Geometryofnumbers methods in the cusp and applications to class groups
Abstract
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 selfadjoint operators for the symmetric bilinear form $\sum_{i = 1}^n x_iy_{n+1i}$, the orbits of which naturally parametrize $2$torsion ideal classes of monogenized degree$n$ orders.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09466
 Bibcode:
 2021arXiv211009466S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11R29;
 11R45;
 11H55;
 11E76
 EPrint:
 38 pages