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 , SymkKl 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 , SymkKl n+1, s). We also prove similar results for \({\otimes^k {\rm Kl}_{n+1}}\) and \({\bigwedge^k {\rm Kl}_{n+1}}\) .
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The research of L. Fu is supported by the NSFC (10525107).
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Fu, L., Wan, D. L-functions of symmetric products of the Kloosterman sheaf over Z. Math. Ann. 342, 387–404 (2008). https://doi.org/10.1007/s00208-008-0240-5
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DOI: https://doi.org/10.1007/s00208-008-0240-5