An Elliptic Curve Based Perspective on the Arithmetic of Pell Conics

Abstract

Franz Lemmermeyer's previous work laid the framework for a description of the arithmetic of Pell conics, which is analogous to that of elliptic curves. He describes a group law on conics and conjectures the existence of an analogous Tate--Shafarevich group with order the squared ideals of the narrow class group. In this thesis, we provide a cohomological definition of the Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured. Furthermore, we extend Lemmermeyer's work by giving a geometric description of the analogous Tamagawa numbers and compute their values. We also develop a Neron differential for the Pell conic and use it to compute the volume of the curve.