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 [14], we obtain structure theorems for spaces of half-integral weight cusp formsSk/2(4N,χ) wherekandNare odd nonnegative integers withk≥ 3, andχis an even quadratic Dirichlet character modulo 4N. We give complete results in the case whereNis 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 onNandχ, by giving conditions under which a newform has an equivalent cusp form inSk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any suchNandχ. 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.