THE EULER SYSTEM OF HEEGNER POINTS

TOM WESTON

Contents

1. Introduction 1 Notation 3 2. Local cohomology groups 4 2.1. Basic facts 4 2.2. Local Selmer structures 4 2.3. Local cohomology of elliptic curves 4 3. Global cohomology groups 5 3.1. Selmer groups 5 3.2. Global cohomology of elliptic curves 6 3.3. Global duality 6 3.4. Bounds on restricted Selmer groups I 7 3.5. Bounds on restricted Selmer groups II 8 4. Galois cohomology calculations 10 4.1. Statements 10 4.2. Preliminaries 10 4.3. The −ε-eigenspace 11 4.4. The ε-eigenspace 13 5. Heegner points 14 5.1. Ring class fields 14 5.2. Complex multiplication 15 5.3. Heegner points on elliptic curves 16 6. The Euler system 17 6.1. Kolyvagin’s derivative operator 17 6.2. Derived cohomology classes 18 6.3. Ramification of the derived classes 20 6.4. Final considerations 22 7. Examples 23 References 24

1. Introduction

Let E be an elliptic curve over Q of conductor N and let K be an imaginary quadratic field in which all primes dividing N split. The theory of complex multiplication and a modular parameterization X_{0}(N) → E can be used to define a point y_{K} ∈ E(K), called a Heegner point. y_{K} depends on some choices, but it is well-defined up to sign and torsion, so that its canonical height ĥ(y_{K}) is well-defined independent of any choices. If L(E/K, s) is the L-function of E over K, one has L(E/K, 1) = 0 for trivial reasons, and Gross and Zagier proved the spectacular formula

L′(E/K, 1) = (\frac{1}{\sqrt{D}} ∫_{E(C)} ω ∧ \overline{iω}) · ĥ(y_{K}).

Here D is the discriminant of K/Q and ω is the differential on E coming from the modular parameterization. In particular, L′(E/K, 1) ≠ 0 if and only if y_{K} has infinite order.

Motivated by this and the conjecture of Birch and Swinnerton-Dyer, Gross and Zagier were led to the following conjecture (their conjecture is somewhat more precise):

Conjecture (Gross–Zagier). Assume that y_{K} has infinite order in E(K). Then E(K) has rank one and the Shafarevich-Tate group � (E/K) has trivial p-primary component for any odd prime p such that:

• E has good reduction at p;

• Gal(Q(E[p])/Q) ≅ GL_{2}(F_{p});

• y_{K} ∉ pE(K).

In this paper we give Kolyvagin’s proof of this conjecture. The key idea is that the Heegner point y_{K} does not come alone: it lies at the bottom of a certain family of points of E defined over anti-cyclotomic extensions of K. These points satisfy relations which make them an Euler system. This allows one to use them and techniques from Galois cohomology to bound the Selmer group Sel(K, E[p]) (and thus to obtain information on E(K)/pE(K) and � (E/K)[p]) for appropriate primes p.

This paper could not exist without the magnificent paper [6]. In particular, Sections 5 and 6 of this paper follow the corresponding sections of [6] very closely. We deviate from Gross in our treatment of the Galois cohomology. Our exposition is intended to relate somewhat more directly to the more recent theories of Euler systems developed by Rubin, Kato and Perrin-Riou.

We present this Galois cohomology in Sections 2 and 3. We first define Selmer groups and recall the relevant duality results. Our treatment of Euler systems is based on the exact sequence (7) which divides the computation of a Selmer group into two parts. The first part is the computation of a certain restricted Selmer group. This can be done quite generally, and we present these results without reference to elliptic curves. The second part is the production of an Euler system. We give no general theory for this; we only explain it in the case of elliptic curves in Sections 6.

Section 4 contains the details of the proof of Kolyvagin’s theorem; it assumes the existence of the cohomology classes constructed via the Euler system in Section 6. Section 4 is mostly a straightforward application of the results of Section 3; we try to emphasize that this part of the proof consists entirely in fairly routine Galois cohomology calculations.

We introduce Heegner points and their basic properties (which make them an Euler system) in Section 5. In Section 6 we explain how to use these properties to prove the results needed in Section 4.

It is our hope that this paper can serve as an introduction to Euler systems for a graduate student with some knowledge of Galois cohomology and the arithmetic of elliptic curves. Although Heegner points do not fit into most of the general theories of Euler systems, we regard it is an especially interesting and striking example of the fundamental ideas involved.

This paper grew out of one of the student projects at the 2001 Arizona Winter School. In the process of helping my student group to prepare their presentation on [6], it became clear to me that the Galois cohomology in Gross’ argument could be simplified in several ways. This paper is my attempt to demonstrate that. Although it may seem strange that the allegedly simpler proof given here is somewhat longer than that given in [6], I believe that much of the extra bulk comes from the general treatment of the Galois cohomology and some elaboration on points Gross treats very quickly. (See [4] and [13] for another example of an intended simplification by this author which ended up being far longer than the original, especially considering that all of the hard parts were omitted.)

My Winter School student group consisted of Kirsten Eisentraeger, Chris Hall, Ling Long, Satya Mohit, Jorge Pineiro, Marat Sadykov and Michael Schein. I would like to express my sincere gratitude to each of them for preparing and giving an outstanding presentation of these ideas; I would also like to thank them for helping me to understand this proof far better than I ever had before. I hope that skipping the Desert Museum seemed at least a little worthwhile in the end. Robert Pollack provided invaluable insight throughout the project, and Mark Dickinson helped me work out some of the details in this write-up. The calculations presented in Section 7 were done by Kirsten Eisentraeger and Peter Green; William Stein helped with some of them as well. I would also like to thank the organizers of the Winter School for giving all of us this opportunity. Lastly, I would like to thank Barry Mazur for inviting me to share in his lectures at the Winter School and for all of the help and inspiration he provided. If nothing else, I hope that this paper can at least serve as the long-awaited conclusion to [7].

Notation. If H is an abelian group with pH = 0, we write H^{∨} for its Pontrjagin dual Hom_{F_{p}}(H, F_{p}). If H is an arbitrary abelian group, we write H[p] for the p-torsion in H.

If K is a perfect field we will write G_{K} for the absolute Galois group Gal(K̄/K). If L is an extension of K and T is a Gal(L/K)-module, we write H^{i}(L/K, T) for the group cohomology H^{i}(Gal(L/K), T); if L = K̄ we will just write this as H^{i}(K, T). (All Galois modules we consider will be discrete so that there is no need to worry about topologies.) If T is a p-torsion G_{K}-module for some prime p, we write T^{∗} for its Cartier dual Hom_{F_{p}}(T, μ_{p}) (where μ_{p} denotes the G_{K}-module of the p^{th} roots of unity) endowed with the adjoint Galois action. We write K(T) for the fixed field of the kernel of the homomorphism G_{K} → Aut(T).

We will identify non-archimedean places of number fields with prime ideals; we will usually use places when working abstractly and ideals when working somewhat more concretely. We will tend to ignore archimedean places; this will not be a problem since we almost always have p ≠ 2.

We will have frequent use for the following construction: let M/L and L/K be Galois extensions and assume that Gal(M/L) is abelian. There is a natural action of Gal(L/K) on Gal(M/L) defined as follows: given τ ∈ Gal(L/K) and σ ∈ Gal(M/L), let τ̃ be any lift of τ to Gal(M/K). Then the action of τ on σ is given by ^{τ} σ = τ̃στ̃^{−1}. (The fact that Gal(M/L) is abelian implies that this is independent of the choice of τ̃.) The action of Gal(L/K) on Gal(M/L) is trivial precisely when M is an abelian extension of K.

We will often be working with eigenspaces for involutions, and we make the following sign convention: whenever ± appears in a formula, it is to be regarded as a fixed choice of sign, and every other ± in that formula should agree with this choice; a ∓ indicates the opposite of this initial choice.

2. Local cohomology groups

2.1. Basic facts. Fix a prime l and let K be a finite extension of Q_{l}. Let K^{ur} denote the maximal unramified extension of K; we write I_{K} for the inertia group Gal(K̄/K^{ur}) and G_{K}^{ur} for G_{K}/I_{K} ≅ Gal(K^{ur}/K). Let T be a finite dimensional F_{p}-vector space with a discrete action of G_{K}. We say that T is unramified if I_{K} acts trivially on T.

We will need the following facts about the Galois cohomology group H^{1}(K, T). Inflation-restriction: There is an exact sequence [10, Chapter 7, Section 6]

(1) 0 → H^{1}(K^{ur}/K, T^{I_{K}}) ⟶^{inf} H^{1}(K, T) ⟶^{res} H^{1}(I_{K}, T)^{G_{K}^{ur}} → 0.

(We will always use inflation to regard H^{1}(K^{ur}/K, T^{I_{K}}) as a subspace of H^{1}(K, T).)

Tate local duality: There is a perfect pairing of F_{p}-vector spaces [8, Chapter I, Corollary 2.3]

H^{1}(K, T) ⊗_{F_{p}} H^{1}(K, T^{∗}) → F_{p}. (2)

Unramified duality: If l ≠ p and T is unramified, then H^{1}(K^{ur}/K, T) and H^{1}(K^{ur}/K, T^{∗}) are exact orthogonal complements under the Tate pairing; see [8, Chapter I, Theorem 2.6].

2.2. Local Selmer structures. By a local Selmer structure F (sometimes called a finite/singular structure) for T we simply mean a choice of F_{p}-subspace H_{f,F}^{1}(K, T) of H^{1}(K, T). We define the singular quotient H_{s,F}^{1}(K, T) as H^{1}(K, T)/H_{f,F}^{1}(K, T), so that there is an exact sequence

0 → H_{f,F}^{1}(K, T) → H^{1}(K, T) → H_{s,F}^{1}(K, T) → 0. (3)

For c ∈ H^{1}(K, T) we write c^{s} for the image of c in H_{s,F}^{1}(K, T).

We say that F is the unramified structure if H_{f,F}^{1}(K, T) = H^{1}(K^{ur}/K, T^{I_{K}}). In this case the sequence (3) identifies with the sequence (1).

Given a local Selmer structure F on T, we define the Cartier dual local Selmer structure F^{∗} on T^{∗} to be the exact orthogonal complement of H_{f,F}^{1}(K, T) under the Tate pairing (2). In particular, there is an induced perfect pairing

H_{s,F}^{1}(K, T) ⊗_{F_{p}} H_{f,F}^{1}(K, T^{∗}) → F_{p}. (4)

Note that if T is unramified, F is the unramified structure and l ≠ p, then T^{∗} is unramified and F^{∗} is the unramified structure.

2.3. Local cohomology of elliptic curves. Let E be an elliptic curve over the local field K. We will be studying the G_{K}-module E[p] = E(K̄)[p]; it is a twodimensional F_{p}-vector space. Recall that the Weil pairing on E[p] is a perfect Galois equivariant pairing

E[p] ⊗_{F_{p}} E[p] → μ_{p};

see [11, Chapter 3, Section 8] It follows that E[p]^{∗} is canonically isomorphic to E[p].

We have an exact sequence

0 → E[p] → E(K̄) ⟶^{p} E(K̄) → 0

of G_{K}-modules. The associated long exact sequence in G_{K}-cohomology yields a short exact sequence

0 → E(K)/pE(K) ⟶^{κ} H^{1}(K, E[p]) → H^{1}(K, E)[p] → 0;

here by H^{1}(K, E) we mean H^{1}(K, E(K̄)). The Kummer map is given explicitly as follows: for P ∈ E(K), fix Q ∈ E(K̄) such that pQ = P. Then the cocycle κ(P) sends σ ∈ G_{K} to σ(Q) − Q ∈ E[p]. (One checks that changing the choice of Q changes this cocycle by a coboundary.)

We define the geometric local Selmer structure F on E[p] by setting

H_{f,F}(K, E[p]) = im κ.

We will need the following two facts (see [1, Chapter 1]):

Duality: The Cartier dual of the geometric structure is the geometric structure.

(This makes sense since E[p]^{∗} ≅ E[p].) Compatibility: If E has good reduction over K and l ≠ p, then E[p] is an un-

ramified E[p]-module and the geometric structure agrees with the unramified structure.

3. Global cohomology groups

3.1. Selmer groups. Let K be a number field. For a place v of K we write K_{v} for the completion of K at v; we write G_{v} and I_{v} for the absolute Galois group and inertia group of K_{v}, respectively. We also write F_{v} for the residue field of O_{K_{v}}. We fix an embedding K̄ ↪ K̄_{v} for every v; this yields a restriction map G_{v} ↪ G_{K}.

Let T be a finite-dimensional F_{p}-vector space with a discrete action of G_{K}. For any place v we can regard T as a G_{v}-module via the injection G_{v} ↪ G_{K}; we say that T is unramified at v if it is unramified as a G_{v}-module. Since T is finite and discrete it is unramified at almost all places v. Note that for any place v we have a restriction map

res_{v} : H^{1}(K, T) → H^{1}(K_{v}, T)
c ↦ c_{v}

which is in fact independent of the choice of embedding G_{v} ↪ G_{K}.

We define a global Selmer structure F on T to be a choice of local Selmer structures for every place of K; we further require these structures to be unramified at almost all places. Thus a global Selmer structure F on T is a choice of F_{p}-subspace

H_{f,F}^{1}(K_{v}, T) ⊆ H^{1}(K_{v}, T)

for every place of v, such that

H_{f,F}^{1}(K_{v}, T) = H^{1}(K_{v}^{ur}/K_{v}, T^{I_{v}})

for almost all v.

We define the Selmer group Sel_{F}(K, T) to be the set of all c ∈ H^{1}(K, T) such that c_{v} lies in H_{f,F}^{1}(K_{v}, T) for every place v of T. That is,

Sel_{F}(K, T) = ker(H^{1}(K, T) → ⊕_{v}H_{s,F}^{1}(K_{v}, T)).

3.2. Global cohomology of elliptic curves. Let E be an elliptic curve over K. Since the coordinates of the p-torsion points E[p] generate a finite extension of K, the Galois module E[p] is discrete; in particular, it is unramified almost everywhere. (In fact, by our local discussion we know that E[p] is unramified at all places of good reduction which do not divide p.) We define the geometric global Selmer structure F on E[p] by letting it agree with the geometric local Selmer structure at every place of K. By our local discussion we know that this really will be a Selmer structure (since it it is the unramified structure at all places of good reduction not dividing p). We also know that the geometric structure is self-dual under the Weil pairing identification E[p]^{∗} ≅ E[p].

We claim that the Selmer group Sel_{F}(K, E[p]) agrees with the usual p-torsion Selmer group Sel(K, E[p]) (see [11, Chapter 10, Section 4]) of E sitting in an exact sequence

0 → E(K)/pE(K) → Sel(K, E[p]) → � (K, E)[p] → 0 (5)

with

� (K, E) = ker(H^{1}(K, E) → ∏_{v}H^{1}(K_{v}, E)).

Indeed, the classical Sel(K, E[p]) is defined to be the subspace of H^{1}(K, E[p]) of classes whose restriction to H^{1}(K_{v}, E[p]) lie in the image of the local Kummer map κ_{v} for every v. This is precisely our definition of Sel_{F}(K, E[p]).

3.3. Global duality. Fix a G_{K}-module T as in Section 3.1 and a global Selmer structure F on T; this induces a Cartier dual Selmer structure F^{∗} on T^{∗}. We will omit these structures from our notation for the remainder of the section.

For any ideal 𝔞 of the ring of integers O_{K}, we define groups

Sel_{𝔞}(K, T) = {c ∈ H^{1}(K, T) | c_{v} ∈ H_{f,F}^{1}(K_{v}, T) for all v ∤ 𝔞};

Sel^{𝔞}(K, T^{∗}) = {c ∈ H^{1}(K, T^{∗}) | c_{v} ∈ H_{f,F}^{1}(K_{v}, T^{∗}) for all v and c_{v} = 0 for v|𝔞}.

From these definitions we have exact sequences

0 → Sel(K, T) → Sel_{𝔞}(K, T) → ⊕_{v|𝔞}H_{s}^{1}(K_{v}, T)

0 → Sel^{𝔞}(K, T^{∗}) → Sel(K, T^{∗}) → ⊕_{v|𝔞}H_{f}^{1}(K_{v}, T^{∗}).

The dualities (2) induce a duality between

⊕_{v|𝔞}H_{s}^{1}(K_{v}, T)  and  ⊕_{v|𝔞} H_{f}^{1}(K_{v}, T^{∗}).

Combining these facts, we obtain a sequence

(6) 0 → Sel(K, T) → Sel_{𝔞}(K, T) → ⊕_{v|𝔞}H_{s}^{1}(K_{v}, T)

→ Sel(K, T^{∗})^{∨} → Sel^{𝔞}(K, T^{∗})^{∨} → 0

which is obviously exact except possibly in the middle.

Proposition 3.1. The sequence (6) is exact.

Proof. A proof is given in [8, Chapter 1, Theorem 4.10]; see also [9, Theorem 1.7.3].

It follows that for any ideal 𝔞 we have a short exact sequence

(7) 0 → ( ⊕_{v|𝔞}H_{s}^{1}(K_{v}, T))/ im Sel_{𝔞}(K, T) → Sel(K, T^{∗})^{∨} → Sel^{𝔞}(K, T^{∗})^{∨} → 0.

In particular, if we can choose 𝔞 so that we can compute both of the flanking terms, we will immediately determine Sel(K, T^{∗}). In the next section we will show that for appropriate 𝔞 it is easy to obtain a bound on Sel^{𝔞}(K, T^{∗}); this part of the argument only depends on the coarse structure of T. The difficult part is to exhibit elements in Sel_{𝔞}(K, T) so as to bound the cokernel of the map

Sel_{𝔞}(K, T) → ⊕_{v|𝔞}H_{s}^{1}(K_{v}, T).

This is accomplished, in a few special cases, via Euler systems.

3.4. Bounds on restricted Selmer groups I. We will prove the result we actually need for our applications to elliptic curves in the next section. In this section we present a less cluttered version which is often useful.

We now assume that p is an odd prime. Let L/K be a finite extension of number fields and let T be a finite-dimensional F_{p}-vector space with an action of Gal(L/K). (Of course, we can also regard T as a discrete G_{K}-module.) We assume that T is irreducible as a Gal(L/K)-module; that is, T has no proper subspaces which are stable under the action of Gal(L/K). We wish to exhibit ideals 𝔞 for which we can bound Sel^{𝔞}(K, T) in terms of L. (The T of this section corresponds to the T^{∗} of the previous section.) In fact, we will use nothing special about the subspace Sel(K, T) of H^{1}(K, T), so we proceed in somewhat more generality.

Suppose that τ ∈ Gal(L/K) is an involution; that is, τ^{2} = 1. Associated to τ we have a decomposition T = T^{+} ⊕ T^{−} where

T^{ε} = {t ∈ T | τt = εt}

for ε ∈ {±}. We say that τ is non-scalar if both T^{+} and T^{−} are non-zero.

Let S be a finite-dimensional F_{p}-subspace of H^{1}(K, T). If 𝔞 is an ideal of O_{K}, we write

S^{𝔞} = {s ∈ S | s_{v} = 0 in H^{1}(K_{v}, T) for all v|𝔞}.

