Systoles of arithmetic surfaces and Markoff spectrum
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.
scientific article; zbMATH DE number 892585
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