Calculation of traces of theta series by means of the Weil representation
Let \(n\) be a positive integer and \(S\) a rational positive definite matrix of degree \(2n\) with determinant equal to a square. Associate to \(S\) a theta series \(\Theta_S(Z)\) defined on the Siegel space \({\mathcal H}_m\). If the level of \(S\) divides the positive integer \(N\) the theta series \(\Theta_S (Z)\) belongs to the space \({\mathcal M}_m(n, N)\) of modular forms of weight \(n\), degree \(m\) and level \(N\). Let \(\Theta_m(n, N)\) be the space generated by such theta series. Obviously if \(q\) divides \(N\), \[ \Theta_m(n, q)\subset \Theta_m(n, N) \cap{\mathcal M}_m(n, q). \] In this paper the author treats the problem of the equality of the two spaces. This problem has been treated by several authors including the reviewer using the global trace operator, the \( \Phi\)-operator and some commutation rules. In this paper the author attacks this problem by defining a local trace operator using the local Weil representation of \(\text{Sp}(m)\). In the final theorem an obstruction to the solution of the problem is indicated. We have to mention that both methods were used in [\textit{S. Böcherer, J. Funke} and \textit{R. Schulze-Pillot}, J. Number Theory 78, 119-139 (1999; Zbl 0936.11029)], where better results are obtained. Recently, in two papers \textit{F.~L. Chiera} considered the trace operator for the main congruence groups [Theta series and trace operators for \(\Gamma_n[q]\) (Preprint, Univ. Mannheim) (2001) and for \(\Gamma_{n,0}[q]\) and \(\Gamma_n[q]\) (Preprint, Univ. Roma ``La Sapienza'') (2001)].
scientific article; zbMATH DE number 1309654