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 a recent 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.

number theory

Audience: researchers in the topic

Five College Number Theory Seminar