A refined conjecture of Mazur-Tate type for Heegner points

Henri Darmon

May 23, 2000

Contents

1 Preliminaries 2

2 Statement of the results 3 2.1 The conjecture of Mazur Tate type . . . . . . . . . . . . . . . 3 2.2 Application to the Galois module structure of Heegner points . 9

3 Restatement of the results 11 3.1 Calculus of abelian group rings . . . . . . . . . . . . . . . . . 11 3.2 Generalities on Heegner points . . . . . . . . . . . . . . . . . . 14 3.3 A divisibility theorem for Heegner points . . . . . . . . . . . . 16

4 The Heegner cohomology classes 18 4.1 More on derivatives . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 The Heegner cohomology classes . . . . . . . . . . . . . . . . . 19 4.3 Tate duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Application of the Chebotarev density theorem . . . . . . . . 24

5 Proof of the main results 27 5.1 Proof of theorem 3.15 . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Proof of theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Proof of theorem 2.6 . . . . . . . . . . . . . . . . . . . . . . . 34

Abstract

In [MT1], B. Mazur and J. Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curve E. This conjecture relates congruences for certain integral homology cycles on E(C) (the modular symbols) to the arithmetic of E over Q. In this paper we formulate an analogous conjecture for E over suitable imaginary quadratic fields, in which the role of the modular symbols is played by Heegner points. A large part of this conjecture can be proved, thanks to the ideas of Kolyvagin on the Euler system of Heegner points. In effect the main result of this paper can be viewed as a generalization of Kolyvagin’s result relating the structure of the Selmer group of E over K to the Heegner points defined in the Mordell-Weil groups of E over ring class fields of K. An explicit application of our method to the Galois module structure of Heegner points is given in section 2.2.

Acknowledgements: I wish to thank Massimo Bertolini with whom I have had many fruitful discussions on the topics of this paper. I am also grateful to my advisor Benedict Gross for guiding me towards this topic. This research was funded in part by a Natural Sciences and Engineering Research Council of Canada (NSERC) ’67 award, and by a Sloan doctoral dissertation fellowship.

1 Preliminaries

Let E be a modular elliptic curve. There is a morphism

ϕ : X_{0}(N) ⟶ E

defined over Q, where N is the arithmetic conductor of E and X_{0}(N) is the algebraic curve which classifies pairs of elliptic curves related by a cyclic N-isogeny.

The pull-back of a Néron differential ω on E is an eigenform f of weight 2 on X_{0}(N) with Fourier expansion given by

ϕ^{∗}(ω) = c(ϕ) ∑_{n=1}^{∞} a_{n}q^{n}dq/q, q = e^{2πiτ}.

The Fourier expansion is normalized so that a_{1} = 1, and c(ϕ) is the Manin constant associated to the modular parametrization ϕ. The Hasse-Weil Lfunction L(E/Q, s) can be identified with the L-series attached to f,

L(f, s) = ∑_{n=1}^{∞} a_{n}n^{−s}.

From Hecke’s theory one knows that L(f, s) has an Euler product and a functional equation relating its value at s to its value at 2 − s. The parity of the order of vanishing of the L-function at the central point s = 1 can be read off from the functional equation. More precisely, let ϵ denote the eigenvalue of the Atkin-Lehner involution w_{N} acting on f. Then L(f, s) vanishes to odd order at s = 1 if ϵ = 1, and to even order if ϵ = −1.

Fix a quadratic imaginary field K of discriminant D in which all primes dividing N are split. If N = p_{1}^{e_{1}} ⋯ p_{k}^{e_{k}}, one may choose for each p_{i} an ideal P_{i} of K above it, and set

N = P_{1}^{e_{1}} ⋯ P_{k}^{e_{k}}.

Given a positive integer T which is relatively prime to ND, let O_{T} denote the order of K of conductor T. Because T is prime to N, the ideal O_{T} ∩ N is invertible, and the natural projection of complex tori

C/O_{T} ⟶ C/(O_{T} ∩ N)^{−1}

corresponds to a cyclic N-isogeny of elliptic curves. Hence it can be identified with a point of X_{0}(N). By the theory of complex multiplication, this point is defined over K_{T}, the ring class field of K of conductor T. Let α(T) denote the image of this point in E(K_{T}) by the modular parametrization ϕ.

2 Statement of the results

2.1 The conjecture of Mazur Tate type

Given a square-free integer S = l_{1} ⋯ l_{t} prime to ND, write

G_{S} = Gal(K_{S}/K_{1}), Γ_{S} = Gal(K_{S}/K).

Define the regularized Heegner points by the formulas

β_{S} = ∑_{T |S} μ(T)α(T), β_{S}^{ω} = ∑_{T |S} μ(T)ω(T)α(T),

where μ is the Möbius function, and ω is the quadratic Dirichlet character associated to K. Let A(E, S) (resp. A^{ω}(E, S)) denote the formal resolvent associated to the Heegner point β_{S} (resp. β_{S}^{ω}):

A(E, S) := ∑_{σ∈Γ_{S}} σβ_{S} ⊗ σ ∈ E(K_{S}) ⊗ Z[Γ_{S}],
A^{ω}(E, S) := ∑_{σ∈Γ_{S}} σβ_{S}^{ω} ⊗ σ^{−1} ∈ E(K_{S}) ⊗ Z[Γ_{S}].

The role of the θ-element of [MT1] will be played by the element

θ′(E, S) := A(E, S) ⊗_{Z[Γ_{S}]} A(E, S)^{ω} = ∑_{σ,τ∈Γ_{S}} σβ_{S} ⊗ τβ_{S}^{ω} ⊗ στ^{−1};

it belongs to the triple tensor product E(K_{S})^{⊗2} ⊗ Z[Γ_{S}].

For technical reasons it will be convenient to replace Z by a subring Z of Q and view θ′(E, S) as belonging to E(K_{S})^{⊗2} ⊗ Z[Γ_{S}] by extending scalars from Z to Z. For the time being we make no assumptions on Z. Let I denote the augmentation ideal in the group ring Z[Γ_{S}], and let r denote the rank of the Mordell-Weil group of E over K. We conjecture the following:

Conjecture 2.1 (order of vanishing) For any Z, the element θ′(E, S) belongs to the subgroup E(K_{S})^{⊗2} ⊗ I^{r−1} of E(K_{S})^{⊗2} ⊗ Z[Γ_{S}].

Remarks:

1. This statement is analogous to the part of the Birch Swinnerton-Dyer conjecture which predicts that the order of vanishing of the complex L-function of E over K is equal to r. Our conjecture involves r − 1, and not r, because of the philosophy that θ′(E, S) should mirror the behavior of L′(E/K, s) at s = 1. A justification for this philosophy is provided by the analytic formula of Gross and Zagier [GZ]. More precisely, let h : E(K_{S})^{⊗2} ⟶ R be the canonical Néron-Tate height over K_{S}, and let χ : Γ_{S} ⟶ C^{∗} be a complex character of Γ_{S}, extended by linearity to the group ring of Z[Γ_{S}]. Combining h and χ gives rise to a natural linear map:

h ⊗ χ : E(K_{S})^{⊗2} ⊗ Z[Γ_{S}] ⟶ C.

Theorem 2.2 (Gross Zagier) Suppose that S = 1 so that K_{S} is the Hilbert class field of K. Then

h ⊗ χ(θ′(E, S)) = c(ϕ)^{2}[K_{S} : K]\sqrt{Disc(K)}\frac{L′(E/K, χ, 1)}{∫∫_{E(C)} ω ∧ ω̄}

2. Conjecture 2.1 is inspired by the refined conjectures of Birch Swinnerton-Dyer type introduced by Mazur and Tate. For an explanation of these conjectures, the reader may consult the fundamental reference [MT1], or [D2]. It seems that such refined conjectures provide a congenial setting for the Euler Systems of Kolyvagin to express themselves: the properties of such Euler systems (relations between special elements and arithmetic) are naturally formulated as conjectures of Mazur Tate type. This program has been carried out in the simpler case of cyclotomic units [D1], where one finds conjectural formulas which are a slight generalization of those of Thaine [Th].

3. What of the original Mazur Tate conjectures? At present, still no proof is known. What one might need in this case is a cyclotomic Euler system, consisting in a compatible system of cohomology classes

c_{n,p} ∈ H^{1}(Q(μ_{n}), T_{p}(E)).

In a remarkable recent development, Kato [Ka] has succeeded in constructing precisely such an Euler system, using elements in K_{2} of modular curves constructed from Steinberg symbols of Siegel units. He has also succeeded in relating his Euler system to the special values of the complex L-function L(f, χ, 1) twisted by Dirichlet characters, and hence to modular symbols. It seems possible that this work of Kato will shed light on the Mazur Tate conjectures.

Assuming conjecture 2.1, we can project θ′(E, S) to an element θ̃′(E, S) in the group E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}). This element plays the role of the leading coefficient in the refined Birch and Swinnerton-Dyer conjecture. To make a conjecture about its value, let

u = \frac{1}{2}#(O_{K}^{∗}), τ = #E(K)_{tor}.

Given a prime p dividing N, let m_{p} denote the order of the group of connected components in the special fiber at p for the Néron model of E over Spec(Z). Let

m = ∏_{p|N} m_{p}.

