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On the Kohnen plus space for Hilbert modular forms of half-integral weight I

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  06 September 2013

Kaoru Hiraga  and
Tamotsu Ikeda
Kaoru Hiraga
Affiliation:
Graduate School of Mathematics, Kyoto University, Kitashirakawa, Kyoto, 606-8502, Japan email hiraga@math.kyoto-u.ac.jpikeda@math.kyoto-u.ac.jp
Tamotsu Ikeda
Affiliation:
Graduate School of Mathematics, Kyoto University, Kitashirakawa, Kyoto, 606-8502, Japan email hiraga@math.kyoto-u.ac.jpikeda@math.kyoto-u.ac.jp
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Abstract

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In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

Keywords

Hilbert modular formsmodular forms of half-integral weightKohnen plus spaceJacobi forms

MSC classification

Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F30: Fourier coefficients of automorphic forms 11F50: Jacobi forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013 , pp. 1963 - 2010
Copyright
© The Author(s) 2013 

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