Closed geodesics on semi-arithmetic Riemann surfaces

2020/04/28 by Gregory Cosac, Cosac, Gregory, Cayo Dória +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric Topology (math.GT) #Meromorphic and Entire Functions #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2004.13683

openalex publication_date 2020/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove that their systoles are dense in the positive real numbers. Furthermore, this leads to a construction, for each genus g ≥ 2, of infinite families of semi-arithmetic surfaces with pairwise distinct invariant trace fields, giving a negative answer to a conjecture of B. Jeon. Finally, for any semi-arithmetic surface we find a sequence of congruence coverings with logarithmic systolic growth and, for the special case of surfaces admitting modular embedding, we are able to exhibit explicit constants.

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• On Systole, Kissing Number and Volume of Arithmetic Manifolds

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