The density of supersingular Fermat varieties
type: Item
instance of: scholarly article
title: The density of supersingular Fermat varieties
zbMATH Open document ID: 0545.10034
DOI: 10.1007/BF01191181
author: William C. Waterhouse
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: Let \(\delta\) (m) be the number of congruence classes mod m that have some power congruent to -1. This number measures the portion of primes modulo which the Jacobian of \(X^ m+Y^ m=1\) is supersingular [see e.g. \textit{N. Yui}, J. Algebra 65, 1-35 (1980; Zbl 0437.14015)]. The author first derives Yui's elementary formula for \(\delta\) (m) as a special case of a theorem on finite Abelian groups. The main theorem then is that, though \(\delta\) (m) takes on values arbitrarily close to 1, the average value goes to 0. This is shown by observing that \(\delta(m)\leq 4/2^{\omega(m)}\), where \(\omega\) (m) is the number of distinct primes dividing m. The size of \((1/N)\sum_{m\leq N}2^{-\omega(m)}\) is then estimated by standard techniques. In a subsequent paper (see the following review), \textit{W. Schwarz} and the author have determined the asymptotic behavior of \(\sum_{m\leq N}\delta(m)\).
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 3867445
zbMATH Keywords: Fermat surface
zbMATH Keywords: Jacobian
zbMATH Keywords: arithmetic function
zbMATH Keywords: number of congruence classes
zbMATH Keywords: finite Abelian groups
zbMATH Keywords: average value
MaRDI profile type: Publication
cites work: Q3856819
cites work: On Fermat varieties
cites work: On the Jacobian variety of the Fermat curve
cites work: Q4745939
full work available at URL: https://doi.org/10.1007/bf01191181
OpenAlex ID: W2027673549
description: scientific article; zbMATH DE number 3867445