1 Introducing �

For an elliptic curve E defined over a number field K, let � (E/K) = ker(H^{1}(K, E) → ∏_{v} H^{1}(K_{v}, E)) be the Shafarevich-Tate group of E/K. Recall that H^{1}(K, E) = \underrightarrow{lim} H^{1}(L/K, E), where the direct limit is taken over all finite Galois extensions L/K; since each one of these groups is torsion, killed by [L : K], so is H^{1}(K, E). Therefore, � (E/K) is a torsion group, a fact we will later to use to conclude that � (E/K) = 0 for curves having � (E/K)[p] = 0 for all primes p. In what follows, we will obtain results of the form � (E/K)[p] = 0 for almost all p, where [K : Q] = 2. The canonical map � (E/Q) → � (E/K) has kernel contained in H^{1}(K/Q, E), a finite abelian 2-group, so � (E/K)[p] = 0 ⇒ � (E/Q)[p] = 0 as long as p ≠ 2. The difficulty in studying the Shafarevich-Tate group is the major obstacle to progress on the Birch and Swinnerton-Dyer conjecture:

Conjecture 1.1. Let E be an elliptic curve defined over Q. Then the order of vanishing at s = 1 of L(EQ, s)equals the rank of E(Q). More precisely, letting r = rk_{Z}E(Q), and phrasing the conjecture in terms of #� (E/Q),

#� (E/Q) = \frac{L^{r}(E/Q, 1)|E(Q)_{tors}|^{2}}{r!R(E/Q) ∏_{v} c_{v}(E)}

R(E/Q) denotes the elliptic regulator, the determinant of the height pairing matrix on a set of free generators. The c_{v} for finite places are the Tamagawa numbers, measuring bad reduction at v (in particular, they are 1 when E has good reduction at v), and c_{∞} = ∫_{E(R)} |ω|, where ω is the invariant differential \frac{dx}{2y+a_{1}x+a_{3}} attached to a global minimal Weierstrass equation.

Note that a theorem of Cassels (see [Cas65]) asserts that, assuming the Shafarevich-Tate group is finite, the full Birch and Swinnerton-Dyer conjecture is invariant under isogeny. Thus, if we can check the conjecture for a single curve in a given isogeny class, we will have verified it for all Q-isogenous curves (as we will see, Kolyvagin has proven that � (E/Q) is finite for curves of analytic rank at most 1).

2 Kolyvagin and Consequences

Let K = Q(\sqrt{D}) be a quadratic extension of discriminant D. We say that K satisfies the Heegner hypothesis for an elliptic curve E/Q of conductor N if all prime factors of N split in K. This allows the construction of a Heegner point y_{K} on E/K. Kolyvagin shows ([Kol90]) that if y_{K} has infinite order (i.e., L′(E/K, 1) ≠ 0), then E(K) has rank 1. Moreover, he proves � (E/K) is finite, with the following bounds (we follow Gross’s notation and organization of Kolyvagin’s results in [Gro91]). Let I_{K} = [E(K) : Zy_{K}]. There exists an integer t_{E/K} divisible only by primes p (shown to be finite by Serre in [Ser72]) such that the representation G(Q̄/Q) → Aut(E[p]) is not surjective; then #� (E/K)|t_{E/K}I_{K}^{2}. The existence of an imaginary quadratic extension K satsifying the Heegner hypothesis and such that L′(E/K, 1) ≠ 0 is assumed in Kolyvagin’s paper; this result was simultaneously proven (using different methods) in [MM91] and [BFH90].

In a recent thesis, Cha ([Cha03]) has in certain cases provided weaker conditions under which Kolyvagin’s results hold. Let K be any finite extension of Q and let ℓ be an odd prime. Cha assumes that

1. There is an unramified prime divisor v of ℓ in K/Q such that E has either good or multiplicative reduction at v.

2. E(K)[ℓ] = 0.

He then proves the following two theorems:

1. H^{1}(K(E[ℓ^{i}])/K, E[ℓ^{i}]) = 0 for all i ≥ 1 unless ℓ = 3 and G(K(E[ℓ])/K) ≅ G_{except}, where G_{except} is the subgroup of GL_{2}(F_{ℓ}) given by

G_{except} = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) |a ∈ F_{ℓ}^{∗}, b ∈ F_{ℓ}\} .

2. When K is an imaginary quadratic extension of Q satisfying the Heegner hypothesis, let y_{K} ∈ E(K) denote the usual Heegner point. Let m be the largest integer such that y_{K} ∈ ℓ^{m}E(K) modulo ℓ-torsion, and assume ℓ does not divide the discriminant of K and E has good or multiplicative reduction at ℓ. Then if the Galois representation

ρ_{ℓ} : G(Q̄/Q) → Aut(E[ℓ])

is irreducible,

ord_{ℓ}|� (E/K)| ≤ 2m.

In particular, the second theorem will imply � (E/K) has no ℓ-torsion if ℓ ∤ [E(K) : Zy_{K}]. This is exactly the sort of result we try to obtain, so that we can reduce the calculation of |� (E/Q)| to dealing with its p-primary components for a few exceptional primes p. These are generally dealt with by p-descent, however, and Cha’s assumption on the reduction of E at a given prime makes the result ineffective for potentially large prime divisors of the conductor of E. The assumption on irreducibility of the Galois representation is, from a computational perspective, a great improvement on Kolyvagin’s original assumption, but we can further improve this hypothesis to one about torsion over K. Thus, while Cha’s results are of interest, in what follows we will prove somewhat stronger refinements of the hypotheses of Kolyvagin’s argument.

3 Weakening the Hypotheses for Triviality of � (E/K)[p]

We now undertake to weaken the hypotheses of Gross’s Proposition 2.1, replacing G(Q(E[p])/Q) = GL_{2}(F_{p}) with the statement that no curve in the Q-isogeny class of E has any p-torsion over K. Gross only uses the hypothesis on G(Q(E[p])/Q) at two points in the paper: section 4, where he constructs Kolyvagin’s cohomology classes and shows that restriction gives an isomorphism H^{1}(K, E[p]) ≅ H^{1}(K_{n}, E[p])^{G_{n}}, where K_{n} denotes the ring class field of conductor n over K and G_{n} = G(K_{n}/K); and section 9, where he requires H^{n}(K(E[p])/K, E[p]) = 0 for all n > 0, for then by the Hochshild-Serre spectral sequence he obtains the isomorphism H^{1}(K, E[p]) ≅ H^{1}(K(E[p]), E[p])^{G(K(E[p])/K)}. It is worth noting, as in [Cha03], that in light of the Birch and Swinnerton-Dyer conjecture, the assumption E(K)[p] = 0 should be enough for this argument to go through, but we do not have a proof of this result.

We will need to introduce an additional condition on K, that K ∩ Q(E[p]) = Q. This ensures that G(Q(E[p])/Q) = G(K(E[p])/K), an isomorphism that also induces an isomorphism on cohomology H^{i}(Q(E[p])/Q, E[p]) = H^{i}(K(E[p])/K, E[p]). Thus, whenever we prove vanishing of the first cohomology group in what follows, we are really interested in the consequence for the second group. The criterion of Néron-Ogg-Shafarevich implies that Q(E[p])/Q is ramified precisely at p and primes dividing the conductor N of E. If we choose K so that K/Q is ramified at some prime not in this list, then K cannot be contained in Q(E[p]). In particular, among those K satisfying the Heegner hypothesis, it suffices to choose one having at least two odd prime divisors of its discriminant: these primes ramify in K/Q, at most one of them can equal p, and the other one cannot ramify in Q(E[p])/Q because primes dividing N must split (and so not ramify) in K, by the Heegner hypothesis.

The following lemma (exercise 6 in [RS01]) will be useful in the next two propositions.

Lemma 3.1. The determinant of the mod ℓ representation attached to E is the cyclotomic character.

Proof. The Weil pairing induces an isomorphism of G(Q̄/Q)-modules E[ℓ] ⋀ E[ℓ] ≅ μ_{ℓ}. Let us fix a basis {e_{1}, e_{2}} of E[ℓ], with respect to which ρ_{ℓ}(σ) has the form

( \begin{matrix}a & b \\ c & d \end{matrix} ). Then

σ(e_{1} ∧ e_{2}) = (ae_{1} + ce_{2}) ∧ (be_{1} + de_{2}) = det(ρ_{ℓ}(σ))e_{1} ∧ e_{2}.

It follows that the above composition gives the cyclotomic character (i.e., the action of G(Q̄/Q) on μ_{ℓ}), which is clearly surjective.

The next two propositions show that the weakened hypothesis is sufficient in section 9 of Gross’s paper.

Proposition 3.2. Suppose E has no Q-rational p-isogeny. Then H^{i}(Q(E[p])/Q, E[p]) = 0 for all i > 0.

Proof. First observe that G(Q(E[p])/Q) = G is precisely the image of the Galois representation ρ_{p} : G(Q̄/Q) → Aut(E[p]) ≅ GL_{2}(F_{p}). By assumption, G is the image of an irreducible representation. There are two cases to consider: if p ∤ #G, the cohomology clearly vanishes because multiplication by #G kills the cohomology but is an isomorphism on the p-group E[p]. Otherwise, let p|#G. By Proposition 15 of [Ser72], G either contains SL_{2}(F_{p}) or is contained in a Borel subgroup of GL_{2}(F_{p}). E[p] is reducible under the action of a Borel subgroup, so by hypothesis SL_{2}(F_{p}) ⊂ G. Thus, G contains a nontrivial scalar (minus the identity); in fact, Lemma 4.1 implies that det : G → F_{p}^{∗} is surjective, so G = GL_{2}(F_{p}) and contains all of the nontrivial scalars. Applying the inflation-restriction exact sequence to the subgroup of G generated by the scalars implies that H^{i}(G, E[p]) = 0 for all i > 0, because this subgroup has order prime to p (p−1, in fact), and it leaves no subgroup of E[p] invariant.

For the above result, we assumed that the representation of G(Q̄/Q) was irreducible, but we in fact improve our hypotheses further with the following proposition:

Proposition 3.3. In the notation of the above proposition, suppose that the image G of the representation is contained in a Borel subgroup (we used our irreducibility hypothesis to avoid this situation before). Then unless E or a p-isogenous curve (over Q) E′ has a Q-rational p-torsion point, H^{i}(Q(E[p]/Q, E[p]) = 0 for all i > 0.

Proof. Choosing a basis, suppose that the action of G on E[p] = F_{p}^{2} is given by

( \begin{matrix}χ & ∗ \\ 0 & ψ \end{matrix} ) for characters χ and ψ. If χ is trivial, all matrices of the above form fix

( \begin{matrix}1 \\ 0 \end{matrix} ). In particular, there is a point of E[p] fixed by the action of G, a contradiction

since we have assumed that E(Q)[p] = 0. Now suppose that ψ is trivial. Matrices of

the above form preserve the line generated by ( \begin{matrix}1 \\ 0 \end{matrix} ), so this line forms a G(Q̄/Q)-

stable subspace of E[p]. In particular, there exists an isogeny over Q to a curve E′

having this line as kernel. The image of the complementary line generated by ( \begin{matrix}0 \\ 1 \end{matrix} ) is a 1-dimensional subspace of E′[p], and if ψ = 1, G(Q̄/Q) clearly acts trivially

on this subspace (we have an isomorphism of Galois modules E/ < ( \begin{matrix}1 \\ 0 \end{matrix} ) >≅ E′).

Thus, E′(Q)[p] is nontrivial, contradicting our assumption.

Let W be the (unique) p-Sylow subgroup of ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) consisting of matrices of

the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). We may assume W ⊂ G, for otherwise G has order prime to

p, and the cohomology clearly vanishes.

We begin by explicitly computing H^{j}(W, E[p]) using the fact that W is cyclic

(generated by w = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), for instance). Recall that for cyclic groups we can

compute cohomology using the particularly simple projective resolution

... → Z[W] → Z[W] → Z → 0

where the boundary maps alternate between w − 1 and Norm = ∑_{i=0}^{p−1} w^{i} (i.e., the maps are given by multiplication in the group ring Z[W]). Then we immediately see that

H^{j}(W, E[p]) = \{ \begin{matrix}ker(1 − w)/im(Norm(w)) =< ( \begin{matrix}1 \\ 0 \end{matrix} ) > & if j is even; \\ ker(Norm(w))/im(1 − w) = F_{p}^{2}/ < ( \begin{matrix}1 \\ 0 \end{matrix} ) > & if j is odd \end{matrix} \} .

Since χ and ψ are nontrivial by assumption, the G/W -invariants for both of these groups are trivial. Thus, H^{j}(W, E[p])^{G/W} = 0 for j ≥ 0. Let us then consider the Hochschild-Serre spectral sequence

H^{i}(G/W, H^{j}(W, F_{p}^{2})) ⇒ H^{i+j}(G, F_{p}^{2}).

For i > 0, since |G/W | is prime to p, and H^{j}(W, F_{p}^{2}) is a p-group (∀j), the group H^{i}(G/W, H^{j}(W, F_{p}^{2})) is trivial. But when i = 0 we have just computed that H^{i}(G/W, H^{j}(W, F_{p}^{2})) = H^{j}(W, F_{p}^{2})^{G/W} = 0, so the entire spectral sequence is trivial, and we conclude that H^{n}(G, E[p]) = 0 for all n ≥ 0.

The next proposition extends the result of section 4 of Gross’s paper. Note that if E has no p-isogeny (defined over Q) for p > 2, then it has no p-torsion over K.

Proposition 3.4. Let E be an elliptic curve with E(K)[p] = 0, where p > 3 or, if p = 3, K ≠ Q(μ_{3}). Let L be a finite abelian extension of K. Then H^{i}(L/K, E(L)[p]) = 0 for all i ≥ 1. (Note that in practice, L = K_{n}, the ring class field of conductor n.)

Proof. We may write the abelian group G(L/K) as a direct sum P ⊕ P′, where P is its Sylow p-subgroup and (p, #P′) = 1. We claim that the subgroup of E(L)[p] invariant under P′ is trivial. Let G = G(L/K)/H, where H is the subgroup of G(L/K) that acts trivially on E(L)[p]. If (#G, p) = 1, P ⊆ H, so P′ surjects onto G. As there is no nontrivial element of E(L)[p] invariant under all of G(L/K) (by the assumption on E(K)[p], the same then holds for P′.

If p|#G, we cannot have E(L)[p] = Z/pZ: the latter group has automorphism group isomorphic to (Z/pZ)^{∗}, of order p − 1, but if p|#G, G would give rise to at least p distinct automorphisms. Thus, E(L)[p] is the full p-torsion subgroup of E, and we can identify G with a subgroup of GL_{2}(Z/pZ) acting on E(L)[p] = (Z/pZ)^{2}.

We can choose a basis of (Z/pZ)^{2} so that G contains the subgroup ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ),

where x ∈ Z/pZ. Being abelian, G must be contained in the normalizer of this sub-

group, so G ⊆ \{( \begin{matrix}a & b \\ 0 & a \end{matrix} ) |a, b ∈ Z/pZ\}, and we claim that G contains an element

with a ≠ 1. Since E[p] = E(L)[p], the representation G(Q̄/K) → Aut(E[p]) factors through G(L/K) (recall that the image of the representation is G(K(E[p]/K)). The determinant of the mod-p representation of G(Q̄/Q) is surjective (onto F_{p}^{∗}), and [K : Q] = 2, so the character G(K̄/K) → F_{p}^{∗} has image of index at most 2 in F_{p}^{∗}. That is, it contains at least \frac{p−1}{2} elements, the squares in F_{p}^{∗}. Thus, for p > 3, G

contains an element with non-trivial determinant having the form ( \begin{matrix}a & b \\ 0 & a \end{matrix} ) with

a ≠ 1. Now,( \begin{matrix}a & b \\ 0 & a \end{matrix} )^{p} = ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) since we’re working mod p, and it follows that

G(L/K) contains an element that acts as a nontrivial scalar. In particular, since the group of scalars of GL_{2}(F_{p}) has p − 1 elements, this nontrivial scalar must be an element of P′. Now, since P′ leaves no nontrivial element of E(L)[p] invariant, the result will follow from an application of the inflation-restriction exact sequence:

0 → H^{1}(P, E(L)[p]^{P′}) → H^{1}(L/K, E(L)[p]) → H^{1}(P′, E(L)[p])

The first group vanishes since E(L)[p]^{P′} = 0, and the third group vanishes since the order of P′ is prime to p, and thus to the order of E(L)[p]. We conclude that the middle group is trivial, as desired. We can therefore extend the sequence to the second cohomology groups, deduce the same triviality result, and by induction conclude that H^{m}(L/K, E(L)[p]) = 0 for all m ≥ 1.

If p = 3, E(L)[3] = E[3] ⇒ μ_{3} ⊂ L ⇒ K = Q(μ_{3}). The last implication holds because G(L/Q) is abelian since G(K_{n}/K) and G(K/Q) are, so it has a unique index 2 subgroup; both K and Q(μ_{3}) correspond to index 2 subgroups by elementary Galois theory). This contradicts our assumption on K, so we must have E(L)[3] = 0, in which case the cohomology is clearly trivial.

