Cycle Indices in Arithmetic Geometry

Abstract

A cycle index is a polynomial that encodes information about the orbits of a finite group action on a finite set, which is used in combinatorics to count various objects up to group actions. In this thesis, we give a general framework for the cycle indices of a sequence of finite group actions on finite sets and study how certain cycle indices can be used in algebraic geometry and number theory.

As motivating examples, we will see how the zeta series of a projective variety over a finite field is related to the cycle indices of symmetric groups and how the distribution of the cokernel of a Haar random matrix over the p-adic integers is related to the cycle indices of conjugation actions of general linear groups on matrices over the finite field of p elements.