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2000 Hecke structure of spaces of half-integral weight cusp forms
Sharon M. Frechette
Nagoya Math. J. 159: 53-85 (2000).

Abstract

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [4], we obtain structure theorems for spaces of half-integral weight cusp forms $S_{k/2}{4N}{\chi}$ where $k$ and $N$ are odd nonnegative integers with $k \geq 3$, and $\chi$ is an even quadratic Dirichlet character modulo $4N$. We give complete results in the case where $N$ is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on $N$ and $\chi$, by giving conditions under which a newform has an equivalent cusp form in $S_{k/2}{4N}{\chi}$. We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such $N$ and $\chi$. In addition, we compare our structure results to Ueda's [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Citation

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Sharon M. Frechette. "Hecke structure of spaces of half-integral weight cusp forms." Nagoya Math. J. 159 53 - 85, 2000.

Information

Published: 2000
First available in Project Euclid: 27 April 2005

zbMATH: 1018.11022
MathSciNet: MR1783564

Subjects:
Primary: 11F37
Secondary: 11F11, 11F32

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal

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Vol.159 • 2000
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