Title: Computing Galois Images of Picard curves and Hilbert modular forms

Abstract: Inspired by Serre's Uniformity Conjecture, Sutherland and Zywina have independently developed algorithms to compute the mod ell Galois images associated to large databases of elliptic curves. Based on this data, they have given conjectures refining that of Serre. For genus two curves, an algorithm of Dieulefait has recently been implemented, leading in turn to algorithms which compute isogeny classes of principally polarised abelian surfaces (see next week's talk!).

In this talk, I will discuss a general, computationally efficient, method to rule out 1 dimensional factors from compatible systems of Galois representations, along with other ingredients which go into such an algorithm for Picard curves and Hilbert modular forms. Time (and technology) dependent, I will give a few live demonstrations and discuss results and interesting examples found so far.

This is partial joint work with Shiva Chidambaram.