Some trace relations of twisting operators on the spaces of cusp forms of half-integral weight
Let N be a positive integer divisible by 4 and \(\chi\) an even character modulo N such that \(\chi^ 2=1\). Assume \(\psi\) is a nontrivial primitive character such that \(\psi^ 2=1\) and the conductor of \(\psi\), say L, is odd and \(L^ 2 | N\). Under this assumption the twisting operator \[ R_{\psi}: f(z)=\sum^{\infty}_{n=1}a(n)e(nz)\quad \mapsto \quad f| R_{\psi}(z)=\sum^{\infty}_{n=1}\psi(n)a(n)e(nz) \] is a linear operator on the space of cusp forms \(S(k+1/2,N,\chi)\) of half- integral weight \(k+1/2\) with the character \(\chi\) on the group \(\Gamma_ 0(N).\) In this note the author announces a result which gives some relations between the trace of the operator \(R_{\psi}\tilde T(n^ 2)\) on \(S(k+1/2,N,\chi)\) and the traces of several operators of the form \(W(N')T(n)\) on the space of cusp forms \(S(2k,N'')\) where \(\tilde T(n^ 2)\) and T(n) are the Hecke operators on corresponding spaces separately, \(W(N')\) is the normalizer of \(\Gamma_ 0(N)\) and \(N',N''\) are suitable integers determined by N and L. A similar relation for the trace of the operator \(R_{\psi}\tilde T(n^ 2)\) on the Kohnen subspace \(S(k+1/2,N,\chi)_ k\) is also found. (See also the following review).
scientific article; zbMATH DE number 4198086