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

• Published in: Research in the mathematical sciences Vol. 3; no. 1; pp. 1 - 21
• Main Author: Mochizuki, Shinichi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 05.06.2016; Springer Nature B.V
• Subjects: Analogies; Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Review; Special Collection in Celebration of the Research of Fedor Bogomolov on the Occasion of his 70th Birthday; Hyperbolic geometry; Multiradial representation; Indeterminacies; Theta function; Bogomolov’s proof; Gaussian distribution; Inter-universal Teichmüller theory; Szpiro Conjecture; Symplectic geometry; Upper half-plane
• ISSN: 2197-9847; 2522-0144; 2197-9847
• DOI: 10.1186/s40687-016-0057-x

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.