Darmon's Program: A survey

TL;DR

This survey reviews Darmon's program for solving families of generalized Fermat equations, summarizing recent advances, background material, and simplified examples to facilitate understanding and application of the methods.

Contribution

It provides a comprehensive overview of Darmon's approach, including recent developments and accessible explanations for applying these techniques to generalized Fermat equations.

Findings

Summarizes recent progress in Darmon's program

Provides background material for understanding the methods

Includes simplified examples of key steps

Abstract

We give an overview of Darmon's program for resolving families of generalized Fermat equations with one varying exponent and survey what is currently known about this approach based on recent work of Billerey-Chen-Dieulefait-Freitas and Chen-Koutsianas. Additionally, we provide background material which is helpful to understand and apply the methods developed in these recent works. In particular, we explain the basic strategy for and simplified examples of each of the steps that is required in order to resolve a family of generalized Fermat equations.