Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights

Abstract

In [4], W. Kohnen constructed explicit Shintani lifts from the space of cusp forms of weight $2k$ on $Γ_0(N)$, which are mapped into the plus space $S^+_{k+1/2}(Γ_0(4N))$, where $N$ is an odd integer. This lifting is adjoint to the (modified) Shimura map defined by him with respect to the Petersson scalar product. Using this construction along with the theory of newforms on $S^+_{k+1/2}(Γ_0(4N))$ (where $N$ is an odd square-free natural number) developed in [3], he derived explicit Waldspurger theorem for the newforms belonging to the space $S^+_{k+1/2}(Γ_0(4N))$. Further in this work, Kohnen considered the $\pm 1$ eigen subspaces corresponding to the Atkin-Lehner involution and showed that the intersection of this $\pm$ spaces with the corresponding newform spaces are isomorphic under the Shimura correspondence. A natural question is whether a similar result can be obtained for the intersection of this $(\pm1)$ subspaces with the space of oldforms. In this direction, S. Choi and C. H. Kim [2] considered the case where $N$ is an odd prime $p$ and constructed similar Shimura and Shintani maps between subspaces of forms of half-integral and integral weights. The subspaces considered by Choi and Kim are nothing but $+1$ eigenspace under the Fricke involution. In this paper, we generalise the work of Choi and Kim to the case where $N$ is an odd square-free natural number.