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 , p. 1-83
Zbl 0645.10028 | MR 89k:11120 | 3 citations dans Numdam
doi : 10.5802/aif.1141
URL stable : http://www.numdam.org/item?id=AIF_1988__38_3_1_0

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 ψξ(n)n k+-2s-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 +2m+1 <f,f>pourlesentiersmtelsque0m<k-1, en regardant tous les ingrédients comme variables, où f,f est le produit de Petersson de f.
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 ψξ(n)n k+-2s-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.

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