Construction of Arithmetic Teichmuller Spaces II$\frac{1}{2}$: Deformations of Number Fields

Abstract

This paper lays the foundation of the Theory of Arithmetic Teichmuller Spaces of Number Fields by explicitly constructing many arithmetically inequivalent avatars of a fixed number field. This paper also constructs a topological space of such avatars and describes its symmetries. Notably amongst these symmetries is a global Frobenius morphism which changes the avatar of the number field! The existence of such avatars has been suggested and used by Shinichi Mochizuki in his work on the arithmetic Vojta and Szpiro conjectures. Important to the global aspect of this theory is the fact that the product formula for a number field defines an arithmetic period mapping (Section 5.9). The key advantage of my approach is that one can quantify the difference between two inequivalent avatars and this renders my theory fundamentally and quantitatively more precise than Mochizuki's approach. In the appendix, I provide a discussion of the proofs of the geometric Szpiro Conjectures due to [Bogomolov et. al. 2000] and [Zhang 2001] from the point of view of this paper. I also discuss applications of my theory to the theory of arithmetic loops and arithmetic knots.