On diagonalization of singular Frobenius operators on Siegel modular forms

Abstract

A Frobenius operator ║Π( m ) maps a Siegel modular form with Fourier coefficients f ( A ), where A runs over matrices of nonnegative definite integral quadratic forms of given order, to the function with Fourier coefficients f ( mA ). In cases of Siegel modular forms or cusp forms of integral weights k for the groups Γ n 0 ( q ) with Dirichlet characters χ modulo q , the Frobenius operators with m dividing a power of q (singular operators) can be interpreted as Hecke operators on the corresponding spaces and so map the spaces into themselves. It is proved, in particular, that if q is square-free and the character χ 2 is primitive modulo q , then each space of cusp forms has a basis of common eigenfunctions for all regular Hecke operators and all singular Frobenius operators, and the absolute values of all eigenvalues of ║Π( m ) are equal to m ½( nk - n ( n +1)/2) . The result for n = 1 is due to W. W. Li; n = 2 and a prime level q was obtained in an earlier paper by the author and A. A. Panchishkin.