Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic ℤ_{p}-extensions

M. Bertolini^{∗} H. Darmon^{†}

February 19, 2003

Contents

1 p-adic L-functions 7 1.1 Modular forms on quaternion algebras . . . . . . . . . . . . . 7 1.2 p-adic Rankin L-functions . . . . . . . . . . . . . . . . . . . . 12

2 Selmer Groups 16 2.1 Galois representations and cohomology . . . . . . . . . . . . . 16 2.2 Finite/singular structures . . . . . . . . . . . . . . . . . . . . 20 2.3 Definition of the Selmer group . . . . . . . . . . . . . . . . . . 23

3 Some Preliminaries 25 3.1 Λ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Controlling the Selmer group . . . . . . . . . . . . . . . . . . . 26 3.3 Rigid pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 The Euler System Argument 35 4.1 The Euler system . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 The argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Shimura curves 43

∗Partially supported by GNSAGA (INdAM), M.U.R.S.T., and the EC.

†Partially supported by CICMA and by an NSERC research grant.

5.1 The moduli definition . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 The Cerednik-Drinfeld theorem . . . . . . . . . . . . . . . . . 45 5.3 Character groups . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.4 Hecke operators and the Jacquet-Langlands correspondence . . 48 5.5 Connected components . . . . . . . . . . . . . . . . . . . . . . 53 5.6 Raising the level and groups of connected components . . . . . 57

6 The theory of complex multiplication 64

7 Construction of the Euler System 66

8 The first explicit reciprocity law 66

9 The second explicit reciprocity law 69

Introduction

Let E be an elliptic curve over ℚ, let p be an ordinary prime for E, and let K be an imaginary quadratic field. Write K_{∞}/K for the anticyclotomic ℤ_{p}-extension of K and set G_{∞} = Gal(K_{∞}/K).

Following a construction of sec. 2 of [BD1] which is recalled in section 1, one attaches to the data (E, K, p) an anticyclotomic p-adic L-function L_{p}(E, K) which belongs to the Iwasawa algebra Λ := ℤ_{p}[[G_{∞}]]. This element, whose construction was inspired by a formula proved in [Gr1], is known, thanks to work of Zhang ([Zh] §1.4), to interpolate special values of the complex L-function of E/K twisted by characters of G_{∞}.

Let Sel(K_{∞}, E_{p^{∞}})^{∨} be the Pontrjagin dual of the p-primary Selmer group attached to E over K_{∞}, equipped with its natural Λ-module structure, as defined in section 2. It is a compact Λ-module; write C for its characteristic power series, which is well-defined up to units in Λ.

Let N_{0} denote the conductor of E, set N = pN_{0} if E has good ordinary reduction at p, and set N = N_{0} if E has multiplicative reduction at p so that p divides N_{0} exactly. It will be assumed throughout that the discriminant of K is prime to N, so that K determines a factorisation

N = pN^{+}N^{−},

where N^{+} (resp. N^{−}) is divisible only by primes different from p which are split (resp. inert) in K.

The main goal of the present work is to prove (under the mild technical assumption 6 on (E, K, p) given at the end of this introduction) theorem 1 below, a weak form of the Main Conjecture of Iwasawa Theory for elliptic curves in the ordinary and anticyclotomic setting.

Theorem 1 Assume that N^{−} is the square-free product of an odd number of primes. The characteristic power series C divides the p-adic L-function L_{p}(E, K).

The hypothesis on N^{−} made in theorem 1 arises naturally in the anticyclotomic setting, and some justification for it is given at the end of the introduction.

Denote by L_{p}(E, K, s) the p-adic Mellin transform of the measure defined by the element L_{p}(E, K) of Λ. Let r be the rank of the Mordell-Weil group E(K). The next result follows by combining theorem 1 with standard techniques of Iwasawa theory.

Corollary 2 ord_{s=1}L_{p}(E, K, s) ≥ r.

A program of study of L_{p}(E, K, s) in the spirit of the work of Mazur, Tate and Teitelbaum [MTT] is outlined in [BD1], and partially carried out in [BD2]– [BD5]. In particular, section 4 of [BD1] formulates a conjecture predicting the exact order of vanishing of L_{p}(E, K, s) at s = 1. More precisely, set E^{+} = E and let E^{−} be the elliptic curve over ℚ obtained by twisting E by K. Write r^{±} for the rank of E^{±}(ℚ), so that r = r^{+} + r^{−}. Set r̃^{±} = r^{±} + δ^{±}, where

δ^{±} = \{ \begin{matrix}1 & if E^{±} has split multiplicative reduction at p, \\ 0 & otherwise. \end{matrix}

Finally set ρ := max(r̃^{+}, r̃^{−}) and r̃ := r̃^{+} + r̃^{−}. Conjecture 4.2 of [BD1] predicts that

ord_{s=1}L_{p}(E, K, s) = 2ρ = r̃ + |r̃^{+} − r̃^{−}|.  (1)

This conjecture indicates that L_{p}(E, K, s) vanishes to order strictly greater than r, if either r̃ > r or if r̃^{+} ≠ r̃^{−}. The first source of extra vanishing is accounted for by the phenomenon of exceptional zeroes arising when p is a prime of split multiplicative reduction for E over K, which was discovered by Mazur, Tate and Teitelbaum in the cyclotomic setting [MTT]. The second source of extra vanishing is specific to the anticyclotomic setting, and may be accounted for by certain predictable degeneracies in the anticyclotomic p-adic height, related to the fact that Sel(K_{∞}, E_{p^{∞}})^{∨} fails to be semisimple as a module over Λ when r^{+} ≠ r^{−}. (Cf. for example [BD\frac{1}{2}].)

A more careful study of the Λ-module structure of Sel(K_{∞}, E_{p^{∞}}), which in the good ordinary reduction case is carried out in [BD0] and [BD\frac{1}{2}], yields the following refinement of corollary 2 which is consistent with the conjectured equality (1).

Corollary 3 If p is a prime of good ordinary reduction for E, then

ord_{s=1}L_{p}(E, K, s) ≥ 2ρ.

Let O be a finite extension of ℤ_{p}, and let χ : G_{∞} ⟶ O^{×} be a finite order character, extended by ℤ_{p}-linearity to a homomorphism of Λ to O. If M is any Λ-module, write

M^{χ} = M ⊗_{χ} O,

where the tensor product is taken over Λ via the map χ.

Let LLI_{p}(E/K_{∞}) denote the p-primary part of the Shafarevich-Tate group of E over K_{∞}. A result of Zhang ([Zh], §1.4) generalising a formula of Gross established in [Gr1] in the special case where N is prime and χ is unramified, relates χ(L_{p}(E, K)) to a non-zero multiple of the classical L-value L(E/K, χ, 1) (where one views χ as a complex-valued character by choosing an embedding of O into ℂ). Theorem 1 combined with Zhang’s formula leads to the following corollary, a result which lends some new evidence for the classical Birch and Swinnerton-Dyer conjecture.

Corollary 4 If L(E/K, χ, 1) ≠ 0, then E(K_{∞})^{χ} and LLI_{p}(E/K_{∞})^{χ} are finite.

Remarks:

1. The restriction that χ be of p-power conductor is not essential for the method that is used in this work, so that it should be possible, with little extra effort, to establish corollary 4 for arbitrary anticyclotomic χ, and for the χ-part of the full Shafarevich-Tate group and not just its p-primary part, using the techniques in the proof of theorem 1.

2. Corollary 4 was also proved in [BD2] by a different, more restrictive method which requires the assumption that p is a prime of multiplicative reduction for E/K which is inert in K. Hence, in contrast with the previous remark, the method of [BD2] cannot be used to obtain the finiteness of the full Shafarevich-Tate group of E, but only of its p-primary part for a finite set of primes p.

3. The non-vanishing of L(E/K, χ, 1) seems to occur fairly often. For example, Vatsal has shown ([Va1], theorem 1.4) that L(E/K, χ, 1) is non-zero for almost all χ when χ varies over the anticyclotomic characters of p-power conductor for a fixed p.

Another immediate consequence of theorem 1 is that Sel(K_{∞}, E_{p^{∞}}) is a cotorsion Λ-module whenever L_{p}(E, K) is not identically 0, so that in particular one has

Corollary 5 If L_{p}(E, K) is non-zero, then the Mordell-Weil group E(K_{∞}) is finitely generated.

Remark: The non-vanishing of L_{p}(E, K) has been established by Vatsal. See for example theorem 1.1 of [Va2] which even gives a precise formula for the associated μ-invariant.

Assumptions:

Let E_{p} be the mod p representation of G_{ℚ} attached to E. For simplicity, it is assumed throughout the paper that (E, K, p) satisfies the following conditions.

Assumption 6 1. The prime p is ≥ 5.

2. The Galois representation attached to E_{p} has image isomorphic to GL_{2}(𝔽_{p}).

3. The prime p does not divide the minimal degree of a modular parametrisation X_{0}(N_{0}) ⟶ E.

4. For all primes ℓ such that ℓ^{2} divides N, and p divides ℓ + 1, the module E_{p} is an irreducible I_{ℓ}-module.

Remarks:

1. Note that these assumptions are satisfied by all but finitely many primes once E is fixed, provided that E has no complex multiplications. They are imposed to simplify the argument and could probably be relaxed. This is unlike the condition in theorem 1 which – although it may appear less natural to the uninitiated – is an essential feature of the situation being studied. Indeed, for square-free N^{−}, the restriction on the parity of the number of primes appearing in its factorisation is equivalent to requiring that the sign in the functional equation of L(E, K, χ, s), for χ a ramified character of G_{∞}, is equal to 1. Without this condition, the p-adic L-function L_{p}(E, K, s) would vanish identically. See [BD1] for a discussion of this case where it becomes necessary to interpolate the first derivatives L′(E, K, χ, 1).

2. The analogue of theorem 1 for the cyclotomic ℤ_{p}-extension has been proved by Kato. Both the proof of theorem 1 and Kato’s proof of the cyclotomic counterpart are based on Kolyvagin’s theory of Euler systems.

3. The original “Euler system” argument of Kolyvagin relies on the presence of a systematic supply of algebraic points on E - the so-called Heegner points defined over K and over abelian extensions of K. As can be seen from corollaries 4 and 5, the situation in which we have placed ourselves precludes the existence of a non-trivial norm-compatible system of points in E(K_{∞}). One circumvents this difficulty by resorting to the theory of congruences between modular forms and the Cerednik-Drinfeld interchange of invariants, which, for each n ≥ 1, realises the Galois representation E_{p^{n}} in the p^{n}-torsion of the Jacobian of certain Shimura curves for which the Heegner point construction becomes available. By varying the Shimura curves, a compatible collection of cohomology classes in H^{1}(K_{∞}, E_{p^{n}}) is produced, a collection which can be related to special values of L-functions and is sufficient to control the Selmer group Sel(K_{∞}, E_{p^{∞}}). It should be noted that this geometric approach to the theory of Euler Systems produces ramified cohomology classes in H^{1}(K_{∞}, E_{p^{n}}) directly without resorting to classes defined over auxiliary ring class field extensions of K_{∞}; in particular, Kolyvagin’s derivative operators make no appearance in the argument. In the terminology of [MR], the strategy of this article produces a “Kolyvagin system” without passing through an Euler system in the sense of [Ru]. This lends some support for the suggestion made in [MR] that Kolyvagin systems are the more fundamental objects of study.

Acknowledgements: It is a pleasure to thank Professor Ihara for some useful information on his work, as well as Kevin Buzzard, Ben Howard and the anonymous referees for many helpful comments which led to some corrections and significant improvements in the exposition.

1 p-adic L-functions

1.1 Modular forms on quaternion algebras

Let N^{−} be an arbitrary square-free integer which is the product of an odd number of primes, and let N^{+} be any integer prime to N^{−}. Let p be a prime which does not divide N^{+}N^{−} and write N = pN^{+}N^{−}. Let B be the definite quaternion algebra ramified at all the primes dividing N^{−}, and let R be an Eichler ℤ[1/p]-order of level N^{+} in B. The algebra B is unique up to isomorphism, and the Eichler order R is unique up to conjugation by B^{×}, by strong approximation (cf. [Vi], chapitre III, §4 and §5).

Denote by T the Bruhat-Tits tree of B_{p}^{×}/ℚ_{p}^{×}, where

B_{p} := B ⊗ ℚ_{p} ≃ M_{2}(ℚ_{p}).

The set V(T ) of vertices of T is indexed by the maximal ℤ_{p}-orders in B_{p}, two vertices being adjacent if their intersection is an Eichler order of level p. Let E^{→}(T ) denote the set of ordered edges of T , i.e., the set of ordered pairs (s, t) of adjacent vertices of T . If e = (s, t), the vertex s is called the source of e and the vertex t is called its target; they are denoted by s(e) and t(e) respectively.

The tree T is endowed with a natural left action of B_{p}^{×}/ℚ_{p}^{×} by isometries corresponding to conjugation of maximal orders by elements of B_{p}^{×}. This

action is transitive on both V(T ) and E^{→}(T ). Let R^{×} denote the group of invertible elements of R. The group Γ := R^{×}/ℤ[1/p]^{×} – a discrete subgroup of B_{p}^{×}/ℚ_{p}^{×} in the p-adic topology – acts naturally on T and the quotient T /Γ is a finite graph.

Definition 1.1 A modular form (of weight two) on T /Γ is a ℤ_{p}-valued function f on E^{→}(T ) satisfying

f(γe) = f(e), for all γ ∈ Γ.

Denote by S_{2}(T /Γ) the space of such modular forms. It is a free ℤ_{p}-module of finite rank. More generally, if Z is any ring, denote by S_{2}(T /Γ, Z) the space of Γ-invariant functions on E^{→}(T ) with values in Z.

Duality. Let e_{1}, …, e_{s} be a set of representatives for the orbits of Γ acting on E^{→}(T ), and let w_{j} be the cardinality of the finite group Stab_{Γ}(e_{j}). The space S_{2}(T /Γ) is endowed with a ℤ_{p}-bilinear pairing defined by

⟨f_{1}, f_{2}⟩ = ∑_{i=1}^{s} w_{i}f_{1}(e_{i})f_{2}(e_{i}).  (2)

This pairing is non-degenerate so that it identifies S_{2}(T /Γ) ⊗ ℚ_{p} with its ℚ_{p}-dual.

Hecke operators. Let ℓ ≠ p be a prime which does not divide p. Choose an element M_{ℓ} of reduced norm ℓ in the ℤ[1/p]-order R that was used to define Γ. The double coset ΓM_{ℓ}Γ decomposes as a disjoint union of left cosets:

ΓM_{ℓ}Γ = γ_{1}Γ ∪ ⋯ ∪ γ_{t}Γ.  (3)

Here t = ℓ + 1 (resp. ℓ, 1) if ℓ does not divide N^{+}N^{−} (resp. divides N^{+}, N^{−}). The function f_{|ℓ} defined on E^{→}(T ) by the rule

f_{|ℓ}(e) = ∑_{i=1}^{t} f(γ^{−1}e)  (4)

is independent of the choice of M_{ℓ} or of the representatives γ_{1}, …, γ_{t}, and the assignment f ↦ f_{|ℓ} is a linear endomorphism of S_{2}(T /Γ), called the ℓ-th Hecke operator at ℓ and denoted T_{ℓ} if ℓ does not divide N, and U_{ℓ} if ℓ divides N^{+}N^{−}.

Associated to the prime p there is a Hecke operator denoted U_{p} and defined by the rule

(U_{p}f)(e) = ∑_{s(e′)=t(e)} f(e′).  (5)

The Hecke operators T_{ℓ} (with ℓ|̸N) are called the good Hecke operators. They are self-adjoint for the pairing on S_{2}(T /Γ) defined in (2):

⟨T_{ℓ}f_{1}, f_{2}⟩ = ⟨f_{1}, T_{ℓ}f_{2}⟩.  (6)

Oldforms and Newforms. Let S_{2}(V/Γ, Z) denote the space of Γ-invariant Z-valued functions on V(T ), equipped with a Z-valued bilinear pairing as in (2) with edges replaced by vertices. There are two natural “degeneracy maps” s^{∗}, t^{∗} : S_{2}(V/Γ) ⟶ S_{2}(T /Γ) defined by

s^{∗}(f)(e) = f(s(e)), t^{∗}(f)(e) = f(t(e)).

A form f ∈ S_{2}(T /Γ, Z) is said to be p-old if there exist Γ-invariant functions f_{1} and f_{2} on V(T ) such that

f = s^{∗}(f_{1}) + t^{∗}(f_{2}).  (7)

A form which is orthogonal to the oldforms (i.e., is orthogonal to the image of s^{∗} and t^{∗}) is said to be p-new. The form f is p-new if and only if f is harmonic in the sense that it satisfies

s_{∗}(f)(v) := ∑_{s(e)=v} f(e) = 0, t_{∗}(f)(v) := ∑_{t(e)=v} f(e) = 0, ∀v ∈ V(T ). (8)

This can be seen by noting that s_{∗} and t_{∗} are the adjoints of the maps s^{∗} and t^{∗} respectively.

p-isolated forms. Let 𝕋 be the Hecke algebra acting on the space S_{2}(T /Γ). A form f in this space is called an eigenform if it is a simultaneous eigenvector for all the Hecke operators, i.e.,

T_{ℓ}(f) = a_{ℓ}(f)f, for all ℓ |̸N,
U_{ℓ}(f) = α_{ℓ}(f)f, for all ℓ|N,

where the eigenvalues a_{ℓ}(f) and α_{ℓ}(f) belong to ℤ_{p}. Such an eigenform determines a maximal ideal 𝔪_{f} of 𝕋 by the rule

𝔪_{f} := ⟨p, T_{ℓ} − a_{ℓ}(f), U_{ℓ} − α_{ℓ}(f)⟩ .

Definition 1.2 The eigenform f is said to be p-isolated if the completion of S_{2}(T /Γ) at 𝔪_{f} is a free ℤ_{p}-module of rank one.

In other words, f is p-isolated if there are no non-trivial congruences between f and other modular forms in S_{2}(T /Γ). Note that this is really a property of the mod p eigenform in S_{2}(T /Γ, 𝔽_{p}) associated to f, or of the maximal ideal 𝔪_{f}, so that it makes sense to say that 𝔪_{f} is p-isolated if it is attached to (the reduction of) a p-isolated eigenform.

The Jacquet-Langlands correspondence. The complex vector space S_{2}(H/Γ_{0}(N)) of classical modular forms of weight 2 on H/Γ_{0}(N)) is similarly endowed with an action of Hecke operators, which will also be denoted by the symbols T_{ℓ}, U_{ℓ} and U_{p} by abuse of notation. Let ϕ be an eigenform on Γ_{0}(N) which arises from a newform ϕ_{0} of level N_{0}. It is a simultaneous eigenfunction for all the good Hecke operators T_{ℓ}. Assume that it is also an eigenfunction for the Hecke operator U_{p}. Write a_{ℓ} for the eigenvalue of T_{ℓ} acting on ϕ, and α_{p} for the eigenvalue of U_{p} acting on ϕ.

Remark: If p does not divide N_{0}, so that ϕ is not new at p, then the eigenvalue α_{p} is a root of the polynomial x^{2} − a_{p}x + p, where a_{p} is the eigenvalue of T_{p} acting on ϕ_{0}. If p divides N_{0}, then ϕ = ϕ_{0} and the eigenvalue α_{p} is equal to 1 (resp −1) if the abelian variety attached to ϕ by the Eichler-Shimura construction has split (resp. non-split) multiplicative reduction at p.

Proposition 1.3 Let ϕ be as above. Then there exists an eigenform f in S_{2}(T /Γ, ℂ) satisfying

T_{ℓ}f = a_{ℓ}(ϕ)f for all ℓ |̸N,

U_{ℓ}f = α_{ℓ}(ϕ)f for all ℓ|N^{+}, U_{p}f = α_{p}(ϕ)f. (9)

The form f with these properties is unique up to multiplication by a nonzero complex number. Conversely, given an eigenform f ∈ S_{2}(T /Γ, ℂ), there exists an eigenform ϕ ∈ S_{2}(H/Γ_{0}(N)) satisfying (9).

Proof: Suppose first that p divides N_{0}, so that ϕ is a newform on Γ_{0}(N). Let R_{0} be an Eichler ℤ-order of level pN^{+} in the definite quaternion algebra of discriminant N^{−}. Write R̂_{0} = R_{0} ⊗ ℤ̂ = ∏_{ℓ} R_{0} ⊗ ℤ_{ℓ}, and B̂ := R̂_{0} ⊗ ℚ. The Jacquet-Langlands correspondence (which, in this case, can be established using the Eichler trace formula as in [Ei]; see also [JL] and the dicussion in chapter 5 of [DT]) implies the existence of a unique function

f : B^{×}\B̂^{×}/R̂_{0}^{×} ⟶ ℂ  (10)

satisfying T_{ℓ}f = a_{ℓ}f for all ℓ|̸N, and U_{p}f = α_{p}f. (Where the operators T_{ℓ} and U_{p} are the general Hecke operators defined in terms of double cosets as in [Sh].) Strong approximation identifies the double coset space appearing in (10) with the space R^{×}\B_{p}^{×}/(R_{0})_{p}^{×}. The transitive action of B_{p}^{×} on the set of maximal orders in B_{p} by conjugation yields an action of B_{p}^{×} on T by isometries, for which the subgroup (R_{0})_{p}^{×} is equal to the stabiliser of a certain

oriented edge. In this way B_{p}^{×}/(R_{0})_{p}^{×} is identified with E^{→}(T ), and f can thus be viewed as an element of S_{2}(T /Γ, ℂ).

If p does not divide N_{0}, let a_{p} denote the eigenvalue of T_{p} acting on ϕ_{0}, and let R_{0} denote now the Eichler order of level N^{+} in the quaternion algebra B. As before, to the form ϕ_{0} is associated a unique function

f_{0} : B^{×}\B̂^{×}/R̂_{0}^{×} ⟶ ℂ  (11)

satisfying T_{ℓ}f = a_{ℓ}f for all ℓ|̸N_{0}. As before, strong approximation makes it possible to identify f_{0} with a Γ-invariant function on V(T ). In this description, the action of T_{p} on f_{0} is given by the formula

T_{p}(f_{0}(v)) = ∑_{w} f_{0}(w),

where the sum is taken over the p + 1 vertices w of T which are adjacent to v. Define functions f_{s}, f_{t} : E^{→}(T ) ⟶ ℂ by the rules:

f_{s}(e) = f_{0}(s(e)), f_{t}(e) = f_{0}(t(e)).

The forms f_{s} and f_{t} both satisfy T_{ℓ}(g) = a_{ℓ}g for all ℓ|̸N, and span the two-dimensional eigenspace of forms with this property. A direct calculation reveals that

U_{p}f_{s} = pf_{t}, U_{p}f_{t} = −f_{s} + a_{p}f_{t}.

The function f = f_{s} − α_{p}f_{t} satisfies U_{p}f = α_{p}f, and is, up to scaling, the unique eigenform in S_{2}(T /Γ, ℂ) with this property.

The converse is proved by essentially reversing the argument above: to an eigenform f ∈ S_{2}(T /Γ, ℂ) is associated a function on the adelic coset space attached to B^{×} as in (10); the Jacquet-Langlands correspondence (applied now in the reverse direction) produces the desired ϕ ∈ S_{2}(H/Γ_{0}(N)).

The Shimura-Taniyama conjecture. Let E be an elliptic curve as in the introduction. For each prime ℓ which does not divide N, set

a_{ℓ} = ℓ + 1 − #E(𝔽_{ℓ}).

If E has good ordinary reduction at p, let α_{p} ∈ ℤ_{p} be the unique root of the polynomial x^{2} −a_{p}x+p which is a p-adic unit. Set α_{p} = 1 (resp. −1) if E has split (resp. non-split) multiplicative reduction at p. The following theorem is a consequence of the Shimura-Taniyama conjecture in view of proposition 1.3.

Proposition 1.4 There exists an eigenform f in S_{2}(T /Γ) satisfying

T_{ℓ}f = a_{ℓ}f, for all ℓ |̸N U_{p}f = α_{p}f,
f ∉ pS_{2}(T /Γ).

The form f with these properties is unique up to multiplication by a scalar in ℤ_{p}^{×}.

Proof: Proposition 1.3 shows that there exists a form f ∈ S_{2}(T /Γ, ℂ) satisfying the conclusion of proposition 1.4. The eigenvalues a_{ℓ} belong to ℤ, and, since E is ordinary at p, the eigenvalue α_{p} belongs to the ring of integers O of a quadratic extension of ℚ in which p splits completely. Hence, the form f can be chosen to lie in S_{2}(T /Γ, O). After applying the unique embedding of O into ℤ_{p} which sends α_{p} to a p-adic unit, and rescaling f appropriately, one obtains a form in S_{2}(T /Γ) satisfying the conclusion of proposition 1.4.

