$\ell $ -adic images of Galois for elliptic curves over $\mathbb {Q}$ (and an appendix with John Voight). (12th August 2022) - Rouse, Jeremy; Sutherland, Andrew V.; Zureick-Brown, David
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 $X_H$ .