Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of inter-universal Teichmüller theory

The purpose of the present paper is to expose, in substantial detail, certain remarkable similarities between inter-universal Teichmüller theory and the theory surrounding Bogomolov’s proof of the geometric version of the Szpiro Conjecture. These similarities are, in some sense, consequences the fact that both theories are closely related to the hyperbolic geometry of the classical upper half-plane. We also discuss various differences between the theories, which are closely related to the conspicuous absence in Bogomolov’s proof of Gaussian distributions and theta functions, i.e., which play a central role in inter-universal Teichmüller theory.