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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.19552 |
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| _version_ | 1866909922334081024 |
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| author | Harris, Michael Hsieh, Ming-Lun Yamana, Shunsuke |
| author_facet | Harris, Michael Hsieh, Ming-Lun Yamana, Shunsuke |
| contents | We construct the five-variable $p$-adic $L$-function attached to Hida families on $\mathrm U(2,1)\times\mathrm U(1,1)$, interpolating the square-root of Rankin-Selberg $L$-values in the \emph{shifted piano} range. Our construction relies on a new theta operator and its $p$-adic variation which plays a role analogous to the classical Ramanujan-Serre theta operator in Hida's $p$-adic Rankin-Selberg method. The interpolation formula, including the modified Euler factors at $p$ and at the real place, is consistent with the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-adic $L$-functions for $\mathrm U(2,1)\times\mathrm U(1,1)$ Harris, Michael Hsieh, Ming-Lun Yamana, Shunsuke Number Theory We construct the five-variable $p$-adic $L$-function attached to Hida families on $\mathrm U(2,1)\times\mathrm U(1,1)$, interpolating the square-root of Rankin-Selberg $L$-values in the \emph{shifted piano} range. Our construction relies on a new theta operator and its $p$-adic variation which plays a role analogous to the classical Ramanujan-Serre theta operator in Hida's $p$-adic Rankin-Selberg method. The interpolation formula, including the modified Euler factors at $p$ and at the real place, is consistent with the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou. |
| title | $p$-adic $L$-functions for $\mathrm U(2,1)\times\mathrm U(1,1)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.19552 |