Darmon's Program: A survey

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.