$ell$-adic images of Galois for elliptic curves over $mathbb{Q}$

We discuss the $ell$-adic case of Mazurs 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 $ellge 13$ 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, compute explicit equations for most of them, and classify the rational points on all of them except $X_{{rm ns}}^{+}(N)$, 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 subgroups that arise for any prime $ell$ and any $E/mathbb{Q}$ without complex multiplication; these exceptional subgroups are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional subgroups is complete and show that any counterexamples must arise from unexpected rational points on $X_{rm ns}^+(ell)$ with $ellge 17$, or one of the six modular curves noted above. This gives us an efficient algorithm to compute the $ell$-adic images of Galois for any non-CM elliptic curve over $mathbb{Q}$. In an appendix with John Voight we generalize Ribets observation that simple abelian varieties attached to newforms on $Gamma_1(N)$ are of ${rm GL}_2$-type; this extends Kolyvagins theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.