Counting Orders in Number Fields
Time and place
1 PM on Thursday, October 9th, 2014; NAC 6/113
Nathan Kaplan (Yale)
Abstract
We focus on several related counting problems in number theory. How many sublattices of Z^n have index k? How many of these sublattices are actually subrings? These questions fit nicely within the theory of zeta functions of groups and rings. We will then transition from Z^n to counting in algebraic number fields, finite extensions of the rational numbers. We study orders in these fields, subrings that have size measured by discriminant. We approach these counting questions by analyzing the zeta functions that count subrings of Z^n and orders in a number field. We derive asymptotic formulas by analyzing the poles of these functions. We use a combination of methods from analytic and algebraic number theory, algebraic geometry, representation theory of finite groups, and combinatorics. This work is joint with Ramin Takloo-Bighash and Jake Marcinek.
