Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves

TL;DR

This paper formalizes the solution of Mordell equations using class groups in Lean 3, including formalizations of number theory concepts and new tactics for efficient computations in quadratic rings.

Contribution

It introduces a formalized approach to solving Mordell equations in Lean 3, incorporating formalizations of ideal norms, quadratic fields, and class number computations, along with new computational tactics.

Findings

Formalized solutions for several Mordell equations with d<0.

Developed new tactics for efficient computations in quadratic rings.

Extended formalizations to include ideal norms, quadratic fields, and class groups.

Abstract

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.