḡ of G mod 𝔪 is an irreducible subgroup of GL_{2}(k), by Proposition 3.7, the reduction of G mod 𝔭 is an irreducible subgroup of GL_{2}(R/𝔭); hence G̃ is an irreducible subgroup of GL_{2}(k̃). Therefore, G̃ generates the k̃-algebra M_{2}(k̃) (see [Isa94], p.145).

For any A ∈ M_{2}(R), let Ã ∈ M_{2}(k̃) denote the element obtained by viewing A as an element of M_{2}(R_{𝔭}) and reducing mod 𝔭. Each Ã ∈ M_{2}(k̃) may be expressed as a k̃-linear combination of elements of G̃, say

Ã = α̃_{0}Ã_{0} + ⋯ + α̃_{r}Ã_{r}.

For each i = 0, …, r, choose A_{i} ∈ G reducing to Ã_{i} mod 𝔭, and α_{i} ∈ R_{𝔭} reducing to α̃_{i} mod 𝔭. Since A_{i} ∈ G = ⟨M_{0}, M_{1}⟩ for each i = 0, …, r, the lift A = α_{0}A_{0} + ⋯ + α_{r}A_{r} ∈ M_{2}(R_{𝔭}) of Ã satisfies MAM^{−1} ∈ M_{2}(R_{𝔭}). Hence MAM^{∗} ∈ det(M)M_{2}(R_{𝔭}), and reducing mod 𝔭 gives M̃ÃM̃^{∗} = 0. Let M̃ = ( \begin{matrix}a & b \\ c & d \end{matrix}). Taking Ã = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) gives 0 = M̃ÃM̃^{∗} = ( \begin{matrix}−ac & a^{2} \\ −c^{2} & ac \end{matrix} ) and hence a = c = 0. Taking Ã = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) similarly gives b = d = 0 and therefore M̃ = 0; that is, M ∈ M_{2}(𝔭), so α divides each entry of M, which gives the desired contradiction. □

We now prove the main result of this section:

Theorem 3.10 Let R be a local UFD with residue field k, and suppose that (M_{0}, M_{1}, M_{2}) is a triple of matrices in SL_{2}(R) satisfying M_{0}M_{1}M_{2} = 1, whose reductions mod 𝔪 together generate an irreducible subgroup of GL_{2}(k). Then (M_{0}, M_{1}, M_{2}) is rigid in GL_{2}(R).

Proof: Let K = Qu(R), and L = K(λ_{0}, λ_{1}), where λ_{0}, λ_{1} are eigenvalues of M_{0}, M_{1} respectively. Choosing a basis for L^{2}, we view M_{0} and M_{1} as linear transformations of L^{2} with respect to this basis. By Proposition 3.7, M_{0} and M_{1} generate an irreducible subgroup of GL_{2}(R), and also of GL_{2}(L); hence choosing eigenvectors v_{0}, v_{1} ∈ L^{2} corresponding to the eigenvalues λ_{0}, λ_{1} gives a basis {v_{0}, v_{1}} for L^{2}. Writing (M_{0}, M_{1}, M_{2}) with respect to this basis gives a triple (M̃_{0}, M̃_{1}, M̃_{2}) globally conjugate to (M_{0}, M_{1}, M_{2}) in

GL_{2}(L) such that M̃_{0} = ( \begin{matrix}α & γ \\ 0 & α^{−1} \end{matrix} ) and M̃_{1} = ( \begin{matrix}β & 0 \\ δ & β^{−1} \end{matrix} ) for some α, β, δ, γ ∈ L^{×}.

Rescaling {v_{0}, v_{1}} if necessary, we may assume that γ = 1.

Let (M′_{0}, M′_{1}, M′_{2}) be any triple of matrices which is locally conjugate to (M_{0}, M_{1}, M_{2}) in GL_{2}(R), and which satisfies M′_{0}M′_{1}M′_{2} = 1. Since M_{0} ∼ M′_{0} and M_{1} ∼ M′_{1}, λ_{0}, λ_{1} ∈ L are eigenvalues of M′_{0}, M′_{1} respectively. Thus by the same reasoning as for (M_{0}, M_{1}, M_{2}) above, there is a triple (M̃′_{0}, M̃′_{1}, M̃′_{2}) globally conjugate to (M′_{0}, M′_{1}, M′_{2}) in GL_{2}(L) such that M̃′_{0} = ( \begin{matrix}α′ & 1 \\ 0 & α^{′−1} \end{matrix} )

and M̃′_{1} = ( \begin{matrix}β′ & 0 \\ δ′ & β^{′−1} \end{matrix} ) for some α′, β′, δ′ ∈ L^{×}. Now M̃_{0} ∼ M̃′_{0} implies that

Tr(M̃_{0}) = Tr(M̃′_{0}); that is, α + α^{−1} = α′ + α^{′−1}. Hence (α − α′)(αα′ − 1) = 0, and therefore α = α′ or α = α^{′−1}. If α = α^{′−1} ≠ α′, then conjugat-

ing (M̃_{0}, M̃_{1}, M̃_{2}) by M = ( \begin{matrix}1 & 0 \\ α′−α^{′−1} & 1 \end{matrix} ) gives MM̃_{0}M^{−1} = ( \begin{matrix}α^{′−1} & 1 \\ 0 & α′ \end{matrix} ) and

MM̃′_{1}M^{−1} = ( \begin{matrix}β′ & 0 \\ ∗ & β^{′−1} \end{matrix} ). Thus replacing (M̃′_{0}, M̃′_{1}, M̃′_{2}) with (M̃′_{0}, M̃′_{1}, M̃′_{2})^{M}

if necessary, and renaming α′, δ′ accordingly, we have α = α′. Similarly, Tr(M̃_{1}) = Tr(M̃′_{1}) gives β = β′ or β = β^{′−1}. If β = β^{′−1} ≠ β′, then tak-

ing M = ( \begin{matrix}δ′ & β^{′−1}−β′ \\ 0 & δ′ \end{matrix} ) gives MM̃′_{0}M^{−1} = M̃′_{0} and MM̃′_{1}M^{−1} = ( \begin{matrix}β^{′−1} & 0 \\ δ′ & β′ \end{matrix} ), so

replacing (M̃′_{0}, M̃′_{1}, M̃′_{2}) with (M̃′_{0}, M̃′_{1}, M̃′_{2})^{M} if necessary gives β = β′, and does not affect M̃′_{0}. Multiplying gives

M̃_{2} = (M̃_{0}M̃_{1})^{−1} = (\begin{matrix}α^{−1}β^{−1} & −β^{−1} \\ −α^{−1}δ & αβ + δ \end{matrix}) ,

and similarly for M̃′_{2}; thus the equation Tr(M̃_{2}) = Tr(M̃′_{2}) becomes

α^{−1}β^{−1} + αβ + δ = α^{′−1}β^{′−1} + α′β′ + δ′,

and therefore δ = δ′. Thus we have shown that (M̃_{0}, M̃_{1}, M̃_{2}) = (M̃′_{0}, M̃′_{1}, M̃′_{2}); in particular, (M_{0}, M_{1}, M_{2}) is globally conjugate to (M′_{0}, M′_{1}, M′_{2}) in GL_{2}(L).

In order to obtain global conjugacy in GL_{2}(R), first note that either L = K(λ_{0}) or L is a quadratic extension of K(λ_{0}). In the latter case, by Proposition 3.7, M_{0}, M_{1} generate an irreducible subgroup of GL_{2}(K(λ_{0})); hence by Lemma 3.8, (M_{0}, M_{1}, M_{2}) is globally conjugate to (M′_{0}, M′_{1}, M′_{2}) in GL_{2}(K(λ_{0})). Applying Lemma 3.8 again if necessary (that is, if K(λ_{0}) ≠ K), we find that (M_{0}, M_{1}, M_{2}) is globally conjugate to (M′_{0}, M′_{1}, M′_{2}) in GL_{2}(K). Since R is a UFD, by Lemma 3.9, (M_{0}, M_{1}, M_{2}) is globally conjugate to (M′_{0}M′_{1}, M′_{2}) in GL_{2}(R). Therefore, (M_{0}, M_{1}, M_{2}) is rigid. □

A similar argument to that in the proof of Theorem 3.10 can be used to prove the result for any local domain R, provided that M_{0} and M_{1} both have eigenvalues in R. In this case, it is not necessary to pass to the field L; the arguments used above can be applied in R itself. In fact, this argument can be extended to prove the result for any local domain R provided that at least one of M_{0} and M_{1} has an eigenvalue in R, although the details are significantly more complicated. New difficulties arise when neither M_{0} nor M_{1} has an eigenvalue in R, and it is not clear whether the assumption of unique factorization is necessary in this case.

3.4 Extending the Universal Deformation

We will now use the rigidity theorem of §3.2 to extend the universal deformations of §2.6 to representations of a larger Galois group. Let K be any algebraic extension of ℚ, and let Π_{K}, γ_{0}, γ_{1}, γ_{∞} be as in §3.2. Let

ρ^{geom} : Π\overline{K} ⟶ SL_{2}(R)

be any representative of one of the following universal deformations:

(1) the {γ_{0}, γ_{1}}-ordinary universal deformation of Theorem 2.31, in which case R = ℤ_{p}[[t]];

(2) the γ_{i}-ordinary universal deformation of either Theorem 2.29 or Corollary 2.30, in which case R = ℤ_{p}[[t_{1}, t_{2}]]; or

(3) the (determinant one) universal deformation of Theorem 2.28, in which case R = ℤ_{p}[[t_{1}, t_{2}, t_{3}]].

Theorem 3.11 The projectivization

ρ̂^{geom} : Π\overline{K} ⟶ PGL_{2}(R) = GL_{2}(R)/R^{×}

of ρ can be extended uniquely to a representation ρ̂ : Π_{K(μ)} ⟶ PGL_{2}(R), where μ is as in Theorem 3.5

Proof: Since ρ̄(γ_{0}), ρ̄(γ_{1}), ρ̄(γ_{∞}) generate an irreducible subgroup of GL_{2}(𝔽_{p}) and R is a local UFD, by Theorem 3.10, (ρ^{geom}(γ_{0}), ρ^{geom}(γ_{1}), ρ^{geom}(γ_{∞})) is rigid in GL_{2}(R). By Proposition 3.7, ρ^{geom}(γ_{0}) and ρ^{geom}(γ_{1}) generate an acentral subgroup of GL_{2}(R), so Z_{GL_{2}(R)}(Imρ^{geom}) = R^{×} = Z(GL_{2}(R)). The result now follows from Theorem 3.5(1). □ Remark. Given a residual representation ρ̄: Π\overline{K} ⟶ SL_{2}(𝔽_{p}), let m denote the prime-to-p part of lcm_{i=0,1,∞}(o(ρ̄(γ_{i}))), where o(ρ̄(γ_{i})) denotes the order of ρ̄(γ_{i}) (in particular, m | p^{2} − 1). Let μ_{m} denote the set of mth roots of unity and μ_{p^{∞}} the set of all p^{n}th roots of unity in \overline{K}. Note that the kernel of the reduction map GL_{2}(R) ⟶ GL_{2}(𝔽_{p}) is equal to 1 + M_{2}(𝔪) ≅ M_{2}(𝔪), and M_{2}(𝔪) = \underleftarrow{lim} M_{2}(𝔪/𝔪^{n}) is an inverse limit of p-groups, so the image of ρ^{geom}(γ_{i}) in any finite quotient of GL_{2}(R) has order dividing p^{n}m for some n. Therefore, K(μ) is contained in K(μ_{m}, μ_{p^{∞}}).

Let ρ̄ denote the residual representation of ρ^{geom}. If ρ̄(γ_{i}) has an eigenvalue in 𝔽_{p} for some i = 0, 1, ∞, then the above result can be strengthened.

Theorem 3.12 If ρ^{geom} is the universal deformation of case (3), suppose that ρ̄(γ_{i}) has distinct eigenvalues in 𝔽_{p} for some i = 0, 1, ∞. Then in all three cases, ρ^{geom} extends to a representation ρ : Π_{K(μ)} ⟶ GL_{2}(R) which is unique up to multiplication by a representation ψ : G_{K(μ)} ⟶ R^{×}.

Proof: Let ρ̂ be as in Theorem 3.11. With the notation of §3.2, if we show that for some i the inclusion

Z(GL_{2}(R)) ↪ r^{−1}(ρ̂ ∘ ϕ_{i}(G_{K(μ)}))

splits, then the result will follow from Theorem 3.5(2). In all three cases, ρ^{geom}(γ_{i}) has a rank one eigenspace V ⊂ R^{2} for some i = 0, 1, ∞ (in case (3), this follows from the argument of Lemma 2.25). Fix such an i, and let N = r^{−1}(ρ̂ ∘ ϕ_{i}(G_{K(μ)})). We claim that N fixes V. From the proof of Lemma 3.3, for each γ ∈ ϕ_{i}(G_{K(μ)}), we have γγ_{i}γ^{−1} = γ_{i}^{χ(γ)}; applying ρ^{geom} gives ρ^{geom}(γγ_{i}γ^{−1}) = ρ^{geom}(γ_{i})^{χ(γ)}. Since K(μ) ⊃ K(μ_{m}, μ_{p^{∞}}), and ρ^{geom}(γ_{i}) has order dividing p^{n}m (for some n) in every finite quotient of GL_{2}(R), we have ρ^{geom}(γ_{i})^{χ(γ)} = ρ^{geom}(γ_{i}), so ρ^{geom}(γγ_{i}γ^{−1}) = ρ^{geom}(γ_{i}). From the definition of ρ̂ in the proof of Theorem 3.5 and the fact that ker(r) = Z(GL_{2}(R)), it follows that ρ^{geom}(γγ_{i}γ^{−1}) = g_{γ}ρ^{geom}(γ_{i})g_{γ}^{−1} for any g_{γ} ∈ r^{−1}(ρ̂(γ)). Every M ∈ N can be obtained as g_{γ} for some γ, so Mρ^{geom}(γ_{i})M^{−1} = ρ^{geom}(γ_{i}) for all M ∈ N.

Let λ be the eigenvalue of ρ^{geom}(γ_{i}) corresponding to V = Rv. Since Mρ^{geom}(γ_{i})M^{−1}v = λv, we have

ρ^{geom}(γ_{i})(M^{−1}v) = λM^{−1}v,

so M^{−1}v is an eigenvector of ρ^{geom}(γ_{i}) with eigenvalue λ. Since V is the full eigenspace of ρ^{geom}(γ_{i}) with eigenvalue λ, we must have M^{−1}v ∈ V. Therefore, N fixes V, as claimed. Thus each M ∈ N induces a linear map on R^{2}/V, which is a free R-module of rank one. Fixing an isomorphism

GL(R^{2}/V) ≅ R^{×} = Z(GL_{2}(R))

gives the desired splitting. □

If the residual representation ρ̄ is γ_{i}-ordinary, the universal property of the determinant one universal deformation ρ^{univ} gives a map R^{univ} ⟶ R_{ord}^{univ} which takes any extension of ρ^{univ} to an extension of ρ_{ord}^{univ}. Similarly, if ρ̄ is S-ordinary, where S = {γ_{0}, γ_{1}}, we obtain maps R^{univ} ⟶ R_{S−ord}^{univ} and R_{γ_{i}−ord}^{univ} ⟶ R_{S−ord}^{univ} for each i = 0, 1, which take extensions of ρ^{univ} and ρ_{γ_{i}−ord}^{univ} respectively to extensions of ρ_{S−ord}^{univ}. Furthermore, the map R^{univ} ⟶ R_{S−ord}^{univ} factors through both maps R_{γ_{i}−ord}^{univ} ⟶ R_{S−ord}^{univ} via the map R^{univ} ⟶ R_{γ_{i}−ord}^{univ} discussed above.

4 Geometric Construction of Universal De-

formations

4.1 Jacobians of Curves

Fix an odd prime p. Let ρ̄: Π_{ℚ} ⟶ GL_{2}(𝔽_{p}) be the representation describing the action of Π_{ℚ} on the p-torsion points of the Legendre family E_{L} of elliptic curves over ℚ(t), given by the equation

E_{L} : y^{2} = x(x − 1)(x − t).

Let ρ̄^{geom} : Π ⟶ GL_{2}(𝔽_{p}) denote the restriction of ρ̄ to Π = Π\overline{ℚ}, and let σ_{0}, σ_{1}, σ_{∞} ∈ Π be generators of inertia groups at 0, 1, ∞ respectively such that σ_{0}σ_{1}σ_{∞} = 1. Then ρ̄^{geom} is an absolutely irreducible representation characterized up to conjugation by the property that ρ̄^{geom}(σ_{0}) and ρ̄^{geom}(σ_{1}) both have order p and ρ̄^{geom}(σ_{∞}) has order 2p (see [Dar00], p.419). Let S = {σ_{0}, σ_{1}}. In this chapter, we give an explicit geometric construction of the S-ordinary universal deformation ρ_{S−ord}^{univ} of ρ̄^{geom}. In fact, the representation that we construct will be a representation of the larger Galois group Π_{ℚ(μ_{p^{∞}})}, and thus will be the extension of ρ_{S−ord}^{univ} given in Theorem 3.12(1).

Let K be a field.

Definition 4.1 An abelian variety A over K is a complete variety over K together with a group law μ : A × A ⟶ A which is a morphism defined over K, and for which the inverse map a ⟼ a^{−1} is also a morphism defined over K.

A morphism of abelian varieties is a morphism of varieties which is also a homomorphism of the underlying groups. The group law on an abelian variety is commutative (see [Lan83], Ch. II, §1, Theorem 1). Note that an abelian variety of dimension one is simply an elliptic curve. When K = ℂ, the classical uniformization of elliptic curves E/ℂ may be generalized to abelian varieties A/ℂ of arbitrary dimension g. A lattice Λ of ℂ^{g} is a free ℤ-module of rank 2g which has a basis which is also an ℝ-basis for ℂ^{g}.

