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

2007/10/16 by Lei Fu, Daqing Wan, Fu, Lei +1
Mathematics · #11L05 #14F20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11L05 #msc:14F20

paper · pdf · doi:10.48550/arxiv.0710.2949

16 pages

arxiv created 2007/10/16 · openalex publication_date 2007/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical n-variable Kloosterman sums over the finite field \bf F_p give rise to a lisse \bf Q_l-sheaf \rm Kl_n+1 on \bf G_m, \bf F_p=\bf P¹_\bf F_p-\0,∞\, which we call the Kloosterman sheaf. Let L_p(\bf G_m,\bf F_p, \rm Sym^k\rm Kl_n+1, s) be the L-function of the k-fold symmetric product of \rm Kl_n+1. We construct an explicit virtual scheme X of finite type over \rm Spec \bf Z such that the p-Euler factor of the zeta function of X coincides with L_p(\bf G_m,\bf F_p, \rm Sym^k\rm Kl_n+1, s). We also prove similar results for ⊗^k \rm Kl_n+1 and \bigwedge^k \rm Kl_n+1.

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