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

摘要

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 of 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.