1 Introduction

For an elliptic curve E defined over a number field K the following diagram defines the Selmer and the Shafarevich-Tate groups as Sel(E/K)_{p} = ker f, � (E/K) = ker g:

\xymatrix{0 \ar[r] & (E(K)/pE(K)) \ar[r] & H^{1}(G(K̄/K), E[p]) \ar[r] \ar[d]_{Res} \ar[dr]^{f} & H^{1}(G(K̄/K), E)[p] \ar[d]^{g_{p}} \\  &  & H^{1}(G(K̄_{λ}/K_{λ}), E[p]) & ∏_{v} H^{1}(G(K̄_{v}/K_{v}), E)[p]}

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. (BSD) Let E be an elliptic curve defined over Q. Then the order of vanishing at s = 1 of L(E/Q, 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 canonical height pairing matrix on a set of free generators. The c_{v} = [E(Q_{v}) : E_{0}(Q_{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 Gross-Zagier Formulae

One of the factors in the BSD formula is the derivative of the L-function. 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.

Then the following limit formula was proved by Gross and Zagier ([GZ86])

Theorem 2.1. (Gross-Zagier)

ĥ(y_{K}) = \frac{u^{2}\sqrt{D}}{∫_{E(ℂ)} ω ∧ iω̄} L′(E/K, 1),

(where ω is the invariant differential associated with the elliptic curve and u is half the number of units of O_{K}).

Complex conjugation acts on E(K) and we get a decomposition

Proposition 2.2. For a quadratic imaginary field K = Q(\sqrt{D}), E(K) decomposes, up to 2-torsion, as a direct sum E(Q) ⊕ E_{D}(Q), where E_{D} is the quadratic twist of E. (Recall that from a Weierstrass equation of E/Q, we obtain a Weierstrass equation of E_{D}/Q as follows:

\begin{matrix}E : y^{2} = x^{3} + ax + b \\ E_{D} : y^{2} = x^{3} + D^{2}ax + D^{3}b \end{matrix} )

Proof. Denote complex conjugation by τ. We will decompose E(K) into its eigenspaces under the action of τ. First, we kill E(K)[2] (by tensoring with Z[\frac{1}{2}], for instance), so that, using the fact that τ^{2} = 1, we can define the projections from E(K) to its +1 and −1 eigenspaces under τ: for P ∈ E(K),

P = \frac{1 + τ}{2} P + \frac{1 − τ}{2} P.

But if P = (x, y) ∈ E(K) satisfies τP = P, then P ∈ E(Q); if τP = −P = (x, −y), then x ∈ Q and y ∈ \sqrt{D}Q. In particular, (Dx, D\sqrt{D}y) ∈ E_{D}(Q), and conversely we can obtain any such point in the −1 eigenspace of E(K) from a point of E_{D}(Q). Thus, having eliminated the 2-torsion, the decomposition of E(K) into its τ-eigenspaces just reads

E(K) = E(Q) ⊕ E_{D}(Q).

In [MSD74] Mazur and Swinnerton-Dyer introduce an analytic and an algebraic p-adic L-function attached to E and they conjecture that the two are equal. The analytic L-function satisfies a Gross-Zagier formula proved by Perrin-Riou ([?]):

Theorem 2.3. Assume that p splits in K. For any ρ ∈ G(K_{∞}/K) (where K_{∞} is the unique Z_{p}^{2}-extension of K) we have

\frac{d}{ds}L_{p}(f, 1)(ρ^{s})|_{s=0} = (1 − \frac{1}{α_{p}(f)})^{2} \frac{ĥ(y_{1,K})}{hu^{2}},

where f is the newform attached to E, h is the class number of K and α_{p}(f) is the unit root of X^{2} − a(p)X + p = 0.

The algebraic p-adic L-function is defined as the characteristic series of the Pontryagin dual of the p^{∞} Selmer group. Rubin proved the Mazur and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by K and primes that split in K so that E has ordinary good reduction at p ([Rub91]).

