Enumerative arithmetic

Abstract

This thesis consists of three chapters. Each chapter is on

a different subject. However, all three chapters address issues that arise in

counting arithmetically interesting objects. Chapter 1 is on the unit

equation in positive characteristic. Chapter 2 is about the statistical behavior of ray class groups, of fixed integral conductor, in families of imaginary quadratic fields. Chapter 3 concerns the study of the unit group of local fields in the category of filtered groups.