Affine symmetric spaces and 2-torsion in the class group of unit-monogenized cubic fields

Davenport’s lemma has been a crucial ingredient in recent applications of geometry of numbers to arithmetic statistics. The lemma estimates, with error-term, the number of lattice points contained in bounded semi-algebraic regions of ℝn in terms of the volumes of these regions and their projections to coordinate subspaces. In this talk, I will describe an application of geometry of numbers to a non-Euclidean setting (in which Davenport’s lemma does not apply) by explaining how to bound the average number of 2-torsion elements in the class group of unit-monogenized cubic fields. The proof proceeds by replacing Davenport’s lemma with counting results for affine symmetric spaces. Joint work with Iman Setayesh, Arul Shankar, and Ashvin Swaminathan.