COMPUTATIONAL VERIFICATION OF THE BIRCH ... - William Stein

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26 GRIGOROV, JORZA, PATRIKIS, STEIN, AND TARNIT¸ Ǎ-PǍTRAS¸CU By hypothesis, the discriminant disc(K) is divisible by a prime other than p and is coprime to N. If K were contained in Q(E[p]), all primes that ramify in K would ramify in Q(E[p]), so they would divide Np, which contradicts our hypothesis. Thus the quadratic field K is linearly disjoint from Q(E[p]), so the restriction map Gal(K(E[p])/K) → Gal(Q(E[p])/Q), is an isomorphism. The action of Gal(K(E[p])/K) on the module E[p] is through Gal(Q(E[p])/Q) so for our application it will suffice to show that for i > 0, H i (Q(E[p])/Q, E[p]) = 0. Let G ⊆ Gal(Q(E[p])/Q) be the image of ρ E,p. If p ∤ #G, then for i > 0 we have H i (G, E[p]) = 0 since E[p] is a p-group. Therefore we may assume that p | #G. By [Ser72, Prop. 15], the image G either contains SL2(Fp) or is contained in a Borel subgroup of GL2(Fp). First consider the case when G contains SL2(Fp). By Lemma 5.6, the determinant det : G → F ∗ p is surjective, so in fact G = GL2(Fp). In this case, we already know Propositions 5.2–5.3. Thus we turn next to the case when G is contained in a Borel subgroup. Lemma 5.7. Assume that G is contained in a Borel subgroup of GL2(Fp). Moreover, assume that there is a basis of E[p] so that G acts as χ ∗ 0 ψ where χ and ψ are nontrivial characters. Then H i (G, E[p]) = 0 for all i ≥ 0. Proof. Let W = ( 1 ∗ ∗ ∗ 0 1 ) be the unique p-Sylow subgroup of ( 0 ∗ ) ⊂ GL2(Fp). We may assume W ⊂ G, for otherwise G has order prime to p, and the cohomology vanishes. We begin by explicitly computing H j (W, E[p]) using the fact that W is cyclic, generated by w = ( 1 1 0 1 ). Recall that for cyclic groups we can compute cohomology using the projective resolution · · · → Z[W ] → Z[W ] → Z → 0 where the boundary maps alternate between multiplication by w−1 and Norm(w) = p−1 i=0 wi . Then we see that Ker(1 − w)/ Im(Norm(w)) = 〈( 1 0 )〉 , if j is even, H j (W, E[p]) = Ker(Norm(w))/ Im(1 − w) = F2 p/ 〈( 1 0 )〉 , if j is odd. 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. Consider the Hochschild- Serre spectral sequence H i (G/W, H j (W, E[p])) ⇒ H i+j (G, E[p]). For i > 0, since #(G/W ) is prime to p, and H j (W, E[p]) is a p-group for all j, the group H i (G/W, H j (W, E[p])) is trivial. But when i = 0 we have just computed that H i (G/W, H j (W, E[p])) = H j (W, E[p]) G/W = 0, so the entire spectral sequence is trivial, and we conclude that H n (G, E[p]) = 0 for all n ≥ 0. In this section we show how vanishing of H i (Q(E[p])/Q, E[p]) follows from a statement about torsion and rational isogenies. Note that E has no Q-rational p-isogeny if and only if ρ E,p is irreducible.
COMPUTATIONAL VERIFICATION OF BSD 27 Proposition 5.8. If p is an odd prime and E has no Q-rational p-isogeny, then H i (Q(E[p])/Q, E[p]) = 0 for all i > 0. Proof. Our hypothesis that E has no Q-rational p-isogeny implies that ρ E,p is irreducible. As we already noted, the problem reduces to the case when either G is contained in a Borel subgroup or G = GL2(Fp). The latter case follows from Proposition 5.3. The former case contradicts the hypothesis since the module E[p] is reducible as a module over a Borel subgroup. For the above result, we used the irreducibility of the representation to deal with the case when G was contained in a Borel subgroup. The following proposition completes the proof of the general case: Proposition 5.9. Suppose p is an odd prime and that E(Q)[p] = 0 and for all elliptic curves E ′ that are p-isogenous to E over Q we have E ′ (Q)[p] = 0. Then H i (Q(E[p])/Q, E[p]) = 0 for all i > 0. Proof. If E admits no p-isogeny, then Proposition 5.8 implies the required vanishing. Thus we may assume that E admits a rational p-isogeny, so E[p] is reducible, and G = Im(ρE,p) is contained in a Borel subgroup. In particular, for some basis of E[p], the image G acts as χ ∗ 0 ψ for characters χ and ψ. If both χ and ψ are nontrivial, then Lemma 5.7 implies the proposition and we are done. Thus assume that either χ or ψ is trivial. First suppose that χ is trivial. Then all matrices of the above form fix ( 1 0 ). Therefore there is a point of E[p] fixed by the action of G, which contradicts the assumption that E(Q)[p] = 0. Next suppose that ψ is trivial. Matrices of the above form preserve the line generated by ( 1 0 ), so this line forms a Gal(Q/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 under this isogeny of the line generated by ( 0 1 ) is a 1-dimensional subspace of E′ [p], and since ψ = 1, Gal(Q/Q) acts trivially on this subspace (we have an isomorphism of Galois modules E/ 〈( 1 0 )〉 ∼ = E ′ ). Thus, E ′ (Q)[p] is nontrivial, contradicting our assumption. References [ABC + ] B. Allombert, K. Belabas, H. Cohen, X. Roblot, and I. Zakharevitch, PARI/GP, http://pari.math.u-bordeaux.fr/. [ARS05] A. Agashe, K. A. Ribet, and W. A. Stein, The Manin constant, congruence primes, and the modular degree, Preprint, http://www.williamstein.org/papers/manin-agashe/, With an appendix by J. Cremona (2005). [AS05] A. Agashe and W. Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp. 74 (2005), no. 249, 455–484 (electronic), With an appendix by J. Cremona and B. Mazur. MR 2085902 [BCDT01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic). MR 2002d:11058 [BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265, Computational algebra and number theory (London, 1993). MR 1 484 478
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