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


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.


Download Citation

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


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

zbMATH: 1018.11022
MathSciNet: MR1783564

Primary: 11F37
Secondary: 11F11 , 11F32

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal

Vol.159 • 2000
Back to Top