Arithmetic of Pell surfaces

TL;DR

This paper introduces a group structure on primitive integer points of Pell surfaces, linking it to the n-torsion subgroup of the narrow ideal class group in quadratic fields, revealing new algebraic relationships.

Contribution

It defines a novel group law on Pell surface points and establishes a surjective homomorphism to the n-torsion class group, connecting geometric and algebraic number theory.

Findings

Established a group structure on Pell surface points.

Constructed a surjective homomorphism to the n-torsion subgroup.

Bridged geometric points with algebraic class groups.

Abstract

We define a group stucture on the primitive integer points (A,B,C) of the algebraic variety Q_0(B,C)=A^n, where Q_0 is the principal binary quadratic form of fundamental discriminant \Delta and n is a fixed integer greater than 1. A surjective homomorphism is given from this group to the $n$-torsion subgroup of the narrow ideal class group of the quadratic number field Q(\sqrt{\Delta}).