$L$-functions of Kloosterman sheaves

TL;DR

This paper extends the understanding of motivic L-functions associated with symmetric powers of Kloosterman sheaves, proving new conjectures and exploring their properties for higher dimensions beyond the known case of n=1.

Contribution

It generalizes previous results on motivic L-functions of Kloosterman sheaves to higher dimensions and proves several conjectures relating traces of sheaves to modular form coefficients.

Findings

Proved meromorphic continuation and functional equations for higher-dimensional motivic L-functions.

Established conjectures linking Kloosterman sheaves traces to Fourier coefficients of modular forms.

Extended the scope of known results from n=1 to n>1 cases.

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 Fres\'an, 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.