A survey on the generalized Fermat equation of various signatures over totally real fields
Satyabrat Sahoo
TL;DR
This survey synthesizes modularity-method results for generalized Fermat equations over totally real fields, focusing on signatures (p,p,p), (p,p,2), (p,p,3), and (r,r,p). It details the construction of Frey elliptic curves, the role of Hilbert modular forms, and the use of level-lowering and S-unit techniques to obtain asymptotic nonexistence or finite-solution results, connecting classical Q-state results to totally real field settings via Eichler–Shimura-type correspondences. The article also clarifies the strategy for proving no-solution results and surveys modularity breakthroughs for elliptic curves over various totally real fields, highlighting the general expectation that most such curves are modular. Collectively, the work foregrounds the interplay between Frey curves, Hilbert modular forms, and S-unit bounds in advancing our understanding of generalized Fermat equations over number fields. The findings have implications for generalized Fermat-type Diophantine problems in arithmetic geometry and illustrate the ongoing unification of Diophantine analysis with automorphic forms over number fields.
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.