Melanie Matchett Wood

Abstract:There are certain finite abelian groups that arise from objects in number theory that are quite mysterious and of great interest, for example the class group arising from a finite extension of the rational numbers, or the Tate-Shafarevich group of an elliptic curve y^2=x^3+ax+b (for some rational numbers a,b). We discuss the question of what a class group of a random extension, or the Tate-Shafarevich group of a random elliptic curve, looks like, and explain heuristics of several authors including Cohen and Lenstra, and Delaunay, for how these random groups behave. Finally we will relate the predictions of these heuristics to phenomena that can be seen and proven about random integral matrices.