## STNB2020(34th edition)

### Images of residual Galois representations of elliptic curves without CM

Pedro Lemos

#### Abstract

It is a fact that when $p$ is a prime larger than 37, the image of the Galois representation modulo $p$ of an elliptic curve defined over the rationals and without complex multiplication is either the whole of ${\rm GL}_2({\mathbb F}_p)$, or is contained in the normaliser of a non-split Cartan subgroup. In this talk, I will show that, in the case where the representation is not surjective, this image must, in fact, be the whole normaliser of a non-split Cartan subgroup for $p$ large enough. This is joint work in progress with Samuel Le Fourn.