Theorem 4.2 Let A/ℂ be an abelian variety of dimension g. There exists a lattice Λ of ℂ^{g} and a complex analytic group isomorphism

ϕ : ℂ^{g}/Λ ⟶ A(ℂ).

Proof: See [Mum70], Ch. I, §1(2). □

We now describe how to associate an abelian variety Jac(C)/K to any complete nonsingular curve C/K of genus g > 0. A divisor D on C is a formal finite sum of \overline{K}-rational points on C, that is D = ∑_{P ∈C(\overline{K})}n_{P} P, where each n_{P} ∈ ℤ and n_{P} = 0 for almost all P ∈ C(\overline{K}). We write Div(C) for the abelian group of divisors on C, where the sum of two divisors D_{1} = ∑_{P ∈C(\overline{K})}n_{P} P and

D_{2} = ∑_{P ∈C(\overline{K})}m_{P} P is given by

D_{1} + D_{2} = ∑_{P ∈C(\overline{K})}(n_{P} + m_{P})P.

Given a rational function f ∈ \overline{K}(C)^{×}, we define the divisor (f) of f to be (f) = ∑_{P ∈C(\overline{K})}ord_{P} (f)P, where ord_{P} (f) is the order of vanishing of f at P. A divisor on C is said to be principal if it is the divisor of some f ∈ \overline{K}(C)^{×}, and the subgroup of Div(C) consisting of all principal divisors is denoted by Pr(C). Two divisors D_{1}, D_{2} ∈ Div(C) are said to be linearly equivalent if D_{1} − D_{2} ∈ Pr(C). The group of linear equivalence classes of divisors on C is called the Picard group of C, and is denoted Pic(C); thus

Pic(C) = Div(C)/Pr(C).

Given any divisor D = ∑_{P ∈C(\overline{K})}n_{P} P on C, the degree deg(D) of D is defined to be deg(D) = ∑_{P ∈C(\overline{K})}n_{P}. The group Pr(C) is contained in the subgroup Div^{0}(C) of Div(C) consisting of the divisors of degree zero (see [Har97], Ch. II, Corollary 6.10), and thus we may define the degree zero part of the Picard group to be Pic^{0}(C) = Div^{0}(C)/Pr(C).

The absolute Galois group G_{K} of K acts naturally on Div(C) by

σ ( ∑_{P ∈C(\overline{K})}n_{P} P ) = ∑_{P ∈C(\overline{K})}n_{P} σ(P)

for σ ∈ G_{K}. Furthermore, for any σ ∈ G_{K}, two divisors D_{1} and D_{2} are linearly equivalent if and only if σ(D_{1}) and σ(D_{2}) are. Thus the action of G_{K} on Div(C) induces actions on Pic(C) and Pic^{0}(C).

For any P ∈ C(\overline{K}), let

f^{P} : C(\overline{K}) ⟶ Pic^{0}(C)

be the map which takes Q ∈ C(\overline{K}) to the linear equivalence class [Q − P] of Q − P.

Theorem 4.3 The group Pic^{0}(C) can be given the structure of the \overline{K}-ra-tional points of an abelian variety Jac(C)/K of dimension equal to the genus of C in such a way that for each P ∈ C(\overline{K}), f^{P} is an embedding, and Jac(C) satisfies the following universal property: if ϕ : C ⟶ A is a morphism from C to an abelian variety A such that ϕ(P) = 0, then there is a unique morphism of abelian varieties ψ : Jac(C) ⟶ A such that the diagram

\xymatrix{C \ar[r]^{f^{P}} \ar[dr]_{ϕ} & Jac(C) \ar[d]^{ψ} \\  & A}

commutes.

Proof: See [Mil86b], Proposition 2.3, Proposition 6.1, and Theorem 1.1. □

The abelian variety Jac(C) is called the Jacobian of C. The universal property of the Jacobian shows that assigning to a curve its Jacobian defines both a covariant and a contravariant functor from the category of complete nonsingular curves with nonconstant morphisms to that of abelian varieties. Given complete nonsingular curves C_{1} and C_{2}, and a nonconstant morphism ϕ : C_{1} ⟶ C_{2}, the map f^{ϕ(P_{0})} ∘ ϕ : C_{1} ⟶ Jac(C_{2}) satisfies f^{ϕ(P_{0})} ∘ ϕ(P_{0}) = 0 for each P_{0} ∈ C_{1}. The universal property of Jac(C_{1}) gives a morphism of abelian varieties ϕ_{∗} : Jac(C_{1}) ⟶ Jac(C_{2}). In terms of divisors, ϕ_{∗} is given by

ϕ_{∗} : [∑ n_{P} P ] ⟼ [∑ n_{P} ϕ(P)] .

In particular, ϕ_{∗} is independent of the choice of P_{0} above. On the other hand, for each Q ∈ C_{2}, let e_{ϕ}(Q) denote the ramification index of ϕ at Q. Fixing a point Q_{0} ∈ C_{2}, there is a morphism f_{ϕ} : C_{2} ⟶ Jac(C_{1}) given by

f_{ϕ} : Q ⟼ [ ∑_{P ∈ϕ^{−1}(Q)}e_{ϕ}(Q)P − ∑_{P ∈ϕ^{−1}(Q_{0})}e_{ϕ}(Q_{0})P ]

which satisfies f_{ϕ}(Q_{0}) = 0. The universal property of Jac(C_{2}) gives a morphism of abelian varieties ϕ^{∗} : Jac(C_{2}) ⟶ Jac(C_{1}), which is given in terms of divisors by

ϕ^{∗} : [ ∑_{Q∈C_{2}}n_{Q}Q] ⟼ [ ∑_{Q∈C_{2}}n_{Q} ∑_{P ∈ϕ^{−1}(Q)}e_{ϕ}(Q)P ] .

Once again, ϕ^{∗} is seen to be independent of the choice of Q_{0} above.

4.2 Tate Modules and ℓ-adic Representations

Let A/K be an abelian variety of dimension g, and fix a prime ℓ not equal to the characteristic of K. For each integer m ∈ ℤ, there is an endomorphism [m] : A(\overline{K}) ⟶ A(\overline{K}) of A defined over K given by multiplication by m. We write A[m] := ker[m], and call the elements of A[m] the m-torsion points of A. For each positive integer m not divisible by the characteristic of K, we have deg[m] = m^{2g} (see [Mil86a], Theorem 8.2). Applying this equality to every positive integer d dividing m shows that A[m] is isomorphic to (ℤ/mℤ)^{2g}. As m ranges through powers of ℓ, the ℓ^{n}-torsion points together with the multiplication-by-ℓ maps [ℓ] : A[ℓ^{n+1}] ⟶ A[ℓ^{n}] form a directed system of groups. The inverse limit

T_{ℓ}(A) := \underleftarrow{lim}_{n∈ℕ} A[ℓ^{n}]

is called the (ℓ-adic) Tate module of A. The Tate module T_{ℓ}(A) is naturally a free ℤ_{ℓ}-module of rank 2g, where the action of α ∈ ℤ_{ℓ} on A[ℓ^{n}] is given by multiplication by the reduction of α mod ℓ^{n}. This action preserves compatibility under the multiplication-by-ℓ maps, and thus defines an action of ℤ_{ℓ} on T_{ℓ}(A).

Since the addition law on A is defined over K, the action of G_{K} on A commutes with [m]. Thus restricting to A[m] gives an action of G_{K} on A[m]. Moreover, since this action commutes with the multiplication-by-ℓ map, we obtain a ℤ_{ℓ}-linear action of G_{K} on T_{ℓ}(A). Choosing a ℤ_{ℓ}-basis for T_{ℓ}(A) gives a homomorphism

ρ_{ℓ} : G_{K} ⟶ GL_{2g}(ℤ_{ℓ}),

called the ℓ-adic representation associated to A.

We define the extended Tate module V_{ℓ}(A) := T_{ℓ}(A) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}, which is a ℚ_{ℓ}-vector space of dimension 2g, together with a ℚ_{ℓ}-linear action of G_{K}. This action gives a representation

ρ_{ℓ} : G_{K} ⟶ GL_{2}(ℚ_{ℓ}),

which may be obtained from the ℓ-adic representation by extending scalars to ℚ_{ℓ}.

Let ψ : A ⟶ B be a morphism of abelian varieties. Since ψ is a group homomorphism, it maps ℓ^{n}-torsion points of A to ℓ^{n}-torsion points of B, and commutes with the multiplication-by-ℓ maps on each side. Thus ψ induces a ℤ_{ℓ}-module homomorphism

ψ_{ℓ} : T_{ℓ}(A) ⟶ T_{ℓ}(B)

by applying ψ componentwise, that is, ψ_{ℓ} : (a_{n})_{n∈ℕ} ⟼ (ψ_{ℓ}(a_{n}))_{n∈ℕ}, where a_{n} ∈ A[ℓ^{n}] and [ℓ]a_{n} = a_{n−1} for each positive integer n. We will also write ψ_{ℓ} for the map V_{ℓ}(A) ⟶ V_{ℓ}(B) obtained by tensoring with ℚ_{ℓ}.

Proposition 4.4 Let ϕ : C_{1} ⟶ C_{2} be a nonconstant morphism of complete nonsingular curves defined over K, and let ϕ_{ℓ}^{∗} : T_{ℓ} (Jac(C_{2})) ⟶ T_{ℓ} (Jac(C_{1})) denote the map induced from ϕ^{∗} : Jac(C_{2}) ⟶ Jac(C_{1}). Then ϕ_{ℓ}^{∗} is injective.

Proof: Let ϕ̃^{∗} : Div(C_{2}) ⟶ Div(C_{1}) denote the map given by

ϕ̃^{∗} : ∑_{Q∈C_{2}}n_{Q}Q ⟼ ∑_{Q∈C_{2}}n_{Q} ∑_{P ∈ϕ^{−1}(Q)}e_{ϕ}(Q)P.

Suppose that D ∈ Div^{0}(C_{2}) is such that ϕ̃^{∗}(D) = (f) for some f ∈ \overline{K}(C_{1}). The map ϕ induces an injection of function fields \overline{K}(C_{2}) ↪ \overline{K}(C_{1}), and we have

Norm_{\overline{K}(C_{2})}^{\overline{K}(C_{1})}(f) = degϕ · D.

In particular, the image of D in Pic^{0}(C_{2}) ≅ Jac(C_{2}) is a degϕ-torsion point. If t ∈ T_{ℓ} (Jac(C_{2})) is such that ϕ_{ℓ}^{∗}(t) = 0, then every component of t is a degϕ-torsion point, and thus t is itself a degϕ-torsion point of T_{ℓ} (Jac(C_{2})). Since T_{ℓ} (Jac(C_{2})) is a free ℤ_{ℓ}-module, we must have t = 0. □ Remark: Since ℚ_{ℓ} is flat over ℤ_{ℓ} (see [Lan93], Ch. XVI, Proposition 3.2), the map ϕ_{ℓ}^{∗} : V_{ℓ} (Jac(C_{2})) ⟶ V_{ℓ} (Jac(C_{1})) is also injective.

4.3 Reduction of Curves

Let K be a field of characteristic zero, C/K a complete nonsingular curve of genus g > 0, and 𝔭 a valuation ideal of K with corresponding valuation ring R. For any valuation ideal 𝔭̂ of \overline{K} above 𝔭, the action of the inertia group I (𝔭̂/𝔭) on the Jacobian of C is closely related to the reduction type of C at 𝔭.

Definition 4.5 A K-model M of a variety V/K is a set of equations with coefficients in K, taken up to multiplication of each equation by elements of K^{×}, such that M defines an element of the K-isomorphism class given by V.

A particular set of equations in the equivalence class M will be called a defining set of equations for M. We will say that a defining set of equations for M is 𝔭-reducible if all of its coefficients lie in R, and each equation has at least one coefficient not in 𝔭. The reduction M̄ of M at 𝔭 is the variety defined over the residue field k = R/𝔭 obtained by reducing mod 𝔭 the coefficients of a 𝔭-reducible set of equations for M. Fixing a valuation ideal 𝔭̂ of \overline{K} above 𝔭 with valuation ring R̂, we obtain a reduction map r : M(\overline{K}) ⟶ M̄(\overline{k}) by choosing for each \overline{K}-rational point of M an expression which has projective coordinates in R̂ but not all in 𝔭̂, and reducing the coordinates mod 𝔭̂.

Definition 4.6 The curve C/K of genus g is said to have good reduction at 𝔭 if it has a K-model whose reduction at 𝔭 is a nonsingular curve of genus g. An abelian variety A/K of dimension g is said to have good reduction at 𝔭 if it has a K-model whose reduction at 𝔭 is an abelian variety of dimension g.

If C (respectively A) has good reduction at 𝔭, we will often identify C (resp. K) with a K-model whose reduction is as in Definition 4.6. In this case, the reduced curve (respectively reduced abelian variety) is independent of the choice of such a K-model. If C or A does not have good reduction at 𝔭, then we say that it has bad reduction at 𝔭.

Jacobians are well-behaved with respect to good reduction in the sense that if C has good reduction at 𝔭 then so does Jac(C), and in this case, the Jacobian of the reduction of C is the reduction of Jac(C). The converse, however, is not true; there exist curves with bad reduction at a valuation ideal 𝔭 whose Jacobians have good reduction at 𝔭 (see [Maz86], p.238 for an example).

Definition 4.7 A representation ρ of G_{K} is said to be unramified at 𝔭 if ρ (I (𝔭̂/𝔭)) = 1 for each valuation ideal 𝔭̂ of \overline{K} above 𝔭. Equivalently, ρ is unramified at 𝔭 if 𝔭 is unramified in the extension of K corresponding to the quotient G_{K}/ker(ρ).

Proposition 4.8 Let ℓ be a rational prime not below 𝔭. If the abelian variety A/K has good reduction at 𝔭, then the ℓ-adic representation ρ_{ℓ} attached to A is unramified at 𝔭.

Proof: It suffices to show that the representation ρ̄_{ℓ^{n}} : G_{K} ⟶ Aut(A[ℓ^{n}]) describing the action of G_{K} on the ℓ^{n}-torsion points of A is unramified for each n. Let Ā denote the reduction of A at 𝔭, and let 𝔭̂ be any valuation ideal of \overline{K} above 𝔭. The reduction map restricts to an isomorphism

A[ℓ^{n}]^{I(𝔭̂/𝔭)} ⟶^{≅} Ā[ℓ^{n}]

(see [ST68], §1, Lemma 2). Since A[ℓ^{n}] and Ā[ℓ^{n}] are both free ℤ/ℓ^{n}ℤ-modules of rank 2 dimĀ = 2 dimA, counting gives A[ℓ^{n}]^{I(𝔭̂/𝔭)} = A[ℓ^{n}]. □ Remark: The converse to Proposition 4.8 is also true, and is known as the criterion of Néron-Ogg-Shafarevich. To be precise, if ρ_{ℓ} is unramified at 𝔭 for some ℓ not below 𝔭, then A has good reduction at 𝔭 (see [ST68], §1).

4.4 Mumford Curves

When a curve C/K has a special type of bad reduction at 𝔭, strong information can be obtained about the action of the inertia groups above 𝔭 on the Tate module of the Jacobian of C. To make this precise, it will be useful to have an alternative description of the inertia group. Let K and F be fields, and let ϕ : K ⟶ F ∪ {∞} be a nontrivial place of K with valuation ring R and valuation ideal 𝔭, as in §2.2. Let K_{𝔭} denote the completion of K at 𝔭, which is the quotient field of the completion of R with respect to the 𝔭-adic topology. Let L be a Galois extension of K, 𝔭̂ a valuation ideal of L above 𝔭, and L𝔭̂ the completion of L at 𝔭̂. The extension L𝔭̂/K_{𝔭} is Galois.

Proposition 4.9 The map r_{L} : Gal (L𝔭̂/K_{𝔭}) ⟶ Gal (L/K) given by restriction to L defines an isomorphism of Gal (L𝔭̂/K_{𝔭}) onto the decomposition group D (𝔭̂/𝔭).

Proof: See [Ser68], Ch. II, §3, Corollaire 4. □

Now let K = k(t), where k is an algebraically closed subfield of ℂ. For each P ∈ ℙ^{1}(k), let ϕ_{P} : K ⟶ k ∪ {∞} be the place given by ϕ_{P} (f) = f(P) for f ∈ K (see §2.2). Identifying ℙ^{1}(k) with k ∪ {∞}, the completion K_{P} of K with respect to ϕ_{P} is given by

K_{P} = \begin{cases}k((t − P)) & if P ∈ k, \\ k((\frac{1}{t})) & if P = ∞. \end{cases}

Let 𝔭 be the valuation ideal of K at P, and 𝔭̂ a valuation ideal of \overline{K} above 𝔭. Then I (𝔭̂/𝔭) = D (𝔭̂/𝔭) may be identified with Gal (\overline{K}_{P} /K_{P} ) as in Proposition 4.9. Moreover, we have

\overline{K}_{P} = \begin{cases}⋃_{n∈ℕ}k (((t − P)^{1/n})) & if P ∈ k \\ ⋃_{n∈ℕ}k (((\frac{1}{t})^{1/n})) & if P = ∞. \end{cases}

Understanding the action of the inertia group I (𝔭̂/𝔭) on the \overline{K}-rational points of an abelian variety A/K is equivalent to understanding the Galois action on the \overline{K_{P}} -rational points of A (as a variety over K_{P}).

Definition 4.10 A complete nonsingular curve C/K_{P} is called a Mumford curve if it has a K_{P} -model whose reduction (at P) is a union of projective lines whose only singularities are ordinary double points.

Mumford proved the existence of the following uniformization, generalizing a theorem of Tate in the case of elliptic curves.

