Motivic action for Siegel modular forms
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912678545457152 |
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| author | Horawa, Aleksander Prasanna, Kartik |
| author_facet | Horawa, Aleksander Prasanna, Kartik |
| contents | We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special cases. For a lift $f$ of a Hilbert modular form $f_0$ to ${\rm GSp}(4)$, we produce elements in the motivic cohomology group for which the conjecture holds, using the results of Ramakrishnan on the Asai $L$-function of $f_0$. For a lift $f$ of a Bianchi modular form $f_0$ to ${\rm GSp}(4)$, we show that our conjecture for $f$ is equivalent to the conjecture of Prasanna-Venkatesh for $f_0$, thus establishing a connection between the motivic action conjectures for locally symmetric spaces of non-hermitian type and those for coherent cohomology of Shimura varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04115 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Motivic action for Siegel modular forms Horawa, Aleksander Prasanna, Kartik Number Theory Algebraic Geometry 19E15, 11G18, 11F46, 11F41, 11F67 We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special cases. For a lift $f$ of a Hilbert modular form $f_0$ to ${\rm GSp}(4)$, we produce elements in the motivic cohomology group for which the conjecture holds, using the results of Ramakrishnan on the Asai $L$-function of $f_0$. For a lift $f$ of a Bianchi modular form $f_0$ to ${\rm GSp}(4)$, we show that our conjecture for $f$ is equivalent to the conjecture of Prasanna-Venkatesh for $f_0$, thus establishing a connection between the motivic action conjectures for locally symmetric spaces of non-hermitian type and those for coherent cohomology of Shimura varieties. |
| title | Motivic action for Siegel modular forms |
| topic | Number Theory Algebraic Geometry 19E15, 11G18, 11F46, 11F41, 11F67 |
| url | https://arxiv.org/abs/2307.04115 |