Since S is finite we can choose a finite Galois extension M of L such that S lies in the image of H^{1}(M/K, T) under inflation. We fix such an M and we assume that τ extends to an involution in Gal(M/K) which we still write as τ. Let {γ_{1}, …, γ_{r}} be a set of generators of Gal(M/L).

Proposition 3.2. With notation as above, let w_{1}, …, w_{r} be places of M such that Frob_{M/K} w_{i} = τγ_{i}. Let v_{i} denote the restriction of w_{i} to K and set 𝔞 = v_{1} ⋯ v_{r}. Then S^{𝔞} lies in the image of H^{1}(L/K, T) under inflation.

The power of this result lies in the fact that in practice H^{1}(L/K, T) is generally easily computable if L is chosen appropriately. For example, let E be an elliptic curve over Q and set T = E[p]. Take τ to be (a choice of) complex conjugation; since p ≠ 2 it follows from the existence of the Weil pairing that τ is non-scalar on E[p]. Set S = Sel(Q, E[p]); it is finite by the weak Mordell-Weil theorem. Let L = Q(E[p]). If Gal(L/Q) is isomorphic to GL_{2}(F_{p}), which is true for almost all primes p, we will see later that H^{1}(L/Q, E[p]) = 0. Thus in this case the proposition says that for appropriately chosen ideals 𝔞 of Z, we have Sel^{𝔞}(Q, E[p]) = 0.

Proof. We begin with the inflation-restriction exact sequence

0 → H^{1}(L/K, T) → H^{1}(M/K, T) → H^{1}(M/L, T)^{Gal(L/K)}.

Since S ⊆ H^{1}(M/K, T), to prove the proposition we must show that the image s̃ ∈ H^{1}(M/L, T) of any s ∈ S^{𝔞} is zero.

Fix an s ∈ S^{𝔞}. Since Gal(M/L) acts trivially on T, we have H^{1}(M/L, T) = Hom(Gal(M/L), T). We may therefore regard s̃ as a homomorphism Gal(M/L) → T. In this language, Gal(L/K)-invariance translates to Gal(L/K)-equivariance of s̃ for the conjugation action on Gal(M/L) and the usual action on T.

Fix one of the places w = w_{i} of M corresponding to γ = γ_{i} as in the statement of the proposition; that is, Frob_{M/K} w = τγ. We will also write w for the induced place of L and v for the induced place of K. We have

Frob_{L/K} w = Frob_{M/K} w|_{L} = τ,

which has order 2. Thus

Frob_{M/L} w = (Frob_{M/K} w)^{2} = (τγ)^{2}.

Since s ∈ S^{𝔞}, we know that the restriction of s to H^{1}(M_{w}/K_{v}, T) is a coboundary. Thus the homomorphism s̃|_{Gal(M_{w}/L_{w})} is zero. In particular, s̃(Frob_{M/L} w) = 0; that is,

s̃(τγτγ) = 0. (8)

Since τ is an involution, τγτ is nothing other than ^{τ} γ. We can thus rewrite (8) as s̃(^{τ} γ) = −s̃(γ). Since s̃ is Gal(L/K)-equivariant, this means that τs̃(γ) = −s̃(γ). We conclude that s̃(γ) lies in the eigenspace T^{−} for the action of τ on T.

Since the γ_{i} generate Gal(M/L), by applying the above argument to each of γ_{1}, …, γ_{r} we see that the F_{p}-span of the image of s̃ : Gal(M/L) → T lies in T^{−}. Let us write this span as W. Since s̃ is Gal(L/K)-equivariant and Gal(M/L) is stable under the conjugation action of Gal(L/K), W is stable under the action of Gal(L/K).

We have now shown that W is Gal(L/K)-stable and lies in T^{−}. However, since τ is non-scalar by hypothesis, we have T^{−} ≠ T; thus W ≠ T. T is irreducible as a Gal(L/K)-module, so this implies that W = 0. Thus s̃ = 0, as desired.

3.5. Bounds on restricted Selmer groups II. As in the previous section, let T be an irreducible F_{p}[Gal(L/K)]-module and let S be a finite-dimensional subspace of H^{1}(M/K, T) for some finite Galois extension M of L. We introduce an intermediate field L_{0} of the extension L/K, Galois over K, and we assume that the action of Gal(L/K) on T factors through Gal(L_{0}/K). We also assume that K is a quadratic extension of a field K_{0} and that the Gal(L_{0}/K)-action on T is the restriction of a Gal(L_{0}/K_{0})-action.

We now let τ denote an involution of Gal(M/K_{0}) which projects to the nontrivial element of Gal(K/K_{0}). (In particular, τ ∉ Gal(M/K).) As before we have a decomposition T = T^{+} ⊕T^{−} and we assume that both of these factors are non-zero. Note that τ also acts on H^{1}(M/K, T) (via conjugation on Gal(M/K) and the usual action on T) and we have a decomposition

H^{1}(M/K, T) = H^{1}(M/K, T)^{+} ⊕ H^{1}(M/K, T)^{−}.

We assume that S ⊆ H^{1}(M/K, T)^{ε} for some ε ∈ {±}.

Finally, let σ be an element of Gal(M/L_{0}) such that τστ^{−1} = σ^{−1}. As before we let {γ_{1}, …, γ_{r}} be a set of generators of Gal(M/L).

Proposition 3.3. With notation as above, let w_{1}, …, w_{r} be places of M such that Frob_{M/K_{0}} w_{i} = τσγ_{i}. Let v_{i} denote the restriction of w_{i} to K and set 𝔞 = v_{1} ⋯ v_{r}. Then S^{𝔞} lies in the image of H^{1}(L/K, T)^{ε}.

The Tchebatorev density theorem guarantees the existence of an infinite number of choices for each of the w_{i} above. In particular, we obtain the following corollary.

Corollary 3.4. There is an ideal 𝔞 of K, divisible only by primes λ lying over primes λ_{0} of K_{0} with Frobenius conjugate to τσ on L, such that S^{𝔞} ⊆ H^{1}(L/K, T)^{ε}.

In our applications to an elliptic curve E over Q, we will take K_{0} = Q and K will be an appropriate imaginary quadratic field. T will be the p-torsion representation E[p], τ will be a complex conjugation and S will be Sel(K, E[p])^{ε} for some ε. We will take L_{0} to be the field K(E[p]) or an extension of it of degree p^{2}; in either case, L will be a certain extension of L_{0} of degree p^{2}. We will need the extra flexibility provided by taking σ to be non-trivial to insure that the Euler system part of our argument succeeds.

Proof. Since S = S^{ε}, it suffices to show that S^{𝔞} lies in the inflation of H^{1}(L/K, T). As before, by the inflation-restriction sequence we must show that for any s ∈ S^{𝔞} the induced Gal(L/K)-homomorphism s̃ : Gal(M/L) → T is zero. Note also that s̃ extends to a homomorphism Gal(M/L_{0}) → T.

Fix one of the places w = w_{i} of M corresponding to γ = γ_{i}. We will also write w for the induced places of L and L_{0} and v for the induced places of K and K_{0}. Note that v is inert in K/K_{0} and splits completely in L_{0}/K. Since Frob_{M/K_{0}} w = τσγ and K/K_{0} is quadratic, we thus have Frob_{M/L_{0}} w = (τσγ)^{2}. Since τ is an involution we have

(τσγ)^{2} = τσττγτσγ = ^{τ} σ ·^{τ} γ · σ · γ.

Since ^{τ} σ = σ^{−1}, we conclude that

Frob_{M/L_{0}} w = σ^{−1} ·^{τ} γ · σ · γ. (9)

As before, since s ∈ S^{𝔞} we must have s̃(Frob_{M/L_{0}} w) = 0. Since s̃ is a homomorphism on Gal(M/L_{0}), (9) now implies that

s̃(σ^{−1}) + s̃(^{τ} γ) + s̃(σ) + s̃(γ) = 0.

Since s̃(σ^{−1}) = −s̃(σ), this implies that

s̃(^{τ} γ) = −s̃(γ). (10)

s̃ need not be equivariant for the action of τ since τ ∉ Gal(L/K). However, since s ∈ H^{1}(L/K, T)^{ε}, we do have

s̃(^{τ} γ) = ετs̃(γ)

by the definition of the τ-action on H^{1}(L/K, T). Combining this with (10) we see that s̃(γ) lies in T^{−ε}.

Let W ⊆ T be the F_{p}-span of s̃(Gal(M/L)); it is Gal(L/K)-stable since s̃ is Gal(L/K)-equivariant. By the above argument applied to each of γ_{1}, …, γ_{r} we see that W ⊆ T^{−ε}. But we are assuming both that T is irreducible and that T^{−ε} ≠ T. Thus W must be zero, and s̃ = 0, as desired.

4. Galois cohomology calculations

4.1. Statements. Let E be an elliptic curve over Q of conductor N and let ε be the negative of the sign of the functional equation of E over Q. We fix a modular parameterization X_{0}(N) → E and we let K be an imaginary quadratic field in which every prime dividing N splits. In the next section we will construct a point y_{K} in E(K) which is mapped to ε · y_{K} (up to torsion) under complex conjugation. (If E(K) has no p-torsion, then this implies that y_{K} ∈ (E(K)/pE(K))^{ε}.) The main result of this paper is the following.

Theorem 4.1. Let p be an odd prime such that:

• E has good reduction at p;

• Gal(Q(E[p])/Q) ≅ GL_{2}(F_{p});

• y_{K} ∉ pE(K). Then Sel(K, E[p]) has order p; it is generated by the image of y_{K} under the Kummer map.

By Lemma 6.3 below, the second condition implies that Gal(K(E[p])/K) ≅ GL_{2}(F_{p}) as well. In particular, E has no K-rational p-torsion.

Note that if y_{K} has infinite order then the conditions of the theorem hold for almost all p (while if y_{K} is torsion, then the conditions of the theorem never hold). In particular, by (5) we have the following corollary.

Corollary 4.2. If y_{K} has infinite order, then E(K) has rank one and � (E/K) has trivial p-primary part for all p satisfying the conditions of the theorem.

In this section we will use Corollary 3.4 to reduce the proof of the theorem to the construction of certain classes in H^{1}(K, E[p]). Before we can apply Corollary 3.4, however, we must compute the relevant Galois cohomology groups.

4.2. Preliminaries. The proof proceeds differently for the different eigenspaces of Sel(K, E[p]) under complex conjugation. Fix a complex conjugation τ ∈ G_{Q} and a prime p as in the theorem. We begin with the following fundamental fact.

Lemma 4.3. dim_{F_{p}} E[p]^{±} = 1.

Proof. Recall that the Weil pairing is a Galois equivariant perfect pairing E[p] ⊗ E[p] → μ_{p}. Since τ acts on μ_{p} as inversion, the Galois equivariance implies that the Weil pairing yields a perfect pairing E[p]^{+} ⊗ E[p]^{−} → μ_{p}. Since at least one of E[p]^{±} is non-zero, the result follows.

The maps of (6) respect the action of complex conjugation; we therefore obtain exact sequences

Sel_{𝔞}(K, E[p])^{±} → ⊕_{v|𝔞}H_{s}^{1}(K_{v}, E[p])^{±} → Sel(K, E[p])^{±}^{∨} → Sel^{𝔞}(K, E[p])^{±}^{∨} → 0. (11)

We will prove the theorem by choosing 𝔞 in such a way that we can compute all of the terms above.

Let l be a prime of Q which is inert in K; set λ = lO_{K}. Then τ yields the non-trivial element of Gal(K_{λ}/Q_{l}), so that we can define a conjugation action of τ on G_{λ} = Gal(K̄_{λ}/K_{λ}) and thus on H^{1}(K_{λ}, E[p]). Furthermore, the action of τ respects H_{f}^{1}(K_{λ}, E[p]) (as always we are using the geometric Selmer structure for E[p]) so that that the eigenspaces H_{s}^{1}(K_{λ}, E[p])^{±} are defined.

Lemma 4.4. Let l be a prime of Q such that:

• E has good reduction at l;

• l ≠ p;

• Frob_{K(E[p])/Q} l is conjugate to τ.

Then H_{s}^{1}(K_{λ}, E[p])^{±} is a one-dimensional F_{p}-vector space.

Proof. Note that l as above is inert in K/Q and splits completely in K(E[p])/K. In particular, E[p] ⊆ E(K_{λ}); it follows by the Weil pairing that μ_{p} ⊆ K_{λ}^{×}. Since the geometric structure at l agrees with the unramified structure and E[p] is unramified at λ, we have

H_{s}^{1}(K_{λ}, E[p]) ≅ H^{1}(I_{λ}, E[p])^{G_{λ}^{ur}} ≅ Hom(I_{λ}, E[p])^{G_{λ}^{ur}}.

By the basic Galois theory of local fields [5, Section 8]

I_{λ}/pI_{λ} ≅ Gal(K_{λ}^{ur}(l^{1/p})/K_{λ}^{ur}) ≅ μ_{p}

as G_{λ}^{ur}-modules. Since G_{λ}^{ur} acts trivially on E[p] and μ_{p} (both are defined over K_{λ}) we conclude that

H_{s}^{1}(K_{λ}, E[p]) ≅ Hom(μ_{p}, E[p]).

This isomorphism respects the action of τ; since τ acts on μ_{p} as inversion, we find that

H_{s}^{1}(K_{λ}, E[p])^{±} ≅ E[p]^{∓}.

Lemma 4.3 now completes the proof.

4.3. The −ε-eigenspace. We now prepare to apply Corollary 3.4. Fix z ∈ E(K̄) with pz = y_{K}. Set K_{0} = Q, L_{0} = K(E[p]) and L = L_{0}(z). By Lemma 4.3, τ is non-scalar on E[p]. Let κ : E(K)/pE(K) → H^{1}(K, E[p]) be the Kummer map. Note that κ(y_{K}) is non-zero (since κ is injective and y_{K} ∉ pE(K) by assumption) and lies in H^{1}(L/K, E[p]). Our first task is to compute the group H^{1}(L/K, E[p]); this is done in the next three lemmas.

Lemma 4.5. H^{i}(L_{0}/K, E[p]) = 0 for all i.

Proof. Since Gal(L_{0}/K) ≅ GL_{2}(F_{p}), the cohomology groups we must calculate are simply H^{i}(GL_{2}(F_{p}), F_{p}^{2}) with the natural action of GL_{2}(F_{p}) on F_{p}^{2}. Let Z = F_{p}^{×} be the normal subgroup of GL_{2}(F_{p}) of scalars; there is a spectral sequence

(12) H^{p}(PGL_{2}(F_{p}), H^{q}(Z, F_{p}^{2})) ⇒ H^{p+q}(GL_{2}(F_{p}), F_{p}^{2}).

Since Z has order p − 1, H^{q}(Z, F_{p}^{2}) = 0 for q > 0; since p ≠ 2, one computes that H^{0}(Z, F_{p}^{2}) = 0 as well. The desired vanishing now follows from (12).

Lemma 4.6. Gal(L/L_{0}) is isomorphic to E[p] as a Gal(L_{0}/K)-module. This isomorphism may not respect the action of τ, but one does have dim_{F_{p}} Gal(L/L_{0})^{±} = 1.

Proof. The cocycle κ(y_{K}) : G_{K} → E[p] is given by σ ↦ σ(z) − z. Thus if we let c : G_{L_{0}} → E[p] be the Gal(L_{0}/K)-equivariant restriction of κ(y_{K}) to G_{L_{0}}, the field L is precisely the kernel of c. We must show that c is surjective.

It follows from Lemma 4.5 that restriction yields an isomorphism

H^{1}(K, E[p]) ≅ Hom(G_{L_{0}}, E[p])^{Gal(L_{0}/K)}.

Since y_{K} ∉ pE(K) we have κ(y_{K}) ≠ 0; thus the homomorphism c is non-zero as well. Since it is Gal(L_{0}/K)-equivariant and Gal(L_{0}/K) acts transitively on E[p], the surjectivity follows.

Lemma 4.7. H^{1}(L/K, E[p]) ≅ F_{p} · κ(y_{K}).

Proof. Lemma 4.5 and the inflation-restriction sequence for the tower L/L_{0}/K yield an isomorphism

H^{1}(L/K, E[p]) ≅ H^{1}(L/L_{0}, E[p])^{Gal(L_{0}/K)}.

Gal(L/L_{0}) acts trivially on E[p], so by Lemma 4.6 this is just the Gal(L_{0}/K)- invariants of Hom(E[p], E[p]). Since Gal(L_{0}/K) ≅ GL_{2}(F_{p}), one computes directly that these invariants consist precisely of the one-dimensional space of scalars in Hom(E[p], E[p]). Thus H^{1}(L/K, E[p]) has dimension one as an F_{p}-vector space. Since κ(y_{K}) is a non-zero element of H^{1}(L/K, E[p]), this completes the proof.

We will give the proof of the next proposition in Section 6.

Proposition 4.8. Assume that y_{K} ∉ pE(K). Let l be a prime of Q which has Frob_{L_{0}/Q} l conjugate to complex conjugation and which does not split completely in L/L_{0}; set λ = lO_{K}. Then there exists a cohomology class c(l) ∈ Sel_{λ}(K, E[p])^{−ε} with c(l)_{λ}^{s} ≠ 0 in H_{s}^{1}(K_{λ}, E[p]).

We can now give the proof of the following portion of the main theorem.

Theorem 4.9. Sel(K, E[p])^{−ε} = 0.

Proof. By Lemma 4.6 we can choose a nontrivial σ ∈ Gal(L/L_{0})^{−}; thus τστ^{−1} = σ^{−1}. We may now apply Corollary 3.4 with S = Sel(K, E[p])^{−ε}. (Note that E[p] is irreducible as a Gal(L_{0}/K)-module since Gal(L_{0}/K) ≅ GL_{2}(F_{p}).) We conclude that there exist primes l_{1}, …, l_{r} of Q with Frob_{L/Q} l_{i} conjugate to τσ and such that

Sel^{𝔞}(K, E[p])^{−ε} ⊆ H^{1}(L/K, E[p])^{−ε};

here 𝔞 = λ_{1} ⋯ λ_{r} with λ_{i} = l_{i}O_{K}. By Lemma 4.7 we know that H^{1}(L/K, E[p]) has dimension one, with generator κ(y_{K}). Since κ commutes with complex conjugation and y_{K} ∈ (E(K)/pE(K))^{ε}, we have κ(y_{K}) ∈ H^{1}(L/K, E[p])^{ε}. Thus

H^{1}(L/K, E[p])^{−ε} = Sel^{𝔞}(K, E[p])^{−ε} = 0.

This takes care of one of the terms in (11).

By Lemma 4.4, the vector space

V = ⊕_{i=1}^{r}H_{s}^{1}(K_{λ_{i}}, E[p])^{−ε}

has dimension r. Since Frob_{L/Q} l_{i} is conjugate to τσ and σ ≠ 1, the primes l_{i} satisfy the conditions of Proposition 4.8. This yields classes c(l_{1}), …, c(l_{r}) ∈ Sel_{𝔞}(K, E[p])^{−ε}. Since the image of c(l_{i}) in V is supported precisely in the onedimensional space H_{s}^{1}(K_{λ}, E[p])^{−ε}, the images of c(l_{1}), …, c(l_{r}) in V are linearly independent. Therefore they span V. Thus the map Sel_{𝔞}(K, E[p])^{−ε} → V is surjective; applying (11) completes the proof.

4.4. The ε-eigenspace. We now consider Sel(K, E[p])^{ε}; the proof here is complicated somewhat by the fact that Sel(K, E[p])^{ε} is manifestly non-trivial, as it contains κ(y_{K}).

We continue with the notation of the previous section. We begin by choosing an auxiliary prime q as in Proposition 4.8; the role it plays will only become clear in Section 6. This yields a class c(q) ∈ Sel_{𝔮}(K, E[p])^{−ε} with c(q)_{𝔮}^{s} ≠ 0, (where 𝔮 is the prime of K above q). Note that κ(y_{K}) and c(q) are linearly independent in H^{1}(K, E[p]) since κ(y_{K})_{𝔮}^{s} = 0. Let L′ be the fixed field of the kernel of the homomorphism G_{L} → E[p] obtained by restricting c(q); L′ is the smallest extension of L such that c(q) ∈ H^{1}(L′/K, E[p]).

Lemma 4.10. Gal(L′/L) is isomorphic to E[p] as a Gal(L/K)-module. Furthermore, we have dim_{F_{p}} Gal(L′/L)^{±} = 1.

Proof. Since c(q) is linearly independent from κ(y_{K}), it follows from inflationrestriction and Lemma 4.7 that the restriction of c(q) to G_{L} is non-zero. From here the proof proceeds as in Lemma 4.6; we omit the details.

Lemma 4.11. H^{1}(L′/K, E[p]) ≅ F_{p} · κ(y_{K}) ⊕ F_{p} · c(q).

Proof. The proof of this is quite similar to the proof of Lemma 4.7: one shows via inflation-restriction for the tower L′/L/K that H^{1}(L′/K, E[p]) has dimension 2 over F_{p}. The elements κ(y_{K}) and c(q) are linearly independent, so they must be a basis.

We give the proof of the next proposition in Section 6. It will require the use of Theorem 4.9.

Proposition 4.12. Assume that y_{K} ∉ pE(K). Let l be a prime of Q which has Frob_{L/Q} l conjugate to complex conjugation and which does not split completely in L′/L; set λ = lO_{K}. Then there exists a cohomology class c(ql) ∈ Sel_{λ}(K, E[p])^{ε} with c(ql)_{λ}^{s} ≠ 0 in H_{s}^{1}(K_{λ}, E[p]).

We can now complete the proof of the main theorem.

Theorem 4.13. Sel(K, E[p])^{ε} = F_{p} · κ(y_{K}).

Proof. By Lemma 4.10 we can choose a nontrivial σ ∈ Gal(L′/L)^{−}. We now apply Corollary 3.4 with S = Sel(K, E[p])^{ε} and the tower of fields L′/L/K/Q. We conclude that there exist primes l_{1}, …, l_{r} of Q such that Frob_{L′/Q} l_{i} is conjugate to τσ and such that

Sel^{𝔞}(K, E[p])^{ε} ⊆ H^{1}(L′/K, E[p])^{ε};

here 𝔞 = λ_{1} ⋯ λ_{r} with λ_{i} = l_{i}O_{K}.

By Lemma 4.11 we know that H^{1}(L′/K, E[p]) has dimension two, with basis κ(y_{K}) and c(q). However, c(q) lies in the −ε-eigenspace, so we conclude that Sel_{𝔞}(K, E[p])^{ε} has dimension at most one.

As before,

V = ⊕_{i=1}^{r}H_{s}^{1}(K_{λ_{i}}, E[p])^{ε}

has dimension r. Proposition 4.12 implies that we have r elements

c(ql_{1}), …, c(ql_{r}) ∈ Sel_{𝔞}(K, E[p])^{ε}

whose images span V. Thus the map Sel_{𝔞}(K, E[p])^{ε} → V is surjective.

We conclude by (11) that

Sel(K, E[p])^{ε} ≅ Sel_{𝔞}(K, E[p])^{ε}

and that these are F_{p}-vector spaces of dimension at most one. Since κ(y_{K}) is a non-zero element of Sel(K, E[p])^{ε}, the theorem follows.

It is perhaps worth noting that we can also prove Theorem 4.13 with the ideal 𝔞𝔮 = λ_{1} ⋯ λ_{r}𝔮; one then has Sel^{𝔞𝔮}(K, E[p])^{ε} = 0, but now the cokernel of

Sel_{𝔞𝔮}(K, E[p])^{ε} → ⊕_{v|𝔞𝔮}H_{s}^{1}(K_{λ}, E[p])^{ε}

has dimension 1.

It remains, then, to prove Propositions 4.8 and 4.12. This will be done via Heegner points.

5. Heegner points

5.1. Ring class fields. The class field theory of the field Q is fairly straightforward: all abelian extensions of Q are contained in cyclotomic extensions of Q. An imaginary quadratic field K, however, has two kinds of abelian extensions: the cyclotomic extensions (which are still abelian over Q) and the anti-cyclotomic extensions (which are not abelian over Q). These anti-cyclotomic extensions are the ring class fields of K.

For proofs of the assertions of this section see [2, Section 9]. Fix an imaginary quadratic field K. For any integer n let O_{n} = Z + nO_{K} be the order of conductor n in O_{K}. Let I(n) denote the group of fractional ideals of O_{K} relatively prime to n and let P (n) be the subgroup generated by principal ideals αO_{K} where α ∈ O_{K} is congruent to a rational integer modulo nO_{K}. It is an elementary fact that the ideal class group Pic(O_{n}) is isomorphic to the quotient I(n)/P (n). This is a generalized ideal class group for K, so class field theory yields an abelian extension K_{n}/K, unramified away from n, such that there is an isomorphism

Pic(O_{n}) ⟶^{≃} Gal(K_{n}/K) (13)

sending a prime ideal λ ∈ I(n) to its Frobenius element. This K_{n} is the ring class field of K of conductor n. Note that if l is a prime of Q, relatively prime to n, which is inert in K, then λ = lO_{K} lies in P (n), so it splits completely in K_{n}/K. As we have said, K_{n} is not abelian over Q: the conjugation action of the non-trivial τ ∈ Gal(K/Q) sends σ ∈ Gal(K_{n}/K) to σ^{−1}.

K_{1} is simply the Hilbert class field of K. We will write G_{n} for Gal(K_{n}/K_{1}); one computes from the definitions that there is a canonical isomorphism

(14) G_{n} ≅ Pic(O_{n})/ Pic(O_{K}) ≅ (O_{K}/nO_{K})^{×}/(Z/nZ)^{×}.

In particular, for an odd prime l which is unramified in K/Q, G_{l} is cyclic and

[K_{l} : K_{1}] = \begin{cases}l − 1 & l splits in K; \\ l + 1 & l inert in K. \end{cases}

In this case K_{l}/K_{1} is totally ramified as well. (We should at least comment that the notation G_{l} could in principle conflict with our earlier notation for Gal(Q̄_{l}/Q_{l}). Since this latter group will never appear below, this should cause no confusion.)

Suppose that n is squarefree. By ramification considerations (or else the Chinese remainder theorem applied to (14)) we find that there is a canonical isomorphism

G_{n} ≅ ∏_{l|n}G_{l} (15)

which will be useful later.

5.2. Complex multiplication. Let N be a positive integer and consider the modular curve X_{0}(N). (See [14] and the references given there for more details on modular curves.) Recall that the non-cuspidal C-valued points of X_{0}(N) classify isomorphism classes of cyclic N-isogenies E → E′ of elliptic curves over C. (A cyclic N-isogeny is simply a map of algebraic curves which respects the group laws and has kernel cyclic of order N.) X_{0}(N) is a smooth projective curve which admits a model over Q and has good reduction at primes not dividing N. If K is a number field, the non-cuspidal K-points of X_{0}(N) correspond to cyclic N-isogenies E → E′ defined over K, but only up to isomorphism over K̄; this subtlety can be safely ignored below.

We now fix a quadratic imaginary field K such that every prime dividing N splits completely in K/Q. It follows that we can choose an ideal N of O_{K} such that O_{K}/N ≅ Z/NZ. Let D denote the discriminant of K/Q.

Let n be an integer which is relatively prime to ND and consider the order O_{n} in K. The ideal N_{n} = N ∩ O_{n} still satisfies O_{n}/N_{n} ≅ Z/NZ. We can therefore consider the cyclic N-isogeny

C/O_{n} → C/N_{n}^{−1}

of elliptic curves over C induced by the identity map on C. We denote the corresponding C-point of X_{0}(N) by x_{n}; it is the Heegner point of conductor n for K. (Note that the point x_{n} depends on the choice of ideal N, so it is not entirely canonical.)

The theory of complex multiplication (see [12, Chapter 2] and the references given there for more details) yields the following information on the action of Aut(C/K) on x_{n}: consider the restriction of σ ∈ Aut(C/K) to Gal(K_{n}/K) with K_{n} the ring class field of K of conductor N. σ|_{K_{n}} corresponds to an ideal class in Pic(O_{n}) under the isomorphism (13); let a_{σ} be any ideal in this class. Then the cyclic N-isogeny corresponding to σx_{n} is

C/𝔞_{σ}^{−1} → C/N_{n}^{−1}𝔞_{σ}^{−1}.

It follows in particular that x_{n} is a K_{n}-rational point of X_{0}(N).

For any field L ⊆ K̄ we let Div X_{0}(N)(L) denote the group of divisors on X_{0}(N)(K̄) which are stable under the action of Gal(K̄/L). For a prime l not dividing N we have the l^{th} Hecke correspondence

T_{l} : Div X_{0}(N)(L) → Div X_{0}(N)(L)

sending a cyclic N-isogeny E → E′ to the formal sum

∑_{C⊆E[l]}(E/C → E′/C)

of l + 1 cyclic N-isogenies.