1.2 p-adic Rankin L-functions

An eigenform f in S_{2}(T /Γ) is said to be ordinary if the eigenvalue α_{p} of U_{p} acting on f is a p-adic unit. This section recalls the definition of the p-adic Rankin L-function attached to an ordinary form on T /Γ and a quadratic algebra K ⊂ B.

If A is any ℤ-algebra, let

A_{ℓ} = A ⊗ ℤ_{ℓ}, Â = A ⊗ ℤ̂ = ∏_{ℓ} A_{ℓ}.  (12)

Let K be a quadratic algebra of discriminant prime to N which embeds in B. Since B is definite of discriminant N^{−}, the algebra K is an imaginary quadratic field in which all prime divisors of N^{−} are inert. Let O_{K} denote the ring of integers of K and let O = O_{K}[1/p] be the maximal ℤ[1/p]-order in K.

Let G̃_{∞} denote the group

G̃_{∞} = K̂^{×}/(ℚ̂^{×} ∏_{ℓ≠p} O_{ℓ}^{×}K^{×})  (13)

Fix an embedding

Ψ : K ⟶ B satisfying Ψ(K) ∩ R = Ψ(O).  (14)

Such a Ψ exists if and only if all the primes dividing N^{+} are split in K. By passing to the adelisation the embedding Ψ induces a map

Ψ̂ : G̃_{∞} ⟶ B^{×}\B̂^{×}/ (ℚ̂^{×} ∏_{ℓ≠p} R_{ℓ}^{×}) .  (15)

By strong approximation ([Vi], chapitre III, §4), the double coset space appearing on the right has a fundamental region contained in B_{p}^{×} ⊂ B̂^{×}. In fact, strong approximation yields a canonical identification

η : B^{×}\B̂^{×}/ (ℚ̂^{×} ∏_{ℓ≠p} R_{ℓ}^{×}) ⟶ Γ\B_{p}^{×}/ℚ_{p}^{×}.  (16)

The modular form f ∈ S_{2}(T /Γ) determines a pairing between G̃_{∞} and E^{→}(T ) by the rule

[σ, e]_{f} := f(ηΨ̂(σ)e) ∈ ℤ_{p}.  (17)

The embedding Ψ induces an embedding of K_{p}^{×} into B_{p}^{×} and hence yields an action of K_{p}^{×}/ℚ_{p}^{×} on T . This action fixes a single vertex if p is inert in K, and no vertex if p is split in K. Let

U_{n} := (1 + p^{n}O_{K} ⊗ ℤ_{p})^{×}/(1 + p^{n}ℤ_{p})^{×}  (18)

denote the standard compact subgroup of K_{p}^{×}/ℚ_{p}^{×} of level n. Choose a sequence e_{1}, e_{2}, …, e_{n}, … of consecutive edges on T satisfying

Stab_{K_{p}^{×}/ℚ_{p}^{×}}(e_{j}) = U_{j}, j = 1, …, n, ….  (19)

Since α_{p} is a p-adic unit, the pairing defined by equation (17) can be used to define a ℤ_{p}-valued distribution ν̃_{f} on G̃_{∞} by the rule

ν̃_{f}(σU_{j}) := α_{p}^{−j}[σ, e_{j}]_{f},  (20)

for all compact open subsets of G̃_{∞} of the form σU_{j} with σ ∈ G̃_{∞}. The distribution relation for ν̃_{f} is ensured by the fact that f is an eigenform for the U_{p} operator with eigenvalue α_{p}. The distribution ν̃_{f} gives rise to an element L̃_{f} in the completed integral group ring ℤ_{p}[[G̃_{∞}]] by the rule

(L̃_{f})_{n} := ∑_{g∈G̃_{n}} ν̃_{f}(gU_{n}) · g,

where G̃_{n} := G̃_{∞}/U_{n} so that G̃_{∞} is the inverse limit of the finite groups G̃_{n}.

Let Δ denote the torsion subgroup of G̃_{∞}, and let

G_{∞} = G̃_{∞}/Δ ≃ ℤ_{p}.  (21)

Write L_{f} for the natural image of L̃_{f} in the Iwasawa algebra

Λ = ℤ_{p}[[G_{∞}]] ≃ ℤ_{p}[[T]],

and denote by ν_{f} the associated measure on G_{∞}. Note that a different choice of edges e_{j} satisfying (19) has the effect of multiplying L_{f} by an element of G_{∞}, so that L_{f} is only well-defined up to multiplication by such elements.

Functional equations. The Iwasawa algebra is equipped with the involution θ ↦ θ^{∗} sending any σ ∈ G_{∞} to σ^{−1}. Let ϵ = ±1 be the sign in the functional equation of the classical L-function L(E/ℚ, s) attached to E/ℚ. Conjecturally, the value of ϵ determines the parity of the rank of E/ℚ. More precisely, this rank should be even if ϵ = 1, and odd if ϵ = −1. Set ϵ_{p} = ϵ if E does not have split multiplicative reduction over ℚ_{p}, and set ϵ_{p} = −ϵ otherwise. The sign ϵ_{p} is interpreted in [MTT] as the sign in the functional equation for the Mazur-Swinnerton-Dyer p-adic L-function attached to E/ℚ. While this L-function differs markedly from the p-adic Rankin L-function considered in this article, one still has

Lemma 1.5 The equality

L_{f}^{∗} = ϵ_{p}L_{f}

holds in Λ, up to multiplication by an element of G_{∞}.

Proof: See proposition 2.13 and equation (11) of [BD1].

Definition 1.6 The anticyclotomic Rankin L-function attached to f and K is the element L_{p}(f, K) of Λ defined by

L_{p}(f, K) = L_{f}L_{f}^{∗}.

Remark: 1. Note that L_{p}(f, K) is a well-defined element of Λ, since multiplying L_{f} by σ ∈ G_{∞} has the effect of multiplying L_{f}^{∗} by σ^{−1}. Thus the ambiguity in the definition of L_{f} arising from the choice of end in T satisfying (19) is cancelled out.

2. Definition 1.6 extends naturally, mutatis mutandis, to any eigenform g in S_{2}(T /Γ, Z), where Z is any ring in which the eigenvalue of U_{p} acting on g is invertible. In this case the anticyclotomic Rankin L-function L_{p}(g, K) is simply an element of the completed group ring Z[[G_{∞}]].

Let μ_{f,K} be the ℤ_{p}-valued measure on G_{∞} associated to L_{p}(f, K). The function L_{p}(f, K, s) is defined to be the p-adic Mellin transform of μ_{f,K}:

L_{p}(f, K, s) := ∫_{G_{∞}} g^{s−1}dμ_{f,K}(g).

where g^{s−1} := exp((s − 1) log(g)), and log : G_{∞} → ℚ_{p} is a choice of p-adic logarithm.

Interpolation properties. Let ϕ be the normalised eigenform on Γ_{0}(N) attached to f via the Jacquet-Langlands correspondence of proposition 1.3, and let Ω_{f} = ⟨ϕ, ϕ⟩ denote the Peterson scalar product of ϕ with itself. It is known (cf. [Zh], sec. 1.4) that the measure μ_{f,K} on G_{∞} satisfies the following p-adic interpolation property:

| ∫G̃_{∞} χ(g)dμ_{f,K}(g)|^{2} =^{.} L(f, K, χ, 1)/(\sqrt{Disc(K)}Ω_{f}),

for all ramified finite order characters χ of G̃_{∞}. Here the values of χ and μ_{f,K} are viewed as complex numbers by fixing an embedding of ℚ̄_{p} in ℂ, and the absolute value taken on the left-hand side is the complex one. The symbol =^{.} indicates an equality up to a simple algebraic fudge factor expressed as a product of terms comparatively less important than the quantities explicitly described in the formulas. Note in particular that dividing L(f, K, χ, 1) by the complex period Ω_{f} yields an algebraic number.

Elliptic curves. If E is an elliptic curve as in the introduction, let f_{E} be the modular form in S_{2}(T /Γ) attached to it by proposition 1.4. The p-adic L-function attached to E and K is defined by:

L_{p}(E, K) := L_{p}(f_{E}, K), L_{p}(E, K, s) := L_{p}(f_{E}, K, s).  (22)

Remark: Note that L_{p}(E, K) is only well-defined up to multiplication by a unit in ℤ_{p}^{×}, since the same is true of the form f_{E} attached to it by proposition 1.4.

2 Selmer Groups

2.1 Galois representations and cohomology

Let f be an ordinary eigenform in S_{2}(T /Γ) with coefficients in ℤ_{p}, and let K be a quadratic imaginary field in which all primes dividing N^{−} (resp. N^{+}) are inert (resp. split). To these two objects a p-adic L-function L_{p}(f, K) was attached in section 1. This section introduces an invariant of a more arithmetic nature – the so called Selmer group attached to f and K.

Galois representations. To f is attached a continous representation of the Galois group G_{ℚ}:

V_{f} ≃ ℚ_{p}^{2},

satisfying

trace((Frob_{ℓ})_{|V_{f}}) = a_{ℓ}(f), for all ℓ |̸N.  (23)

This representation is constructed by invoking proposition 1.3 to associate to f a classical eigenform ϕ ∈ S_{2}(H/Γ_{0}(N)) with the same Hecke eigenvalues as f at the good primes. The representation V_{f} arises in the Jacobian of J_{0}(N) using the well-known constrction of Eichler and Shimura ([DDT], sec. 3.1).

The action of the compact group G_{ℚ} is continous for the p-adic topology on V_{f} and hence preserves a ℤ_{p}-lattice T_{f}. Let

A_{f} = V_{f}/T_{f} ≃ (ℚ_{p}/ℤ_{p})^{2}  (24)

be the divisible G_{ℚ}-module attached to f, and let A_{f,n} := A_{f}[p^{n}] denote the p^{n}-torsion submodule of A_{f}. It will be occasionally convenient to denote A_{f} by A_{f,∞}. Likewise write T_{f,n} = T_{f}/p^{n}T_{f} and set T_{f,∞} := T_{f}. Note that for n < ∞, the modules A_{f,n} and T_{f,n} are isomorphic as G_{ℚ}-modules, but the A_{f,n} fit naturally into an inductive system while the T_{f,n} are part of a projective system. It is therefore useful to maintain the notational distinction between the two.

The fact that f is ordinary at p implies that A_{f,n} is ordinary, in the sense that it has a quotient A_{f,n}^{(1)} which is free of rank one over ℤ/p^{n}ℤ and on which the inertia group I_{p} at p acts trivially. The kernel of the natural projection A_{f,n} ⟶ A_{f,n}^{(1)} is a free module of rank one over ℤ/p^{n}ℤ, denoted A_{f,n}^{(p)}, on which I_{p} acts via the the p-adic cyclotomic character

ϵ : G_{ℚ} ⟶ ℤ_{p}^{×}

describing the action of G_{ℚ} on the p-power roots of unity.

In our treatment of the Selmer group attached to f and K, it is convenient to make the following technical assumption on f:

Assumption 2.1 The Galois representation A_{f,1} is surjective. Furthermore, for all ℓ dividing N_{0} exactly, the Galois representation A_{f,1} has a unique one-dimensional subspace A_{f,1}^{(ℓ)} on which Gal(ℚ̄_{ℓ}/ℚ_{ℓ}) acts via ϵ or −ϵ.

Remark: 1. Note that assumption 2.1 is automatically satisfied for ℓ if A_{f,1} is ramified at ℓ, because A_{f,n} arises from the Tate module of an abelian variety which acquires purely toric reduction over the quadratic unramified extension of ℚ_{ℓ}. If A_{f,1} is unramified at ℓ, then the Frobenius element at ℓ acts on A_{f,1} with eigenvalues ±1 and ±ℓ, and the condition in assumption 2.1 stipulates that p should not divide ℓ^{2} − 1.

2. For the same reason as explained in remark 1, the maximal submodule A_{f,n}^{(ℓ)} on which G_{ℚ_{ℓ}} acts via ±ϵ is free of rank one over ℤ/p^{n}ℤ.

Lemma 2.2 Suppose that E satisfies assumption 6 of the introduction. Then assumption 2.1 is satisfied by the modular form f attached to E.

Proof: Note that in this case T_{f} is simply isomorphic to the Tate module of E. The assumption that p does not divide the degree of the modular parametrisation of E implies that the newform on Γ_{0}(N_{0}) attached to E is p-isolated. By Ribet’s level-lowering theorem [Ri2], it follows that the Galois representation attached to A_{f,1} is ramified at all primes dividing N_{0}, and hence lemma 2.2 follows from remark 1 after the statement of assumption 2.1.

ℤ_{p}-extensions. Class field theory identifies the group G̃_{∞} of (13) with the Galois group of the maximal abelian extension K̃_{∞} of K which is unramified outside of p and which is of “dihedral type” over ℚ. The subfield K_{∞} := K̃_{∞}^{Δ} is called the anticyclotomic ℤ_{p}-extension of K. Its Galois group over K is identified with the group G_{∞} ≃ ℤ_{p} of equation (21). Let K_{m} be the m-th layer of K_{∞}/K, so that Gal(K_{m}/K) ≃ ℤ/p^{m}ℤ.

Galois Cohomology. For each m ∈ ℕ and n ∈ ℕ ∪ {∞}, let H^{1}(K_{m}, A_{f,n}) and H^{1}(K_{m}, T_{f,n}) denote the usual continuous Galois cohomology groups of Gal(K̄_{m}/K_{m}) with values in these modules. (Note that

H^{1}(K_{m}, A_{f}) := lim_{n^{⟶}} H^{1}(K_{m}, A_{f,n}), H^{1}(K_{m}, T_{f}) := lim_{n^{⟵}} H^{1}(K_{m}, T_{f,n}).)

To study the behaviour of these groups as K_{m} varies over the finite layers of the anticyclotomic ℤ_{p}-extension, it is convenient to introduce the groups

H^{1}(K_{∞}, A_{f,n}) := lim_{m^{⟶}} H^{1}(K_{m}, A_{f,n}),  Ĥ^{1}(K_{∞}, T_{f,n}) = lim_{m^{⟵}} H^{1}(K_{m}, T_{f,n}),

where the direct limit is taken with respect to the natural restriction maps, and the inverse limit is taken with respect to the norm (corestriction) maps. The compatible actions of the group rings ℤ_{p}[G_{m}] on the groups H^{1}(K_{m}, A_{f,n}) and H^{1}(K_{m}, T_{f,n}) yield an action of the Iwasawa algebra Λ = ℤ_{p}[[G_{∞}]] on both of the groups H^{1}(K_{∞}, A_{f,n}) and Ĥ^{1}(K_{∞}, T_{f,n}).

Local cohomology groups. For each rational prime ℓ, set

K_{m,ℓ} := K_{m} ⊗ ℚ_{ℓ} = ⊕_{λ|ℓ}K_{m,λ},

where the direct sum is taken over all primes λ of K_{m} dividing ℓ, and write for any G_{K_{m}}-module X:

H^{1}(K_{m,ℓ}, X) := ⊕_{λ|ℓ}H^{1}(K_{m,λ}, X).

Set

H^{1}(K_{∞,ℓ}, A_{f,n}) = lim_{m^{⟶}} H^{1}(K_{m,ℓ}, A_{f,n}), Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) = lim_{m^{⟵}} H^{1}(K_{m,ℓ}, T_{f,n})

for the local counterparts of H^{1}(K_{∞}, A_{f,n}) and Ĥ^{1}(K_{∞}, T_{f,n}). The Iwasawa algebra Λ acts naturally on these modules in a manner which is compatible with the restriction maps. For each rational prime ℓ, write

H^{1}(I_{m,ℓ}, A_{f,n}) := ⊕_{λ|ℓ}H^{1}(I_{m,λ}, A_{f,n}),

where I_{m,λ} denotes the inertia group at λ.

Tate duality. Let ℓ be a rational prime, and let n ∈ ℕ ∪ {∞}. The finite Galois modules T_{f,n} = A_{f,n} are isomorphic to their own Kummer duals: the Weil pairing gives rise to a canonical G_{ℚ}-equivariant pairing

T_{f,n} × A_{f,n} ⟶ ℤ/p^{n}ℤ(1) = μ_{p^{n}}.

Combining this with the cup product pairing in cohomology gives rise to the collection of local Tate pairings at the primes above ℓ over the finite layers K_{m} in K_{∞}:

⟨ , ⟩_{m,ℓ} : H^{1}(K_{m,ℓ}, T_{f,n}) × H^{1}(K_{m,ℓ}, A_{f,n}) ⟶ ℚ_{p}/ℤ_{p},  (25)

which gives rise, after passing to the limit with m, to a perfect pairing

⟨ , ⟩_{ℓ} : Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) × H^{1}(K_{∞,ℓ}, A_{f,n}) ⟶ ℚ_{p}/ℤ_{p}.

These pairings satisfy the rule

⟨λκ, s⟩_{ℓ} = ⟨κ, λ^{∗}s⟩_{ℓ},

for all λ ∈ Λ, and hence give an isomorphism of Λ-modules

Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) ⟶ H^{1}(K_{∞,ℓ}, A_{f,n})^{∨},

where the Pontrjagin dual X^{∨} of a Λ-module X is itself endowed with a Λ-module structure by the rule

λf(x) := f(λ^{∗}x), for all λ ∈ Λ, f ∈ X^{∨}, x ∈ X.

2.2 Finite/singular structures

Let ℓ |̸N be a rational prime. The singular part of H^{1}(K_{m,ℓ}, A_{f,n}) is the group

H_{sing}^{1}(K_{m,ℓ}, A_{f,n}) := H^{1}(I_{ℓ}, A_{f,n})^{G_{K_{ℓ}}}.

There is a natural map arising from restriction – the so-called residue map –

∂_{ℓ} : H^{1}(K_{m,ℓ}, A_{f,n}) ⟶ H_{sing}^{1}(K_{m,ℓ}, A_{f,n}).

Let H_{fin}^{1}(K_{m,ℓ}, A_{f,n}) denote the kernel of ∂_{ℓ}. The classes in H_{fin}^{1}(K_{m,ℓ}, A_{f,n}) are sometimes called the finite or unramified classes.

Of course, identical definitions can be made in which A_{f,n} is replaced by T_{f,n}. By passing to the limit as m ⟶ ∞ (taking either a direct or an inverse limit) the definition of the residue map ∂_{ℓ} extends both to H^{1}(K_{∞,ℓ}, A_{f,n}) and to Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) and the groups

H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}),  Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}),
H_{sing}^{1}(K_{∞,ℓ}, A_{f,n}),  Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n})

are defined in the natural way.

Let ℓ be a prime dividing N_{0} exactly. Recall in this case (in view of assumption 2.1) the distinguished line A_{f,n}^{(ℓ)} consisting of elements on which G_{ℚ_{ℓ}} acts via ±ϵ. The ordinary part of H^{1}(K_{∞,ℓ}, A_{f,n}) is defined to be the group

H_{ord}^{1}(K_{∞,ℓ}, A_{f,n}) := H^{1}(K_{∞,ℓ}, A_{f,n}^{(ℓ)}).

Finally, at the prime p, set

H_{ord}^{1}(K_{∞,p}, A_{f,n}) := res_{p}^{−1} (H^{1}(I_{∞,p}, A_{f,n}^{(p)})) ,

where res_{p} : H^{1}(K_{∞,p}, A_{f,n}) ⟶ H^{1}(I_{∞,p}, A_{f,n}) is induced from the restriction maps at the (finitely many) primes of K_{∞}/K above p.

Proposition 2.3 If ℓ is a prime not dividing N, the groups H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) and Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}) are annihilators of each other under the local Tate pairing ⟨ , ⟩_{ℓ}. The same is true of H_{ord}^{1}(K_{∞,ℓ}, A_{f,n}) and Ĥ_{ord}^{1}(K_{∞,ℓ}, T_{f,n}) for ℓ||N. In particular, H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) and Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) are the Pontryagin duals of each other.

Proof: The result over the finite layers K_{m} follows from standard properties of the local Tate pairing, and is then deduced over K_{∞} by passage to the limit.

Proposition 2.3 yields a perfect pairing of Λ-modules (denoted by the same symbols ⟨ , ⟩_{ℓ} by abuse of notation)

⟨ , ⟩_{ℓ} : Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) × H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) ⟶ ℚ_{p}/ℤ_{p}.

The following lemma makes explicit the structure of the local cohomology groups Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) and H^{1}(K_{∞,ℓ}, A_{f,n}).

Lemma 2.4 Suppose that ℓ is a rational prime which does not divide N. If ℓ is split in K/ℚ, then

Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) = 0, H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) = 0.

Proof: Because (ℓ) = λ_{1}λ_{2} is split in K/ℚ, the frobenius element attached to λ_{1} topologically generates a subgroup of finite index in G_{∞}. Hence K_{∞,ℓ} is isomorphic to a direct sum of a finite number of copies of the unramified ℤ_{p}-extension of ℚ_{ℓ}. Since A_{f,n} is of exponent p^{n}, any unramified cohomology class in H^{1}(K_{m,ℓ}, A_{f,n}) becomes trivial after restriction to H^{1}(K_{m′,ℓ}, A_{f,n}) for m′ sufficiently large. This implies the second assertion; the first follows from the non-degeneracy of the local Tate pairing.

The primes ℓ which are inert in K/ℚ exhibit a markedly different behaviour, because they split completely in the anticyclotomic tower. It is the presence of such primes which accounts for some of the essential differences between the anticyclotomic theory and the more familiar Iwasawa theory of the cyclotomic ℤ_{p}-extension.

Lemma 2.5 If ℓ does not divide N and is inert in K/ℚ, then

Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) ≃ H_{sing}^{1}(K_{ℓ}, T_{f,n}) ⊗ Λ,

and

H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) ≃ Hom(H_{sing}^{1}(K_{ℓ}, T_{f,n}) ⊗ Λ, ℚ_{p}/ℤ_{p}).

Proof: Since ℓ is inert in K and K_{∞}/ℚ is an extension of dihedral type, the frobenius element at ℓ in Gal(K_{∞}/ℚ) is of order two and hence ℓ splits completely in K_{∞}/K. The choice of a prime λ_{m} of K_{m} above ℓ thus determines an isomorphism H^{1}(K_{m,ℓ}, T_{f,n}) ⟶ H^{1}(K_{ℓ}, T_{f,n}) ⊗ ℤ_{p}[G_{m}]. Choosing a compatible sequence of primes λ_{m} of K_{m} which lie above each other, one obtains an isomorphism

Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) ≃ H^{1}(K_{ℓ}, T_{f,n}) ⊗ Λ,

from the definition of the completed group ring Λ. The first isomorphism of the lemma now follows by passing to the singular parts of the cohomology, while the second is a consequence of proposition 2.3.

Admissible primes. A rational prime ℓ is said to be n-admissible relative to f if it satisfies the following conditions:

1. ℓ does not divide N = pN^{+}N^{−};

2. ℓ is inert in K/ℚ;

3. p does not divide ℓ^{2} − 1;

4. p^{n} divides ℓ + 1 − a_{ℓ} or ℓ + 1 + a_{ℓ}.

A 1-admissible prime will simply be called admissible (so that in particular any n-admissible prime is admissible).

Note that if ℓ is n-admissible, the module T_{f,n} is unramified at ℓ and the Frobenius element over ℚ at ℓ acts semisimply on this module with eigenvalues ±ℓ and ±1 which are distinct modulo p. From this a direct calculation shows that:

Lemma 2.6 The local cohomology groups H_{sing}^{1}(K_{ℓ}, T_{f,n}) and H_{fin}^{1}(K_{ℓ}, T_{f,n}) are both isomorphic to ℤ/p^{n}ℤ.

Proof: The group H_{sing}^{1}(K_{ℓ}, T_{f,n}) is identified with H^{1}(I_{ℓ}, T_{f,n})^{G_{K_{ℓ}}}. Since T_{f,n} is unramified at ℓ, this first cohomology group is identified with a group of homomorphisms, which necessarily factor through the tame inertia group at ℓ. The Frobenius element over K at (ℓ) acts on this tame inertia group as multiplication by ℓ^{2}, while it acts on T_{f,n} with eigenvalues ℓ^{2} and 1. The result follows from this, in light of the fact that p does not divide ℓ^{2} − 1. Similarly, the group H_{fin}^{1}(K_{ℓ}, A_{f,n}) is identified with the G_{K_{ℓ}}-coinvariants of A_{f,n} which are also isomorphic to ℤ/p^{n}ℤ.

Lemma 2.7 The local groups Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) and Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}) are each free of rank one over Λ/p^{n}Λ.

Proof: Since ℓ is inert in K/ℚ, lemma 2.5 implies that Ĥ^{1}(K_{ℓ,∞}, T_{f,n}) is isomorphic to H^{1}(K_{ℓ}, T_{f,n}) ⊗ Λ. The result now follows from lemma 2.6.

Remark:

1. Note that the n-admissible primes are not the primes appearing in Kolyvagin’s study of the Selmer groups of elliptic curves, where the condition that p^{n} divides ℓ + 1 was imposed.

