Serre weights, Galois deformation rings, and local models

TL;DR

This paper surveys recent advances in Serre weight conjectures, linking Galois deformation rings to local models, and reports on new results in tame generic cases involving the geometry of deformation rings and local models.

Contribution

It introduces new constructions of local models that mirror the singularities of Galois deformation rings in tame generic settings, advancing understanding of Serre weight conjectures.

Findings

Established conjectures in tame generic contexts.

Constructed local models with singularities matching deformation rings.

Linked geometry of local models to Galois deformation theory.

Abstract

We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and M\'ezard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights.