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A \(p\)-adic measure attached to the zeta functions associated with two elliptic modular forms. II. (English) Zbl 0645.10028
Let \(f=\sum^{\infty}_{n=1} a(n)q^n\) and \(g=\sum^{\infty}_{n=1}b(n)q^n\) be holomorphic common eigenforms of all Hecke operators for the congruence subgroup \(\Gamma _0(N)\) of \(\mathrm{SL}_2(\mathbb Z)\) with “Nebentypus” character \(\psi\) and \(\xi\) and of weight \(k\) and \(\ell\), respectively. Define the Rankin product of \(f\) and \(g\) by
\[ \mathcal D_N(s,f,g) = \Bigl(\sum^{\infty}_{n=1}\psi \xi (n)n^{k+\ell - 2s-2}\Bigr)\Bigl(\sum^{\infty}_{n =1}a(n)b(n) n^{-s}\Bigr).\]
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 \(\frac{\mathcal D_ N(\ell +m,f,g)}{\pi^{\ell +2m+1}<f,f>}\) for integers \(m\) with \(0\leq m<k-\ell\), by regarding all the ingredients \(m\), \(f\) and \(g\) as variables. Here \(<f,f>\) is the Petersson self-inner product of \(f\).
[For part I, cf. Invent. Math. 79, 159–195 (1985; Zbl 0573.10020).]
Reviewer: Haruzo Hida

MSC:
11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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