A \(p\)-adic measure attached to the zeta functions associated with two elliptic modular forms. II
type: Item
instance of: scholarly article
title: A \(p\)-adic measure attached to the zeta functions associated with two elliptic modular forms. II
zbMATH Open document ID: 0645.10028
author: Haruzo Hida
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74808
review text: 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).]
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 4051732
zbMATH Keywords: Hecke operators
zbMATH Keywords: congruence subgroup
zbMATH Keywords: Rankin product
zbMATH Keywords: p-adically analytic
zbMATH Keywords: p-adic interpolation
zbMATH Keywords: special values
MaRDI profile type: Publication
OpenAlex ID: W1974465117
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DOI: 10.5802/AIF.1141
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 4051732