This work derives an asymptotic formula, as a norm bound grows, for the number of integral matrices over a number field having a prescribed irreducible characteristic polynomial, writing the count in terms of orbital integrals of gl_n and extending the Eskin-Mozes-Shah counting theorem. Students learn how a lattice-point counting problem is translated into adelic orbital integrals, how local Brauer evaluations are read through local class field theory, and how those evaluations correspond to local endoscopic data for SL_n via the Langlands-Shelstad fundamental lemma.
For an irreducible polynomial over the ring of integers of a number field, this paper counts the n-by-n integral matrices whose characteristic polynomial equals that polynomial and whose norm is boun…