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Mathematische Annalen

, Volume 342, Issue 2, pp 387–404 | Cite as

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

  • Lei FuEmail author
  • Daqing Wan
Article

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 , 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 \({\otimes^k {\rm Kl}_{n+1}}\) and \({\bigwedge^k {\rm Kl}_{n+1}}\) .

Mathematics Subject Classification (2000)

14F20 11L05 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of CaliforniaIrvineUSA

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