To summarize, we can now apply Kolyvagin’s arguments (as given in [Gro91]) to find � (E/Q)[p] = 0 for all odd primes p such that all curves in the Q-isogeny class of E have no K-rational p-torsion and p ∤ I_{K}. If E(K)[p] ≠ 0 the subsequent situation is particularly easy to deal with because p-descent is much more easily implemented with a known rational p-torsion point.

4 Existence of K

Here we collect the results that imply the existence of quadratic imaginary extensions K/Q such that K satisfies the Heegner hypothesis and ord_{s=1}L(E/K) = 1. Then the Heegner point y_{K} has infinite order, so we may apply the work of Kolyvagin to conclude that � (E/K) is finite and E(K) has rank 1. There are two cases to consider:

1. If ord_{s=1}L(E/Q) = 0, then the papers [MM91] and [BFH90] both imply the existence of infinitely many distinct K (that is, with different fundamental discriminants) satisfying our two hypotheses.

2. If ord_{s=1}L(E/Q) = 1, then a result of Waldspurger ([Wal85]) does the trick, as does the above result of Bump-Friedberg-Hoffstein.

See also [Dar04] for a clear overview of how all of these results– along with the work of Gross-Zagier and Kolyvagin– fit together to settle the rank 0 and 1 BSD conjecture.

In our computations, however, we do not merely take any K satisfying the Heegner hypothesis and the analytic rank hypothesis. We instead choose K to be linearly disjoint from the fields Q(E[p]) for all primes p, and we have observed that a simple way to ensure this is to require the discriminant of K (:= D) to be divisible by at least 2 distinct odd primes. A conjecture of Goldfeld says that the density of discriminants D such that K = Q(\sqrt{D}) satisfies our hypotheses is roughly \frac{1}{2} (see [Gol79]). Thusfar the best proven result is due to Ono and Skinner, who showed ([OS98]) that, in the case ord_{s=1}L(E/Q) = 0, the number of such discriminants has density at least on the order of magnitude of \frac{1}{logX}. Unfortunately, this is precisely the density of the prime numbers, so a density argument will not help us here (Ono and Skinner’s result also does not distinguish between discriminants that give rise to the same imaginary quadratic extension). Note that Ono later improved this theorem (by a small power of logX) under the assumption that E/Q[2] = 0 (see [Ono01]).

In the worst case, the only K that exist and satisfy both of our hypotheses have only a single odd prime divisor p of their discriminants. But then for ℓ ≠ p, K is linearly disjoint from Q(E[ℓ]), so we can run our algorithm as before, only adding p to the list of “bad primes” on which we have to perform descents.

This bad prime p might be large, however, making the p-descent cumbersome. In that case, it would be better in practice to produce a second field K′ satisfying our hypotheses (recall that infinitely many exist). Unless K′ = K, K′ is linearly disjoint from Q(E[p]) since p will not ramify in K′. There is a good chance that we might be able to show � (E/K)[p] = 0, but it is possible that a curve in E’s isogeny class has p-torsion or that y_{K} is a multiple of p in E(K). There are universal bounds on the possible p-torsion for quadratic fields, so the problematic primes resulting from torsion will still be ’small,’ but the ever-mysterious index of the Heegner point may force us to perform descents on large primes. In practice, though, the bad primes are small.

5 p-descents

Traditionally, performing a p-descent on E/Q means computing the quotient group E(Q)/pE(Q) by first computing the p-Selmer group and then somehow trying to get a handle on � (E/Q)[p]. Our task is much simpler since we already know the rank of any curve we are working with (by Kolyvagin’s theorem). In particular, dim_{F_{p}}E(Q)/pE(Q) = rk_{Z}E(Q) + dim_{F_{p}}E(Q)[p]; we know all of these quantities, so we can compute � (E/Q)[p] (our ultimate goal) by simply finding the order of the p-Selmer group and applying the fundamental exact sequence

0 → E(Q)/pE(Q) → Sel(E/Q)_{p} → � (E/Q)[p] → 0.

Sel(E/Q)_{p} is effectively computable (see [Sil92]), so this poses no problem for the validity of our algorithm.

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