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Zbl 0645.10028
Hida, Haruzo
A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II. (English)
[J] Ann. Inst. Fourier 38, No.3, 1-83 (1988). ISSN 0373-0956; ISSN 1777-5310/e

[For part I, cf. Invent. Math. 79, 159-195 (1985; Zbl 0573.10020).] \par Let $f=\sum \sp{\infty}\sb{n=1}a(n)q$ n and $g=\sum \sp{\infty}\sb{n=1}b(n)q$ n be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $\Gamma \sb 0(N)$ of $SL\sb 2({\bbfZ})$ with ``Nebentypus'' character $\psi$ and $\xi$ and of weight k and $\ell$, respectively. Define the Rankin product of f and g by $$ {\cal D}\sb N(s,f,g)=(\sum \sp{\infty}\sb{n=1}\psi \xi (n)n\sp{k+\ell - 2s-2})(\sum \sp{\infty}\sb{n\quad =1}a(n)b(n)n\sp{-s}). $$ 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 $\frac{{\cal D}\sb N(\ell +m,f,g)}{\pi \sp{\ell +2m+1}<f,f>}$ for integers m with $0\le m<k-\ell,$ by regarding all the ingredients m, f and g as variables. Here $<f,f>$ is the Petersson self-inner product of f.
[H.Hida]

MSC 2000:: *11F33 Congruences for (p-adic) modular forms
11F67 Special values of automorphic L-series, etc
11F12 Automorphic forms, one variable

Keywords: Hecke operators; congruence subgroup; Rankin product; p-adically analytic; p-adic interpolation; special values

Citations: Zbl 0573.10020

Cited in: Zbl 0721.11024 Zbl 0705.11033 Zbl 0675.10019

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