On the possible adelic indices of certain families of elliptic curves

Abstract: A well-known theorem of Serre bounds the largest prime $\ell$ for which the mod $\ell$ Galois representation of a non-CM elliptic curve $E/\mathbb{Q}$ is nonsurjective. Serre asked whether a universal bound on the largest nonsurjective prime might exist. Significant partial progress has been made toward this question. Lemos proved that it has an affirmative answer for all $E$ admitting a rational cyclic isogeny. Zywina offered a more ambitious conjecture about the possible adelic indices that can occur as $E$ varies. We will discuss an ongoing project (joint with Tyler Genao, Jacob Mayle, and Rakvi) that extends Lemos's result to prove Zywina's conjecture for certain families of elliptic curves.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar