$p$-adic measures attached to Siegel modular forms

Böcherer, Siegfried ; Schmidt, Claus-Günther

Annales de l'Institut Fourier, Tome 50 (2000), p. 1375-1443 / Harvested from Numdam

Résumé

On étudie les valeurs critiques de la fonction $L$ complexe standard, associée à une forme modulaire de Siegel holomorphe et des fonctions $L$ tordues des caractères de Dirichlet. Notre objet principal est, pour un nombre premier rationnel $p$ donné, l’interpolation $p$-adique des valeurs critiques essentiellement algébriques en laissant varier les caractères de Dirichlet afin d’obtenir un contrôle systématique des dénominateurs des valeurs critiques par des congruences de Kummer généralisées. Pour organiser cette information on montre l’existence de mesures $p$-adiques telles que l’intégration d’un caractère de Dirichlet de conducteur une $p$-puissance sur la mesure, donne la valeur critique normalisée de la fonction $L$ complexe tordue du caractère de Dirichlet. D’une manière standard les mesures $p$-adiques définissent des fonctions $L\,p$-adiques qui par conséquent interpolent $p$-adiquement les valeurs critiques normalisées.

Abstract

We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of $p$-adic measures such that integration of any Dirichlet character of $p$-power conductor over the measure yields the suitably normalized critical value of the complex $L$-function twisted by the Dirichlet character. In a standard manner the $p$-adic measures naturally define $p$-adic $L$-functions which hence $p$-adically interpolate the normalized critical values.

Détail

Publié le : 2000-01-01

MR 2001k:11082 | Zbl 0962.11023

DOI : https://doi.org/10.5802/aif.1796

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