Theorem 4.11 Let C/K_{P} be a Mumford curve of genus g, and J its Jacobian. Then there is a surjective group homomorphism

υ : (\overline{K}_{P}^{×})^{g} ⟶ J (\overline{K}_{P} )

commuting with the action of G_{K_{P}} on each side, whose kernel is a discrete subgroup of (\overline{K}_{P}^{×})^{g} freely generated by elements q_{1}, …, q_{g} ∈ (K_{P}^{×})^{g}.

Proof: See [Gv80], Ch. VI, §§1.3,1.4. That a Mumford curve in our sense is indeed a Mumford curve in the sense of [Gv80] may be found in [Gv80], Ch. IV, Theorem 3.10. □ Remark: Theorem 4.11 remains true if K_{P} is replaced with any field which is complete with respect to a non-archimedean valuation.

Let 𝔭 and 𝔭̂ be as above. Let ℓ be a rational prime, and let ρ_{ℓ} be the ℓ-adic representation associated to the Jacobian of a curve C/K of genus g.

Corollary 4.12 Suppose that C becomes a Mumford curve over K_{P}. Then for each σ ∈ I (𝔭̂/𝔭), we have

ρ_{ℓ}(σ) ∼ (\begin{matrix}Id_{g} & ∗ \\ 0 & Id_{g} \end{matrix}) ,

where Id_{g} denotes the g × g identity matrix.

Proof: Since the isomorphism υ of Theorem 4.11 commutes with the action of G_{K_{P}}, it suffices to consider the action of G_{K_{P}} on the ℓ^{n}th roots of the identity in (\overline{K}_{P}^{×})^{g}/⟨q_{1}, …, q_{g}⟩. Choosing a primitive ℓ^{n}th root of unity ζ_{n} ∈ \overline{K}_{P}, the subgroup of ℓ^{n}th roots of the identity in (\overline{K}_{P}^{×})^{g}/⟨q_{1}, …, q_{g}⟩ is generated by the cosets of (ζ_{n}, 1, …, 1), (1, ζ_{n}, 1, …, 1), …, (1, …, 1, ζ_{n}), together with ℓ^{n}th roots of q_{1}, …, q_{g}. Fix ℓ^{n}th roots q_{1}^{1/ℓ^{n}}, …, q_{g}^{1/ℓ^{n}} of q_{1}, …, q_{g}. Each σ ∈ G_{K_{P}} fixes each of (ζ_{n}, 1, …, 1), …, (1, …, 1, ζ_{n}), and since each q_{j} is an element of (K_{P}^{×})^{g}, σ takes q_{j}^{1/ℓ^{n}} to another ℓ^{n}th root of q_{j}. Hence

σq_{j}^{1/ℓ^{n}} = (ζ_{n}^{j_{1}}, …, ζ_{n}^{j_{g}})q_{j}^{1/ℓ^{n}}

for some j_{1}, …, j_{g} ∈ ℤ/ℓ^{n}ℤ. Therefore, with respect to the ℤ/ℓ^{n}ℤ-basis {(ζ_{n}, 1, …, 1), …, (1, …, 1, ζ_{n}), q_{1}, …, q_{g}} for the ℓ^{n}th roots of the identity in (\overline{K}_{P}^{×})^{g}/⟨q_{1}, …, q_{g}⟩, σ acts as ( \begin{matrix}Id_{g} & ∗ \\ 0 & Id_{g} \end{matrix} ). The result now follows from the discussion preceding Theorem 4.11. □

4.5 Hypergeometric Families of Curves

Fix an odd prime p, and consider the so-called hypergeometric family of curves over ℚ(t) given by

C_{n} : y^{2} = x (x^{2p^{n}} + (4t − 2)x^{p^{n}} + 1)

for each n ≥ 0. Let ζ_{n} be a generator of the group μ_{p^{n}} of p^{n}th roots of unity in \overline{ℚ}. The group μ_{p^{n}} acts as a group of automorphisms on C_{n} by

ζ_{n} · (x, y) = (ζ_{n}x, ζn^{\frac{p^{n}+1}{2}} y).

We will denote by γ_{n} the automorphism of C_{n} given by the action of ζ_{n} in order to distinguish the group ring ℤ_{p}[γ_{n}] = ℤ_{p}[μ_{p^{n}}] from the subring ℤ_{p}[ζ_{n}] of the field ℚ_{p}(ζ_{n}). We will also identify μ_{p^{n}} with the automorphism group generated by γ_{n}.

In addition to the hyperelliptic involution (x, y) ⟼ (x, −y) and the action of μ_{p^{n}}, there is also an involution τ_{n} of C_{n} given by

τ_{n} : (x, y) ⟼ (\frac{1}{x}, \frac{y}{x^{p^{n}+1}} ) .

Note that τ_{n}∘γ_{n} = γ_{n}^{−1}∘τ_{n}, so the automorphism group ⟨τ_{n}, γ_{n}⟩ is isomorphic to the dihedral group of order 2p^{n}. Tautz, Top, and Verberkmoes studied C_{n} and its quotient C_{n}^{−} = C_{n}/⟨τ_{n}⟩ in [TTV91] (see in particular Theorem 1 and Proposition 3). For any odd prime r, the Galois representation on the r-torsion points of the Jacobian of C_{1}^{−} was subsequently studied by Darmon in connection with the equation x^{r}+y^{r} = z^{p} (see [Dar00], Theorem 1.10), as well as by Darmon and Mestre to construct a regular extension of ℚ(ζ_{n} + ζ_{n}^{−1})(t) with Galois group PSL_{2}(𝔽_{q}) for certain finite fields 𝔽_{q} (see [DM00], §§2,3).

Proposition 4.13 The quotient curve C_{n}^{−} is birationally equivalent over ℚ(t) to the curve given by

y^{2} = xg_{n}(x^{2} − 2) + 4t − 2,

where g_{n}(x) = ^{∏_{j=1}^{\frac{p^{n}−1}{2}}}(x + ζ_{n}^{j} + ζ_{n}^{−j}).

Idea of Proof: The subfield of the function field of C_{n} consisting of those elements fixed by τ_{n} is generated by x + x^{−1} and \frac{y}{x^{\frac{p^{n}+1}{2}}} . Using the formal relation

X^{p^{n}} + X^{−p^{n}} = (X + X^{−1})g_{n}(X^{2} + X^{−2})  (4.14)

gives the desired equation. See [TTV91], Proposition 3 for details. □ For each m ≤ n, there is a morphism ϕ_{n,m} : C_{n} ⟶ C_{m} given by

ϕ_{n,m} : (x, y) ⟼ (x^{p^{n−m}}, x^{\frac{p^{n−m}−1}{2}} y) .

If the generators ζ_{n} for each μ_{p^{n}} are chosen to be compatible in the sense that ζ_{n}^{p} = ζ_{n−1} for all n, then ϕ_{n,m} ∘ γ_{n} = γ_{m} ∘ ϕ_{n,m}. Also, each ϕ_{n,m} satisfies τ_{m} ∘ ϕ_{n,m} = ϕ_{n,m} ∘ τ_{n}, so ϕ_{n,m} induces a morphism ϕ_{n,m}^{−} : C_{n}^{−} ⟶ C_{m}^{−} given explicitly by composing the maps

ϕ_{n,n−1}^{−} : (x, y) ⟶ (\frac{1}{2^{p−1}} p−1 ; ∑_{k=0}\binom{p}{2k}x^{p−2k}(x^{2} − 4)^{k}, y) .

Note that the action of the Galois group G_{ℚ(t)} commutes with the maps ϕ_{n,m}, ϕ_{n,m}^{−}. Letting J_{n} and J_{n}^{−} denote the Jacobians of C_{n} and C_{n}^{−} respectively, the induced maps (ϕ_{n,m})_{∗} : V_{p}(J_{n}) ⟶ V_{p}(J_{m}) make the extended Tate modules into a compatible system of G_{ℚ(t)}-modules (similarly for V_{p}(J_{n}^{−}) with the maps (ϕ_{n,m}^{−})_{∗}).

Since γ_{n} does not commute with τ_{n}, it does not give rise to an automorphism of C_{n}^{−}. However, the endomorphism γ_{n} +γ_{n}^{−1} of J_{n} does commute with τ_{n}, and gives rise to endomorphisms of J_{n}^{−}. Let π_{n} : C_{n} ⟶ C_{n}^{−} be the natural map. From the proof of Proposition 4.4, the kernel of π_{n}^{∗} : J_{n}^{−} ⟶ J_{n} is contained in J_{n}^{−}[ degπ_{n}] = J_{n}^{−}[2].

Proposition 4.15 For each γ ∈ μ_{p^{n}}, there is an endomorphism (γ + γ^{−1})^{−} of J_{n}^{−} such that π_{n}^{∗} ∘ (γ + γ^{−1})^{−} = (γ + γ^{−1}) |_{Imπ_{n}^{∗}}.

Proof: See [TTV91], §3.1. □

When it is clear from the context that we are referring to endomorphisms of J_{n}^{−}, we will write γ + γ^{−1} in place of (γ + γ^{−1})^{−}.

The action of the full Galois group G_{ℚ(t)} does not commute with the action of μ_{p^{n}}; however, if we restrict to the subgroup G_{ℚ(μ_{p^{∞}},t)}, then these actions do commute, so the action of G_{ℚ(μ_{p^{∞}},t)} on V_{p}(J_{n}) is ℚ_{p}[μ_{p^{n}}]-linear. In order to obtain 2-dimensional representations of Π_{ℚ(μ_{p^{∞}})} on V_{p}(J_{n}), we must show that V_{p}(J_{n}) is a free ℚ_{p}[μ_{p^{n}}]-module of rank two. First we will show that if V_{p}(J_{n}) is indeed a free ℚ_{p}[μ_{p^{n}}]-module, then it must have rank two.

Proposition 4.16 The dimension of J_{n} is p^{n}, and the dimension of J_{n}^{−} is \frac{p^{n}−1}{2}.

Proof: Since the dimension of the Jacobian of a curve is equal to the genus of the curve, we must calculate the genera of C_{n} and C_{n}^{−}. Both may be computed using the Riemann-Hurwitz formula. For example, let

h : C_{n} (\overline{ℚ(t)}) ⟶ ℙ^{1} (\overline{ℚ(t)})

be the degree two map taking (x, y) to x. Then h is ramified only at ∞ and the roots of x (x^{2p^{n}} + (4t − 2)x^{p^{n}} + 1), which are distinct. Thus h has 2p^{n} +2 ramification points, each having index two, so the Riemann-Hurwitz formula gives

2 genus(C_{n}) − 2 = 2 (2 genus(ℙ^{1}) − 2) + 2p^{n} + 2,

and therefore, genus(C_{n}) = p^{n}. □

To show that V_{p}(J_{n}) is free over ℚ_{p}[μ_{p^{n}}], we will need the following lemmas:

Lemma 4.17 Let K be a field whose characteristic is not equal to p, and let C/K be a curve with Jacobian J. Suppose that ξ is a nontrivial automorphism of C having a fixed point P_{ξ} ∈ C. Then the automorphism of T_{p}(J) induced from ξ is also nontrivial.

Proof: Let f^{P_{ξ}} : C ⟶ J be the embedding of Theorem 4.3. For any point Q ∈ C not fixed by ξ, we have

ξf^{P_{ξ}}(Q) = ξ_{∗}[Q − P_{ξ}] = [ξ_{∗}Q] − [P_{ξ}] = f^{P_{ξ}}(ξQ),

and f^{P_{ξ}}(ξQ) ≠ f^{P_{ξ}}(Q) since f^{P_{ξ}} is injective. Therefore ξ_{∗} is a nontrivial automorphism of J. The result now follows from the fact that for any abelian varieties A and B over a field K of characteristic not equal to p, the natural map

Hom(A, B) ⟶ Hom_{ℤ_{p}−mod} (T_{p}(A), T_{p}(B))

is injective (see [Mil86a], Lemma 12.2). □

Lemma 4.18 Let G be a finite group acting as automorphisms on a curve C/ℂ(t) with Jacobian J. Then for some ℚ_{p}[G]-module M, V_{p}(J) is isomorphic to M^{2} as a ℚ_{p}[G]-module.

Proof: Let α ∈ ℂ be a point at which C has good reduction, and let J_{α} denote the reduction of J at t = α. Since char(ℂ) = 0, for each m ∈ ℕ the reduction map r_{α} : J ⟶ J_{α} restricts to an isomorphism from the m-torsion points of J fixed by any given inertia group above t = α to the m-torsion points of J_{α} (see [ST68], Lemma 2). Since J has good reduction at t = α, each inertia group above t = α acts trivially on T_{p}(J), and therefore r_{α} induces an isomorphism T_{p}(J) ≅ T_{p}(J_{α}). Thus it suffices to prove the result when J is replaced with J_{α}, where the action of G on J_{α} is induced from J via r_{α}.

Let g be the dimension of J_{α}, and let Λ be a lattice of ℂ^{g} such that ℂ^{g}/Λ ≅ J_{α} as in Theorem 4.2. The action of G on J_{α} lifts to a linear action on ℂ^{g} fixing Λ. Let {λ_{1}, …, λ_{2g}} be a ℤ-basis for Λ. Reordering the λ_{j}’s if necessary, we may assume that {λ_{1}, …, λ_{g}} and {λ_{g+1}, …, λ_{2g}} are ℂ-bases for ℂ^{g}. Since {λ_{1} ⊗ 1, …, λ_{2g} ⊗ 1} is a ℂ-basis for Λ ⊗ ℂ, the representation of G on Λ ⊗ ℂ is isomorphic to two copies of that on ℂ^{g}.

On the other hand, identifying J_{α}[ p^{n}] with \frac{1}{p^{n}} Λ/Λ, there is a canonical ℤ_{p}-module isomorphism T_{p}(J_{α}) ≅ Λ ⊗ ℤ_{p} commuting with the action of G, given by

( 2g ; ∑_{j=1}a_{j,n} \frac{λ_{j}}{p^{n}} )_{n∈ℕ} ⟼ 2g ; ∑_{j=1}λ_{j} ⊗ (a_{j,n})_{n∈ℕ} ,

where each a_{j,n} ∈ ℤ/p^{n}ℤ. Let χ_{V} be the character corresponding to the representation of G on V := V_{p}(J_{α}) ≅ ℚ_{p}^{2g}, and χ_{W} the character corresponding to the representation of G on W := ℂ^{g}. From above, there is a ℂ[G]-module isomorphism Λ ⊗ ℂ ≅ W^{2}, so the character corresponding to the representation of G on Λ ⊗ ℂ is 2χ_{W}. Since G acts on the free ℤ-module Λ, χ_{W} must take values in ℚ, and thus 2χ_{W} is the character obtained from the representation of G on Λ ⊗ ℚ, and hence also from that on Λ ⊗ ℚ_{p} ≅ V. Therefore, χ_{V} is equal to 2χ_{W}. □

Proposition 4.19 The extended Tate module V_{p}(J_{n}) is a free module of rank two over ℚ_{p}[μ_{p^{n}}].

Proof: Let χ be an irreducible character of μ_{p^{n}} over ℚ_{p}. Over L = ℚ_{p}(ζ_{n}), χ decomposes as a sum of 1-dimensional characters χ_{1}, …, χ_{r}. The characters χ_{1}, …, χ_{r} form a Galois conjugacy class over ℚ_{p}, and each appears with multiplicity one (see [Isa94], Theorem 9.21). On the other hand, given any irreducible character χ̃ of μ_{p^{n}} over L, there is a unique irreducible character χ of μ_{p^{n}} over ℚ_{p} which has χ̃ as a constituent when lifted to L (see [Isa94], Corollary 9.7). Therefore, the irreducible characters χ of μ_{p^{n}} over ℚ_{p} are precisely those of the form

χ = ∑_{σ∈Gal(ℚ_{p}(ζ_{k})/ℚ_{p})}χ̃^{σ},

where χ̃ is a 1-dimensional character of μ_{p^{n}} over ℚ_{p}(ζ_{k}) which is not defined over ℚ_{p}(ζ_{k−1}). Fixing a generator γ_{n} of μ_{p^{n}}, such a character χ̃ is determined by χ̃(γ_{n}), which is a primitive p^{k}th root of 1. Moreover, if χ̃′ is any other character of μ_{p^{n}} defined over ℚ_{p}(ζ_{k}) but not over ℚ_{p}(ζ_{k−1}), then χ̃′(γ_{n}) is also a primitive p^{k}th root of 1, and hence there is some σ ∈ Gal (ℚ_{p}(ζ_{k})/ℚ_{p}) for which χ̃′ = χ̃^{σ}. Therefore, the irreducible characters of μ_{p^{n}} over ℚ_{p} are in one-to-one correspondence with the factor groups of μ_{p^{n}}, and are given by χ_{0}, …, χ_{n}, where χ_{0} is the trivial character, and χ_{j} has dimension p^{j} − p^{j−1} for each j = 1, …, n.

When n = 0, V_{p}(J_{0}) has rank two over ℚ_{p} by Proposition 4.16. Suppose for induction that V_{p}(J_{n−1}) ≅ ℚ_{p}[μ_{p^{n−1}}]^{2} as ℚ_{p}[μ_{p^{n−1}}]-modules. By Proposition 4.4, the map ϕ_{n,n−1} : C_{n} ⟶ C_{n−1} induces an injection

(ϕ_{n,n−1}^{∗})_{p} : V_{p}(J_{n−1}) ↪ V_{p}(J_{n}).

