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

Abstract

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

16 pages