A survey on the generalized Fermat equation of various signatures over totally real fields

TL;DR

This survey reviews recent advances in solving the generalized Fermat equation over totally real fields using the modularity method, covering various signatures and highlighting key results and techniques.

Contribution

It compiles and discusses numerous recent results on the solutions of generalized Fermat equations over totally real fields employing the modularity approach.

Findings

Solutions are largely classified for various signatures over totally real fields.

The modularity method has been successfully applied to extend Fermat's Last Theorem results.

New techniques have been developed for handling different signatures in these equations.

Abstract

Following the famous proof of Fermat's Last Theorem by Andrew Wiles using the modularity of elliptic curves over $\mathbb{Q}$, significant developments have been made in the study of Diophantine equations using the modularity method. This article presents a survey of numerous results on the solutions of the generalized Fermat equation of signatures $(p,p,p)$, $(p,p,2)$, $(p,p,3)$, and $(r,r,p)$ over totally real number fields using the modularity method.