Construction of Arithmetic Teichmuller Spaces III: A `Rosetta Stone' and a proof of Mochizuki's Corollary 3.12

Abstract

This is a continuation of my work on Arithmetic Teichmuller Spaces (arXiv:2106.11452, arXiv:2210.11635, arXiv:2303.01662, arXiv:2305.10398). This paper establishes a number of important results including (1) a proof Mochizuki's Corollary 3.12 (2) establishes a `Rosetta Stone' for a parallel reading of Mochizuki's Inter-Universal Teichmuller Theory and my Theory of Arithmetic Teichmuller Spaces, and (3) a proof that Mochizuki's gluing of Hodge-Theaters, Frobenioids along prime-strips as described in his theory is naturally provided by the existence of Arithmetic Teichmuller Spaces. (4) Includes the geometric case of Mochizuki's Corollary 3.12 in §12.

163 Pages; Substantially expanded with several additions and other improvements: Intro is re-written; §2.1 is rewritten. New additions are as follows: §1.3, 1.7,1.9; §4.6 (Relationship with (global) period mapping); §8.2.2 (Role of geometric base-points). §12 Geometric case of Mochizuki's Corollary 3.12. Comments and correction are welcome!