3 Kolyvagin and Consequences

Assume that E does not have complex multiplication. 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]. Gross notes that if E is semistable then for all primes p ≥ 11 the p-adic representation is surjective.

Kolyvagin uses an Euler system of Heegner points to construct certain cohomology classes. Then he applies Tate local duality to study the local cohomology classes associated with Sel(E/K)_{p} and finally uses some density theorems (Chebotarev) to bound Sel(E/K)_{p}. Then the fundamental exact sequence

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

gives triviality of � (E/K)[p] in all but finitely many cases.

The main assumption on the p where Kolyvagin does prove triviality of p-torsion of � (i.e., the surjectivity of the p-adic representation) is used in two places in the argument.

1. First, the construction of the cohomology classes requires that restriction Res : H^{1}(K, E[p]) → H^{1}(K_{n}, E[p])^{G(K_{n}/K)} is an isomorphism (where K_{n} is the ring class field of conductor n over K). The inflation restriction sequence yields

H^{1}(K_{n}/K, E(K_{n})[p]) → H^{1}(K, E[p]) → H^{1}(K_{n}, E[p])^{G(K_{n}/K)} → H^{2}(K_{n}/K, E(K_{n})[p]).

The result follows from the fact that E(K_{n})[p] = 0 if the representation is surjective.

2. The second part where surjectivity is essential is to define a nondegenerate

pairing

H^{1}(K, E[p]) ⊗ G(K(E[p])) → E[p].

For simplicity of notation write L = K(E[p]). Again, the inflation restriction sequence yields

H^{1}(L/K, E(L)[p]) → H^{1}(K, E[p]) → H^{1}(L, E[p])^{G(L/K)} → H^{2}(L/K, E(L)[p]).

Since G(L) acts trivially on E[p], it is enough to prove that H^{i}(L/K, E[p]) = 0. The surjectivity of the p-adic representation implies that G = G(L/K) = G(Q(E[p])/Q) ≡ GL_{2}(F_{p}). If Z is the group of scalars the Hochschild-Serre spectral sequence H^{i}(G/Z, H^{j}(Z, E[p])) ⟹ H^{i+j}(L/K, E[p]) will give the result since |Z| = p − 1 and E[p] is a p-group.

The main computational problem of this method is that there is no universal bound on p so that the p-adic representation is surjective for all elliptic curves and primes larger than the bound. Moreover, the surjectivity condition is hard to verify computationally.

4 Weakening the Hypotheses of Kolyvagin’s Method

4.1 Cha’s work

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.

Kolyvagin proved the following effective version of his theorem (see [Kol90]):

Theorem 4.1 (Kolyvagin). Let R be the ring of endomorphisms of E, with F its field of fractions. Suppose the Heegner point y_{K} has infinite order. Then if ℓ is an odd prime unramified in F such that G(F (E[ℓ])/F) = Aut_{R}(E[ℓ]),

ord_{ℓ}|� (E/K)| ≤ 2ord_{ℓ}[E(K) : Zy_{K}].

Assuming E does not have complex multiplication, the hypotheses of both these theorems imply E(K)[ℓ] = 0, so, in particular, � (E/K) has no ℓ-torsion if ℓ ∤ [E(K) : Zy_{K}] (for a proof that irreducibility of the mod-ℓ representation over Q implies E(K)[ℓ] = 0, see Lemma 10.1.

Mazur proves in [Maz78] that for p > 37 the representation is irreducible since E does not have complex multiplication. This is exactly the sort of result we try to obtain, since we can reduce the calculation of |� (E/Q)| to dealing with its pprimary components for a few exceptional primes p(p ≤ 37 in general and p ≤ 7 for semistable curves as well as the primes dividing the conductor and I_{K} = [E(K) : Zy_{K}]). 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.

