A local sign decomposition for symplectic self-dual Galois representations of rank two
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918129970446336 |
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| author | Burungale, Ashay Kobayashi, Shinichi Nakamura, Kentaro Ota, Kazuto |
| author_facet | Burungale, Ashay Kobayashi, Shinichi Nakamura, Kentaro Ota, Kazuto |
| contents | We prove the existence of a new structure on the first Galois cohomology of generic families of symplectic self-dual $p$-adic representations of $G_{\mathbb{Q}_p}$ of rank two (a local sign decomposition): a functorial decomposition into free rank one Lagrangian submodules which encodes the $p$-adic variation of Bloch--Kato subgroups via completed epsilon constants, mirroring a symplectic structure.
The local sign decomposition has diverse local as well as global arithmetic consequences. This includes compatibility of the Mazur--Rubin arithmetic local constant and completed epsilon constants, answering a question of Mazur and Rubin. The compatibility leads to new cases of the $p$-parity conjecture for Hilbert modular forms at supercuspidal primes $p$. We also formulate and prove an analogue of Rubin's conjecture over ramified quadratic extensions of $\mathbb{Q}_p$. Using it, we construct an integral $p$-adic $L$-function for anticyclotomic deformation of a CM elliptic curve at primes $p$ ramified in the CM field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A local sign decomposition for symplectic self-dual Galois representations of rank two Burungale, Ashay Kobayashi, Shinichi Nakamura, Kentaro Ota, Kazuto Number Theory We prove the existence of a new structure on the first Galois cohomology of generic families of symplectic self-dual $p$-adic representations of $G_{\mathbb{Q}_p}$ of rank two (a local sign decomposition): a functorial decomposition into free rank one Lagrangian submodules which encodes the $p$-adic variation of Bloch--Kato subgroups via completed epsilon constants, mirroring a symplectic structure. The local sign decomposition has diverse local as well as global arithmetic consequences. This includes compatibility of the Mazur--Rubin arithmetic local constant and completed epsilon constants, answering a question of Mazur and Rubin. The compatibility leads to new cases of the $p$-parity conjecture for Hilbert modular forms at supercuspidal primes $p$. We also formulate and prove an analogue of Rubin's conjecture over ramified quadratic extensions of $\mathbb{Q}_p$. Using it, we construct an integral $p$-adic $L$-function for anticyclotomic deformation of a CM elliptic curve at primes $p$ ramified in the CM field. |
| title | A local sign decomposition for symplectic self-dual Galois representations of rank two |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.17776 |