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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2504.00821 |
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| _version_ | 1866909560716918784 |
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| author | Zhao, Ruishen |
| author_facet | Zhao, Ruishen |
| contents | We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)/\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a non-very-Eisenstein point $ϕ$ on the old component (generically parametrizing forms old at $l$) satisfying $T_{l}(ϕ)=l(l^3+1)$, then this point also lies in the new component (generically parametrizing forms new at $l$). This provides a $p$-adic analogue of Bellaïche and Graftieaux's mod $p$ level raising for classical automorphic forms on $U(3)$, and also generalizes James Newton's $p$-adic level raising results for definite quaternion algebras. Key ingredients include abelian Ihara lemma (proved for any definite unitary group $U(n)$) and some duality arguments about certain Hecke modules. Finally we also discuss some methods to construct such points explicitly and further development. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | p-adic level raising on the eigenvariety for U(3) Zhao, Ruishen Number Theory Representation Theory We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)/\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a non-very-Eisenstein point $ϕ$ on the old component (generically parametrizing forms old at $l$) satisfying $T_{l}(ϕ)=l(l^3+1)$, then this point also lies in the new component (generically parametrizing forms new at $l$). This provides a $p$-adic analogue of Bellaïche and Graftieaux's mod $p$ level raising for classical automorphic forms on $U(3)$, and also generalizes James Newton's $p$-adic level raising results for definite quaternion algebras. Key ingredients include abelian Ihara lemma (proved for any definite unitary group $U(n)$) and some duality arguments about certain Hecke modules. Finally we also discuss some methods to construct such points explicitly and further development. |
| title | p-adic level raising on the eigenvariety for U(3) |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2504.00821 |