Tohoku Mathematical Journal

On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions

Hisashi Kojima and Yasushi Tokuno

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Abstract

The purpose of this paper is to derive a generalization of Kohnen-Zagier's results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces, and to refine our previous results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. Employing kernel functions, we construct a correspondence $\varPsi$ from modular forms of half integral weight $k+1/2$ belonging to Kohnen's spaces to modular forms of weight $2k$. We explicitly determine the Fourier coefficients of $\varPsi(f)$ in terms of those of $f$. Moreover, under certain assumptions about $f$ concerning the multiplicity one theorem with respect to Hecke operators, we establish an explicit connection between the square of Fourier coefficients of $f$ and the critical value of the zeta function associated with the image $\varPsi(f)$ of $f$ twisted with quadratic characters, which gives a further refinement of our results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 1 (2004), 125-145.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246384

Digital Object Identifier
doi:10.2748/tmj/1113246384

Mathematical Reviews number (MathSciNet)
MR2028921

Zentralblatt MATH identifier
1093.11030

Subjects
Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms
Secondary: 11F30: Fourier coefficients of automorphic forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

Keywords
Modular forms of half integral weight Fourier coefficients of modular forms special value of zeta function

Citation

Kojima, Hisashi; Tokuno, Yasushi. On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions. Tohoku Math. J. (2) 56 (2004), no. 1, 125--145. doi:10.2748/tmj/1113246384. https://projecteuclid.org/euclid.tmj/1113246384


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