Conics - a Poor Man's Elliptic Curves

TL;DR

This paper explores the arithmetic of Pell conics, demonstrating that it parallels elliptic curves with a theory of 2-descent, Selmer groups, and conjectural BSD analogs, suggesting deep structural similarities.

Contribution

It introduces a framework for understanding Pell conics analogous to elliptic curves, including 2-descent and conjectural BSD-type statements.

Findings

Arithmetic of Pell conics admits a structure similar to elliptic curves.

Development of a 2-descent theory for Pell conics.

Proposal of an analog of the Birch and Swinnerton-Dyer conjecture for Pell conics.

Abstract

The aim of this article is to show that the arithmetic of Pell conics admits a description which is completely analogous to that of elliptic curves: there is a theory of 2-descent with associated Selmer and Tate-Shafarevich groups, and there should be an analog of the conjecture of Birch and Swinnerton-Dyer.