Symboles modulaires et produit de Petersson

Bernardi, Dominique; Perrin-Riou, Bernadette

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 795-859.

On revisite des articles de Eichler et de Shimura afin de donner une formule algébrique (basée sur les symboles de Farey) pour le produit d’intersection sur l’espace des symboles modulaires tel qu’il est décrit par Pollack et Stevens. On définit l’homomorphisme de périodes d’une série d’Eisenstein (symbole d’Eisenstein–Dedekind–Stevens) et on étend le produit d’intersection à ces objets. On construit une base adaptée à un traitement algorithmique de l’espace des séries d’Eisenstein de période rationnelle pour ${\Gamma }_0\left(N\right)$. On donne un algorithme pour construire un symbole de Farey d’un sous-groupe d’indice fini d’un groupe donné par un symbole de Farey.

We revisit some papers by Eichler and Shimura in order to give an algebraic formulation (based on Farey symbols) for the intersection product on the space of modular symbols, as described by Pollack and Stevens. We define the period homomorphism of an Eisenstein series (Eisenstein–Dedekind–Stevens symbol) and extend the definition of the intersection product to these objects. We construct a computationally convenient basis for the space of Eisenstein series for ${\Gamma }_0\left(N\right)$ with rational periods. Given a Farey symbol for a subgroup $\Gamma$ of the modular group and a subgroup ${\Gamma }^{\prime }$ of finite index of $\Gamma$, we give an algorithmic construction for a Farey symbol for ${\Gamma }^{\prime }$.

Classification : 11F03, 11F11, 11F67, 11F30
Mots clés : Eisenstein series, Petersson product, Farey symbol, modular curve, modular symbol

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