L-functions of symmetric products of the Kloosterman sheaf over Z

Abstract

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