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

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _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 ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).