N.B. Some sections here are written up formally; others are just notes. Note also I need to learn how to do bibliographies.

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.

2 Known Results for CM Elliptic Curves

NEED MORE SPECIFICALLY WHAT RUBIN GIVES– The result of Rubin in [6] implies, for elliptic curves with CM by the ring of integers in a quadratic imaginary field K, the full BSD conjecture up to primes dividing the number of units in the ring of integers (at worst requiring p-descent checks for p = 2or3). But does this hold for CM by any order, or just the ring of integers???

We will now summarize and give references for the main theoretical results that give information about � and the rank of elliptic curves over Q. Progress was first made in the case of elliptic curves with complex multiplication: J. Coates and A. Wiles showed in [1] that for such an elliptic curve E/Q, L(E/Q, 1) ≠ 0 ⇒ E(Q) is finite. Improving on their techniques, K. Rubin proved two major results in his paper [7] about elliptic curves over Q with complex multiplication. Rubin’s Theorem A implies that whenever L(E/Q, 1) ≠ 0, � (E/Q) is finite. Theorem A also includes a local result, providing evidence for the full Birch and Swinnerton-Dyer conjecture, describing for which primes p � (E/Q)[p] can be non-trivial. Note that before Rubin’s result, not a single Shafarevich-Tate group was known to be finite; his local result makes � (E/Q) effectively computable in some cases. Under the same hypotheses, Theorem B of Rubin’s paper states that rk_{Z}(E/Q) ≥ 2 ⇒ ord_{s=1}L(E/Q, s) ≥ 2. The previous year B. Gross and D. Zagier proved ([2]) their limit formula

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

where y_{K} denotes the usual Heegner point. Combining Rubin’s result with the work of Coates-Wiles and the theorem, of Gross-Zagier, one obtains the

Theorem 2.1. Let E/Q be an elliptic curve with complex multiplication by an order in a quadratic imaginary field. Then

ord_{s=1}L(E/Q, s) ≤ 1 ⇒ rk_{Z}(E/Q) = ord_{s=1}L(E/Q, s).

3 Kolyvagin and Consequences

Kolyvagin shows ([4]) 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. Writing C_{D} = [E(K) : Zy_{K}], and letting c_{3} and c_{4} be constants I don’t understand yet that are mired in Kolyvagin’s paper, c_{3}C_{D} kills � (E/K), and #� (E/K)|c_{4}C_{D}^{2}.

Under the same hypothesis, he obtains the corresponding result for E/Q and E_{D}/Q: 2c_{3}C_{D} kills � (E/Q) and � (E_{D}/Q), and both groups have order dividing c_{4}2^{a}C_{D}^{2}, where a is the 2-rank of both groups. Moreover, he gives a condition for determining whether E/Q or its twist E_{D}/Q contains the point of infinite order. If E has modular parametrization γ : X_{0}(N) → E, then γ ∘ w_{N} = εγ + γ(0), where w_{N} is the principal (Fricke) involution on the modular curve (alternatively, ε is the eigenvalue on the normalized newform associated to E of the involution w_{N}). Kolyvagin deduces that ε = 1 ⇒ E(Q) contains the point of infinite order, and ε = −1 ⇒ E_{D}(Q) contains the point of infinite order. (Note that our calculations for E/Q will give us some info free of charge about E_{D}/Q) ε = 1 makes our calculations easier, because while we’ll be using low-level E/Q, we don’t know how high the conductor of E_{D}/Q might be.

