Polygones fondamentaux d'une courbe modulaire

TL;DR

This paper revisits Siegel's construction of fundamental polygons for Riemann surfaces, focusing on congruence subgroups of SL(2,Z), and derives minimal generating systems and module presentations.

Contribution

It specializes Siegel's general method to congruence subgroups, providing explicit minimal generators and module presentations for these cases.

Findings

Constructs fundamental polygons for congruence subgroups

Provides minimal hyperbolic element generating systems

Describes presentations of associated modules

Abstract

A few pages in Siegel describe how, starting with a fundamental polygon for a compact Riemann surface, one can construct a symplectic basis of its homology. This note retells that construction, specializing to the case where the surface is associated to a congruence subgroup $\Gamma$ of $SL_2(Z)$. One then obtains by classical procedures a generating system for $\Gamma$ with a minimal number of hyperbolic elements and a presentation of the $Z[\Gamma]$-module $Z[P^1(Q)]_0$.