Since γ_{n} acts on the image of V_{p}(J_{n−1}) as a generator γ_{n−1} of μ_{p^{n−1}}, (ϕ_{n,n−1}^{∗})_{p} gives an inclusion ℚ_{p}[μ_{p^{n−1}}]^{2} ↪ V_{p}(J_{n}) of ℚ_{p}[μ_{p^{n}}]-modules. Since γ_{n}^{p^{n−1}} acts nontrivially on C_{n}, by Lemma 4.17, the automorphism induced from γ_{n}^{p^{n−1}} on V_{p}(J_{n}) also acts nontrivially, and therefore μ_{p^{n}} acts faithfully on V_{p}(J_{n}). From above, there is only one irreducible representation of μ_{p^{n}} over ℚ_{p} which does not factor through μ_{p^{n−1}}, namely that having the character χ_{n}; therefore, the ℚ_{p}[μ_{p^{n}}]-module M corresponding to χ_{n} must appear as a summand of V_{p}(J_{n}) (as a ℚ_{p}[μ_{p^{n}}]-module). By Lemma 4.18, two copies of M must appear, so there is an isomorphic copy of ℚ_{p}[μ_{p^{n−1}}]^{2} ⊕ M^{2} contained in V_{p}(J_{n}). Now ℚ_{p}[μ_{p^{n−1}}] ⊕ M is a direct sum of all irreducible ℚ_{p}[μ_{p^{n}}]-modules, and hence is isomorphic to ℚ_{p}[μ_{p^{n}}]. By Proposition 4.16, V_{p}(J_{n}) has ℚ_{p}-dimension 2p^{n}, so the inclusion of ℚ_{p}[μ_{p^{n}}]^{2} in V_{p}(J_{n}) is an isomorphism. □ Remark: For each choice of n-tuple

ζ = (1, ζ_{1}, …, ζ_{n}) ∈ {1} × μ_{p} × ⋯ × μ_{p^{n}}

in which each ζ_{j} is a primitive p^{j}th root of unity in \overline{ℚ}_{p}, there is a ring isomorphism

ℚ_{p}[μ_{p^{n}}] ≅ ℚ_{p} ⊕ ℚ_{p}(ζ_{1}) ⊕ ⋯ ⊕ ℚ_{p}(ζ_{n})

given by mapping γ_{n} to ζ. This isomorphism arises through the isomorphism ℚ_{p}[μ_{p^{n}}] ≅ ℚ_{p}[T]/ (T^{p^{n}} − 1) which takes γ_{n} to T. Factoring T^{p^{n}}−1 and applying the Chinese remainder theorem gives the isomorphism above. Choosing a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}), we obtain an isomorphism

V_{p}(J_{n}) ≅ ℚ_{p}^{2} ⊕ ℚ_{p}(ζ_{1})^{2} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{n})^{2}

of ℚ_{p}[μ_{p^{n}}]-modules.

By Proposition 4.4, we may view V_{p}(J_{n}^{−}) and V_{p}(J_{k}) as lying inside V_{p}(J_{n}) whenever k < n.

Lemma 4.20 The intersection of V_{p}(J_{n}^{−}) with V_{p}(J_{0}) in V_{p}(J_{n}) is trivial.

Proof: Since ϕ_{n,0}(P) = ϕ_{n,0}(Q) if and only if P = γ_{n}^{j}Q for some j, V_{p}(J_{0}) is contained in the submodule V_{p}(J_{n})^{μ_{p^{n}}} of elements of V_{p}(J_{n}) fixed by μ_{p^{n}}. Similarly, V_{p}(J_{n}^{−}) is contained in V_{p}(J_{n})^{⟨τ_{n}⟩}. Now ϕ_{n,0} ∘ τ_{n} = τ_{0} ∘ ϕ_{n,0}, so the action of τ_{n} on V_{p}(J_{n}) restricts to the action of τ_{0} on V_{p}(J_{0}). Note that τ_{0} acts nontrivially on every point (x, y) ∈ C_{0} for which x ≠ ±1, and fixes the points where x = ±1. By Lemma 4.17, ⟨τ_{0}⟩ and hence also ⟨τ_{n}⟩ act faithfully on V_{p}(J_{0}). Let D_{n} = ⟨γ_{n}, τ_{n}⟩, so that V_{p}(J_{n})^{D_{n}} = V_{p}(J_{n})^{μ_{p^{n}}} ∩V_{p}(J_{n})^{⟨τ_{n}⟩}. Suppose that V_{p}(J_{n})^{D_{n}} is nontrivial; then by Lemma 4.18, it has ℚ_{p}-dimension at least two. On the other hand, by Proposition 4.19, V_{p}(J_{n})^{μ_{p^{n}}} has ℚ_{p}-dimension two, so we must have V_{p}(J_{n})^{D_{n}} = V_{p}(J_{n})^{μ_{p^{n}}}, contradicting that ⟨τ_{n}⟩ acts faithfully on V_{p}(J_{0}) ⊂ V_{p}(J_{n})^{μ_{p^{n}}}. □

Proposition 4.21 The ℚ_{p}-vector space V_{n} := V_{p}(J_{n}^{−})⊕V_{p}(J_{0}) ⊂ V_{p}(J_{n}) is a free ℚ_{p}[γ_{n} + γ_{n}^{−1}]-module of rank two. Moreover, two elements b_{0}, b_{1} ∈ V_{p}(J_{n}) form a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-basis for V_{n} if and only if they are elements of V_{n} and they form a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}).

Proof: Suppose that {b_{0}, b_{1}} ⊂ V_{n} is a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}). Then the set B = \{(γ_{n}^{j} + γ_{n}^{−j})b_{l}\}_{j=0,... , \frac{p^{n}−1}{2} ; l=0,1} ⊂ V_{n} is linearly independent over ℚ_{p}, and thus generates a ℚ_{p}-vector space of dimension p^{n} + 1 contained in V_{n}. By Proposition 4.16, V_{n} has ℚ_{p}-dimension 2 (\frac{p^{n}−1}{2}) + 2 = p^{n} + 1, so B is a ℚ_{p}-basis for V_{n}; in particular, b_{0}, b_{1} generate V_{n} over ℚ_{p}[γ_{n} + γ_{n}^{−1}]. Furthermore, since b_{0}, b_{1} are linearly independent over ℚ_{p}[μ_{p^{n}}], they must also be linearly independent over ℚ_{p}[γ_{n} + γ_{n}^{−1}]. Therefore, {b_{0}, b_{1}} forms a basis for V_{n} over ℚ_{p}[γ_{n} + γ_{n}^{−1}]. Thus to prove the first statement it suffices to show that there exists a ℚ_{p}[μ_{p^{n}}]-basis {b_{0}, b_{1}} for V_{p}(J_{n}) which is contained in V_{n}.

Note that D_{n} = ⟨γ_{n}, τ_{n}⟩ is isomorphic to the dihedral group of order 2p^{n}. The irreducible characters of D_{n} over \overline{ℚ} consist of the trivial character, the nontrivial irreducible character of D_{n}/⟨τ_{n}⟩, and \frac{p^{n}−1}{2} characters of dimension two each taking the value 0 at τ_{n} (see [JL93], §18.3). Since τ_{n} has order 2, it must have eigenvalues 1 and −1 under each of the two-dimensional irreducible representations. From Lemma 4.20, the representation of D_{n} on V_{p}(J_{0}) = V_{p}(J_{n})^{μ_{p^{n}}} consists of two copies of the nontrivial one-dimensional representation. Each irreducible summand of the representation of D_{n} on V_{p}(J_{n})/V_{p}(J_{0}) decomposes over \overline{ℚ}_{p} as a sum of the two-dimensional representations of D_{n}, and the subspace of τ_{n}-fixed points of each of these has dimension one. Thus the subspace (V_{p}(J_{n})/V_{p}(J_{0}))^{τ_{n}} of V_{p}(J_{n})/V_{p}(J_{0}) has dimension \frac{2p^{n}−2}{2} = p^{n} − 1, and contains V_{p}(J_{n}^{−}). Since V_{p}(J_{n}^{−}) itself has dimension p^{n} − 1, we have V_{p}(J_{n}^{−}) = V_{p}(J_{n})^{τ_{n}}. Now

V_{p}(J_{n−1}) ∩ V_{p}(J_{n}^{−}) = V_{p}(J_{n−1})^{τ_{n}} = V_{p}(J_{n−1})^{τ_{n−1}} = V_{p}(J_{n−1}^{−}),

so V_{p}(J_{n}^{−}) ∩ V_{n} = V_{n−1}. Fix a ℚ_{p}[μ_{p^{n}}]-module isomorphism

V_{p}(J_{n}) ≅ ℚ_{p}^{2} ⊕ ℚ_{p}(ζ_{1})^{2} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{n})^{2},

and identify each subspace with its image in ℚ_{p}^{2} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{n})^{2}. Since V_{p}(J_{n−1}) = ℚ_{p}^{2} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{n−1})^{2}, we have V_{n} = V_{n−1} ⊕ (ℚ_{p}(ζ_{n})^{2} ∩ V_{n}) as a ℚ_{p}-vector space, and hence

dim_{ℚ_{p}} (ℚ_{p}(ζ_{n})^{2} ∩ V_{n}) = dim_{ℚ_{p}}V_{n} − dim_{ℚ_{p}}V_{n−1} = p^{n} − p^{n−1}.

We now proceed by induction. When n = 0, V_{0} = V_{p}(J_{0}), and therefore contains a ℚ_{p}-basis for V_{p}(J_{0}). Suppose for induction that V_{n−1} contains a ℚ_{p}[μ_{p^{n−1}}]-basis {b_{0,n−1}, b_{1,n−1}} for V_{p}(J_{n−1}). If (ℚ_{p}(ζ_{n})^{2} ∩ V_{n}) contains a ℚ_{p}(ζ_{n})-basis {b_{0,n}, b_{1,n}} for ℚ_{p}(ζ_{n})^{2}, then {b_{0,n} + b_{0,n−1}, b_{1,n} + b_{1,n−1}} is a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}) contained in V_{n}, as desired. If not, then since

dim_{ℚ_{p}} (ℚ_{p}(ζ_{n})^{2} ∩ V_{n}) = p^{n} − p^{n−1} = dim_{ℚ_{p}}ℚ_{p}(ζ_{n}),

there must be some b ∈ ℚ_{p}(ζ_{n})^{2} ∩ V_{n} for which ℚ_{p}(ζ_{n})^{2} ∩ V_{n} = ℚ_{p}(ζ_{n}) b. But then ℚ_{p}(ζ_{n})^{2} ∩ V_{n} is an irreducible faithful ℚ_{p}[μ_{p^{n}}]-module on which τ_{n} acts trivially, which contradicts the fact that the only irreducible representation of D_{n} having τ_{n} in its kernel is the trivial one.

All that remains is to prove that if B′ = {b′_{0}, b′_{1}} is a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-basis for V_{p}(J_{n}^{−}) then it is a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}). Let B = {b_{0}, b_{1}} ⊂ V_{p}(J_{n}^{−}) be a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}). From above, B is also a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-basis for V_{p}(J_{n}^{−}), and therefore writing b′_{0} = α_{0,0}b_{0} + α_{1,0}b_{1}, b′_{1} = α_{0,1}b_{0} + α_{1,1}b_{1} with α_{i,j} ∈ ℚ_{p}[γ_{n} + γ_{n}^{−1}], the matrix M = (α_{i,j})_{0≤i,j≤1} is invertible. Extending scalars to ℚ_{p}[μ_{p^{n}}], M remains invertible, so B′ is indeed a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}). □

Since the action of G_{ℚ(μ_{p^{∞}},t)} on V_{p}(J_{n}) and V_{p}(J_{n}^{−}) commutes with the actions of ℚ_{p}[μ_{p^{n}}] and ℚ_{p}[γ_{n} + γ_{n}^{−1}] respectively, choosing a ℚ_{p}[μ_{p^{n}}]-basis B for V_{p}(J_{n}) contained in V_{p}(J_{n}^{−}), the action of G_{ℚ(μ_{p^{∞}},t)} on V_{p}(J_{n}) and V_{p}(J_{n}^{−}) respectively gives representations

ρ_{n} : G_{ℚ(μ_{p^{∞}},t)} ⟶GL_{2} (ℚ_{p}[μ_{p^{n}}])
and ρ_{n}^{−} : G_{ℚ(μ_{p^{∞}},t)} ⟶GL_{2} (ℚ_{p}[γ_{n} + γ_{n}^{−1}]) .

Since the basis is the same for both representations, ρ_{n} is simply the representation obtained from ρ_{n}^{−} by extending scalars to ℚ_{p}[μ_{p^{n}}]. Since the action of G_{ℚ(μ_{p^{∞}},t)} commutes with ϕ_{n,m}, the representations ρ_{n} are compatible with respect to the maps ℚ_{p}[μ_{p^{n}}] ⟶ ℚ_{p}[μ_{p^{m}}] taking γ_{n} to γ_{m} for each m ≤ n. We will show that, with respect to an appropriate basis, the image of ρ_{n} is contained in GL_{2} (ℤ_{p}[μ_{p^{n}}]), and thus we obtain a representation ρ^{hg} : G_{ℚ(μ_{p^{∞}},t)} ⟶ GL_{2} (\underleftarrow{lim} ℤ_{p}[μ_{p^{n}}]) ≅ GL_{2} (ℤ_{p}[[T]]).

4.6 The Reduction Type of C_{n}, C_{n}^{−} and the Associated Galois Representation

In order to understand the local behaviour of the Galois representation

ρ_{n} : G_{ℚ(μ_{p^{∞}},t)} ⟶ GL_{2} (ℚ_{p}[μ_{p^{n}}]) ,

we will consider the reduction type of C_{n} and C_{n}^{−} at various places. By the above discussion, ρ_{n} may be viewed as the representation associated to C_{n}^{−} by composing with the natural inclusion GL_{2} (ℚ_{p}[γ_{n} + γ_{n}^{−1}]) ↪ GL_{2} (ℚ_{p}[μ_{p^{n}}]), so we will no longer distinguish between the two representations. Thus we may obtain information about ρ_{n} by considering either C_{n} or C_{n}^{−}.

Proposition 4.22 As a curve over \overline{ℚ}(t), C_{n} has good reduction outside t = 0, 1, ∞.

Proof: For t ∈ \overline{ℚ}, the curve C_{n}(t)/\overline{ℚ} given by

C_{n}(t) : y^{2} = f(x) = x (x^{2p^{n}} + (4t − 2)x^{p^{n}} + 1)

is singular if and only if f(x) has a repeated root. The roots of f(x) are given by

x = 0, ζ_{n}^{j} (\frac{−(4t − 2) ± \sqrt{(4t − 2)^{2} − 4}}{2}) j = 0, …, p^{n} − 1.

Thus f(x) has a repeated root if and only if

−(4t − 2) + \sqrt{(4t − 2)^{2} − 4} = ζ_{n}^{j} (−(4t − 2) − \sqrt{(4t − 2)^{2} − 4})

for some j. Solving gives t = 0 or 1.

Whenever t ≠ 0, 1, ∞, one may apply the argument of the proof of Proposition 4.16 to show that the genus of C_{n}(t) is the same as that of C_{n}. □

Corollary 4.23 The representation ρ_{n} factors through Π_{ℚ(μ_{p^{∞}},t)}.

Proof: By Proposition 4.8, ρ_{n}^{geom} = ρ_{n}|_{Π\overline{ℚ}} is unramified outside t = 0, 1, ∞, and hence factors through the Galois group of the maximal algebraic extension ˆ \overline{ℚ}(t) of \overline{ℚ}(t) unramified outside t = 0, 1, ∞. The result now follows from Corollary 3.4 with K = ℚ(μ_{p^{∞}}). □

The residual representation ρ̄ of each ρ_{n} is the representation of Π_{ℚ(μ_{p^{∞}})} describing the action on the p-torsion points of the elliptic curve C_{0}. Let E_{L}/ℚ(t) denote the Legendre family of elliptic curves

E_{L} : y^{2} = x(x − 1)(x − t).

There is a 2-isogeny ϕ : C_{0} ⟶ E_{L} given by

ϕ : (x, y) ⟼ (−\frac{y^{2}}{4x^{2}} + t, \frac{iy(1 − x^{2})}{8x^{2}});

in particular, ρ̄^{geom} = ρ̄|_{Π\overline{ℚ}} is also the representation of Π\overline{ℚ} attached to the p-torsion points of E_{L}. In order to determine ρ̄^{geom} explicitly, we first need a lemma:

Lemma 4.24 Let E_{1} : y^{2} = x^{3} + ax^{2} + bx + c be an elliptic curve defined over a field K, and let ρ_{1} : G_{K} ⟶ GL_{2}(𝔽_{p}) be the Galois representation associated to E_{1}. For any d ∈ K^{×}, the twist

E_{2} : dy^{2} = x^{3} + ax^{2} + bx + c

of E_{1} has the associated Galois representation ρ_{2} = ρ_{1} ⊗ χ_{K(\sqrt{d})/K}, where

χ_{K(\sqrt{d})/K}(σ) = \begin{cases}1 & if σ fixes \sqrt{d}, \\ −1 & otherwise. \end{cases}

Proof: Fix an 𝔽_{p}-basis {(x_{0}, y_{0}), (x_{1}, y_{1})} for the p-torsion points of E_{1}. The map ϕ\sqrt{d} : E_{1} ⟶ E_{2} defined by ϕ\sqrt{d}(x, y) = (x, \frac{y}{\sqrt{d}}) is an isomorphism of elliptic curves, so \{(x_{0}, \frac{y_{0}}{\sqrt{d}}) , (x_{1}, \frac{y_{1}}{\sqrt{d}})\} is an 𝔽_{p}-basis for the p-torsion points of E_{2}; moreover, if σ · (x_{i}, y_{i}) = a_{0}(x_{0}, y_{0}) + a_{1}(x_{1}, y_{1}), then for σ ∈ G_{K},

σ · (x_{i}, \frac{y_{i}}{\sqrt{d}}) = a_{0} (x_{0}, \frac{y_{0}}{σ(\sqrt{d})}) + a_{1} (x_{1}, \frac{y_{1}}{σ(\sqrt{d})}) .

