Hecke structure of spaces of half-integral weight cusp forms

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.