2. The notion of admissible prime introduced here is similar to the one introduced in [BD0], def. 2.20, the main difference arising from the fact that the local cohomology groups H_{fin}^{1}(K_{ℓ}, A_{f,n}) and H_{sing}^{1}(K_{ℓ}, T_{f,n}) are both free of rank one (and not two) over ℤ/p^{n}ℤ.

2.3 Definition of the Selmer group

Let ℓ be a prime not dividing N. Composing restriction from K_{∞} to K_{∞,ℓ} with ∂_{ℓ} yields residue maps on the global cohomology groups, still denoted ∂_{ℓ} by an abuse of notation,

∂_{ℓ} : H^{1}(K_{∞}, A_{f,n}) → H_{sing}^{1}(K_{∞,ℓ}, A_{f,n}),  (26)
∂_{ℓ} : Ĥ^{1}(K_{∞}, T_{f,n}) → Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}).  (27)

Note that if ℓ is split in K/ℚ, the residue map of (27) is 0 by lemma 2.4. If ∂_{ℓ}(κ) = 0 for κ ∈ H^{1}(K_{∞}, A_{f,n}) (resp. Ĥ^{1}(K_{∞}, T_{f,n})), let

v_{ℓ}(κ) ∈ H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}) (resp. Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}))

denote the natural image of κ under the restriction map at ℓ.

Definition 2.8 The Selmer group Sel_{f,n} attached to f, n and K_{∞} is the group of elements s in H^{1}(K_{∞}, A_{f,n}) satisfying

1. ∂_{ℓ}(s) = 0 for all rational primes ℓ not dividing N.

2. The class s is ordinary at the primes ℓ|N^{−}p.

3. The class s is trivial at the primes ℓ|N^{+}.

Caveat: Note that the group Sel_{f,n} depends on the value of N, hence on the modular form f itself, and not just on the Galois representation A_{f,n} attached to it. The same remark holds for the compactified Selmer group Ĥ_{S}^{1}(K_{∞}, T_{f,n}) defined below:

Definition 2.9 Let S be a square-free integer which is relatively prime to N. The compactified Selmer group Ĥ_{S}^{1}(K_{∞}, T_{f,n}) attached to f, S and K_{∞} is the group of elements κ in Ĥ^{1}(K_{∞}, T_{f,n}) satisfying

1. ∂_{ℓ}(κ) = 0 for all rational primes ℓ not dividing SN;

2. The class κ is ordinary at the primes ℓ|N^{−}p.

3. The class κ is arbitrary at the primes ℓ dividing N^{+}, and at the primes dividing S.

Global reciprocity. Let κ ∈ Ĥ^{1}(K_{∞}, T_{f,n}) and let s ∈ H^{1}(K_{∞}, A_{f,n}) be global cohomology classes. For each rational prime q, let κ_{q} and s_{q} denote the restrictions of these cohomology classes to the (semi-)local cohomology group attached to the prime q. The global reciprocity law of class field theory implies that

∑_{q}⟨κ_{q}, s_{q}⟩_{q} = 0,  (28)

where the sum is taken over all the rational primes. In particular, if κ belongs to Ĥ_{S}^{1}(K_{∞}, T_{f,n}) and s belongs to Sel_{f,n}, then since the local conditions defining these two groups are orthogonal at the primes not dividing S, and since s has trivial residue at the primes dividing S, formula (28) becomes:

∑_{q|S}⟨∂_{q}(κ), v_{q}(s)⟩_{q} = 0.

Of particular interest is the following special case:

Proposition 2.10 Suppose that κ belongs to Ĥ_{ℓ}^{1}(K_{∞}, A_{f,n}). Then

⟨∂_{ℓ}(κ), v_{ℓ}(s)⟩_{ℓ} = 0,

for all s ∈ Sel_{f,n}.

The strategy of the proof of theorem 1 is to produce, for sufficiently many primes ℓ that are inert in K, cohomology classes κ(ℓ) ∈ Ĥ_{ℓ}^{1}(K_{∞}, A_{f,n}) whose residue ∂_{ℓ}(κ(ℓ)) can be related to the p-adic L-function L_{p}(f, K) constructed in section 1. Thanks to proposition 2.10, the elements ∂_{ℓ}(κ(ℓ)) yield relations in a presentation for Sel_{f,n}^{∨}.

3 Some Preliminaries

3.1 Λ-modules

If X is any module over a ring R, let Fitt_{R}(X) denote the Fitting ideal of X over R. If R = Λ and X is finitely generated, let Char(X) denote the characteristic ideal attached to X.

Proposition 3.1 Let X be a finitely generated Λ-module and let L be an element of Λ. Suppose that φ(L) belongs to Fitt_{O}(X ⊗_{φ} O), for all homomorphisms φ : Λ ⟶ O, where O is a discrete valuation ring. Then L belongs to Char(X).

Proof: If X is not Λ-torsion, then Fitt_{Λ}(X) = 0. Since

Fitt_{O}(X ⊗_{φ} O) = φ(Fitt_{Λ}(X)),

it follows that φ(L) = 0 for all φ. This implies (by the Weierstrass preparation theorem, for example) that L = 0. Hence one may assume without loss of generality that X is a Λ-torsion module. In that case the structure theory of Λ-modules ensures the existence of an exact sequence of Λ-modules:

X ⟶^{j} ⊕_{i}Λ/(g_{i}) ⟶ C ⟶ 0,  (29)

where C and ker j are finite Λ-modules and the g_{i} are non-zero distinguished polynomials or powers of p. By definition, g := ∏_{i} g_{i} is a generator of Char(X). Since C is finite, its Λ-Fitting ideal can be generated by two elements ι_{1} and ι_{2} having no common irreducible factors. By tensoring the exact sequence (29) with O one finds that

φ(ι_{i})Fitt_{O}(X ⊗_{φ} O) ⊂ (φ(g)), for i = 1, 2.

It follows by assumption that φ(g) divides φ(ι_{i}L) for all φ. Hence (as can be seen by using the Weierstrass preparation theorem) g divides ι_{i}L for i = 1, 2, and therefore g divides L.

3.2 Controlling the Selmer group

Suppose now that A_{f,1} satisfies the irreducibility condition 2 of assumption 6.

Theorem 3.2 Let s be a non-zero element of H^{1}(K, A_{f,1}). There exist infinitely many n-admissible primes ℓ relative to f such that ∂_{ℓ}(s) = 0 and v_{ℓ}(s) ≠ 0.

Proof: Let ℚ(A_{f,n}) be the extension of ℚ fixed by the kernel of the Galois representation A_{f,n}. It is unramified at the primes not dividing N. Since the discriminant of K is assumed to be prime to N, the fields ℚ(A_{f,n}) and K are linearly disjoint. Letting M denote the compositum of these fields, there is therefore a natural inclusion

Gal(M/ℚ) = Gal(K/ℚ) × Gal(ℚ(A_{f,n})/ℚ) ⊂ {1, τ} × Aut_{ℤ/p^{n}ℤ}(A_{f,n}),

so that elements of Gal(M/ℚ) can be labelled by certain pairs (τ^{j}, T) with j ∈ {0, 1} and T ∈ Aut_{ℤ/p^{n}ℤ}(A_{f,n}). Let M_{s} be the extension of M cut out by the image s̄ of s under restriction to H^{1}(M, A_{f,1}) = Hom(Gal(M̄/M), A_{f,1}). Assume without loss of generality that s belongs to a specific eigenspace for the action of τ, so that

τs = δs, for some δ ∈ {1, −1}.

Under this assumption, the extension M_{s} is Galois over ℚ, not merely over K. In fact, by the assumption that A_{f,1} is an irreducible G_{ℚ}-module, Gal(M_{s}/ℚ) is identified with the semi-direct product

Gal(M_{s}/ℚ) = A_{f,1} ⋊ Gal(M/ℚ),  (30)

where the quotient Gal(M/ℚ) acts on the abelian normal subgroup A_{f,1} of Gal(M_{s}/ℚ) by the rule

(τ^{j}, T)(v) = δ^{j}T̄v.  (31)

Here T̄ denotes the natural image of T in Aut_{𝔽_{p}}(A_{f,1}). By part 2 of assumption 6 on the Galois representation A_{f,1}, the group Gal(M_{s}/ℚ) contains an element of the form (v, τ, T), where 1. The automorphism T has eigenvalues δ and λ, where λ ∈ (ℤ/p^{n}ℤ)^{×} is not equal to ±1 mod p and has order prime to p. (Note that here the assumption that p > 3 is needed.)

2. The vector v ∈ A_{f,1} is non-zero and belongs to the δ-eigenspace for T̄.

Let ℓ |̸N be a rational prime which is unramified in M_{s}/ℚ and satisfies

Frob_{ℓ}(M_{s}/ℚ) = (v, τ, T).  (32)

By the Chebotarev density theorem, there exist infinitely many such primes. In fact, the set of such primes has positive density. The fact, immediate from (32), that Frob_{ℓ}(M/ℚ) = (τ, T) implies that ℓ is an admissible prime. To see that v_{ℓ}(s) ≠ 0, choose a prime λ of M above ℓ, and let d be the (necessarily even) degree of the corresponding residue field extension. Then

Frob_{λ}(M_{s}/M) = (v, τ, T)^{d} = v + δT̄v + T̄^{2}v + ⋯ + δT̄^{d−1}v = dv.

Let s̄ denote the image of s in

H^{1}(M, A_{f,1}) = Hom(Gal(M̄/M), A_{f,1})

under restriction. Since d is prime to p by property 1 of T, it follows that s̄(Frob_{λ}(M_{s}/M)) = ds̄(v) ≠ 0, so that the restriction at λ of s̄ is non-zero. Hence, so is v_{ℓ}(s), a fortiori.

Global cohomology groups. Following [BD0], definition 2.22, a finite set S of primes is said to be n-admissible relative to f if

1. All ℓ ∈ S are n-admissible relative to f.

2. The map Sel(K, A_{f,n}) ⟶ ⊕_{ℓ∈S}H_{fin}^{1}(K_{ℓ}, A_{f,n}) is injective.

A direct argument based on theorem 3.2 shows that n-admissible sets exist. (See also the proof of lemma 2.23 of [BD0].) In fact, any finite collection of n-admissible primes can be enlarged to an n-admissible set.

Proposition 3.3 If S is an n-admissible set, then the group Ĥ_{S}^{1}(K_{∞}, T_{f,n}) is free of rank #S over Λ/p^{n}Λ.

Proof: The fact that H_{S}^{1}(K_{m}, T_{f,n}) is free over ℤ/p^{n}ℤ[G_{m}] is essentially theorem 3.2 of [BD0], whose proof carries over, mutatis mutandis, to the present context with its slightly modified notion of admissible prime. Proposition 3.3 follows by passing to the limit as m ⟶ ∞.

Theorem 3.4 Let 𝔪_{Λ} denote the maximal ideal of Λ. Then

1. The natural map from H^{1}(K, A_{f,1}) → H^{1}(K_{∞}, A_{f,n})[𝔪_{Λ}] induced by restriction is an isomorphism.

2. If S is an n-admissible set, the natural map from Ĥ_{S}^{1}(K_{∞}, T_{f,n})/𝔪_{Λ} to H^{1}(K, T_{f,1}) induced by corestriction is injective.

Proof: Let I_{Λ} denote the augmentation ideal of Λ. The inflation-restriction sequence from K to K_{m} gives the exact sequence

H^{1}(K_{m}/K, A_{f,n}^{G_{Km}}) ⟶ H^{1}(K, A_{f,n}) ⟶^{j} H^{1}(K_{m}, A_{f,n})[I_{Λ}] ⟶
⟶ H^{2}(K_{m}/K, A_{f,n}^{G_{Km}}).

By part 2 of assumption 6 in the introduction, the module A_{f,n}^{G_{Km}} is trivial. (Otherwise, the Galois representation attached to A_{f,n} would have solvable image, contradicting the hypotheses that were made in the introduction.) Hence the map j is an isomorphism. Taking the G_{K}-cohomology of the exact sequence

0 ⟶ A_{f,1} ⟶ A_{f,n} ⟶^{p} A_{f,n−1} ⟶ 0

and using the fact that A_{f,1}^{G_{K}} = 0 once again, shows that the natural map

H^{1}(K, A_{f,1}) ⟶ H^{1}(K, A_{f,n})[p]  (33)

is an isomorphism. It follows that the natural map

H^{1}(K, A_{f,1}) ⟶ H^{1}(K_{m}, A_{f,n})[𝔪_{Λ}]

is an isomorphism as well. Part 1 of theorem 3.4 follows by taking the direct limit as m ⟶ ∞.

Part 2 of theorem 3.4 follows directly from proposition 3.3.

3.3 Rigid pairs

Let W_{f} := ad_{0}(A_{f,1}) = Hom_{0}(A_{f,1}, A_{f,1}) be the adjoint representation attached to A_{f,1}, i.e., the vector space of trace zero endomorphisms of A_{f,1}. It is a three-dimensional 𝔽_{p}-vector space endowed with a natural action of G_{ℚ} arising from the conjugation of endomorphisms. Write W_{f}^{∗} := Hom(W_{f}, μ_{p}) for the Kummer dual of W_{f}.

Recall that A_{f,1} is ordinary at p, so that there is an exact sequence of I_{p}-modules

0 ⟶ A_{f,1}^{(p)} ⟶ A_{f,1} ⟶ A_{f,1}^{(1)} ⟶ 0

where A_{f,1}^{(p)} represents the subspace on which I_{p} acts via the cyclotomic character ϵ, and A_{f,1}^{(1)} represents the I_{p}-coinvariants of A_{f,1}. Let

W_{f}^{(p)} := Hom(A_{f,1}^{(1)}, A_{f,1}^{(p)}).

It is an I_{p}-submodule of W_{f}; let W_{f}^{(1)} := W_{f}/W_{f}^{(p)}. The classes in H^{1}(ℚ_{p}, W_{f}) whose restriction at p belong to H^{1}(I_{p}, W_{f}^{(p)}) are called ordinary at p.

Likewise, if ℓ is a prime which divides N exactly, recall the submodules A_{f,1}^{(ℓ)} and A_{f,1}^{(1)} on which G_{ℚ_{ℓ}} acts by ±ϵ and ±1 respectively. (These submodules are well-defined, by virtue of assumption 2.1.) Set

W_{f}^{(ℓ)} := Hom(A_{f,1}^{(1)}, A_{f,1}^{(ℓ)}).

The classes in H^{1}(ℚ, W_{f}) whose restriction at ℓ belongs to H^{1}(ℚ_{ℓ}, W_{f}^{(ℓ}) are called ordinary at ℓ.

If ℓ is an admissible prime for f, the eigenvalues of Frob_{ℓ} acting on the Galois representation A_{f,1} are ±1 and ±ℓ. Recall also that ℓ^{2} ≠ 1 belongs to 𝔽_{p}^{×}. Therefore, the eigenvalues of Frob_{ℓ} acting on W_{f} (resp. W_{f}^{∗}) are the distinct elements 1, ℓ, and ℓ^{−1} (resp. ℓ, 1 and ℓ^{2}) of 𝔽_{p}^{×}. Let W_{f}^{(ℓ)} and W_{f}^{∗(ℓ)} denote the one-dimensional 𝔽_{p}-subspace on which Frob_{ℓ} acts with eigenvalue ℓ. The classes in H^{1}(ℚ, W_{f}) whose restriction at ℓ belong to H^{1}(ℚ_{ℓ}, W_{f}^{(ℓ)}) are called ordinary at ℓ. (In [Ram], section 3, these classes are referred to as null cocycles.) Note that H^{1}(ℚ_{ℓ}, W_{f}) decomposes as a direct sum of two one-dimensional 𝔽_{p}-subspaces,

H^{1}(ℚ_{ℓ}, W_{f}) = H_{fin}^{1}(ℚ_{ℓ}, W_{f})⊕H_{ord}^{1}(ℚ_{ℓ}, W_{f}^{(ℓ)}),

where

H_{fin}^{1}(ℚ_{ℓ}, W_{f}) := H^{1}(ℚ_{ℓ}^{nr}/ℚ_{ℓ}, W_{f})

is the space of unramified cocycles. A similar remark holds for W_{f}^{∗}.

Let S be a square-free product of admissible primes for f.

Definition 3.5 The S-Selmer group attached to W_{f}, denoted Sel_{S}(ℚ, W_{f}), is the subspace of cohomology classes ξ ∈ H^{1}(ℚ, W_{f}) satisfying

1. For all ℓ which do not divide NS, the image of ξ in H^{1}(ℚ_{ℓ}, W_{f}) belongs to H_{fin}^{1}(ℚ_{ℓ}, W_{f}).

2. The class ξ is ordinary at the primes ℓ dividing NS exactly.

3. The class ξ belongs to the kernel of the restriction to H^{1}(I_{ℓ}, W_{f}) if ℓ is a prime such that ℓ^{2} divides N^{+}.

Similar definitions can be made for Sel_{S}(ℚ, W_{f}^{∗}). Note that H^{1}(ℚ_{ℓ}, W_{f}^{(ℓ)}) and H^{1}(ℚ_{ℓ}, W_{f}^{∗(ℓ)}) are orthogonal to each other under the local Tate pairing.

Proposition 3.6 The modular form f is p-isolated if and only if Sel_{1}(ℚ, W_{f}) is trivial.

Proof: Let R denote the universal ring attached to deformations ρ of the Galois representation A_{f,1}, satisfying

1. The determinant of ρ is the cyclotomic character describing the action of G_{ℚ} on the p-power roots of unity.

2. ρ is unramified outside NS.

3. ρ is ordinary at p, i.e., the restriction of ρ to I_{p} is of the form ( \begin{matrix}ϵ & ∗ \\ 0 & 1 \end{matrix} ).

4. For all ℓ dividing N^{+}N^{−}S exactly, the restriction of ρ to a decomposi-

tion group at ℓ is ordinary, i.e., is of the form ( \begin{matrix}ϵ & ∗ \\ 0 & 1 \end{matrix} ).

The ring R is a complete local Noetherian ℤ_{p}-algebra with residue field 𝔽_{p}. Let 𝔪 denote the maximal ideal of R. Standard results of deformation theory (cf. lemma 2.39 and secs. 2.6 and 2.7 of [DDT], ) identify 𝔪/(p, 𝔪^{2}) with the Pontryagin dual of Sel_{S}(ℚ, W_{f}). It follows that R = ℤ_{p} if and only if Sel_{S}(ℚ, W_{f}) is trivial. Taking S = 1, a calculation as in [W], sec. 3 shows that the ring R surjects onto the ring 𝕋_{f} of Hecke operators acting on the space S_{2}(T /Γ), completed at the maximal ideal attached to f. A deep result of Wiles ([W], [DDT] theorem 3.42) asserts that this surjection is an isomorphism. Hence the fact that R = ℤ_{p} is equivalent to the fact that 𝕋_{f} = ℤ_{p}, which in turn is equivalent to the fact that the modular form f is p-isolated.

If S is a square-free product of admissible primes for A_{f,1}, let Sel_{(S)}(ℚ, W_{f}) denote the Selmer group defined in the same way as Sel_{S}(ℚ, W_{f}) above, but with no condition imposed at the primes of S. Let Sel_{[S]}(ℚ, W_{f}) denote the subgroup of Sel_{S}(ℚ, W_{f}) consisting of classes that are trivial at the primes in S. These notations can be combined: thus, if S_{1}, S_{2}, S_{3} are pairwise coprime square-free products of admissible primes, the group Sel_{S_{1}(S_{2})[S_{3}]}(ℚ, W_{f}) is given the obvious meaning. Similar definitions can be made with W_{f} replaced by W_{f}^{∗}. Note that the Selmer groups Sel_{(S)}(ℚ, W_{f}) and Sel_{[S]}(ℚ, W_{f}^{∗}) are dual Selmer groups in the sense of [DDT], section 2.3, and the same is true of Sel_{S}(ℚ, W_{f}) and Sel_{S}(ℚ, W_{f}^{∗}).

Proposition 3.7 If f is p-isolated, and ℓ is an admissible prime for f, then Sel_{(ℓ)}(ℚ, W_{f}) and Sel_{(ℓ)}(ℚ, W_{f}^{∗}) are one-dimensional 𝔽_{p}-vector spaces.

Proof: It follows from a direct calculation of orders of local cohomology groups, combined with theorem 2.18 of [DDT], that the groups Sel_{1}(ℚ, W_{f}) and Sel_{1}(ℚ, W_{f}^{∗}) have the same cardinality. By proposition 3.6, both these groups are trivial. Applying theorem 2.18 of [DDT] once more, and using the fact that p divides (ℓ + 1)^{2} − a_{ℓ}^{2}, one finds that

#Sel_{(ℓ)}(ℚ, W_{f})/#Sel_{[ℓ]}(ℚ, W_{f}^{∗}) = p.

Hence Sel_{(ℓ)}(ℚ, W_{f}) is one-dimensional over 𝔽_{p}. The same argument, with W_{f} and W_{f}^{∗} interchanged, shows that Sel_{(ℓ)}(ℚ, W_{f}^{∗}) is one-dimensional as well.

Suppose that f is p-isolated so that the conclusion of proposition 3.7 holds. If ℓ ≠ ℓ_{1} is any admissible prime, write

v_{ℓ} : Sel_{(ℓ_{1})}(ℚ, W_{f}) ⟶ H_{fin}^{1}(ℚ_{ℓ}, W_{f}), v_{ℓ}^{∗} : Sel_{(ℓ_{1})}(ℚ, W_{f}^{∗}) ⟶ H_{fin}^{1}(ℚ_{ℓ}, W_{f}^{∗})

for the natural maps induced from restriction at ℓ.

Proposition 3.8 1. If Sel_{ℓ_{1}}(ℚ, W_{f}) ≠ 0, and v_{ℓ_{2}} and v_{ℓ_{2}}^{∗} are both nonzero, then Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) = 0.

2. If Sel_{ℓ_{1}}(ℚ, W_{f}) = 0, and v_{ℓ_{2}} is 0, then either Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) = 0 or Sel_{ℓ_{2}}(ℚ, W_{f}) is one-dimensional.

Proof: 1. Since ℓ_{1} is admissible, we may invoke proposition 3.7 and let ξ_{1} and ξ_{1}^{∗} be generators of Sel_{(ℓ_{1})}(ℚ, W_{f}) and Sel_{(ℓ_{1})}(ℚ, W_{f}^{∗}) respectively. Note that ξ_{1} belongs to Sel_{ℓ_{1}}(ℚ, W_{f}) (ie., the restiction of ξ_{1} at ℓ_{1} is ordinary) since Sel_{ℓ_{1}}(ℚ, W_{f}) ≠ 0. A calculation using theorem 2.18 of [DDT] shows that Sel_{ℓ_{1}}(ℚ, W_{f}) and Sel_{ℓ_{1}}(ℚ, W_{f}^{∗}) have the same dimension, and hence ξ_{1}^{∗} also belongs to H_{ℓ_{1}}^{1}(ℚ, W_{f}^{∗}).

By theorem 2.18 of [DDT],

#Sel_{ℓ_{1}(ℓ_{2})}(ℚ, W_{f})/#Sel_{ℓ_{1}[ℓ_{2}]}(ℚ, W_{f}^{∗}) = p.

The assumption that v_{ℓ_{2}}(ξ_{1}^{∗}) ≠ 0 implies that #Sel_{ℓ_{1}[ℓ_{2}]}(ℚ, W_{f}^{∗}) = 0. Hence Sel_{ℓ_{1}(ℓ_{2})}(ℚ, W_{f}) = Sel_{ℓ_{1}}(ℚ, W_{f}) is one-dimensional and spanned by ξ_{1}. Now the assumption that v_{ℓ_{2}}(ξ_{1}) ≠ 0 shows that Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) = 0.

2. It is convenient to distinguish two cases:

a) If Sel_{ℓ_{2}}(ℚ, W_{f}) = 0, then the space Sel_{(ℓ_{1})ℓ_{2}} is one-dimensional by theorem 2.18 of [DDT]. The assumption that v_{ℓ_{2}}(ξ_{1}) = 0 shows that in fact ξ_{1} generates Sel_{(ℓ_{1})ℓ_{2}}(ℚ, W_{f}). But ξ_{1} does not belong to Sel_{ℓ_{1}(ℓ_{2})}(ℚ, W_{f}) since if it did it would also belong to Sel_{ℓ_{1}}(ℚ, W_{f}) which is trivial by assumption. Hence

Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) = Sel_{(ℓ_{1})ℓ_{2}}(ℚ, W_{f}) ∩ Sel_{ℓ_{1}(ℓ_{2})}(ℚ, W_{f}) = 0.

b) If Sel_{ℓ_{2}}(ℚ, W_{f}) ≠ 0, it is necessarily one-dimensional since it lies in the one-dimensional space Sel_{(ℓ_{2})}(ℚ, W_{f}).

Definition 3.9 A pair (ℓ_{1}, ℓ_{2}) of admissible primes is said to be a rigid pair if the Selmer group Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) is trivial.

