On differential operators and congruences for Siegel modular forms of degree two
Let r be a positive integer. Let f and g be Siegel modular forms of degree two and weight w and \(w+2r\) respectively which are nonzero common eigen functions of all Hecke operators. We denote by \(\lambda\) (m,f) the eigenvalue of the m-th Hecke operator on f. Under some conditions, e.g. all the Fourier coefficients of f and g belong to an algebraic number field K, the author proves the congruences \(m^{2r} \lambda (m,f)\equiv \lambda (m,g) mod {\mathfrak p}\) for all \(m\geq 1\), where \({\mathfrak p}\) is a certain prime ideal of K. The proof uses the Maass operator and Sturm's holomorphic projection to avoid treating the Eisenstein series of weight two. The key point is an integrability property of a \(C^{\infty}\)-map (differentiation followed by holomorphic projection). \{The full details will be published in J. Math. Soc. Japan (to appear).\}
scientific article; zbMATH DE number 3910487