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

Abstract

For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace S-k+1/2(new) (N) subset of Sk+1/2(N), and S-k+1/2(new) (N) and S-2k(new)(N) are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product ag(m)<(a(g)(n))over bar> of two arbitrary Fourier coefficients of a Hecke eigenform g of half-integral 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-2k(+)(p) and Sk+1/2(p). We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].