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), p. 1-83 / Harvested from Numdam

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

𝒟N(s,f,g)=(n=1ψξ(n)nk+-2s-2)(n=1a(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

𝒟N(+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

𝒟N(s,f,g)=(n=1ψξ(n)nk+-2s-2)(n=1a(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.

@article{AIF_1988__38_3_1_0,
     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},
     volume = {38},
     year = {1988},
     pages = {1-83},
     doi = {10.5802/aif.1141},
     mrnumber = {89k:11120},
     zbl = {0645.10028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1988__38_3_1_0}
}
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) pp. 1-83. doi : 10.5802/aif.1141. http://gdmltest.u-ga.fr/item/AIF_1988__38_3_1_0/

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