A rigid analytic Gross-Zagier formula and arithmetic applications

Massimo Bertolini^{1,}^{2}
Henri Darmon^{3,}^{4}

(With an Appendix by B. Edixhoven)

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Gross’ formula for special values of L-series . . . . . . . . . . . . . . . 4 2 Bad reduction of Shimura curves . . . . . . . . . . . . . . . . . . . 5 3 Heegner points and connected components . . . . . . . . . . . . . . . 7 4 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 A rigid analytic Gross-Zagier formula . . . . . . . . . . . . . . . . 11 6 Kolyvagin cohomology classes . . . . . . . . . . . . . . . . . . . . 13 7 Bounding Mordell-Weil groups . . . . . . . . . . . . . . . . . . . 20 8 Mordell-Weil groups in anticyclotomic towers . . . . . . . . . . . . . 24

References . . . . . . . . . . . . . . . . . . . . . . . . . 25

Introduction

Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring O_{f} whose fraction field K_{f} has finite degree over ℚ. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character ϵ. The L-series L(f/K, s) = L(f, s)L(f ⊗ ϵ, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 − s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p.

The field K determines a factorization N = N^{+}N^{−} of N by taking N^{+}, resp. N^{−} to be the product of all the prime factors of N which are split, resp. inert in K. Given a ring class field extension H of K of conductor c prime to N, write H_{n} for the ring class field of conductor cp^{n}. It is an extension of H of degree e_{n} := 2u^{−1}(p + 1)p^{n−1}, where u is the order of the group of roots of unity in the order O of K of conductor c. Recall that p splits in H/K, and the primes of H above p are totally ramified in H_{n}. For n ≥ 1, a construction explained in [BD1], sec. 2.5 allows us to define a compatible collection of Heegner points P_{n} over H_{n} on a certain Shimura curve X. In the notations of [BD1], sec. 1.3, X is the curve X_{N^{+}p,N^{−}/p} over ℚ attached to an Eichler order of level N^{+}p in the indefinite quaternion algebra of discriminant N^{−}/p.

1Partially supported by grants from MURST, CNR and EC.

2Università di Pavia, Pavia, Italy.

3Partially supported by CICMA and by grants from CNR, FCAR, and NSERC.

4McGill University, Montreal, Canada.

Let J be the jacobian of X, J_{n} the Néron model of J over H_{n}, and Φ_{n} the group of connected components at p of J_{n}. More precisely,

Φ_{n} := ⊕_{𝔭|p}Φ_{𝔭},

where Φ_{𝔭} is the group of connected components of the fiber at 𝔭 of J_{n} and the sum is extended over all the primes 𝔭 of H_{n} above p.

Define a Heegner divisor α_{n} := (P_{n}) − (w_{N}P_{n}), where w_{N} is the Atkin-Lehner involution denoted w_{N^{+}p,N^{−}/p} in [BD1], sec. 1.8. We view α_{n} as an element of J_{n}, and let ᾱ_{n} be its natural image in Φ_{n}.

We have found that the position of ᾱ_{n} in Φ_{n} is encoded in the special values of the L-functions attached to cusp forms of weight 2 on X twisted by characters of Δ := Gal(H/K).

More precisely, observe that the Galois group Gal(H_{n}/K) acts on J(H_{n}) and on J_{n}. Since the primes above p are totally ramified in H_{n}/H, the induced action on Φ_{n} factors through Δ. Define e_{χ} := ∑_{g∈Δ} χ^{−1}(g)g ∈ ℤ[χ][Δ], and let ᾱ_{n}^{χ} := e_{χ}ᾱ_{n}.

The ring 𝕋 generated over ℤ by the Hecke correspondences on X acts in a compatible way on J(H_{n}), J_{n} and Φ_{n}. Write ϕ_{f} : 𝕋 → O_{f} for the homomorphism associated to f by the Jacquet-Langlands correspondence (cf. [BD1], sec. 1.6), and let π_{f} ∈ 𝕋 ⊗ K_{f} be the idempotent corresponding to ϕ_{f}. Fix n_{f} ∈ O_{f} so that η_{f} := n_{f}π_{f} belongs to 𝕋 ⊗ O_{f}, and define ᾱ_{n}^{f,χ} := η_{f}ᾱ_{n}^{χ}.

The group Φ_{n} is equipped with a canonical monodromy pairing

[ , ]_{n} : Φ_{n} × Φ_{n} → ℚ/ℤ,

which we extend to a hermitian pairing on Φ_{n} ⊗ O_{f}[χ] with values in K_{f}[χ]/O_{f}[χ], denoted in the same way by abuse of notation. Our main result is:

Theorem A Suppose that χ is a primitive character of Δ. Then

[ᾱ_{n}^{χ}, ᾱ_{n}^{f,χ}]_{n} = \frac{1}{e_{n}} \frac{L(f/K, χ, 1)}{(f, f)}\sqrt{d} · u^{2} · n_{f} (mod O_{f}[χ]),

where (f, f) is the Petersson scalar product of f with itself, and d denotes the discriminant of O.

The proof is based on Grothendieck’s description of Φ_{n} [Groth], on the work of Edixhoven on the specialization map from J_{n} to Φ_{n} [Edix], and on a slight generalization [Dag] of Gross’ formula for special values of L-series [Gr1]. Theorem A can be viewed as a p-adic analytic analogue of the Gross-Zagier formula, and was suggested by the conjectures of Mazur-Tate-Teitelbaum type formulated in [BD1], ch.

5. It is considerably simpler to prove than the Gross-Zagier formula, as it involves neither derivatives of L-series nor global heights of Heegner points.

The above formula has a number of arithmetic applications. Let A_{f} be the abelian variety quotient of J associated to ϕ_{f} by the Eichler-Shimura construction. Following the methods of Kolyvagin, we can use the Heegner points α_{n} to construct certain cohomology classes in H^{1}(H, (A_{f})_{e_{n}}), whose local behaviour is related via theorem A to L(A_{f}/K, χ, 1) = ∏_{σ} L(f^{σ}/K, χ, 1), where σ ranges over the set of embeddings of K_{f} in ℚ̄. This can be used to study the structure of the χ-isotypical component A_{f}(H)^{χ} := e_{χ}A_{f}(H) ⊂ A_{f}(H)⊗ℤ[χ] of the Mordell-Weil group A_{f}(H). In particular, we show:

Theorem B If L(A_{f}/K, χ, 1) is non-zero, then A_{f}(H)^{χ} is finite.

When χ = χ̄, this result also follows from the work of Gross-Zagier [GZ] and Kolyvagin-Logachev [KL], but if χ is non-quadratic the previous techniques cannot be used to study these questions.

It is worth stating the following two corollaries of theorem B.

Corollary C Let E/ℚ be a semistable elliptic curve, and assume that L(E/ℚ, 1) is non-zero. Then E(ℚ) is finite.

Proof. By the fundamental work of Wiles and Taylor-Wiles (cf. [W] and [TW]), E is modular. A theorem of Waldspurger [Wald] ensures the existence of an imaginary quadratic field K such that L(E/K, 1) is non-zero. Then theorem B implies that E(K), and hence E(ℚ), is finite.

The previously known proof invokes an analytic result of Bump-Friedberg-Hoffstein [BFH] and Murty-Murty [MM], according to which there exists an auxiliary imaginary quadratic field K such that all the primes dividing N are split in K and the first derivative L′(E/K, 1) is non-zero. In this setting, there is a Heegner point in E(K), arising from the modular curve parametrization X_{0}(N) → E. This point has infinite order by the formula of Gross-Zagier [GZ]. Then Kolyvagin’s theorem [Ko] implies that E(K) has rank one, and that E(ℚ) is finite. This proof is more general than ours, since it applies to all modular elliptic curves, and also yields the finiteness of the Shafarevich-Tate group of E. Our proof depends crucially on the existence of a prime p of multiplicative reduction, and only establishes the finiteness of the p-primary part of the Shafarevich-Tate group.

Theorem B allows us to control the growth of Mordell-Weil groups over anticyclotomic ℤ_{ℓ}-extensions, addressing a conjecture of Mazur [Ma2]. Let f and K be as at the beginning of this section. Let ℓ_{1}, …, ℓ_{k} be primes not dividing N, and let K_{∞} denote the compositum of all the ring class field extensions of K of conductor of the form ℓ_{1}^{n_{1}} ⋯ ℓ_{k}^{n_{k}}, where n_{1}, …, n_{k} are non-negative integers. Thus, the Galois group of K_{∞}/K is isomorphic to the product of a finite group by ℤ_{ℓ_{1}} × ⋯ × ℤ_{ℓ_{k}}.

Corollary D Assume that L(A_{f}/K, χ, 1) ≠ 0 for almost all finite order characters of Gal(K_{∞}/K). Then the Mordell-Weil group A_{f}(K_{∞}) is finitely generated.

(See the details of the proof in section 8.) Computations of root numbers show that L(f/K, χ, s) vanishes to even order at s = 1 for all χ as above, and it is expected that L(f/K, χ, 1) be non-zero for almost all χ. (For results in this direction, see [Ro1] and [Ro2].) We remark that a result similar to corollary D for the cyclotomic ℤ_{ℓ}-extension of ℚ has been announced recently by K. Kato.

Theorems A and B provide a technique to study “analytic rank-zero situations” in terms of Heegner points of conductor divisible by powers of a prime p of multiplicative reduction for A_{f} and inert in K. What makes this possible, ultimately, is a “change of sign” phenomenon: if L(f/K, s) vanishes to even order, and χ is an anticyclotomic character of conductor cp^{n} with c prime to N, then L(f/K, χ, s) vanishes to odd order, and there are Heegner points on A_{f} defined over the extension cut out by χ. The previous applications of the theory of Heegner points, such as the analytic formula of Gross-Zagier and the methods of Kolyvagin, occur in situations where L(f/K, s) and L(f/K, χ, s) both vanish to odd order.

Aknowledgements We are very grateful to Bas Edixhoven for writing an appendix to this paper, which provides a crucial step towards proving our results. We also thank the referee for useful remarks.

1 Gross’ formula for special values of L-series

We keep the notations of the introduction. Let B be the definite quaternion algebra ramified at the primes dividing N^{−}. Let R_{1}, …, R_{t} denote the oriented Eichler orders of level N^{+} in B (see [Rob], sec. 1.6 and [BD1], sec. 1.1). Define

𝕄 := ℤ · R_{1} ⊕ ⋯ ⊕ ℤ · R_{t}

to be the free ℤ-module of formal ℤ-linear combinations of the R_{i}. (This is the module denoted by J_{N^{+},N^{−}} in [BD1].) Let

⟨ , ⟩ : 𝕄 × 𝕄 → ℤ

be the pairing defined by the rule ⟨R_{i}, R_{j}⟩ = δ_{ij}w_{i}, where w_{i} is one half the order of R_{i}^{×}.

Let O be a fixed oriented order of K of conductor c (see [BD1], sec. 2.2), with c prime to N. A Gross point of conductor c is an optimal embedding

ψ : O → R_{i},

i = 1, …, t, preserving the orientations on O and R_{i}. Here, two R_{i}-valued embeddings are identified if they are conjugate under the natural action of R_{i}^{×}. The set of the Gross points of conductor c is endowed with a natural free and transitive action of the group Δ = Pic(O) (see [BD1], sec. 2.3).

Fix a Gross point ψ. For g ∈ Δ, write ξ, resp. ξ^{g} for the natural image of ψ, resp. ψ^{g} in 𝕄. Let ξ^{χ} be ∑_{g∈Δ} χ^{−1}(g)ξ^{g} ∈ 𝕄 ⊗ ℤ[χ] and let ξ^{f,χ} be the element η_{f}ξ^{χ} of 𝕄 ⊗ O_{f}[χ]. With the notations of the introduction, we have:

Theorem 1.1 Suppose that χ is primitive. Then

⟨ξ^{χ}, ξ^{f,χ}⟩ = \frac{L(f/K, χ, 1)}{(f, f)}\sqrt{d} · (u/2)^{2} · n_{f}.

The above formula is proved in [Gr1] only when N is a prime number and c = 1, so that χ is an unramified character. One can check that the methods of Gross extend directly to this more general setting: see the work in progress [Dag].

2 Bad reduction of Shimura curves

We review results on the bad reduction at p of X and J_{n}, due to Deligne-Rapoport, Drinfeld, Grothendieck and Raynaud. In [Dr], Drinfeld constructs a model X_{ℤ} of X over ℤ, i.e., a projective scheme over ℤ whose generic fiber is equal to X. The definition of X_{ℤ} is via moduli: X_{ℤ} coarsely represents the moduli functor which associates to a scheme S the set of isomorphism classes of abelian schemes of dimension 2 over S, endowed with quaternionic multiplication and a suitable level N^{+}p-structure. (The definition of the functor is explained in [Dr] and [Rob].)

Consider the model X := X_{ℤ} ⊗ ℤ_{p} of X over ℤ_{p}. Let X_{p} denote the special fiber of X. By the work of Deligne-Rapoport [DeRa] and Katz-Mazur [KaMa] (see also [Rob], ch. 4, in particular the remark at the end of the introduction of §4), the following holds. Let X′ be the Shimura curve denoted X_{N^{+},N^{−}/p} in [BD1], sec. 1.3. Denote by X′ its Drinfeld model over ℤ_{p}, and by X′_{p} the special fiber at p of X′. The fiber X_{p} is the union of two copies of X′_{p}, crossing transversally at the supersingular points of X′_{p}. A point on X′_{p} is called supersingular if it corresponds to an abelian surface over 𝔽̄_{p} together with its additional structure, having endomorphism ring equal to an Eichler order in a quaternion algebra. This order is necessarily an Eichler order of level N^{+} in the definite quaternion algebra of discriminant N^{−}, by the work of Waterhouse [Wa]. By abuse of language, we call supersingular also the (ordinary) double points of X_{p}. Let s ∈ X_{p} be any such point. It is defined either over 𝔽_{p} or over 𝔽_{p^{2}}. Viewing s as a point on X over the ring W (𝔽̄_{p}) of Witt vectors of 𝔽̄_{p}, its strict localization is of the form

W (𝔽̄_{p})[[x, y]]/(xy − p^{k}),

where the “width of singularity” k = k_{s} is ≥ 1. The model X is regular at s if and only if k_{s} is equal to 1.

We need also consider the behaviour under base-change of X. Given a local field F of residue characteristic p, write R for its ring of integers. Let π be a uniformizer of R, and let e be the ramification index of R over ℤ_{p}. Consider the base change X_{R} = X ⊗ R of X to R. The singularities of the special fiber X_{π} of X_{R} correspond bijectively to the ordinary double points of X_{p}. Suppose that the strict localization of a double point of X_{p} is of the form W (𝔽̄_{p})[[x, y]]/(xy − p^{k}). Then the strict localization of the corresponding point on X_{π} is

W (𝔽̄_{p})[[x, y]]/(xy − π^{ek}).

Thus the singularities of the special fiber of X_{R} are ordinary double points, corresponding to the double points of X_{p}, and their width gets multiplied by the ramification index e.

It is important for us to have a description of the group of connected components of the fiber at π of the Néron model J over F of the jacobian of X. Let s_{1}, …, s_{t} be the supersingular points of X_{π}, with respective width k_{1}, …, k_{t} (relative to the uniformizer π). The work of Raynaud [Ray 1] relates the Picard scheme Pic(X) to J. Building on this, Grothendieck [Groth] gives the following description of the group of connected components Φ = Φ_{π} = J_{π}/J_{π}^{0} of the special fiber J_{π} of J. There is a canonical identification

J_{π}^{0} = Pic^{0}(X_{π}).

Since the singularities of X are ordinary double points, Pic^{0}(X), and hence J, has semistable reduction at π. The character group 𝕄_{0} of the maximal torus of J_{π} is equal to the group of degree zero divisors with ℤ-coefficients supported on s_{1}, …, s_{t}. The work of Waterhouse (see [Wa], and [Rob], thm. 4.2.2.) shows that the map sending s_{i} to its endomorphism ring R_{i} induces a bijection between the set of supersingular points and the set of oriented Eichler orders R_{1}, …, R_{t} introduced in the previous section. Hence 𝕄_{0} is identified with the kernel of the degree map 𝕄 → ℤ, where 𝕄 is the module of section 1. Given a ℤ- module Λ, we write Λ^{∨} for its ℤ-dual Hom(Λ, ℤ). Let

⟨ , ⟩ : 𝕄_{0} × 𝕄_{0} → ℤ

be defined as the restriction to 𝕄_{0} of the pairing of the previous section, and let

ϕ : 𝕄_{0} → 𝕄_{0}^{∨}

be the induced map. One can show that the width of singularity k_{i} is equal to ew_{i}. (Recall that w_{i} is equal to \frac{1}{2}#(R_{i}^{×}).)

Theorem 2.1 (Grothendieck)

1. The pairing e⟨ , ⟩ is equal to the monodromy pairing on 𝕄_{0}.

2. There is a canonical identification

Φ = coker(𝕄_{0} ⟶^{eϕ} 𝕄_{0}^{∨}),

where ϕ is the map induced by the pairing ⟨ , ⟩.

Proof.

1. See Grothendieck, loc. cit., thm. 12.5, and [Ray 2], pp. 16-17.

2. Grothendieck, loc. cit., thm. 11.5.

The monodromy pairing e⟨ , ⟩ on 𝕄_{0} gives rise to a pairing

[ , ] : Φ × Φ → ℚ/ℤ

(which we still call the monodromy pairing by abuse of terminology), in the following way. The map eϕ induces an isomorphism from 𝕄_{0} ⊗ ℚ to Hom(𝕄_{0}, ℚ), which allows us to extend the pairing e⟨ , ⟩ to a ℚ-valued pairing on 𝕄_{0}^{∨} ⊂ Hom(𝕄_{0}, ℚ). The reader will check that passing to the quotient gives rise to a well defined pairing on Φ = 𝕄_{0}^{∨}/eϕ(𝕄_{0}), with values in ℚ/ℤ.

As a corollary we obtain:

Corollary 2.2

1. There is an exact sequence

0 → 𝕄_{0} ⟶^{e} 𝕄_{0} ⟶^{κ} Φ → 𝕄_{0}^{∨}/ϕ(𝕄_{0}) → 0.

2. Given v_{1} and v_{2} in 𝕄_{0}, we have

[κv_{1}, κv_{2}] = \frac{1}{e}⟨v_{1}, v_{2}⟩ (mod ℤ).

An alternate description of Φ, also based on the work of Raynaud, is given in [MaRa]. Example. Given a discrete valuation ring extension R of ℤ_{11}, with absolute ramification degree e, and letting π denote a uniformizer of R, as an example we compute in terms of the above description the group of connected components of the fiber at π of the modular curve X = X_{0}(11) = X_{11,1}. There are two singular points s_{1} and s_{2} on X, with width of singularity k_{1} = 2e and k_{2} = 3e. (They correspond to supersingular elliptic curves in characteristic 11, having j-invariant equal to 1728 and 0, respectively.) The fiber X_{π} has two components C and C′, crossing at s_{1} and s_{2}. The character group 𝕄_{0} of the maximal torus is equal to ℤ(s_{1} − s_{2}). The pairing ⟨ , ⟩ is characterized by ⟨s_{1} − s_{2}, s_{1} − s_{2}⟩ = 5e, so that Φ is a cyclic group of order 5e.

3 Heegner points and connected components

In this section, we describe the natural image in Φ_{n} of the divisors of degree zero on X(H_{n}) supported on the Heegner points of conductor cp^{n}. Let G_{n} be Gal(H_{n}/H). The group G_{n} is cyclic of order e_{n}. Let 𝔭 be a prime of H_{n} above p, and let X_{𝔭} be the fiber at 𝔭 of the base change of X to the ring of integers of the completion of H_{n} at 𝔭. Let P be a Heegner point of conductor cp^{n}. (Cf. [BD1], sec. 2.1 for the definition of these points.)

Lemma 3.1 The Heegner point P reduces modulo 𝔭 to a supersingular point in X_{𝔭}.

Proof. In view of the modular interpretation of X ([Rob]), the Heegner point P corresponds to an abelian surface with quaternionic multiplication and level structure, and the ring of endomorphisms of the modulus P is equal to the order O_{n} of K of conductor cp^{n}. This abelian surface is isogenous to a product E × E, where E is an elliptic curve whose ring of endomorphisms is equal to an order of K. Since p is inert in K, the curve E has supersingular reduction at 𝔭. The claim follows.

Identify in the natural way the module of divisors, resp. of divisors of degree zero supported on the supersingular points of X_{𝔭} with 𝕄, resp. 𝕄_{0}. Let Div^{hp}, resp. Div_{0}^{hp} denotes the module of formal divisors, resp. degree zero divisors supported on the Heegner points of conductor cp^{n}.

Lemma 3.1 allows us to define a reduction map

ρ : Div^{hp} → ⊕_{𝔭|p}𝕄.

Corollary 2.2 gives the exact sequence

0 → ⊕_{𝔭|p}𝕄_{0} ⟶^{e_{n}} ⊕_{𝔭|p}𝕄_{0} ⟶^{κ} Φ_{n} → ⊕_{𝔭|p}(𝕄_{0}^{∨}/ϕ(𝕄_{0})) → 0,

where by abuse of notation we denote by κ also the map on ⊕_{𝔭|p}𝕄_{0}.

Theorem 3.2 Let D be a divisor in Div_{0}^{hp}, and let D̄ be the natural image of D in Φ_{n}. Then

D̄ = κρ(D).

Proof. Given a prime 𝔭 of H_{n} above p, let ρ_{𝔭} : Div^{hp} → 𝕄, resp. κ_{𝔭} : 𝕄_{0} → Φ_{𝔭} denote the 𝔭-component of the map ρ, resp. κ. Note that κ_{𝔭} factors as

𝕄_{0} ⊂ 𝕄 ⟶^{ϕ̃} 𝕄^{∨} ⟶^{π} Φ_{𝔭},

where ϕ̃ is induced by the pairing on 𝕄 defined in the previous section, and π is equal to the dual of 𝕄_{0} ⊂ 𝕄 composed with the projection of M_{0}^{∨} onto Φ_{𝔭} determined by theorem 2.1. Identify 𝕄 with the free ℤ-module generated by the supersingular points s_{1}, …, s_{t} of X_{𝔭}, and write s_{1}^{∨}, …, s_{t}^{∨} for the dual basis of 𝕄^{∨} relative to the standard scalar product. If ρ_{𝔭}(D) is equal to ∑_{i=1}^{t} n_{i} · s_{i}, then its image ϕ̃ρ_{𝔭}(D) in 𝕄^{∨} is ∑_{i=1}^{t} w_{i}n_{i} ·s_{i}^{∨}. Thus, by the formula (2.3) of the Appendix, theorem 3.2 is equivalent to the equality w_{i} = m(s_{i}), where the numbers m(s_{i}) are defined in section 2 of the Appendix. Let H_{𝔭} be the completion of H_{n} at 𝔭. Write Ĥ_{𝔭}^{unr} for the completion of the maximal unramified extension of H_{𝔭}, R for the ring of integers of Ĥ_{𝔭}^{unr}, and π_{R} for a uniformizer of R. Observe that in order to apply the results of the Appendix, we have to pull-back our objects to R.

Let now P be a Heegner point of conductor cp^{n}. We first consider the case of modular curves. Here P corresponds to a diagram (E ⟶^{α} E′, β), where we may assume that E, resp. E′ is an elliptic curve defined over H_{n−1}, resp. H_{n}, and where α : E → E′ is an isogeny of degree p and β denotes the prime-to-p level structure carried by P. Viewing E and E′ as elliptic curves over R/pR = R/π_{R}^{e_{n}}R, let F : E → E^{(p)} be the Frobenius morphism. By the formula (3.3) of the Appendix and by the above remarks, we are reduced to showing that E^{(p)} and E′ are not isomorphic over R/π_{R}^{2}R. This is a consequence of Lubin-Tate’s theory of formal moduli and of Gross’ work on quasi-canonical liftings of formal groups. More precisely, let 𝔪(E′) ∈ π_{R}R, resp. 𝔪(E^{(p)}) ∈ π_{R}(R/pR) denote the formal modulus of E′, resp. E^{(p)}, defined as in [LT], sec. 3. By proposition 5.3 of [Gr3], part 3, ord_{π_{R}}𝔪(E′) is equal to 1. On the other hand, since E is defined over H_{n−1} and H_{n} is totally ramified over H_{n−1}, we find that 𝔪(E^{(p)}) belongs to π_{R}^{[H_{n}:H_{n−1}]}(R/pR). Since formal moduli classify liftings of formal groups up to isomorphism (see [LT], thm. 3.1 and prop. 3.3), this proves theorem 3.2 in the case of modular curves.

In the general case, the local study of Heegner points on Shimura curves reduces to similar considerations on deformations of formal groups of dimension one: see section 4 of the Appendix, and in particular proposition 4.2. This concludes the proof of theorem 3.2.

4 Proof of theorem A

Let 𝔻 be the free ℤ-module of formal linear combinations of Gross points of conductor c. Given a Heegner point P of conductor cp^{n} and a prime 𝔭 of H_{n} above p, the reduction modulo 𝔭 of endomorphisms determines an embedding O_{n} → R_{i}, where R_{i} is one of the Eichler orders above. This embedding extends to an embedding ψ : O → R_{i}, since there are no optimal embeddings of orders of conductor divisible by p into the R_{i} (cf. [BD1], lemma 2.1).

Proposition 4.1 The embedding ψ : O → R_{i} is optimal.

Proof. Let P_{∗} ∈ X(H_{1}) be the Heegner point such that the divisors U_{p}^{n−1}P_{∗} and Norm_{H_{n}/H_{1}}P on X are equal. (See the discussion in [BD1], sec. 2.4.) Let P′ ∈ X′(H) be the image of P_{∗} by the natural projection. (Recall that X′ is the Shimura curve introduced in section 2.) Note that P′ has endomorphism ring O, and the embedding ψ is equal to the reduction modulo 𝔭 of the endomorphisms of P′. It is a consequence of [GZ], proposition 7.3 that ψ is optimal.

Proposition 4.1 allows us to define a map

Ψ : Div^{hp} → ⊕_{𝔭|p}𝔻.

We define the action of the Galois group Δ on ⊕_{𝔭|p}𝔻 by permutation of the summands: ⊕_{𝔭|p}𝔻 := Ind_{(1)}^{Δ}(𝔻) as a Δ-module. With this definition, note that Ψ is Δ-equivariant. Recall also from section 1 that Δ acts on 𝔻. We may extend this action diagonally to ⊕_{𝔭|p}𝔻.

Lemma 4.2 The two actions of Δ on ⊕_{𝔭|p}𝔻 agree on Im(Ψ).

Proof. Fix a prime 𝔭 of H above p, and let Ψ_{𝔭} : Div^{hp} → 𝔻 be the natural map obtained by composing Ψ with the projection on the component at 𝔭. With P′ as in the proof of proposition 4.1, the claim amounts to showing that for all σ in Δ

Ψ_{𝔭}((P′)^{σ}) = Ψ_{𝔭}(P′)^{σ},

where the action of Δ on the right hand side is the one considered in section 1. Let s ∈ 𝕄 be the reduction modulo 𝔭 of P′, so that Ψ_{𝔭}(P′) corresponds to the reduction modulo 𝔭 of endomorphisms

ψ : End(P′) → End(s).

Let 𝔞 be the element of Pic(O) representing σ such that

𝔞 = Hom((P′)^{σ}, P′).

It follows from proposition 7.3 of [GZ] that Hom(s^{σ}, s) is equal to End(s)𝔞, where we let s^{σ} denote the reduction modulo 𝔭 of (P′)^{σ}. Observe that End(s^{σ}) is the right order of Hom(s^{σ}, s), so that reduction modulo 𝔭 of endomorphisms gives rise to an optimal embedding

ψ′ : O = End((P′)^{σ}) → R′ = End(s^{σ}).

But ψ′ is equal to ψ^{σ} by definition: see [Gr1], p. 134.

Let ω be the natural map from ⊕_{𝔭|p}𝔻 to ⊕_{𝔭|p}𝕄. Then we have a natural commutative diagram of Galois and Hecke equivariant maps

(∗)  \begin{matrix}Div^{hp} & ⟶^{ρ} & ⊕_{𝔭|p}𝕄 \\ ^{Ψ}↓ & & ^{=}↓ \\ ⊕_{𝔭|p}𝔻 & ⟶^{ω} & ⊕_{𝔭|p}𝕄. \end{matrix}

By tensoring with O_{f}[χ], we extend κ to a map from ⊕_{𝔭|p}𝕄_{0}⊗O_{f}[χ] to Φ_{n}^{f,χ}⊗O_{f}[χ], and denote it by the same symbol.

Proof of Theorem A. Noting that w_{N} acts as −1 on f ([BD1], sec. 1.8 and 2.8), so that w_{N}η_{f} = −η_{f}, we have the chain of equalities

[ᾱ_{n}^{χ}, ᾱ_{n}^{f,χ}]_{n} = [κρ((P_{n}) − (w_{N}P_{n}))^{χ}, κρ((P_{n}) − (w_{N}P_{n}))^{f,χ}]_{n}  (by theorem 3.2)

≡ \frac{4}{e_{n}} ⟨(ρP_{n})^{χ}, (ρP_{n})^{f,χ}⟩ (by corollary 2.2)

= \frac{4}{e_{n}} ⟨ω(ΨP_{n})^{χ}, ω(ΨP_{n})^{f,χ}⟩ (by the commutative diagram (∗) )

= \frac{1}{e_{n}} \frac{L(f/K, χ, 1)}{(f, f)}\sqrt{d} · u^{2} · n_{f} (mod O_{f}[χ]),

where the last equality follows from theorem 1.1.

Remark. We may combine the formulae of theorem A corresponding to the various n in a single statement. Let

Φ_{∞} := lim_{n^{⟵}} Φ_{n},

where the inverse limit is with respect to the maps of multiplication by p. By theorem 2.1, there is a surjection (which is well-defined up to sign) from Φ_{∞} to ⊕_{𝔭|p}𝕄_{0}^{∨} ⊗ ℤ_{p}. The monodromy pairings [ , ]_{n} give rise to a canonical pairing

[ , ]_{∞} : Φ_{∞} × Φ_{∞} → ℤ_{p}.

