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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 Q̄l-sheaf Kl n+1 on Gm, Fp}=P1Fp-

, which we call the Kloosterman sheaf. Let L p (G m, Fp , 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, Fp , Sym k Kl n+1, s). We also prove similar results for ⊗k Kln+1 and k Kln+1. © 2008 Springer-Verlag.

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