4.2 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. We have used the fact that the only ramified primes in Q(Eℓ])/Q are, by the criterion of Néron-Ogg-Shafarevich (see [Sil92]), primes dividing the conductor N of E and ℓ itself.

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 particular, the contributions from nontrivial elements of the Shafarevich-Tate group). In practice, though, the bad primes are small.

4.3 Reducing the hypotheses to statements about torsion

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

Lemma 4.2. 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 proposition shows the weakened hypothesis is sufficient to replace surjectivity at 2.

Proposition 4.3. 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 4.4. 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 provides a weaker hypothesis for the use of surjectivity at 1.

Proposition 4.5. 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. (In the actual proof for 1 we use L = K_{n}, the ring class field of conductor n which is abelian.)

Proof. The proof is the now standard use of Sylow groups and the Hochschild-Serre spectral sequence.

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′. Therefore E(L)[p]^{P′} = 0.

The result will now follow from the Hochschild-Serre spectral sequence H^{i}(P′, H^{j}(P, E(L)[p]^{P′})) ⟹ H^{i+j}(L/K, E(L)[p]). Then we must have H^{n}(L/K, E(L)[p]) = 0 for n > 0.

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} (of course we may still discard very easily primes > 37 in general and > 7 in the semistable case. Remark 4.6. 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.

5 Elliptic Curves with Complex Multiplication

The previous discussion fails in the case of elliptic curves E with complex multiplication. Alternative results are obtained by Rubin in [Rub91].

Perrin-Riou proves [?] the following corollaries to the p-adic Gross-Zagier type formula:

Proposition 5.1. 1. If f is the normalized newform attached to an elliptic curve with complex multiplication then L_{p} has analytic rank 1 if and only if L has analytic rank 1.

2. If the rank of L_{p} is 1 then the p-part of the Birch and Swinnerton-Dyer conjecture is true up to a unit of Z_{p}.

This follows from the Gross-Zagier type formula and a similar formula for the algebraic p-adic L-series obtained in [?].

Rubin proves the following theorem ([Rub91])

Theorem 5.2. Let E have complex multiplication by K. (Since the Birch and Swinnerton-Dyer conjecture is isogeny-invariant we may assume that End(E) = O_{K}.)

1. If L(E/K, 1) ≠ 0 then for all primes 𝔭 not dividing |O_{K}^{×}| we have

|� [𝔭^{∞}| = ℕ(𝔭)^{m(𝔭)},

where m(𝔭) = ord_{𝔭} (|E(K)|\frac{L(ψ̄,1)}{Ω}), where ψ is the Hecke character associated to E.

2. If the analytic rank of E is 1 then Perrin-Riou’s work together with the conjecture of Mazur and Swinnerton-Dyer implies that � (E/Q)[p] is as expected from the BSD conjecture whenever p > 2 splits in K.

6 Computations

The following algorithm will prove the full BSD for elliptic curves without complex multiplication up to conductor 20000.

Algorithm 6.1. Let E be an elliptic curve over Q of analytic rank at most 1. The following algorithm computes |� (E/Q)[p]| for all primes p.

1. [Choose K] Choose the first quadratic imaginary field K that satisfies the Heegner hypothesis, such that E/K has analytic rank 1, and whose discriminant is divisible by at least two primes.

2. [Find p-torsion] Decide for which primes p, there is a curve E′ that is Q-isogenous to E such that E′(K)[p] ≠ 0. (We enumerate the Q-isogeny class of E using [standard method]. Then for each E′ in the Q-isogeny class, compute the torsion subgroup of E′(K) using [standard method], and see whether or not p divides its order.) Let B be the product of 2 and these primes.

3. [Root number] Compute the root number of E.

4. [Compute Mordell-Weil]

(a) If the root number is −1, compute E(Q) and let z be a generator modulo torsion.

(b) If the root number is +1, compute E^{D}(Q), and let z be a generator modulo torsion.

