A lower bound for the discrepancy in a Sato-Tate type measure
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916495449128960 |
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| author | Das, Jishu |
| author_facet | Das, Jishu |
| contents | Let $S_k(N)$ denote the space of cusp forms of even integer weight $k$ and level $N$. We prove an asymptotic for the Petersson trace formula for $S_k(N)$ under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato-Tate distribution for levels not divisible by $8$. This generalizes a result of Jung and Sardari for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda for the distribution of eigenvalues $λ_{p^2}(f)$ where $f$ is a Hecke eigenform and $p$ is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights $k_n$ such that discrepancy in the analogue distribution obtained by Omar and Mazhouda has a lower bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_18798 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A lower bound for the discrepancy in a Sato-Tate type measure Das, Jishu Number Theory 11F25, 11F72 (Primary), 11L05 (Secondary) Let $S_k(N)$ denote the space of cusp forms of even integer weight $k$ and level $N$. We prove an asymptotic for the Petersson trace formula for $S_k(N)$ under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato-Tate distribution for levels not divisible by $8$. This generalizes a result of Jung and Sardari for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda for the distribution of eigenvalues $λ_{p^2}(f)$ where $f$ is a Hecke eigenform and $p$ is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights $k_n$ such that discrepancy in the analogue distribution obtained by Omar and Mazhouda has a lower bound. |
| title | A lower bound for the discrepancy in a Sato-Tate type measure |
| topic | Number Theory 11F25, 11F72 (Primary), 11L05 (Secondary) |
| url | https://arxiv.org/abs/2311.18798 |