If σ(\sqrt{d}) = \sqrt{d}, then σ · (x_{i}, \frac{y_{i}}{\sqrt{d}}) = a_{0} (x_{0}, \frac{y_{0}}{\sqrt{d}}) + a_{1} (x_{1}, \frac{y_{1}}{\sqrt{d}}) for i = 0, 1. Otherwise, σ(\sqrt{d}) = −\sqrt{d}, and thus

σ · (x_{i}, \frac{y_{i}}{\sqrt{d}}) = a_{0} (x_{0}, −\frac{y_{0}}{\sqrt{d}}) + a_{1} (x_{1}, −\frac{y_{1}}{\sqrt{d}})
= −a_{0} (x_{0}, \frac{y_{0}}{\sqrt{d}}) − a_{1} (x_{1}, \frac{y_{1}}{\sqrt{d}}) ,

as desired. □ Proposition 4.25 The representation ρ̄^{geom} satisfies

ρ̄^{geom}(σ_{0}) ∼ (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and ρ̄^{geom}(σ_{1}) ∼ (\begin{matrix}1 & 0 \\ −4 & 1 \end{matrix}) .

Proof: At t = 0, C_{0} reduces to the curve

C_{0}(0) : y^{2} = x(x − 1)^{2},

whose only singularity is a node at the point (1, 0). Let N_{0} be the genus 0 curve defined by N_{0} : y^{2} = x. There is a birational map ϕ_{0} : C_{0}(0) ⟶ N_{0} given by

ϕ_{0} : (x, y) ⟶ (x, \frac{y}{x − 1}) ,

so C_{0}(0) is birationally equivalent to a projective line. Therefore, C_{0} is a Mumford curve at t = 0, and Corollary 4.12 gives ρ̄^{geom}(σ_{0}) ∼ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). A similar argument shows that ρ̄^{geom}(σ_{1}) ∼ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) as well.

For the reduction at t = ∞, consider the twist C′_{0} of C_{0} given by

C′_{0} : ty^{2} = x^{3} + (4t − 2)x^{2} + x.

Letting u = \frac{1}{t} and replacing x with \frac{u}{x} and y with \frac{uy}{x^{2}} gives the model

y^{2} = x^{3} + (4 − 2u)x^{2} + u^{2}x

for C′_{0}. At u = 0 (that is, at t = ∞), this model reduces to

y^{2} = x^{2}(x + 4),

which is a projective line with a nodal singularity at the point (0, 0); therefore, C′_{0} is a Mumford curve at t = ∞. Since σ_{∞}(\sqrt{t}) = −\sqrt{t}, Lemma 4.24 gives

ρ̄^{geom}(σ_{∞}) ∼ ( \begin{matrix}−1 & −1 \\ 0 & −1 \end{matrix} ).

Fixing an 𝔽_{p}-basis for the p-torsion points of C_{0} with respect to which ρ̄^{geom}(σ_{0}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), we cannot have ρ̄^{geom}(σ_{1}) = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) since ρ̄^{geom}(σ_{∞}) has order 2p. Thus changing basis if necessary, we may assume that ρ̄^{geom}(σ_{1}) = ( \begin{matrix}1 & 0 \\ α & 1 \end{matrix} ) for some α ∈ 𝔽_{p}^{×}. Multiplying gives

ρ̄^{geom}(σ_{∞}) = (ρ̄^{geom}(σ_{0})ρ̄^{geom}(σ_{1}))^{−1} = (\begin{matrix}1 & −1 \\ −α & 1 + α \end{matrix});

in particular, tr ρ̄^{geom}(σ_{∞}) = 2 + α. On the other hand, tr ρ̄^{geom}(σ_{∞}) = −2 since ρ̄^{geom}(σ_{∞}) ∼ ( \begin{matrix}−1 & −1 \\ 0 & −1 \end{matrix} ), and thus α = −4. □

To examine the reduction type of C_{n}^{−} at t = 0, 1, we will use some identities which appear in [Dar00], p.420.

Lemma 4.26 Let g_{n}(x) be as in Proposition 4.13. Then

xg_{n}(x^{2} − 2) = g_{n}(−x)^{2}(x − 2) + 2 = g_{n}(x)^{2}(x + 2) − 2.

Proof: First we will show that xg_{n}(x^{2}−2)−2 = g_{n}(−x)^{2}(x−2). From (4.14), we have

x^{p^{n}} + x^{−p^{n}} = (x + x^{−1})g_{n}(x^{2} + x^{−2}).  (4.27)

Thus putting x = ζ_{n}^{j} + ζ_{n}^{−j} into xg(x^{2} − 2) − 2 gives

(ζ_{n}^{j} + ζ_{n}^{−j})g_{n}(ζ_{n}^{2j} + ζ_{n}^{−2j}) − 2 = (ζ_{n}^{j})^{p^{n}} + (ζ_{n}^{j})^{−p^{n}} − 2 = 0,

so ζ_{n}^{j} + ζ_{n}^{−j} is a root of xg_{n}(x^{2} − 2) − 2 for each j = 0, …, \frac{p^{n}−1}{2}. Since

g_{n}(x) = ^{∏_{j=1}^{\frac{p^{n}−1}{2}}}(x + ζ_{n}^{j} + ζ_{n}^{−j}), each ζ_{n}^{j} + ζ_{n}^{−j} is also a root of g_{n}(−x)^{2}(x − 2).

Taking x = −ζ_{n}^{j} − ζ_{n}^{−j} gives

−(ζ_{n}^{j} + ζ_{n}^{−j})g_{n}(ζ_{n}^{2j} + ζ_{n}^{−2j}) − 2 = −4,

and replacing j with j + p^{n} if necessary so that j is even,

g_{n}(ζ_{n}^{j} + ζ_{n}^{−j})^{2}(−ζ_{n}^{j} − ζ_{n}^{−j} − 2) = g_{n} ((ζ_{n}^{j/2})^{2} + (ζ_{n}^{j/2})^{−2})^{2} (−ζ_{n}^{j} − ζ_{n}^{−j} − 2)

= (\frac{(ζ_{n}^{j/2})^{p^{n}} + (ζ_{n}^{j/2})^{−p^{n}}}{ζ_{n}^{j/2} + ζ_{n}^{−j/2}} )^{2} (−ζ_{n}^{j} − ζ_{n}^{−j} − 2)

= \frac{4}{ζ_{n}^{j} + ζ_{n}^{−j} + 2}(−ζ_{n}^{j} − ζ_{n}^{−j} − 2) = −4,

so xg_{n}(x^{2} −2)−2 and g_{n}(−x)^{2}(x−2) also agree at the \frac{p^{n}+1}{2} points −ζ_{n}^{j} −ζ_{n}^{−j} for j = 0, …, \frac{p^{n}−1}{2}. Thus xg_{n}(x^{2} −2)−2 and g_{n}(−x)^{2}(x−2) are polynomials of degree p^{n} which agree at p^{n} + 1 points, and hence are equal.

Since xg_{n}(x^{2} − 2) is odd,

xg_{n}(x^{2} − 2) = −(−x)g_{n}((−x)^{2} − 2)
= − (g_{n} (−(−x))^{2} (−x − 2) + 2)
= g_{n}(x)^{2}(x + 2) − 2,

as desired. □

Proposition 4.28 The curve C_{n}^{−} is a Mumford curve at t = 0 and t = 1.

Proof: Using the identity xg_{n}(x^{2} − 2) = g_{n}(−x)^{2}(x − 2) + 2, we have

C_{n}^{−} : y^{2} = g_{n}(−x)^{2}(x − 2) + 4t.

At t = 0, this reduces to the curve C_{n}^{−}(0) : y^{2} = g_{n}(−x)^{2}(x − 2), whose singularities consist of ordinary double points at (ζ_{n}^{j} + ζ_{n}^{−j}, 0) for each j = 1, …, \frac{p^{n}−1}{2}. The map from C_{n}^{−}(0) to N : y^{2} = x − 2 which takes (x, y) to (x, \frac{y}{g(−x)}) defines a birational equivalence between C_{n}^{−}(0) and a curve of genus zero, that is, a projective line.

The argument for t = 1 is similar, except that one uses instead the identity xg_{n}(x^{2} − 2) = g_{n}(x)^{2}(x + 2) − 2. □

To calculate the image of the inertia group at t = ∞, we view C_{n}^{−} as being defined over the field \overline{ℚ} ((\frac{1}{t})).

Proposition 4.29 The curve C_{n}^{−} acquires good reduction at t = ∞ over the field \overline{ℚ} (((\frac{1}{t})^{1/2p^{n}})).

Proof: Let u = (\frac{1}{t})^{1/2p^{n}}. Consider the curve C̃_{n}^{−}/\overline{ℚ}((u)) given by

C̃_{n}^{−} : y^{2} = x^{∏_{j=1}^{\frac{p^{n}−1}{2}}} (1 + (ζ_{n}^{j} + ζ_{n}^{−j} − 2)u^{4}x^{2}) + (4 − 2u^{2p^{n}})x^{p^{n}+1}.

There is an isomorphism ϕ : C̃_{n}^{−} ⟶ C_{n}^{−} defined over \overline{ℚ}((u)) given by

ϕ : (x, y) ⟼ (\frac{1}{u^{2}x}, \frac{y}{u^{p^{n}}x^{\frac{p^{n}+1}{2}}} ) .

Reducing C̃_{n}^{−} at u = 0 (that is, at t = ∞) gives the nonsingular curve

C̃_{n}^{−}(∞) : y^{2} = 4x^{p^{n}+1} + x,

which has genus \frac{p^{n}−1}{2}. □

Corollary 4.30 The inertia group I_{∞} ⊂ Π\overline{ℚ} at t = ∞ is mapped by ρ_{n} to a subgroup of GL_{2} (ℚ_{p}[μ_{p^{n}}]) of order dividing 2p^{n}.

Proof: By Proposition 4.8, the restriction of ρ_{n} to I_{∞} factors through the Galois group Gal (\overline{ℚ} (((\frac{1}{t})^{1/2p^{n}})) / \overline{ℚ} ((\frac{1}{t}))), which has order 2p^{n}. □

We were not able to give an elementary proof that ρ_{n}(I_{∞}) has order exactly 2p^{n}, because of the difficulty in understanding in general when a curve with bad reduction at a particular place may have a Jacobian with good reduction at that place. This result will follow, however, from a general construction of Katz.

4.7 A Theorem of Katz

Let α_{1}, α_{2}, β_{1}, β_{2} ∈ ℚ be such that α_{i} −β_{j} is not an integer for any i, j = 1, 2. Suppose that

κ : Π\overline{ℚ} ⟶ GL_{2} (ℚ_{p}(ζ_{n}))

is such that κ(σ_{0}) has eigenvalues e^{2πiα_{1}}, e^{2πiα_{2}}, κ(σ_{1}) has repeated eigenvalue 1, and κ(σ_{∞}) has eigenvalues e^{2πiβ_{1}}, e^{2πiβ_{2}}. According to a theorem of Belyĭ, such a representation is unique up to conjugation by an element of GL_{n} (ℚ_{p}(ζ_{n})). To be precise, Belyĭ’s theorem asserts that for any field k, if M_{0}, M_{1} ∈ GL_{n}(k) generate an irreducible subgroup of GL_{n}(k), and one of M_{0}, M_{1}, or (M_{0}M_{1})^{−1} differs from a scalar matrix by a matrix of rank one, then (M_{0}, M_{1}, (M_{0}M_{1})^{−1}) is rigid in GL_{n}(k) (see [Bel80], Theorem 2).

Note that if A/\overline{ℚ}(t) is an abelian variety of dimension p^{n} − p^{n−1} which contains ℤ[ζ_{n}] in its endomorphism ring, then V_{p}(A) is a vector space of dimension 2 over ℚ_{p}(ζ_{n}); if, moreover, A has good reduction outside t = 0, 1, ∞, then the action of Π\overline{ℚ} on V_{p}(A) gives rise to a 2-dimensional representation of Π\overline{ℚ} over ℚ_{p}(ζ_{n}). Katz’ theorem realizes κ as the representation associated to such an abelian variety A defined over ℚ(t).

Let N be a common denominator for α_{1}, α_{2}, β_{1}, β_{2}, and let A(j) = Nα_{j}, B(j) = Nβ_{j} ∈ ℤ for each j = 1, 2. The nonsingular curve D̃/ℚ(t) defined by

D̃ : \begin{cases}x_{1}^{N} = y_{1}^{A(1)}(1 − y_{1})^{B(1)−A(1)} \\ x_{2}^{N} = y_{2}^{A(2)}(1 − y_{2})^{B(2)−A(2)} \\ y_{1}y_{2} = t \end{cases}

possesses a natural action of μ_{N} × μ_{N} ⊂ ℚ̄ × ℚ̄ by

(ζ_{N}^{j}, ζ_{N}^{l}) · (x_{1}, y_{1}, x_{2}, y_{2}) = (ζ_{N}^{j}x_{1}, y_{1}, ζ_{N}^{l}x_{2}, y_{2})

for each j, l ∈ ℤ/p^{n}ℤ, (x_{1}, y_{1}, x_{2}, y_{2}) ∈ D̃(\overline{ℚ(t)}), where ζ_{N} is a primitive Nth root of unity. Defining a character

χ : μ_{N} × μ_{N} ⟶ \overline{ℚ}^{×}
(ζ_{N}^{j}, ζ_{N}^{l}) ⟼ ζ_{N}^{j+l},

ker(χ) is the subgroup of μ_{N} ×μ_{N} consisting of elements of the form (ζ_{N}^{j}, ζ_{N}^{−j}). Let D be the quotient of D̃ by the group of automorphisms ker(χ).

Theorem 4.31 The Jacobian of D has a quotient A of dimension p^{n} − p^{n−1} whose endomorphism ring contains ℤ[ζ_{n}], and whose associated representation of Π\overline{ℚ} is κ.

Proof: See [Kat90], Theorem 5.4.4. □ Remark: Theorem 5.4.4 of [Kat90] more generally gives geometric constructions of n-dimensional representations

κ : Π\overline{ℚ} ⟶ GL_{n} (ℚ_{p}(ζ_{n}))

for any choice of eigenvalues e^{2πiα_{1}}, …, e^{2πiα_{n}} of κ(σ_{0}), and e^{2πiβ_{1}}, …, e^{2πiβ_{n}} of κ(σ_{∞}), where α_{1}, …, α_{n}, β_{1}, …, β_{n} ∈ ℚ satisfy the condition that α_{i} − β_{j} is not an integer for any i, j; again κ(σ_{1}) has repeated eigenvalue 1.

In order to show that ρ_{n}(σ_{∞}) has order 2p^{n}, we use Theorem 4.31 with α_{1} = α_{2} = 0, β_{1} = \frac{1}{2p^{n}}, and β_{2} = −\frac{1}{2p^{n}}; thus we will construct a representation

κ_{n} : Π\overline{ℚ} ⟶ GL_{2} (ℚ_{p}(ζ_{n}))

such that κ_{n}(σ_{0}), κ_{n}(σ_{1}) each have repeated eigenvalue 1, and κ_{n}(σ_{∞}) has eigenvalues −ζ_{n}, −ζ_{n}^{−1}, where ζ_{n} is a primitive p^{n}th root of unity. Taking N = 2p^{n}, we obtain A(1) = A(2) = 0, B(1) = 1, and B(2) = −1. Let D̃_{n}/\overline{ℚ}(t) be the curve defined by

D̃_{n} : \begin{cases}X_{1}^{2p^{n}} = 1 − Y_{1} \\ X_{2}^{2p^{n}} = (1 − Y_{2})^{−1} \\ Y_{1}Y_{2} = t. \end{cases}  (4.32)

With χ as above, let D_{n} denote the quotient curve D̃_{n}/ker(χ). The subfield of the function field of D̃_{n} consisting of those elements invariant under ker(χ) is generated over \overline{ℚ(t)} by

y = Y_{1} = t/Y_{2} and x = X_{1}X_{2}.

From (4.32), we see that a model for D_{n} is given by

D_{n} : x^{2p^{n}} = (1 − y) (1 − \frac{t}{y} )^{−1} .

For each n, let D_{n}^{∘} be the curve defined by

D_{n}^{∘} : x^{p^{n}} = (1 − y) (1 − \frac{t}{y} )^{−1} .

Let K_{n} and K_{n}^{∘} denote the Jacobians of D_{n} and D_{n}^{∘} respectively. Let π_{n} : D_{n} ⟶ D_{n−1} be the morphism mapping (x, y) to (x^{p}, y), and let π_{n}^{∘} : D_{n} ⟶ D_{n}^{∘} be the morphism mapping (x, y) to (x^{2}, y).

There is a natural action of μ_{p^{n}} on D_{n} and on D_{n}^{∘} by γ_{n}(x, y) = (ζ_{n}x, y), which satisfies the relations

π_{n} ∘ γ_{n} = γ_{n−1} ∘ π_{n} and π_{n}^{∘} ∘ γ_{n} = γ_{n}^{2} ∘ π_{n}^{∘},

provided that the generators γ_{n} of each μ_{p^{n}} are chosen so that ζ_{n}^{p} = ζ_{n−1}. Thus V_{p}(K_{n}), V_{p}(K_{n}^{∘}), and V_{p}(K_{n−1}) are ℚ_{p}[μ_{p^{n}}]-modules, and the morphisms π_{n}, π_{n}^{∘} induce ℚ_{p}[μ_{p^{n}}]-module inclusions

π_{n}^{∗} : V_{p}(K_{n−1}) ↪ V_{p}(K_{n}) and (π_{n}^{∘})^{∗} : V_{p}(K_{n}^{∘}) ↪ V_{p}(K_{n}).

On the other hand, D_{n} is related to C_{n} by the morphism

ψ_{n} : D_{n} ⟶ C_{n}
(x, y) ⟼ (x^{2}, x^{p^{n}+1} (\frac{2(y + ty^{−1} − 1) + x^{2p^{n}} + x^{−2p^{n}}}{x^{p^{n}} + x^{−p^{n}}} )) .

We define abelian varieties

A_{n} := K_{n}/ (π_{n}^{∗}(K_{n−1}) + (π_{n}^{∘})^{∗}(K_{n}^{∘})) ,
J_{n}^{new} := J_{n}/ϕ_{n,n−1}^{∗}(J_{n−1}).

Let p_{n} : K_{n} ⟶ A_{n} be the natural projection.

Proposition 4.33 The abelian variety A_{n} is the quotient of K_{n} of Theorem 4.31, and the map p_{n} ∘ ψ_{n}^{∗} : J_{n} ⟶ A_{n} induces a ℚ_{p}(ζ_{n})-vector space isomorphism V_{p}(J_{n}^{new}) ≅ V_{p}(A_{n}) which commutes with the action of Π\overline{ℚ}. In particular, the eigenvalues of σ_{∞} as a ℚ_{p}(ζ_{n})-linear map on V_{p}(J_{n}^{new}) are −ζ_{n}, −ζ_{n}^{−1}.

Proof: A computation using the Riemann-Hurwitz formula shows that the genus of D_{n} is 2p^{n} − 1, and that of D_{n}^{∘} is p^{n} − 1. If we show that

(π_{n}^{∘})^{∗} (V_{p}(K_{n}^{∘})) ⋂ ψ_{n}^{∗} (V_{p}(J_{n})) = {0},

then counting ℚ_{p}-dimensions, we must have

V_{p}(K_{n}) ≅ V_{p}(K_{n}^{∘}) ⊕ V_{p}(J_{n}).  (4.34)

Let σ be the involution of D_{n} which maps (x, y) to (−x, y), so that D_{n}^{∘} = D_{n}/⟨σ⟩. Then σ_{∗} : K_{n} ⟶ K_{n} fixes each point in (π_{n}^{∘})^{∗}(K_{n}^{∘}). On the other hand, letting h denote the hyperelliptic involution h : (x, y) ⟼ (x, −y) on C_{n}, we have the relation ψ_{n} ∘σ = h∘ψ_{n}. Since h_{∗} acts as −1 on J_{n}, σ_{∗} acts as −1 on ψ_{n}^{∗}(J_{n}); in particular, σ_{∗} acts nontrivially on every element of ψ_{n}^{∗}(J_{n}) which does not have order 2. Therefore, (π_{n}^{∘})^{∗}(K_{n}^{∘}) ∩ ψ_{n}^{∗}(J_{n}) is contained in K_{n}[2], and in particular (π_{n}^{∘})^{∗} (V_{p}(K_{n}^{∘})) ∩ ψ_{n}^{∗} (V_{p}(J_{n})) = {0}, as desired.

A similar analysis to that in the proof of Proposition 4.19 shows that there is a ℚ_{p}[μ_{p^{n}}]-module isomorphism

V_{p}(K_{n}^{∘}) ≅ ℚ_{p}(ζ_{n})^{2} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{1})^{2}.

Thus by (4.34), we have

V_{p}(K_{n}) ≅ ℚ_{p}(ζ_{n})^{4} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{1})^{4} ⊕ ℚ_{p}^{2}

for each n ≥ 1. Since π_{n}^{∗} and (π_{n}^{∘})^{∗} are ℚ_{p}[μ_{p^{n}}]-module inclusions,

π_{n}^{∗} (V_{p}(K_{n−1})) + (π_{n}^{∘})^{∗} (V_{p}(K_{n}^{∘})) ≅ ℚ_{p}(ζ_{n})^{2} ⊕ ℚ_{p}(ζ_{n−1})^{4} ⊕ ⋯ ⊕ ℚ_{p}(ζ_{1})^{4} ⊕ ℚ_{p}^{2},

and therefore V_{p}(A_{n}) ≅ ℚ_{p}(ζ_{n})^{2}. Let K_{n}^{new} := K_{n}/π_{n}^{∗}(K_{n−1}), and let (K_{n}^{∘})^{new} := K_{n}^{∘}/(π_{n,n−1}^{∘})^{∗}(K_{n−1}^{∘}), where π_{n,n−1}^{∘} : D_{n}^{∘} ⟶ D_{n−1}^{∘} is given by π_{n,n−1}^{∘}(x, y) = (x^{p}, y); thus V_{p}(K_{n}^{new}) ≅ ℚ_{p}(ζ_{n})^{4} and V_{p} ((K_{n}^{∘})^{new}) ≅ ℚ_{p}(ζ_{n})^{2}. Note that D_{n}^{∘} is the curve constructed by Theorem 4.31 when α_{1} = α_{2} = 0, β_{1} = \frac{1}{p^{n}}, and β_{2} = \frac{−1}{p^{n}}. The abelian variety (K_{n}^{∘})^{new} is the only quotient of K_{n}^{∘} of dimension p^{n} − p^{n−1} which contains ℤ[ζ_{n}] in its endomorphism ring, so by Theorem 4.31, σ_{∞} has eigenvalues ζ_{n}, ζ_{n}^{−1} as a ℚ_{p}(ζ_{n})-linear automorphism of V_{p} ((K_{n}^{∘})^{new}).

Let K ⊂ K_{n}^{new} be such that K_{n}^{new}/K is the quotient of Theorem 4.31. Then V_{p}(K) ⊂ V_{p}(K_{n}^{new}) must contain the eigenvectors of σ_{∞} corresponding to the eigenvalues ζ_{n}, ζ_{n}^{−1}, for otherwise σ_{∞} would have at least three distinct eigenvalues as a ℚ_{p}(ζ_{n})-linear automorphism of the 2-dimensional vector space V_{p}(K_{n}^{new}). Therefore, V_{p}(K) = (π_{n}^{∘})^{∗} (V_{p}(K_{n}^{∘})), so A_{n} is the quotient of Theorem 4.31. Moreover, the inclusion ψ_{n}^{∗} composed with the natural map V_{p}(K_{n}) ⟶ V_{p}(A_{n}) induces a ℚ_{p}(ζ_{n})-vector space isomorphism

(ψ_{n}^{∗})^{new} : V_{p}(J_{n}^{new}) ⟶ V_{p}(A_{n}).

Since the isomorphism of (4.34) commutes with the action of Π\overline{ℚ}, so does (ψ_{n}^{∗})^{new}. □

Proposition 4.35 The ℚ_{p}[γ_{n} + γ_{n}^{−1}]-module V_{n} = V_{p}(J_{n}^{−}) ⊕ V_{p}(J_{0}) has a basis {b_{0}, b_{1}} with respect to which

ρ_{n}(σ_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and ρ_{n}(σ_{1}) = (\begin{matrix}1 & 0 \\ α_{n} & 1 \end{matrix})

for some α_{n} ∈ ℚ_{p}[γ_{n} + γ_{n}^{−1}]^{×}.

Proof: By Proposition 4.25, the representation ρ_{0}^{geom} = ρ_{0}|_{Π\overline{ℚ}} is given by

ρ_{0}(σ_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , ρ_{0}(σ_{1}) = (\begin{matrix}1 & 0 \\ α_{0} & 1 \end{matrix})

for some α_{0} ∈ ℤ_{p}^{×} reducing to −4 mod p. Thus by Proposition 4.28 and Corollary 4.12, the p-adic representation ρ̃_{n} : Π\overline{ℚ} ⟶ GL_{p^{n}+1}(ℚ_{p}) satisfies

ρ̃_{n}(σ_{i}) ∼ ( \begin{matrix}1 & 0 & … & 0 & ∗ & ∗ & … & ∗ & 0 & 0 \\ 0 & 1 & … & 0 & ∗ & ∗ & … & ∗ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & 1 & ∗ & ∗ & … & ∗ & 0 & 0 \\ 0 & 0 & … & 0 & 1 & 0 & … & 0 & 0 & 0 \\ 0 & 0 & … & 0 & 0 & 1 & … & 0 & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & 0 & 0 & 0 & … & 1 & 0 & 0 \\ 0 & 0 & … & 0 & 0 & 0 & … & 0 & 1 & 1 \\ 0 & 0 & … & 0 & 0 & 0 & … & 0 & 0 & 1 \end{matrix} )  (4.36)

for each i = 0, 1.

We now proceed by induction on n. When n = 0, the result follows from above. Assume that there is such a basis for V_{n−1}. Choosing a p^{n}th root of unity ζ_{n} ∈ \overline{ℚ}_{p} gives rise to an isomorphism

ℚ_{p}[γ_{n} + γ_{n}^{−1}] ≅ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) ⊕ ℚ_{p}[γ_{n−1} + γ_{n−1}^{−1}],

and thus also a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-module isomorphism

V_{n} ≅ ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2} ⊕ V_{n−1}.

By assumption, there is a basis {b_{0,n−1}, b_{1,n−1}} for V_{n−1} which satisfies σ_{i} · b_{i,n−1} = b_{i,n−1} for i = 0, 1. From (4.36), σ_{0} fixes a ℚ_{p}-subspace of V_{n} of dimension \frac{p^{n}+1}{2}, and since there is a nontrivial subspace of V_{p}(J_{0}) not fixed by σ_{0}, σ_{0} fixes a subspace of V_{n−1} of dimension at most p^{n−1}. Since p > 2 > 2 − \frac{1}{p^{n−1}} , we have \frac{p^{n}+1}{2} > p^{n−1}, so there some nonzero element b̂_{0} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2} fixed by σ_{0}. Let b_{0} = b̂_{0} + b_{0,n−1}. Since b_{0,n−1} generates a free module over ℚ_{p}[γ_{n−1} + γ_{n−1}^{−1}] and ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) is a field, b_{0} generates a free module over ℚ_{p}[γ_{n} + γ_{n}^{−1}]. By the same argument, there is an element b_{1} ∈ V_{n} fixed by σ_{1} which generates a free module over ℚ_{p}[γ_{n} + γ_{n}^{−1}].

We claim that {b_{0}, b_{1}} is a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-basis for V_{n}. Let b̂_{1} = b_{1} − b_{1,n−1}, so b̂_{1} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}. It suffices to show that σ_{0} and σ_{1} have no common nontrivial fixed point in ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}, for then in particular we have

ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) b̂_{0} ⋂ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) b̂_{1} = {0}.

Let

ρ̂_{n} : Π\overline{ℚ} ⟶ GL_{2} (ℚ_{p}(ζ_{n} + ζ_{n}^{−1}))

be the representation obtained by composing ρ_{n} with the natural projection ℚ_{p}[γ_{n} + γ_{n}^{−1}] ⟶ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}). Let W be a subspace of ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2} satisfying ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2} = ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) b̂_{0} ⊕ W. Given any nonzero w ∈ W, {b̂_{0}, w} is a basis for ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}, and from (4.36), σ_{0} · w = w + w_{0}, where w_{0} ∈ ℚ_{p}(ζ_{n}+ζ_{n}^{−1})^{2} is fixed by σ_{0}. If ρ̂_{n}(σ_{0}) is nontrivial, then ℚ_{p}(ζ_{n}+ζ_{n}^{−1}) b̂_{0} is the subspace of all σ_{0}-fixed points, so w_{0} = β_{0}b̂_{0} for some β_{0} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}), and therefore ρ̂_{n}(σ_{0}) = ( \begin{matrix}1 & β_{0} \\ 0 & 1 \end{matrix} ) with respect to the basis \{b̂_{0}, w\}. Similarly, ρ̂_{n}(σ_{1}) ∼ ( \begin{matrix}1 & β_{1} \\ 0 & 1 \end{matrix} ) for some β_{1} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}). If σ_{0}, σ_{1} have a common fixed point in ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}, then with respect to some basis we have

ρ̂_{n}(σ_{0}) = (\begin{matrix}1 & β_{0} \\ 0 & 1 \end{matrix}) , ρ̂_{n}(σ_{1}) = (\begin{matrix}1 & β_{1} \\ 0 & 1 \end{matrix})

for some β_{0}, β_{1} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}. Then ρ̂_{n}(σ_{∞}) = ( \begin{matrix}1 & −(β_{0}+β_{1}) \\ 0 & 1 \end{matrix} ), contradicting

that ρ̂_{n}(σ_{∞}) has exact order 2p^{n}. Therefore, σ_{0} and σ_{1} have no common nontrivial fixed point in ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}, and {b_{0}, b_{1}} is indeed a basis for V_{n}.

By induction, we have

ρ_{n}(σ_{0}) = (\begin{matrix}1 & δ_{0} \\ 0 & 1 \end{matrix}) , ρ_{n}(σ_{1}) = (\begin{matrix}1 & 0 \\ δ_{1} & 1 \end{matrix})

for some δ_{0}, δ_{1} ∈ ℚ_{p}[γ_{n} + γ_{n}^{−1}]. All that remains is to show that δ_{0} and δ_{1} are units. Suppose that δ_{0} ∉ ℚ_{p}[γ_{n} + γ_{n}^{−1}]^{×}. Writing δ_{0} = δ_{0,n} + δ_{0,n−1} where δ_{0,n} ∈ ℚ_{p}(ζ_{n} + ζ_{n}^{−1}) and δ_{0,n−1} ∈ ℚ_{p}[γ_{n−1} + γ_{n−1}^{−1}], it follows from the inductive hypothesis that δ_{0,n−1} ∈ ℚ_{p}[γ_{n−1} + γ_{n−1}^{−1}]^{×}, so δ_{0,n} must not be a unit of ℚ_{p}(ζ_{n} + ζ_{n}^{−1}). Hence δ_{0,n} = 0, contradicting that σ_{0} acts nontrivially on ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}. Therefore, δ_{0} ∈ ℚ_{p}[γ_{n} + γ_{n}^{−1}]^{×}. The same argument shows that δ_{1} ∈ ℚ_{p}[γ_{n} + γ_{n}^{−1}]^{×} as well. □

4.8 The Universal Deformation

We now consider the representation obtained by taking the inverse limit of the various representations ρ_{n}.

Proposition 4.37 There is a ℚ_{p}[μ_{p^{n}}]-basis for V_{p}(J_{n}) with respect to which Imρ_{n} ⊂ GL_{2} (ℤ_{p}[μ_{p^{n}}]).

Proof: By Proposition 4.35, there is a ℚ_{p}[μ_{p^{n}}]-basis B for V_{p}(J_{n}) with respect to which

ρ_{n}(σ_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , ρ_{n}(σ_{1}) = (\begin{matrix}1 & 0 \\ α_{n} & 1 \end{matrix}) .

Multiplying gives

ρ_{n}(σ_{∞}) = (\begin{matrix}1 & −1 \\ −α_{n} & 1 + α_{n} \end{matrix}) .

Since −γ_{n} is an eigenvalue of ρ_{n}(σ_{∞}) and ρ_{n} has determinant one, we have

tr ρ_{n}(σ_{∞}) = 2 + α_{n} = −γ_{n} − γ_{n}^{−1},

so α_{n} = −(γ_{n} + γ_{n}^{−1} + 2) ∈ ℤ_{p}[μ_{p^{n}}]. □

To obtain a representation ρ : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T]]), we need the following result from Iwasawa theory:

Proposition 4.38 For each compatible system (γ_{n})_{n∈ℕ} of generators γ_{n} of μ_{p^{n}}, the map sending (γ_{n})_{n∈ℕ} to 1 + T defines an isomorphism

\underleftarrow{lim} ℤ_{p}[μ_{p^{n}}] ≅ ℤ_{p}[[T]].

Proof: See [Was82], Theorem 7.1. □

For each n, let γ_{n} be a generator of μ_{p^{n}} such that −γ_{n} is an eigenvalue of ρ_{n}(σ_{∞}). The compatible system (γ_{n})_{n∈ℕ} of generators of μ_{p^{n}} corresponds to an isomorphism \underleftarrow{lim} ℤ_{p}[μ_{p^{n}}] ≅ ℤ_{p}[[T]]. Let

ρ : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T]])

be the representation obtained with respect to this isomorphism by the compatibility of the various representations ρ_{n}. Since Imρ_{n} ⊂ GL_{2} (ℤ_{p}[γ_{n} + γ_{n}^{−1}]), the image of ρ lies in GL_{2} (ℤ_{p}[[(1 + T) + (1 + T)^{−1}]]). In fact, we have

ρ(σ_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , ρ(σ_{1}) = (\begin{matrix}1 & 0 \\ α & 1 \end{matrix})  (4.39)

for some α ∈ ℤ_{p}[[T]]^{×}; moreover, from the proof of Proposition 4.37, α satisfies 2 + α = −(1 + T) − (1 + T)^{−1}, and hence

\begin{aligned}α & = −3 − T − (1 + T)^{−1} \\ & = −3 − T − (1 − T + T^{2} − ⋯) \\ & = −4 − T^{2} + T^{3} − · · · . \end{aligned}  (4.40)

We claim that there is a ℤ_{p}-algebra automorphism ψ of ℤ_{p}[[T]] which takes ℤ_{p}[[T^{2}]] to ℤ_{p}[[(1 + T) + (1 + T)^{−1}]] = ℤ_{p}[[T^{2} − T^{3} + T^{4} − ⋯]]. Let f(T) = a_{1}T + a_{2}T^{2} + ⋯ ∈ ℤ_{p}[[T]] be a square root of T^{2} − T^{3} + T^{4} − ⋯; for example, let a_{1} = 1, and define each a_{n} recursively by

a_{n} = \frac{1}{2} ((−1)^{n+1} −_{∑_{2≤i,j≤n−1} ; i+j=n+1}a_{i}a_{j}) .

Let ψ be the ℤ_{p}-algebra endomorphism taking T to f(T). Then ψ is injective, and induces a surjective map on cotangent spaces. Therefore, ψ is an isomorphism, as claimed.

Composing ρ with the automorphism of GL_{2} (ℤ_{p}[[T]]) induced from ψ^{−1} gives a representation

ρ′ : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T^{2}]])

which satisfies

ρ′(σ_{0}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , ρ′(σ_{1}) = (\begin{matrix}1 & 0 \\ −4 − T^{2} & 1 \end{matrix}) .

