A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II

Hida, Haruzo

Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 1-83.

Soient $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n$ et $g={\sum }_{n=1}^{\infty }b\left(n\right)q^n$ deux formes paraboliques pour le sous-groupe ${\Gamma }_0\left(N\right)$ de $S{L}_2\left(\mathbf{Z}\right)$, propre pour tous les opérateurs de Hecke, de caractère respectivement $\psi$ et $\xi$, de poids $k$ et $\ell$. Définissons le produit de Rankin de $f$ et $g$ par la formule

$${𝒟}_N(s,f,g)=(\sum _{n=1}^{\infty }\psi \xi \left(n\right)n^{k+\ell -2s-2}\left)\right(\sum _{n=1}^{\infty }a\left(n\right)b\left(n\right)n^{-s}).$$

En supposant que $f$ et $g$ sont ordinaires en $p$, nombre premier $\ge 5$, nous allons construire une fonction $L$ analytique $p$-adique de trois variables qui interpole les valeurs

$$\frac{{𝒟}_N(\ell +m,f,g)}{{\pi }^{\ell +2m+1}<f,f>}\text{pour}\text{les}\text{entiers}m\text{tels}\text{que}0\le m<k-1,$$

en regardant tous les ingrédients comme variables, où $\langle f,f\rangle$ est le produit de Petersson de $f$.

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n$ and $g={\sum }_{n=1}^{\infty }b\left(n\right)q^n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup ${\Gamma }_0\left(N\right)$ of $S{L}_2\left(\mathbf{Z}\right)$ with “Nebentypus” character $\psi$ and $\xi$ and of weight $k$ and $\ell$, respectively. Define the Rankin product of $f$ and $g$ by

$${𝒟}_N(s,f,g)=(\sum _{n=1}^{\infty }\psi \xi \left(n\right)n^{k+\ell -2s-2}\left)\right(\sum _{n=1}^{\infty }a\left(n\right)b\left(n\right)n^{-s}).$$

Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values $\frac{{𝒟}_N(\ell +m,f,g)}{{\pi }^{\ell +2m+1}<f,f>}$ for integers $m$ with $0\le m<k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $\langle f,f\rangle$ is the Petersson self-inner product of $f$.

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