Breuil–Mézard conjectures for central division algebras

Dotto, Andrea

doi:10.2140/ant.2025.19.213

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Breuil–Mézard conjectures for central division algebras

Andrea Dotto

Vol. 19 (2025), No. 2, 213–246

DOI: 10.2140/ant.2025.19.213

Abstract

We formulate an analogue of the Breuil–Mézard 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 ${GL}_{n}$ . 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 $ℓ$ -adic coefficients.

Keywords

Galois deformation theory, smooth representations of $p$-adic groups

Mathematical Subject Classification

Primary: 11S37, 22E50

Milestones

Received: 23 March 2021

Revised: 14 February 2022

Accepted: 11 April 2022

Published: 31 January 2025

Authors

Andrea Dotto
Department of Mathematics King’s College London London United Kingdom

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Vol. 19, No. 2, 2025