-adic images of Galois for elliptic curves over (and an appendix with John Voight)

Abstract

We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ . Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ . In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$ .

Computing the endomorphism ring of an elliptic curve over a number field

“…In our implementation of Algorithm 2 we did not incorporate the suggestions in Remarks 11 and 13, both of which might improve the practical performance. We used the intrinsic EndomorphismRingData implemented in the repository [35] described in [34] to perform the endomorphism ring computation, and relied on Magma's builtin IsSupersingular, ClassNumber, and HilbertClassPolynomial intrinsics. The time spent on supersingular testing and class number computation in Algorithm 2 is negligible, but the time spent computing H D is not (it will typically dominate) and our implementation could be substantially improved by using a faster method to compute Hilbert class polynomials, such as [45], or the implementation in Arb [22] used by SageMath.…”

2-adic Galois images of non-CM isogeny-torsion graphs

“…On the other hand, in the tables, the groups are denoted of the form A = N.i.g.n where A is the label of some group in the RZB database and N.i.g.n is the label of that same group coming from the RSVZB database. The term N denotes the level of the group, i the index of the group in GL(2, Z/N Z), g denotes the genus of the modular curve generated by the group, and n is a tiebreaker (see pages 9-10 in [9] for how the groups are organized). Finally, for an elliptic curve E/Q and a positive integer N we denote the image of the 2-adic Galois representation attached to E and the image of the mod-N Galois representation attached to E as ρ E,2 ∞ (G Q ) and ρ E,N (G Q ), respectively.…”

“…The author would like to thank David Zureick-Brown for help on classifying the rational points on the fiber product of H 7 and B 0 (13). We take inspiration from the study of nonhyperelliptic curves of genus 3 in Subsection 8.3 of [9]; especially the first and second examples in the subsection. The modular curve X generated by the group H 7 × B 0 (13) is denoted 52.56.3.1 in the LMFDB.…”

“…Let G be an isogeny-torsion graph associated to a Q-isogeny class of non-CM elliptic curves defined over Q. Then (1) if G is of L 1 type, then the 2-adic Galois Image attached to G is one of the 22 arrangements in Table 10, (2) if G is of L 2 (2) type, then the 2-adic Galois Image attached to G is one of the 80 arrangements in Table 11,(3) if G is of T 4 type, then the 2-adic Galois Image attached to G is one of the 60 arrangements in Table 12, (4) if G is of T 6 type, then the 2-adic Galois Image attached to G is one of the 81 arrangements in Table 13, (5) if G is of T 8 type, then the 2-adic Galois Image attached to G is one of the 53 arrangements in Table 14, (6) if G is of S type, then the 2-adic Galois Image attached to G is one of the 5 arrangements in Table 15, (7) if G is of R 6 type, then the 2-adic Galois Image attached to G is one of the 2 arrangements in Table 16, (8) if G is of R 4 type, then the 2-adic Galois Image attached to G is one of the 13 arrangements in Table 17, (9) if G is of L 3 (9) or L 3 (25) type, then the 2-adic Galois Image attached to G is conjugate to GL(2, Z 2 ) (see Table 18), (10) if G is of L 2 (p) type where p is an odd prime, then the 2-adic Galois Image attached to G is one of the 34 arrangements in Table 19.…”

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“…(2) Relatively prime division fields with non-trivial intersection force the image of ρ E to not be the full direct product of the images of the -adic representations. Recently, there has been great progress in understanding and classifying the ways in which an elliptic curve over Q can fail to have surjective -adic Galois representation; see for example [RZB15,SZ17,RSZB]. On the other hand, there has been recent progress [Mor19,DGJ20,CNLM + ] in determining what composite images can occur.…”