ARITHMETIC DEFORMATION THEORY VIA
ARITHMETIC FUNDAMENTAL GROUPS AND NONARCHIMEDEAN THETA-FUNCTIONS,
NOTES ON THE WORK OF SHINICHI MOCHIZUKI

IVAN FESENKO

ABSTRACT. These notes survey the main ideas, concepts and objects
of the work by Shinichi Mochizuki on interuniversal Teichmüller
theory [31], which might also be called arithmetic deformation
theory, and its application to diophantine geometry. They provide
an external perspective which complements the review texts [32]
and [33].Some important developments which preceded [31] are
presented in the first section. Several important aspects of
arithmetic deformation theory are discussed in the second section.
Its main theorem gives an inequality–bound on the size of volume
deformation associated to a certain log-theta-lattice.
The application to several fundamental conjectures in number theory
follows from a further direct computation of the right hand side of
the inequality. The third section considers additional related
topics, including practical hints on how to study the theory.

https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf