A lower bound for classical Kloosterman sums and an application
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915705166757888 |
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| author | Baier, Stephan Das, Jishu Mahajan, Jewel |
| author_facet | Baier, Stephan Das, Jishu Mahajan, Jewel |
| contents | We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound for classical Kloosterman sums and an application Baier, Stephan Das, Jishu Mahajan, Jewel Number Theory Primary 11L05, 11F72 We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$. |
| title | A lower bound for classical Kloosterman sums and an application |
| topic | Number Theory Primary 11L05, 11F72 |
| url | https://arxiv.org/abs/2406.13013 |