Pacific Institute for the Mathematical Sciences Colloquium: "Diophantine equations, geometry, and Brauer groups"

Abstract

The set of solutions to a diophantine equation is strongly influenced by the geometry of the associated algebraic variety. This paradigm, for example, suggests that an elementary proof of Fermat’s Last Theorem is unlikely to exist. I will describe what the paradigm suggests about diophantine equations whose geometric avatars are surfaces. In particular, I will introduce the Grothendieck-Brauer group of an algebraic variety and show how it explains the shape of the set of solutions to certain diophantine equations.