A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety
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arXiv
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| Format: | Preprint |
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2020
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| author | Balasubramanyam, Baskar Bergdall, John Longo, Matteo |
| author_facet | Balasubramanyam, Baskar Bergdall, John Longo, Matteo |
| contents | Let $F$ be a totally real field and $\mathscr{E}$ the middle-degree eigenvariety for Hilbert modular forms over $F$, constructed by Bergdall--Hansen. We study the ramification locus of $\mathscr{E}$ in relation to the $p$-adic properties of adjoint $L$-values. The connection between the two is made via an analytic twisted Poincaré pairing over affinoid weights, which interpolates the classical twisted Poincaré pairing for Hilbert modular forms, itself known to be related to adjoint $L$-values by works of Ghate and Dimitrov. The overall strategy connecting the pairings to ramification is based on the theory of $L$-ideals, which was used by Bellaïche and Kim in the case where $F = \mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_05237 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety Balasubramanyam, Baskar Bergdall, John Longo, Matteo Number Theory 11F85, 11F41 (11F67, 11F33, 11G18) Let $F$ be a totally real field and $\mathscr{E}$ the middle-degree eigenvariety for Hilbert modular forms over $F$, constructed by Bergdall--Hansen. We study the ramification locus of $\mathscr{E}$ in relation to the $p$-adic properties of adjoint $L$-values. The connection between the two is made via an analytic twisted Poincaré pairing over affinoid weights, which interpolates the classical twisted Poincaré pairing for Hilbert modular forms, itself known to be related to adjoint $L$-values by works of Ghate and Dimitrov. The overall strategy connecting the pairings to ramification is based on the theory of $L$-ideals, which was used by Bellaïche and Kim in the case where $F = \mathbb{Q}$. |
| title | A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety |
| topic | Number Theory 11F85, 11F41 (11F67, 11F33, 11G18) |
| url | https://arxiv.org/abs/2011.05237 |