Denote by ᾱ_{∞}^{f} the natural image in Φ_{∞} of the norm-compatible sequence of Heegner divisors (α_{n}^{f}). Then, theorem A can be restated as follows.

Theorem 4.3 Suppose that χ is a primitive character of Δ. Then we have the equality in O_{f}[χ]

[ᾱ_{∞}^{f,χ}, ᾱ_{∞}^{f,χ}]_{∞} = \frac{L(f/K, χ, 1)}{(f, f)}\sqrt{d} · u^{2} · n_{f}^{2}.

5 A rigid analytic Gross-Zagier formula

In [BD1], we formulate conjectures for elliptic curves with values in anticyclotomic towers, which are analogues of the conjectures of Mazur, Tate and Teitelbaum for the cyclotomic ℤ_{p}-extension of ℚ [MTT], and which contain new arithmetic features that have no counterpart in the setting of [MTT]. Here we point out that the results contained in this paper give a proof of an important special case of our conjectures, precisely conjecture 5.5 of section 5.2.

Suppose that c = 1 and that f has rational Fourier coefficients, so that it corresponds to an isogeny class of elliptic curves defined over ℚ. Any E in this isogeny class has multiplicative reduction at our fixed prime p (which is inert in K). By the Jacquet-Langlands correspondence there are maps

π_{E∗} : J → E, π_{E}^{∗} : E → J

(cf. [BD1], sec. 1.9). Here the map π_{E}^{∗} is the dual of the Shimura curve parametrization π_{E∗}. We assume that π_{E∗} has connected kernel, or equivalently that π_{E}^{∗} is injective. We define the degree of π_{E∗} to be the positive integer d_{X} corresponding to the endomorphism π_{E∗} ∘ π_{E}^{∗} of E. There is also a classical modular curve parametrization of minimal degree d_{X_{0}(N)} from X_{0}(N) to the strong Weil curve E′ in the isogeny class of E. Let δ_{X} be the ratio d_{X}/d_{X_{0}(N)}, and let Ω be the complex period associated to E′.

Since p is inert in K, the curve E/K_{p} has split multiplicative reduction. Let

Φ_{n}^{E} = ⊕_{𝔭|p}Φ_{𝔭}^{E}

be the group of connected components at p of the Néron model of E over H_{n}. By Tate’s theory, Φ_{𝔭}^{E} is isomorphic (up to sign) to ℤ/e_{n}c_{p}ℤ, where c_{p} := ord_{p}(q_{E}). Let H_{n,p} stand for H_{n} ⊗ ℚ_{p}. Define

Ê(H_{∞,p}) := lim_{n^{⟵}} E(H_{n,p}), Φ_{∞}^{E} := lim_{n^{⟵}} Φ_{n}^{E},

where the inverse limits are taken with respect to the norm and the multiplication by p maps respectively. Note that there is a surjection from Φ_{∞}^{E} to ℤ_{p}, which induces a map

λ_{q_{E}} : Ê(H_{∞,p}) → ℤ_{p}

by specializing to the group of connected components.

Let y_{n} := π_{E∗}(α_{n}) ∈ E(H_{n}). The Heegner points y_{n} are norm-compatible and, by [BD1], sec. 2.5, Norm_{H_{n}/K}y_{n} = 0. The reader should think of λ_{q_{E}}((y_{n})) as the leading coefficient of the p-adic L-function associated to the sequence (y_{n}). (See [BD1], sec. 2.7 for more details.)

Theorem 5.1 We have

λ_{q_{E}}((y_{n}))^{2} = \frac{L(E/K, 1)}{Ω}\sqrt{d} · u^{2} · δ_{X} · c_{p}.

Proof. The group Φ_{n}^{E} is equipped with the monodromy pairing

[ , ]_{E,n} : Φ_{n}^{E} × Φ_{n}^{E} → ℚ/ℤ,

whose 𝔭-th component [ , ]_{E,𝔭} is characterized by [1, 1]_{E,𝔭} = \frac{1}{e_{n}c_{p}}. Let

π̄_{E∗} : Φ_{n} → Φ_{n}^{E}

be the map on connected components induced by π_{E∗}. Write 1 for the trivial character of Δ. The number λ_{q_{E}}((y_{n}))^{2} is equal to [π̄_{E∗}ᾱ_{n}^{1}, π̄_{E∗}ᾱ_{n}^{1}]_{E,n} · (e_{n}c_{p}) modulo e_{n}. Hence

\frac{1}{c_{p}e_{n}} λ_{q_{E}}((y_{n}))^{2} ≡ [π̄_{E∗}ᾱ_{n}^{1}, π̄_{E∗}ᾱ_{n}^{1}]_{E,n} = [ᾱ_{n}^{1}, π̄_{E}^{∗}π̄_{E∗}ᾱ_{n}^{1}]_{n}.

Observe that π̄_{E}^{∗}π̄_{E∗} is equal to d_{X}π_{f} acting on Φ_{n}. This can be seen by noting that π_{E}^{∗}π_{E∗} is necessarily an integer multiple of π_{f}, and

(π_{E}^{∗}π_{E∗})^{2} = d_{X}π_{E}^{∗}π_{E∗}.

Define now η_{f} to be d_{X}π_{f}, so that n_{f} = d_{X}. Then by theorem A we have

[ᾱ_{n}^{1}, π̄_{E}^{∗}π̄_{E∗}ᾱ_{n}^{1}]_{n} = [ᾱ_{n}^{1}, ᾱ_{n}^{f,1}]_{n} = \frac{1}{e_{n}} \frac{L(f/K, 1)}{(f, f)}\sqrt{d} · u^{2} · d_{X} (mod ℤ).

Using the fact that (f, f) is equal to d_{X_{0}(N)}Ω, we find

λ_{q_{E}}((y_{n}))^{2} = \frac{L(E/K, 1)}{Ω}\sqrt{d} · u^{2} · δ_{X} · c_{p},

as was to be shown.

Remarks

1. The formula of theorem 5.1 may be seen as an analogue of the formula of Gross-Zagier, in the rigid analytic setting, and of the theorem of Greenberg-Stevens [GS], in the anticyclotomic setting.

2. Theorem A shows that the sequence of Heegner points (y_{n}) maps non-trivially to the group of connected components Φ_{∞}^{E} precisely when L(E/K, 1) is non-zero. On the other hand, it is always easy to construct norm compatible sequences of local points on E whose norm to K_{p} is equal to zero, and whose image in Φ_{∞}^{E} is non-trivial. This follows from the fact that the period q_{E} is a local universal norm in the anticyclotomic ℤ_{p}-extension of K_{p}.

3. The classical Birch and Swinnerton-Dyer conjecture predicts that

\frac{L(E/K, 1)}{Ω} = #� (E/K) ∏_{ℓ|N^{+}} c_{ℓ}^{2} ∏_{ℓ|N^{−}} c_{ℓ} · (#(E(K)_{tors})^{2} · \sqrt{d} · (u/2)^{2})^{−1},

where c_{ℓ} is the number of connected components of the fiber at ℓ of the Néron model of E over ℚ. Combining this with theorem 5.1 suggests that

(δ_{X})^{−1} = ∏_{ℓ|(N^{−}/p)} c_{ℓ} (mod (ℚ^{×})^{2}).

It would be interesting to investigate when this equality holds not just up to squares. In this connection, see the forthcoming work of Ribet and Takahashi [RT].

6 Kolyvagin cohomology classes

Let A denote the abelian variety A_{f} of the introduction. We construct cohomology classes in H^{1}(H, A_{e_{n}}) from Heegner points on X defined over ring class field extensions of H_{n}, and we study their ramification properties. These classes will be used in the next section to bound the Mordell-Weil group A(H). Preliminaries. If L is a number field, we write G_{L} for its absolute Galois group, and G_{M/L} for the Galois group of a finite Galois field extension M/L. We denote by Frob_{ℓ}(M/L) the Frobenius element attached to a prime ℓ of L which is unramified in M. It is a well-defined conjugacy class in G_{M/L}.

From now on we let 𝕋 denote the algebra generated by the Hecke operators acting on A. The map ϕ_{f} of the introduction induces an isomorphism of 𝕋 onto the ring O_{f} generated over ℤ by the Fourier coefficients of f. We will write T_{ℓ} to denote the image of the ℓ-th Hecke operator in 𝕋. The ring 𝕋 need not be integrally closed: let 𝕋̃ be the integral closure of 𝕋 in its fraction field, and let ι be the exponent of 𝕋 in 𝕋̃. The adelic Tate module T (A) := lim_{m^{⟵}} A_{m} is endowed with a natural action

of G_{ℚ}, and T (A) ⊗_{𝕋} 𝕋̃ is free of rank 2 over 𝕋̃ ⊗ ℤ̂. Choosing a basis gives Galois representations

ρ : G_{ℚ} → GL_{2}(𝕋̃ ⊗ ℤ̂), and ρ_{m} : G_{ℚ} → GL_{2}(𝕋̃/ιm𝕋̃),

for all positive integers m. Let F_{m} denote the smallest field extension of H through which ρ_{m} factors. Note that A_{m} is defined over F_{m}.

Given a squarefree product t = ∏ ℓ of primes ℓ such that (cp, ℓ) = 1, let H_{n}[t] denote the compositum of H_{n} with the ring class field K[t] of conductor t. Write G_{t} for Gal(H_{n}[t]/H_{n}) and G̃_{t} for Gal(H_{n}[t]/H). Thus we have canonical identifications G̃_{t} = G_{n} × G_{t} and G_{t} = ∏_{ℓ} G_{ℓ}.

Definition. A prime ℓ is a Kolyvagin prime (relative to n) if ℓ does not divide 2c(p + 1)N, and satisfies

Frob_{ℓ}(F_{e_{n}}/ℚ) = [τ],

where τ denotes a fixed complex conjugation in G_{ℚ}.

