L-functions of Symmetric Products of the Kloosterman Sheaf over Z

TL;DR

This paper constructs a geometric scheme over integers whose zeta function's local factors match the L-functions of symmetric and tensor powers of the Kloosterman sheaf, linking number theory and algebraic geometry.

Contribution

It explicitly relates L-functions of symmetric products of Kloosterman sheaves to the zeta functions of a constructed algebraic scheme over integers.

Findings

Constructed a scheme over Z matching L-functions of symmetric Kloosterman sheaves.

Established similar results for tensor and wedge powers of the sheaf.

Connected number-theoretic L-functions with geometric zeta functions.

Abstract

The classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$ give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m, {\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloosterman sheaf. Let $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$ be the $L$-function of the $k$-fold symmetric product of ${\rm Kl}_{n+1}$. We construct an explicit virtual scheme $X$ of finite type over ${\rm Spec} {\bf Z}$ such that the $p$-Euler factor of the zeta function of $X$ coincides with $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$. We also prove similar results for $\otimes^k {\rm Kl}_{n+1}$ and $\bigwedge^k {\rm Kl}_{n+1}$.