Towards the $p$-adic derived Hecke algebra for weight one forms
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908403814629376 |
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| author | Zhang, Robin |
| author_facet | Zhang, Robin |
| contents | This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris and Venkatesh, we formulate a conjecture relating the action of $p$-adic derived Hecke operators on cusp forms of weight $1$ and level $Γ_1(N)$ to the $p$-adic logarithm of the Stark unit for the corresponding adjoint Deligne-Serre representation. This new $p$-adic conjecture can be viewed as complementary to the Harris-Venkatesh conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards the $p$-adic derived Hecke algebra for weight one forms Zhang, Robin Number Theory 11R42 (Primary) 11G18, 11F41, 11F67, 11F75, 11F85, 11R27 (Secondary) This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris and Venkatesh, we formulate a conjecture relating the action of $p$-adic derived Hecke operators on cusp forms of weight $1$ and level $Γ_1(N)$ to the $p$-adic logarithm of the Stark unit for the corresponding adjoint Deligne-Serre representation. This new $p$-adic conjecture can be viewed as complementary to the Harris-Venkatesh conjecture. |
| title | Towards the $p$-adic derived Hecke algebra for weight one forms |
| topic | Number Theory 11R42 (Primary) 11G18, 11F41, 11F67, 11F75, 11F85, 11R27 (Secondary) |
| url | https://arxiv.org/abs/2506.09139 |