$L$-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.

Corrections and improvements based on referee reports