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

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 $i=1,\dots,n$.