Tangential localization for Selmer varieties

Abstract

This paper proposes a tangential version of the theory of Selmer varieties together with a formulation of cohomological duality in families of Lie algebras indexed by nonabelian cohomology. This theory allows one to consider deformations of cohomology classes as one moves over the Selmer variety and suggests an approach for generalizing to number fields the homotopical techniques for proving Diophantine finiteness that were developed over $\mathbb{Q}$. The utility of this perspective is demonstrated by way of a new proof of Siegel’s theorem on finiteness of $S$-integral points for the projective line minus three points over a totally real field.