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 $$\overline{\rho }_{E,p}$$ 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 $$E_1,\cdots ,E_n$$ 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 $$\overline{\rho }_{E,p}$$ is surjective, or $$\overline{\rho }_{E,p} \sim \overline{\rho }_{E_i,p}$$ for some $$i=1,\dots ,n$$ .