Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups

[Les séries d’Eisenstein ordinaires $p$-adiques et la fonction $L$ $p$-adique pour les groupes unitaires]

Hsieh, Ming-Lun

Annales de l'Institut Fourier, Tome 61 (2011), p. 987-1059 / Harvested from Numdam

Résumé

Le but de ce travail est d’accomplir le premier pas de notre programme vers la conjecture principale pour $\text{GL}_2\times {\mathcal{K}}^\times $, par la methode de congruences entre séries d’Eisenstein sur $GU(3, 1)$, où ${\mathcal{K}}$ est d’un corps quadratique imaginaire. Nous construisons une famille $p$-adique de séries d’Eisenstein ordinaires sur le groupe de similitudes unitaires avec le terme constant optimal qui est essentiellement le produit de la fonction $L$ $p$-adique de Kubota-Leopoldt et d’une fonction $L$ $p$-adique pour $\text{GL}_2\times {\mathcal{K}}^\times $. Cette construction donne ainsi un nouveau point de vue sur la fonction $L$ $p$-adique de $\text{GL}_2\times {\mathcal{K}}^\times $.

Abstract

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for $\text{GL}_2\times {\mathcal{K}}^\times $ by the method of Eisenstein congruence on $GU(3,1)$, where ${\mathcal{K}}$ is an imaginary quadratic field. We construct a $p$-adic family of ordinary Eisenstein series on the group of unitary similitudes $GU(3,1)$ with the optimal constant term which is basically the product of the Kubota-Leopodlt $p$-adic $L$-function and a $p$-adic $L$-function for $\text{GL}_2\times {\mathcal{K}}^\times $. This construction also provides a different point of view of $p$-adic $L$-functions of $\text{GL}_2\times {\mathcal{K}}^\times $.

Détail

Publié le : 2011-01-01

MR 2918724 | Zbl 1271.11051 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2635
Classification:  11F33,  11F70,  11R23
Mots clés: Les séries d’Eisenstein sur les groupes unitaires, la conjecture principale de Iwasawa-Greenberg

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