A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908680068268032 |
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| author | Comtat, Félicien |
| author_facet | Comtat, Félicien |
| contents | We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $\rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $\rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $\rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P \backslash G / H$, for each parabolic subgroup $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06027 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$ Comtat, Félicien Number Theory 11F72 (primary) 11F46, 11F30, 11F70 (secondary) We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $\rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $\rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $\rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P \backslash G / H$, for each parabolic subgroup $P$. |
| title | A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$ |
| topic | Number Theory 11F72 (primary) 11F46, 11F30, 11F70 (secondary) |
| url | https://arxiv.org/abs/2409.06027 |