Let ℓ be a Kolyvagin prime relative to n. Then, the group G_{ℓ} is cyclic of order ℓ+1.

Lemma 6.1 If ℓ is a Kolyvagin prime relative to n, then T_{ℓ} belongs to e_{n}𝕋 and ℓ + 1 is divisible by e_{n}.

Proof. By the Eichler-Shimura relation, the characteristic polynomial of ρ_{e_{n}}(Frob_{ℓ}) is equal to x^{2} − T_{ℓ}x + ℓ. The characteristic polynomial of ρ_{e_{n}}(τ) is equal to x^{2} − 1. It follows that ℓ + 1 is divisible by ιe_{n}, and the image of T_{ℓ} in 𝕋̃ ⊗ ℤ̂ is divisible by ιe_{n}, so that T_{ℓ} belongs to e_{n}𝕋.

Fix a generator σ, resp. σ_{ℓ} for G_{n}, resp. G_{ℓ}, and write Norm, resp. Norm_{ℓ} for the associated norm operators. Define Kolyvagin’s derivative operators

D = ∑_{i=1}^{e_{n}−1} iσ^{i}, D_{ℓ} = ∑_{i=1}^{ℓ} iσ_{ℓ}^{i}.

The following equations hold:

(1)  (σ − 1)D = e_{n} − Norm, (σ_{ℓ} − 1)D_{ℓ} = (ℓ + 1) − Norm_{ℓ}.

Given a squarefree product t of Kolyvagin primes, let D_{t} ∈ ℤ[G_{t}], resp. D̃_{t} ∈ ℤ[G̃_{t}] stand for ∏_{ℓ} D_{ℓ}, resp. D ∏_{ℓ} D_{ℓ}. When t = 1 is the empty product, we mean that D̃_{t} is equal to D.

For all the t as before, the results of [BD1], sec. 2.5 allow us to define Heegner points α_{n}(t) ∈ A(H_{n}[t]) satisfying the relations

(2)  Norm(α_{n}(t)) = 0, Norm_{ℓ}(α_{n}(t)) = T_{ℓ}α_{n}(t/ℓ).

Let α_{n} denote the Heegner point corresponding to the empty product. (The point α_{n} is the image in A of the Heegner point α_{n} of the previous sections.)

Let Φ_{n}^{A}, resp. Φ_{n,t}^{A} be the group of connected components at p of the Néron model of A over H_{n}, resp. H_{n}[t]. Since the primes of H_{n} over p are unramified in H_{n}[t], we have (Φ_{n,t}^{A})^{G_{t}} = Φ_{n}^{A}. Let D_{t}ᾱ_{n}(t) be the image of D_{t}α_{n}(t) in Φ_{n,t}^{A}.

Lemma 6.2

1. The natural image Q_{n}(t) of D̃_{t}α_{n}(t) in (A(H_{n}[t])/e_{n}A(H_{n}[t])) is fixed by G̃_{t}.

2. The element D_{t}ᾱ_{n}(t) belongs to Φ_{n}^{A}.

Proof. Part 1 follows from equations (1) and (2), combined with lemma 6.1. To prove part 2, it is enough to show that D_{t}ᾱ_{n}(t) is fixed by G_{t}. Let ℓ be a prime dividing t. Then we have

(σ_{ℓ} − 1)D_{t}ᾱ_{n}(t) = (ℓ + 1)D_{t}ᾱ_{n}(t) − T_{ℓ}D_{t}ᾱ_{n}(t/ℓ).

Since Norm acts as multiplication by e_{n} on Φ_{n,t}^{A} and kills α_{n}(t) and α_{n}(t/ℓ), we find that ᾱ_{n}(t) and ᾱ_{n}(t/ℓ) belong to (Φ_{n,t}^{A})_{e_{n}}. The result follows from lemma 6.1.

Lemma 6.3 The order of the group A(H_{n}[t])_{tors} is bounded independently of n and t.

Proof. Choose two primes q_{1} and q_{2} which are inert in K and do not divide N. The residue field of H_{n}[t] at any prime above q_{1} is 𝔽_{q_{1}^{2}}, and likewise for q_{2}. Since A(H_{n}[t])_{tors} injects in A(𝔽_{q_{1}^{2}}) ⊕ A(𝔽_{q_{2}^{2}}), the claim follows.

In view of lemma 6.3, define an absolute integer constant a which annihilates A_{e_{n}}(H_{n}[t]) for all n and t, and the groups Φ_{[ℓ]}^{A} of connected components at ℓ of A/ℚ, for all primes ℓ which divide N and are split in K.

Construction of the Kolyvagin classes. Recall the e_{n}-descent exact sequence over H_{n}[t]:

0 → A(H_{n}[t])/e_{n}A(H_{n}[t]) ⟶^{δ} H^{1}(H_{n}[t], A_{e_{n}}) → H^{1}(H_{n}[t], A)_{e_{n}} → 0,

and let c_{n}(t)^{oo} = −δQ_{n}(t) be the natural image of −D̃_{t}α_{n}(t) in H^{1}(H_{n}[t], A_{e_{n}}). By lemma 6.2, the class c_{n}(t)^{oo} belongs to H^{1}(H_{n}[t], A_{e_{n}})G̃_{t}. The Hochschild-Serre spectral sequence H^{i}(G̃_{t}, H^{j}(H_{n}[t], A_{e_{n}})) ⟹ H^{i+j}(H, A_{e_{n}}) gives rise to the exact sequence (inflation-restriction):

H^{1}(G̃_{t}, A_{e_{n}}(H_{n}[t])) ⟶^{infl} H^{1}(H, A_{e_{n}}) ⟶^{res} H^{1}(H_{n}[t], A_{e_{n}})G̃_{t} → H^{2}(G̃_{t}, A_{e_{n}}(H_{n}[t])).

Hence there exists a class c_{n}(t)^{o} in H^{1}(H, A_{e_{n}}) such that res(c_{n}(t)^{o}) = ac_{n}(t)^{oo}. This class is well-defined in H^{1}(H, A_{e_{n}}), modulo the image of the inflation map. Thus the class c_{n}(t) = ac_{n}(t)^{o} is well-defined in H^{1}(H, A_{e_{n}}).

We call the class c_{n}(t) ∈ H^{1}(H, A_{e_{n}}) the Kolyvagin cohomology class associated to n and to the product t of Kolyvagin primes (relative to n). The class c_{n}(1) corresponding to the empty product t = 1 will also be denoted by c_{n}. The remainder of this section is devoted to a study of the Kolyvagin classes c_{n}(t).

Explicit description of c_{n}(t). We will have use for the following explicit formula for the class c_{n}(t). Let [\frac{D̃_{t}α_{n}(t)}{e_{n}}] be a (fixed) point in A(H̄) such that e_{n}[\frac{D̃_{t}α_{n}(t)}{e_{n}}] = D̃_{t}α_{n}(t). For all γ ∈ G_{H}, the point (γ − 1)D̃_{t}α_{n}(t) belongs to e_{n}A(H_{n}[t]) by lemma 6.2. Let \frac{(γ−1)D̃_{t}α_{n}(t)}{e_{n}} be a point in A(H_{n}[t]) such that e_{n} \frac{(γ−1)D̃_{t}α_{n}(t)}{e_{n}} = (γ − 1)D̃_{t}α_{n}(t). This point is well-defined modulo A_{e_{n}}(H_{n}[t]), and hence the point a\frac{(γ−1)D̃_{t}α_{n}(t)}{e_{n}} is uniquely defined. Define a cochain c_{n}(t)′ with values in A_{e_{n}} by the formula

c_{n}(t)′(γ) = −a(γ − 1)[\frac{D̃_{t}α_{n}(t)}{e_{n}}] + a\frac{(γ − 1)D̃_{t}α_{n}(t)}{e_{n}}.

The cochain c_{n}(t)′ is not a cocycle in general. One checks, however, that its coboundary dc_{n}(t)′(γ_{1}, γ_{2}) = c_{n}(t)′(γ_{1}γ_{2}) − (c_{n}(t)′(γ_{1}) + γ_{1}c_{n}(t)′(γ_{2})) takes values in A_{e_{n}}(H_{n}[t]), so that the class ac_{n}(t)′ is a cocycle. As McCallum [McC] has remarked, we have:

Lemma 6.4 The Kolyvagin class c_{n}(t) is represented by the cocycle ac_{n}(t)′.

Proof. A direct computation.

Action of complex conjugation. Define a sign w to be −1 if A/ℚ_{p} has split multiplicative reduction, and to be +1 if A/ℚ_{p} has non-split multiplicative reduction.

Proposition 6.5 There exists γ ∈ Δ such that τc_{n}(t) = (−1)^{#{ℓ|t}} · w · γc_{n}(t).

Proof. By [BD1], prop. 2.5, we have τα_{n}(t) = −wγ′α_{n}(t), for some element γ′ of Gal(H_{n}[t]/K). Using equation (2) and lemma 6.1, a direct calculation proves the claim. (See [Gr2], prop. 5.4.)

Descent Modules. Given a rational prime ℓ and a number field F, we let F_{ℓ} be ⊕_{λ|ℓ}F_{λ}, where the sum is taken over the primes λ of F above ℓ and F_{λ} denotes the completion of F at λ. We extend additively functors defined on finite extensions of ℚ_{ℓ}. Thus, for instance, H^{1}(H_{ℓ}, A) := ⊕_{λ|ℓ}H^{1}(H_{λ}, A), etc.

Recall the descent exact sequence:

0 → A(H)/e_{n}A(H) → H^{1}(H, A_{e_{n}}) → H^{1}(H, A)_{e_{n}} → 0.

For each prime ℓ of K there is also a corresponding exact sequence of local cohomology groups, obtained by replacing H by H_{ℓ}. Both of these sequences respect the natural actions of the Hecke algebra 𝕋 and the Galois group Δ.

Let W ⊂ H^{1}(H, A_{e_{n}}) be the image of A(H)/e_{n}A(H), and let X = H^{1}(H, A)_{e_{n}} be the cokernel. Denote by W_{ℓ} and X_{ℓ} the local counterparts of these groups, for each prime ℓ.

We will need an explicit description of the modules W_{ℓ} and X_{ℓ}, at least when ℓ is a Kolyvagin prime relative to n and when ℓ = p.