Finally, let B denote the “Birch Swinnerton-Dyer constant”

B = c(ϕ) · u · m · \sqrt{#\underline{III}(E/K)} · τ^{−1}.

It is conjectured (cf. [GZ], p. 311) that B is an integer and that, when E(K)/E(K)_{tor} is generated by a single element P, the following identity is true in E(K)/E(K)_{tor}:

Trace_{K_{1}/K}(α(1)) = ±BP.  (1)

This conjecture follows by comparing the Gross Zagier formula with the classical Birch and Swinnerton-Dyer conjecture.

Now we define a regulator term belonging to E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}). Let

E_{S}(K) = ker (E(K) ⟶ (⊕_{v|S}E(k_{v})) ⨁(⊕_{v}E/E_{0}(K_{v}))) ,

and let J_{S} be the order of the cokernel of this map. In [MT1], [MT2], Mazur and Tate define a height pairing

⟨ ⟩_{S} : E(K) × E_{S}(K) ⟶ I/I^{2}.

(In fact, their height pairing takes values in G_{S}, but we use here the isomorphism I/I^{2} ≃ G_{S}.)

Suppose first that E(K) is free over Z, and let P_{1}, …, P_{r} (resp. Q_{1}, …, Q_{r}) denote integral bases for E(K) (resp. E_{S}(K)) which induce compatible orientations. The partial regulator R_{ij} in I^{r−1}/I^{r} is defined to be the determinant of the ijth minor of the pairing matrix (⟨P_{i}, Q_{j}⟩_{S}) with entries in I/I^{2}. The regulator R_{S} is given by the formula

R_{S} = ∑_{i,j=1}^{r}(−1)^{i+j}P_{i} ⊗ Q_{j} ⊗ R_{ij}.  (2)

When E(K) is not free, one normalizes this definition as in [MT1], p. 735: choose finite index subgroups A and B of E(K) and E_{S}(K) which are free, and define the regulator R(A, B) by picking bases P_{1}, …, P_{r} and Q_{1}, …, Q_{r} for A and B, and using the formula (2). If the multiplication by the product of indexes j = [E(K) : A][E_{S}(K) : B] induces an isomorphism on the abelian group E(K)^{⊗2} ⊗ I^{r−1}/I^{r}, then one defines

R_{S} = R(A, B)j^{−1}.

This quantity, when it is defined, does not depend on the choice of A and B. Furthermore, suitable A and B for which j is invertible exist, say, if r > 1 and τ is prime to the order of Γ_{S}, or if τ is invertible in the ring Z. From now on to simplify we assume that Z contains Z[τ^{−1}]. Under this assumption we can state the main conjecture:

Conjecture 2.3 Assume that τ^{−1} ∈ Z. Then

1. The element θ′(E, S) belongs to E(K_{S})^{⊗2} ⊗ I^{r−1}.

2. The leading coefficient θ̃′(E, S) ∈ E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}) belongs to the image of the natural map

t : E(K)^{⊗2} ⊗ (I^{r−1}/I^{r}) ⟶ E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}).

3. θ̃′(E, S) = t(c(ϕ)^{2} · u^{2} · #\underline{III}(E/K) · J_{S}R_{S}).

Remark:

1. When r = 1 and S = 1, we have:

J_{1}R_{1} = m^{2}τ^{−2}P ⊗ P,

where P is a generator (modulo torsion) for E(K). Note that this equation is true in E(K) ⊗ Z regardless of the choice of P, since τ is invertible in Z. Hence conjecture 2.3 follows in this case from the conjectured equation (1), which is itself a consequence of the classical Birch and Swinnerton-Dyer conjecture.

2. The conjecture we have formulated is compatible under the norm from K_{S} to K_{T}, when T is a divisor of S (cf. section 3.2). This is the motivation for working with the regularized Heegner points.

We now state the main results of this paper which give evidence for conjecture 2.3.

To do this we suppose that the following primes are invertible in Z:

1. The primes 2 and 3.

2. All primes p < (r − 1)/2.

3. All primes p such that Gal(Q(E_{p^{∞}})/Q) is not isomorphic to GL_{2}(Z_{p}).

4. All primes p which divide m.

Note that assumption 3 forces τ to be invertible in Z. The set of primes satisfying condition 3 (and hence, all four of the above) is a finite set if and only if E has no complex multiplications, by a result of Serre [Se].

Complex conjugation acts on the Mordell-Weil group E(K). Let r^{+} and R^{−} denote the ranks of the + and − eigencomponents E(K)^{+} and E(K)^{−} of E(K) under this involution, and let

ρ = max(r^{+}, r^{−}) − 1, if r^{+} ≠ r^{−},
ρ = r^{+} = r^{−} = r/2, if r^{+} = r^{−}.

Note that the order of vanishing of L(E/K, s) is odd; hence by the Birch Swinnerton-Dyer conjecture, one expects that r is odd, so that r^{+} and r^{−} should have opposite parities and equality r^{+} = r^{−} should never hold in our situation.

Theorem 2.4 (Main result) Suppose that S is a product of primes which are inert in K. Then θ′(E, S) belongs to the subgroup E(K_{S})^{⊗2} ⊗ I^{2ρ} of E(K_{S})^{⊗2} ⊗ Z[G].

Since 2ρ ≥ r − 1 (with equality holding if and only if |r^{+} − r^{−}| = 1), theorem 2.4 implies part 1 of conjecture 2.3 about the order of vanishing, slightly weakened because of the assumptions which were made on Z.

If |r^{+} − r^{−}| > 1, then 2ρ > r − 1, and the theorem 2.4 proves more than what is predicted by conjecture 2.3. Can one give a conceptual account of this extra vanishing? At least one can show that when |r^{+} − r^{−}| > 1, the regulator term R_{S} vanishes (cf. prop. 5.12). However, in that case, the leading coefficient should be defined to be the projection of θ′(E, S) in the group E(K_{S})^{⊗2} ⊗ (I^{2ρ}/I^{2ρ+1}). We were not able to supply a prediction, even a conjectural one, for the value of this leading coefficient, except when 2ρ = r − 1.

Assume now that |r^{+} − r^{−}| = 1. Given a prime p, let θ̃′(E, S)_{(p)} denote the reduction modulo p of θ̃′(E, S), i.e., the natural image of θ̃′(E, S) in the group E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}) ⊗ Z/pZ. Let t_{p} be the natural map induced by t,

t_{p} : E(K)^{⊗2} ⊗ (I^{r−1}/I^{r}) ⊗ Z/pZ ⟶ E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}) ⊗ Z/pZ.

Note that the module (I^{r−1}/I^{r}) ⊗ Z/pZ is trivial unless p divides the order of Γ_{S} and p is not invertible in the ring Z. A p-descent argument establishes the following:

Theorem 2.5 .

1. θ̃′(E, S)_{(p)} belongs to the image of t_{p}.

2. If p divides #\underline{III}(E/K)J_{S}, then θ̃′(E, S)_{(p)} = 0.

2.2 Application to the Galois module structure of Heegner points

In stating the following result, we do not strive for the greatest generality of what can be shown by our methods, but only present an illustrative special case.

An abelian extension L of K is said to be of dihedral type if it is normal over Q and the involution τ in Gal(K/Q) acts on Gal(L/K) by τστ^{−1} = σ^{−1}. Let L be a dihedral type extension of K with Galois group G = Z/pZ, where p is a prime which does not divide 6m, and satisfies

Gal(Q(E_{p^{∞}})/Q) = Aut(T_{p}(E)).

Assume that L is ramified only at primes of Q which are inert in K/Q. Then L can be embedded in a ring class field of K, whose conductor over K is a product of inert primes. Let L̃ denote the smallest such field, and let α ∈ E(L) be the trace of the Heegner point in E(L̃) defined in section 1. The Z[G]-module E(L) generated by α is a quotient of a free Z[G]-submodule of rank 1 of E(L). The work of Kolyvagin tells us that the position of the module E(L) in E(L) is strongly related to the arithmetic of E(L), a fact which was foreshadowed by the analytic formula of Gross and Zagier.

Denote by E(L)_{C} the complex representation of G attached to E(L), i.e., the image of E(L) ⊗ C in E(L) ⊗ C. Given a prime l ≠ p, let E(L)_{l} denote the image of E(L) in E(L)⊗F̄_{l}, where F̄_{l} is the algebraic closure of the finite field F_{l} with l elements.

The representation E(L)_{C} splits into a direct sum of eigencomponents E(L)_{C}^{χ} attached to complex characters χ of G. By applying the methods of Kolyvagin one can show (cf. [BD]) that

if E(L)_{C}^{χ} ≠ 0 then dim_{C}(E(L) ⊗ C)^{χ} = 1.

Likewise one has a decomposition of the representation E(L)_{l} into eigencomponents E(L)_{l}^{χ} associated this time to F̄_{l}-valued characters of G. From the methods of Kolyvagin one expects (at least if l is large enough) that

if E(L)_{l}^{χ} ≠ 0 then dimF̄_{l}(Sel_{l}(E/L))^{χ} = 1,

where Sel_{l}(E/L) is the l-Selmer group. Evidence for this is given in [BD].

