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Mathematics > Number Theory

arXiv:2203.02045 (math)
[Submitted on 3 Mar 2022]

Title:Serre weights, Galois deformation rings, and local models

Authors:Daniel Le, Bao Viet Le Hung
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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 Mézard 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.
Comments: 24 pages
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Representation Theory (math.RT)
MSC classes: 11F33, 11F80
Cite as: arXiv:2203.02045 [math.NT]
  (or arXiv:2203.02045v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.2203.02045

Submission history

From: Daniel Le [view email]
[v1] Thu, 3 Mar 2022 22:29:33 UTC (37 KB)
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