Solutions to many Diophantine equations in number theory can be studied using the theory of algebraic stacks. In this talk, I will describe a collection of "generalized Fermat equations" whose integer solutions can be found using an appropriate family of stacky curves. These curves are interesting in their own right: they provide potential counterexamples to the local-global principle for integral points of stacky curves. In fact, results of Darmon-Granville and joint work in progress with Duque-Rosero, Keyes, Roy, Sankar and Wang indicate there are infinitely many of these counterexamples. No background on stacks is required for this talk.
Stacky Curves and Generalized Fermat Equations
Andrew Kobin, Emory UniversityAuthors: Juanita Duque-Rosero, Chris Keyes, Andrew Kobin, Manami Roy, Soumya Sankar, Yidi Wang
2023 AWM Research Symposium
Rethinking Number Theory [Organized by Deewang Bhamidipati, Eva Goedhart, and Amita Malik]