In both cases the module of Heegner points tells us whether the ranks of certain eigenspaces in Mordell Weil groups or l-Selmer groups are one or not.

The situation changes greatly when one considers the module E(L)_{p}, the image of E(L) in E(L) ⊗ F_{p}. This module no longer decomposes into eigenspaces for the G-action, since the representation is a modular representation: the group ring F_{p}[G] is isomorphic to the local ring F_{p}[ϵ]/(ϵ^{p}). Let

r_{p}^{±} = dim_{F_{p}} Sel_{p}(E/K)^{±},

where the superscripts of + and − denote the + and − eigenspaces for the action of complex conjugation on Sel_{p}(E/K). Let

ρ_{p} = max(r_{p}^{+}, r_{p}^{−}) − 1, if r_{p}^{+} ≠ r_{p}^{−},
ρ_{p} = r_{p}^{+} = r_{p}^{−} = r_{p}/2, if r_{p}^{+} = r_{p}^{−}.

Theorem 2.6 dim_{F_{p}}(E(L)_{p}) ≤ p − ρ_{p}.

Thus the F_{p}-dimension of E(L)_{p} reflects some quantitative information about the rank of the p-Selmer group of E over K. This result leads to several natural questions:

1. Is the bound of theorem 2.6 sharp? We can only provide a conjectural answer when \underline{III}_{p} = 1 and when |r_{p}^{+} − r_{p}^{−}| = 1 by relying on the philosophy of conjecture 2.3.

2. What is the nature of the module of G-invariants (E(L)_{p})^{G}? This module is necessarily non-trivial and one-dimensional if (E(L)_{p}) is non-trivial. In many cases one can show that it gives rise to elements in the Selmer group Sel_{p}(E/K). Can one predict what these elements are?

The remainder of this paper is devoted to the proofs of theorems 2.4, 2.5, and 2.6. In section 3 we state an explicit result about congruences for certain combinations of Heegner points over ring class fields (theorem 3.15 of section 3.3), and show that this result implies theorem 2.4. Section 4 is devoted to the construction and study of certain cohomology classes made from Heegner points which generalize those that were studied by Kolyvagin. Finally, the section 5 is devoted to a proof of theorem 3.15.

3 Restatement of the results

3.1 Calculus of abelian group rings

Let G be a finite abelian group. Given an element σ of order n_{σ} in G, define the derivative operator in the group ring Z[G] by the formula:

D_{σ}^{k} = ∑_{i=0}^{n_{σ}−1} \binom{i}{k}σ^{i}.

If G = ⟨σ⟩ is cyclic, then D_{σ}^{0} is the norm element in the group ring and D_{σ}^{1} is the derivative operator used by Kolyvagin.

One can decompose G as a product of cyclic groups

G = G_{1} × ⋯ × G_{t},

where the order n_{i} of G_{i} is a multiple of the order n_{j} of G_{j} whenever i < j. This decomposition is not unique, but the orders n_{i} are well-defined. Choose a generator σ_{i} for each G_{i}, and view these generators as elements of G. Given a multi-index \underline{k} = (k_{1}, …, k_{t}) of integers, the partial derivative operator D\underline{k} in the group ring Z[G] is defined to be

D\underline{k} = D_{σ_{1}}^{k_{1}} ⋯ D_{σ_{t}}^{k_{t}}.

Let M be a Z[G]-module, and let a be an element of M. We wish to study the resolvent element

∑_{σ∈G} σa ⊗ σ ∈ M ⊗_{Z} Z[G].

The following gives a Taylor expansion formula for this resolvent element around the augmentation ideal.

Theorem 3.1 (Taylor formula)

θ = ∑_{\underline{k}} D\underline{k}a ⊗ (σ_{1} − 1)^{k_{1}} ⋯ (σ_{t} − 1)^{k_{t}},

where the sum is taken over all t-tuples \underline{k} = (k_{1}, …, k_{t}) of positive integers.

The proof is a routine computation, and we omit it. Observe that although the sum is taken over an infinite set, all but finitely many of the terms are zero: the partial derivative D\underline{k} vanishes once one of the k_{i} is greater than n_{i}.

Let p be a prime that is not invertible in Z. The natural inclusion of Q in Q_{p} induces a map Z ⟶ Z_{p}. Let I_{p} denote the augmentation ideal in the group ring Z_{p}[G], and let θ_{p} denote the image of θ in M ⊗ Z_{p}[G].

Lemma 3.2 The element θ belongs to M ⊗ I^{r} if and only if θ_{p} belongs to M ⊗ I_{p}^{r} for all primes p which are not invertible in Z.

Proof: The successive quotients I^{k}/I^{k+1} are finite abelian groups whose orders are divisible only by the primes which are not invertible in Z. Since an element in a finite abelian group is trivial if and only if it maps to zero in each p-primary part of the group, the result follows.

Let ϵ_{p} : Z_{p}[G] ⟶ Z_{p} denote the augmentation map on the group ring.

Lemma 3.3 Assume that x and y belong to the group ring Z_{p}[G] and that the product xy belongs to I_{p}^{r}. If ϵ_{p}(y) is invertible in Z_{p}, then x belongs to I_{p}^{r}.

Proof: Multiplication by y induces an isomorphism on I_{p}^{k}/I_{p}^{k+1} for all k.

Lemma 3.4 If σ is of order prime to p, then (σ − 1) belongs to I_{p}^{r} for all r.

Proof: Let n be the order of σ. Then

0 = σ^{n} − 1 = n(σ − 1) + \binom{n}{2}(σ − 1)^{2} + ⋯ + (σ − 1)^{2}.

The right-hand side is equal to

(σ − 1) (n + \binom{n}{2}(σ − 1) + ⋯ + (σ − 1)^{n}) .

Since the second factor maps to n by ϵ_{p}, and n belongs to Z_{p}^{∗}, the result follows from lemma 3.3.

Lemma 3.5 If the order of σ is q, a power of p, then q(σ − 1) belongs to I_{p}^{p}.

Proof: As in the proof of lemma 3.4 one finds

q(σ − 1) + \binom{q}{2}(σ − 1)^{2} + ⋯ + (σ − 1)^{q} = 0.

Hence

q(σ − 1) (1 + \binom{q}{2}/q(σ − 1) + \binom{q}{3}/q(σ − 1)^{2} + ⋯ + \binom{q}{p − 1}/q(σ − 1)^{p−2})

= −\binom{q}{p}(σ − 1)^{p} + ⋯ + (σ − 1)^{q}.

Applying lemma 3.3, one finds that q(σ − 1) belongs to I_{p}^{p}.

Lemma 3.6 Let q be the maximal power of p dividing the order of σ. Then q(σ − 1) belongs to I_{p}^{p}.

Proof: Write σ = σ_{1}σ_{2}, where σ_{1} is of order q and σ_{2} is of order prime to p. The result follows from the identity

(σ − 1) = (σ_{1} − 1)(σ_{2} − 1) + (σ_{1} − 1) + (σ_{2} − 1)

combined with lemmas 3.4 and 3.5.

Given \underline{k} = (k_{1}, …, k_{t}), let

n(\underline{k}) = gcd_{k_{i}>0} n_{i}.

(Recall that the n_{i} are the orders of the σ_{i}). Let n_{p}(\underline{k}) denote the maximal power of p dividing n(\underline{k}).

Lemma 3.7 Suppose r ≤ p. The element θ_{p} belongs to I_{p}^{r} if for all t-tuples \underline{k} = (k_{1}, …, k_{t}) with k_{1} + ⋯ k_{t} < r, we have

D\underline{k}a ≡ 0 (mod n_{p}(\underline{k})).

Proof: This follows from the Taylor formula 3.1 together with lemma 3.6.

We say that an element a in a Z-module M is divisible by an integer n if there exists a′ in M with a = na′.

Lemma 3.8 Suppose that all primes which are strictly less than r are invertible in Z. Then the element θ in Z[G] belongs to I^{r} if

n(\underline{k}) divides D\underline{k}a for all \underline{k} = (k_{1}, …, k_{t}) with k_{1} + ⋯ + k_{t} < r.

Proof: Combine lemmas 3.2 and 3.7.

We conclude this section with a property of the derivatives D_{σ}^{k} that will be useful later:

Lemma 3.9 If σ is of order n, then

(σ − 1)D_{σ}^{k} = \binom{n}{k} − σD_{σ}^{k−1}.

This is proved by a straightforward computation.

3.2 Generalities on Heegner points

We give ourselves fixed embeddings Q̄ ⟶ C and Q̄ ⟶ Q̄_{p} for every prime p. Complex conjugation Frob_{∞} ∈ Gal(C/R) acts by Galois automorphisms on any Galois extension of Q. Similarly, the Frobenius element at p, Frob_{p} acts on any Galois extension of Q which is unramified at p.

We recall some standard facts on Heegner points over ring class fields of K. We do not strive for the greatest generality, but only state the results in the form which we shall need in the proofs. A more thorough discussion can be found in [Gr1] or [Gr4].

Let S be the set of square-free integers prime to ND which are products of primes which are inert in K. For all T ∈ S we are given the following data:

1. An abelian extension K_{T} of K, the ring class field of K associated to the order of conductor T. It is ramified only at the places of K which lie above the primes dividing T. Thus K_{1} is the Hilbert class field of K. One writes G = Gal(K_{1}/K), G_{T} = Gal(K_{T} /K_{1}) and Γ_{T} = Gal(K_{T} /K).

2. The Heegner point α(T) in E(K_{T}).

Writing T = l_{1} ⋯ l_{s}, the extension K_{T} is a compositum of the fields K_{l_{i}} which are linearly disjoint over K_{1}. Hence there is a canonical direct product decomposition

G_{T} = G_{l_{1}} × ⋯ × G_{l_{s}}

which gives inclusions G_{S} ⊂ G_{T} and Γ_{S} ⊂ Γ_{T} for all divisors S of T. We will implicitly identify elements of G_{S} with their images in G_{T}. For any S dividing T, the partial norm operator N_{S} in the group ring Z[G_{T}] is defined by

N_{S} = N_{G_{S}} = ∑_{σ∈G_{S}} σ.

For each prime l, choose a generator σ_{l} of G_{l}, and write D_{l}^{k} for the partial derivative D_{σ_{l}}^{k}. Thus D_{l}^{1} is the derivative operator studied by Kolyvagin. These operators act on the field K_{T} and on the Mordell-Weil group E(K_{T}) in the natural way.

Given S ∈ S and l a prime in S which is prime to S, let λ denote a prime of K above l and let σ_{λ,S} ∈ Γ_{S} be the Frobenius automorphism associated to λ.

Proposition 3.10 N_{l}(α(Sl)) = a_{l}α(S).

Proof: See [Gr4], p. 240, prop. 3.7.

Proposition 3.11 α(Sl) ≡ σ_{λ,S}α(S) (mod λ′), where λ′ is any prime of K_{Sl} above λ.

Proof: See [Gr4], p. 240, prop. 3.7.

Propositions 3.10 and 3.11 make up the axioms of an Euler system for Heegner points in the sense of Kolyvagin [Ko3].

The action of complex conjugation Frob_{∞} on the Heegner points is given by the following proposition:

Proposition 3.12 Frob_{∞}α(T) = ϵσ_{0}α(T) + (torsion) for some σ_{0} ∈ Γ_{T}.

(Recall that ϵ is the eigenvalue for the operator w_{N} acting on f; it is opposite to the sign in the functional equation for L(f, s).) Proof: See [Gr4], p. 243, prop. 5.3.

We make a brief digression concerning the compatibility of conjecture 2.3 under norms. Writing T = Sl, let μ_{l} denote the map

μ_{l} : E(K_{T})^{⊗2} ⊗ Z[Γ_{T}] ⟶ E(K_{T})^{⊗2} ⊗ Z[Γ_{S}]

induced by the homomorphism Γ_{T} ⟶ Γ_{S}.

Proposition 3.13 μ_{l}(θ̃′(E, T)) = #E(k_{l}) · θ̃′(E, S), where k_{l} = O_{K}/lO_{K}.

Proof: By prop. 3.10 combined with a direct computation, the behavior of the regularized Heegner points β_{T} and β_{T}^{ω} under norms is given by:

N_{l}β_{T} = (l + 1 − a_{l})β_{S}, β_{T}^{ω} = (l + 1 + a_{l})β_{S}^{ω}.

The result now follows from the formula

#E(k_{l}) = (l + 1 − a_{l})(l + 1 + a_{l}).

On the other hand the naturality of the Mazur Tate pairing implies that

μ_{l} ∘ ⟨ , ⟩_{T} = ⟨ , ⟩_{S}

on E(K) × E_{T} (K), so that

μ_{l}(J_{T} R_{T}) = #E(k_{l}) · J_{S}R_{S}.

Hence conjecture 2.3 is compatible with the map μ_{l} when l is inert in K/Q. A similar compatibility can be shown when l is split in K/Q. Hence in particular, conjecture 2.3 in the case r = 1 follows from the classical Birch Swinnerton-Dyer conjecture, thanks to the formula of Gross and Zagier.

The compatibility under norms is the reason for using the regularized Heegner points β_{S} instead of the simpler points α(S) in the definition of θ′(E, S).

3.3 A divisibility theorem for Heegner points

In this section, we state a theorem on congruences for certain combinations of Heegner points over ring class fields. Using the results of sections 3.1 and 3.2, we show that it implies theorem 2.4.

Let q be a power of a prime p which is not invertible in the ring Z. Let

L_{q} = {l rational prime, Frob_{l} = Frob_{∞} in K(μ_{q})/Q}.

Lemma 3.14 The prime l belongs to L_{q} if and only if it is inert in K/Q and q divides l + 1.

Any finitely generated Z/qZ-module M can be decomposed as

M = (Z/qZ)^{r_{q}(M)} × M′,

where the exponent of M′ divides q strictly. The integer r_{q}(M) does not depend on the decomposition.

Let Sel_{q}(M) be the q-Selmer group for E/K which arises out of the descent for the isogeny of multiplication by q. Complex conjugation Frob_{∞} acts on Sel_{q}(M) and splits it into + and − eigenspaces since q is odd. Let r_{q}^{+} and r_{q}^{−} denote the values of r_{q}(Sel_{q}(E/K)^{+}) and r_{q}(Sel_{q}(E/K)^{−}) respectively. Let

ρ_{q} = max(r_{q}^{+}, r_{q}^{−}) − 1, if r_{q}^{+} ≠ r_{q}^{−},
ρ_{q} = r_{q}^{+} = r_{q}^{−} = r_{q}/2, if r_{q}^{+} = r_{q}^{−}.

The class group C of K can be decomposed (non-canonically) as a product

C = C′ × ⟨ξ_{1}⟩ × ⋯ × ⟨ξ_{a}⟩,

where q does not divide the exponent of C′ and each ξ_{i} is of order a power of p which is greater or equal to q.

Given S = l_{1} ⋯ l_{s} a product of distinct primes in L_{q}, let D\underline{k} be the partial derivative operator in the group ring Z[Γ_{S}] of the form:

D\underline{k} = N_{C′}D_{ξ_{1}}^{j_{1}} ⋯ D_{ξ_{a}}^{j_{a}}D_{l_{1}}^{k_{1}} ⋯ D_{l_{s}}^{k_{s}},

where \underline{k} = (j_{1}, …, j_{a}, k_{1}, …, k_{s}) is an (a + s)-tuple of positive integers. One defines the support of D\underline{k} to be the integer S, its conductor S′ to be the product of the l_{i} with k_{i} > 0, and its order to be k = j_{1}+⋯+j_{a}+k_{1}+⋯+k_{s}. There is an obvious partial ordering on the set of partial derivatives with support S. Given \underline{k}′ = (j1^{′}, …, ja^{′}, k1^{′}, …, ks^{′}), one says that D\underline{k}′ is less than D\underline{k} if

jt^{′} ≤ j_{t}, kt^{′} ≤ k_{t} ∀t.

Theorem 3.15 Let q be a power of a prime which is not invertible in Z. If order(D\underline{k}) < ρ_{q}, then

D\underline{k}α(S) ≡ 0 (mod q).

Claim 3.16 Theorem 3.15 implies theorem 2.4.

Proof: Let S = l_{1} ⋯ l_{t} be a product of primes which are inert in K. Let D\underline{k} be a partial derivative of support S and conductor S′ which is of order < ρ. We can write

D\underline{k} = D′N_{S/S′},

where D′ has support S′. By proposition 3.10,

D\underline{k}α(S) = ( ∏_{l|S/S′} a_{l})D′α(S′).

Fix a prime p which is not invertible in Z and let q be the largest power of p dividing n(\underline{k}). By definition all the primes dividing S′ belong to L_{q}. Since ρ ≤ ρ_{q}, we can apply theorem 3.15 to conclude that D′α(S′) ≡ 0 (mod n_{p}(\underline{k})). Hence n(\underline{k}) divides D\underline{k}α(S) whenever order(D\underline{k}) < ρ. By lemma 3.8 it follows that the resolvent element

θ_{S} = ∑_{σ∈Γ_{S}} σα(S) ⊗ σ ∈ E(K_{S}) ⊗ Z[Γ_{S}]

belongs to E(K_{S}) ⊗ I^{ρ}. Similarly the elements θ_{T} for all T dividing S belong to E(K_{S}) ⊗ I^{ρ}, as well as the elements θ_{T}^{∗} which are obtained by applying to θ_{T} the involution sending σ ∈ Γ_{S} to σ^{−1}. The elements A(E, S) and A(E, S)^{ω} introduced in section 2.1 can be expressed as combinations of the θ_{T} and the θ_{T}^{∗} over the integral group ring Z[Γ_{S}]. Hence they belong to E(K_{S}) ⊗ I^{ρ} as well. Therefore

θ′(E, S) belongs to E(K_{S}) ⊗ I^{2ρ}.

4 The Heegner cohomology classes

4.1 More on derivatives

Let q = p^{M} be a power of a prime p which is not invertible in Z.

Lemma 4.1 .

1. (σ_{l} − 1)D_{l}^{k} = \binom{l+1}{k} − σ_{l}D_{l}^{k−1}.

2. For all 0 < k < p,

(σ_{l} − 1)D_{l}^{k} ≡ −σ_{l}D_{l}^{k−1} (mod q),

and a similar formula hold for the D_{ξ_{j}}^{k}.

Proof: Part 1 is a restatement of lemma 3.9. For part 2, one uses the fact that q divides l + 1 (lemma 3.14), and hence the binomial coefficient \binom{l+1}{k}.

The group Gal(K_{S}/Q) is a semi-direct product of Gal(K/Q) = ⟨Frob_{∞}⟩ with Γ_{S}. Complex conjugation Frob_{∞} acts on Γ_{S} by the formula:

Frob_{∞}σFrob_{∞}^{−1} = σ^{−1}.

Extending this action to the group ring Z/qZ[Γ_{S}], one has the following action of Frob_{∞} on D\underline{k}:

Lemma 4.2 Frob_{∞}D\underline{k}Frob_{∞}^{−1} = (−1)^{k}D\underline{k} + (lower order derivatives).

Proof: It suffices to show this for a partial derivative operator of the form D_{l}^{k}. In this case, one has

Frob_{∞}D_{l}^{k}Frob_{∞}^{−1} − (−1)^{k}D_{l}^{k} = ∑_{i=0}^{#G_{l}} f(i)σ_{l}^{i},  (3)

where f(i) = \binom{#G_{l}−i}{k} − (−1)^{k}\binom{i}{k}. The function f(i) is the reduction mod q of a polynomial with rational coefficients taking integral values at integral arguments. Moreover the degree of f is strictly less than k. The Z-module of all such polynomial functions is spanned by the \binom{i}{k′} with k′ < k. Hence the left hand side of (3) can be expressed as an integral combination of partial derivatives of lower order.

4.2 The Heegner cohomology classes

Fis a product S of primes in L_{q}, and let D\underline{k} be a fixed partial derivative in the group ring Z[Γ_{S}]. Define a set L_{q,E} of rational primes as follows:

L_{q,E} = {l rational prime | l |̸NDp and Frob_{l} = Frob_{∞} in K(E_{q})/Q}.

Lemma 4.3 A prime l not dividing NDp belongs to L_{q,E} if and only if it belongs to L_{q} and in addition q divides a_{l}.

Proof: If l ∈ L_{q,E}, we have the equalities of the minimal polynomials of Frob_{l} and Frob_{∞} acting on E_{q}:

x^{2} − a_{l}x + l = x^{2} − 1 (mod q).

The lemma follows by equating coefficients of these two polynomials.

Lemma 4.4 If L is a solvable extension of Q, then E_{q}(L) = 0.

Proof: Suppose E_{p}(L) ≠ 0. Since Gal(Q̄/Q) acts transitively on the ptorsion in E, this implies that L contains all of the p-torsion points, which is implossible since GL_{2}(Z/pZ) is not solvable when p > 3.

Choose a prime l which satisfies the following condition:

l |̸ND · p · S, Frob_{l} = Frob_{∞} in K_{S}(E_{q})/Q.

Clearly this implies that l belongs to L_{q,E}. Let P denote the class of D\underline{k}α(S) in E(K_{S})/qE(K_{S}). Let T = Sl, and let P (l) denote the class of D\underline{k} · D_{l}^{1}α(T) in E(K_{T})/qE(K_{T}). We make the following assumption:

Hypothesis 4.5 For all the partial derivatives D′ which are strictly less than D\underline{k}D_{l}^{1}, the class of D′α(T) is 0 in the group E(K_{T})/qE(K_{T}).

Under hypothesis 4.5, one has:

Lemma 4.6 The class of P (l) is fixed under the action of Γ_{T}.

Proof: Let σ = σ_{l} for some l dividing T or σ = ξ_{j} for some j. Then (1 − σ)D\underline{k}D_{l}^{1} = D′, where D′ is some partial derivative which is strictly less than D\underline{k}D_{l}^{1}, by lemma 4.1. Hence (1 − σ)P (l) = 0, by hypothesis 4.5.

From lemma 4.4, the group E_{q}(K_{T}) is trivial and hence the following sequence is exact:

0 ⟶ E(K_{T}) ⟶^{q} E(K_{T}) ⟶ E(K_{T})/qE(K_{T}) ⟶ 0.

Taking Γ_{T} -invariants yields an exact cohomology sequence

0 ⟶ E(K)/qE(K) ⟶ (E(K_{T})/qE(K_{T}))^{Γ_{T}} ⟶^{δ} H^{1}(Γ_{T} , E(K_{T}))_{q}.

Let d(l) = δP (l). We will identify d(l) with its image by inflation in H^{1}(K, E)_{q}.

The class d(l) is the global cohomology class which plays a key role in Kolyvagin’s method. We now undertake to analyze its properties.

Behavior of the class d(l) under complex conjugation: Complex conjugation Frob_{∞} acts on the group H^{1}(K, E)_{q} in the natural way. Let ϵ_{k} = (−1)^{k+1}ϵ be the parity of the order of D\underline{k}, times the sign in the functional equation.

Proposition 4.7 The class d(l) is in the ϵ_{k}–eigenspace for the action of Frob_{∞}.

Proof: By lemma 4.4, the group E(K_{T}) has no q-torsion, and hence the torsion subgroup E(K_{T})_{tor} is killed in E(K_{T})/qE(K_{T}). Hence by lemma 3.12 we have

Frob_{∞}[α(T)] = ϵσ_{0}[α(T)],

where [α(T)] denotes the image of α(T) in E(K_{T})/qE(K_{T}). Combining lemma 4.2 with the hypothesis 4.5 that all lower order partial derivatives of α(T) vanish, we find:

Frob_{∞}P (l) = ϵ_{k}σ_{0}P (l) = ϵ_{k}P (l).

The last equality follows from lemma 4.6. Since the map δ used to construct the class d(l) from P (l) is equivariant with respect to the Galois actions, it follows that d(l) is in the ϵ_{k}-eigenspace for Frob_{∞}.

Local behavior of the class d(l): Given a place v of K, let d(l)_{v} denote the restriction of d(l) in H^{1}(K_{v}, E)_{q}. The prime l is inert in K/Q. Let λ be the place of K above it. The prime λ splits completely in K_{S}/K; choose a place λ′ of K_{S} above λ. Finally, the extension K_{T} /K_{S} is completely ramified at λ′; let λ′′ denote the unique place of K_{T} above λ′. The localization d(l)_{λ} belongs to

H^{1}(Gal((K_{T})_{λ′′}/K_{λ}), E((K_{T})_{λ′′}))_{q} = H^{1}(G_{l}, E((K_{T})_{λ′′})).

Fact 4.8 The choice of generator σ_{l} of G_{l} determines a canonical inclusion of H^{1}(G_{l}, E((K_{T})_{λ′′}))_{q} inside E_{q}(k_{λ}).

Proof: Since the kernel of the reduction map E((K_{T})_{λ′′}) ⟶ E(k_{λ}) is a pro-l group, the group H^{1}(G_{l}, E((K_{T})_{λ′′}))_{q} injects into H^{1}(G_{l}, E(k_{λ}))_{q} = hom(G_{l}, E(k_{λ}))_{q}. The latter group can be identified with E(k_{λ})_{q} thanks to the chosen generator σ_{l} of G_{l}.

Theorem 4.9 .

1. For all archimedean places and all places which do not divide l or the conductor S′ of D\underline{k}, the class d(l)_{v} is equal to 0.

2. The image of d(l)_{λ} in E(k_{λ})_{q} is equal to

\frac{(l + 1)Frob_{∞} − a_{l}}{q} P̃,

where P̃ is the reduction of P mod λ′.

Proof:

1. If v is archimedean, then K_{v} = C and E/K_{v} has trivial Galois cohomology. Suppose now that v is a non-archimedean place not lying above S′l. By construction the class d(l)_{v} is inflated from a class in H^{1}(K_{S′l}/K, E)_{q}. Since the extension K_{S′l} is unramified at v, the class d(l)_{v} comes from a class in H^{1}(K_{v}^{unram}/K_{v}, E)_{q}. Let E^{0} denote the connected component of the Néron model at v, and let J_{v} = E/E_{0}. The group H^{1}(K_{v}^{unram}/K_{v}, E^{0}) vanishes, (cf. [Mi] ch. I, prop. 3.8), and hence H^{1}(K_{v}^{unram}/K_{v}, E)_{q} injects into H^{1}(K_{v}^{unram}/K_{v}, J_{v})_{q}. By the assumption that q is not invertible in Z, we know that q is prime to m and hence to the order of J_{v}. Hence the group H^{1}(K_{v}^{unram}/K_{v}, J_{v})_{q} is trivial, and the class d(l)_{v} vanishes.

2. The image of d(l)_{λ} in E(k_{λ}) by the isomorphism defined above is the point

red_{λ′′} (\frac{(σ_{l} − 1)P (l)}{q}) = red_{λ′′} (\frac{(σ_{l} − 1)D_{l}D\underline{k}α(T)}{q}) ,

where red_{λ′′} : E((K_{T})_{λ′′}) ⟶ E(k_{λ}) is the reduction map. But

(σ_{l} − 1)D_{l}D\underline{k}α(T) = (l + 1 − N_{l})D\underline{k}α(T) (by lemma 4.1)
= (l + 1)D\underline{k}α(T) − a_{l}D\underline{k}α(S) (by prop. 3.10).

Hence,

red_{λ′′} (\frac{(σ_{l} − 1)P (l)}{q}) = red_{λ′′} (\frac{l + 1}{q}Frob_{∞}D\underline{k}α(S) − \frac{a_{l}}{q} D\underline{k}α(S)) ,

by prop. 3.11 combined with the fact that Frob_{λ′} = Frob_{∞}. Since P̃ = red_{λ′′}(D\underline{k}α(S)), the lemma is proved.

4.3 Tate duality

The cup product in cohomology combined with the Weil pairing

E_{q} ⊗ E_{q} ⟶ μ_{q}

give rise to an alternating pairing

H^{1}(K_{λ}, E_{q}) × H^{1}(K_{λ}, E_{q}) ⟶ H^{2}(K_{λ}, μ_{q}) = Z/qZ.

(The identification H^{2}(K_{λ}, μ_{q}) = Z/qZ is provided by the map

inv_{λ} : H^{2}(K_{λ}, μ_{q}) ⟶ Z/qZ

of local class field theory.) It is a result of Tate that this local pairing is non-degenerate (cf. [Mi], ch. I, cor. 2.3).

The local q-descent exact sequence

0 ⟶ E(K_{λ})/qE(K_{λ}) ⟶ H^{1}(K_{λ}, E_{q}) ⟶ H^{1}(K_{λ}, E)_{q} ⟶ 0

allows us to view E(K_{λ})/qE(K_{λ}) as a submodule of H^{1}(K_{λ}, E_{q}). This subspace is maximal isotropic for the local Tate pairing (cf. [Gr4], p. 247, prop. 7.5, or [Mi], ch. I, thm. 2.6.). Therfore one gets a perfect pairing

⟨ , ⟩ : E(K_{λ})/qE(K_{λ}) × H^{1}(K_{λ}, E)_{q} ⟶ Z/qZ,

i.e., an isomorphism

H^{1}(K_{λ}, E)_{q} ⟶ (E(K_{λ})/qE(K_{λ}))^{∗}.  (4)

Here the superscript ∗ denotes Pontryagin dual, i.e., for a Z/qZ-module M,

M^{∗} = hom(M, Z/qZ).

By composing the dual of the natural map Sel_{q}(E/K) ⟶ E(K_{λ})/qE(K_{λ}) with the isomorphism (4), one obtains a map

ϕ_{λ} : H^{1}(K_{λ}, E)_{q} ⟶ Sel_{q}(E/K)^{∗}.

Similarly, if S is any submodule of the Selmer group Sel_{q}(E/K), one obtains by restriction a map H^{1}(K_{λ}, E)_{q} ⟶ S^{∗}, which by abuse of notation we denote by the same letter ϕ_{λ}. The map ϕ_{λ} commutes with the action of complex conjugation on the modules H^{1}(K_{λ}, E_{q}) and Sel_{q}(E/K), and hence preserves the decomposition into eigenspaces of the modules.

We will be exploiting the cohomology class d(l) in the following way. Let Sel_{q}(S′) ⊂ Sel_{q}(E/K) be the kernel of the map

Sel_{q}(E/K) ⟶ ⊕_{v|S′}E(K_{v})/qE(K_{v}).

Proposition 4.10 The local class d(l)_{λ} is in the kernel of the map

ϕ_{λ} : H^{1}(K_{λ}, E)_{q}^{ϵ_{k}} ⟶ (Sel_{q}(S′)^{∗})^{ϵ_{k}}.

Proof: Let s belong to Sel_{q}(S′), and let s_{λ} denote its image in E(K_{λ})/qE(K_{λ}). We need to show that

⟨s_{λ}, d(l)_{λ}⟩ = 0.

Let d̃(l) denote a lift of d(l) to the group H^{1}(K, E_{q}). The cup-product s∪d̃(l) belongs to the global Brauer group H^{1}(K, μ_{q}). By the definition of the local pairing, we have:

∑_{v}⟨s_{v}, d(l)_{v}⟩ = ∑_{v} inv_{v}(s ∪ d̃(l)).

The latter sum is 0, by the reciprocity law of global class field theory. On the other hand, if the place v does not divide S′λ, then d(l)_{v} = 0, by theorem 4.9. If the place v divides S′, then s_{v} = 0, since s ∈ Sel_{q}(S′). Hence all of the terms in the first sum are zero, with the possible exception of ⟨s_{λ}, d(l)_{λ}⟩. It follows that ⟨s_{λ}, d(l)_{λ}⟩ = 0.

4.4 Application of the Chebotarev density theorem

We make the following hypotheses:

Hypothesis 4.11 The point P = D\underline{k}α(S) is not the q–th power of a point in E(K_{S}).

Hypothesis 4.12 r_{q}(Sel_{q}(S′)^{ϵ_{k}}) ≥ 1.

Set F = K_{S}(E_{q}). We start with a few cohomological lemmas.

Lemma 4.13 The fields K_{S} and K(E_{q}) are linearly disjoint over K.

Proof: The intersection of the K_{S} and K(E_{q}) is a subfield of K(E_{q}) which is abelian over K and hence is contained in K(μ_{q}), since Gal(K(E_{q})/K) = GL_{2}(Z/qZ). But K_{S} ∩ K(μ_{q}) = K, since S and q are relatively prime.

Lemma 4.14 Let (Z/qZ)^{2} be equipped with the natural action of GL_{2}(Z/qZ). Then

H^{p}(GL_{2}(Z/qZ), (Z/qZ)^{2}) = 0.

Proof: Let C ≃ (Z/qZ)^{∗} be the center of GL_{2}(Z/qZ) consisting of the scalar matrices. The Hochschild-Serre spectral sequence

H^{p}(PGL_{2}(Z/qZ), H^{q}(C, (Z/qZ)^{2})) ⇒ H^{p+q}(GL_{2}(Z/qZ), (Z/qZ)^{2})

shows that H^{p}(GL_{2}(Z/qZ), (Z/qZ)^{2}) = 0, since C has order prime to q, and H^{0}(C, (Z/qZ)^{2}) = 0 (here we use the fact that q is odd).

Lemma 4.15 The restriction map H^{1}(K, E_{q}) ⟶ H^{1}(K_{S}, E_{q}) is injective.

Proof: Its kernel is the group H^{1}(K_{S}/K, E_{q}(K_{S})) which is trivial since E_{q}(K_{S}) = 0 by lemma 4.13.

Lemma 4.16 The restriction map H^{1}(K_{S}, E_{q}) ⟶ H^{1}(F, E_{q}) is injective.

Proof: By lemma 4.13, we have Gal(F/K_{S}) = GL_{2}(Z/qZ), and the kernel of the restriction map is the group

H^{1}(F/K_{S}, E_{q}) = H^{1}(GL_{2}(Z/qZ), E_{q}).

This group is trivial by lemma 4.14, and the result follows.

Lemma 4.17 The restriction map H^{1}(K, E_{q}) ⟶ H^{1}(F, E_{q}) is injective.

Proof: Combine lemmas 4.15 and 4.16.

Let l be a rational prime satisfying the condition

Condition 4.18 l |̸ND · p · S, Frob_{l} = Frob_{∞} in F/Q.

In this case, l is inert in K/Q. Let λ be the unique prime of K above l. The prime λ splits completely in F/K. Choose a prime λ_{F} of F above it. The residue field of F at λ_{F} is identified with k_{λ}.

Proposition 4.19 There exists a prime l satisfying the condition 4.18 such that

1. The image of P̃ in E(k_{λ})/qE(k_{λ}) is non-zero.

2. The map Sel_{q}(S′)^{ϵ_{k}} ⟶ (E(k_{λ})/qE(k_{λ}))^{ϵ_{k}} is surjective.

Proof: By the hypothesis 4.11, the class P in E(K_{S})/qE(K_{S}) is non-trivial. Complex conjugation acts on E(K_{S})/qE(K_{S}) in a natural way, and P can be written uniquely as a sum of projections onto the + and − eigencomponents for this action. At least one of these projections is non-trivial: call it P′. The cohomology class in H^{1}(K_{S}, E_{q}) corresponding to P′ is also non trivial; hence, so is its restriction in H^{1}(F, E_{q}), by lemma 4.16. Let ζ_{1} denote this restriction.

By hypothesis 4.12, we may choose an element of order exactly q in Sel_{q}(S′)^{ϵ_{k}}, and the image ζ_{2} of this element in H^{1}(F, E_{q}) is still of order exactly q, by lemma 4.17.

Both ζ_{1} and ζ_{2} are homomorphisms from Gal(F^{ab}/F) into E_{q}. Let F̃ be the smallest extension of F which is cut out by ζ_{1} and ζ_{2} and is Galois over Q. Let U = Gal(F̃/F). There is an exact sequence

1 ⟶ Gal(F̃/F) ⟶ Gal(F̃/K_{S}) ⟶ Gal(F/K_{S}) ⟶ 1
||  ||
U  GL_{2}(Z/qZ)

which determines a GL_{2}(Z/qZ)-action on U. Similarly, complex conjugtion Frob_{∞} acts on U by inner automorphisms. The cohomology classes ζ_{1}, and ζ_{2} are fixed under the action of GL_{2}(Z/qZ) on hom(U, E_{q}), since they come from classes in H^{1}(K_{S}, E_{q}) by inflation. Let U^{+} denote the subspace of U which is fixed by Frob_{∞}. The class ζ_{1} belongs to a fixed eigenspace of hom(U, E_{q}) for the action of Frob_{∞}, by construction. The class ζ_{2} belongs to the ϵ_{k}–eigenspace, since it comes from a class in Sel_{q}(S′)^{ϵ_{k}}. Hence both ζ_{1} and p^{M−1}ζ_{2} are non-zero on U^{+}. Otherwise, they would map U onto a given eigenspace of E_{q} for the Frob_{∞}-action, contradicting the GL_{2}(Z/qZ)- invariance of the image. Thus, we can find γ ∈ U^{+} such that

ζ_{1}(γ) ≠ 0, ζ_{2}(γ) is of order exactly q.

Now choose l such that

Frob_{l} = Frob_{∞}γ in F̃/Q,

where the equality is one of conjugacy classes in the group Gal(F̃/Q). One can find such a prime, by the Chebotarev density theorem. Clearly, l satisfies the condition 4.18. In addition,

ζ_{1}(Frob_{λ_{F}}) = ζ_{1}(Frob_{l}^{2}) = ζ_{1}(γ^{Frob^{∞}} · γ) = ζ_{1}(γ^{2}) ≠ 0.

Hence, P is not a q-th power in E(k_{λ})/qE(k_{λ}), and condition 1 is satisfied. By the same computation, one shows that ζ_{2}(Frob_{λ_{F}}) is of order exactly q in E_{q}, which implies that the image of ζ_{2} in H^{1}(K_{λ}, E_{q})^{ϵ_{k}} is itself of order exactly q, so that condition 2 is satisfied as well, since

(E(K_{λ})/qE(K_{λ}))^{ϵ_{k}} ≃ Z/qZ.

This proves the proposition.

5 Proof of the main results

5.1 Proof of theorem 3.15

Given a Z/qZ-module M, define r_{p}(M) to be

r_{p}(M) = dim_{F_{p}}(M ⊗ F_{p}).

Lemma 5.1 If 0 ⟶ A′ ⟶ A ⟶^{f} A′′ is an exact sequence of Z/qZmodules, then

r_{q}(A) ≤ r_{q}(A′) + r_{p}(A′′).

Proof: We may assume without loss of generality that f is surjective. Let N denote the image of A_{p^{M−1}} in A′′. Because of the surjectivity assumption, the module A′′/N is annihilated by p. There is a natural exact sequence of F_{p}-vector spaces:

\begin{aligned}0 ⟶ p^{M−1}A′ ⟶ & p^{M−1}A & ⟶ & A′′/N \\ & p^{M−1}a & ↦ & f(a). \end{aligned}

Hence

dim_{F_{p}}(p^{M−1}A) ≤ dim_{F_{p}}(p^{M−1}A′) + dim_{F_{p}}(A′′/N).

The lemma now follows from the fact that dim_{F_{p}}(p^{M−1}A) = r_{q}(A) (and likewise for A′) and from the inequality dim_{F_{p}}(A′′/N) ≤ r_{p}(A′′).

Lemma 5.2 For any prime v of K which lies above a prime of L_{q},

r_{p}(E(K_{v})/qE(K_{v})^{ϵ_{k}}) ≤ 1.

Proof: We have

r_{p}(E(K_{v})/qE(K_{v})) = r_{p}(E(K_{v})/pE(K_{v})) = r_{p}(E(k_{v})/pE(k_{v})),

because the norm of v is prime to p. Since E(k_{v}) is a finite group, the group E(k_{v})/pE(k_{v}) is isomorphic to E_{p}(k_{v}) which is at most 2 dimensional. Moreover, we know that

#E(k_{v})^{±} = l + 1 ∓ a_{l}.

Since p divides l + 1, the order of E(k_{v}) is divisible by p if and only if p divides a_{l}, and then p divides the order of each eigenspace so that

r_{p}(E(k_{v})/pE(k_{v})^{±}) = 1.

Let D\underline{k} be as in the previous section a partial derivative of order k with support S and conductor S′, and let ϵ_{k} = (−1)^{k+1}ϵ. Consider the modules

Sel_{q}(S′), A(S′) = ⊕_{v|S′}E(K_{v})/qE(K_{v})

which fit into the exact sequence

0 ⟶ Sel_{q}(S′)^{±} ⟶ Sel_{q}(E/K)^{±} ⟶ A(S′)^{±}.

Let Sel_{q}^{±} denote the plus and minus eigenspaces for the action of complex conjugation on Sel_{q}(E/K).

Theorem 5.3 If order(D\underline{k}) < r_{q}(Sel_{q}(S′)^{ϵ_{k}}) + r_{p}(A(S′)^{ϵ_{k}}), then

D\underline{k}α(S) ≡ 0 (mod q).

Proof: The weight of D\underline{k} is defined to be

wt(D\underline{k}) = order(D\underline{k}) − #\{l|S such that l belongs to L_{q,E} \}.

We will show theorem 5.3 by induction on wt(D\underline{k}).

Step 1: Case where wt(D\underline{k}) < 0. In that case, D\underline{k} contains a factor of the form D_{l}^{0}, with l ∈ L_{q,E}. But then, by proposition 3.10,

D_{l}^{0}α(S) = a_{l} · α(S/l),

which is 0, since q divides a_{l} by lemma 4.3. Thus one always has D\underline{k}α(S) = 0, without assuming any inequality for the order of D\underline{k}.

Step 2: Proof for wt(D\underline{k}) = w ≥ 0: We make the induction hypothesis that theorem 5.3 holds in weight strictly less than w. We argue by contradiction, assuming that P = D\underline{k}α(S) is non-zero but that

order(D\underline{k}) < r_{q}(Sel_{q}(S′)^{ϵ_{k}}) + r_{p}(A(S′)^{ϵ_{k}}).

Lemma 5.4 r_{q}(Sel_{q}(S′)^{ϵ_{k}}) > 0.

Proof: Otherwise we would have

order(D\underline{k}) < r_{p}(A(S′)^{ϵ_{k}}).

The right hand side in this inequality is less than or equal to the number of primes dividing S′, by lemma 5.2, and thus cannot be greater than order(D\underline{k}).

Invoking proposition 4.19, we choose a prime l satisfying the conditions

Conditions 5.5 .

1. Frob_{l} = Frob_{∞} in K_{S}(E_{q})/Q.

2. P̃ ≠ 0 in E(k_{λ})/qE(k_{λ}).

3. The map Sel_{q}(S′)^{ϵ_{k}} ⟶ (E(K_{λ})/qE(K_{λ}))^{ϵ_{k}} is surjective (or, dually,

the map

ϕ_{λ} : H^{1}(K_{λ}, E)_{q}^{ϵ_{k}} ⟶ (Sel_{q}(S′)^{ϵ_{k}})^{∗}

is injective).

The crucial observation is that the hypothesis 4.5 of section 4.2 is still satisfied in our new setting.

Lemma 5.6 The partial derivative D\underline{k}D_{l}^{1} satisfies the hypothesis 4.5 of section 4.2.

Proof: Let D′ be a partial derivative which is strictly less than D\underline{k}D_{l}^{1}. We assume without loss of generality that the order of D′ is equal to k, the order of D\underline{k}. Lemma 5.1, applied to the exact sequence

1 ⟶ Sel_{q}(S′l)^{ϵ_{k}} ⟶ Sel_{q}(S′)^{ϵ_{k}} ⟶ (E(K_{λ})/qE(K_{λ}))^{ϵ_{k}}

shows that

r_{q}(Sel_{q}(S′)^{ϵ_{k}}) ≤ r_{q}(Sel_{q}(S′l)^{ϵ_{k}}) + r_{p}(E(K_{λ})/qE(K_{λ})^{ϵ_{k}})
= r_{q}(A′(S′l)^{ϵ_{k}}) + 1 (by lemma 5.2).

Also, by lemma 5.2,

r_{p}(A(S′)^{ϵ_{k}}) = r_{p}(A(S′l)^{ϵ_{k}}) − 1.

Combining these two inequalities, we find that

order(D′) = k < r_{q}(Sel_{q}(S′l)^{ϵ_{k}}) + r_{p}(A(S′l)^{ϵ_{k}}).

Since the support of D′ is divisible by an extra prime in L_{q,E},

wt(D′) < wt(D\underline{k}).

Thus we may apply the induction hypothesis to conclude that D′α(S) = 0.

Because of this lemma, we can apply the construction of section 4.2, to obtain a class d(l) in H^{1}(K, E)_{q}. Combining 1 and 2 of conditions 5.5 satisfied by l with theorem 4.9, we find

d(l)_{λ} ≠ 0.

By proposition 4.10, it follows that the map

ϕ_{λ} : H^{1}(K_{λ}, E)_{q}^{ϵ_{k}} ⟶ (Sel_{q}(S′)^{ϵ_{k}})^{∗}

fails to be injective, contradicting the third of the conditions 5.5.

We now derive some consequences of the main result.

