Systoles of arithmetic surfaces and Markoff spectrum
type: Item
instance of: scholarly article
title: Systoles of arithmetic surfaces and Markoff spectrum
zbMATH Open document ID: 0853.11054
DOI: 10.1007/BF01444218
published in: Mathematische Annalen
publication date: 1996-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165440
review text: Previous authors have used the geometry of certain low-index covers of the modular surface to study the Markoff numbers. The author uses all principal congruence covers; he shows that the infamous uniqueness conjecture is equivalent to the uniqueness -- up to isometry -- of systoles (shortest non-boundary geodesics) which are sufficiently far from any cusp. The results for the Markoff numbers are typified by: If \((x, y, z)\), with \(z\geq x, y\), is a solution in positive integers of \(x^2+ y^2+ z^2= 3xyz\), and \(z\) is a prime power, then \(z\) uniquely determines \((x, y, z)\). One might confer [\textit{A. Baragar}, Can. Math. Bull. 39, No. 1, 3-9 (1996; Zbl 0846.11020)] for related results by way of rather different techniques.
reviewed by: Thomas A. Schmidt
Mathematics Subject Classification ID: 11J06
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 892585
zbMATH Keywords: Markoff spectrum
zbMATH Keywords: Markoff numbers
zbMATH Keywords: principal congruence covers
zbMATH Keywords: systoles
zbMATH Keywords: geodesics
author: Paul Schmutz Schaller
MaRDI profile type: Publication
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OpenAlex ID: W1989764769
description: scientific article; zbMATH DE number 892585