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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.09843 |
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| _version_ | 1866909137760157696 |
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| author | Le, Daniel Hung, Bao Viet Le Morra, Stefano |
| author_facet | Le, Daniel Hung, Bao Viet Le Morra, Stefano |
| contents | Let $F/F^+$ be a CM extension and $H_{/F^+}$ a definite unitary group in three variables that splits over $F$. We describe Hecke isotypic components of mod $p$ algebraic modular forms on $H$ at first principal congruence level at $p$ and "minimal" level away from $p$ in terms of the restrictions of the associated Galois representation to decomposition groups at $p$ when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod $p$ Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $K_1$-invariants in the mod $p$ cohomology of $U(3)$ arithmetic manifolds Le, Daniel Hung, Bao Viet Le Morra, Stefano Number Theory Representation Theory Let $F/F^+$ be a CM extension and $H_{/F^+}$ a definite unitary group in three variables that splits over $F$. We describe Hecke isotypic components of mod $p$ algebraic modular forms on $H$ at first principal congruence level at $p$ and "minimal" level away from $p$ in terms of the restrictions of the associated Galois representation to decomposition groups at $p$ when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod $p$ Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support. |
| title | $K_1$-invariants in the mod $p$ cohomology of $U(3)$ arithmetic manifolds |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2403.09843 |