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

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