Let ρ̄ be the representation obtained from ρ′ by reducing mod (p, T^{2}), so that ρ̄ is the representation associated to the p-torsion points of the Legendre family of elliptic curves, and let S = {σ_{0}, σ_{1}}.

Theorem 4.41 As a representation over ℤ_{p}[[T^{2}]] ≅ ℤ_{p}[[T]], ρ′|_{Π\overline{ℚ}} is a representative of the S-ordinary universal deformation [ρ_{S−ord}^{univ}] of ρ̄.

Proof: The universal property of [ρ_{S−ord}^{univ}] gives a ℤ_{p}-algebra homomorphism

ϕ : ℤ_{p}[[T]] ⟶ ℤ_{p}[[T^{2}]]

which takes [ρ_{S−ord}^{univ}] to [ρ′|_{Π\overline{ℚ}}]. From the proof of Theorem 2.31, ϕ(T) = T^{2}, so ϕ is an isomorphism. □ Remark: The representation ρ′ arises naturally as a representation of the larger Galois group Π_{ℚ(μ_{p^{∞}})}. Since the image of ρ̄(σ_{i}) has order dividing 2p for each i = 0, 1, ∞, by Theorem 3.12, ρ_{S−ord}^{univ} can be extended to a representation of Π_{ℚ(μ_{p^{∞}})}; therefore, up to multiplication by a representation χ : G_{ℚ(μ_{p^{∞}})} ⟶ ℤ_{p}[[T^{2}]]^{×}, ρ′ is the composition of this extension of ρ_{S−ord}^{univ} with the map induced from ϕ.

5 Relation to Ihara’s Cocycle

5.1 Ihara’s Construction

We define a ℤ_{p}-algebra A by

A := ℤ_{p}[[t_{0}, t_{1}, t_{∞}]]/ ((t_{0} + 1)(t_{1} + 1)(t_{∞} + 1) − 1) .

In [Iha86b], Ihara constructs a cocycle

F : G_{ℚ} ⟶ A^{×}

which describes, for each n, the action of G_{ℚ(μ_{p^{n}})} on the primitive quotients of the Jacobian of the Fermat curve

F_{n} : x^{p^{n}} + y^{p^{n}} = 1.

We briefly describe Ihara’s original construction.

Fix a prime p, and let F be the pro-p completion of the free group on two generators g_{0}, g_{1}. Let g_{∞} = (g_{0}g_{1})^{−1}, and let M be the maximal algebraic pro-p extension of ℚ(t) unramified outside {0, 1, ∞}. Fix an isomorphism ι : F ⟶ Gal (M/\overline{ℚ}(t)) such that for each i = 0, 1, ∞, g_{i} is mapped to a topological generator of an inertia group above i. The choice of such an isomorphism ι gives rise to a representation of G_{ℚ} in the group of outer automorphisms of F. More precisely, let A be the subgroup of Aut(F) consisting of those automorphisms σ for which there is some α ∈ ℤ_{p}^{×} satisfying σ(g_{i}) ∼ g_{i}^{α} for each i = 0, 1, ∞. An automorphism of a group G is said to be an inner automorphism if it arises as conjugation by some element of G. We denote by Int(G) the group of inner automorphisms of G, and by Out(G) the group Aut(G)/Int(G) of outer automorphisms of G.

Definition 5.1 The pro-p braid group (of degree 2) is the group

Φ := A/Int(F).

Given γ ∈ G_{ℚ} ≅ Gal (M/ℚ(t)) /Gal (M/\overline{ℚ}(t)), choose a lift γ̃ of γ to Gal (M/ℚ(t)). Conjugation by γ̃ defines an automorphism of Gal (M/ℚ(t)) whose reduction modulo Int (Gal (M/ℚ(t))) depends only on γ. By the isomorphism ι, we obtain an outer automorphism σ_{γ} of F. Moreover, by Theorem 3.1, σ_{γ} is an element of Φ; thus the assignment γ ↦ σ_{γ} defines a representation

ϕ : G_{ℚ} ⟶ Φ.

Let F′′ = [[F, F], [F, F]] denote the double commutator subgroup of F. Let Ψ denote the image of Φ in Out (F/F′′) under the canonical homomorphism r : Out(F) ⟶ Out (F/F′′). In [Iha86b], Ihara studies the representation

ψ : G_{ℚ} ⟶ Ψ

obtained by composing ϕ with r.

The quotient F/F′ is isomorphic to the pro-p completion of the abelianization of the free group on two generators; that is, F/F′ is isomorphic to the pro-p completion ℤ_{p} × ℤ_{p} of the free abelian group ℤ × ℤ on two generators.

Since F′/F′′ is abelian, the automorphism of F′/F′′ given by conjugation by any element g ∈ F depends only on the reduction of g mod F′; thus F/F′ acts by conjugation on F′/F′′. The group F′/F′′ is an abelian pro-p group, and hence is endowed with a canonical action of ℤ_{p}. Therefore, F′/F′′ is a module over

ℤ_{p}[[F/F′]] ≅ ℤ_{p}[[ℤ_{p} × ℤ_{p}]] ≅ ℤ_{p}[[u, v]] ≅ A.

Fixing the isomorphism ℤ_{p}[[F/F′]] ⟶ A which maps g_{i} to t_{i} + 1 for each i = 0, 1, ∞, F′/F′′ obtains the structure of an A-module in such a way that multiplication by t_{i} + 1 is given by conjugation by g_{i} for each i = 0, 1, ∞.

Theorem 5.2 This action of A makes F′/F′′ into a free A-module of rank one generated by [g_{0}, g_{1}].

Proof: See [Iha86b], §II, Theorem 2. □

Let χ_{p} : G_{ℚ} ⟶ ℤ_{p}^{×} denote the p-cyclotomic character, which describes the action of G_{ℚ} on μ_{p^{∞}} ⊂ \overline{ℚ}. The group G_{ℚ} acts as ℤ_{p}-algebra automorphisms on A by

γ · (1 + t_{i}) = (1 + t_{i})^{χ_{p}(γ)}

for each γ ∈ G_{ℚ}, and each i = 0, 1, ∞. For γ ∈ G_{ℚ}, let F_{γ}(t_{0}, t_{1}, t_{∞}) ∈ A^{×} be the unique element satisfying

ψ(γ) ([g_{0}, g_{1}]) = F_{γ}(t_{0}, t_{1}, t_{∞}) · [g_{0}, g_{1}].

Proposition 5.3 The assignment γ ⟼ F_{γ} defines a continuous 1-cocycle F : G_{ℚ} ⟶ A^{×}.

Proof: See [Iha86b], §II, Theorem 3B(ii). □

Since G_{ℚ(μ_{p^{∞}})} acts trivially on A, the restriction of F to G_{ℚ(μ_{p^{∞}})} is a homomorphism, which we also denote by F.

Let a, b ∈ ℤ/p^{n}ℤ \ {0} be such that at least one of a, b is a unit in ℤ/p^{n}ℤ, and let c = −(a + b). Let a_{n} (respectively b_{n}, c_{n}) denote the integer in {0, 1, …, p^{n} − 1} reducing to a (resp. b, c) mod p^{n}. The Jacobian J_{n}^{(a,b,c)} of the complete nonsingular model of the curve

C_{n}^{(a,b,c)} : x^{p^{n}} = y^{a_{n}}(y − 1)^{b_{n}}

has a quotient A_{n}^{(a,b,c)} which contains ℤ[ζ_{n}] in its endomorphism ring, where the action of ζ_{n} on A_{n}^{(a,b,c)} arises from the action of μ_{p^{n}} on C_{n}^{(a,b,c)} given by ζ_{n} · (x, y) = (ζ_{n}x, y) for a generator ζ_{n} of μ_{p^{n}} (see [Iha86b], p.76). In fact, the Tate module T_{p} (A_{n}^{(a,b,c)}) is a free module of rank one over ℤ_{p}[ζ_{n}], and the action of G_{ℚ(μ_{p^{n}})} on T_{p} (A_{n}^{(a,b,c)}) commutes with the action of ℤ_{p}[ζ_{n}]. Therefore, the action of γ ∈ G_{ℚ(μ_{p^{n}})} is given by multiplication by some element F_{γ,n}^{(a,b,c)} ∈ ℤ_{p}[ζ_{n}]^{×}.

Theorem 5.4 (Ihara, 1986) For each γ ∈ G_{ℚ(μ_{p^{n}})}, F_{γ,n}^{(a,b,c)} is equal to F_{γ} (ζ_{n}^{a} − 1, ζ_{n}^{b} − 1, ζ_{n}^{c} − 1).

Proof: See [Iha86b], §II, Theorem 4 and its corollary. □

For each n, let J_{n} denote the Jacobian of the Fermat curve F_{n}. Then J_{n} is isogenous to the sum of J_{n−1} together with each A_{n}^{(a,b,c)}, where exactly one triple (a, b, c) is chosen from each set {(ka, kb, kc)}_{k∈(ℤ/p^{n}ℤ)^{×}} (see [Iha86b], p.78). On the other hand, under the isomorphism ι, the commutator subgroup F′ corresponds to the subfield M′ = ⋃_{n}K_{n}, where K_{n} is the function field of F_{n}, and the subgroup F′′ corresponds to M′′ = ⋃_{n}K_{n}^{unrab}, where K_{n}^{unrab} denotes the maximal unramified abelian pro-p extension of K_{n}. Thus we have the following tower of extensions:

M_{F′′} M′′ = ⋃_{n∈ℕ}K_{n}^{unrab}_{F′/F′′}_{F/F′′} M′ = ⋃_{n∈ℕ}K_{n}_{F/F′} \overline{ℚ}(t)

An isogeny of abelian varieties is a surjective homomorphism of abelian varieties whose kernel is finite. Subgroups H of the Tate module T_{p}(J_{n}) of finite index are in one-to-one correspondence with isogenies f_{H} : J ⟶ J_{n} in such a way that f_{H}(T_{p}(J_{n})) = H. By geometric class field theory, such isogenies are in one-to-one correspondence with finite unramified abelian coverings C_{H} ⟶ F_{n} in such a way that

Gal (\overline{ℚ}(C_{H})/\overline{ℚ}(F_{n})) ≅ T_{p}(J_{n})/H,

where \overline{ℚ}(C_{H}), \overline{ℚ}(F_{n}) are the function fields over \overline{ℚ} of C_{H} and F_{n} respectively (see [Ser88], Ch. VI, §2, Proposition 10 and the corollary to Proposition 11). Therefore, letting S_{n} denote the set of finite unramified abelian extensions K_{n}, we have

Gal (K_{n}^{unrab}/K_{n}) = \underleftarrow{lim}_{L∈S_{n}}Gal (L/K_{n})

= \underleftarrow{lim}_{H⊂T_{p}(J_{n}) ; finite index}Gal (\overline{ℚ}(C_{H})/K_{n})

≅ \underleftarrow{lim}_{H⊂T_{p}(J_{n}) ; finite index} (T_{p}(J_{n})/H) = T_{p}(J_{n}),

so T_{p}(J_{n}) is isomorphic to Gal (K_{n}^{unrab}/K_{n}). Thus one might expect the homomorphism F with the property of Theorem 5.4 to arise from the representation Ψ.

5.2 The Inertia Group at Infinity

Let ρ : Π_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T]]) be as in §4.8, and for any uniformizer π at ∞, let ℚ_{π}(μ_{p^{∞}}) = ℚ(μ_{p^{∞}})((π)). Restricting ρ to the inertia subgroup I_{∞} at ∞ gives a representation ρ_{∞} of I_{∞} = Gal (M_{∞}/ℚ_{π}(μ_{p^{∞}})), where

M_{∞} := ⋃_{n} \overline{ℚ(μ_{p^{∞}})} ((π^{1/n})). The tower

M_{∞}

G_{ℚ(μp∞)}

⋃_{n}ℚ(μ_{p^{∞}}) ((π^{1/n}))

ℚ_{π}(μ_{p^{∞}})

gives an inclusion ι_{π} : G_{ℚ(μ_{p^{∞}})} ↪ I_{∞} which depends on the choice of uniformizer π. Restricting ρ_{∞} to the image of ι_{π} thus gives a representation

ρ_{∞,π} : G_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[[T]]) .

Let p_{n} : GL_{2} (ℤ_{p}[[T]]) ⟶ GL_{2} (ℤ_{p}[μ_{p^{n}}]) be the map induced from the ring homomorphism ℤ_{p}[[T]] ⟶ ℤ_{p}[μ_{p^{n}}] taking T to γ_{n} − 1; since ρ was obtained by the isomorphism \underleftarrow{lim} ℤ_{p}[μ_{p^{n}}] ≅ ℤ_{p}[[T]] taking (γ_{n} − 1)_{n∈ℕ} to T, we have ρ_{n}^{−} ⊗ ℤ_{p}[μ_{p^{n}}] = p_{n} ∘ ρ.

Fixing the uniformizer π = 1/16t, and letting u = (1/16t)^{1/2p^{n}}, C_{n}^{−} is isomorphic over ℚ((u)) to the curve

C̃_{n}^{−} : y^{2} = x^{∏^{\frac{p^{n}−1}{2}} ; j=1} (1 + (ζ_{n}^{j} + ζ_{n}^{−j} − 2)u^{4}x^{2}) + \frac{x^{p^{n}+1}}{4} − 2u^{2p^{n}}x^{p^{n}+1}

by the map C̃_{n}^{−} ⟶ C_{n}^{−} taking (x, y) to (\frac{1}{u^{2}x}, \frac{y}{u^{p^{n}}x^{\frac{p^{n}+1}{2}}} ). The curve C̃_{n}^{−} has good reduction at u = 0, and gives the reduced curve

C̄_{n}^{−} : y^{2} = \frac{x^{p^{n}+1}}{4} + x.

On the other hand, the curve C_{n}^{(1,1,p^{n}−2)} considered by Ihara when a = b = 1 is given by

C_{n}^{(1,1,p^{n}−2)} : y(y − 1) = x^{p^{n}},

and there is an isomorphism ψ : C̄_{n}^{−} ⟶ C_{n}^{(1,1,p^{n}−2)} given by

ψ : (x, y) ⟼ (\frac{1}{x}, \frac{y}{x^{\frac{p^{n}+1}{2}}} + \frac{1}{2}) .

The endomorphism γ_{n} + γ_{n}^{−1} of J_{n}^{−} gives rise to a corresponding endomorphism of the Jacobian J̄_{n}^{−} of C̄_{n}^{−}, and the reduction map

J_{n}^{−} (\overline{ℚ_{π}(μ_{p^{∞}})}) ⟶ J̄_{n}^{−} (\overline{ℚ(μ_{p^{∞}})})

induces a ℚ_{p}[γ_{n} + γ_{n}^{−1}]-module isomorphism V_{p}(J_{n}^{−}) ≅ V_{p}(J̄_{n}^{−}). Counting ℚ_{p}-dimensions shows that under the isomorphism of Jacobians induced from ψ, the quotient A_{n}^{(1,1,p^{n}−2)} of J_{n}^{(1,1,p^{n}−2)} must correspond to a quotient A_{n} of J̄_{n}^{−} such that the extended Tate module V_{p}(A_{n}) corresponds to the unique ℚ_{p}[γ_{n} + γ_{n}^{−1}]-module quotient of V_{p}(J̄_{n}^{−}) isomorphic to ℚ_{p}(ζ_{n} + ζ_{n}^{−1})^{2}.

The endomorphism γ_{n} + γ_{n}^{−1} of the Jacobian of C̃_{n}^{−} arises from the action of μ_{p^{n}} on C_{n} via the map

C_{n} ⟶ C̃_{n}^{−}
(x, y) ⟼ (\frac{1}{(x + x^{−1})u^{2}} , \frac{y}{ux^{\frac{p^{n}+1}{2}} (x + x^{−1})^{\frac{p^{n}+1}{2}}} ) ,

where u^{2p^{n}} = π. If P_{1} = (x, y) ∈ C_{n} is a preimage of Q = (w, z) ∈ C̃_{n}^{−}, then so is P_{2} = (\frac{1}{x}, \frac{y}{x^{p^{2}+1}} ); applying γ_{n} to P_{1}, P_{2} and mapping to C̃_{n}^{−} gives the points

Q_{1} = (\frac{ζ_{n}x}{(ζ_{n}^{2}x^{2} + 1)u^{2}} , \frac{y}{u(ζ_{n}x^{2} + ζ_{n}^{−1})^{\frac{p^{n}+1}{2}}} ) ,
Q_{2} = (\frac{ζ_{n}x}{(ζ_{n}^{2} + x^{2})u^{2}} , \frac{y}{u(ζ_{n} + ζ_{n}^{−1}x^{2})^{\frac{p^{n}+1}{2}}} ) .

Projectivizing these points and specializing at u = 0 gives the point at infinity (0 : 1 : 0) on C̄_{n}^{−} if v_{u}(x) < 2, where v_{u} denotes the u-adic valuation. If v_{u}(x) ≥ 2, then v_{u}(y) ≥ 1. Letting x′ = x/u^{2}, y′ = y/u, specializing gives

the points Q̄_{1} = (ζ_{n}x̄′, ζn^{\frac{p^{n}+1}{2}} ȳ′) and Q̄_{2} = (ζ_{n}^{−1}x̄′, ζ_{n}^{− \frac{p^{n}+1}{2}} ȳ′) respectively,

where x̄′, ȳ′ denote the reductions of x, y mod u. Therefore, the action of γ_{n} + γ_{n}^{−1} on J̄_{n}^{−} obtained from that on J̃_{n}^{−} is precisely that considered by Ihara arising from the action of μ_{p^{n}} on C̄_{n}^{−} by γ_{n}(x, y) = (ζ_{n}x, ζn^{\frac{p^{n}+1}{2}} y). In particular, T_{p}(A_{n}) is a free ℤ_{p}[ζ_{n}]-module of rank one, and thus also a free ℤ_{p}[ζ_{n} + ζ_{n}^{−1}]-module of rank two. Moreover, writing

ρ_{∞,π}(σ) = M_{σ}(T) ∈ GL_{2} (ℤ_{p}[[(1 + T) + (1 + T)^{−1}]])

for each σ ∈ G_{ℚ(μ_{p^{∞}})}, the representation

ρ_{∞,n} : G_{ℚ(μ_{p^{∞}})} ⟶ GL_{2} (ℤ_{p}[ζ_{n} + ζ_{n}^{−1}])

given by ρ_{∞,n}(σ) = M_{σ}(ζ_{n} − 1) is the Galois representation associated to T_{p}(A_{n}) as a ℤ_{p}[ζ_{n} + ζ_{n}^{−1}]-module.