In addition to theorem 3.2 guaranteeing the existence of a plentiful supply of n-admissible primes sufficient to control the Selmer group Sel_{f,n}, there arises the need for the somewhat more technical theorems 3.10 and 3.11 below guaranteeing the existence of a large supply of rigid pairs of n-admissible primes in certain favorable circumstances.

Theorem 3.10 Suppose that f is a p-isolated eigenform in S_{2}(T /Γ), and let ℓ_{1} be an admissible prime for f. Let s be a non-zero class in H^{1}(K, A_{f,1}). For any n, there exist infinitely many n-admissible primes ℓ_{2} such that 1. ∂_{ℓ_{2}}(s) = 0 and v_{ℓ_{2}}(s) ≠ 0.

2. Either (ℓ_{1}, ℓ_{2}) is a rigid pair, or Sel_{ℓ_{2}}(ℚ, W_{f}) is one-dimensional.

Proof: Assume as in the proof of theorem 3.2 that s belongs to a fixed eigenspace for complex conjugation, so that τs = δs for some δ ∈ {1, −1}. Write M = K(A_{f,n}) and let M_{s} be the Galois extension of M cut out by s as in the proof of theorem 3.2.

The fact that f is p-isolated implies, by proposition 3.7, that the Selmer groups Sel_{(ℓ_{1})}(ℚ, W_{f}) and Sel_{(ℓ_{1})}(ℚ, W_{f}^{∗}) are one-dimensional over 𝔽_{p}. Let ξ and ξ^{∗} denote as before generators of these spaces. The image ξ̄, ξ̄^{∗} of ξ, ξ^{∗} in H^{1}(M, W_{f}) = Hom(G_{M}, W_{f}) and H^{1}(M, W_{f}^{∗}) cuts out extensions M_{ξ} and M_{ξ^{∗}} of M whose Galois groups are identified, via ξ̄ and ξ̄^{∗}, with W_{f} and W_{f}^{∗} respectively.

Let M_{s,ξ,ξ^{∗}} denote the compositum of M_{s}, M_{ξ}, and M_{ξ^{∗}} over M. Since the Galois representations A_{f,1}, W_{f} and W_{f}^{∗} are absolutely irreducible and pairwise non-isomorphic, the Galois group of M_{s,ξ,ξ^{∗}} over ℚ is isomorphic to the semi-direct product

Gal(M_{s,ξ,ξ^{∗}}/ℚ) = (A_{f,1} × W_{f} × W_{f}^{∗}) ⋊ Gal(M/ℚ),

where the action of the quotient Gal(M/ℚ) on the abelian normal subgroup (A_{f,1} × W_{f} × W_{f}^{∗}) is given by

(τ^{j}, T)(v, w, w^{∗}) = (δ^{j}T̄v, T̄wT̄^{−1}, T̄w^{∗}T̄^{−1} det(T)).

Case 1: Suppose that ξ belongs to Sel_{ℓ_{1}}(ℚ, W_{f}), so that ξ^{∗} belongs also to Sel_{ℓ_{1}}(ℚ, W_{f}^{∗}). The group Gal(M_{s,ξ,ξ^{∗}}/ℚ) contains an element of the form (v, w, w^{∗}, τ, T), where

1. The transformation T acting on A_{f,n} has eigenvalues δ and λ, were λ is an element of (ℤ/p^{n}ℤ)^{×} of order prime to p which is ≠ ±1.

2. The vector v belongs to the unique line in A_{f,1} on which T acts by δ.

3. The vector w (resp. w^{∗}) belongs to the unique line in W_{f} (resp. W_{f}^{∗}) which is fixed by T.

Let ℓ_{2} be a rational prime satisfying

Frob_{ℓ_{2}}(M_{s,ξ,ξ^{∗}}/ℚ) = (v, w, w^{∗}, τ, T).  (34)

There are infinitely many such primes ℓ_{2}, by the Chebotarev density theorem. Now, note that

1. By the same reasoning as in the proof of theorem 3.2, the prime ℓ_{2} is n-admissible and v_{ℓ_{2}}(s) ≠ 0.

2. A similar argument shows that v_{ℓ_{2}}(ξ) ≠ 0 and v_{ℓ_{2}}(ξ^{∗}) ≠ 0. From this it follows, by part 1 of proposition 3.8, that Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) = 0. Hence (ℓ_{1}, ℓ_{2}) is a rigid pair, and theorem 3.10 follows.

Case 2: Suppose that ξ does not belong to Sel_{ℓ_{1}}(ℚ, W_{f}), so that ξ^{∗} also does not belong to Sel_{ℓ_{1}}(ℚ, W_{f}^{∗}). Keeping the notations of case 1, let ℓ_{2} be a rational prime satisfying

Frob_{ℓ_{2}}(M_{s,ξ,ξ^{∗}}/ℚ) = (v, 0, 0, τ, T).  (35)

There are infinitely such primes ℓ_{2}, by the Chebotarev density theorem. Note that the prime ℓ_{2} is n-admissible and v_{ℓ_{2}}(s) ≠ 0, and that v_{ℓ_{2}}(ξ) and v_{ℓ_{2}}(ξ^{∗}) both vanish. It follows from part 2 of proposition 3.8 that either Sel_{ℓ_{1}ℓ_{2}}(ℚ, W_{f}) is trivial – i.e., (ℓ_{1}, ℓ_{2}) is a rigid pair – or that Sel_{ℓ_{2}}(ℚ, W_{f}) is one-dimensional.

Theorem 3.11 Suppose that f is a p-isolated eigenform in S_{2}(T /Γ), and let ℓ_{1} be an admissible prime for f. Let s be a non-zero class in H^{1}(K, A_{f,1}). Suppose further that Sel_{ℓ_{1}}(ℚ, W_{f}) is one-dimensional over 𝔽_{p}. Then there exist infinitely many n-admissible primes ℓ_{2} such that

1. ∂_{ℓ_{2}}(s) = 0 and v_{ℓ_{2}}(s) ≠ 0.

2. (ℓ_{1}, ℓ_{2}) is a rigid pair.

Proof: This follows directly from the analysis of case 1 in the proof of theorem 3.10.

Congruences between modular forms. Let ℓ_{1}, ℓ_{2} be distinct n-admissible primes relative to f, such that p^{n} divides ℓ_{1}+1−ϵ_{1}a_{ℓ_{1}}(f) and ℓ_{2}+1−ϵ_{2}a_{ℓ_{2}}(f), for ϵ_{1} and ϵ_{2} equal to ±1. Let B′ be the definite quaternion algebra of discriminant Disc(B)ℓ_{1}ℓ_{2}, let R′ be an Eichler ℤ[1/p]-order of level N^{+} in B′ and let Γ′ := (R′)^{×}/ℤ[1/p]^{×}. The theory of congruences between modular forms yields the following proposition:

Proposition 3.12 There exists an eigenform g ∈ S_{2}(T /Γ′, ℤ/p^{n}ℤ) such that the equalities modulo p^{n} hold:

T_{q}g ≡ a_{q}(f)g (q |̸Nℓ_{1}ℓ_{2}), U_{q}g ≡ a_{q}(f)g (q|N),  (36)
U_{ℓ_{1}}g ≡ ϵ_{1}g, U_{ℓ_{2}}g ≡ ϵ_{2}g.

If furthermore the pair (ℓ_{1}, ℓ_{2}) is a rigid pair, then g can be lifted to an eigenform with coefficients in ℤ_{p} satisfying (36) above. This form is p-isolated.

Proof: The existence of the mod p^{n} eigenform g, which relies on the concepts and notations introduced in sections 5 and 9, is given in theorem 9.3. This g corresponds to a surjective algebra homomorphism f_{ℓ_{2},ℓ_{1}} : 𝕋_{ℓ_{2},ℓ_{1}} ⟶ ℤ/p^{n}ℤ, where 𝕋_{ℓ_{2},ℓ_{1}} is the Hecke algebra defined in the proof of theorem 9.3. If (ℓ_{1}, ℓ_{2}) is a rigid pair, this algebra is isomorphic to ℤ_{p} and therefore f_{ℓ_{2},ℓ_{1}} lifts to characteristic 0 so that g can be lifted (uniquely) to a form in S_{2}(T /Γ′).

4 The Euler System Argument

4.1 The Euler system

Section 7 describes the construction of certain global cohomology classes

κ(ℓ) ∈ Ĥ_{ℓ}^{1}(K_{∞}, T_{f,n}),

indexed by the n-admissible primes ℓ attached to f. The proof of theorem 1 relies crucially on the existence of these classes and on their behaviour under localisation described in theorems 4.1 and 4.2 below. Both theorems are instances of explicit reciprocity laws relating these explicit cohomology classes to special values of L-functions, and form the technical heart of the proof of theorem 1.

Observe that when ℓ is an n-admissible prime, the local cohomology group Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) decomposes as a direct sum

Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) = Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n})⊕Ĥ_{ord}^{1}(K_{∞,ℓ}, T_{f,n})
= Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n})⊕Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}).

The map ∂_{ℓ} is simply the projection onto the second factor, while v_{ℓ} can be extended naturally to a map

v_{ℓ} : Ĥ^{1}(K_{∞,ℓ}, T_{f,n}) ⟶ Ĥ_{fin}^{1}(K_{∞,ℓ}, T_{f,n}) = Ĥ^{1}(K_{∞,ℓ}, T_{f,n})/Ĥ_{ord}^{1}(K_{∞,ℓ}, T_{f,n})

defined as the projection onto the first factor.

Theorem 4.1 If ℓ is an n-admissible prime, then v_{ℓ}(κ(ℓ)) = 0. The equality

∂_{ℓ}(κ(ℓ)) = L_{f} (mod p^{n})

holds in Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) ≃ Λ/p^{n}Λ, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

Note that the ambiguity in the statement of theorem 4.1 is unavoidable, since the identification of H_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) with Λ/p^{n}Λ, and the element L_{f}, are both only defined up to multiplication by elements in ℤ_{p}^{×} and G_{∞}.

Theorem 4.1 is proved in section 8.

The second theorem describes the localisation of κ(ℓ_{1}) at an n-admissible prime ℓ_{2} which is different from ℓ_{1}. Recall the discrete subgroup Γ′ of PSL_{2}(ℚ_{p}) and the ℤ/p^{n}ℤ-valued eigenform g in S_{2}(T /Γ′, ℤ/p^{n}ℤ) attached to f and (ℓ_{1}, ℓ_{2}) in proposition 3.12.

Theorem 4.2 The equality

v_{ℓ_{2}}(κ(ℓ_{1})) = L_{g}

holds in Ĥ_{fin}^{1}(K_{∞,ℓ_{2}}, T_{f,n}) ≃ Λ/p^{n}Λ, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

Theorem 4.2 is proved in section 9.

Since the definition of g is symmetric in ℓ_{1} and ℓ_{2}, one obtains the following reciprocity formula for the classes κ(ℓ):

Corollary 4.3 For all pairs of n-admissible primes ℓ_{1}, ℓ_{2} attached to f, the equality

v_{ℓ_{1}}(κ(ℓ_{2})) = v_{ℓ_{2}}(κ(ℓ_{1}))

holds in Λ/p^{n}Λ, up to multiplication by elements of ℤ_{p}^{×} and G_{∞}.

4.2 The argument

To an ordinary eigenform f ∈ S_{2}(T /Γ) with coefficients in ℤ_{p} one has associated two invariants: the p-adic L-function L_{p}(f, K) ∈ Λ (section 1) and the Selmer group Sel_{f,n} (section 2). This section explains the proof of theorem 1. In our approach based on congruences between modular forms, it is indispensable to prove the following generalisation which is stronger insofar as it applies to all p-isolated modular eigenforms in S_{2}(T /Γ) with coefficients in ℤ_{p} satisfying assumption 2.1.

Theorem 4.4 Let f be an ordinary eigenform in S_{2}(T /Γ) with coefficients in ℤ_{p} which is p-isolated, and satisfies assumption 2.1. The characteristic power series of Sel_{f,∞}^{∨} divides the p-adic L-function L_{p}(f, K).

Proof: By proposition 3.1, it suffices to show that

φ(L_{f})^{2} belongs to Fitt_{O}(Sel_{f,∞}^{∨} ⊗_{φ} O),  (37)

for all homomorphisms φ of Λ into a discrete valuation ring O. For this it is enough to show that

φ(L_{f})^{2} belongs to Fitt_{O}(Sel_{f,n}^{∨} ⊗_{φ} O), for all n ≥ 1.  (38)

Fix O, φ, and n. Write π for a uniformiser of O, and let e := ord_{π}(p) be the ramification degree of O over ℤ_{p}. Write

t_{f} := ord_{π}(φ(L_{f})).

Assume without loss of generality that

1. t_{f} < ∞. (Otherwise, φ(L_{f}) = 0 and (38) is trivially verified.)

2. The group Sel_{f,n}^{∨} ⊗ O is non-trivial. (Otherwise, its Fitting ideal is equal to O and (38) is trivially verified.)

Theorem 4.4 (or rather, equation (38)) is proved by induction on t_{f}.

We begin by describing the construction of certain cohomology classes attached to an admissible prime ℓ. Let ℓ be any (n + t_{f})-admissible prime, and enlarge {ℓ} to an (n + t_{f})-admissible set S. Let

κ(ℓ) ∈ Ĥ_{ℓ}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊂ Ĥ_{S}^{1}(K_{∞}, T_{f,n+t_{f}})

be the cohomology class attached to ℓ as in section 4.1, and denote by κ_{φ}(ℓ) the natural image of this class in

M := Ĥ_{S}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊗_{φ} O.

Note that this module is free over O/p^{(n+t_{f})}, by proposition 3.3. By theorem 4.1,

ord_{π}(κ_{φ}(ℓ)) ≤ ord_{π}(∂_{ℓ}κ_{φ}(ℓ)) = ord_{π}(φ(L_{f})) = t_{f},

so that t := ord_{π}(κ_{φ}(ℓ)) ≤ t_{f}. Choose an element κ̃_{φ}(ℓ) ∈ M satisfying

π^{t}κ̃_{φ}(ℓ) = κ_{φ}(ℓ).

Note that κ̃_{φ}(ℓ) is well defined modulo the π^{t}-torsion subgroup of M, which is contained in the kernel of the natural homomorphism

Ĥ_{S}^{1}(K_{∞}, T_{f,n+t_{f}}) ⊗_{φ} O ⟶ Ĥ_{S}^{1}(K_{∞}, T_{f,n}) ⊗_{φ} O.

To remove this ambiguity, let κ′_{φ}(ℓ) be the natural image of the class κ̃_{φ}(ℓ) in Ĥ_{S}^{1}(K_{∞}, T_{f,n}) ⊗ O. The key properties of the class κ′_{φ}(ℓ) are summarised in lemmas 4.5 and 4.6 below.

Lemma 4.5 The class κ′_{φ}(ℓ) enjoys the following properties:

1. ord_{π}(κ′_{φ}(ℓ)) = 0.

2. ∂_{q}κ′_{φ}(ℓ) = 0, for all q |̸ℓN^{−}.

3. v_{ℓ}(κ′_{φ}(ℓ)) = 0.

4. ord_{π}(∂_{ℓ}κ′_{φ}(ℓ)) = t_{f} − t.

Proof: The first property follows from the fact that ord_{π}(κ_{φ}(ℓ)) = t. The second is a direct consequence of the fact that κ(ℓ) belongs to Ĥ_{ℓ}^{1}(K_{∞}, T_{f,n+t_{f}}), while the third and fourth follow from theorem 4.1.

Lemma 4.6 The element ∂_{ℓ}(κ′_{φ}(ℓ)) belongs to the kernel of the natural homomorphism

η_{ℓ} : Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) ⊗_{φ} O ⟶ Sel_{f,n}^{∨} ⊗_{φ} O.

Proof: Let I_{φ} denote the kernel of φ. By the global reciprocity law of class field theory, the class κ̃_{φ}(ℓ) satisfies

∑_{q|S}⟨∂_{q}(κ̃_{φ}(ℓ)), s_{q}⟩_{q} = 0,  (39)

for all s ∈ Sel_{f,n+t_{f}} [I_{φ}]. (Here, s_{q} simply denotes the natural image of s in H_{fin}^{1}(K_{q,∞}, A_{f,n+t_{f}}).) On the other hand, π^{t}κ̃_{φ}(ℓ) = κ_{φ}(ℓ) has trivial residue at all the primes q ≠ ℓ. Hence, for such primes, the element ∂_{q}(κ̃_{φ}(ℓ)) annihilates

π^{t}H_{fin}^{1}(K_{∞,q}, A_{f,n+t_{f}})[I_{φ}] ⊃ H_{fin}^{1}(K_{∞,q}, A_{f,n})[I_{φ}].

Hence, if s belongs to Sel_{f,n}[I_{φ}], the terms in the sum (39) corresponding to the primes q ≠ ℓ are zero. Hence so is the term corresponding to ℓ. It follows that ∂_{ℓ}(κ′_{φ}(ℓ)) annihilates the image of Sel_{f,n}[I_{φ}] in H_{fin}^{1}(K_{∞,ℓ}, A_{f,n}), as was to be shown.

We now turn to the proof of (38), in the case where t_{f} = 0, which provides the basis for the induction argument.

Proposition 4.7 If t_{f} = 0, (i.e., L_{f} is a unit) then Sel_{f,n}^{∨} is trivial.

Proof: Since L_{f} is a unit, theorem 4.1 implies that ∂_{ℓ}(κ_{φ}(ℓ)) generates Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}) ⊗_{φ} O, for all n-admissible primes ℓ. Hence the map η_{ℓ} of lemma 4.6 is trivial for all such primes. This is enough to conclude that Sel_{f,n}^{∨} is trivial. For otherwise, Nakayama’s lemma implies that the group

Sel_{f,n}^{∨}/𝔪_{Λ} = (Sel_{f,n}[𝔪_{Λ}])^{∨}

is non-zero. Let s be a non-trivial element of Sel_{f,n}[𝔪_{Λ}]. By part 1 of theorem 3.4, s can be viewed as an element of H^{1}(K, A_{f,1}). Invoking theorem 3.2, choose an n-admisible prime ℓ such that v_{ℓ}(s) ≠ 0. The non-degeneracy of the local Tate pairing implies that η_{ℓ} is non-zero, a contradiction.

Turning now to the general case of equation (38), let Π be the set of rational primes ℓ satisfying the following conditions:

1. ℓ is (n + t_{f})-admissible.

2. The quantity t = ord_{π}(κ_{φ}(ℓ)) is minimal, among all primes satisfying condition 1.

Note that the set Π is non-empty, by theorem 3.2. Let t be the common value of ord_{π}(κ_{φ}(ℓ)) for ℓ ∈ Π.

Lemma 4.8 One has t < t_{f}.

Proof: Suppose not. Then ord_{π}(κ_{φ}(ℓ)) = t_{f}, for all (n+t_{f})-admissible primes ℓ. Let s be a non-zero element of H^{1}(K, A_{f,1})∩Sel_{f,n}, which exists by theorem 3.4. Invoking theorem 3.2, choose an (n + t_{f})-admissible prime ℓ such that v_{ℓ}(s) ≠ 0. By lemma 4.5, the natural image of ∂_{ℓ}(κ′_{φ}(ℓ)) in H^{1}(K_{ℓ}, T_{f,1})⊗_{φ} O is non zero. By lemma 4.6, it is also orthogonal to v_{ℓ}(s) with respect to the local Tate pairing, contradicting the fact that the vectors ∂_{ℓ}(κ′_{φ}(ℓ)) and v_{ℓ}(s) are both supposed to be non-zero and that the Tate pairing is a perfect duality between these two one-dimensional vector spaces over O/π.

Lemma 4.9 There exist primes ℓ_{1}, ℓ_{2} ∈ Π such that (ℓ_{1}, ℓ_{2}) is a rigid pair.

Proof: Choose any ℓ_{1} in Π, and let s denote the natural image of κ′_{φ}(ℓ_{1}) in

Ĥ_{S}^{1}(K_{∞}, T_{f,n}) ⊗_{φ} O/(π) = (Ĥ_{S}^{1}(K_{∞}, T_{f,n})/𝔪_{Λ}) ⊗ (O/π)
⊂ H^{1}(K, T_{f,1}) ⊗ (O/π),

where the last inclusion (cf. part 2 of theorem 3.4) is induced from the corestriction map and the natural projection T_{f,n} ⟶ T_{f,1}. Observe that the class s is a non-zero element of H^{1}(K, T_{f,1}) ⊗ (O/π) which satisfies ∂_{q}(s) = 0 for all q |̸ℓ_{1}N. Invoking theorem 3.10, choose an n + t_{f}-admissible prime ℓ_{2} such that

1. v_{ℓ_{2}}(s) ≠ 0, and

2. either (ℓ_{1}, ℓ_{2}) is a rigid pair, or Sel_{ℓ_{2}}(ℚ, W_{f}) is one-dimensional.

The reader will note that:

t = ord_{π}(κ_{φ}(ℓ_{1})) ≤ ord_{π}(κ_{φ}(ℓ_{2})) ≤ ord_{π}(v_{ℓ_{1}}(κ_{φ}(ℓ_{2}))).  (40)

The first inequality holds by the minimality assumption made in the choice of the prime ℓ_{1}. The second inequality is a consequence of the fact that v_{ℓ_{1}} is a homomorphism. By the reciprocity law of corollary 4.3, and the choice of ℓ_{2},

ord_{π}(v_{ℓ_{1}}(κ_{φ}(ℓ_{2}))) = ord_{π}(v_{ℓ_{2}}(κ_{φ}(ℓ_{1}))) = ord_{π}(κ_{φ}(ℓ_{1})).  (41)

(To see the second equality, note that the inequality

ord_{π}(v_{ℓ_{2}}(κ_{φ}(ℓ_{1}))) ≥ ord_{π}(κ_{φ}(ℓ_{1}))

is clear, and that strict inequality holds precisely when v_{ℓ_{2}}(s) = 0.) Combining (40) and (41), it follows that that the inequalities must be equalities throughout, so that

t = ord_{π}(κ_{φ}(ℓ_{1})) = ord_{π}(κ_{φ}(ℓ_{2})).

Hence ℓ_{2} belongs to Π. If (ℓ_{1}, ℓ_{2}) is a rigid pair, we are done. Otherwise, the group Sel_{ℓ_{2}}(ℚ, W_{f}) is one-dimensional. In that case one can repeat the argument above, with ℓ_{1} replaced by ℓ_{2}, invoking this time theorem 3.11 instead of 3.10 to obtain a pair (ℓ_{2}, ℓ_{3}) satisfying the conclusion of lemma 4.9. This completes the proof of the lemma.

Let (ℓ_{1}, ℓ_{2}) be a rigid pair of (n + t_{f})-admissible primes in Π, whose existence is guaranteed by lemma 4.9. By theorem 4.2, note that t = t_{g} = ord_{π}(φ(L_{g})), where g is the p-isolated eigenform in S_{2}(T /Γ′) attached to f and (ℓ_{1}, ℓ_{2}) through proposition 3.12.

Let Sel_{[ℓ_{1}ℓ_{2}]} denote the subgroup of Sel_{f,n} consisting of classes which are locally trivial at the primes dividing ℓ_{1} and ℓ_{2}. By definition, there is a natural exact sequence of Λ-modules

0 ⟶ S_{ℓ_{1}ℓ_{2}}^{f} ⟶ Sel_{f,n}^{∨} ⟶ Sel_{[ℓ_{1}ℓ_{2}]}^{∨} ⟶ 0,  (42)

where S_{ℓ_{1}ℓ_{2}}^{f} denotes the kernel of the natural surjection of duals of Selmer groups. Note the natural surjection given by local Tate duality:

η_{f} : (Ĥ_{sing}^{1}(K_{∞,ℓ_{1}}, A_{f,n}) ⊕ Ĥ_{sing}^{1}(K_{∞,ℓ_{2}}, A_{f,n})) ⟶ S_{ℓ_{1}ℓ_{2}}^{f}

induced from the inclusion

(S_{ℓ_{1}ℓ_{2}}^{f})^{∨} ⊂ H_{fin}^{1}(K_{∞,ℓ_{1}}, A_{f,n}) ⊕ H_{fin}^{1}(K_{∞,ℓ_{2}}, A_{f,n}).

The domain of η_{f} is isomorphic to (Λ/p^{n}Λ)^{2}, by lemma 2.7. Let η_{f}^{φ} denote the map induced from η_{f} after tensoring by O via φ. The domain of η_{f}^{φ} is isomorphic to (O/p^{n}O)^{2}. By lemma 4.6, the kernel of η_{f}^{φ} contains the vectors (∂_{ℓ_{1}}κ′_{φ}(ℓ_{1}), 0) and (0, ∂_{ℓ_{2}}κ′_{φ}(ℓ_{2})) in

(Ĥ_{sing}^{1}(K_{∞,ℓ_{1}}, A_{f,n}) ⊕ Ĥ_{sing}^{1}(K_{∞,ℓ_{2}}, A_{f,n})) ⊗_{φ} O ≃ (O/p^{n}O)^{2}.

