Trace formula and trace identity of twisted Hecke operators on the spaces of cusp forms of weight \(k+1/2\) and level \(32M\)
In a sequence of papers [J. Math. Kyoto Univ. 28, No. 3, 505--555 (1988; Zbl 0673.10021) and Jap. J. Math., New Ser. 17, No. 1, 83--135 (1991; Zbl 0742.11031)] among others, the author studied trace formulae for Hecke operators on spaces of cusp forms of half-integral weight \(k + \frac12\), level \(N\) and character \(\chi\). In the present note a case is discussed which was not treated before: For level \(N = 32 M\) with \(M\) odd, these spaces are not generally closed under twisted Hecke operators \(R_{\psi} \tilde{T}(n^{2})\). However, they are closed under certain conditions on the conductors of \(\chi\) and \(\psi\), and the author presents explicit trace formulae for these cases. The details and proofs are announced to appear elsewhere.
scientific article; zbMATH DE number 5547614