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

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 XH , compute explicit equations for all but three of them and classify the rational points on all of them except Xₙₛ⁺(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 Q̄ -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 Xₙₛ⁺( ) 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 GL₂ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of XH .