By part 3 of lemma 4.5,

t_{f} − t_{g} = ord_{π}(∂_{ℓ_{1}}κ′_{φ}(ℓ_{1})) = ord_{π}(∂_{ℓ_{2}}κ′_{φ}(ℓ_{2})).

Hence

π^{2(t_{f} −t_{g})} belongs to the Fitting ideal of S_{ℓ_{1}ℓ_{2}}^{f} ⊗_{φ} O.  (43)

One may repeat the above argument with the modular form g. Thus we have an exact sequence similar to (42) but involving g instead of f:

0 ⟶ S_{ℓ_{1}ℓ_{2}}^{g} ⟶ Sel_{g,n}^{∨} ⟶ Sel_{[ℓ_{1}ℓ_{2}]}^{∨} ⟶ 0,  (44)

as well as a surjection given by local Tate duality:

η_{g} : (Ĥ_{fin}^{1}(K_{∞,ℓ_{1}}, A_{f,n}) ⊕ Ĥ_{fin}^{1}(K_{∞,ℓ_{2}}, A_{f,n})) ⟶ S_{ℓ_{1}ℓ_{2}}^{g}.

By global reciprocity, the kernel of the map η_{g}^{φ} obtained from η_{g} after tensoring by O via φ contains the elements

(v_{ℓ_{1}}κ′_{φ}(ℓ_{1}), v_{ℓ_{2}}κ′_{φ}(ℓ_{1})) = (0, v_{ℓ_{2}}κ′_{φ}(ℓ_{1}))

as well as (v_{ℓ_{1}}κ′_{φ}(ℓ_{2}), 0). But

ord_{π}(v_{ℓ_{2}}κ′_{φ}(ℓ_{1})) = ord_{π}(v_{ℓ_{1}}κ′_{φ}(ℓ_{2})) = t_{g} − t = 0.

It follows from this that the module S_{g} ⊗_{φ} O is trivial, and the natural surjection

Sel_{g,n}^{∨} ⊗_{φ} O ⟶ Sel_{[ℓ_{1}ℓ_{2}]}^{∨} ⊗_{φ} O is an isomorphism.  (45)

Recall that, by lemma 4.8,

t_{g} < t_{f}, and that the eigenform g satisfies all the hypotheses of theorem 4.4, including assumption 2.1, in light of the fact that ℓ_{1} and ℓ_{2} are admissible. One is thus in a position to invoke the induction hypothesis to conclude that

φ(L_{g})^{2} belongs to the Fitting ideal of Sel_{g,n}^{∨} ⊗_{φ} O.  (46)

The theory of Fitting ideals implies that

π^{2t_{f}} = π^{2(t_{f} −t_{g})}π^{2t_{g}}
∈ Fitt_{O}(S_{ℓ_{1}ℓ_{2}}^{f} ⊗ O)Fitt_{O}(Sel_{g,n}^{∨} ⊗ O), by (43) and (46)
= Fitt_{O}(S_{ℓ_{1}ℓ_{2}}^{f} ⊗ O)Fitt_{O}(Sel_{[ℓ_{1}ℓ_{2}]}^{∨} ⊗ O), by (45)
⊂ Fitt_{O}(Sel_{f,n}^{∨} ⊗ O), by (42).

Hence (38) is proved: φ(L_{f})^{2} belongs to the Fitting ideal of Sel_{f,n}^{∨} ⊗_{φ} O. Theorem 4.4 follows.

5 Shimura curves

The construction and the properties of the Euler System used in the argument of section 4 are based on the geometry over ℤ of certain Shimura curves, which is reviewed in this section.

Let M be a positive integer, and let M = M^{+}M^{−} be an integer decomposition of M such that M^{−} is a squarefree product (possibly empty) of an even number of prime factors. Following [Ro] and [BD1], one may attach to such a decomposition a Shimura curve X_{M^{+},M^{−}}. If M^{−} is equal to 1, X_{M^{+},M^{−}} is the classical modular curve X_{0}(M) of level M. In the general case, X_{M^{+},M^{−}} is the Shimura curve with level M^{+}-structure associated to the indefinite quaternion algebra of discriminant M^{−}.

5.1 The moduli definition

The curve X = X_{M^{+},M^{−}} has the following moduli interpretations. Models over ℤ[\frac{1}{M}] Let B be the indefinite quaternion algebra over ℚ of discriminant M^{−}. Fix a maximal order R_{max} in B, and an Eichler order R of level M^{+} contained in R_{max}. Write F = F_{ℤ[ \frac{1}{M}]} for the functor from the category of schemes over ℤ[\frac{1}{M}] to the category of sets which associates to a scheme S the set of isomorphism classes of triples (A, ι, C), where:

1. A is an abelian scheme over S of relative dimension 2,

2. ι : R_{max} → End(A) is an action of R_{max} on A,

3. C is a level M^{+}-structure on A, that is, a subgroup scheme of A of order (M^{+})^{2} which is stable and cyclic for the action of R_{max}.

If M^{−} is strictly greater than 1, the functor F is coarsely representable by a smooth projective scheme X_{ℤ[ \frac{1}{M}]} over ℤ[\frac{1}{M}], with smooth fibers. Let H_{∞} be the complex upper half plane, and let R_{1}^{×} be the group of norm 1 elements in R. Fix an embedding of B in M_{2}(ℝ), and write Γ_{∞,1} for the natural image of R_{1}^{×} in PGL_{2}(ℝ). The group Γ_{∞,1} acts discontinuously (on the right) on H_{∞}, and the complex points X(ℂ) of the generic fiber X = X_{ℚ} are identified with the Riemann surface H_{∞}/Γ_{∞,1}. For more information, see [BC], chapter III and [Bu].

If M^{−} is equal to 1, then B is isomorphic to the split quaternion algebra M_{2}(ℚ), and one may assume that R_{max} corresponds to the standard maximal order M_{2}(ℤ). A triple (A, i, C) as above is then of the form (E×E, ι, C_{E}×C_{E}) where E is an elliptic curve, C_{E} is a level M-structure on E, and the action ι is the natural matrix action of M_{2}(ℤ) on E × E. Thus, the functor F is coarsely representable by the affine modular curve Y_{0}(M)_{ℤ[ \frac{1}{M}]}. The projective completion X_{0}(M)_{ℤ[ \frac{1}{M}]} of Y_{0}(M)_{ℤ[ \frac{1}{M}]}, obtained by adding a finite set of cusps, is a moduli space for generalized elliptic curves with level M-structure. See [DR] and [Bu]. Models over ℤ_{ℓ} for ℓ | M^{+} Assume that ℓ is a prime dividing exactly M^{+}. (This is the only case which is relevant to the arguments of this paper.)

The reader is referred to [Ed], sections 3 and 4, [DR], [KM], and [Bu] for the definition of the variant of the moduli functor F which can be used in this case. The resulting canonical model X_{ℤ_{ℓ}} is a nodal model of X, in the sense of [Ed]. That is:

1. X_{ℤ_{ℓ}} is proper and flat over ℤ_{ℓ}, and its generic fiber is X (viewed as a curve over ℚ_{ℓ}), 2. the irreducible components of the special fiber X_{𝔽_{ℓ}} of X_{ℤ_{ℓ}} are smooth,

and the only singularities of X_{𝔽_{ℓ}} are ordinary double points.

More precisely, the special fiber X_{𝔽_{ℓ}} consists of two copies of the irreducible curve (X_{M^{+}/ℓ,M^{−}})_{𝔽_{ℓ}} intersecting transversally at the supersingular points, which are identified via the Frobenius morphism at ℓ. (Note that X_{M^{+}/ℓ,M^{−}} has good reduction at ℓ, and a description of (X_{M^{+}/ℓ,M^{−}})_{𝔽_{ℓ}} follows from the model (X_{M^{+}/ℓ,M^{−}})_{ℤ[ \frac{1}{M/ℓ}]} given above.)

Models over ℤ_{ℓ} for ℓ | M^{−} Assume that M^{−} is strictly greater than 1. Fix a prime ℓ dividing M^{−}. As before, one may define a model X_{ℤ_{ℓ}} of X over ℤ_{ℓ} via moduli. The new moduli functor F_{ℤ_{ℓ}} on schemes over ℤ_{ℓ} is defined similarly to F_{ℤ[ \frac{1}{M}]}, except for the requirement that the action ι be special in the sense of [BC], section III.3. The functor F_{ℤ_{ℓ}} is coarsely representable by a scheme X_{ℤ_{ℓ}}, which is a nodal model of X. A more precise description of X_{ℤ_{ℓ}} is given in section 5.2 where, in particular, it is explained that the irreducible components of X_{𝔽_{ℓ}} are rational curves.

5.2 The Cerednik-Drinfeld theorem

(References: see [JoLi1] and [BC] for details, and [BD3], section 4 for an exposition.) Assume that M^{−} is strictly greater than 1, and let ℓ be a prime dividing M^{−}. Let B be the definite quaternion algebra over ℚ of discriminant M^{−}/ℓ, and let R be an Eichler order in B of level M^{+}ℓ. Set B_{ℓ} = B ⊗ ℤ_{ℓ} and R_{ℓ} = R ⊗ ℤ_{ℓ}, and fix an isomorphism κ_{ℓ} : B_{ℓ} ⟶^{∼} M_{2}(ℚ_{ℓ}) mapping R_{ℓ} onto the standard Eichler order of level ℓ in M_{2}(ℤ_{ℓ}), consisting of matrices which are upper triangular modulo ℓ. Write Γ_{ℓ}, respectively, Γ_{ℓ,1} for the image of R[\frac{1}{ℓ}]^{×}, respectively, R[\frac{1}{ℓ}]_{1}^{×} in PGL_{2}(ℚ_{ℓ}), where R[\frac{1}{ℓ}]_{1}^{×} is the subgroup of norm 1 elements in R[\frac{1}{ℓ}]^{×}.

Let ℂ_{ℓ} be a completion of an algebraic closure of ℚ_{ℓ}, and let Ĥ_{ℓ} be the ℓ-adic upper half plane, viewed as a formal scheme over ℤ_{ℓ}. The generic fiber of Ĥ_{ℓ} is identified with a rigid analytic space H_{ℓ} over ℚ_{ℓ} whose ℂ_{ℓ}-valued points are

H_{ℓ}(ℂ_{ℓ}) = ℙ^{1}(ℂ_{ℓ}) − ℙ^{1}(ℚ_{ℓ}) = ℂ_{ℓ} − ℚ_{ℓ}.

The special fiber of Ĥ_{ℓ} consists of an infinite sequence of projective lines, intersecting at ordinary double points. The ℓ-adic group Γ_{ℓ,1} acts discontinously on Ĥ_{ℓ} and H_{ℓ} on the right. The action on H_{ℓ}(ℂ_{ℓ}) is given by the rule

zγ = \frac{az + b}{cz + d}, where γ^{−1} is represented by (\begin{matrix}a & b \\ c & d \end{matrix}) .

The quotients Ĥ_{ℓ}/Γ_{ℓ,1} and H_{ℓ}/Γ_{ℓ,1} exist in the category of formal schemes and of rigid analytic spaces, respectively. Since the irreducible components of the special fiber of Ĥ_{ℓ} have finite stabilizer in Γ_{ℓ,1}, it follows that the irreducible components of the special fiber of H_{ℓ}/Γ_{ℓ,1} are rational curves.

Write X̂_{ℤ_{ℓ}} for the formal completion of X_{ℤ_{ℓ}} along its special fiber, where X_{ℤ_{ℓ}} is the model of X over ℤ_{ℓ} introduced in section 5.1. Let X^{an} be the rigid analytic space over ℚ_{ℓ} associated to X. Fix a quadratic unramified extension ℚ_{ℓ^{2}} of ℚ_{ℓ}, and denote by ℤ_{ℓ^{2}} the ring of integers of ℚ_{ℓ^{2}} and by 𝔽_{ℓ^{2}} its residue field.

Theorem 5.1 The formal schemes X̂_{ℤ_{ℓ}} and Ĥ_{ℓ}/Γ_{ℓ,1} are naturally isomorphic over ℤ_{ℓ^{2}}. In particular, X^{an} is isomorphic to H_{ℓ}/Γ_{ℓ,1} over ℚ_{ℓ^{2}}, and X_{𝔽_{ℓ}} is isomorphic to the special fiber of Ĥ_{ℓ}/Γ_{ℓ,1} over 𝔽_{ℓ^{2}}.

Theorem 5.1 can be proved by comparing the moduli definition of X_{ℤ_{ℓ}} given in section 5.1 with Drinfeld’s interpretation [Dr] of Ĥ_{ℓ}, where the base-change of Ĥ_{ℓ} to the maximal unramified extension of ℤ_{ℓ} is identified with the classifying space of certain formal groups of dimension 2 and height 4 which are endowed with an action of the local order R_{max} ⊗ ℤ_{ℓ}. To obtain the version of the Cerednik-Drinfeld theorem given here, one must descend the isomorphism obtained from the above comparison to ℤ_{ℓ}, using the arguments of Jordan and Livné in [JoLi1]; see also [BC], chapter III.

5.3 Character groups

Let ℓ be a prime dividing exactly M, and let X_{ℤ_{ℓ^{2}}} be a fixed nodal model of X over the unramified quadratic extension ℤ_{ℓ^{2}} of ℤ_{ℓ}. (It is convenient to extend scalars from ℤ_{ℓ} to ℤ_{ℓ^{2}}, also in view of the isomorphism of theorem 5.1.) The dual graph G_{ℓ} = G_{ℓ}(X) of X at ℓ is defined to be the finite graph determined by the following properties:

1. the set of vertices V(G_{ℓ}) is the set of irreducible (geometric) components of the special fiber X_{𝔽_{ℓ^{2}}}, 2. the set of (unoriented) edges E(G_{ℓ}) is the set of singular points of X_{𝔽_{ℓ^{2}}},

3. two vertices v and v′ are joined by an edge e if v and v′ intersect at the singular point e.

Let xy = ℓ^{ν_{e}} be a local equation for X_{ℤ_{ℓ^{2}}} at the double point e. The assignment e ↦ ν_{e} equips the graph G_{ℓ} with a weight function ν : E(G_{ℓ}) → ℕ.

Fix an orientation on G_{ℓ}, that is, a pair of maps s, t : E(G_{ℓ}) → V(G_{ℓ}) such that s(e) and t(e) are the ends of the edge e, called the source and the target of e, respectively.

Let ℤ[V(G_{ℓ})] and ℤ[E(G_{ℓ})] be the module of formal divisors with integer coefficients supported on V(G_{ℓ}) and E(G_{ℓ}), respectively. Let ℤ[V(G_{ℓ})]^{0} and ℤ[E(G_{ℓ})]^{0} be the submodule of degree zero divisors in ℤ[V(G_{ℓ})] and ℤ[E(G_{ℓ})], respectively. Let

∂_{∗} : ℤ[E(G_{ℓ})] → ℤ[V(G_{ℓ})]^{0}

be the ℤ-linear boundary map defined by the rule ∂_{∗}(e) = t(e) − s(e).

Let J = J(X) be the jacobian of X, and let J_{ℤ_{ℓ^{2}}} be the Néron model of J over ℤ_{ℓ^{2}}. Write J_{𝔽_{ℓ^{2}}} for the special fiber of J, J_{𝔽_{ℓ^{2}}}^{0} for the connected component of the origin in J_{𝔽_{ℓ^{2}}}, and Φ_{ℓ} = Φ_{ℓ}(X) for the group of connected (geometric) components J_{𝔽_{ℓ^{2}}}/J_{𝔽_{ℓ^{2}}}^{0}. Denote by X_{ℓ} = X_{ℓ}(X) the character group Hom(T_{𝔽_{ℓ^{2}}}, 𝔾_{m}) of J at ℓ, where T_{𝔽_{ℓ^{2}}} is the maximal torus of J_{𝔽_{ℓ^{2}}}^{0}.

Proposition 5.2 The module X_{ℓ} is isomorphic to the kernel of ∂_{∗}, and hence fits into the exact sequence

0 → X_{ℓ} → ℤ[E(G_{ℓ})] ⟶^{∂_{∗}} ℤ[V(G_{ℓ})]^{0} → 0.

Sketch of proof: The module X_{ℓ} is canonically identified with the integral homology group H_{1}(G_{ℓ}, ℤ). Since the kernel of the map ∂_{∗} (which depends on the choice of orientation on G_{ℓ}) computes H_{1}(G_{ℓ}, ℤ), the proposition follows. (For more details, see [Ed], section 1.)

The character group X_{ℓ} for ℓ | M^{+} Let X_{ℤ_{ℓ^{2}}} be (the base change of) the model of X at a prime ℓ dividing exactly M^{+}, introduced in section 5.1. Let S_{ℓ} = S_{ℓ}(X) be the set of supersingular points of the special fiber X_{𝔽_{ℓ^{2}}}. By the facts recalled in section 5.1, S_{ℓ} is equal to E(G_{ℓ}); furthermore, V(G_{ℓ}) contains two elements, say v_{1} and v_{2}. Fix an orientation on G_{ℓ} so that s(e) = v_{1} and t(e) = v_{2} for all edges e.

Proposition 5.3 The character group X_{ℓ} is identified with the group ℤ[S_{ℓ}]^{0} of degree zero divisors with ℤ-coefficients supported on S_{ℓ}.

Proof: In view of the above remarks, proposition 5.3 follows directly from proposition 5.2.

The character group X_{ℓ} for ℓ | M^{−} Let T_{ℓ} be the Bruhat-Tits tree of PGL_{2}(ℚ_{ℓ}).

Proposition 5.4 The dual graph G_{ℓ} is identified with T_{ℓ}/Γ_{ℓ,1}.

Proof: It follows from theorem 5.1, using the fact that T_{ℓ} is identified with the dual graph of the special fiber of the formal scheme Ĥ_{ℓ} (see [BC], ch. I).

From now on, fix the following orientation on E(G_{ℓ}). Let v_{0} be the vertex of T_{ℓ} corresponding to the local maximal order M_{2}(ℤ_{ℓ}). Say that a vertex of T_{ℓ} is even or odd depending on whether its distance from v_{0} is even or odd, respectively. Since the elements of Γ_{ℓ,1} have determinant 1, they send even vertices of T_{ℓ} to even ones, and odd vertices to odd ones. Thus, there is a well defined notion of even and odd vertex for the quotient graph G_{ℓ}. Define the source and target maps s, t : E(G_{ℓ}) → V(G_{ℓ}) so that s(e) is the even vertex of e and t(e) is the odd vertex of e.

Write δ_{∗} for the restriction to ℤ[E(G_{ℓ})]^{0} of the map ∂_{∗} (relative to the above choice of orientation).

Proposition 5.5 The module X_{ℓ} fits into the exact sequence

0 → X_{ℓ} → ℤ[E(G_{ℓ})]^{0} ⟶^{δ_{∗}} ℤ[V(G_{ℓ})]^{0}.

Proof: By the choice of orientation made above, the elements of H_{1}(G_{ℓ}, ℤ) belong to ℤ[E(G_{ℓ})]^{0}. Proposition 5.5 follows directly from proposition 5.2.

5.4 Hecke operators and the Jacquet-Langlands correspondence

(References: see [Ri1], [Ri2] and [JoLi3].) Assume that M^{−} is strictly greater than 1, and let ℓ be a prime dividing M^{−}. This section is concerned with the study of natural families of Hecke operators acting on the terms of the exact sequence of proposition 5.5. The natural Hecke algebra acting by Picard functoriality on the jacobian J = Pic^{0}(X) induces an action on the character group X_{ℓ}. On the other hand, in order to define an action of Hecke operators on ℤ[E(G_{ℓ})]^{0} and Im(δ_{∗}), one must use the interpretation of these modules in terms of double coset spaces provided by lemma 5.6.

Remark: A Hecke correspondence T on X induces endomorphisms T and ξ of J via Picard (contravariant) and Albanese (covariant) functoriality. The reader is referred to the discussion in [Ri2], pp. 445-6 for details on the definitions. Unless stated otherwise, the Hecke actions considered in this section and in the following ones will be induced from the Hecke action on J obtained from Picard functioriality. If w_{M^{+},1} denotes the Atkin-Lehner involution defined in [BD1], section 1.5, the relation

w_{M^{+},1}T w_{M^{+},1} = ξ

holds. (This can be seen by imitating the arguments in proposition 3.54 of [Sh].) In particular, the Hecke operators corresponding to the primes which do not divide M^{+} induce the same endomorphism via Picard and Albanese functoriality. (For the primes dividing M^{−}, this can also be checked by observing that the corresponding Hecke operators are involutions; in this case, a general property of curves shows that the two functorialities induce the same endomorphism.) In view of the above remarks, the detailed discussion of Hecke actions contained in chapters 3 and 4 of [Ri2] extends to the more general setting of this paper.

Let B, R and κ_{ℓ} be as in section 5.2, and let \underline{R} be the Eichler order of level M^{+} which contains R and is mapped by κ_{ℓ} to M_{2}(ℤ_{ℓ}). Use the notations of formula (12).

Lemma 5.6 1. The set E(G_{ℓ}) is identified with the double coset space R̂^{×}\B̂^{×}/B^{×}.

2. The set V(G_{ℓ}) is identified with the disjoint union ( ˆ\underline{R}^{×}\B̂^{×}/B^{×})×\{0, 1\} of the double coset space ˆ \underline{R}^{×}\B̂^{×}/B^{×} with itself.

Proof: Strong approximation (see [Vi], p. 61) yields the identifications

R̂^{×}\B̂^{×}/B^{×} = R_{ℓ}^{×}\B_{ℓ}^{×}/R[\frac{1}{ℓ}]^{×}, ˆ \underline{R}^{×}\B̂^{×}/B^{×} = \underline{R}_{ℓ}^{×}\B_{ℓ}^{×}/R[\frac{1}{ℓ}]^{×}.

Let G̃_{ℓ} be the graph T_{ℓ}/Γ_{ℓ}, and let E^{→}(G̃_{ℓ}) be the set of oriented edges of G̃_{ℓ}. Using the map κ_{ℓ}, one obtains the identifications

R̂^{×}\B̂^{×}/B^{×} = E^{→}(G̃_{ℓ}), ˆ \underline{R}^{×}\B̂^{×}/B^{×} = V(G̃_{ℓ}).

To conclude, observe that E(G_{ℓ}) is identified with E^{→}(G̃_{ℓ}) by mapping the unoriented edge {v, w} (mod Γ_{ℓ,1}) to the oriented edge (v, w) (mod Γ_{ℓ}) if v is even, and to (w, v) (mod Γ_{ℓ}) if v is odd (see [BD4], lemma 2.2 for more details). Moreover, the disjoint union V(G̃_{ℓ}) × {0, 1} is identified with V(G_{ℓ}) by mapping (ṽ, 0) to the even lift of ṽ and (ṽ, 1) to the odd lift of ṽ, under the natural projection V(G_{ℓ}) → V(G̃_{ℓ}).

Let

α : R̂^{×}\B̂^{×}/B^{×} → ˆ \underline{R}^{×}\B̂^{×}/B^{×}

be the natural projection induced by the inclusion R ⊂ \underline{R}. Denote by ŵ the element of B̂^{×} whose local components away from ℓ are equal to 1, and whose local component at ℓ maps by κ_{ℓ} to the diagonal matrix diag(1, ℓ). Since the Eichler order S = ŵ ˆ\underline{R}^{×}ŵ^{−1} ∩ B of level M^{+} contains R, there is a natural projection from R̂^{×}\B̂^{×}/B^{×} to Ŝ^{×}\B̂^{×}/B^{×}. Call β the composition of this projection with the identification of Ŝ^{×}\B̂^{×}/B^{×} with ˆ \underline{R}^{×}\B̂^{×}/B^{×} induced by the assignment b̂ ↦ ŵ^{−1}b̂.

Let

X′_{ℓ} = ℤ[R̂^{×}\B̂^{×}/B^{×}]^{0}, respectively, X′′_{ℓ} = ℤ[ ˆ\underline{R}^{×}\B̂^{×}/B^{×}]^{0}

be the module of degree zero formal divisors with ℤ-coefficients supported on R̂^{×}\B̂^{×}/B^{×}, respectively, on ˆ \underline{R}^{×}\B̂^{×}/B^{×}.

Define a degeneracy map

d_{∗} : X′_{ℓ} → (X′′_{ℓ})^{2}

by extending by linearity the rule d_{∗}(x) = (β(x), 0) − (0, α(x)) for x in R̂^{×}\B̂^{×}/B^{×}.

Proposition 5.7 The modules ℤ[E(G_{ℓ})]^{0} and Im(δ_{∗}) of proposition 5.5 are identified with X′_{ℓ} and (X′′_{ℓ})^{2}, respectively. Moreover, the map δ_{∗} corresponds under these identifications to d_{∗} (up to sign).

