Explicit Matrices for Hecke Operators on Siegel Modular Forms

We present an explicit set of matrices giving the action of the Hecke operators T (p), Tj(p ) on Siegel modular forms. Introduction It is well-known that the space of elliptic modular forms of weight k has a basis of simultaneous eigenforms for the Hecke operators, and the Fourier coefficients of an eigenform (and hence the eigenform) are completely determined by its eigenvalues and first Fourier coefficient. In the theory of Siegel modular forms, the role of the Hecke operators is not yet completely understood, thus there are many avenues open for conjecture and exploration, including computational exploration. The purpose of this note is to present an explicit set of matrices giving the action of the Hecke operators on Siegel modular forms, with the goal of facilitating computational exploration. (This construction also yields an explicit set of matrices giving the action of Hecke operators on Jacobi modular forms; we remark on this further at the end of this note.) Definitions and results For F a Siegel modular form of degree n and p a prime, we define the Hecke operator T (p) by