[Paper Review] Conics - a Poor Man's Elliptic Curves

Franz Lemmermeyer|ArXiv.org|Nov 18, 2003

Algebraic Geometry and Number Theory13 references20 citations

TL;DR

This paper establishes a precise arithmetic analogy between Pell conics—defined by $X^2 - \Delta Y^2 = 4$—and elliptic curves, showing that conics serve as a 'poor man's' version of elliptic curves. It develops a full 2-descent theory for Pell conics, including Selmer and Tate-Shafarevich groups, and formulates an analog of the Birch–Swinnerton-Dyer conjecture, with a regulator and L-function matching the structure of the BSD conjecture for elliptic curves.

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.

Motivation & Objective

• To establish a precise arithmetic analogy between Pell conics and elliptic curves, treating conics as a 'poor man's' version of elliptic curves.
• To develop a 2-descent theory for Pell conics, including Selmer and Tate-Shafarevich groups.
• To formulate and verify an analog of the Birch–Swinnerton-Dyer conjecture for Pell conics, matching the structure of the original conjecture.
• To explore the cohomological and analytic structures of conics, including L-functions, regulators, and Tamagawa numbers.
• To investigate whether the group of integral points on Pell conics admits a geometric group law analogous to that on elliptic curves.

Proposed method

• Uses the geometric group law on conics via lines through a fixed rational point, with explicit formulas for addition on Pell conics: $(r,s)+(t,u) = \left(\frac{rt + \Delta su}{2}, \frac{ru + st}{2}\right)$.
• Establishes a group isomorphism between rational points on Pell conics and units of norm 1 in real quadratic fields $\mathbb{Q}(\sqrt{\Delta})$.
• Applies the 2-descent method to Pell conics, defining Selmer and Tate-Shafarevich groups via local conditions and cohomology.
• Introduces a regulator for Pell conics as $R(\mathcal{C}) = 2^{1-u} \log \eta$, where $\eta$ is a fundamental unit.
• Derives an L-function for Pell conics via zeta functions and relates it to the class number and regulator of the associated quadratic field.
• Constructs an analog of the BSD conjecture by matching the formula $\frac{2hR}{w} = \Omega \cdot \#\cyrshape(\mathcal{C}) \cdot R(\mathcal{C}) \cdot \prod c_p$, with $h$ the narrow class number and $w$ the number of roots of unity.

Experimental results

Research questions

• RQ1 Can a 2-descent theory be developed for Pell conics analogous to that for elliptic curves, including Selmer and Tate-Shafarevich groups?
• RQ2 Is there a meaningful analog of the Birch–Swinnerton-Dyer conjecture for Pell conics, with matching invariants like regulator, L-function, and Tamagawa factors?
• RQ3 What is the cohomological interpretation of the Selmer and Tate-Shafarevich groups for Pell conics?
• RQ4 How do the analytic invariants (regulator, L-function, class number) of Pell conics relate to the arithmetic of the associated real quadratic field?
• RQ5 Can the group of integral points on Pell conics be given a geometric group law, and does it satisfy properties like associativity via Pascal’s theorem?

Key findings

• The group of rational points on a Pell conic $\mathcal{C}: X^2 - \Delta Y^2 = 4$ with neutral element $N = (2,0)$ forms a group under the geometric addition law, isomorphic to the group of norm-1 units in $\mathbb{Q}(\sqrt{\Delta})$.
• For finite fields $\mathbb{F}_q$, the group $\mathcal{C}(\mathbb{F}_q) \simeq \mathbb{Z}/m\mathbb{Z}$ with $m = q - \left(\frac{\Delta}{p}\right)^f$, mirroring the structure of elliptic curve groups over finite fields.
• Over $p$-adic integers $\mathbb{Z}_p$, the group $\mathcal{C}(\mathbb{Z}_p)$ is isomorphic to $\mathbb{Z}/(p \pm 1) \oplus \mathbb{Z}_p$ depending on the Legendre symbol $\left(\frac{\Delta}{p}\right)$, analogous to the structure of $E(\mathbb{Q}_p)$.
• The regulator of a Pell conic is $R(\mathcal{C}) = 2^{1-u} \log \eta$, where $\eta$ is a fundamental unit and $u = 0$ if $\Delta < 0$, $u = 1$ if $\Delta > 0$.
• The analog of the BSD conjecture for Pell conics is $\frac{2hR}{w} = \Omega \cdot \#\cyrshape(\mathcal{C}) \cdot R(\mathcal{C}) \cdot \prod c_p$, with $h$ the narrow class number, $R$ the regulator, $w$ the number of roots of unity, and $c_p$ local factors.
• The paper shows that $\#\cyrshape(\mathcal{C}) \simeq \text{Cl}^+(k)^2$, suggesting a deep connection between the Tate-Shafarevich group of conics and the narrow class group of the associated quadratic field.