Corollary 5.7 If order(D\underline{k}) < r_{q}(Sel_{q}(S′)^{−ϵ_{k}}) + r_{p}(A(S′)^{−ϵ_{k}}) − 1, then

D\underline{k}α(S) ≡ 0 (mod q).

Proof: Suppose that P = D\underline{k}α(S) ≠ 0 (mod q). Then, by proposition 4.19, we can choose a prime l such that

1. Frob_{l} = Frob_{∞} in K_{S}(E_{q})/Q.

2. P̃ ≠ 0 in E(k_{λ})/qE(k_{λ}).

We observe that the point D\underline{k}D_{l}^{1}α(T) = P (l) is non-zero in E(K_{T})/qE(K_{T}). For, either there is a partial derivative D′ strictly less than D\underline{k}D_{l}^{1} such that D′α(T) is non-zero mod q, in which case D\underline{k}D_{l}^{1} is also non-zero mod q by lemma 4.1; or D\underline{k}D_{l}^{1} satisfies the hypothesis 4.5, in which case we can apply the general construction of section 4.2 to obtain a cohomology class d(l) in H^{1}(K, E)_{q}. By theorem 4.9, this class is non-zero locally at λ, and hence a fortiori globally. Hence the point P (l) from which it comes is non-zero as well. By the assumption,

order(D\underline{k}D_{l}^{1}) < r_{q}(Sel_{q}(S′)^{−ϵ_{k}}) + r_{p}(A(S′)^{−ϵ_{k}}).

On the other hand,

r_{q}(Sel_{q}(S′)^{−ϵ_{k}}) + r_{p}(A(S′)^{−ϵ_{k}}) ≤ r_{q}(Sel_{q}(S′l)^{−ϵ_{k}}) + r_{p}(A(S′l)^{−ϵ_{k}}),

by the same calculation as in the proof of lemma 5.6. Combining the two inequalities together, we find

order(D\underline{k}D_{l}^{1}) < r_{q}(Sel_{q}(S′l)^{−ϵ_{k}}) + r_{p}(A(S′l)^{−ϵ_{k}}).

Let k′ = order(D\underline{k}D_{l}^{1}) = k + 1. Since (−1)^{k′} = −ϵ_{k}, we can apply theorem 5.3 to conclude that D_{k}D_{l}^{1} = 0 mod p, which is a contradiction.

Corollary 5.8 .

1. If order(D\underline{k}) < r_{q}(Sel_{q}^{ϵ_{k}}), then D\underline{k}α(S) ≡ 0 (mod q).

2. If order(D\underline{k}) < r_{q}(Sel_{q}^{−ϵ_{k}}) − 1, then D\underline{k}α(S) ≡ 0 (mod q).

Proof: By lemma 5.1,

r_{q}(Sel_{q}^{ϵ_{k}}) ≤ r_{q}(A′(S′)^{ϵ_{k}}) + r_{p}(A′′(S′)^{ϵ_{k}}).

Hence part 1 follows from theorem 5.3. Part 2 of the corollary follows similarly from corollary 5.7.

We finally come to the proof of theorem 3.15 whose statement we recall:

Theorem 3.15: Let q be a power of a prime which is not invertible in Z, and let D\underline{k} be a partial derivative whose support S is a product of primes in L_{q}. If order(D\underline{k}) < ρ_{q}, then

D\underline{k}α(S) ≡ 0 (mod q).

Proof: If the inequality is true, then either

order(D\underline{k}) < r_{q}(Sel_{q}^{ϵ_{k}}) or order(D\underline{k}) < r_{q}(Sel_{q}^{−ϵ_{k}}) − 1.

The result then follows from corollary 5.8.

5.2 Proof of theorem 2.5

We now turn to the proof of theorem 2.5. We assume that |r^{+} − r^{−}| = 1, so that 2ρ = r − 1. To study the leading coefficient θ̃′(E, S)_{(p)}, we must study the images of the elements D\underline{k}α(S) in E(K_{S})/pE(K_{S}), where D\underline{k} is a partial derivative of order ρ and support S.

Part 2 of theorem 2.5 will follow from:

Proposition 5.9 If D\underline{k}α(S) ≠ 0 mod p, then \underline{III}_{p}(E/K) = 0, and the map

Sel_{p}(E/K) ⟶ ⊕_{v|S}E(K_{v})/pE(K_{v})

is surjective (i.e., J_{S} ⊗ F_{p} = 0).

Proof: If D\underline{k}α(S) ≠ 0 mod p, then by theorem 5.3 and corollary 5.7, we have:

ρ ≥ r_{p}(Sel_{p}(S)^{ϵ_{k}}) + r_{p}(A(S)^{ϵ_{k}}) ≥ r_{p}(Sel_{p}(E/K)^{ϵ_{k}}) ≥ r^{ϵ_{k}},

ρ ≥ r_{p}(Sel_{p}(S)^{−ϵ_{k}}) + r_{p}(A(S)^{−ϵ_{k}}) − 1 ≥ r_{p}(Sel_{p}(E/K)^{−ϵ_{k}}) − 1 ≥ r^{−ϵ_{k}} − 1.

Since 2ρ = r − 1, we have equalities everywhere, and hence \underline{III}_{p}(E/K) = 0. Also, since r_{p}(Sel_{p}(E/K)) = r_{p}(Sel_{p}(S)) + r_{p}(A(S)), the map

Sel_{p}(E/K) ⟶ A(S)

is surjective.

To show part 1, we must show that

Proposition 5.10 D\underline{k}α(S) is in the image of the natural map

E(K)/pE(K) ⟶ E(K_{S})/pE(K_{S}).

Proof: Consider the exact sequence

0 ⟶ E(K)/pE(K) ⟶ E(K_{S})/pE(K_{S}) ⟶ H^{1}(K, E)_{p}

(cf. section 4.2). Let P be the image of D\underline{k}α(S) in E(K_{S})/pE(K_{S}), and let d be the image of P in H^{1}(K, E)_{p}. The cohomology class d is the obstruction for the point P to come from E(K)/pE(K); we want to show that it vanishes. By theorem 4.9, the class d is trivial locally except possibly at the places dividing S. Hence by the definition of the local Tate pairing and the reciprocity law of global class field theory (cf. sec. 4.3), the image of d in ⊕_{v|S}H^{1}(K_{v}, E)_{p} maps to 0 in Sel_{p}(E/K)^{∗}. But by prop. 5.9, the map Sel_{p}(E/K) ⟶ A(S) ⊗ F_{p} is surjective, and hence dually the map

⊕_{v|S}H^{1}(K_{v}, E)_{p} ⟶ Sel_{p}(E/K)^{∗}

is injective. Therefore the class d is locally trivial everywhere; it belongs to \underline{III}_{p}(E/K). By prop. 5.9, \underline{III}_{p}(E/K) = 0, and hence d = 0.

To conclude, we make some remarks concerning the Mazur Tate height pairing

⟨ , ⟩_{S} : E(K) × E_{S}(K) ⟶ (I/I^{2}),

where I denotes the augmentation ideal in the group ring Z[Γ_{S}].

Claim 5.11 If P ∈ E(K) and Q ∈ E_{S}(K) belong to the same eigenspaces for the action of complex conjugation, then ⟨P, Q⟩ = 0.

Proof: Let τ = Frob_{∞} denote complex conjugation. By the linearity of the Mazur Tate pairing and the fact that P and Q belong to the same eigenspace for τ, we have:

⟨τP, τQ⟩ = ⟨P, Q⟩.

On the other hand, the behavior of the Mazur Tate pairing under Galois action (cf. [MT2], p. 216, (3.4.2)),

⟨τP, τQ⟩_{S} = τ⟨τP, Q⟩_{S}τ^{−1} = −⟨P, Q⟩_{S},

and hence since I/I^{2} is of odd order, ⟨P, Q⟩_{S} = 0.

Because of this claim, the pairing matrix has all of its (r − 1) × (r − 1)- minors equal to 0 whenever |r^{+} − r^{−}| > 1. Hence the regulator R_{S} vanishes. Since the leading coefficient θ̃′(E, S) is also zero in E(K_{S})^{⊗2} ⊗ I^{r−1}/I^{r} (because 2ρ > r − 1), we have shown:

Proposition 5.12 If |r^{+} − r^{−}| > 1, then both the regulator R_{S} and the leading coefficient θ̃′(E, S) are zero in E(K_{S})^{⊗2} ⊗ (I^{r−1}/I^{r}), where I denotes the augmentation ideal in the group ring Z[Γ_{S}]. Hence conjecture 2.3 is true in this case, after tensoring with Z.

5.3 Proof of theorem 2.6

We finish with the proof of theorem 2.6 (cf. section 2.2).

Theorem 2.6 dim_{F_{p}}(E(L)_{p}) ≤ p − ρ_{p}.

Proof: Let σ be a generator for the group G, and let α be the Heegner point in E(L). By applying lemma 4.1, one sees that the non-zero vectors among

D_{σ}^{0}α, D_{σ}^{1}α, …, D_{σ}^{p−1}α

give a basis for the vector space E_{p} over F_{p}. By theorem 3.15 the partial derivatives of α of order < ρ_{p} are 0 mod p, and the result follows.

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