Explicit action of Hecke operators on Hilbert-Siegel modular forms.

摘要

We develop an algorithm for determining an explicit set of coset representatives (indexed by lattices) for the action of the Hecke operators T( p), T_j(p2) on Hilbert-Siegel modular forms of fixed degree and weight. This algorithm associates each coset representative with a particular lattice $W$, $pL\subseteq W\subseteq 1pL$ where $L$ is a fixed reference lattice. We then evaluate the action of the Hecke operators on Fourier series. Since this evaluation yields incomplete character sums for T_j(p2), we complete these sums by replacing this operator with a linear combination of T_ℓ(p2), 0 ≤ ℓ ≤ j. In all cases, this yields a clean and simple description of the action on Fourier coefficients.