The Breuil--Mezard conjecture for quaternion algebras

TL;DR

This paper extends the Breuil--Mezard conjecture to quaternion algebras, linking it to the GL_2 case, and establishes a mod p Jacquet--Langlands correspondence for finite fields.

Contribution

It formulates a quaternion algebra version of the Breuil--Mezard conjecture and proves it assuming the GL_2 case, also establishing a mod p Jacquet--Langlands correspondence.

Findings

Reduction of the quaternion algebra conjecture to the GL_2 case

Establishment of a mod p Jacquet--Langlands correspondence

New insights into representations over finite fields

Abstract

We formulate a version of the Breuil--Mezard conjecture for quaternion algebras, and show that it follows from the Breuil--Mezard conjecture for GL_2. In the course of the proof we establish a mod p analogue of the Jacquet--Langlands correspondence for representations of GL_2(k), k a finite field of characteristic p.