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.

2 Known Results for CM Elliptic Curves

NEED MORE SPECIFICALLY WHAT RUBIN GIVES– 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 [J. Coates, A. Wiles, “On the Conjecture of Birch and Swinnerton-Dyer”, Invent. Math., 39 (1977), 223-251.] 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 [K. Rubin, “Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication”, Invent. Math., 89 (1987), 527-560.] 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 ([B. H. Gross, D. B. Zagier, “Heegner points and derivatives of L-series”, Invent. Math., 84 (1986), 225-320.])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 ([V. A. Kolyvagin, “Euler Systems”, in The Grothendieck Festschrift. Prog. in Math., Boston, Birkhauser (1990), 435-483.]) 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 [V. A. Kolyvagin, “On the Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves”, WHAT AMERICAN JOURNAL DOES THE TRANSLA-TION COME FROM?] 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 next proposition extends the result of section 4 of Gross’s paper. Note that if E has no p-isogeny (defined over K), then certainly it has no p-torsion over K.

Proposition 4.1. Let E be an elliptic curve with E(K)[p] = 0, where p > 3 or, if p = 3, K ≠ Q(μ_{3}). Then H^{m}(K_{n}/K, E(K_{n})[p]) = 0 for all m ≥ 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 subgroup 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(K_{n}) contains all E’s full p-torsion subgroup, μ_{p} ⊂ K_{n} (this follows from basic properties of the Weil pairing). Then G(K_{n}/K) acts on μ_{p}, inducing a well-defined action of G on μ_{p}, for the kernel of G(K_{n}/K) → → G consists of automorphisms acting trivially on E(K_{n})[p], and therefore acting trivially on μ_{p} as well. Choosing a primitive p^{th} root of unity ζ, we claim that the action of G on μ_{p} is given by

g · ζ = ζ^{det(g)}

for all g ∈ G. That is, det(g) is the cyclotomic character. NOW WHAT?

Having shown that P′ leaves no subgroup of E(K_{n})[p] invariant, 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.

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

Proposition 4.2. Suppose E has no p-isogeny. Then H^{1}(K(E[p])/K, E[p]) = 0.

Proof.

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 C_{D} = [E(K) : Zy_{K}]. 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 C_{D} = \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.