Proof: Lemma 5.6 implies directly that ℤ[E(G_{ℓ})]^{0} and Im(δ_{∗}) are identified with X′_{ℓ} and with a submodule of ℤ[ ˆ\underline{R}^{×}\B̂^{×}/B^{×}]^{2}. A more careful study shows that d_{∗} is surjective, and corresponds to δ_{∗} under the above identifications. Proposition 5.7 follows.

Let 𝕋 be the Hecke algebra acting faithfully on X_{ℓ}, induced by the Hecke algebra acting on the jacobian J (by Picard functoriality). The module ℤ[E(G_{ℓ})]^{0} is also endowed with a faithful action of an algebra 𝕋′ of Hecke correspondences, coming from its double coset description given in lemma 5.6: see [BD1], section 1.5 and also section 1.1. Similarly, the identification of Im(δ_{∗}) with (X′′_{ℓ})^{2} equips Im(δ_{∗}) with the diagonal action of Hecke operators T′′_{q} for q not dividing M/ℓ, and U′′_{q} for q dividing M/ℓ. Moreover, the quotient Im(δ_{∗}) of ℤ[E(G_{ℓ})]^{0} is stable for the induced action of 𝕋′: denote by 𝕋′′ the algebra quotient of 𝕋′ acting faithfully on it. Write 𝕋̃ for the polynomial ring with ℤ-coefficients generated by the indeterminates T̃_{q}, for primes q not dividing M, and Ũ_{q}, for primes q dividing M. Note that the Hecke algebras 𝕋, 𝕋′ and 𝕋′′ are natural quotients of 𝕋̃.

Proposition 5.8 1. The exact sequence of proposition 5.5 is equivariant for the natural actions of 𝕋̃ on X_{ℓ}, ℤ[E(G_{ℓ})]^{0}, and Im(δ_{∗}) defined above.

2. For q ≠ ℓ, the q-th Hecke operator T′_{q} ∈ 𝕋′ (q |̸M) or U′_{q} ∈ 𝕋′ (q|M) acts on Im(δ_{∗}) ≃ (X′′_{ℓ})^{2} via the diagonal action induced by the natural q-th Hecke operator T′′_{q} (q |̸M) or U′′_{q} ∈ 𝕋′ (q|M), respectively, acting on the double coset space X′′_{ℓ}. Moreover, the induced action of U′_{ℓ} ∈ 𝕋′ on Im(δ_{∗}) is given by the formula (x, y) ↦ (T′′_{ℓ}x − y, ℓx), where T′′_{ℓ} is the ℓ-th Hecke operator acting on X′′_{ℓ}.

3. The Hecke algebra 𝕋′ is isomorphic to the Hecke algebra acting on modular forms of level M which are new at M^{−}/ℓ. Its quotient 𝕋, respectively, 𝕋′′ is isomorphic to the Hecke algebra acting on modular forms of level M which are new at M^{−}, respectively, which are new at M^{−}/ℓ and old at ℓ.

Sketch of proof: Step 1: Recalling that M^{−} is divisible by an even number of primes, fix a prime m such that m ≠ ℓ and m | M^{−}. Denote by X′, respectively, X′′ the Shimura curve X_{M^{+}ℓm,M^{−}/ℓm}, respectively, X_{M^{+}m,M^{−}/ℓm}. Write X_{m}(X′), respectively, X_{m}(X′′) for the character group of X′, respectively, X′′ at m. These character groups are described in proposition 5.3: one has

X_{m}(X′) = ℤ[S_{m}(X′)]^{0}, X_{m}(X′′) = ℤ[S_{m}(X′′)]^{0}.

Let a be the natural projection from X′ to X′′, and let b be the composition of the Atkin-Lehner involution at ℓ acting on X′ with a. They induce a surjective degeneracy map

b_{∗} − a_{∗} : X_{m}(X′) → X_{m}(X′′)^{2}.

The module X_{m}(X′), respectively, X_{m}(X′′) is equipped with the action of a Hecke algebra induced by the Hecke algebra acting (by Picard functoriality) on J′ = Pic^{0}(X′), respectively, J′′ = Pic^{0}(X′′). Moreover, X_{m}(X′′)^{2} is endowed with a quotient Hecke action, induced by the map b_{∗} − a_{∗}. Step 2: The assignment sending a supersingular modulus to its ring of endomorphisms equipped with an orientation sets up a bijection of S_{m}(X′), respectively, S_{m}(X′′) onto the set of B^{×}-conjugacy classes of oriented Eichler orders in B of level M^{+}ℓ, respectively, M^{+} (see [BD2], section 2). These conjugacy classes are classified by the elements of the double coset space R̂^{×}\B̂^{×}/B^{×}, respectively, ˆ \underline{R}^{×}\B̂^{×}/B^{×} (see [BD3], section 1). One obtains the isomorphisms

X_{m}(X′) ≃ X′_{ℓ}, X_{m}(X′′) ≃ X′′_{ℓ}.

It can be checked that the above identifications are Hecke equivariant, and that the degeneracy maps d_{∗} and b_{∗} − a_{∗} coincide (up to sign) under these identifications. Step 3: The previous steps give a geometric interpretation of the sequence of proposition 5.5 in terms of the Shimura curves X, X′ and X′′. Then, the proof of proposition 5.8 in the special case where M^{−} = ℓm is a product of two primes is contained in Ribet’s paper [Ri2]: see theorem 3.19, 3.20 and 3.21. The details on the general case are provided in [Ri1].

Let us state for future use the exact sequence of character groups considered in the proof of proposition 5.8.

Proposition 5.9 Let m be a prime ≠ ℓ which divides M^{−}. There is a Hecke equivariant exact sequence

0 → X_{ℓ}(X_{M^{+},M^{−}}) → X_{m}(X_{M^{+}ℓm,M^{−}/ℓm}) → X_{m}(X_{M^{+}m,M^{−}/ℓm})^{2} → 0.

Remark: The identification of the group ℤ[E(G_{ℓ})]^{0} with X_{m}(X_{M^{+}ℓm,M^{−}/ℓm}) and of Im(δ_{∗}) with X_{m}(X_{M^{+}m,M^{−}/ℓm})^{2} can be made entirely canonical. The reader should consult [Ri1] for a thorough discussion of this issue.

Corollary 5.10 (The Jacquet-Langlands correspondence) The subring of the endomorphism ring of J generated by the natural Hecke correspondences on X is identified with the Hecke algebra acting on cusp forms of level M which are new at M^{−}.

Proof: By theorem 5.1, combined with the theory of ℓ-adic uniformization of Mumford and Tate (see [GvdP], chapters VI and VIII), J = Pic^{0}(X) has purely toric reduction at ℓ. Hence, the Hecke algebra 𝕋 acting on X_{ℓ}(X) is canonically identified with the Hecke algebra acting on J. The result follows from the interpretation of 𝕋 given in proposition 5.8.

5.5 Connected components

Let ℓ be a prime dividing M^{−}, and let Φ_{ℓ} be the group of connected components of J at ℓ, defined in section 5.3.

Let us begin by reviewing Grothendieck’s description of Φ_{ℓ}. Let

⟨ , ⟩ : X_{ℓ} × X_{ℓ} → ℤ

be the natural monodromy pairing on X_{ℓ}. It is defined to be the restriction to X_{ℓ}, by the embedding of proposition 5.2, of the diagonal pairing on ℤ[E(G_{ℓ})] given by the rule

⟨e, e′⟩ = ν_{e}δ_{e,e′},

where ν_{e} is the weight of the edge e defined in section 5.3. Let

j : X_{ℓ} → X_{ℓ}^{∨}

be the map induced by the monodromy pairing, where X_{ℓ}^{∨} denotes the ℤdual of X_{ℓ}. Using the notations introduced before proposition 5.8, note that the map j is 𝕋-equivariant, provided that the action of 𝕋 on the source of j is induced by Albanese functoriality and the action of 𝕋 on the target of j is induced by Picard functoriality. The reader is referred to the remark at the beginning of section 5.4, where these two actions are compared, and to [Ri2], pp. 448-9 for an extended discussion.

Proposition 5.11 The group Φ_{ℓ} is canonically identified with the cokernel of j, and hence fits into the Hecke-equivariant exact sequence

0 → X_{ℓ} ⟶^{j} X_{ℓ}^{∨} ⟶^{τ_{ℓ}} Φ_{ℓ} → 0.

Proof: See theorem 11.5 and 12.5 of [Groth], and section 1 and 2 of [Ed].

Corollary 5.12 There is a natural map

ℤ[V(G_{ℓ})]^{0} ⟶^{ω_{ℓ}} Φ_{ℓ}.

Proof: Recall the exact sequence

0 → X_{ℓ} ⟶^{i} ℤ[E(G_{ℓ})] ⟶^{∂_{∗}} ℤ[V(G_{ℓ})] → ℤ → 0

of proposition 5.2, where the map from ℤ[V(G_{ℓ})] to ℤ is the degree map. The free ℤ-modules ℤ[E(G_{ℓ})] and ℤ[V(G_{ℓ})] can be identified canonically with their ℤ-duals, by using their distinguished bases. Taking the ℤ-dual of the above sequence yields

0 → ℤ → ℤ[V(G_{ℓ})] ⟶^{∂^{∗}} ℤ[E(G_{ℓ})] ⟶^{i^{∨}} X_{ℓ}^{∨} → 0,

where the map from ℤ to ℤ[V(G_{ℓ})] is the diagonal embedding, and ∂^{∗} sends a vertex v to − ∑_{s(e)=v} e if v is even, and to ∑_{t(e)=v} e if v is odd. Let j_{0} denote the map from ℤ[E(G_{ℓ})] to itself induced by the pairing ⟨ , ⟩ defined above. Note that the map j of proposition 5.11 is equal to i^{∨} ∘ j_{0} ∘ i. Since Φ_{ℓ} is identified with the cokernel of j, the assignment sending an element x of ℤ[V(G_{ℓ})]^{0} to the element (τ_{ℓ} ∘ i^{∨} ∘ j_{0})(y) of Φ_{ℓ}, where y is chosen so that ∂_{∗}(y) = x, defines the sought-for map ω_{ℓ}.

Using the notations of proposition 5.8, say that a 𝕋̃-module C is Eisenstein if the relations

T̃_{q}c = (q + 1)c for q ∤ M

hold for all c in C.

Proposition 5.13 The restriction of the map ω_{ℓ} of prop. 5.12 to Im(δ_{∗}), viewed as a submodule of ℤ[V(G_{ℓ})]^{0} via prop. 5.5, induces a 𝕋̃-equivariant map

ω̄_{ℓ} : Im(δ_{∗})/((U′_{ℓ})^{2} − 1) → Φ_{ℓ}.

Furthermore, the kernel and cokernel of ω̄_{ℓ} are Eisenstein.

Remark: The proof of proposition 5.13 follows the argument in the proof of theorem 4.3 of [Ri2], where the result is proved in the special case where M^{−} is the product of two primes. See also the previous work of Jordan and Livné [JoLi2]. For the details on the Hecke compatibility in the statement of proposition 5.13, see chapters 3 and 4 of [Ri2], particularly remark 3.24, and the remark at the beginning of section 5.4.

Sketch of proof: Let j′ be the map from ℤ[E(G_{ℓ})]^{0} to its ℤ-dual (ℤ[E(G_{ℓ})]^{0})^{∨} induced by the pairing ⟨ , ⟩ defined before. Similarly, let

⟨⟨ , ⟩⟩ : ℤ[V(G_{ℓ})] × ℤ[V(G_{ℓ})] → ℤ

be the pairing defined by the rule ⟨⟨v, v′⟩⟩ = ν_{v}δ_{v,v′}, where the weight ν_{v} of the vertex v is defined to be the order of the stabilizer in Γ_{ℓ,1} of any lift of v to V(T_{ℓ}). (Note that the quantity ν_{e}, defined in a more geometric fashion in section 5.3, could equivalently be defined to be the order of the stabilizer in Γ_{ℓ,1} of any lift of e to E(T_{ℓ}).) Let

j′′ : Im(δ_{∗}) → Im(δ_{∗})^{∨}

be the map induced by ⟨⟨ , ⟩⟩. Fix a prime divisor m ≠ ℓ of M^{−}, and let Φ′_{m}, respectively, Φ′′_{m} denote the group of connected components of the Néron model over ℤ_{m^{2}} of the jacobian of X_{M^{+}ℓm,M^{−}/ℓm}, respectively, of X_{M^{+}m,M^{−}/ℓm}. The groups Φ′_{m} and Φ′′_{m} are Eisenstein. This follows from a generalization to Shimura curves of theorem 3.12 of [Ri2], which can be obtained from the results of [Bu] and [JoLi3]. The analogue of proposition 5.11 applied to the character groups at m of X_{M^{+}ℓm,M^{−}/ℓm} and X_{M^{+}m,M^{−}/ℓm}, respectively (see theorem 11.5 and 12.5 of [Groth], and section 1 and 2 of [Ed]), combined with proposition 5.9 and the proof of proposition 5.8, yields the identifications

Φ′_{m} = (ℤ[E(G_{ℓ})]^{0})^{∨}/j′(ℤ[E(G_{ℓ})]^{0}), Φ′′_{m} × Φ′′_{m} = Im(δ_{∗})^{∨}/j′′(Im(δ_{∗})).

Let

j′_{0} : Im(δ_{∗}) → \frac{(ℤ[E(G_{ℓ})]^{0})^{∨}}{j′(i(X_{ℓ}))}

be the composition of the isomorphism of Im(δ_{∗}) with ℤ[E(G_{ℓ})]^{0}/i(X_{ℓ}), induced by δ_{∗}, with the map induced by j′. Set

σ : Im(δ_{∗}) → Im(δ_{∗}), (x, y) ↦ ((ℓ + 1)x + T′′_{ℓ}y, T′′_{ℓ}x + (ℓ + 1)y).

One obtains a commutative diagram

0 ⟶ Im(δ_{∗}) ⟶^{j′′} Im(δ_{∗})^{∨} ⟶ Φ′′_{m} × Φ′′_{m} ⟶ 0
↓^{σ}  ↓^{δ_{∗}^{∨}}  ↓
0 ⟶ Im(δ_{∗}) ⟶^{j′_{0}} \frac{(ℤ[E(G_{ℓ})]^{0})^{∨}}{j′(i(X_{ℓ}))} ⟶  Φ′_{m}  ⟶ 0,

where δ_{∗}^{∨} is induced by the ℤ-dual of δ_{∗}. Note that δ_{∗}^{∨} is injective, and its cokernel is identified with Φ_{ℓ}, in view of proposition 5.11. Moreover, the cokernel of σ is identified with Im(δ_{∗})/((U′_{ℓ})^{2} − 1). For, a calculation shows that the composition of the isomorphism

(x, y) ↦ (−x, T′′_{ℓ}x − y)

of Im(δ_{∗}) with σ gives the action of (U′_{ℓ})^{2} − 1. Proposition 5.13 follows from the snake lemma applied to the above diagram.

Let X_{ℤ_{ℓ^{2}}} be (the base change of) the model of X introduced in section 5.1, and let X_{𝔽_{ℓ^{2}}} be the special fiber of X_{ℤ_{ℓ^{2}}}. Write

D = ∑_{P} n_{P} P ∈ Div^{0}(X)

for a degree zero divisor on X with integer coefficients, and Supp(D) for the support of D. Assume that D satisfies the following assumptions:

1. each P ∈ Supp(D) is defined over ℚ_{ℓ^{2}},

2. each P ∈ Supp(D) reduces modulo ℓ to a point r_{ℓ}(P) which belongs to a unique irreducible component of X_{𝔽_{ℓ^{2}}} (that is, r_{ℓ}(P) is not a double point in X_{𝔽_{ℓ^{2}}}).

Thus, D determines an element

r_{ℓ}(D) = ∑_{P} n_{P} r_{ℓ}(P)

in ℤ[V(G_{ℓ})]^{0}. Consider the specialization map

∂_{ℓ} : J(ℚ_{ℓ^{2}}) → Φ_{ℓ}.

Proposition 5.14 Let D ∈ Div^{0}(X) be a divisor satisfying the above assumptions, and let [D] ∈ J(ℚ_{ℓ^{2}}) be the class of D. Then

∂_{ℓ}([D]) = ω_{ℓ}(r_{ℓ}(D)).

Proof: It follows from Edixhoven’s results in [Ed], section 2. (These results are built on Raynaud’s description of ∂_{ℓ} in terms of the minimal model of X over ℤ_{ℓ^{2}}, given in [Ray].)

5.6 Raising the level and groups of connected components

Let f : E^{→}(T )/Γ → ℤ_{p} be a form on T /Γ, defined as in section 1. Thus, f is attached to an integer factorization N = N^{+}N^{−}, where N^{−} is squarefree and divisible by an odd number of primes and N^{+} is divisible by p. (Note that this differs slightly from the notation of section 1, where N was written as pN^{+}N^{−} and N^{+} was assumed to be prime to p.)

Let m be a distinguished prime divisor of N^{−}. In what follows, ℓ denotes a prime number which does not divide N. Write 𝕋 = 𝕋_{N^{+},N^{−}}, respectively, 𝕋_{ℓ} = 𝕋_{N^{+},N^{−}ℓ} for the Hecke algebra acting on cusp forms on Γ_{0}(N), respectively, on Γ_{0}(Nℓ), which are new at N^{−}, respectively, at N^{−}ℓ. Denote by t_{q} and u_{q}, respectively, T_{q} and U_{q} the Hecke operators in 𝕋, respectively, in 𝕋_{ℓ}.

The form f yields a surjective homomorphism

f : 𝕋 → ℤ/p^{n}ℤ

(still denoted by f by an abuse of notation). Write I_{f} for the kernel of f, and 𝔪_{f} for the unique proper maximal ideal of 𝕋 containing I_{f}.

In view of corollary 5.10, 𝕋 can be identified with the m-new quotient of the Hecke algebra 𝕋_{N^{+}m,N^{−}/m} acting faithfully on X_{N^{+}m,N^{−}/m}. Since the character group X_{m}(X_{N^{+}m,N^{−}/m}) arises from the m-new part of J_{N^{+}m,N^{−}/m}, it follows that 𝕋 acts naturally on it (by Picard functoriality).

Theorem 5.15 Assume:

1. 𝔪_{f} is residually irreducible, 2. the completion X_{m}(X_{N^{+}m,N^{−}/m})_{𝔪_{f}} of X_{m}(X_{N^{+}m,N^{−}/m}) at 𝔪_{f} is free of

rank 1 over the completed Hecke algebra 𝕋_{𝔪_{f}},

3. ℓ is a n-admissible prime relative to f.

Then:

1. There exists a surjective homomorphism

f_{ℓ} : 𝕋_{ℓ} → ℤ/p^{n}ℤ

such that f_{ℓ}(T_{q}) = f(t_{q}) for all q ∤ Nℓ, f_{ℓ}(U_{q}) = f(u_{q}) for all q | N, and f_{ℓ}(U_{ℓ}) = ϵ, where ϵ = ±1 is such that p^{n} divides ℓ + 1 − ϵf(t_{ℓ}).

2. Let I_{f_{ℓ}} ⊂ 𝕋_{ℓ} denote the kernel of the homomorphism f_{ℓ}, and let Φ_{ℓ} denote the group of connected components of the Néron model of J_{N^{+},N^{−}ℓ} over ℤ_{ℓ^{2}}, defined in section 5.3. There is a group isomorphism

Φ_{ℓ}/I_{f_{ℓ}} ≃ ℤ/p^{n}ℤ.

Remark. Compare the proof of theorem 5.15 with section 7 of [Ri2], in which the special case where M^{−} is a product of two primes and I_{f} = 𝔪_{f} is a maximal ideal is treated. Proof: The second assumption implies the existence of an isomorphism

X_{m}(X_{N^{+}m,N^{−}/m})^{2}/I_{f} ≃ (ℤ/p^{n}ℤ)^{2}.

Write T′_{q} and U′_{q} for the Hecke operators in 𝕋_{N^{+}ℓ,N^{−}}. By proposition 5.9 and proposition 5.8, there is an action of 𝕋_{N^{+}ℓ,N^{−}} on X_{m}(X_{N^{+}m,N^{−}/m})^{2}, induced by the diagonal action of t_{q} for q ∤ Nℓ and u_{q} for q | N, and such that the Hecke operator U′_{ℓ} acts via the formula

(x, y) ↦ (t_{ℓ}x − y, ℓx).

Since ℓ is n-admissible, t_{ℓ} is equal modulo I_{f} to ϵ(ℓ + 1). By combining this relation with the above formula for the action of U′_{ℓ} and with the fact that p ≥ 5, one obtains that U′_{ℓ} +ϵ is invertible on X_{m}(X_{N^{+}m,N^{−}/m})^{2}/I_{f}, and that the isomorphisms

X_{m}(X_{N^{+}m,N^{−}/m})^{2}/⟨I_{f}, U′_{ℓ} − ϵ⟩ ≃ X_{m}(X_{N^{+}m,N^{−}/m})^{2}/⟨I_{f}, (U′_{ℓ})^{2} − 1⟩ ≃ ℤ/p^{n}ℤ hold. Hence, the action of 𝕋_{N^{+}ℓ,N^{−}} on the above quotient is via a surjective homomorphism

f_{ℓ}′ : 𝕋_{N^{+}ℓ,N^{−}} → ℤ/p^{n}ℤ.

Write I_{f_{ℓ}}′ for the kernel of f_{ℓ}′. By proposition 5.13 and the residual irreducibility of 𝔪_{f}, combined with proposition 5.9 and the remark after it, one finds an isomorphism

Φ_{ℓ}/I_{f_{ℓ}}′ ≃ X_{m}(X_{N^{+}m,N^{−}/m})^{2}/⟨I_{f}, (U′_{ℓ})^{2} − 1⟩.

It follows that f_{ℓ}′ factors through 𝕋_{ℓ}, giving the sought for character f_{ℓ}, and that Φ_{ℓ}/I_{f_{ℓ}} is isomorphic to ℤ/p^{n}ℤ.

Write X^{(ℓ)} for the Shimura curve X_{N^{+},N^{−}ℓ}, J^{(ℓ)} for the jacobian of X^{(ℓ)}, and Pic(X^{(ℓ)}) for the Picard variety of X^{(ℓ)}. Denote by X_{ℓ} the character group of J^{(ℓ)} at ℓ, and by Ta_{p}(J^{(ℓ)}) the p-adic Tate module of J^{(ℓ)}. Note that the Hecke algebra 𝕋_{ℓ} acts faithfully on J^{(ℓ)}.

Lemma 5.16 Under the assumptions of theorem 5.15, the exponent of the ℤ-module Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is equal to p^{n}.

Proof: Since Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is naturally a 𝕋_{ℓ}/I_{f_{ℓ}}-module, and 𝕋_{ℓ}/I_{f_{ℓ}} is isomorphic to ℤ/p^{n}ℤ, it follows that the exponent of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is at most p^{n}. On the other hand, the Mumford-Tate theory of ℓ-adic uniformization, combined with the Cerednik-Drinfeld theorem, (see section 5.2 and [GvdP], chapters VI and VIII) shows the existence of a symmetric pairing

[ , ] : X_{ℓ} × X_{ℓ} → ℚ_{ℓ}^{×}  (47)

such that there exists an exact sequence of 𝕋_{ℓ}[Gal(ℚ̄_{ℓ^{2}}/ℚ_{ℓ^{2}})]-modules

0 → X_{ℓ} ⟶^{j̃} X_{ℓ}^{∨} ⊗ ℚ̄_{ℓ^{2}}^{×} → J^{(ℓ)}(ℚ̄_{ℓ^{2}}) → 0,  (48)

where X_{ℓ}^{∨} denotes the ℤ-dual Hom(X_{ℓ}, ℤ) of X_{ℓ}, and where the map j̃ is induced by [ , ]. The sequence (48) is Hecke-equivariant, provided that the action of 𝕋_{ℓ} on the first term is induced by Albanese functoriality, and the action on the second and third term is induced by Picard functoriality. Note also that the need of extending scalars to ℚ_{ℓ^{2}} in (48) arises from the fact that the isomorphism of rigid spaces stated in theorem 5.1 is only defined over ℚ_{ℓ^{2}} and not over ℚ_{ℓ}. The pairing [ , ] is related to monodromy pairing of section 5.5 by the rule

ord_{ℓ} ∘ [ , ] = ⟨ , ⟩  (49)

(see [M], theorem 7.6). In view of theorem 5.15, choose an element c of Φ_{ℓ}/I_{f_{ℓ}} of order p^{n}, and lift c to an element c̃ of Φ_{ℓ} of p-power order p^{n′}, with n′ ≥ n. In view of proposition 5.11, fix an element b ∈ X_{ℓ}^{∨} such that τ_{ℓ}(b) = c̃, and let a be the element of X_{ℓ} such that p^{n′}b = j(a). Formula (49) shows that ord_{ℓ} of the period j̃(a) ∈ X_{ℓ}^{∨} ⊗ ℚ_{ℓ}^{×} is divisible by p^{n′}. Thanks to the sequence (48), the choice of a p^{n′}-root of j̃(a) determines an element t̃ of J^{(ℓ)}[p^{n′}] defined over an unramified extension of ℚ_{ℓ}, whose natural image in Φ_{ℓ} is equal to c̃. Writing t for the image of t̃ in J^{(ℓ)}[p^{n′}]/I_{f_{ℓ}}, then the natural image of t in Φ_{ℓ}/I_{f_{ℓ}} is equal to c. Since

Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} = J^{(ℓ)}[p^{n′}]/I_{f_{ℓ}},

it follows that t is an element of order ≥ p^{n} in Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}. Since the exponent of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is at most p^{n}, this proves lemma 5.16.