In [3] Kolyvagin shows that 23 elliptic curves over Q have trivial Shafarevich-Tate group, and he verifies the full Birch and Swinnerton-Dyer conjecture for 5 of them, reducing the verification in other cases to a computation we will check. Note, however, that all of these curves have rank 0: they are quadratic twists of the rank 1 curve E : y^{2} = 4x^{3} −4x+1 having ε = 1. In particular, he shows for the 23 curves E_{D} : −Dy^{2} = 4x^{3}−4x+1 with D ∈ {7, 11, 47, 71, 83, 84, 127, 159, 164, 219, 231, 263, 271, 287, 292, 303, 308, 359, 371, 404, 443, 447, 471}, � (E_{D}/Q) = 0. These are the twists with D < 500 (the extent of the tables of computations Kolyvagin had available), D prime to the conductor of E (37), and as usual ruling out D = 3, 4, for which C_{D} = 1. This suffices to conclude that � (E_{D}/Q) = 0 because in Theorem B of his paper, Kolyvagin shows that for whichever of E/Q and E_{D}/Q that does not contain the point of infinite order, � is killed by C_{D} (as opposed to c_{3}C_{D}, which works for both curves).

By explicitly working with the curve E, Kolyvagin reduces the full BSD conjecture in this case to showing r_{2}(D) = 0, where r_{2}(D) = #{q an odd prime : q|D, (\frac{q}{37}) = 1, and a_{q} is even\}. (\frac{q}{37}) denotes the Legendre symbol, and a_{q} is the Hecke eigenvalue. NOTE Cremona’s tables won’t be good for handling these, because conductors may get very high. His outputs of a_{q} also only go up to 100, and some of our D’s have prime divisors > 100.

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

S. Donnelly has explained a way to weaken the hypotheses of Gross’s Proposition 2.1, replacing G(Q(E[p])/Q) = GL_{2}(/Z/Z) with the statement that E admits no p-isogenies. 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)}.

The following lemma (exercise 6 in [5]) will be useful in the next two propositions.

Lemma 4.1. There are maps

G(Q̄/Q) → Aut(E[ℓ]) ≅ GL_{2}(F_{ℓ}) → F_{ℓ}^{∗}

given by the representation ρ_{ℓ}, a choice of basis, and the determinant. Their composition is surjective, giving the cyclotomic character χ_{ℓ} : G(Q̄/Q) → F_{ℓ}^{∗}.

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.

Lemma 4.2. Let p be an odd prime and E an elliptic curve over Q. If E has no Q-rational p-isogeny, then E has no K-rational isogeny, where [K : Q] = 2.

Proof. The existence of a p-isogeny over Q is equivalent to the reducibility of the representation

ρ_{p} : G(Q̄/Q) → Aut(E[p]) ≅ GL_{2}(F_{p}).

Let the image of this representation be G, and consider the corresponding representation

ρ_{p} : G(Q̄/K) → → H ⊂ Aut(E[p]) ≅ GL_{2}(F_{p}).

Then [G : H] ≤ 2 (letting α ∈ G(Q̄/Q) be a lift of complex conjugation in G(K/Q), ρ_{p}(α) must generate G/H, but clearly ρ_{p}(α)^{2} ∈ H). In particular, if E[p] has a subgroup P of order p invariant under H, the action of α on P can generate a subgroup of order either p or 2p. The latter is impossible, since 2p ∤ p^{2}, and it follows that the action of all of G leaves P invariant. Thus, if the representation over K is reducible, so is the representation over Q, and we conclude that if E has no p-isogenies over Q, it cannot have any over K.

The next proposition shows the weakened hypothesis is sufficient in section 9 of Gross’s paper.

Proposition 4.3. Suppose E has no Q-rational p-isogeny, and let [K : Q] = 2. Then H^{i}(K(E[p])/K, E[p]) = 0 for all i > 0.

Proof. First observe that G(K(E[p])/K) = H is precisely the image of the Galois representation ρ_{p} : G(Q̄/K) → Aut(E[p]) ≅ GL_{2}(F_{p}). Let G and H be as in the preceding lemma. By the lemma, H is the image of an irreducible representation. There are two cases to consider: if p ∤ #H, the cohomology clearly vanishes because multiplication by #G kills the cohomology but is an isomorphism on the p-group E[p]. Otherwise, let p|#H. By Proposition 15 of [8], H 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}) ⊂ H. Thus, H 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. As H is a subgroup of index at most 2, H must contain at least half of the scalars (in particular, it must contain the squares in F_{p}^{∗}. 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.

