A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II
Annales de l'Institut Fourier, Volume 38 (1988) no. 3, pp. 1-83.

Let f= n=1 a(n)q n and g= n=1 b(n)q n be holomorphic common eigenforms of all Hecke operators for the congruence subgroup Γ 0 (N) of SL 2 (Z) with “Nebentypus” character ψ and ξ and of weight k and , respectively. Define the Rankin product of f and g by

𝒟 N ( s , f , g ) = ( n = 1 ψ ξ ( n ) n k + - 2 s - 2 ) ( n = 1 a ( n ) b ( n ) n - s ) .

Supposing f and g to be ordinary at a prime p5, we shall construct a p-adically analytic L-function of three variables which interpolate the values 𝒟 N (+m,f,g) π +2m+1 <f,f> for integers m with 0m<k-1, by regarding all the ingredients m, f and g as variables. Here f,f is the Petersson self-inner product of f.

Soient f= n=1 a(n)q n et g= n=1 b(n)q n deux formes paraboliques pour le sous-groupe Γ 0 (N) de SL 2 (Z), propre pour tous les opérateurs de Hecke, de caractère respectivement ψ et ξ, de poids k et . Définissons le produit de Rankin de f et g par la formule

𝒟 N ( s , f , g ) = ( n = 1 ψ ξ ( n ) n k + - 2 s - 2 ) ( n = 1 a ( n ) b ( n ) n - s ) .

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

𝒟 N ( + m , f , g ) π + 2 m + 1 < f , f > pour les entiers m tels que 0 m < k - 1 ,

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

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     title = {A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. {II}},
     journal = {Annales de l'Institut Fourier},
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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, Volume 38 (1988) no. 3, pp. 1-83. doi : 10.5802/aif.1141. http://www.numdam.org/articles/10.5802/aif.1141/

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