Theorem 5.17 Under the assumptions of theorem 5.15, the Galois representations Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} and T_{f,n} are isomorphic.

Proof: Let 𝔪_{f_{ℓ}} be the maximal ideal in 𝕋_{ℓ} containing I_{f_{ℓ}}. (Thus, 𝕋_{ℓ}/𝔪_{f_{ℓ}} is isomorphic to ℤ/pℤ.) The proof is naturally divided into two steps. In the first step, one works modulo 𝔪_{f_{ℓ}}, and a known result of [BoLeRi] on the structure of Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}} is invoked. Step 1. This first step shows that Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}} is isomorphic to T_{f,1}. Taking p-torsion in the sequence (48) yields the exact sequence of 𝕋_{ℓ}[Gal(ℚ̄_{ℓ^{2}}/ℚ_{ℓ^{2}})]- modules

0 → X_{ℓ}^{∨} ⊗ μ_{p} → J^{(ℓ)}[p] → X_{ℓ}/p → 0.  (50)

After tensoring (50) with 𝕋_{ℓ}/𝔪_{f_{ℓ}}, one finds

0 → ((X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y) ⊗ μ_{p} → J^{(ℓ)}[p]/𝔪_{f_{ℓ}} → X_{ℓ}/𝔪_{f_{ℓ}} → 0,  (51)

where Y is a certain submodule of X_{ℓ}^{∨}/𝔪_{f_{ℓ}}. Taking Galois cohomology over ℚ_{ℓ^{2}} of (51) yields an exact sequence

X_{ℓ}/𝔪_{f_{ℓ}} → H^{1}(ℚ_{ℓ^{2}}, ((X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y) ⊗ μ_{p}) →  (52)
→ H^{1}(ℚ_{ℓ^{2}}, J^{(ℓ)}[p]/𝔪_{f_{ℓ}}) → H^{1}(ℚ_{ℓ^{2}}, X_{ℓ}/𝔪_{f_{ℓ}}).

Note the identifications

H^{1}(ℚ_{ℓ^{2}}, ((X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y) ⊗ μ_{p}) = ((X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y) ⊗ H^{1}(ℚ_{ℓ^{2}}, μ_{p}) =

((X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y) ⊗ ℚ_{ℓ^{2}}^{×}/(ℚ_{ℓ^{2}}^{×})^{p} = (X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y,  (53)

where the last equality follows from the fact that ℓ is an admissible prime and hence p |̸ℓ^{2} − 1. Moreover, J^{(ℓ)}[p]/𝔪_{f_{ℓ}} = Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}, and

H^{1}(ℚ_{ℓ^{2}}, X_{ℓ}/𝔪_{f_{ℓ}}) = Hom_{unr}(Gal(ℚ̄_{ℓ^{2}}/ℚ_{ℓ^{2}}), X_{ℓ}/𝔪_{f_{ℓ}}) = Hom(ℤ/pℤ, X_{ℓ}/𝔪_{f_{ℓ}})

by local class field theory, since p |̸ℓ^{2} − 1. Thus, (52) can be re-written as

X_{ℓ}/𝔪_{f_{ℓ}} → (X_{ℓ}^{∨}/𝔪_{f_{ℓ}})/Y → H^{1}(ℚ_{ℓ^{2}}, Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}) → H_{unr}^{1}(ℚ_{ℓ^{2}}, X_{ℓ}/𝔪_{f_{ℓ}}). (54)

The first map in (54) is induced by the monodromy pairing on X_{ℓ}: this follows from the definition of the sequence (48) which induces (54); see also the proposition on page IV-32 of [Se2], which covers a special case. By proposition 5.11, one obtains the exact sequence

0 → Φ̄_{ℓ}/𝔪_{f_{ℓ}} → H^{1}(ℚ_{ℓ^{2}}, Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}) → H_{unr}^{1}(ℚ_{ℓ^{2}}, X_{ℓ}/𝔪_{f_{ℓ}}), (55)

where Φ̄_{ℓ}/𝔪_{f_{ℓ}} is a quotient of Φ_{ℓ}/𝔪_{f_{ℓ}}. By the main result of [BoLeRi], the module Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}} is semisimple over 𝔽_{p}[Gal(ℚ̄/ℚ)], and hence is isomorphic to k ≥ 1 copies of T_{f,1} by the Eichler-Shimura relations. Hence, H^{1}(ℚ_{ℓ^{2}}, Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}) is isomorphic to H^{1}(ℚ_{ℓ^{2}}, T_{f,1})^{k}. In view of the results of section 2.2, the 𝔽_{p}-vector space H^{1}(ℚ_{ℓ^{2}}, T_{f,1})^{k} is 2k-dimensional; furthermore, it can be decomposed as the direct sum of two k-dimensional subspaces, one of which generated by unramified cohomology classes, and the other by ramified cohomology classes. Since Φ_{ℓ}/𝔪_{f_{ℓ}} is 1-dimensional by theorem 5.15, it follows from (55) and the proof of lemma 2.6 that Φ̄_{ℓ}/𝔪_{f_{ℓ}} is equal to Φ_{ℓ}/𝔪_{f_{ℓ}}, and that k is equal to 1. Hence Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}} is isomorphic to T_{f,1}. Step 2. It remains to show that Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is isomorphic to T_{f,n}. There is a natural G_{ℚ}-equivariant projection

Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} ⟶^{π} Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}.  (56)

In view of lemma 5.16, let t be an element of order p^{n} in Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}, and let t̄= π(t). Since T_{f,1} is irreducible, there is an element g ∈ G_{ℚ} such that t̄ and t̄^{g} are a basis for Ta_{p}(J^{(ℓ)})/𝔪_{f_{ℓ}}. Nakayama’s lemma shows that Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is generated by t and t^{g}. Moreover, since g acts as an automorphism of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}, the order of t^{g} is equal to the order of t. Finally, using the assumption made in the introduction that the Galois representation T_{f,1} has image isomorphic to GL_{2}(𝔽_{p}), one checks directly that the submodules generated by t and t^{g} have trivial intersection, so that Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is isomorphic to (ℤ/p^{n}ℤ)^{2}. This implies that Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is isomorphic to T_{f,n}.

Remark:

1. Repeating the argument in the proof of the first step of theorem 5.17, with p^{n} replacing p and the ideal I_{f_{ℓ}} replacing 𝔪_{f_{ℓ}}, yields the exact sequence (analogue of (55))

0 → Φ̄_{ℓ}/I_{f_{ℓ}} → H^{1}(K_{ℓ}, Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}) → H_{unr}^{1}(K_{ℓ}, X_{ℓ}/I_{f_{ℓ}}),  (57)

where Φ̄_{ℓ}/I_{f_{ℓ}} is a quotient of Φ_{ℓ}/I_{f_{ℓ}}. This exact sequence will be of use in the proof of corollary 5.18.

2. In view of remark 1, equations (48) and (53), with I_{f_{ℓ}} replacing 𝔪_{f_{ℓ}}, show that the natural Kummer map from J^{(ℓ)}(K_{ℓ}) to H^{1}(K_{ℓ}, Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}) factors through the map of specialization to connected components from J^{(ℓ)}(K_{ℓ}) to Φ̄_{ℓ}/I_{f_{ℓ}}.

Recall the quadratic imaginary field K introduced in the previous sections. Note that the n-admissible prime ℓ is inert in K by definition, so that the completion K_{ℓ} is isomorphic to ℚ_{ℓ^{2}}. Given m ≥ 0, let O_{m} = ℤ + p^{m+1}O_{K} be the order of K of conductor p^{m+1}. Let K̃_{m} be the ring class field of K of conductor p^{m+1}, and let K̃_{∞} be the union of the K̃_{m}. (The field K̃_{∞} was introduced in section 2.1.) The field K̃_{m} can be constructed by adjoining to K the value j(O_{m}) of the modular function j (viewed as a function of lattices) on O_{m}. By the theory of complex multiplication, K̃_{m} is an abelian extension of K, which contains the Hilbert class field K̃ of K. The Galois group of K̃_{m} over K̃ is cyclic, of order p^{m}(p − ϵ(p))/u, where ϵ(p) is equal to 1 or −1 depending on whether p is split or inert in K, respectively, and where u denotes half the order of the group of units of K. Write G̃_{m} for the Galois group of K̃_{m} over K, and G̃_{∞} for the Galois group of K̃_{∞} over K. By class field theory, the Galois group G̃_{∞} is identified with the group G̃_{∞} defined in section 1.2 in terms of the ideles of K.

Write Φ_{ℓ,m}, respectively, Φ′_{ℓ,m} for ⊕_{λ|ℓ}Φ_{λ}, where the sum is taken over the primes λ of K_{m} and of K̃_{m} dividing ℓ, respectively, and Φ_{λ} denotes the group of connected components of J^{(ℓ)} at λ. Define Φ̂_{ℓ}, respectively, Φ̃_{ℓ} to be the inverse limit of the groups Φ_{ℓ,m}, respectively, Φ′_{ℓ,m} with respect to the norm maps. Since the prime ℓ is inert in K, it splits completely in K̃_{∞}/K. Hence, the choice of a prime of K̃_{∞} above ℓ identifies Φ̂_{ℓ} with Φ_{ℓ} ⊗ ℤ[[G_{∞}]], and Φ̃_{ℓ} with Φ_{ℓ} ⊗ ℤ[[G̃_{∞}]]. It follows that the identification of Φ_{ℓ}/I_{f_{ℓ}} with ℤ/p^{n}ℤ of theorem 5.15 yields isomorphisms

Φ̂_{ℓ}/I_{f_{ℓ}} ≃ Λ/p^{n}Λ, Φ̃_{ℓ}/I_{f_{ℓ}} ≃ ℤ/p^{n}[[G̃_{∞}]].

Write Ĵ^{(ℓ)}(K_{∞})/I_{f_{ℓ}}, respectively, Ĵ^{(ℓ)}(K̃_{∞})/I_{f_{ℓ}} for the inverse limit of the modules J^{(ℓ)}(K_{m,ℓ})/I_{f_{ℓ}}, respectively, J^{(ℓ)}(K̃_{m,ℓ})/I_{f_{ℓ}} with respect to the norm maps. The inverse limit of the maps of specialization to connected components yields the maps

ˆ∂_{ℓ} : Ĵ^{(ℓ)}(K_{∞})/I_{f_{ℓ}} → Φ̂_{ℓ}/I_{f_{ℓ}} ≃ Λ/p^{n},  (58)

˜∂_{ℓ} : Ĵ^{(ℓ)}(K̃_{∞})/I_{f_{ℓ}} → Φ̃_{ℓ}/I_{f_{ℓ}} ≃ ℤ/p^{n}[[G̃_{∞}]].  (59)

Corollary 5.18 1. There is an isomorphism

Φ_{ℓ}/I_{f_{ℓ}} → H_{sing}^{1}(K_{ℓ}, T_{f,n}),

which is canonical up to the choice of an identification of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} with T_{f,n}.

2. There is an isomorphism

Φ̂_{ℓ}/I_{f_{ℓ}} → Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}),

which is canonical up to the choice of an identification of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} with T_{f,n}.

3. There is a commutative diagram

Ĵ^{(ℓ)}(K_{∞})/I_{f_{ℓ}} ⟶ Ĥ^{1}(K_{∞}, T_{f,n})
↓ ˆ∂_{ℓ}  ↓^{∂_{ℓ}}
Φ̂_{ℓ}/I_{f_{ℓ}}  ⟶ Ĥ_{sing}^{1}(K_{∞,ℓ}, T_{f,n}),

where the top horizontal arrow arises from the natural Kummer map, the lower horizontal arrow is the isomorphism defined above, and ∂_{ℓ} is the residue map defined in section 2.2. Furthermore, there is a similar commutative diagram having the group Ĵ^{(ℓ)}(K̃_{∞})/I_{f_{ℓ}} as its source.

Proof: By theorem 5.17, Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} is isomorphic to T_{f,n}. Hence, by the results of section 2.2, H^{1}(K_{ℓ}, Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}) is free of rank two over ℤ/p^{n}ℤ; moreover, its submodule of unramified classes has rank one. By theorem 5.15, Φ_{ℓ}/I_{f_{ℓ}} is isomorphic to ℤ/p^{n}ℤ. The exact sequence (57) shows that all the elements of Φ̄_{ℓ}/I_{f_{ℓ}} map to ramified classes. It follows that Φ̄_{ℓ}/I_{f_{ℓ}} is equal to Φ_{ℓ}/I_{f_{ℓ}}. Furthermore, the residue map induces a surjection of Φ_{ℓ}/I_{f_{ℓ}} onto the rank one group H_{sing}^{1}(K_{ℓ}, T_{f,n}). This defines the isomorphism stated in part one of corollary 5.18. The second part is a formal consequence of the first. The third part follows from the definition of the isomorphism in part two, combined with the remark 2 after the proof of theorem 5.17.

6 The theory of complex multiplication

(Reference: [BD1], section 2.) Fix a positive integer M, a rational prime p, and an imaginary quadratic field K of discriminant −D, such that p | M but p^{2} ∤ M, and (M, D) = 1. Define an integer decomposition

M = M^{+}M^{−}

such that (M^{+}, M^{−}) = 1, and M^{+} is divisible by p and by the primes divisors of M/p which are split in K. Assume that:

1. M^{−} is squarefree,

2. M^{−} is the product of an even number of primes.

Let X = X_{M^{+},M^{−}} be the Shimura curve attached in section 5.1 to an Eichler order R of level M^{+} in the indefinite quaternion algebra B of discriminant M^{−}. For all m ≥ 0, there is a point P_{m} on X, which has complex multiplication by the order O_{m} of K of conductor p^{m+1} (introduced after the proof of theorem 5.17). More precisely, let (A_{m}, ι_{m}, C_{m}) be a triple corresponding to P_{m} via the moduli interpretation of X given in section 5.1. Write End(P_{m}) for the ring of endomorphisms of A_{m} which commute with the action ι_{m} and respect the level structure C_{m}. Then,

End(P_{m}) ≃ O_{m}.

The point P_{m} is called a Heegner point of level m. By the theory of complex multiplication, P_{m} is defined over the field K̃_{m} (defined after the proof of theorem 5.17). Choose the P_{m} so that they are compatible, in the sense that for all m ≥ 0, there is an isogeny of degree p^{2} from A_{m} to A_{m+1}, which is equivariant for the actions ι_{m} and ι_{m+1}, and preserves the level structures C_{m} and C_{m+1}.

One has the following interpretation of the sequence {P_{m}} in terms of the Bruhat-Tits tree at p. Given a point P on the Shimura curve X_{M^{+}/p,M^{−}}, corresponding to a triple (A, ι, C), let T^{(P)} denote the tree of p-isogenies of P. The vertices of T^{(P)} correspond to points of X_{M^{+}/p,M^{−}} representing moduli related to (A, ι, C) by an isogeny of p-power degree. Two vertices of T^{(P)} are adjacent if the corresponding moduli are related by an isogeny of degree p^{2}. Thus, the oriented edges of T^{(P)} are naturally identified with points on X. The tree T^{(P)} is isomorphic to the Bruhat-Tits tree T , and has a distinguished vertex v_{P} corresponding to P. There is a unique point P on X_{M^{+}/p,M^{−}} such that:

1. End(P) ≃ O_{K},

2. P_{0} corresponds to an edge of the tree T^{(P)}, with origin in P.

Then, the points P_{m} determine a half line of T^{(P)} originating from P, with no back-trackings.

7 Construction of the Euler System

Notations and assumptions being as in section 5.6, the construction of section 6 yields a compatible family of Heegner points

P_{m} ∈ X^{(ℓ)}(K̃_{m})

for all m ≥ 0 (the notation X^{(ℓ)} was defined right before lemma 5.16). View P_{m} as an element of the Picard group Pic(X^{(ℓ)})(K̃_{m}). Since the ideal I_{f_{ℓ}} is not Eisenstein, the natural inclusion

J^{(ℓ)}(K̃_{m})/I_{f_{ℓ}} → Pic(X^{(ℓ)})(K̃_{m})/I_{f_{ℓ}}

is an isomorphism. Let α_{p} be the unit root of Frobenius at p. Write P_{m}^{∗} for the image of α_{p}^{−m}P_{m} in J^{(ℓ)}(K̃_{m})/I_{f_{ℓ}}. The points P_{m}^{∗} are norm-compatible. Hence their images by the coboundary maps

J^{(ℓ)}(K̃_{m})/I_{f_{ℓ}} → H^{1}(K̃_{m}, Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}})

yield a sequence of cohomology classes which are compatible under the corestriction maps. The choice of an isomorphism of Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}} with T_{f,n}, which exists by theorem 5.17, gives a class κ̃(ℓ) in Ĥ^{1}(K̃_{∞}, T_{f,n}), where Ĥ^{1}(K̃_{∞}, T_{f,n}) denotes the inverse limit under the corestriction maps of the groups H^{1}(K̃_{m}, T_{f,n}). Define κ(ℓ) in Ĥ^{1}(K_{∞}, T_{f,n}) to be the corestriction from K̃_{∞} to K_{∞} of κ̃(ℓ). In other words, writing Q_{m} for the norm from K̃_{∞} to K_{∞} of P_{m}^{∗}, the class κ(ℓ) is the natural image in Ĥ^{1}(K_{∞}, T_{f,n}) of the sequence Q_{m} via the coboundary maps.

8 The first explicit reciprocity law

This section is devoted to the proof of theorem 4.1, notations being as in the previous sections. Recall that the class κ(ℓ) is constructed from a family of points on the Shimura curve X^{(ℓ)}; hence, it can be viewed as an element of the usual (compactified) Selmer group of J^{(ℓ)} over K_{∞} relative to the Galois module Ta_{p}(J^{(ℓ)})/I_{f_{ℓ}}. This shows that κ(ℓ) belongs to Ĥ_{ℓ}^{1}(K_{∞}, T_{f,n}).

Recall the groups Φ̂_{ℓ} and Φ̃_{ℓ}, and the maps ˆ∂_{ℓ} and ˜∂_{ℓ}, which were defined in section 5.6 (see equations (58) and (59)).

Lemma 8.1 Theorem 4.1 is implied by the equality

˜∂_{ℓ}({P_{m}^{∗}}) = L̃_{f} (mod p^{n}).

Remark: Note that the terms in the equality of lemma 8.1 are well defined only up to multiplication by elements of ℤ_{p}^{×} and of G̃_{∞}.

Proof: The element ˆ∂_{ℓ}({Q_{m}}) is mapped to ∂_{ℓ}(κ(ℓ)) by the isomorphism of corollary 5.18. Since ˆ∂_{ℓ}({Q_{m}}) is the norm of ˜∂_{ℓ}({P_{m}^{∗}}), and L_{f} is the natural image of L̃_{f} in Λ, the claim follows.

In view of lemma 8.1, one is reduced to studying the specialization of the Heegner points P_{m} to connected components. To begin with, it is necessary to recall the ℓ-adic description of the point P_{m} given in section 5 of [BD3], and based on Drinfeld’s moduli interpretation of the ℓ-adic upper half plane. Let P̄_{m} denote the reduction of P_{m} modulo a fixed prime above ℓ, and let

End(P_{m}) → End(P̄_{m})

be the map obtained by reduction of endomorphisms. Recall that End(P_{m}) is isomorphic to the quadratic order O_{m}. Moreover, the ring End(P̄_{m})[\frac{1}{ℓ}] is isomorphic to \underline{R}[\frac{1}{ℓ}], where \underline{R} denotes an Eichler order of level N^{+} in the definite quaternion algebra B of discriminant N^{−}, which can be chosen to be independent of m. Extending scalars by ℤ[\frac{1}{ℓ}], one obtains an injection

Ψ_{m}^{0} : O_{m}[\frac{1}{ℓ}] → \underline{R}[\frac{1}{ℓ}].

It can be shown that Ψ_{m}^{0} is well defined up to conjugation by elements in \underline{R}[\frac{1}{ℓ}]_{1}^{×}. Therefore, Ψ_{m}^{0} can be identified with an element of the space

(Hom(K, B) × V(T_{ℓ}))/Γ_{ℓ,1},

by mapping Ψ_{m}^{0} to the pair (Ψ_{m}, v_{m}), where Ψ_{m} is the extension of scalars of Ψ_{m}^{0}, and v_{m} is the vertex of T_{ℓ} corresponding to the unique maximal order of B_{ℓ} containing Ψ_{m}(O_{m}). The embedding Ψ_{m} induces an action of K_{ℓ}^{×} on H_{ℓ}, having two fixed points which belong to K_{ℓ} − ℚ_{ℓ}, and which are conjugate by the generator of Gal(K_{ℓ}/ℚ_{ℓ}). Then, P_{m} is identified with the image in H_{ℓ}/Γ_{ℓ,1} of one of these two points, via the isomorphism of theorem 5.1. (A suitable condition of normalization specifies which point corresponds to P_{m}, but this will not be of use here.) Let

r_{ℓ} : H_{ℓ}(ℂ_{ℓ})/Γ_{ℓ,1} ⟶ V(G_{ℓ}) ∪ E(G_{ℓ}) = (V(T_{ℓ}) ∪ E(T_{ℓ}))/Γ_{ℓ,1}

be the reduction map (see for example chapter I of [BC], and also sections 1 and 6 of [BD3]). Here G_{ℓ} is the dual graph T_{ℓ}/Γ_{ℓ,1} of X^{(ℓ)} at ℓ. Given P in X^{(ℓ)}(ℂ_{ℓ}), viewed as a point in H_{ℓ}(ℂ_{ℓ})/Γ_{ℓ,1}, r_{ℓ}(P) is equal to a vertex v if the reduction of P modulo ℓ lands in the single irreducible component of the fiber X_{𝔽_{ℓ^{2}}}^{(ℓ)} corresponding to v, and r_{ℓ}(P) is equal to an edge e if P reduces to the singular point corresponding to e. The above description of P_{m} in terms of an ℓ-adic argument in H_{ℓ} shows that the image of P_{m} by r_{ℓ} is equal to the vertex v_{m} (mod Γ_{ℓ,1}). This follows from the GL_{2}(ℚ_{ℓ})-equivariance of the reduction map, combined with the fact that P_{m} corresponds to a fixed point for the action of Ψ_{m}(K_{ℓ}^{×}) on H_{ℓ}(ℂ_{ℓ}) and v_{m} is the unique fixed point for the action of Ψ_{m}(K_{ℓ}^{×}) on V(T_{ℓ}) ∪ E(T_{ℓ}). Hence, the reduction of P_{m} modulo ℓ lands in the single irreducible component of the fiber X_{𝔽_{ℓ^{2}}}^{(ℓ)} defined by v_{m} (mod Γ_{ℓ,1}). Furthermore, by proposition 5.14, v_{m} (mod Γ_{ℓ,1}) computes the image of P_{m} in the group of connected components Φ_{ℓ,m}/I_{f_{ℓ}}.

On the other hand, recalling that p divides the level of the order \underline{R}, strong approximation (see [Vi], p. 61) gives an identification

(Hom(K, B) × V(T_{ℓ}))/Γ_{ℓ,1} = (Hom(K, B) × E^{→}(T_{p}))/Γ_{p}

where, in the notations of section 1.1, T_{p} = T and Γ_{p} = Γ. The compatibility condition on the Heegner points P_{m} translates in the condition that they may be represented by pairs (Ψ, e_{m}) in Hom(K, B) × E^{→}(T_{p}), where Ψ arises from an embedding

Ψ^{0} : O_{K}[1/p] → \underline{R}[1/p]

which does not depend on m, and where the e_{m} determine a half line with no back-trackings in T_{p}. The results of [BD3], section 5 show that the action of G̃_{∞} on the P_{m} is compatible with the action of G̃_{∞} on the edges e_{m}, which was defined in section 1.2 using the embedding Ψ. Fix a prime λ_{∞} of K̃_{∞} above ℓ, and set λ_{m} = λ_{∞} ∩ K̃_{m}. For σ ∈ G̃_{∞}, write ∂_{λ_{m}}(P_{m}^{σ}) for the natural image of P_{m}^{σ} in the group of connected components Φ_{λ_{m}}/I_{f_{ℓ}} ≃ ℤ/p^{n}ℤ. The use of proposition 5.14 described above, together with the identification coming from strong approximation, shows that the map ∂_{λ_{m}} on the points P_{m}^{σ} can be viewed as a function

E^{→}(T_{p})/Γ_{p} ⟶ ℤ/p^{n}ℤ.

