Remark on Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. II
type: Item
instance of: scholarly article
title: Remark on Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. II
zbMATH Open document ID: 1012.11034
author: Hisashi Kojima
published in: Kodai Mathematical Journal
publication date: 2003-06-23T00:00:00Z
review text: In [J. Math. Pures Appl. 60, 375--484 (1981; Zbl 0431.10015)] \textit{J.-L. Waldspurger} first found that the square of Fourier coefficients \(a(n)\) at a square-free integer \(n\) of modular forms \(f(z) =\sum^\infty_{n=1} a(n)e[nz]\) of half integral weight is essentially proportional to the central value of the zeta function at a certain integer attached to the modular form \(F\) if \(f\) corresponds to \(F\) by the Shimura correspondence \(\Psi\) and \(f\) is an eigenfunction of Hecke operators. \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, 175--198 (1981; Zbl 0468.10015), Math. Ann. 271, 237--268 (1985; Zbl 0542.10018)] determined explicitly the constant of the proportionality in the case of modular forms belonging to Kohnen's spaces \(S_{(2k+1)/2}(N,\chi)\) of weight \((2k+1)/2\) and of square-free level \(N\) with character \(\chi\) which is a subspace of \(S_{(2k+1)/2}(4N,\chi_1)\), where \(S_{(2k+1)/2}(4N,\chi_1)\) means the space of modular cusp forms of half integral weight given in [\textit{G. Shimura}, Ann. Math. (2) 97, 440--481 (1973; Zbl 0266.10022)]. Kohnen-Zagier treated the case where \(N =1\) (resp. \(N\) is an odd square-free integer and \(\chi\) is the trivial character of level \(N\)). In [Duke Math. J. 71, 501--557 (1993; Zbl 0802.11017)], \textit{G. Shimura} intended to generalize such formulas to the case of Hilbert modular forms \(f\) of half integral weight and succeeded in obtaining many general interesting formulas. Among these, some explicit and useful formulas about the proportionality constant were formulated under the assumption that \(f\) satisfies the multiplicity one theorem. For modular forms belonging to Kohnen's spaces \(S_{(2k+1)/2}(N,\chi)\), Shimura did not give the same explicit formula as that of Kohnen and Zagier. In Part I [see Zbl 1012.11035 below)] the author represented explicitly the ratio of the square of Fourier coefficients \(a(n)\) at a fundamental discriminant \(n\) of modular forms \[ f(z)=\sum_…
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 1333302
zbMATH Keywords: modular cusp forms of half integral weight
zbMATH Keywords: theta series
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Rankin's convolution
zbMATH Keywords: Hecke operators
zbMATH Keywords: Kohnen's space
zbMATH Keywords: central value of the zeta function
zbMATH Keywords: Shimura correspondence
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043991
OpenAlex ID: W1992391748
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DOI: 10.2996/KMJ/1138043991
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description: scientific article; zbMATH DE number 1333302