$L$-functions of Kloosterman sheaves
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909340569436160 |
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| author | Qin, Yichen |
| author_facet | Qin, Yichen |
| contents | In this article, we study a family of motives $\mathrm{M}_{n+1}^k$ associated with the symmetric power of Kloosterman sheaves, as constructed by Fresán, Sabbah, and Yu. They demonstrated that for $n=1$, the motivic $L$-functions of $\mathrm{M}_{2}^k$ extend meromorphically to $\mathbb{C}$ and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the motivic $L$-functions of some of the motives $\mathrm{M}_{n+1}^k$, with $n>1$, as well as other related $2$-dimensional motives. In particular, we prove several conjectures of Evans type, which relate traces of Kloosterman sheaves and Fourier coefficients of modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04882 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L$-functions of Kloosterman sheaves Qin, Yichen Algebraic Geometry Number Theory In this article, we study a family of motives $\mathrm{M}_{n+1}^k$ associated with the symmetric power of Kloosterman sheaves, as constructed by Fresán, Sabbah, and Yu. They demonstrated that for $n=1$, the motivic $L$-functions of $\mathrm{M}_{2}^k$ extend meromorphically to $\mathbb{C}$ and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the motivic $L$-functions of some of the motives $\mathrm{M}_{n+1}^k$, with $n>1$, as well as other related $2$-dimensional motives. In particular, we prove several conjectures of Evans type, which relate traces of Kloosterman sheaves and Fourier coefficients of modular forms. |
| title | $L$-functions of Kloosterman sheaves |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2305.04882 |