Extremal weights and a tameness criterion for mod $p$ Galois representations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917867673354240 |
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| author | Le, Daniel Hung, Bao Viet Le Levin, Brandon Morra, Stefano |
| author_facet | Le, Daniel Hung, Bao Viet Le Levin, Brandon Morra, Stefano |
| contents | We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of our conjecture. We then introduce a new class of weights associated to $n$-dimensional local mod $p$ representations which we call \emph{extremal weights}. Using a ``Levi reduction" property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre's conjecture for unit groups of some division algebras in generic situations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_06442 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extremal weights and a tameness criterion for mod $p$ Galois representations Le, Daniel Hung, Bao Viet Le Levin, Brandon Morra, Stefano Number Theory Representation Theory We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of our conjecture. We then introduce a new class of weights associated to $n$-dimensional local mod $p$ representations which we call \emph{extremal weights}. Using a ``Levi reduction" property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre's conjecture for unit groups of some division algebras in generic situations. |
| title | Extremal weights and a tameness criterion for mod $p$ Galois representations |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2206.06442 |