Modular Symbols for Q-Rank One Groups and Voronı Reduction

AbstractLetGbe a reductive algebraic group of Q-rank one associated to a self-adjoint homogeneous cone defined over Q, and letΓ⊂Gbe a torsion-free arithmetic subgroup. Letdbe the cohomological dimension ofΓ. We present an algorithm to compute the action of the Hecke operators onHd(Γ;Z). This generalizes the classical modular symbol algorithm, whenΓ⊂SL2(Z), 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 Q-rank