5. [Height of Heegner point] Compute the height h_{K}(y_{K}) relative to K of the Heegner point associated to K using the Gross-Zagier formula:

ĥ_{K}(y_{K}) = α · L′(E/K, 1) = \begin{cases}α · L′(E_{D}/Q, 1) · L(E/Q, 1) & if E has rank 0 \\ α · L(E_{D}/Q, 1) · L′(E/Q, 1) & if E has rank 1, \end{cases}

where α = \frac{u^{2}\sqrt{|D|}}{∥ ∫_{E(ℂ)} ω∧\overline{iω}∥}. Here u is half the number of units in the ring of integers

of K. We may compute

∫_{E(ℂ)} ω ∧ \overline{iω} = ∫_{E(ℂ)} \frac{dx}{y} ∧ \overline{\frac{dx}{y}} = −i ∫_{ℂ/Λ} ℘′(z)\frac{dz}{℘′(z)} ∧ \overline{℘′(z)\frac{dz}{℘′(z)}}
= −i ∫_{ℂ} dz ∧ dz̄ = ∫ (dx + idy) ∧ (−idx − dy) = −2 ∫ dx ∧ dy = −2a,

where a is the volume of the lattice ℂ/Λ.

6. [Index of Heegner point] Compute

I_{K} = \sqrt{h_{K}(y_{K})/h_{K}(z)} = [E(K) : Zy_{K}],

up to primes that divide the B from step 2. Note that h_{K}(z) = 2 · h_{Q}(z), and that we do not compute E(K) or Zy_{K} directly, but instead compute the index using properties of heights.

7. [Annihilate � ] Then � (E/Q)[p] = 0 for all primes p ∤ B · I_{K}.

8. [p-descent] For each prime p | B · I_{K}, do a p-descent and compute � (E/Q)[p]. (Note that this is likely not too difficult because there is a p-torsion point over K on a curve F that is Q-isogenous to E. Ideas: If an isogeny from E to F has degree divisible by p, then E has a rational p-isogeny, which makes p-descent eiser. If an isogeny from E to F has degree coprime to p, then � (F/Q)[p] ≅ � (E/Q)[p], and F has a K-rational p-torsion point, so p-descent on F should be relatively easy.) To reduce the number of p for which one must do a p-descent, use several K.

Proof. Step 1 guarantees that G(K(E[p])/K) ≡ G(Q(E[p])/Q). This step will always find such a K due to the density theorem mentioned. Step 2 will determine the primes for which the weakened hypothesis fails and so the primes for which we must do descent.

Since the root number and the rank have the same parity, the fact that the rank is at most 1 implies that the root number determines the rank of the curve /Q. Therefore, by computing the Mordell-Weil group of E or E^{D} over Q (but only one of them) we can find a generator of E(K). Step 6 computes the last set of primes at which we need descent.

Finally, the last step takes care of the exceptional primes.

For elliptic curves with complex multiplication by K, if E has rank 0 we may just do descent for the primes dividing |O_{K}^{×}|. If E has rank 1, then we need to take care of the primes that do not split in K. We may also apply Kolyvagin’s method with our weakened (computationally viable) hypotheses to eliminate some of these primes. Then we may do descent on those.

7 General Algorithms for Elliptic Curves

1. For root number cite the paper that Cremona said.

2. For Mordell-Weil ?

3. For descent there is a more than sufficient article by our own Michael Stoll online.

4. Perhaps how to evaluate the L series.

5. In general, what are algorithms to generate the isogeny classes of E.

8 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.

