Higher Period Integrals and Derivatives of L-functions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918425859719168 |
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| author | Liu, Shurui Wang, Zeyu |
| author_facet | Liu, Shurui Wang, Zeyu |
| contents | We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin--Selberg $L$-function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Period Integrals and Derivatives of L-functions Liu, Shurui Wang, Zeyu Number Theory Algebraic Geometry Representation Theory We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality à la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin--Selberg $L$-function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties. |
| title | Higher Period Integrals and Derivatives of L-functions |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2504.00275 |