Lemma 6.6 Suppose that ℓ is a Kolyvagin prime. Then there is a canonical Δ and 𝕋-equivariant isomorphism

ψ_{ℓ} : X_{ℓ} → Hom(G_{ℓ}, W_{ℓ}).

Proof. Let H_{ℓ}^{unr} = ⊕_{λ|ℓ}H_{λ}^{unr} be the maximal unramifield extension of H_{ℓ}. Since ℓ is a Kolyvagin prime, we have by the inflation-restriction sequence

X_{ℓ} = H^{1}(H_{ℓ}^{unr}, A_{e_{n}})^{Frob^{ℓ}} = Hom(Gal(H̄_{ℓ}/H_{ℓ}^{unr}), A_{e_{n}})^{Frob^{ℓ}} = Hom(G_{ℓ}, A_{e_{n}}(H_{ℓ})).

Finally, we identify A_{e_{n}}(H_{ℓ}) = ⊕_{λ|ℓ}A_{e_{n}}(H_{λ}) with W_{ℓ} = ⊕_{λ|ℓ}A(H_{λ})/e_{n}A(H_{λ}) via the map ⊕_{λ|ℓ}((ℓ + 1)Frob_{λ}(H/ℚ) − T_{ℓ})/e_{n}. See [Gr2] for more details.

Note that the proof of lemma 6.6 shows that the 𝕋[Δ]-modules X_{ℓ} and W_{ℓ} are both isomorphic to A_{e_{n}}(H_{ℓ}) = Ind_{(1)}^{Δ}A_{e_{n}}.

We now turn to ℓ = p. Let Y_{p} denote H^{1}(G_{n}, A(H_{n,p}))_{e_{n}}, identified by inflation with a submodule of X_{p}.

Lemma 6.7 There is a canonical Δ and 𝕋-equivariant injection

ψ_{p} : Y_{p} → Hom(G_{n}, Φ_{n}^{A}).

Proof. The Mumford-Tate theory of p-adic uniformization gives rise to an exact sequence

0 → U_{H_{n,p}} ⊗ (𝕄_{0}^{A})^{∨} → A(H_{n,p}) → Φ_{n}^{A} → 0,

where U_{H_{n,p}} is the group of units in H_{n,p}^{×} and 𝕄_{0}^{A} is the character group of A at p. Taking G_{n}-cohomology yields

0 → Φ^{A} → Φ_{n}^{A} → H^{1}(G_{n}, U_{H_{n,p}} ⊗ (𝕄_{0}^{A})^{∨}) → Y_{p} → Hom(G_{n}, Φ_{n}^{A}),

where Φ^{A} denotes the group of connected components of A over H_{p}. The G_{n}- cohomology of the natural sequence

0 → U_{H_{n,p}} ⊗ (𝕄_{0}^{A})^{∨} → H_{n,p}^{×} ⊗ (𝕄_{0}^{A})^{∨} ⟶^{ord} (𝕄_{0}^{A})^{∨} → 0

shows that H^{1}(G_{n}, U_{H_{n,p}} ⊗ (𝕄_{0}^{A})^{∨}) is isomorphic to (ℤ/e_{n}ℤ)^{dim(A)}. Hence ψ_{p} is injective.

Residues and duality. Let ξ be a global cohomology class in H^{1}(H, A_{e_{n}}). If ℓ is a rational prime, the residue at ℓ of ξ, denoted ∂_{ℓ}ξ, is defined to be the natural image of ξ in X_{ℓ}. If ∂_{ℓ}ξ is zero, then the image of ξ in H^{1}(H_{ℓ}, A_{e_{n}}) belongs to W_{ℓ}. We denote by v_{ℓ}ξ this image, which we call the value of ξ at ℓ.

The class ξ has only finitely many non-trivial residues, since it is unramified for almost all primes. Define the support Supp(ξ) of ξ to be the set of primes of ℚ at which ξ has non-trivial residue.

Choose a polarization of A, i.e., a ℚ-isogeny from A to its dual abelian variety A^{∨}. This choice combined with the canonical non- degenerate Weil pairing A_{e_{n}} ⊗A_{e_{n}}^{∨} → μ_{e_{n}} gives rise to a pairing A_{e_{n}} ⊗A_{e_{n}} → μ_{e_{n}}, whose left and right radical have order bounded independently of n. Cup product followed by this modified Weil pairing gives a symmetric Galois and Hecke-equivariant local Tate pairing

⟨ , ⟩_{ℓ} : H^{1}(H_{ℓ}, A_{e_{n}}) × H^{1}(H_{ℓ}, A_{e_{n}}) → H^{2}(H_{ℓ}, μ_{e_{n}}) ⟶^{inv_{ℓ}} ℤ/e_{n}ℤ,

where inv_{ℓ} denotes the sum of the invariants at the primes dividing ℓ ([Mi]). By results of Tate, the radical of ⟨ , ⟩_{ℓ} has order bounded independently of n. The submodule W_{ℓ} is isotropic for ⟨ , ⟩_{ℓ}, and hence the local Tate pairing gives rise to

⟨ , ⟩_{ℓ} : W_{ℓ} × X_{ℓ} → ℤ/e_{n}ℤ,

which by abuse of notation we denote in the same way.

The following reciprocity law of Poitou and Tate is fundamental.

Proposition 6.8 (Global reciprocity) Suppose that ξ_{1} and ξ_{2} are global classes in H^{1}(H, A_{e_{n}}) with disjoint support. Then we have:

∑_{ℓ∈Supp(ξ_{2})}⟨v_{ℓ}ξ_{1}, ∂_{ℓ}ξ_{2}⟩_{ℓ} = − ∑_{ℓ∈Supp(ξ_{1})}⟨v_{ℓ}ξ_{2}, ∂_{ℓ}ξ_{1}⟩_{ℓ}.

Proof. By the formula for the local Tate pairing we have ⟨v_{ℓ}ξ_{1}, ∂_{ℓ}ξ_{2}⟩_{ℓ} = inv_{ℓ}(ξ_{1}∪ξ_{2}), for all ℓ in the support of ξ_{2}, and likewise for ξ_{1}. Hence

∑_{ℓ∈Supp(ξ_{2})}⟨v_{ℓ}ξ_{1}, ∂_{ℓ}ξ_{2}⟩_{ℓ} + ∑_{ℓ∈Supp(ξ_{1})}⟨v_{ℓ}ξ_{2}, ∂_{ℓ}ξ_{1}⟩_{ℓ} = ∑_{ℓ} inv_{ℓ}(ξ_{1} ∪ ξ_{2}).

The last sum is 0, by the global reciprocity law of class field theory ([Mi]).

Local behaviour of the Kolyvagin classes. We turn to the behaviour of the classes c_{n}(t) under localization. It might help the reader already familiar with Kolyvagin’s theory to notice that the classes c_{n}(t) differ from the cohomology classes considered by Kolyvagin (see for example [Gr2]) for the fact that they may be non-trivial not only at tamely ramified (Kolyvagin) primes, but also at the wildly ramified prime p.

Proposition 6.9

1. If ℓ does not divide pt, then ∂_{ℓ}c_{n}(t) = 0.

2. If ℓ divides t, then:

(a) (Induction formula) ψ_{ℓ}(∂_{ℓ}c_{n}(t))(σ_{ℓ}) = v_{ℓ}c_{n}(t/ℓ).

(b) (Orthogonality relation) ⟨v_{ℓ}𝕋[Δ]c_{n}(t/ℓ), ∂_{ℓ}c_{n}(t)⟩_{ℓ} = 0.

3. The residue ∂_{p}c_{n}(t) belongs to Y_{p}, and

ψ_{p}(∂_{p}c_{n}(t))(σ) = a^{2}D_{t}ᾱ_{n}(t),

where D_{t}ᾱ_{n}(t) denotes the image of D_{t}α_{n}(t) in Φ_{n}^{A}.

Proof.

1. Observe that the extension H_{n}[t]/H is unramified outside pt. Then, by [Ma1], prop. 4.3, the group H^{1}(Gal((H_{n}[t])_{ℓ}/H_{ℓ}), A((H_{n}[t])_{ℓ})) is naturally isomorphic to H^{1}(Gal(𝔽_{(H_{n}[t])_{ℓ}}/𝔽_{H_{ℓ}}), Φ_{[ℓ]}^{A}). If ℓ is a prime of good reduction for A, then Φ_{[ℓ]}^{A} is trivial, and hence ∂_{ℓ}c_{n}(t) = 0. If ℓ is inert in K, then H_{n}[t]/H has trivial residue field extension at ℓ, and thus c_{n}(t) restricts to zero at ℓ. Finally, if ℓ splits in K and divides N, then our choice of a guarantees that ∂_{ℓ}c_{n}(t) = 0.

2. By the explicit description of c_{n}(t), and the description of X_{ℓ} given in the proof of lemma 6.6, we have the equality

∂_{ℓ}c_{n}(t)(σ_{ℓ}) = a^{2} \frac{(σ_{ℓ} − 1)D̃_{t}α_{n}(t)}{e_{n}}

in A_{e_{n}}(H_{ℓ}). The equations (1) and (2) yield

\frac{(σ_{ℓ} − 1)D̃_{t}α_{n}(t)}{e_{n}} = \frac{(ℓ + 1)D̃_{t/ℓ}α_{n}(t) − T_{ℓ}D̃_{t/ℓ}α_{n}(t/ℓ)}{e_{n}}.

Part (a) now follows from the description of the map ψ_{ℓ} given in proposition 6.6, and the congruence

α_{n}(t) ≡ Frob_{λ′}(H_{n}[t]/ℚ)α_{n}(t/ℓ) (mod λ′)

for all primes λ′ of H_{n}[t] above ℓ. (The above congruence is consequence of equation (2), combined with the Eichler-Shimura relation. See [Gr2], prop. 3.7 and 6.2 for more details.)

The proof of (b) in a special case in contained in proposition 3.6, (b) of [B2]. The general case is proved along similar lines.

3. It follows from lemma 6.4, combined with equation (1).

Given a character χ : Δ → ℤ[χ]^{×}, let [χ] be the Galois orbit of χ, and define the operator e_{[χ]} ∈ ℤ[Δ] to be e_{[χ]} := ∑_{χ^{σ}∈[χ]} e_{χ^{σ}}. Let V be a ℤ[Δ]-module. Unlike the previous sections, we now define V^{χ} := e_{[χ]}V. It is a ℤ[Δ]-submodule of V. The action of the group ring ℤ[Δ] on V^{χ} factors through the map ℤ[Δ] → ℤ[χ] induced by χ. In this way, V^{χ} can and will be viewed as a ℤ[χ]-module. Suppose in addition that V is a ℤ[G_{H/ℚ}]-module. Then the complex conjugation τ acts on V^{χ} and this action is skew-linear with respect to the action of ℤ[χ], i.e., ταv = ᾱτv for all v ∈ V^{χ} and α ∈ ℤ[χ]. Let V^{χ,±} be the ±-eigenspace for τ acting on V^{χ}.

Let c_{n}(t)^{χ} be the class e_{[χ]}c_{n}(t) in H^{1}(H, A_{e_{n}})^{χ}.

Proposition 6.10 If L(A/K, χ, 1) ≠ 0, then the 𝕋[χ]-submodule of Y_{p}^{χ} generated by ∂_{p}c_{n}^{χ} has index bounded independently of n.

Proof. Since L(A/K, χ, 1) is non-zero, then L(f^{σ}/K, χ, 1) is non-zero for all the Galois conjugate forms f^{σ} of f. Proposition 6.10 follows from theorem A combined with part 3 of proposition 6.9.

7 Bounding Mordell-Weil groups

Assuming that L(A/K, χ, 1) ≠ 0, we show in this section that the image of A(H)/p^{n}A(H) in H^{1}(H, A_{p^{n}})^{χ} is bounded independently of n, thus proving theorem B.

We will make a shift in notation, letting W^{χ}, resp. X^{χ} be (A(H)/p^{n}A(H))^{χ}, resp. H^{1}(H, A)_{p^{n}}^{χ}, and likewise for their local counterparts W_{ℓ}^{χ} and X_{ℓ}^{χ}. Moreover, we replace the class c_{n}(t) defined in section 6 with its natural image in H^{1}(H, A_{p^{n}}).

Given a ℤ[χ]-module V, let V^{dual} be the Pontryagin dual Hom(V, ℚ/ℤ) of V, viewed as a ℤ[χ]-module via the rule α(f(v)) = f(αv) for all α ∈ ℤ[χ] and v in V. Consider the natural map

ν_{n} : W^{χ} → (Y_{p}^{χ})^{dual},

equal to the composite map W^{χ} ⟶^{v_{p}} W_{p}^{χ} → (X_{p}^{χ})^{dual} → (Y_{p}^{χ})^{dual}, where the second map is induced by the local Tate duality, and the third map is the dual of the natural inclusion Y_{p}^{χ} ↪ X_{p}^{χ}. Our proof of theorem B divides naturally in two steps: first we bound uniformly the image of ν_{n}, and then its kernel.

Proposition 7.1 The order of the image of ν_{n} is bounded independently of n.

Proof. Let C be the submodule of H^{1}(H, A_{p^{n}})^{χ} generated over 𝕋[χ] by the Kolyvagin class c_{n}^{χ}. Let P be a point of W^{χ}. By proposition 6.9, the classes in C have support only above p, and by definition P has empty support. Hence by proposition 6.8, we have

⟨v_{p}P, x⟩_{p} = 0 ∀x ∈ ∂_{p}(C).

By proposition 6.10, the index of ∂_{p}(C) in Y_{p}^{χ} is bounded independently of n. The result follows.

Recall that w is the sign defined in section 6.

Corollary 7.2 Suppose that χ = χ̄. Then A(H)^{χ,w} is finite.

Proof. When χ = χ̄ the module Y_{p}^{χ} is identified with a submodule of X_{p}^{χ,w}, whose index in X_{p}^{χ,w} is bounded independently of n. Since ν_{n} is τ-equivariant, proposition 7.1 shows that the map W^{χ,w} → (X_{p}^{χ,w})^{dual} induced by ν_{n} has image bounded independently of n. Hence, by the τ-equivariance of the local Tate duality, the natural image of A(H)^{χ,w} in A(H_{p})^{χ,w} ⊗ ℤ_{p} is finite. Since the natural map A(H)^{χ,w} → A(H_{p})^{χ,w} ⊗ ℤ_{p} has finite kernel, the claim follows.

It is worth recording the following consequence of corollary 7.2.

Theorem 7.3 Let E be a semistable elliptic curve, having a prime p of non-split multiplicative reduction. If L(E/ℚ, 1) is non-zero, then E(ℚ) is finite.

Proof. By [Wi] and [TW], E is modular. One chooses an auxiliary imaginary quadratic field K such that p is inert in K and L(E/K, 1) is non-zero. This is possible, by a theorem of Waldspurger [Wald]. In this case, the sign w is 1 and E(H)^{1,w} = E(ℚ) is finite by corollary 7.2.

Remark. Note that the proofs of proposition 7.1, corollary 7.2 and theorem 7.3 do not use the Kolyvagin primes ℓ, but only the wildly ramified prime p. Bounding the kernel of ν_{n} requires a more involved argument, based on the use of Kolyvagin primes and the Chebotarev density theorem.

Preliminaries. We begin with proving two lemmas, which are used to establish the important technical proposition 7.6, stating that a global cohomology class is essentially determined by its restriction at the Kolyvagin primes. The key ingredient in the proof of proposition 7.6 is the Chebotarev density theorem.

From now on we denote the field F_{n} defined in the previous section by F. Recall that A_{p^{n}} is defined over F.

Lemma 7.4 The order of H^{1}(G_{F/H}, A_{p^{n}}) divides an integer b_{1} independent of n.

Proof. This is proved in [KL], prop. 5.10 along the following lines. By a result of Serre [Se], the image Π of the Galois representation ρ_{p^{n}} contains a group Π_{0} of scalar matrices equal to the natural image of 1 + b′_{1}ℤ_{p} in 𝕋̃/p^{n}𝕋̃, where b′_{1} is a non-zero integer independent of n. The Hochschild-Serre spectral sequence for Π_{0} ◁ Π gives the exact sequence

0 → H^{1}(Π/Π_{0}, A_{p^{n}}^{Π_{0}}) → H^{1}(Π, A_{p^{n}}) → H^{1}(Π_{0}, A_{p^{n}}).

Now A_{p^{n}}^{Π_{0}} is contained in A_{b′_{1}}, and H^{1}(Π_{0}, A_{p^{n}}) is contained in A_{p^{n}}/b′_{1}A_{p^{n}}, so that the order of H^{1}(Π, A_{p^{n}}) divides the order of A_{b′_{1}}^{2}. Since b′_{1} does not depend on n, the result follows upon taking b_{1} = #A_{b′_{1}}^{2}.

Lemma 7.5 There exists a constant b_{2} independent of n such that the following holds.

(a) Let U be a submodule of A_{p^{n}} which is stable under the action of G_{F/H}. Then we can find u ∈ U^{+} so that b_{2} annihilates the quotient U/ℤ[G_{F/H}]u.

(b) Let U be a submodule of Hom_{𝕋}(𝕋[χ]^{2}, A_{p^{n}}) which is stable under the action of ℤ[χ][G_{F/H}]. Then we can find u ∈ U^{+} so that b_{2} annihilates the quotient U/ℤ[χ][G_{F/H}]u.

Proof.

(a) By replacing U by its pre-image under the natural projection T_{p}(A) → A_{p^{n}}, where T_{p}(A) denotes the p-adic Tate module of A, we are reduced to proving the claim for a submodule U of T_{p}(A), which is stable under the action of ℤ_{p}[G_{H}].

The Galois group G_{H} acts naturally on T_{p}(A): let R be the image of the group ring ℤ_{p}[G_{H}] in End(T_{p}(A)). By a result of Serre [Se], R has finite index in End(T_{p}(A)). Let 𝕋̃ be as before the integral closure of 𝕋 in its fraction field. Note that T_{p}(A) ⊗_{𝕋} 𝕋̃ is isomorphic to (𝕋̃ ⊗ ℤ_{p})^{2}. Since 𝕋 has finite index in 𝕋̃, the natural inclusion of R in End(T_{p}(A)) ⊗ 𝕋̃ ≃ M_{2}(𝕋̃ ⊗ ℤ_{p}) has finite cokernel of order independent of n: let b_{2} ∈ ℤ_{p} be an annihilator.

Let Ū be M_{2}(𝕋̃⊗ℤ_{p})U, viewed as a submodule of T_{p}(A)⊗_{𝕋}𝕋̃. Then b_{2} annihilates Ū/U, and Ū = M_{2}(𝕋̃ ⊗ ℤ_{p})ū, for ū ∈ Ū^{+}. Defining u to be b_{2}ū concludes the proof of part (a).

(b) As before, we may replace U by its pre-image in Hom_{𝕋}(𝕋[χ]^{2}, T_{p}(A)). Let Ū be M_{2}(𝕋̃ ⊗ ℤ_{p})U, viewed as a submodule of

Hom_{𝕋}(𝕋[χ]^{2}, T_{p}(A)) ⊗_{𝕋} 𝕋̃ ≃ Hom𝕋̃(𝕋̃[χ]^{2}, 𝕋̃^{2}) ⊗ ℤ_{p}.

The last module is identified with (𝕋̃^{2} ⊗ ℤ_{p}[χ]^{∨}) ⊕ (𝕋̃^{2} ⊗ ℤ_{p}[χ]^{∨}) ≃ T^{2} ⊕ T^{2}, where T is the ring 𝕋̃ ⊗ ℤ_{p}[χ], and the action of M_{2}(T ) on T^{2} is by left multiplication, viewing T^{2} as column vectors. Note that T is a semilocal principal ideal ring, equal to the product of discrete valuation rings which are finite extensions of ℤ_{p}. By working component by component, we may, and will from now on, assume that T is a discrete valuation ring. Projection onto the second factor gives an exact sequence of M_{2}(T )-modules

0 → Ū′ → Ū → Ū′′ → 0.

