Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki

摘要

These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on inter-universal Teichmüller theory (Inter-universal Teichmüller theory I–IV, 2012–2015) which might also be called arithmetic deformation theory, and its application to diophantine geometry. They provide an external perspective which complements the review texts of Mochizuki (Invitation to inter-universal Teichmüller theory (lecture note version), 2015) and (Algebraic Number Theory and Related Topics 2012. RIMS Kôkyûroku Bessatsu, vol B51, pp. 301–346, 2014). Some important developments which preceded (Inter-universal Teichmüller theory I–IV, 2012–2015) 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.