Hecke Structure of Spaces of Half-integral Weight Cusp Forms

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 forms S k=2 (4N;) where k and N are odd nonnegative integers with k 3, and 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 , by giving conditions under which a newform has an equivalent cusp form in S k=2 (4N;). 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. 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. 1. Introduction A vital part of the theory of integral weight modular forms is the study of simultaneous Hecke eigenforms, in particular newforms. The classical \multiplicity-one" result says that a newform is explicitly determined up to constant multiple by its eigenvalues for almost all the Hecke operators T k (p), p a prime, k a positive integer. If we attempt to deene \half-integral weight newforms" using a deenition analogous to that for integral weight, the theory breaks down rapidly, the crucial point being the lack of a multiplicity-one result. There are however signiicant connections between integral weight Hecke eigenforms and half-integral weight Hecke eigenforms, most notably the Shimura correspondence 12]. This correspondence maps Hecke eigenforms to Hecke eigenforms, which suggests that our knowledge of the integral weight structure can be \transported" to knowledge about half-integral weight forms. Through a representation-theoretic approach, Shintani 13] provides a mapping which is an adjoint to the Shimura lift and also preserves Hecke eigenforms. Unfortunately, the image of the Shintani map may be trivial, so it does not necessarily aaord a practical method of transporting the Hecke structure back.