Modular Symbols for image-Rank One Groups and Voronı Reduction Original Research Article

LetGbe a reductive algebraic group of image-rank one associated to a self-adjoint homogeneous cone defined over image, and letΓsubset ofGbe a torsion-free arithmetic subgroup. Letdbe the cohomological dimension ofΓ. We present an algorithm to compute the action of the Hecke operators onHd(Γ;image). This generalizes the classical modular symbol algorithm, whenΓsubset ofSL2(image), to a setting including Bianchi groups and Hilbert modular groups. In addition, we generalize some results of Voronoı for real positive-definite quadratic forms to self-adjoint homogeneous cones of arbitrary image-rank.