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

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dc.creatorFu, Lei
dc.creatorWan, Daqing
dc.date2007-10-16
dc.date.accessioned2026-07-07T08:36:34Z
dc.date.available2026-07-07T08:36:34Z
dc.descriptionThe classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$ give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m, {\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloosterman sheaf. Let $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$ 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 $\otimes^k {\rm Kl}_{n+1}$ and $\bigwedge^k {\rm Kl}_{n+1}$.
dc.description16 pages
dc.identifierhttps://arxiv.org/abs/0710.2949
dc.identifierhttp://arxiv.org/abs/0710.2949
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/140107
dc.subjectAlgebraic Geometry
dc.subjectNumber Theory
dc.subject14F20, 11L05
dc.titleL-functions of Symmetric Products of the Kloosterman Sheaf over Z
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