Algebra Seminar - dr. G. Turcas University of Babeș-Bolyai

Title:

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

Abstract:

For every integer d ≥ 1, there exists an explicit constant B_d such that the following holds. Let K be a number field of degree d, let q > max{d − 1, 5} be any rational prime that is totally inert in K, and let E be any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime above q. Then, for every rational prime p > B_d, E has an irreducible mod p Galois representation. In this talk, we will discuss the ingredients that go into the proof of the aforementioned result and we will present an application to Diophantine equations. All the results presented are in joint work with Filip Najman.