Mathematical Research Letters

Volume 29 (2022)

Number 4

Closed geodesics on semi-arithmetic Riemann surfaces

Pages: 961 – 1001



Gregory Cosac (Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil)

Cayo Dória (Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, Brazil)


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 $g \geq 2$, 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.

The first author was supported by CNPq-Brazil research grant 141204/2016-8.

Dória is grateful for the support of FAPESP grant 2018/15750-9.

Received 30 August 2020

Received revised 3 June 2021

Accepted 26 October 2021

Published 23 February 2023