Let F : G_{ℚ(μ_{p^{∞}})} ⟶ A^{×} be Ihara’s representation. There is a ℤ_{p}-algebra isomorphism θ : A ⟶ ℤ_{p}[[u, v]] which takes t_{0} to u and t_{1} to v. Let

r : ℤ_{p}[[u, v]] ⟶ ℤ_{p}[[T]]

be the ℤ_{p}-algebra homomorphism such that r(u) = r(v) = T, and let F̄ = r ∘ θ ∘ F. Since r ∘ θ(t_{∞}) = (T + 1)^{−2} − 1, we have

F_{σ}(ζ_{n} − 1, ζ_{n} − 1, ζ_{n}^{−2} − 1) = F̄_{σ}(ζ_{n} − 1)

for each σ ∈ G_{ℚ(μ_{p^{∞}})}; hence letting p̄_{n} : ℤ_{p}[[T]] ⟶ ℤ_{p}[ζ_{n}] denote the homomorphism taking T to ζ_{n} − 1, p̄_{n} ∘ F̄ is the representation of G_{ℚ(μ_{p^{∞}})} associated to A_{n}^{(1,1,p^{n}−2)}.

In order to obtain the representation F̄ from ρ_{∞,π}, we first need some lemmas:

Lemma 5.5 Let V be a free ℤ_{p}[ζ_{n}]-module of rank one, and let σ be the automorphism of V given by multiplication by α ∈ ℤ_{p}[ζ_{n}]^{×}. Let δ be the nontrivial element of Gal (ℚ_{p}(ζ_{n})/ℚ_{p}(ζ_{n} + ζ_{n}^{−1})). Then the eigenvalues of σ ⊗_{ℤ_{p}[ζ_{n}+ζ_{n}^{−1}]} ℤ_{p}[ζ_{n}] are α and α^{δ}.

Proof: Let {v} be a ℤ_{p}[ζ_{n}]-basis for V. Fix the ℤ_{p}[ζ_{n} + ζ_{n}^{−1}]-basis {v, ζ_{n}v} for V, and let α_{0}, α_{1} ∈ ℤ_{p}[ζ_{n} + ζ_{n}^{−1}] be such that α = α_{0} + ζ_{n}α_{1}. Since

σ(v) = αv = α_{0}v + α_{1}ζ_{n}v

\begin{aligned}and σ(ζ_{n}v) = αζ_{n}v & = ζ_{n}^{2}α_{1}v + α_{0}ζ_{n}v \\ & = −α_{1}v + (α_{0} + (ζ_{n} + ζ_{n}^{−1})α_{1}) ζ_{n}v, \end{aligned}

σ is given by ( \begin{matrix}α_{0} & −α_{1} \\ α_{1} & α_{0}+(ζ_{n}+ζ_{n}^{−1})α_{1} \end{matrix} ) relative to the basis \{v, ζ_{n}v\}. In particular, the characteristic polynomial f_{σ} of σ is given by

f_{σ}(X) = X^{2} − (2α_{0} + (ζ_{n} + ζ_{n}^{−1})α_{1}) X + (α_{0}^{2} + (ζ_{n} + ζ_{n}^{−1})α_{0}α_{1} + α_{1}^{2})
= (X − (α_{0} + ζ_{n}α_{1})) (X − (α_{0} + ζ_{n}^{−1}α_{1}))
= (X − α)(X − α^{δ}),

as desired. □

Lemma 5.6 The representation ρ_{∞,π} is conjugate to an upper-triangular representation.

Proof: Let χ denote the cyclotomic character. For γ ∈ I_{∞}, we have γσ_{∞}γ^{−1} = σ_{∞}^{χ(γ)}, and thus ρ(γ)ρ(σ_{∞})ρ(γ)^{−1} = ρ(σ_{∞})^{χ(γ)}. Since ρ(σ_{∞}) has order dividing 2p^{k} for some k in every finite quotient of GL_{2} (ℤ_{p}[[T]]), we have ρ(σ_{∞})^{χ(γ)} = ρ(σ_{∞})^{χ_{p}(γ)}, where χ_{p} : G_{ℚ} ⟶ ℤ_{p}^{×} denotes the p-cyclotomic character which describes the action of G_{ℚ} on μ_{p^{∞}} ⊂ \overline{ℚ}. The group I_{∞} is contained in Π_{ℚ(μ_{p^{∞}})}, so γ fixes μ_{p^{∞}} pointwise; thus χ_{p}(γ) = 1, and hence ρ(γ)ρ(σ_{∞})ρ(γ)^{−1} = σ_{∞}. In particular, the image of ρ_{∞,π} is contained in the centralizer Z_{GL_{2}(ℤ_{p}[[T]])} (ρ(σ_{∞})) of ρ(σ_{∞}) in GL_{2} (ℤ_{p}[[T]]). If we show that the non-scalar matrix ρ(σ_{∞}) is upper-triangular, then its centralizer must also be upper-triangular, thus proving the lemma.

From 4.39 and 4.40, we have

ρ(σ_{∞}) = (\begin{matrix}1 & −1 \\ 2 + (1 + T) + (1 + T)^{−1} & −1 − (1 + T) − (1 + T)^{−1} \end{matrix}) .

For any g(T) ∈ ℤ_{p}[[T]]^{×}, applying Hensel’s lemma to X^{2} − g(T) shows that g(T) is a square in ℤ_{p}[[T]] if and only if its reduction mod (p, T) is a square in 𝔽_{p}. Thus g_{1}(T) := 2+(1+T)+(1+T)^{−1} and g_{2}(T) := g_{1}(T)−4 = T^{2}(1+T)^{−1} are both squares in ℤ_{p}[[T]] since g_{1}(T) reduces to 4 ∈ 𝔽_{p} and g_{2}(T)/T^{2} reduces to 1 ∈ 𝔽_{p}. Let g(T) ∈ ℤ_{p}[[T]] be a square root of g_{1}(T)g_{2}(T), and let

h(T) := \frac{1}{2} (g_{1}(T)^{2} + g(T)) .

Conjugating ρ(σ_{∞}) by the matrix M = ( \begin{matrix}1 & 0 \\ h(T) & 1 \end{matrix} ) gives

Mρ(σ_{∞})M^{−1} = (\begin{matrix}h(T) + 1 & −1 \\ 0 & 1 − h(T) − g_{1}(T) \end{matrix}) ,

as desired. □

Identify ρ_{∞,π} with any one of its upper-triangular conjugates, and let f_{1}, f_{2} : G_{ℚ(μ_{p^{∞}})} ⟶ ℤ_{p}[[T]]^{×} be such that for each σ ∈ G_{ℚ(μ_{p^{∞}})},

ρ_{∞,π}(σ) = (\begin{matrix}(f_{1})_{σ}(T) & ∗ \\ 0 & (f_{2})_{σ}(T) \end{matrix}) .

Theorem 5.7 One of f_{1} or f_{2} is equal to F̄; the other is uniquely determined by the property that the image of each σ ∈ G_{ℚ(μ_{p^{∞}})} gives F̄(ζ_{n} − 1)^{δ} when evaluated at ζ_{n} − 1.

Proof: Given σ ∈ G_{ℚ(μ_{p^{∞}})}, by Lemmas 5.5 and 5.6, the action of σ on T_{p}(A_{n}) as a ℤ_{p}[ζ_{n}]-module is given by multiplication by (f_{j(n)})σ(ζ_{n} − 1) for some j(n) = 1 or 2. On the other hand, since A_{n} is isomorphic to A_{n}^{(1,1,p^{n}−2)} over ℚ(μ_{p^{∞}}), σ acts on T_{p}(A_{n}) by multiplication by F̄_{σ}(ζ_{n} − 1). Therefore, for some j = 1 or 2, (F̄_{σ} − (f_{j})_{σ})(ζ_{n} − 1) = 0 for infinitely many n. It follows from the Weierstrass Preparation Theorem that a nonzero power series can have only a finite number of zeroes z ∈ \overline{ℚ}_{p} satisfying |z| < 1, where | · | is the p-adic norm (see [Was82], Corollary 7.2). Therefore, F̄_{σ} = (f_{j})_{σ}. Since F̄, f_{1}, f_{2} are homomorphisms, we have F̄ = f_{k} for some k = 1 or 2. The final statement follows from Lemma 5.5 together with the corollary of the Weierstrass Preparation Theorem used above. □

Remark: Which f_{k} is equal to F̄ depends on the choice of conjugate of ρ_{∞,π}. Since ρ_{∞,π} describes the action of G_{ℚ(μ_{p^{∞}})} on T_{p}(A_{n}) as a ℤ_{p}[ζ_{n}+ζ_{n}^{−1}]-module, our construction does not distinguish which f_{k} is equal to F̄.

6 Conclusion

In the preceding chapters, we have described a new construction of a specialization of Ihara’s cocycle. This construction arises from the {σ_{0}, σ_{1}}-ordinary universal deformation of the residual representation ρ̄ which describes the action of Π\overline{ℚ} on the Legendre family of elliptic curves. This universal deformation was extended by the rigidity theorem to a representation ρ of Π_{ℚ(μ_{p^{∞}})}. Using a geometric construction of ρ, we showed that a specialization of Ihara’s cocycle appears when ρ is specialized at infinity (given a particular choice of uniformizer).

This work suggests a number of directions for further research. First of all, the σ_{0}-ordinary universal deformation ring of the residual representation ρ̄ is ℤ_{p}[[u, v]] ≅ A; thus we are led to the following question: Question 1 Does the extended σ_{0}-ordinary universal deformation of ρ̄ of Theorem 3.12 give rise to Ihara’s full cocycle when specialized at infinity?

Let k be any field, and let M_{0}, M_{1}, M_{2} ∈ GL_{n}(k) be matrices satisfying M_{0}M_{1}M_{2} = Id_{n} which generate an irreducible subgroup of GL_{n}(k). By a theorem of Belyĭ, if one of M_{0}, M_{1}, or M_{2} differs from a scalar matrix by a matrix of rank one, then the triple (M_{0}, M_{1}, M_{2}) is rigid in GL_{n}(k). Thus one would expect that subject to an appropriate “ordinariness” condition, the universal deformation (R^{univ}, ρ^{univ}) of a residual representation

ρ̄: Π\overline{ℚ} ⟶ GL_{n}(𝔽_{p})

would be rigid; that is, the triple (ρ^{univ}(σ_{0}), ρ^{univ}(σ_{1}), ρ^{univ}(σ_{∞})) would be rigid in GL_{n}(R^{univ}). Therefore, one expects to be able to extend this ρ^{univ} to a representation ρ of Π_{K(t)}, where K is a given cyclotomic extension of ℚ(μ_{p^{∞}}). Furthermore, since Katz’ construction applies to rigid representations of arbitrary dimension, it should be possible to construct ρ geometrically, in a similar manner to the construction of Chapter 4. Question 2 What effect would increasing the dimension of the residual representation have on our construction? In particular, would Ihara’s cocycle still appear, or some (possibly nonabelian) variant?

If the group Π\overline{ℚ} is replaced with another algebraic fundamental group Π in our construction, the universal deformation seems much less likely to be rigid. Since the number of topological generators of Π is in general greater than 2, it may be necessary to increase the dimension of the residual representation in order to obtain a rigid situation. Also, a further study of rigid m-tuples in GL_{n}(R^{univ}) would be required if this generalization is to succeed. Question 3 Under what conditions could our construction be carried out if Π\overline{ℚ} is replaced with some other algebraic fundamental group? Under those conditions, what cocycles appear?

Another direction arises from Ihara’s generalization of his own construction of his cocycle. In [Iha86a], he considers different towers of étale coverings of ℙ^{1}(\overline{ℚ}) \ \{0, 1, ∞\} having certain properties, and for each such tower constructs a “universal” cocycle

ϕ : G_{ℚ} ⟶ A^{×},

where A is a completed group ring ℤ_{p}[[𝔤]], the group 𝔤 depending on the tower of coverings.

Question 4 Is it possible to generalize our construction to give other cocycles of Ihara?

In general, the algebra A is not a power series ring; thus it would be necessary to begin with an obstructed deformation problem, which could not arise from a residual representation of an algebraic fundamental group. Therefore, significant difficulties already appear in the first step of such a generalization.

References

[Arm97] M.A. Armstrong. Basic Topology. Springer-Verlag, New York, 1997.

[Bel80] G.V. Belyĭ. On Galois extensions of a maximal cyclotomic field. Math. USSR Izvestija, 14(2), 1980.

[Dar00] Henri Darmon. Rigid local systems, Hilbert modular forms and Fermat’s last theorem. Duke Math. J., 102(3):413–449, 2000.

[dL97] Bart de Smit and Hendrick W. Lenstra, Jr. Explicit construction of universal deformation rings. In G. Cornell, G. Stevens, and J.H. Silverman, editors, Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995 ), pages 313–326, New York, 1997. Springer-Verlag.

[DM00] Henri Darmon and Jean-François Mestre. Courbes hyperelliptiques à multiplications réelles et une construction de Shih. Canad. Math. Bull., 43(3):304–311, 2000.

[Gou] Fernando Q. Gouvêa. Deformations of Galois representations. Notes from a course given by the author during the summer of 1999, available at www.colby.edu/personal/fqgouvea/web.html.

[GS71] S. Greco and P. Salmon. Topics in 𝔪-adic Topologies. Springer-Verlag, Berlin, 1971.

[Gv80] Lothar Gerritzen and Marius van der Put. Schottky Groups and Mumford Curves. Number 817 in Lecture notes in mathematics. Springer-Verlag, Berlin, 1980.

[Har97] Robin Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1997.

[Iha86a] Yasutaka Ihara. On Galois representations arising from towers of coverings of ℙ^{1} \ {0, 1, ∞}. Invent. Math., 86(3):427–459, 1986.

[Iha86b] Yasutaka Ihara. Profinite braid groups, Galois representations and complex multiplications. Ann. of Math. (2), 123(1):43–106, 1986.

[Isa94] I. Martin Isaacs. Character Theory of Finite Groups. Dover Publications, New York, 1994.

[JL93] Gordon James and Martin Liebeck. Representations and Characters of Finite Groups. Cambridge University Press, Cambridge, 1993.

[Kat90] Nicholas M. Katz. Exponential Sums and Differential Equations. Number 124 in Annals of Mathematics Studies. Princeton University Press, Princeton, 1990.

[Lan64] Serge Lang. Introduction to Algebraic Geometry. Interscience Publishers, New York, 1964.

[Lan83] Serge Lang. Abelian Varieties. Springer-Verlag, New York, 1983.

[Lan93] Serge Lang. Algebra. Addison-Wesley, Reading, MA, 3rd edition, 1993.

[Mat87] Heinrich Matzat. Konstruktive Galoistheorie. Number 1284 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987.

[Maz86] B. Mazur. Arithmetic on curves. Bull. Amer. Math. Soc. (N.S.), 14(2):207–259, 1986.

[Maz89] B. Mazur. Deforming Galois representations. In Y. Ihara, K. Ribet, and J.-P. Serre, editors, Galois Groups over ℚ, number 16 in Math. Sci. Res. Inst. Publ., New York, 1989. Springer-Verlag.

[Maz97] B. Mazur. An introduction to the deformation theory of Galois representations. In G. Cornell, G. Stevens, and J.H. Silverman, editors, Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995 ), pages 243–311, New York, 1997. Springer-Verlag.

[Mil86a] J.S. Milne. Abelian varieties. In G.Cornell and J.H. Silverman, editors, Arithmetic Geometry, pages 103–150, New York, 1986. Springer-Verlag.

[Mil86b] J.S. Milne. Jacobian varieties. In G.Cornell and J.H. Silverman, editors, Arithmetic Geometry, pages 167–212, New York, 1986. Springer-Verlag.

[MM99] Gunter Malle and Heinrich Matzat. Inverse Galois Theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999.

[Mor96] Patrick Morandi. Field and Galois Theory. Springer-Verlag, New York, 1996.

[Mum70] David Mumford. Abelian Varieties. Oxford University Press, London, 1970.

[Ser68] Jean-Pierre Serre. Corps Locaux. Hermann, Paris, 2nd edition, 1968.

[Ser88] Jean-Pierre Serre. Algebraic Groups and Class Fields. Springer-Verlag, New York, 1988. Translation of the French edition.

[Ser92] Jean-Pierre Serre. Topics in Galois Theory. Jones and Bartlett, Boston, 1992.

[Sha72] Stephen S. Shatz. Profinite Groups, Arithmetic, and Geometry. Princeton University Press, Princeton, NJ, 1972.

[ST68] Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Ann. of Math.(2), 88:492–517, 1968.

[TTV91] W. Tautz, J. Top, and A. Verberkmoes. Explicit hyperelliptic curves with real multiplication and permutation polynomials. Can. J. Math., 43(5):1055–1064, 1991.

[Vol96] Helmut Volklein. Groups as Galois Groups. Cambridge University Press, Cambridge, 1996.

[Was82] Lawrence C. Washington. Introduction to Cyclotomic Fields. Springer-Verlag, New York, 1982.