On Serre's conjecture for mod l Galois representations over totally real fields

TL;DR

This paper generalizes Serre's conjecture from the rationals to totally real fields for mod l Galois representations, proposing a local-global principle based on a mod l Langlands philosophy.

Contribution

It formulates a new analogue of Serre's conjecture for totally real fields and introduces a mod l local-global principle using ideas from Emerton and Vigneras.

Findings

Formulated a generalization of Serre's conjecture for totally real fields.

Proposed a mod l local-global principle for quaternion algebra groups.

Showed how the principle implies the conjecture.

Abstract

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the "weight" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a "mod l Langlands philosophy". Using ideas of Emerton and Vigneras, we formulate a mod l local-global principle for the group D^*, where D is a quaternion algebra over a totally real field, split above l and at 0 or 1 infinite places, and show how it implies the conjecture.