MIT Number Theory Seminar

Speaker:  Ashvin Swaminathan (Harvard University)

Title:  Geometry-of-numbers in the cusp, and class groups of orders in number fields

Abstract: In this talk, we discuss the distributions of class groups of orders in number fields. We explain how studying such distributions is related to counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations.  We introduce two new methods to solve this counting problem, and as an example, we demonstrate how one of these methods can be used to determine the average size of the 2-torsion in the class groups of totally real or complex cubic orders,  when such orders are enumerated by discriminant.  Much of this work is joint with Arul Shankar, Artane Siad, and Ila Varma.

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