Modularity and Generalized Fermat's Equation

This volume contains the detailed exposition of virtual lectures on the theme ""Modularity and Generalized Fermat's Equation"", which were delivered in January-April 2022 during the program organized by the Bhaskaracharya Pratishthan Research Institute in Mathematics, Pune, India. Modularity of elliptic curves over the field of rational numbers plays a central role in the proof of the celebrated Fermat's Last Theorem (FLT) by Wiles and Taylor-Wiles in 1995. Around the same time Henri Darmon and his collaborators initiated the study of Generalized Fermat Equations (GFE) in a series of papers. This work leads to an ambitious program to study the generalized Fermat equation $aX^m bY^n=cZ^r$ using the modularity of abelian varieties of $\mathrm{GL}_2$-type over totally real fields. This volume covers some of the background material and presents the current state of the art in this area.