Systolic length of triangular modular curves

Michael M. Schein, Amir Shoan

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)462-488
Number of pages27
JournalJournal of Number Theory
StatePublished - Oct 2022

Bibliographical note

Funding Information:
The first author was supported by the Germany-Israel Foundation for Scientific Research and Development under grant 1246/2014 during part of this work.

Publisher Copyright:
© 2021 Elsevier Inc.


  • Congruence subgroups
  • Quaternion algebras
  • Semiarithmetic groups
  • Systoles
  • Triangle groups


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