On $p$-adic $L$-functions of $GL\left(2\right)\times GL\left(2\right)$ over totally real fields

Hida, Haruzo

Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 311-391.

Soit $D(s,f,g)$ le produit de Rankin de deux formes paraboliques hilbertiennes $f$ et $g$. Cette fonction $L$ est en fait la fonction $L$ standard d’une représentation automorphe du groupe algébrique $GL\left(2\right)\times GL\left(2\right)$ défini sur un corps totalement réel. En supposant que $f$ et $g$ sont ordinaires en $p$, nombre premier, nous allons construire une fonction analytique $p$-adique de plusieurs variables qui interpole la partie algébrique de $D(m,f,g)$ pour des entiers $m$ critiques en considérant $m$, $f$ et $g$ comme variables.

Let $D(s,f,g)$ be the Rankin product $L$-function for two Hilbert cusp forms $f$ and $g$. This $L$-function is in fact the standard $L$-function of an automorphic representation of the algebraic group $GL\left(2\right)\times GL\left(2\right)$ defined over a totally real field. Under the ordinarity assumption at a given prime $p$ for $f$ and $g$, we shall construct a $p$-adic analytic function of several variables which interpolates the algebraic part of $D(m,f,g)$ for critical integers $m$, regarding all the ingredients $m$, $f$ and $g$ as variables.

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