On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

TL;DR

This paper establishes a correspondence between solutions of a family of cubic Thue equations and isomorphism classes of simplest cubic fields, providing explicit solutions and a new proof of their non-isomorphism in most cases.

Contribution

It introduces a novel correspondence linking cubic Thue equation solutions to simplest cubic field classifications and determines all solutions for certain divisors, offering new insights into field isomorphisms.

Findings

66 non-trivial solutions for specific divisors

Proof that most simplest cubic fields are non-isomorphic

Explicit correspondence between solutions and field classes

Abstract

Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=\lambda \] where $\lambda>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $\lambda$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$.