## STNB2020(34th edition)

### The Serre filtration on mod p Hilbert modular forms of level p

Fred Diamond

#### Abstract

A result of Serre relates the space of mod $p$ modular forms of level $\Gamma_1(Np)$ and weight 2 to the spaces of mod $p$ modular forms of level $\Gamma_1(N)$ and weight between 2 and $p$+1. I’ll explain a generalization of this to the context of Hilbert modular forms which is motivated by the interplay between "algebraic" and "geometric" Serre weight conjectures for mod $p$ Galois representations. The resulting filtration on mod p Hilbert modular forms of parallel weight 2 and pro-$p$ Iwahori level mirrors, via a mod p geometric Jacquet-Langlands correspondence, the more evident filtration on cohomology coming from the mod $p$ representation theory of ${\rm GL}_2$. This is joint work with P. Kassaei and S. Sasaki.