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 4.4. Let E be an elliptic curve with E(K)[p] = 0, where p > 3 or, if p = 3, K ≠ Q(μ_{3}). Then H^{i}(K_{n}/K, E(K_{n})[p]) = 0 for all i ≥ 1.

Proof. We may write the abelian group G(K_{n}/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(K_{n})[p] invariant under P′ is trivial. Let G = G(K_{n}/K)/H, where H is the subgroup of G(K_{n}/K) that acts trivially on E(K_{n})[p]. If (#G, p) = 1, P ⊆ H, so P′ surjects onto G. As there is no nontrivial element of E(K_{n})[p] invariant under all of G(K_{n}/K), the same then holds for P′.

If p|#G, we cannot have #E(K_{n})[p] = p, for then a subgroup of G of order p would act trivially on E(K_{n})[p], contradicting the definition of G. Thus, E(K_{n})[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(K_{n})[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

subgroup, 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(K_{n})[p], the representation G(Q̄/K) → Aut(E[p]) factors through G(K_{n}/K) (recall that the image of the representation is G(K(E[p]/K)). We argued before that this image contained the scalars corresponding to squares in F_{p}^{∗}, so P′ contains at least \frac{p−1}{2} (which is > 1 for p > 3) elements that leave no subgroup of E(K_{n})[p] invariant. Now, the result will follow from an application of the inflation-restriction exact sequence:

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

The first group vanishes since E(K_{n})[p]^{P′} = 0, and the third group vanishes since the order of P′ is prime to p, and thus to the order of E(K_{n})[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}(K_{n}/K, E(K_{n})[p]) = 0 for all m ≥ 1.

If p = 3, E(K_{n})[3] = E[3] ⇒ μ_{3} ⊂ K_{n} ⇒ K = Q(μ_{3}). The last implication holds because G(K_{n}/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(K_{n})[3] = 0, in which case the cohomology is clearly trivial.

5 The Decomposition of E(K) under Complex Conjugation

To compute the index I_{K} = #E(K)/Zy_{K} needed to verify the full Birch and Swinnerton-Dyer conjecture, we compare the canonical height of the Heegner point y_{K} with the height of a generator of E(K), which we know to have algebraic rank 1 by Kolyvagin’s theorem. To assist in finding the (infinite order) generator of E(K), we use the Proposition 5.1. 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).

6 Application of Gross-Zagier to our calculations

In light of Donnelly’s observation, we can apply Kolyvagin’s result that � (E/K)[p] = 0 for all odd primes p not dividing the Heegner point y_{K} (i.e., p does not divide C_{D}) and such that E/K has no p-isogenies. By the inflation-restriction sequence, � (E/Q)[p] = for all of these primes as well, since � (E/Q)[p] = maps to � (E/K)[p] with kernel contained in H^{1}(K/Q, E(K), a finite 2-group. Thus, it will be essential to compute the constant I_{K} = [E(K) : Zy_{K}] (actually, we only want this index for E(K)/E(K)_{tors}. First we will have to find a generator P of the free part of E(K) and compute its canonical height ĥ(P). We can use another form of the Gross-Zagier formula to compute ĥ(y_{K}):

ĥ(y_{K}) = \frac{u^{2}\sqrt{D}}{∥ω_{f}∥} L′(E_{D}/Q, 1)L(E/Q, 1),

where u is half the number of units in K = Q(\sqrt{−D}), and ∥ω_{f}∥ is the Peterson norm of the newform f corresponding to E (note that for D ≠ 3, 4, u = 1).

With these preliminaries, we can compute the index I_{K} = \sqrt{ĥ(y_{K})/ĥ(P)}. Note that we are free to choose K = Q(\sqrt{−D}) as long as D ≠ 3, 4 and all the prime factors of N (the conductor of E) split in K. To find D given N amounts to solving a finite number of congruences, so we should be able to implement this.

References

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

[3] V. A. Kolyvagin, The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1154–1180, 1327. MR 90f:11035

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