The cuspidal cohomology of GL3/ℚ and cubic fields

We investigate the subspace of the homology of a congruence subgroup [Formula: see text] of [Formula: see text] with coefficients in the Steinberg module [Formula: see text] which is spanned by certain modular symbols formed using the units of a totally real cubic field E. By Borel–Serre duality, [Formula: see text] is isomorphic to [Formula: see text]. The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology [Formula: see text]. Their span is a naturally defined subspace [Formula: see text] of [Formula: see text]. Using a computer, we study where [Formula: see text] sits between 0 and [Formula: see text]. On the basis of our computations, we conjecture that [Formula: see text], and we raise the question as to whether for each E individually it might always be true that [Formula: see text].