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

Hida, Haruzo

doi:10.5802/aif.1141

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.

Résumé
Abstract

Soient $f = \sum_{n = 1}^{\infty} a (n) q^{n}$ et $g = \sum_{n = 1}^{\infty} b (n) q^{n}$ deux formes paraboliques pour le sous-groupe $Γ_{0} (N)$ de $S L_{2} (Z)$ , propre pour tous les opérateurs de Hecke, de caractère respectivement $ψ$ et $ξ$ , de poids $k$ et $ℓ$ . Définissons le produit de Rankin de $f$ et $g$ par la formule

𝒟_{N} (s, f, g) = (\sum_{n = 1}^{\infty} ψ ξ (n) n^{k + ℓ - 2 s - 2}) (\sum_{n = 1}^{\infty} a (n) b (n) n^{- s}) .

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

\frac{𝒟_{N} (ℓ + m, f, g)}{π^{ℓ + 2 m + 1} < f, f >} pour les entiers m tels que 0 \leq m < k - 1,

en regardant tous les ingrédients comme variables, où $〈 f, f 〉$ est le produit de Petersson de $f$ .

Let $f = \sum_{n = 1}^{\infty} a (n) q^{n}$ and $g = \sum_{n = 1}^{\infty} b (n) q^{n}$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $Γ_{0} (N)$ of $S L_{2} (Z)$ with “Nebentypus” character $ψ$ and $ξ$ and of weight $k$ and $ℓ$ , respectively. Define the Rankin product of $f$ and $g$ by

𝒟_{N} (s, f, g) = (\sum_{n = 1}^{\infty} ψ ξ (n) n^{k + ℓ - 2 s - 2}) (\sum_{n = 1}^{\infty} a (n) b (n) n^{- s}) .

Supposing $f$ and $g$ to be ordinary at a prime $p \geq 5$ , we shall construct a $p$ -adically analytic $L$ -function of three variables which interpolate the values $\frac{𝒟_{N} (ℓ + m, f, g)}{π^{ℓ + 2 m + 1} < f, f >}$ for integers $m$ with $0 \leq m < k - 1,$ by regarding all the ingredients $m$ , $f$ and $g$ as variables. Here $〈 f, f 〉$ is the Petersson self-inner product of $f$ .

MR Zbl

DOI : 10.5802/aif.1141

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     author = {Hida, Haruzo},
     title = {A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. {II}},
     journal = {Annales de l'Institut Fourier},
     pages = {1--83},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {3},
     year = {1988},
     doi = {10.5802/aif.1141},
     zbl = {0645.10028},
     mrnumber = {89k:11120},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1141/}
}

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Hida, Haruzo. A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II. Annales de l'Institut Fourier, Tome 38 (1988) no. 3, pp. 1-83. doi : 10.5802/aif.1141. https://aif.centre-mersenne.org/articles/10.5802/aif.1141/

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