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

Hsieh, Ming-Lun

Ordinary

p

-adic Eisenstein series and

p

-adic

L

-functions for unitary groups

Hsieh, Ming-Lun

Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 987-1059

Abstract
Résumé

The purpose of this work is to carry out the first step in our four-step program towards the main conjecture for ${GL}_{2} \times 𝒦^{\times}$ by the method of Eisenstein congruence on $G U (3, 1)$ , where $𝒦$ is an imaginary quadratic field. We construct a $p$ -adic family of ordinary Eisenstein series on the group of unitary similitudes $G U (3, 1)$ 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 ${GL}_{2} \times 𝒦^{\times}$ . This construction also provides a different point of view of $p$ -adic $L$ -functions of ${GL}_{2} \times 𝒦^{\times}$ .

Le but de ce travail est d’accomplir le premier pas de notre programme vers la conjecture principale pour ${GL}_{2} \times 𝒦^{\times}$ , par la methode de congruences entre séries d’Eisenstein sur $G U (3, 1)$ , où $𝒦$ 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 ${GL}_{2} \times 𝒦^{\times}$ . Cette construction donne ainsi un nouveau point de vue sur la fonction $L$ $p$ -adique de ${GL}_{2} \times 𝒦^{\times}$ .

MR 2918724 | Zbl 1271.11051 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.2635
Classification: 11F33, 11F70, 11R23
Keywords: Eisenstein series on unitary groups, Iwasawa-Greenberg main conjectures

@article{AIF_2011__61_3_987_0,
     author = {Hsieh, Ming-Lun},
     title = {Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {987-1059},
     doi = {10.5802/aif.2635},
     mrnumber = {2918724},
     zbl = {1271.11051},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_987_0}
}

Hsieh, Ming-Lun. Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 987-1059. doi : 10.5802/aif.2635. http://www.numdam.org/item/AIF_2011__61_3_987_0/

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