Let n be an integer relatively prime to ND and let l be a prime not dividing nND. There is a natural trace map

Tr_{l} : Div X_{0}(N)(K_{nl}) → Div X_{0}(N)(K_{n})

sending a point of X_{0}(N)(K_{nl}) to the formal sum of its Gal(K_{nl}/K_{n})-conjugates. It follows from the main theorem of complex multiplication as given above and the definition of the Hecke correspondence that Tr_{l} x_{nl} = T_{l}x_{n}.

Heegner points also satisfy a certain congruence which we now explain. If L/Q is any extension and λ is a prime of L with residue field F_{λ}, we write

red_{λ} : E(L_{λ}) → E(F_{λ})

for reduction modulo λ.

Let l be inert in K and set λ = lO_{K}; λ splits completely in K_{n}/K and is totally ramified in K_{nl}/K_{n}. Let λ_{n} be a prime of K_{n} above λ and let λ_{nl} be the unique prime of K_{nl} over λ_{n}. The residue fields F_{λ_{n}} and F_{λ_{nl}} at these primes are canonically isomorphic to the residue field F_{λ} = O_{K}/λ; F_{λ} is isomorphic to F_{l^{2}}, although not canonically.

We consider the image of the Heegner points x_{n}, x_{nl} under the reduction maps red_{λ_{n}} and red_{λ_{nl}}; note that both maps land in E(F_{λ}). The congruence (which is a fairly direct consequence of the Eichler-Shimura relation) is

red_{λ_{nl}}(x_{nl}) = red_{λ_{n}}(Frob_{K_{n}/K} λ_{n} · x_{n})).

See [6, Proposition 3.7] for details.

We summarize these results in the following proposition.

Proposition 5.1. Let K be as above. For every integer n relatively prime to ND there is a point x_{n} ∈ X_{0}(N)(K_{n}) such that:

• If l is a prime not dividing nND, then we have an equality Tr_{l} x_{nl} = T_{l}x_{n} of divisors on X_{0}(N)(K_{nl});

• If l is a prime not dividing nND which is inert in K, then we have

red_{λ_{nl}}(x_{nl}) = red_{λ_{n}}(Frob_{K_{n}/K} λ_{n} · x_{n}).

for any prime λ_{n} of K_{n} over λ.

5.3. Heegner points on elliptic curves. Let E be an elliptic curve of conductor N and let

φ : X_{0}(N) → E

be a modular parameterization of E; φ corresponds to a normalized newform f = ∑ a_{n}q^{n} on Γ_{0}(N). Recall that for a prime l not dividing N we have

a_{l} = l + 1 − #E(F_{l}).

Let ε be the negative of the sign of the functional equation of E over Q; it is also the eigenvalue of f for the Atkin-Lehner involution w_{N}.

For any field L, φ induces a map

X_{0}(N)(L) → E(L)

and we define the Heegner points of E by

y_{n} = φ(x_{n}) ∈ E(K_{n}).

The translation of Proposition 5.1 is the following. Note that we can regard the trace map Tr_{l} as a map E(K_{nl}) → E(K_{n}) (rather than as a map of divisors) by applying the group law on E.

Proposition 5.2. Let E be an elliptic curve of conductor N (endowed with a fixed modular parameterization as above) and let K be an imaginary quadratic field of discriminant D in which every prime dividing N splits. Then for any integer n not dividing ND there is a point y_{n} ∈ E(K_{n}) such that:

• If l is a prime not dividing nND, then Tr_{l} y_{nl} = a_{l}y_{n} in E(K_{n});

• If l is a prime not dividing nND which is inert in K, then we have

red_{λ_{nl}}(y_{nl}) = red_{λ_{n}}(Frob_{K_{n}/K} λ_{n} · y_{n})

for any prime λ_{n} of K_{n} over λ. Furthermore, if τ is a complex conjugation, then there exists σ ∈ Gal(K_{n}/K) such that τy_{n} = εσy_{n} in E(K)/E(K)_{tors}.

Proof. The fact that φ ∘ T_{l} = a_{l}φ is a consequence of Eichler-Shimura theory; see [3, Section 13] for a discussion. Given this, the first two statements follow from Proposition 5.1. The last statement follows from the behavior of x_{n} under the Atkin-Lehner involution w_{N}; see [6, Proposition 5.3].

We conclude this section by defining the basic Heegner point y_{K}. It is simply the image of y_{1} ∈ E(K_{1}) under the trace map

Tr_{K_{1}/K} : E(K_{1}) → E(K).

We let complex conjugation act on E(K)/pE(K) in the usual way.

Lemma 5.3. Assume that E(K) has no p-torsion. Then y_{K} ∈ (E(K)/pE(K))^{ε}.

Proof. Since E(K) has no p-torsion, Proposition 5.2 implies that there is a σ ∈ Gal(K_{1}/K) such that τy_{1} = εσy_{1} ∈ E(K)/pE(K). Let Tr denote the trace from K_{1} to K. We have

Tr τy_{1} = ε Tr σy_{1} ∈ E(K)/pE(K). (16)

Since the conjugation action of τ on Gal(K_{1}/K) is inversion and Tr is stable under this operation, we have Tr τ = τ Tr. Since σ ∈ Gal(K_{1}/K), we also have Tr σ = Tr. Thus (16) becomes

τ Tr y_{1} = ε Tr y_{1} ∈ E(K)/pE(K),

which is the statement of the lemma.

6. The Euler system

6.1. Kolyvagin’s derivative operator. We now fix an elliptic curve E of conductor N, a modular parameterization X_{0}(N) → E, an imaginary quadratic field K and a prime p as in Section 4. This data defines a basic Heegner point y_{K} ∈ E(K) and Heegner points y_{n} ∈ E(K_{n}) for every n relatively prime to ND, with D the discriminant of K/Q. We are also assuming that E has good reduction at p and that Gal(Q(E[p])/Q) ≅ GL_{2}(F_{p}). We will not invoke the assumption y_{K} ∉ pE(K) until the very end of the argument.

We will now convert the Heegner points y_{n} ∈ E(K_{n})/pE(K_{n}) into cohomology classes in H^{1}(K, E[n]). The most brutal way to do this is to simply take the trace of y_{n} from K_{n} down to K and then apply the Kummer map. This approach, however, does not yield interesting cohomology classes. Instead we will apply a certain derivative operator to the y_{n} to obtain Gal(K_{n}/K_{1})-invariant elements. Applying the trace from K_{1} to K will yield the desired cohomology classes.

Let R be the set of squarefree integers which are relatively prime to pND and which are products of primes l such that Frob_{K(E[p])/Q} l is conjugate to complex conjugation. Note that such an l is inert in K/Q and splits completely in K(E[p])/K. We will only need to consider Heegner points y_{n} for n ∈ R.

Lemma 6.1. For every prime l ∈ R, p divides l + 1 and a_{l} = l + 1 − #E(F_{l}).

Proof. The characteristic polynomial of Frob l on E[p] is x^{2} − a_{l}x + l and the characteristic polynomial of complex conjugation on E[p] is x^{2} − 1. Since Frob l is conjugate to complex conjugation on E[p], these polynomials must be congruent modulo p; the lemma follows.

For a prime l ∈ R, the group G_{l} = Gal(K_{l}/K_{1}) is cyclic of order l + 1. Fix a generator σ_{l} and define operators

D_{l} = ∑_{i=1}^{l} iσ_{l}^{i}, Tr_{l} = ∑_{i=0}^{l} σ_{l}^{i} ∈ Z[G_{l}].

Tr_{l} is just the trace operator for K_{l}/K_{1}. These are related by the telescoping identity

(σ_{l} − 1)D_{l} = l + 1 − Tr_{l} . (17)

For general n ∈ R, we have G_{n} ≅ ∏_{l|n} G_{l}, and we define

D_{n} = ∏_{l|n}D_{l} ∈ Z[G_{n}].

We will need two more operators. Fix a set of coset representatives for the subgroup Gal(K_{n}/K_{1}) of Gal(K_{n}/K); let Tr (resp. Tr^{−1}) be the sum of these representatives (resp. of their inverses). If M is a Z[Gal(K_{n}/K)]-module and m ∈ M^{G_{n}}, then Tr m lies in M^{Gal(K_{n}/K)} and is independent of the choice of coset representatives above; in particular, Tr m = Tr^{−1} m.

We can regard all of the above operators as elements of the commutative group ring Z[Gal(K_{n}/K)]. These operators do not commute with complex conjugation τ; indeed, since conjugation by τ is inversion on Gal(K_{n}/K), one finds, for example, that

τ Tr = Tr^{−1} τ; τD_{l} = l Tr_{l} −σ_{l}D_{l}τ. (18)

6.2. Derived cohomology classes. There is a natural action of the group ring Z[Gal(K_{n}/K)] on E(K_{n})/pE(K_{n}). The following lemma is the main step in the construction of our desired cohomology classes.

Lemma 6.2. For any n ∈ R we have D_{n}y_{n} ∈ (E(K_{n})/pE(K_{n}))^{G_{n}}.

Proof. Since G_{n} is generated by the σ_{l} for l dividing n, the statement of the lemma is equivalent to the statement that

(σ_{l} − 1)D_{n}y_{n} ∈ pE(K_{n})

for all l dividing n. Using (17) we compute that

(σ_{l} − 1)D_{n}y_{n} = D_{n/l}(σ_{l} − 1)D_{l}y_{n}
= D_{n/l}(l + 1 − Tr_{l})y_{n}

in E(K_{n}). By Proposition 5.2 this equals D_{n/l}((l + 1)y_{n} − a_{l}y_{n/l}). By Lemma 6.1 this lies in pE(K_{n}), which completes the proof.

We now define

P_{n} = Tr D_{n}y_{n} ∈ E(K_{n});

by Lemma 6.2, the image of P_{n} in E(K_{n})/pE(K_{n}) is Gal(K_{n}/K)-invariant and is independent of the choice of Tr. We will use the next two lemmas to produce an element of H^{1}(K, E[p]) from P_{n}.

Lemma 6.3. E has no K_{n}-rational p-torsion for any n ∈ R.

Proof. This is basically the statement that the groups GL_{2}(F_{p}) and Gal(K_{n}/Q) are not very compatible; see [6, Lemma 4.3] for details.

Lemma 6.4. The restriction map

res : H^{1}(K, E[p]) → H^{1}(K_{n}, E[p])^{Gal(K_{n}/K)}

is an isomorphism.

Proof. By inflation-restriction the kernel and cokernel of this map are the groups H^{i}(K_{n}/K, E[p]^{Gal(K_{n}/K)}) for i = 1, 2 respectively. Both of these groups vanish by Lemma 6.3, so the restriction map is an isomorphism.

Now consider the image of P_{n} under the Kummer map

κ : E(K_{n})/pE(K_{n}) → H^{1}(K_{n}, E[p]).

The Kummer map is Galois equivariant, so that κ(P_{n}) ∈ H^{1}(K_{n}, E[p])^{Gal(K_{n}/K)}, and we define c(n) ∈ H^{1}(K, E[p]) to be the unique class such that res c(n) = κ(P_{n}).

These are the cohomology classes which we need to complete our proof. We begin our investigation of them by determining how they behave under complex conjugation.

Lemma 6.5. Let n ∈ R have k prime factors. Then c(n) ∈ H^{1}(K, E[p])^{(−1)^{k}ε}.

Proof. Since κ and res respect the action of complex conjugation, it suffices to show that τP_{n} = (−1)^{k}εP_{n} in E(K_{n})/pE(K_{n}). In the calculations below we always work in E(K_{n})/pE(K_{n}). By (18) we have

τP_{n} = τ Tr ∏_{l|n}D_{l}y_{n} = Tr^{−1} ∏_{l|n}(l Tr_{l} −σ_{l}D_{l})τy_{n}.

By Proposition 5.2 there is a σ ∈ Gal(K_{n}/K) such that τy_{n} = (−1)^{k}εσy_{n} in E(K_{n})/pE(K_{n}); thus

τP_{n} = (−1)^{k}εσ Tr^{−1} ∏_{l|n}(l Tr_{l} −σ_{l}D_{l})y_{n}.

For each l we have Tr_{l} y_{n} = a_{l}y_{n/l}; by Proposition 5.2 this is zero in E(K_{n})/pE(K_{n}). It follows that only one term survives in the above product and

τP_{n} = (−1)^{k}εσ(∏_{l|n}σ_{l}) Tr^{−1} D_{n}y_{n}. (19)

But D_{n}y_{n} is G_{n}-invariant, so that Tr^{−1} D_{n}y_{n} = Tr D_{n}y_{n} = P_{n}. Since P_{n} is Gal(K_{n}/K)-invariant, the other group operations in (19) are trivial; we conclude that τP_{n} = (−1)^{k}εP_{n}, as claimed.

6.3. Ramification of the derived classes. We begin with the fact that the class c(n) is unramified away from n.

Lemma 6.6. Fix n = l_{1} ⋯ l_{r} ∈ R and set λ_{i} = l_{i}O_{K}. Then

c(n) ∈ Sel_{λ_{1}···λ_{r}}(K, E[p]).

Proof. Let v be a place of K distinct from λ_{1}, …, λ_{r}; we must show that c(n)_{v} ∈ H_{f}^{1}(K_{v}, E[n]), or equivalently that c(n)_{v}^{s} = 0 in H_{s}^{1}(K_{v}, E[n]). This is straightforward if E has good reduction at v. In this case we have H_{s}^{1}(K_{v}, E[p]) = Hom(I_{v}, E[p])^{G_{v}^{ur}}, so that c(n)_{v}^{s} is zero exactly when the restriction of c(n) to I_{v} is trivial.

Let w be a place of K_{n} over v. Since v does not divide n, K_{n,w}/K_{v} is unramified. In particular, the inertia group of K_{n,w} is also I_{v}. We therefore have a commutative diagram

[figure: H^{1}(K_{v}, E[p]); res; Hom(I_{v}, E[p]); E(K_{n,w})/pE(K_{n,w}); κ; H^{1}(K_{n,w}, E[p]); Hom(I_{v}, E[p])]

in which the bottom row is exact. (Both rows are just portions of the exact sequence (3) over K_{v} and K_{n,w}, respectively.) We know that res c(n)_{v} = κ(P_{n}), so that exactness of the bottom row shows that (res c(n)_{v})^{s} is trivial. But then c(n)_{v}^{s} is trivial as well, as claimed.

The proof at places v of bad reduction for E is somewhat more subtle and involves an analysis of the Néron model of E. We refer to [6, Proposition 6.2] for the details.

We now turn to the local behavior of a class c(n) at primes dividing n. The key computation is given in Lemma 6.7 below; we first set some notation. Let nl ∈ R with l prime and set λ = lO_{K}. Since P_{nl} ∈ E(K_{nl}) is Gal(K_{nl}/K)-invariant in E(K_{nl})/pE(K_{nl}), we know that (σ_{l} − 1)P_{nl} ∈ pE(K_{nl}). Since also E(K_{nl}) has no p-torsion, this implies that there is a unique Q_{n,l} ∈ E(K_{nl}) with pQ_{n,l} = (σ_{l}−1)P_{nl}. Indeed, it is clear from the proof of Lemma 6.2 that Q_{n,l} is given by the formula

Q_{n,l} = \frac{l + 1}{p} Tr D_{n}y_{nl} − \frac{a_{l}}{p} P_{n}; (20)

this makes sense since l + 1 and a_{l} are divisible by p.

Recall that λ splits completely in K_{n}/K and is totally ramified in K_{nl}/K_{n}. For each prime λ_{n} of K_{n} over λ, we write λ_{nl} for the unique prime of K_{nl} over λ_{n}. We will consider the image of Q_{n,l} under the reduction map

red_{λ_{nl}} : E(K_{nl,λ_{nl}}) → E(F_{λ});

recall that F_{λ} is the residue field of each of λ, λ_{n} and λ_{nl}. Note that we also have K_{n,λ_{n}} = K_{λ}, so that we can consider P_{n} as an element of E(K_{λ}).

Lemma 6.7. red_{λ_{nl}}(Q_{n,l}) is trivial in E(F_{λ}) if and only if P_{n} ∈ pE(K_{λ}).

Proof. We first show that

(21) red_{λ_{n}}(Tr D_{n}y_{nl}) = red_{λ_{n}}(Frob_{K_{n}/K} λ_{n} · Tr D_{n}y_{n}).

This reduces immediately to showing that

(22) red_{λ_{n}}(σy_{nl}) = red_{λ_{n}}(Frob_{K_{n}/K} λ_{n} · σy_{n})

for any σ ∈ Gal(K_{n}/K). Note that the case σ = 1 is given in Proposition 5.2. In general, we begin with the congruence of Proposition 5.2 for the ideal σ^{−1}λ_{n}:

red_{σ^{−1}λ_{nl}}(y_{nl}) = red_{σ^{−1}λ_{n}}(Frob_{K_{n}/K}(σ^{−1}λ_{n}) · y_{n}).

Since red_{σ^{−1}λ_{n}} = red_{λ_{n}} ∘σ, this is equivalent to

red_{λ_{nl}}(σy_{nl}) = red_{λ_{n}}(σ Frob_{K_{n}/K}(σ^{−1}λ_{n}) · y_{n}).

But Frob_{K_{n}/K}(σ^{−1}λ_{n}) = σ^{−1} Frob_{K_{n}/K}(λ_{n})σ, which now completes the proof of (22) and thus of (21) as well.

Combining (21) with (20) and the fact that Tr D_{n}y_{n} = P_{n}, we conclude that

(23) red_{λ_{nl}}(Q_{n,l}) = red_{λ_{n}} ((\frac{l + 1}{p} Frob −\frac{a_{l}}{p} ) P_{n})

where we now simply write Frob for Frob_{K_{n}/K} λ_{n}.

We claim that (l + 1) Frob −a_{l} annihilates E(F_{λ}). To see this, recall that F_{λ} ≅ F_{l^{2}}; thus Frob is an involution and we have a decomposition E(F_{λ}) = E(F_{λ})^{+} ⊕ E(F_{λ})^{−}. We have E(F_{λ})^{+} = E(F_{l}), which has order l + 1 − a_{l} by definition. The Weil conjectures for elliptic curves imply that E(F_{λ})^{−} then has order l + 1 + a_{l}. In either case, we see that ±(l + 1) − a_{l} annihilates E(F_{λ})^{±}, so that (l + 1) Frob −a_{l} annihilates E(F_{λ}).

Since Frob is the reduction of a complex conjugation, by the proof of Lemma 6.5 we have Frob P_{n} = νP_{n} + pQ for some ν ∈ {±1} and some Q ∈ E(K_{n}). Since (l + 1) Frob −a_{l} annihilates E(F_{λ}), it follows that

red_{λ_{nl}}(Q_{n,l}) = \frac{(l + 1)ν − a_{l}}{p} red_{λ_{n}}(P_{n})

in E(F_{λ})^{ν}. Since E(F_{λ})^{ν} has order equal to the absolute value of (l + 1)ν − a_{l}, it follows that red_{λ_{nl}}(Q_{n,l}) = 0 if and only if red_{λ_{n}}(P_{n}) ∈ pE(K_{n,λ_{n}}). Since K_{λ} = K_{n,λ_{n}} and the kernel of the map E(K_{λ}) → E(F_{λ}) is pro-l, this is equivalent to P_{n} ∈ pE(K_{λ}), as claimed.

