-adic images of Galois for elliptic curves over Q (and an appendix with John Voight). (12th August 2022) - Rouse, Jeremy; Sutherland, Andrew V.; Zureick-Brown, David
Abstract: We discuss the -adic case of Mazur's 'Program B' over Q : the problem of classifying the possible images of -adic Galois representations attached to elliptic curves E over Q, equivalently, classifying the rational points on the corresponding modular curves. The primes =2 and 13 are addressed by prior work, so we focus on the remaining primes = 3, 5, 7, 11 . For each of these , we compute the directed graph of arithmetically maximal -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_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 -adic images that are known to arise for infinitely many Q -isomorphism classes of elliptic curves E/Q, we find only 22 exceptional images that arise for any prime and any E/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_ns^+() with 19, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the -adic images of Galois for any elliptic curve over Q . In an appendix with John Voight, we generalise Ribet's observation that simple abelian varieties attached to newforms on Γ _1(N) are of 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 .