Explicit action of Hecke operators on Siegel modular forms

Abstract

We develop an algorithm for determining an explicit set of coset representatives (indexed by lattices) for the action of the Hecke operators T(p), Tj(p2) on Siegel modular forms of fixed degree and weight. This algorithm associates each coset representative with a particular lattice Ω, pΛ ⊆ Ω ⊆ 1/p Λ where Λ 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 Tj(p2), we complete these sums by replacing this operator with a linear combination of Tℓ(p2), 0 ≤ ℓ ≤ j. In all cases, this yields a clean and simple description of the action on Fourier coefficients. © 2002 Elsevier Science (USA).