Abstract/Summary

Let T be an algebraic torus over ℚ such that T(ℝ) is compact. Assuming the generalized Riemann hypothesis, we give a lower bound for the size of the class group of T modulo its n-torsion in terms of a small power of the discriminant of the splitting field of T. As a corollary, we obtain an upper bound on the n-torsion in that class group. This generalizes known results on the structure of class groups of complex multiplication fields.