Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

Abstract

.N / and S new 2k .N / are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product a g .m/a g .n/ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces S .p/. We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].