Avner Ash (Boston College)

This is joint work with Dan Yasaki.  Given the units in a totally real field of degree k, you can form k-dimensional tori in the locally symmetric space for a congruence subgroup G of GL(k)/Q.  What is the span of their fundamental classes in the homology H_k(G,Q)?  We investigate a similar problem with certain modular symbols replacing the fundamental classes.   I will discuss some conjectures in this regard when k = 2 and 3.  Computer calculations support our conjectures.  We can prove one of our conjectures for GL(2) if we assume the Generalized Riemann Hypothesis.