Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weight

Abstract

Let $R_{\psi}$ be a twisting operator for a quadratic primitive character $\psi$ and $\tilde{T}(n^2)$ the $n^2$-th Hecke operator of half-integral weight. When $\psi$ has an odd conductor, we already found trace identities between twisted Hecke operators $R_{\psi} \tilde{T}(n^2)$ of half-integral weight and certain Hecke operators of integral weight for almost all cases (cf. [U1--3]). In this paper, the restriction is removed and we give similar trace identities for every quadratic primitive character $\psi$, including the case that $\psi$ has an even conductor.