A consequence of this lemma is the following result which is the key to the entire proof. It relates the ramification of c(n) at a prime l dividing n to the behavior of the local divisibility of the point P_{n/l} at l.

Lemma 6.8. Let nl ∈ R with l prime and set λ = lO_{K}. Then c(nl)_{λ}^{s} = 0 if and only if P_{n} ∈ pE(K_{λ}).

Proof. Since H_{s}^{1}(K_{λ}, E[p]) = Hom(I_{λ}, E[p]), we can regard c(nl)_{λ}^{s} as a homomorphism I_{λ} → E[p]. We claim that it factors through I_{λ}/I_{λ_{nl}} for any prime λ_{nl} of K_{nl} above λ. To see this, let res : H^{1}(K, E[p]) → H^{1}(K_{nl}, E[p]) be the restriction map. By definition res c(nl) lies in the image of the Kummer map, so that res c(nl) ∈ Sel(K_{nl}, E[p]). In particular, c(nl)_{λ_{nl}}^{s} = 0. Since H_{s}^{1}(K_{nl,λ_{nl}}, E[p]) equals Hom(I_{λ_{nl}}, E[p]), we see that c(nl)(I_{λ_{nl}}) = 0, as claimed.

Since G_{l} is the inertia group of Gal(K_{nl}/K) at λ, we have I_{λ}/I_{λ_{nl}} ≅ G_{l}. In particular, we now see that c(nl)_{λ}^{s} = 0 if and only if c(nl)(σ_{l}) = 0.

We now need a formula for c(nl). Fix Q ∈ E(K̄) with pQ = P_{nl}. We claim that it is represented by the cocycle G_{K} → E[p] given by

σ ↦ σQ − Q − \frac{1}{p}(σ − 1)P_{nl} ∈ E[p].

Here we know that (σ − 1)P_{nl} ∈ pE(K_{nl}) by Lemma 6.2 and \frac{1}{p}(σ − 1)P_{nl} is the unique (by Lemma 6.3) p^{th} root in E(K_{nl}). It is easy to see that the expression above is in E[p], and to check that it represents c(nl) one simply needs to check that res c(nl) = κ(P_{nl}); this is clear since (σ − 1)P_{nl} = 0 for σ ∈ G_{K_{nl}}.

Note that c(nl)(σ_{l}) equals

σ_{l}Q − Q − Q_{n,l}. (24)

Fix a prime λ̄ of K̄ over λ and consider the corresponding reduction map

red : E(K̄) → E(k̄_{λ}).

Since E has good reduction at p, we know that this map is injective on p-torsion. In particular, c(nl)(σ_{l}) = 0 if and only if the reduction of (24) is trivial.

Note that red(σ_{l}Q − Q) and red(Q_{n,l}) both lie in E[p] even though neither point does prior to reduction. In fact, red(σ_{l}Q − Q) = 0 since σ_{l} lies in inertia and thus acts trivially on the residue fields. Thus

red(σ_{l}Q − Q − Q_{n,l}) = red(Q_{n,l}).

We conclude that c(nl)(σ_{l}) = 0 (and thus c(nl)_{λ}^{s} = 0) if and only if red(Q_{n,l}) = 0. Lemma 6.7 now completes the proof.

6.4. Final considerations. We are now in a position to prove Propositions 4.8 and 4.12. Recall that L_{0} = K(E[p]) and L = K(E[p], \frac{1}{p}y_{K}).

Proposition. Assume that y_{K} ∉ pE(K). Let l be a prime of Q with Frob_{L_{0}/Q} l conjugate to complex conjugation and which does not split completely in L/L_{0}; set λ = lO_{K}. Then there exists a cohomology class c(l) ∈ Sel_{λ}(K, E[p])^{−ε} with c(l)_{λ}^{s} ≠ 0 in H_{s}^{1}(K_{λ}, E[p]).

Proof. We consider the class c(l) defined above via Heegner points. The fact that c(l) ∈ Sel_{λ}(K, E[p])^{−ε} follows from Lemmas 6.6 and 6.5. It remains to check that c(l)_{λ}^{s} ≠ 0 in H_{s}^{1}(K_{λ}, E[p]). By Lemma 6.8 this is true if and only if P_{1} ∉ pE(K_{λ}). Note that P_{1} ∈ E(K) is precisely the point y_{K}. Since L is the minimal extension of L_{0} in which y_{K} is globally divisible by p, y_{K} is divisible by p in E(K_{λ}) precisely if λ splits completely in L/L_{0}. Since we are assuming that λ does not split completely in L/L_{0}, we have y_{K} ∉ pE(K_{λ}), which implies that c(l)_{λ}^{s} ≠ 0.

For the next result, recall that we have chosen a non-trivial c(q) ∈ H^{1}(K, E[p]) and we let L′/L be the minimal extension over which c(q) is defined. We need the following lemma.

Lemma 6.9. Let l be a prime with Frob_{L/Q} l conjugate to complex conjugation; set λ = lO_{K}. Then P_{q} ∈ pE(K_{λ}) if and only if l splits completely in L′/L.

Proof. We show that both conditions are equivalent to the vanishing of c(q)_{λ} in H^{1}(K_{λ}, E[p]). Note first that if l has Frob_{L/Q} l conjugate to τ, then λ splits completely in L. In particular, L_{λ_{L}} = K_{λ} where λ_{L} is any prime of L above λ. Fix such an l and λ_{L} and let λ_{L′} be a prime of L′ above λ_{L}.

Recall that c(q)|_{G_{L}} yields an isomorphism c : Gal(L′/L) ⟶^{≃} E[p]. c(q)_{λ} is therefore trivial precisely when Gal(L′_{λ_{L′}} /L_{λ_{L}}) is trivial. This in turn is the same as λ_{L} splitting completely in L′, as claimed.

For the other equivalence, recall that λ splits completely in the ring class field K_{q}. In particular, the restriction map H^{1}(K, E[p]) → H^{1}(K_{λ}, E[p]) factors through H^{1}(K_{q}, E[p]). We therefore have a commutative diagram

\xymatrix{ & H^{1}(K, E[p]) \ar[d] \\ E(K_{q})/pE(K_{q}) \ar@{^{(}->}[r] \ar[d] & H^{1}(K_{q}, E[p]) \ar[d] \\ E(K_{λ})/pE(K_{λ}) \ar@{^{(}->}[r] & H^{1}(K_{λ}, E[p])}

It now follows from the definitions that c(q)_{λ} is the image of P_{q} ∈ E(K_{λ})/pE(K_{λ}) under the Kummer map. The second asserted equivalence follows.

Proposition. Assume that y_{K} ∉ pE(K). Let l be a prime of Q which has Frob_{L/Q} l conjugate to complex conjugation and which does not split completely in L′/L; set λ = lO_{K}. Then there exists a cohomology class c(ql) ∈ Sel_{λ}(K, E[p])^{ε} with c(ql)_{λ}^{s} ≠ 0 in H_{s}^{1}(K_{λ}, E[p]).

Proof. By Lemmas 6.6 and 6.5 we have c(ql) ∈ Sel_{𝔮λ}(K, E[p])^{ε}. The fact that c(ql)_{λ}^{s} ≠ 0 follows immediately from Lemmas 6.9 and 6.8.

It remains to check that c(ql)_{𝔮}^{s} = 0. Consider the class c(l) ∈ Sel_{λ}(K, E[p])^{−ε}. Since l splits completely in L/L_{0}, we have y_{K} ∈ pE(K_{λ}) and thus by Lemma 6.8 we have c(l)_{λ}^{s} = 0. Thus c(l) ∈ Sel(K, E[p])^{−ε}. But this group is zero by Theorem 4.9. Thus c(l) = 0. By the construction of c(l) this implies that P_{l} ∈ pE(K_{l}). But then certainly P_{l} ∈ pE(K_{l,𝔮}), so that by Lemma 6.8 we have c(ql)_{𝔮}^{s} = 0, as claimed.

7. Examples

In this section we give some examples for the elliptic curve

y^{2} + y = x^{3} − x^{2} − 10x − 20,

(otherwise known as X_{0}(11); see [14]) and the imaginary quadratic field K = Q(\sqrt{−7}). Note that K has class number 1, so that K_{1} = K. E has bad reduction only at 11, and it is known that Gal(Q(E[p])/Q) ≅ GL_{2}(F_{p}) for p ≠ 5.

Set α = \frac{1+\sqrt{−7}}{2}. In the table below we give the minimal polynomials (over K) of the Heegner points x_{n} = y_{n} for 1 ≤ n ≤ 4. (We suppress the various choices required to define the Heegner points.)

n x-coordinate of y_{n}  y-coordinate of y_{n}
1 X − α  X + (4 − 4α)
2 X + (14 − 17α)  X + (−120 + 34α)
3 X^{4} + (93 − 46α)X^{3} + ; (−1530 − 519α)X^{2} + ; (−1816 + 5545α)X + ; (1943 − 14460α)  X^{4} +(−347−475α)X^{3} + ; (29550 − 38190α)X^{2} + ; (−95394 − 782135α)X + ; (−6593671 + 715920α)

4 {X^{2} + (216 − 290α)X + \atop (−1649 + 3745ω)} {X^{2} + (8414 − 2690α)X + \atop (304382 − 191230α)}

In particular, the basic Heegner point y_{K} is (\frac{1+\sqrt{−7}}{2}, −2 + 2\sqrt{−7}). It can be shown (via height computations) that y_{K} has infinite order and is not divisible by any odd primes in E(K). In particular, Theorem 4.1 thus shows that E(K) has rank 1 and that � (E/K) has trivial p-primary part for p ≠ 2, 5, 11.

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[14] T. Weston, The modular curves X_{0}(11) and X_{1}(11), available from the author.

Department of Mathematics, University of Michigan, 48109-1109

E-mail address: weston@math.lsa.umich.edu