The modules Ū′ and Ū′′ are either zero or isomorphic to T^{2} as M_{2}(T )-modules. If Ū′′ is zero, then Ū = Ū′. Assume that Ū′′ is non-zero, and let (ξ, η) be an element of Ū such that η generates Ū′′ as an M_{2}(T )-module. If the vectors ξ and η are not multiples of each other by an element of T ⊗ ℚ, we may find a matrix A ∈ M_{2}(T ) such that A(ξ, η) = (0, η). Thus, M_{2}(T )(0, η) is a submodule of Ū isomorphic to Ū′′. If ξ and η are proportional by an element of T ⊗ℚ, the submodule M_{2}(T )(ξ, η) is isomorphic to Ū′′. In all cases, we have

Ū = Ū′ ⊕ Ū′′

as M_{2}(T )-modules. A direct computation now shows that Ū is generated over M_{2}(T ) by an element of Ū^{+}. The result follows as in the proof of part (a).

Let b denote the integer (independent of n) b_{1}#(𝕋[χ]/2b_{2}𝕋[χ])^{2}.

Proposition 7.6 Let ξ_{1} and ξ_{2} be cohomology classes in H^{1}(H, A_{p^{n}})^{χ}, and let C be the 𝕋[χ]-module they generate. Assume that C is stable under the action of τ, and, if χ = χ̄, suppose further that ξ_{1} and ξ_{2} belong to different eigenspaces for the action of τ. Then there exist infinitely many Kolyvagin primes ℓ such that ∂_{ℓ}ξ_{1} = ∂_{ℓ}ξ_{2} = 0 and such that the order of the kernel of the natural map

v_{ℓ} : C → W_{ℓ}^{χ}

divides the constant b.

Proof. It is convenient to treat the case when χ = χ̄ separately from the case when χ ≠ χ̄.

Suppose first that χ = χ̄, so that ξ_{1} and ξ_{2} belong to different eigenspaces for τ. This enables us to prove the proposition for the 𝕋-modules generated by ξ_{1} and ξ_{2}, considered one at a time. Let ξ ∈ H^{1}(H, A_{p^{n}})^{χ} be one of the cohomology classes ξ_{1} and ξ_{2}, and let ξ′ ∈ Hom(G_{F} , A_{p^{n}})^{χ} be its natural image by restriction. Call L_{ξ} the Galois extension of F cut out by ξ′, and U_{ξ} = Gal(L_{ξ}/F) its Galois group. The map ξ′ is a ℤ[G_{F/H}]-equivariant homomorphism, and it identifies U_{ξ} with a submodule of A_{p^{n}}. Let u be the element of U_{ξ}^{+} produced by lemma 7.5 (a) applied to the module U_{ξ}. By the Chebotarev density theorem, there exist infinitely many primes ℓ such that

Frob_{ℓ}(L_{ξ}/ℚ) = [τu].

Observe that ℓ is a Kolyvagin prime relative to n, and that

Frob_{ℓ}(L_{ξ}/K) = [τuτu] = [u^{2}].

If an element φ of 𝕋ξ is in the kernel of the map v_{ℓ}, then its restriction φ′ to G_{F} satisfies φ′(Frob_{λ}(L_{ξ}/F)) = 0 for all primes λ of F above ℓ, and hence vanishes on ℤ[G_{F/H}]u^{2}. The claim when χ = χ̄ now follows from lemma 7.4 and the choice of u.

Suppose that χ ≠ χ̄. Let C be the module generated over 𝕋[χ] by the classes ξ_{1} and ξ_{2}, and let C′ ⊂ Hom(G_{F} , A_{p^{n}}) be the natural restriction of C. Let L be the extension cut out by C′, and let U = G_{L/F}. Consider the left and right non-degenerate Kummer pairing

( , ) : C′ × U → A_{p^{n}}.

The modules appearing in this pairing are each endowed with various structures coming from the natural action of G_{F/K} and from Hecke operators. More precisely, C′ is a 𝕋[χ]-module; the module U is a module over ℤ[χ][G_{F/H}]; and A_{p^{n}} is equipped with a natural action of 𝕋[G_{F/H}]. The pairing ( , ) obeys the following compatibilities with respect to these actions:

(1) (T c, u) = T (c, u), for all T ∈ 𝕋.

(2) (c, gu) = g(c, u), for all g ∈ G_{F/H}.

(3) (αc, u) = (c, ᾱu), for all α ∈ ℤ[χ]. Hence the Kummer pairing induces an injection of ℤ[χ][G_{F/H}]- modules

U ↪ Hom_{𝕋}(C′, A_{p^{n}}) ↪ Hom_{𝕋}(𝕋[χ]^{2}, A_{p^{n}}).

The last injection is induced by our choice of the two 𝕋[χ]- module generators of C′ coming from ξ_{1} and ξ_{2}. Let u be the element of U^{+} produced by lemma 7.5 (b) applied to the module U. By the Chebotarev density theorem, there exist infinitely many primes ℓ such that

Frob_{ℓ}(L_{ξ}/ℚ) = [τu].

Observe that ℓ is a Kolyvagin prime relative to n. The reader will check as in the case χ = χ̄ that the kernel of the map from C to W_{ℓ} has order dividing b.

Proposition 7.7 The order of the kernel of ν_{n} is bounded independently of n.

Proof. Let P be an element in ker(ν_{n}). Let C be the submodule of H^{1}(H, A_{p^{n}})^{χ} generated over 𝕋[χ] by P and the Kolyvagin class c_{n}^{χ}. Observe that if χ is a quadratic character, P and c_{n}^{χ} belong to different eigenspaces for the action of τ. Choose a Kolyvagin prime ℓ satisfying the conclusion of proposition 7.6 applied to our module C. By Kolyvagin’s induction formula of proposition 6.9 and our choice of ℓ, combined with proposition 6.10, it follows that the ratio of the orders of 𝕋[χ]∂_{ℓ}c_{n}(ℓ)^{χ} and Y_{p}^{χ} is bounded independently of n. Since ∂_{p}c_{n}(ℓ)^{χ} belongs to Y_{p}^{χ} and P belongs to ker(ν_{n}), it follows that ⟨v_{p}P, ∂_{p}(𝕋[χ]c_{n}(ℓ)^{χ})⟩_{p} = 0. Hence, by proposition 6.8 we have

⟨v_{ℓ}P, ∂_{ℓ}(𝕋[χ]c_{n}(ℓ)^{χ})⟩_{ℓ} = 0.

By the Kolyvagin orthogonality relation of proposition 6.9,

⟨v_{ℓ}c_{n}^{χ}, ∂_{ℓ}(𝕋[χ]c_{n}(ℓ)^{χ})⟩_{ℓ} = 0.

Since by our choice of ℓ the orders of 𝕋[χ]v_{ℓ}c_{n}^{χ} and Y_{p}^{χ} differ by an integer independent of n, the claim follows from a counting argument.

The proof of Theorem B now follows by combining proposition 7.1 and 7.7.

Remark. (Suppose for simplicity that A is an elliptic curve.) If the sign of the functional equation of L(A/K, χ, s) is −1, then one can construct a canonical Heegner point in A(H)^{χ}, and it is expected that this point has infinite order precisely when L′(A/K, χ, 1) ≠ 0. Assuming this, it is shown in [BD2] that the rank over ℤ[χ] of A(H)^{χ} is equal to 1. The methods of [BD2] build directly on the fundamental ideas of Kolyvagin, which were used in [Ko] to handle the case χ = χ̄.

8 Mordell-Weil groups in anticyclotomic towers

In this section, we prove that the Mordell-Weil group of A over very general anticyclotomic towers is finitely generated, under the assumption that “generically” A has analytic rank equal to zero. The precursor of this kind of investigations is Mazur’s conjecture stating that A(ℚ_{∞}) is finitely generated, ℚ_{∞} being the cyclotomic ℤ_{p}-extension of ℚ. A proof of this conjecture, based on the use of cohomology classes made from Steinberg symbols of modular units, has been announced recently by K. Kato [Ka].

As before, let K be an imaginary quadratic field such that L(A/K, s) vanishes to even order at s = 1. Fix primes ℓ_{1}, …, ℓ_{k} of good reduction for A, and let K_{∞} denote the compositum of all the ring class field extensions of K of conductor of the form ℓ_{1}^{n_{1}} · . . . · ℓ_{k}^{n_{k}}, where n_{1}, …, n_{k} are non-negative integers. Thus the Galois group of K_{∞}/K is equal to the product of a finite group by ℤ_{ℓ_{1}} × ⋯ × ℤ_{ℓ_{k}}. We now prove Corollary D of the introduction.

Theorem 8.1 Assume that L(A/K, χ, 1) is non-zero for all but finitely many finite order characters χ of Gal(K_{∞}/K). Then the Mordell-Weil group A(K_{∞}) is finitely generated.

Proof. Given χ factoring through a finite extension H of K and such that the special value L(A/K, χ, 1) is non-zero, theorem B shows that A(H)^{χ} is finite. This implies that rank_{ℤ}A(K_{∞}) is finite. Theorem 8.1 now follows from lemma 6.3.

Remark

1. It is expected that the non-vanishing assumption on the L(A/K, χ, 1) always holds in our setting. See [Ro1] and [Ro2] for computations germane to our study.

2. Note that it is not necessary to assume, as customary in Iwasawa theory, that the ℓ_{i} are primes of ordinary reduction for A.

3. Let ℓ be a prime of good ordinary reduction for A, and let K_{∞} be the anticyclotomic ℤ_{ℓ}-extension of an imaginary quadratic field K. Suppose that all the primes dividing N are split in K, so that the L(A/K, χ, s) vanishes to odd order at s = 1 for all finite order characters χ of Gal(K_{∞}/K). The results of [B1] (where A is an elliptic curve) show, under a mild non-triviality assumption on a Iwasawa module built up from Heegner points, that the Pontryagin dual of A(K_{∞}) ⊗ ℚ_{ℓ}/ℤ_{ℓ} and of Sel_{ℓ^{∞}}(A/K_{∞}) has rank 1 over the Iwasawa algebra ℤ_{ℓ}[[Gal(K_{∞}/K)]]. The method of proof of these results builds on Kolyvagin’s theory.

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