Boundaries of modular symbols

Abstract

Let G G be a connected reductive algebraic group defined over Q \mathbb {Q} . We determine the intersection of a minimal modular symbol with a face in the boundary of the Borel-Serre bordification associated to G G . We give an explicit formula for this in the case of G L n GL_n . We apply the formula to modular symbols for G L 3 ( Q ) GL_3(\mathbb {Q}) , constructed from the units of a totally real cubic field, to show that such symbols lie in the cuspidal cohomology.