Polygones fondamentaux d’une courbe modulaire
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2020), pp. 27-59.

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 Γ de PSL 2 (). On en déduit par des procédés classiques un système de générateurs indépendants de Γ ayant un nombre minimal d’éléments hyperboliques et une présentation du [Γ]-module des éléments de [ 1 ()] de degré 0.

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 Γ of finite index in PSL 2 (). One then obtains by classical procedures a generating system for Γ with a minimal number of hyperbolic elements and a presentation of the [Γ]-module of the elements of [ 1 ()] of degree 0.

Publié le :
DOI : https://doi.org/10.5802/pmb.40
@article{PMB_2020____27_0,
     author = {Karim Belabas and Dominique Bernardi and Bernadette Perrin-Riou},
     title = {Polygones fondamentaux d{\textquoteright}une courbe modulaire},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
     pages = {27--59},
     publisher = {Presses universitaires de Franche-Comt\'e},
     year = {2020},
     doi = {10.5802/pmb.40},
     language = {fr},
     url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.40/}
}
Karim Belabas; Dominique Bernardi; Bernadette Perrin-Riou. Polygones fondamentaux d’une courbe modulaire. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2020), pp. 27-59. doi : 10.5802/pmb.40. https://pmb.centre-mersenne.org/articles/10.5802/pmb.40/

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