Geometry-of-numbers methods in the cusp and applications to class groups.

In this article, we compute the mean number of $2$-torsion elements in class groups of monogenized cubic orders, when such orders are enumerated by height. In particular, we show that the average size of the $2$-torsion subgroup in the class group increases when one ranges over all monogenized cubic orders instead of restricting to the family of monogenized cubic fields (or equivalently, monogenized maximal cubic orders) as determined in [8]. In addition, for each fixed odd integer $n \geq 3$, we bound the mean number of $2$-torsion elements in the class groups of monogenized degree-$n$ orders, when such orders are enumerated by height. To obtain such results, we develop a new method for counting integral orbits having bounded invariants and satisfying congruence conditions that lie inside the cusps of fundamental domains for coregular representations -- i.e., representations of semisimple groups for which the ring of invariants is a polynomial ring. We illustrate this method for the representation of the split orthogonal group on self-adjoint operators for the symmetric bilinear form $\sum_{i = 1}^n x_iy_{n+1-i}$, the orbits of which naturally parametrize $2$-torsion ideal classes of monogenized degree-$n$ orders.