Systolic length of triangular modular curves

2020/12/16 by Michael M. Schein, Schein, Michael M., Amir Shoan +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Rings and Algebras (math.RA) #math.DG #math.RA

paper · pdf · doi:10.48550/arxiv.2012.08796

23 pages; to appear in Journal of Number Theory

openalex publication_date 2020/12/16 · arxiv created 2022/02/22 · arxiv updated 2022/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.

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• Small genus, small index critical points of the systole function

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