## Barcelona Fall Workshop on Number Theory II

### A geometric Serre weight conjecture

#### Presenters

Fred Diamond (King's College London)

#### Abstract

Serre’s Conjecture, now a theorem of Khare and Wintenberger, asserts that every odd, irreducible two-dimensional mod p representation of the absolute Galois group of Q arises from a modular form, and the “weight part” of the conjecture determines the minimal weight of such a form. More general Serre weight conjectures have been formulated for Galois representations associated to automorphic forms, but only in the context of regular algebraic weights. I’ll discuss a geometric variant for Hilbert modular forms that allows irregular weight. I’ll explain some new features, and the reason a naive version of the conjecture is false. I’ll also discuss the proof in some cases involving partial weight one, and some connections with the geometry of Hilbert modular varieties. This is joint work with Shu Sasaki.