Notes on \(G_{2}\), determinants, and equidistribution
type: Item
instance of: scholarly article
title: Notes on \(G_{2}\), determinants, and equidistribution
zbMATH Open document ID: 1065.11060
author: Nicholas M. Katz
published in: Finite Fields and their Applications
publication date: 2004-08-06T00:00:00Z
review text: The paper examines the distribution law for the one-parameter family of exponential sums over \({\mathbb F}_p\): \[ \sum_{x\pmod p, x\neq 0}\chi_2(x)\exp(2\pi i(x^7+tx)/p) \] where \(\chi_2(x)=(x/p)\) is the quadratic character and \(p\) is any prime other than 2 or 7. The first observation is that for large characteristic (\(p>15\)), after dividing the sums by a suitable normalising factor, they are distributed like the traces of random elements of the exceptional group \(G_2\). The author shows how to make the distribution law explicit in all characteristics and explains the exceptional cases of characteristic 3 and 13 where a different group intervenes. In fact, the approach deals with the families of sums in which 7 is replaced by an odd integer \(n\geq 3\), and then the group \(G_2\) is replaced by \(\text{SO}(n)\). The argument is a tour de force based on the author's work on exponential sums and monodromy groups.
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 11T23
zbMATH DE Number: 2083586
zbMATH Keywords: exponential sums
zbMATH Keywords: equidistribution
zbMATH Keywords: monodromy
reviewed by: John H. Loxton
MaRDI profile type: Publication
full work available at URL: j.ffa.2003.11.002
OpenAlex ID: W1984836893
cites work: Lectures on exceptional Lie groups. Ed. by Zafer Mahmud and Mamoru Mimura
cites work: Chevalley groups of type \(G_ 2\) as the group of a trilinear form
cites work: Finite subgroups of G2,(c)
cites work: Q3684278
cites work: Sums of Betti numbers in arbitrary characteristic.
cites work: Exponential Sums and Differential Equations. (AM-124)
cites work: Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116)
cites work: \(L\)-functions and monodromy: four lectures on Weil II
cites work: On the monodromy groups attached to certain families of exponential sums
cites work: A semicontinuity result for monodromy under degeneration
cites work: Q4226082
cites work: Fourier transformat, constants of the functional equations and Weil conjecture
cites work: Increasing subsequences and the classical groups
cites work: Linear algebraic groups.
DOI: 10.1016/J.FFA.2003.11.002
description: scientific article; zbMATH DE number 2083586