$ell$-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 $X_H$.

Diophantine and tropical geometry, and uniformity of rational points on curves

We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis of zeros of integrals on domains admitting monodromy.

Uniform bounds for the number of rational points on curves of small Mordell--Weil rank

Let $X$ be a curve of genus $ggeq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{rm tors},dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumford, and was proved to be so by Raynaud [Ray83]. We give an explicit uniform bound on $#X(F)$ when $X$ has Mordell--Weil rank $rleq g-3$. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of $J$ lying on the image of $X$ under an Abel--Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on $X$ when the reduction type of $X$ is highly degenerate. Our methods combine Chabauty--Colemans $p$-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.