\(L\)-functions of symmetric products of the Kloosterman sheaf over \(Z\)
type: Item
instance of: scholarly article
title: \(L\)-functions of symmetric products of the Kloosterman sheaf over \(Z\)
zbMATH Open document ID: 1151.11041
author: Lei Fu
author: Daqing Wan
published in: Mathematische Annalen
publication date: 2008-09-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.2949
review text: The authors construct explicit virtual schemes \(X\), \(Y\) of finite type over \(\operatorname{Spec}{\mathbb Z}\) such that, for each prime \(p\), the \(p\)-Euler factor of their zeta functions (that is, the zeta function of their reduction modulo \(p\)) is the \(L\)-function of the \(k\)-th tensor power (respectively the \(k\)-th symmetric power) of the Kloosterman sheaf \(\text{Sym}^k{\text{Kl}}_{n+1}\), which is known to be a polynomial [the authors, J. Reine Angew. Math. 589, 79--103 (2005; Zbl 1165.11331)]. This seems surprising at first, since the Kloosterman sheaf is an object which is intrinsically defined in characteristic \(p\) by using an additive character \(\psi\) of the finite field \({\mathbb F}_p\), and can not be lifted to characteristic zero. However, it was shown in [loc. cit.] by the same authors that the global \(L\)-function of its symmetric powers (resp its tensor powers) is a polynomial with \textit{integer} coefficients, and it does not depend on the character chosen. Here they construct a geometric object in characteristic zero which gives rise to these \(L\)-functions for each prime \(p\). This fact adds evidence to the conjecture (known to be true in some cases for \(n=1\)) that the infinite product \[ \prod_p L_p({\mathbb G}_{m,{\mathbb F}_p},\text{Sym}^k{\text{Kl}}_{n+1},s) \] is automorphic and thus extends to a meromorphic function in \(s\in{\mathbb C}\).
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 5323303
reviewed by: Antonio Rojas-León
MaRDI profile type: Publication
OpenAlex ID: W2082276218
arXiv ID: 0710.2949
cites work: CONGRUENCES FOR SUMS OF POWERS OF KLOOSTERMAN SUMS
cites work: Q4119314
cites work: Seventh power moments of Kloosterman sums
cites work: L-functions for symmetric products of Kloosterman sums
cites work: Trivial factors for \(L\)-functions of symmetric products of Kloosterman sheaves
cites work: Q4002278
cites work: Sur quelques points d'algèbre homologique
cites work: On families of pure slope \(L\)-functions
cites work: The modularity of the Barth-Nieto quintic and its relatives
cites work: Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116)
cites work: Fourier transformat, constants of the functional equations and Weil conjecture
cites work: Q4013269
cites work: Symmetric powers of the p-adic Bessel equation.
cites work: Dwork's conjecture on unit root zeta functions
DOI: 10.1007/S00208-008-0240-5
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description: scientific article