Polygones fondamentaux d’une courbe modulaire

Belabas, Karim; Bernardi, Dominique; Perrin-Riou, Bernadette

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2020), pp. 27-59.

A few pages in Siegel [9, p. 115 (§2)] 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 subgroup $\Gamma$ of finite index in ${\mathrm{PSL}}_2\left(\mathbb{Z}\right)$. One then obtains by classical procedures a generating system for $\Gamma$ with a minimal number of hyperbolic elements and a presentation of the $\mathbb{Z}\left[\Gamma \right]$-module of the elements of $\mathbb{Z}\left[{\mathbb{P}}^1\left(\mathbb{Q}\right)\right]$ of degree 0.

Quelques pages de Siegel [9, p. 115 (§2)] décrivent la construction d’une base symplectique de l’homologie d’une surface de Riemann compacte à partir d’un polygone fondamental. Cette note reprend cette construction en l’appliquant au cas de la surface de Riemann associée à un sous-groupe d’indice fini $\Gamma$ de ${\mathrm{PSL}}_2\left(\mathbb{Z}\right)$. On en déduit par des procédés classiques un système de générateurs indépendants de $\Gamma$ ayant un nombre minimal d’éléments hyperboliques et une présentation du $\mathbb{Z}\left[\Gamma \right]$-module des éléments de $\mathbb{Z}\left[{\mathbb{P}}^1\left(\mathbb{Q}\right)\right]$ de degré 0.

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