9 Computing the Mordell-Weil group

While in general an unsolved problem, finding a complete set of (free) Mordell-Weil generators when the rank of the curve is already known is a fairly simple, if sometimes time-consuming, problem. Basically, one searches for points by naive height until a point of infinite order is found (the easiest way to check whether a point on E(Q)Q has infinite order is to compute its multiples up to 12; if none of these is zero, the point cannot have finite order by Mazur’s theorem on torsion subgroups of elliptic curves over Q). The first point of infinite order found (call it P) may not be a generator, however: it may only generate a finite-index subgroup of E(Q). But if it is a nontrivial multiple of a generator, the canonical height of the generator is at most \frac{1}{4} ĥ(P) (the canonical height). By a result of Silverman, the naive height of the generator may then be bounded, and an exhaustive computation will then turn it up if it exists. For more details on both the theory and implementation of this method, see [Cre97] or the updated version available on Cremona’s webpage (http://www.maths.nott.ac.uk/personal/jec/book/fulltext/index.html).

10 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^{p}(E/Q) → � (E/Q)[p] → 0.

Sel^{p}(E/Q) is effectively computable, so this poses no problem for the validity of our algorithm. In all cases this calculation can be reduced to “standard” computations over number fields. In particular, for S a set of “bad primes” (traditionally p and places of bad reduction for E/Q), we have to determine the p-part of the S-class group and a basis of the S-units modulo p^{th}-powers. For a full discussion and improvements to the basic approach, see [SS04]. The method is fool-proof when p = 2 or p = 3 (given that we are working over Q), but for larger primes current limitations in computational number theory may make the theoretically possible calculations infeasible. Fortunately, many of our examples are exceptional cases having K-rational p-torsion. This implies they have rational p-isogenies (for a proof, see below), and Schaefer and Stoll, for instance, perform a successful 13-descent on a curve using the fact that it has a rational 13-isogeny.

Lemma 10.1. Let E/Q be an elliptic curve, K a quadratic extension of Q, and p > 3 an odd prime such that E(K)[p] ≠ 0. Then the mod-p representation over Q is reducible. In particular, E has a Q-rational p-isogeny.

Proof. Let P be a nontrivial element of E(K)[p], and let τ be a lift of complex conjugation in G(K/Q) to G(Q̄/Q). If τP is a multiple of P, then the one-dimensional subspace of E[p] generated by P is G(Q̄/Q)-stable, so the representation over Q is reducible. Else, P and τP generate all of E[p]. By definition, τP ∈ E(K), so E(K)[p] = E[p], and because of the Weil pairing, μ_{p} ⊂ K. For p > 3, this is a contradiction.

References

[BFH90] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Nonvanishing theorems for L-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543–618. MR 92a:11058

[Cas65] J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199. MR 31 #3420

[Cha03] Byungchul Cha, Vanishing of some cohomology groups and bounds for the shafarevich-tate groups of elliptic curves, Ph.D. thesis, Johns Hopkins University, 2003.

[Cre97] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[Dar04] Henri Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004. MR 2 020 572

[Gol79] Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108–118. MR 81i:12014

[Gro91] B. H. Gross, Kolyvagin’s work on modular elliptic curves, L-functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge, 1991, pp. 235–256.

[GZ86] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 87j:11057

[Kol90] V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 92g:11109 [Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D.

Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.

[MM91] M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular L-series, Ann. of Math. (2) 133 (1991), no. 3, 447–475. MR 92e:11050

[MSD74] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61. MR 50 #7152

[Ono01] Ken Ono, Nonvanishing of quadratic twists of modular L-functions and applications to elliptic curves, J. Reine Angew. Math. 533 (2001), 81–97. MR 2002a:11051

[OS98] Ken Ono and Christopher Skinner, Non-vanishing of quadratic twists of modular L-functions, Invent. Math. 134 (1998), no. 3, 651–660. MR 2000a:11077

[RS01] K. A. Ribet and W. A. Stein, Lectures on Serre’s conjectures, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 143–232. MR 2002h:11047

[Rub91] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. MR 92f:11151

[Ser72] J-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.

[Sil92] J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original.

[SS04] Edward F. Schaefer and Michael Stoll, How to do a p-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), no. 3, 1209–1231 (electronic). MR 2004g:11045

[Wal85] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242. MR 87g:11061b