Breuil-M\'ezard conjectures for central division algebras

TL;DR

This paper extends the Breuil-Mézard conjecture to the units of central division algebras over p-adic fields, establishing a transfer of inertial types and Serre weights via Deligne-Lusztig theory and proving compatibility with mod p reduction.

Contribution

It formulates an analogue of the Breuil-Mézard conjecture for division algebra units and constructs a transfer of inertial types and Serre weights, linking it to the conjecture for GL_n.

Findings

Established the conjecture for division algebra units assuming GL_n conjecture.

Constructed a transfer of inertial types and Serre weights using Deligne-Lusztig theory.

Proved compatibility with mod p reduction and extended results to adic coefficients.

Abstract

We formulate an analogue of the Breuil-M\'ezard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for $\mathrm{GL}_n$. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne-Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet-Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $\ell$-adic coefficients.