Sato-Tate equidistribution of Kurlberg-Rudnick sums
type: Item
instance of: scholarly article
zbMATH Open document ID: 1011.11058
DOI: 10.1155/S107379280100037X
author: Nicholas M. Katz
publication date: 2002-04-07T00:00:00Z
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 1656647
zbMATH Keywords: exponential sums
zbMATH Keywords: random matrices
zbMATH Keywords: \(\ell\)-adic representations
zbMATH Keywords: mathematical physics
MaRDI profile type: Publication
full work available at URL: s107379280100037x
OpenAlex ID: W39093169
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published in: IMRN. International Mathematics Research Notices
title: Sato-Tate equidistribution of Kurlberg-Rudnick sums
review text: Let \(\psi\) and \(\chi\) be nontrivial additive and multiplicative complex valued characters on a finite field \(k\). The Kurlberg-Rudnick sums of the title are the complex functions on \(k\) defined by \(H(\psi,\chi;t)= \sum_{x\in k}\psi(x^2+tx) \chi(x)\). After a normalization, the author proves that they are traces of conjugacy classes is \(\text{SU}_2\) and therefore define angles \(\theta(\psi,\chi;t)\in [0,\pi]\). He further shows that when \((k,\psi,\chi)\) vary with \(|k|\to\infty\), these angles are approximately equidistributed with respect to the Sato-Tate measure, and moreover, that given any \(r\)-tuple of such sequences of (nonequal and noncontragredient) parameters, the corresponding measures are statistically independent. The proof goes through a reinterpretation of the sums in terms of \(\ell\)-adic sheaves on the affine line, the study of their Fourier transforms, and some (clever) group theory.
reviewed by: Daniel Bertrand
description: scientific article; zbMATH DE number 1656647