A Homological Definition for the Tate--Shafarevich Group of a Pell Conic

TL;DR

This paper provides a cohomological definition of the Tate--Shafarevich group for Pell conics, confirming conjectures about its order related to the narrow class group, thus advancing the arithmetic understanding of these conics.

Contribution

It introduces a new cohomological framework for the Tate--Shafarevich group of Pell conics, confirming conjectured properties about its order.

Findings

Confirmed the order of the Tate--Shafarevich group matches conjectures

Provided a cohomological definition for the group

Linked the group's order to the narrow class group

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 article, we provide a cohomological definition of the Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured.