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

Hsieh, Ming-Lun

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

Le but de ce travail est d’accomplir le premier pas de notre programme vers la conjecture principale pour ${\text{GL}}_2\times {𝒦}^{\times }$, par la methode de congruences entre séries d’Eisenstein sur $GU(3,1)$, 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 ${\text{GL}}_2\times {𝒦}^{\times }$. Cette construction donne ainsi un nouveau point de vue sur la fonction $L$ $p$-adique de ${\text{GL}}_2\times {𝒦}^{\times }$.

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

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