Formes modulaires de Hilbert modulo p et valeurs d'extensions galoisiennes

TL;DR

This paper investigates the relationship between Hilbert modular forms modulo p and Galois extension values, providing a method to recover local Galois extensions from the action of GL_2 on cohomology.

Contribution

It introduces a new approach to determine local Galois extension classes from the mod p cohomology of Hilbert modular forms under certain conditions.

Findings

Recovery of local Galois extension classes from cohomology actions

Extension between characters determined by GL_2(F_v) action

Applicable to generic reducible mod p Galois representations

Abstract

Let F be a totally real field, v an unramified place of F dividing p and rho a continuous irreducible two-dimensional mod p representation of G_F such that the restriction of rho to G_{F_v} is reducible and sufficiently generic. If rho is modular (and satisfies some weak technical assumptions), we show how to recover the corresponding extension between the two characters of G_{F_v} in terms of the action of GL_2(F_v) on the cohomology mod p.