$L$-functions of Kloosterman sheaves

Abstract

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.

arXiv commentCorrections and improvements based on referee reports

Cited by in corpus (1)

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