Polygones fondamentaux d’une courbe modulaire

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 $Z[P^1(Q)]_0$.