A Modular Curve of Level 9 and the Class Number One Problem

TL;DR

This paper provides an explicit parametrization of a specific modular curve related to level 9 and uses it to identify all integral points, offering an alternative approach to solving the class number one problem.

Contribution

It introduces an explicit parametrization of the modular curve for the normalizer of a non-split Cartan subgroup at level 9 and applies it to solve the class number one problem.

Findings

Explicit parametrization of the modular curve at level 9.

Complete determination of integral points on this curve.

An alternative solution to the class number one problem.

Abstract

In this note we give an explicit parametrization of the modular curve associated to the normalizer of a non-split Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem.