Systolic length of triangular modular curves

Michael M. Schein; Amir Shoan

doi:10.1016/j.jnt.2021.11.010

Systolic length of triangular modular curves

Michael M. Schein, Amir Shoan

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	462-488
Number of pages	27
Journal	Journal of Number Theory
Volume	239
DOIs	https://doi.org/10.1016/j.jnt.2021.11.010
State	Published - 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.

Keywords

Congruence subgroups
Quaternion algebras
Semiarithmetic groups
Systoles
Triangle groups

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10.1016/j.jnt.2021.11.010

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title = "Systolic length of triangular modular curves",

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 D{\'o}ria to show that the systolic length grows logarithmically with respect to the genus.",

keywords = "Congruence subgroups, Quaternion algebras, Semiarithmetic groups, Systoles, Triangle groups",

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AU - Shoan, Amir

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N2 - 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.

AB - 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.

KW - Congruence subgroups

KW - Quaternion algebras

KW - Semiarithmetic groups

KW - Systoles

KW - Triangle groups

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VL - 239

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EP - 488

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Systolic length of triangular modular curves

Abstract

Bibliographical note

Keywords

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