On Serre's uniformity conjecture for semistable elliptic curves over totally real fields

TL;DR

This paper proves an effective uniformity result for Galois representations of semistable elliptic curves over totally real fields, showing they are either surjective or closely related to CM curves for large primes.

Contribution

It combines modularity, level lowering, and existing bounds to explicitly classify Galois representations of semistable elliptic curves over totally real fields for large primes.

Findings

Existence of an effectively computable constant C_{K,S}

Classification of Galois representations for primes > C_{K,S}

Reduction to a finite set of CM elliptic curves

Abstract

Let $K$ be a totally real field, and let $S$ be a finite set of non-archimedean places of $K$. It follows from the work of Merel, Momose and David that there is a constant $B_{K,S}$ so that if $E$ is an elliptic curve defined over $K$, semistable outside $S$, then for all $p>B_{K,S}$, the representation $\bar{\rho}_{E,p}$ is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant $C_{K,S}$, and an effectively computable set of elliptic curves over $K$ with CM $E_1,\dotsc,E_n$ such that the following holds. If $E$ is an elliptic curve over $K$ semistable outside $S$, and $p>C_{K,S}$ is prime, then either $\bar{\rho}_{E,p}$ is surjective, or $\bar{\rho}_{E,p} \sim \bar{\rho}_{E_i,p}$ for some $i=1,\dots,n$.