Transformation equations and the special values of Shimura's zeta functions
type: Item
instance of: scholarly article
title: Transformation equations and the special values of Shimura's zeta functions
zbMATH Open document ID: 0565.10023
author: Koji Doi
author: Haruzo Hida
author: Yoshitaka Maeda
published in: Hokkaido Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let a(n) resp. b(n) be the Fourier coefficients of a primitive cusp form f of weight k on \(\Gamma_ 0(N)\) resp. of a modular form g of weight \(\ell <k\) on \(\Gamma_ 0(N)\). For \(D(s,f,g)=\sum^{\infty}_{n=1}a(n) b(n) n^{-s}\) \textit{G. Shimura} [Commun. Pure Appl. Math. 29, 783-804 (1976; Zbl 0348.10015)] proved that the values D(m,f,g) at integers m, \((k+\ell)-1<m<k\), are algebraic numbers times the Petersson self inner product of f and a power of \(\pi\). For a cusp form h on \(\Gamma_ 0(N)\) the trace Tr(h) is the sum of the transforms of f by a set of representatives of \(\Gamma_ 0(N)\) modulo \(\Gamma =SL_ 2({\mathbb{Z}})\) or, equivalently, the sum of all roots of the transformation equation for h. The authors consider the traces \(Tr(gE^*_{\lambda,N})^{\mu}\) where g is a cusp form and \(E^*_{\lambda,N}\) is an Eisenstein series of weight \(\lambda >2\) on \(\Gamma_ 0(N)\). They prove that these traces are linear combinations of the primitive cusp forms f of weight \(k\mu =(\ell +\lambda)\mu\) on \(\Gamma\) with coefficients which are essentially the above mentioned algebraic numbers at \(m=k\mu -1\).
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3900861
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: algebraic numbers
zbMATH Keywords: traces
zbMATH Keywords: cusp form
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757707
OpenAlex ID: W2061035421
DOI: 10.14492/HOKMJ/1381757707
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description: scientific article; zbMATH DE number 3900861