By multiplicity one, this function is equal modulo p^{n} to the eigenform f introduced in section 5.6 (up to multiplication by an element of (ℤ/p^{n}ℤ)^{×}). By comparing with definition (17), one finds that the equality

∂_{λ_{m}}(P_{m}^{σ}) = [σ, e_{m}]_{f} (mod p^{n})  (60)

holds for a suitable choice of λ_{∞}. (Note that the possible choices of λ_{∞} are permuted by G̃_{∞}. Similarly, the choices of embedding Ψ and of a sequence of edges in the definition of the p-adic L-function given in section 1.2 show that L̃_{f} is well defined only up to multiplication by elements of G̃_{∞}.) It follows from (60) that

∂_{λ_{m}}(σP_{m}^{∗}) = α_{p}^{−m}[σ, e_{m}]_{f} (mod p^{n}).

In view of equation (20) and of the definition of L̃_{f}, this concludes the proof.

9 The second explicit reciprocity law

This section is devoted to the proof of theorem 4.2. Recall that ℓ_{1} and ℓ_{2} are distinct n-admissible primes relative to f, such that p^{n} divides ℓ_{1} + 1 − ϵ_{1}a_{ℓ_{1}}(f) and ℓ_{2} + 1 − ϵ_{2}a_{ℓ_{2}}(f), with ϵ_{1} and ϵ_{2} equal to ±1. Let 𝕋_{ℓ_{1}} be the Hecke algebra acting on X^{(ℓ_{1})}. As in the previous sections, equip J^{(ℓ_{1})} with the action of 𝕋_{ℓ_{1}} induced by Picard functoriality. Since f is p-isolated, the assumptions of theorem 5.15 are satisfied in the current setting. Thus, by theorem 5.17, Ta_{p}(J^{(ℓ_{1})})/I_{f_{ℓ_{1}}} is isomorphic to T_{f,n} as a Galois module. Fix such an isomorphism. Then, the map

J^{(ℓ_{1})}(K_{ℓ_{2}})/I_{f_{ℓ_{1}}} → H^{1}(K_{ℓ_{2}}, Ta_{p}(J^{(ℓ_{1})})/I_{f_{ℓ_{1}}})

arising from Kummer theory yields a map

J^{(ℓ_{1})}(K_{ℓ_{2}})/I_{f_{ℓ_{1}}} → H^{1}(K_{ℓ_{2}}, T_{f,n}).  (61)

The image of (61) is equal to H_{fin}^{1}(K_{ℓ_{2}}, T_{f,n}), since T_{f,n} is unramified at ℓ_{2} and ℓ_{2} is a prime of good reduction for J^{(ℓ_{1})}. Since p ≠ ℓ_{2}, the map induced by reduction modulo ℓ_{2}

J^{(ℓ_{1})}(K_{ℓ_{2}})/I_{f_{ℓ_{1}}} → J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}})/I_{f_{ℓ_{1}}}  (62)

is an isomorphism. Hence, by composing the inverse of (62) with (61), and fixing an identification of H_{fin}^{1}(K_{ℓ_{2}}, T_{f,n}) with ℤ/p^{n}ℤ (use lemma 2.6), one obtains a surjective map

J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}})/I_{f_{ℓ_{1}}} → ℤ/p^{n}ℤ.  (63)

Let S_{ℓ_{2}} ⊂ X^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}}) denote the set of supersingular points of X^{(ℓ_{1})} in characteristic ℓ_{2}, and let Div(S_{ℓ_{2}}), respectively, Div^{0}(S_{ℓ_{2}}) be the module of formal divisors, respectively, degree zero formal divisors with ℤ-coefficients supported on S_{ℓ_{2}}.

One may define two different actions of 𝕋_{ℓ_{1}} on Div(S_{ℓ_{2}}) and Div^{0}(S_{ℓ_{2}}), by using either Picard or Albanese functoriality (see the remark at the beginning of section 5.4, which explains that these two actions differ by an Atkin-Lehner involution w_{N^{+},1}). The latter action is the usual action defined on supersingular points, and is more natural if one views the supersingular points as being points in X^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}}). The former action is more natural when viewing divisors on supersingular points as giving rise to points in Pic(X^{(ℓ_{1})})(𝔽_{ℓ_{2}^{2}}) and J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}}), on which the action of 𝕋_{ℓ_{1}} was defined via Picard functoriality. Since f_{ℓ_{1}} is an eigenform for w_{N^{+},1}, which acts via multiplication by ±1, the choice of a specific Hecke action makes no difference for the purpose of establishing the 𝕋_{ℓ_{1}}-equivariance of the maps defined below; let us make the convention that the Hecke action on supersingular points be the Albanese one.

Since the inclusion of Div^{0}(S_{ℓ_{2}}) in Div(S_{ℓ_{2}}) induces an identification of Div^{0}(S_{ℓ_{2}})/I_{f_{ℓ_{1}}} with Div(S_{ℓ_{2}})/I_{f_{ℓ_{1}}} (use the fact that I_{f_{ℓ_{1}}} is not Eisenstein), one obtains a natural map

Div(S_{ℓ_{2}}) → J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}})/I_{f_{ℓ_{1}}}.  (64)

The composition of (64) with the surjection (63) yields a map

γ : Div(S_{ℓ_{2}}) → ℤ/p^{n}ℤ.

Write T_{q} (q ∤ Nℓ_{1}) and U_{q} (q | Nℓ_{1}) for the q-th Hecke operator in 𝕋_{ℓ_{1}}, and T̄_{q} and Ū_{q} for the natural image of T_{q} and U_{q}, respectively, in 𝕋_{ℓ_{1}}/I_{f_{ℓ_{1}}} = ℤ/p^{n}ℤ. Thus, the following equalities modulo p^{n} hold: T̄_{q} ≡ a_{q}(f) for q ∤ Nℓ_{1}, Ū_{q} ≡ a_{q}(f) for q | N, and Ū_{ℓ_{1}} ≡ ϵ_{1}.

Lemma 9.1 The relations

γ(T_{q}x) = T̄_{q}γ(x) (q ∤ Nℓ_{1}ℓ_{2}), γ(U_{q}x) = Ū_{q}γ(x) (q | Nℓ_{1}),
γ(T_{ℓ_{2}}x) = T̄_{ℓ_{2}}γ(x), γ(Frob_{ℓ_{2}}x) = ϵ_{2}γ(x)

hold for x ∈ Div(S_{ℓ_{2}}).

Proof: As observed in the proof of lemma 2.6, H_{fin}^{1}(K_{ℓ_{2}}, T_{f,n}) is identified with the module T_{f,n}/(Frob_{ℓ_{2}}^{2} − 1)T_{f,n} of G_{K_{ℓ_{2}}}-coinvariants of T_{f,n}. Therefore, the map γ is defined by sending a point x to the image of ((Frob_{ℓ_{2}}^{2} − 1)/p^{n})x in T_{f,n}/(Frob_{ℓ_{2}}^{2} − 1)T_{f,n}. The first two identities are a direct consequence of this description of γ. As for the last two identities, note that the Eichler-Shimura relations identify T_{ℓ_{2}} with the correspondence Frob_{ℓ_{2}} + Frob_{ℓ_{2}}^{∨}, where Frob_{ℓ_{2}}^{∨} denotes the transpose of Frob_{ℓ_{2}}. Furthermore, on points defined over 𝔽_{ℓ_{2}^{2}}, one has Frob_{ℓ_{2}}^{∨}x = ℓ_{2}Frob_{ℓ_{2}}x, and hence T_{ℓ_{2}}x = (ℓ_{2} + 1)Frob_{ℓ_{2}}x. The claim follows from the fact that Frob_{ℓ_{2}} acts on T_{f,n} with eigenvalues ϵ_{2} and ϵ_{2}ℓ_{2}, which implies that Frob_{ℓ_{2}} acts as ϵ_{2} on the quotient T_{f,n}/(Frob_{ℓ_{2}}^{2} − 1)T_{f,n}.

Proposition 9.2 The map γ is surjective.

Proof: Recall that p divides exactly N^{+}, and that the Shimura curve X = X^{(ℓ_{1})} was defined in section 5.1 in terms of an Eichler order R of level N^{+} in the indefinite quaternion algebra B of discriminant N^{−}ℓ_{1}. Let J = J^{(ℓ_{1})} be the jacobian of X. Write J(𝔽_{ℓ_{2}^{2}})^{ss} for the subgroup of J(𝔽_{ℓ_{2}^{2}}) generated by divisors supported on supersingular points. Since the map (63) is surjective, it is enough to show that the natural image of J(𝔽_{ℓ_{2}^{2}})^{ss} in the group J(𝔽_{ℓ_{2}^{2}})/I_{f_{ℓ_{1}}} is equal to the whole group. Let Γ^{(ℓ_{2})} denote the group of norm one elements in R[1/ℓ_{2}]^{×}/{±1}. Let X̃ be the Shimura curve defined in the same way as X but imposing an extra Γ_{1}(p)-level structure. Let Γ̃^{(ℓ_{2})} be the finite index subgroup of Γ^{(ℓ_{2})} consisting of elements which are congruent to the standard unipotent matrices modulo p. Write J̃(𝔽_{ℓ_{2}^{2}}) for the 𝔽_{ℓ_{2}^{2}}-points of the jacobian of X̃, and J̃(𝔽_{ℓ_{2}^{2}})^{ss} for the subgroup generated by divisors supported on supersingular points. Since Γ̃^{(ℓ_{2})} is torsion-free, the results of [I2] (see in particular remark G, page 19) establish a canonical isomorphism

J̃(𝔽_{ℓ_{2}^{2}})/J̃(𝔽_{ℓ_{2}^{2}})^{ss} ≃ (Γ̃^{(ℓ_{2})})^{ab},  (65)

where (Γ̃^{(ℓ_{2})})^{ab} denotes the abelianization of Γ̃^{(ℓ_{2})}. By fixing an embedding of B in M_{2}(ℚ_{ℓ_{2}}), one obtains an action of Γ̃^{(ℓ_{2})} on the Bruhat-Tits tree T_{ℓ_{2}}. Let v_{0} be the vertex of T_{ℓ_{2}} such that the stabilizer Γ̃_{v_{0}}^{(ℓ_{2})} of v_{0} in Γ̃^{(ℓ_{2})} is the subgroup of integral elements (relative to the fixed embedding of Γ̃^{(ℓ_{2})} in M_{2}(ℚ_{ℓ_{2}})). Let e_{0} be the edge originating from v_{0} such that the stabilizer Γ̃_{e_{0}}^{(ℓ_{2})} of e_{0} is the subgroup of Γ̃_{v_{0}}^{(ℓ_{2})} whose elements are upper triangular modulo ℓ_{2} relative to this embedding. Write v_{1} for the target of e_{0}. Note that Γ̃_{v_{0}}^{(ℓ_{2})}, respectively, Γ̃_{e_{0}}^{(ℓ_{2})} can be identified with the discrete subgroup of SL_{2}(ℝ) which defines the Shimura curve X̃, respectively, the Shimura curve X̃_{0}(ℓ_{2}) defined in the same way as X̃, but with an extra Γ_{0}(ℓ_{2})-level structure imposed. The group Γ̃^{(ℓ_{2})} acts on the tree T_{ℓ_{2}} with the closed edge attached to e_{0} as a fundamental region. Hence the exact cohomology sequence in proposition 13 of sec. II.2.8 of [Se1], in the case i = 1, M = 𝔽_{p} and G = Γ̃^{(ℓ_{2})} becomes

0 ⟶ Hom(Γ̃^{(ℓ_{2})}, 𝔽_{p}) ⟶ Hom(Γ̃_{v_{0}}^{(ℓ_{2})}, 𝔽_{p}) ⊕ Hom(Γ̃_{v_{1}}^{(ℓ_{2})}, 𝔽_{p})
⟶^{d} Hom(Γ̃_{e_{0}}^{(ℓ_{2})}, 𝔽_{p}). (66)

The modules appearing as the source and target of d are identified by duality with two copies or one copy of the p-torsion in the jacobian of X̃ and X̃_{0}(ℓ_{2}), respectively. Moreover, the map d corresponds under these identifications to the map denoted α_{p} in the statement of theorem 2, p. 451 of [DT]. This theorem (the analogue of Ihara’s lemma in the setting of Shimura curves) states that the action of G_{ℚ} on the kernel of d factors through an abelian quotient of G_{ℚ}. View f_{ℓ_{1}} as a mod p^{n} modular form (with trivial character) on X̃, and write Ĩ_{f_{ℓ_{1}}} for the associated ideal in the Hecke algebra 𝕋̃_{ℓ_{1}}. Let 𝔪̃_{f_{ℓ_{1}}} be the maximal ideal of 𝕋̃_{ℓ_{1}} containing Ĩ_{f_{ℓ_{1}}}. Since 𝔪̃_{f_{ℓ_{1}}} corresponds to an irreducible mod p Galois representation, it follows from the sequence (66) that Hom(Γ̃^{(ℓ_{2})}, 𝔽_{p})[𝔪_{f_{ℓ_{1}}}] is trivial, and therefore that (Γ̃^{(ℓ_{2})})^{ab}/𝔪̃_{f_{ℓ_{1}}} is trivial. By Nakayama’s lemma, this shows that

(Γ̃^{(ℓ_{2})})^{ab}/Ĩ_{f_{ℓ_{1}}} = 0.  (67)

Equation (65) then implies that the natural image of J̃(𝔽_{ℓ_{2}^{2}})^{ss} in J̃(𝔽_{ℓ_{2}^{2}})/Ĩ_{f_{ℓ_{1}}} fills the whole group. Finally, the cokernel of the natural map

J̃^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}}) → J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}})  (68)

can be identified with an abelian quotient of the (finite) image of Γ_{0}(p) in SL_{2}(ℤ/pℤ), and hence has order dividing p−1; see for example the results of chapter 7 of [Co], particularly in pages 107 and 110. It follows in particular that the composition of the map (68) with the projection from J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}}) to J^{(ℓ_{1})}(𝔽_{ℓ_{2}^{2}})/p^{n} is surjective. When combined with the fact that J̃(𝔽_{ℓ_{2}^{2}})^{ss} maps surjectively to J̃(𝔽_{ℓ_{2}^{2}})/Ĩ_{f_{ℓ_{1}}} (as noted after equation (67)), this implies that J(𝔽_{ℓ_{2}^{2}})^{ss} maps surjectively to J(𝔽_{ℓ_{2}^{2}})/I_{f_{ℓ_{1}}}, as was to be shown.

Remark: The proof of proposition 9.2 relies crucially on the analogue of Ihara’s lemma proved in [DT]. In turn, proposition 9.2 provides the key step in the proof of theorem 9.3, a raising the level result which is a theorem of [DT] (extended to mod p^{n} modular forms).

Following the notations of proposition 3.12, let B′ be the definite quaternion algebra of discriminant N^{−}ℓ_{1}ℓ_{2}, R′ an Eichler ℤ[1/p]-order of level N^{+} in B, and Γ′ the group (R′)^{×}/ℤ[1/p]^{×}. The next result contains the part of the statement of proposition 3.12 that remains to be proved.

Theorem 9.3 There exists an eigenform g ∈ S_{2}(T /Γ′, ℤ/p^{n}ℤ) such that

T_{q}g = a_{q}(f)g (q |̸Nℓ_{1}ℓ_{2}), U_{q}g = a_{q}(f)g (q|N),
U_{ℓ_{1}}g = ϵ_{1}g, U_{ℓ_{2}}g = ϵ_{2}g.

Proof: Write 𝕋_{ℓ_{1}} = 𝕋_{N^{+},N^{−}ℓ_{1}} for the Hecke algebra acting on cusp forms on Γ_{0}(Nℓ_{1}) which are new at N^{−}ℓ_{1}, and 𝕋_{ℓ_{2},ℓ_{1}} = 𝕋_{N^{+}ℓ_{2},N^{−}ℓ_{1}} for the Hecke algebra acting on cusp forms on Γ_{0}(Nℓ_{1}ℓ_{2}) which are new at N^{−}ℓ_{1}. By theorem 5.15, there is a mod p^{n} modular form

f_{ℓ_{1}} : 𝕋_{ℓ_{1}} → ℤ/p^{n}ℤ

such that f_{ℓ_{1}}(T_{q}) = f(t_{q}) for all q ∤ Nℓ_{1}, f_{ℓ_{1}}(U_{q}) = f(u_{q}) for all q | N, and f_{ℓ_{1}}(U_{ℓ_{1}}) = ϵ_{1}f_{ℓ_{1}}. Recall from the proof of theorem 5.15 that the homomorphism f_{ℓ_{1}} arises from the group of connected components at ℓ_{1} attached to the Shimura curve X^{(ℓ_{1})} = X_{N^{+},N^{−}ℓ_{1}}. Consider now the Shimura curve X^{(ℓ_{2},ℓ_{1})} := X_{N^{+}ℓ_{2},N^{−}ℓ_{1}}, on which the elements of 𝕋_{ℓ_{2},ℓ_{1}} act as correspondences. By proposition 5.3, the character group X_{ℓ_{2}} of X^{(ℓ_{2},ℓ_{1})} at ℓ_{2} is identified with the module Div^{0}(S_{ℓ_{2}}) defined at the beginning of this section; here S_{ℓ_{2}} is viewed as the set of supersingular points in X^{(ℓ_{2},ℓ_{1})}(𝔽_{ℓ_{2}^{2}}) by the results of section 5.1. Furthermore, the action of 𝕋_{ℓ_{2},ℓ_{1}} on X_{ℓ_{2}} induced from the action on Pic^{0}(X^{(ℓ_{2},ℓ_{1})}) defined by Picard functoriality is compatible with the standard (Albanese) action of 𝕋_{ℓ_{2},ℓ_{1}} via correspondences on the set of supersingular points (see [Ri2], p. 445). The results of [Wa] on the endomorphisms of supersingular abelian surfaces, combined with strong approximation, yield a 𝕋_{ℓ_{2},ℓ_{1}}- compatible identification of S_{ℓ_{2}} with E^{→}(T )/Γ′. (Note that X_{ℓ_{2}} = Div^{0}(S_{ℓ_{2}}) pertains to the ℓ_{2}-new part of the jacobian J^{(ℓ_{2},ℓ_{1})} of X^{(ℓ_{2},ℓ_{1})}, being X_{ℓ_{2}} the character group of the maximal torus of J^{(ℓ_{2},ℓ_{1})} over 𝔽_{ℓ_{2}^{2}}.) Therefore, the

map γ defined above can also be viewed as a ℤ/p^{n}ℤ-valued map on E^{→}(T )/Γ′, whose values are not contained in any proper subgroup of ℤ/p^{n}ℤ, by proposition 9.2. This map defines the sought-for modular form g on Γ′, as can be seen by appealing to lemma 9.1. More precisely, let T_{q}^{∗} (for q not dividing Nℓ_{1}ℓ_{2}) and U_{q}^{∗} (for q dividing Nℓ_{1}ℓ_{2}) denote the Hecke operators in 𝕋_{ℓ_{2},ℓ_{1}}. (Note the discrepancy between the notations for the Hecke operators used in this proof and those used in the statement of proposition 3.12, which are also reproduced in the statement of theorem 9.3; in this proof, and throughout the section, the symbols T_{q} and U_{q} are used to indicate the Hecke operators in 𝕋_{ℓ_{1}}.) Since the Hecke operators at q ≠ ℓ_{2} in 𝕋_{ℓ_{1}} and 𝕋_{ℓ_{2},ℓ_{1}} act in the same way on γ and g, respectively, lemma 9.1 implies directly the relations of theorem 9.3 for all q ≠ ℓ_{2}:

T_{q}^{∗}g = T_{q}γ = a_{q}(f)g (q |̸Nℓ_{1}ℓ_{2}), U_{q}g = U_{q}γ = a_{q}(f)g (q|Nℓ_{1}),

(in particular one has T_{ℓ_{1}}^{∗}g = ϵ_{1}g). As for the operator U_{ℓ_{2}}^{∗}, it is known that U_{ℓ_{2}}^{∗}x = Frob_{ℓ_{2}}x for x ∈ S_{ℓ_{2}} (see [Ri2], proposition 3.8). Hence, lemma 9.1 yields

(U_{q}^{∗}g)(x) = γ(F rob_{ℓ_{2}}x) = ϵ_{2}g(x).

This concludes the proof.

The proof of theorem 9.3 implies the following result.

Corollary 9.4 Under the indentification of S_{ℓ_{2}} with E^{→}(T )/Γ′, the map γ corresponds to an eigenform g ∈ S_{2}(T /Γ′, ℤ/p^{n}ℤ) satisfying the conclusions of theorem 9.3.

Consider the sequence {P_{m}} of Heegner points P_{m} ∈ X^{(ℓ_{1})}(K̃_{m}), constructed in section 6. Fix a prime λ_{∞} of K̃_{∞} above ℓ_{2}, and let λ_{m} = λ_{∞} ∩ K̃_{m}. Since ℓ_{2} is inert in K, the point P_{m} reduces modulo λ_{m} to a supersingular point P̄_{m} ∈ X^{(ℓ_{1})}(𝔽_{λ_{m}}). Identifying 𝔽_{λ_{m}} with 𝔽_{ℓ_{2}^{2}}, P̄_{m} can be viewed as an element

of S_{ℓ_{2}}. Identifying S_{ℓ_{2}} with E^{→}(T )/Γ′, the sequence {P̄_{m}} can be described by a sequence of consecutive edges {e_{m}} in E^{→}(T ), modulo Γ′, in such a way that the map End(P_{m}) → End(P̄_{m}) of reduction of endomorphisms modulo λ_{m} induces by extension of scalars an embedding

Ψ : K → B′,

which is independent of m. Then, the natural Galois action of G̃_{∞} on the P_{m} is compatible with the action of G̃_{∞} on the e_{m} via Ψ, which was defined in section 1.2. Writing

L̃_{g,m} := α_{p}^{−m} ∑_{σ∈G̃_{m}} g(\overline{σP}_{m}) · σ^{−1} ∈ ℤ/p^{n}ℤ[G̃_{m}],

it follows that the sequence {L̃_{g,m}} defines an element of ℤ/p^{n}ℤ[[G̃_{∞}]], equal to L̃_{g}. Define the local cohomology groups

H_{fin}^{1}(K̃_{m,ℓ_{2}}, T_{f,n}) := ⊕_{λ|ℓ_{2}}H_{fin}^{1}((K̃_{m})_{λ}, T_{f,n}),

where the sum is taken over all the primes of K̃_{m} dividing ℓ_{2}, and

Ĥ_{fin}^{1}(K̃_{∞,ℓ_{2}}, T_{f,n}) := lim_{m^{⟵}} H_{fin}^{1}(K̃_{m,ℓ_{2}}, T_{f,n}),

where the inverse limit is taken with respect to the natural corestriction maps. The fixed identification of H_{fin}^{1}(K_{ℓ_{2}}, T_{f,n}) with ℤ/p^{n}ℤ, together with the choice of the prime λ_{∞}, yields the identifications

H_{fin}^{1}(K̃_{m,ℓ_{2}}, T_{f,n}) = ℤ/p^{n}[G̃_{m}], Ĥ_{fin}^{1}(K̃_{∞,ℓ_{2}}, T_{f,n}) = ℤ/p^{n}[[G̃_{∞}]].

In view of corollary 9.4 and the definition of the map γ, the image of P_{m}^{∗} in H_{fin}^{1}(K̃_{m,ℓ_{2}}, T_{f,n}) corresponds to L̃_{g,m} (mod p^{n}), and the image of the compatible sequence {P_{m}^{∗}} in Ĥ_{fin}^{1}(K̃_{∞,ℓ_{2}}, T_{f,n}) corresponds to L̃_{g} (mod p^{n}), under the above identifications. Recall the class κ̃(ℓ_{1}), defined in section 7 as the image of the sequence {P_{m}^{∗}} in Ĥ^{1}(K̃_{∞}, T_{f,n}) by the coboundary map. The value v_{ℓ_{2}}(κ̃(ℓ_{1})) at ℓ_{2} of κ̃(ℓ_{1}) is naturally an element of Ĥ_{fin}^{1}(K̃_{∞,ℓ_{2}}, T_{f,n}), and is equal to the image of {P_{m}^{∗}}, and hence to L̃_{g} (mod p^{n}). Since L_{g} is the image in Λ of L̃_{g}, and κ(ℓ_{1}) is the corestriction from K̃_{∞} to K_{∞} of κ̃(ℓ_{1}), theorem 4.2 follows.

Remark: The result proved in this section can be viewed as a generalization of the main result of [BD5]. The proof given here follows closely the approach in [Va2], avoiding the study of certain groups of connected components which was involved in the methods of [BD5].

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