A Genus Two Curve Related to the Class Number One Problem

TL;DR

This paper presents a new approach to the class number one problem by linking imaginary quadratic fields to rational points on a specific genus two curve, and provides methods to find all such points.

Contribution

It introduces a novel genus two curve related to the class number one problem and demonstrates how to determine all rational points on this curve.

Findings

All rational points on the curve  are found.

The curve  is connected to classical Heegner curves and coverings.

The approach offers a new perspective on class number one problem solutions.

Abstract

We give another solution to the class number one problem by showing that imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve $\mscrK_3$. In fact one can find all rational points on $\mscrK_3$. The curve $\mscrK_3$ arises naturally via certain coverings of curves:\ $\mscrK_3\rg\mscrK_6$,\ $\mscrK_1\rg\mscrK_2$\ with $\mscrK_2\colon y^2=2x(x^3-1)$ denoting the Heegner curve, also in connection with the so-called Heegner-Stark covering $\mscrK_1\rg\mscrK_s$.