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This paper lays the foundation of the Theory of Arithmetic Teichmuller Spacesof Number Fields by explicitly constructing many arithmetically inequivalentavatars of a fixed number field. This paper also constructs a topological spaceof such avatars and describes its symmetries. Notably amongst these symmetriesis a global Frobenius morphism which changes the avatar of the number field!The existence of such avatars has been suggested and used by Shinichi Mochizukiin his work on the arithmetic Vojta and Szpiro conjectures. The key advantageof my approach is that one can quantify the difference between two inequivalentavatars and this renders my theory fundamentally and quantitatively moreprecise than Mochizuki's approach. In the appendix, I provide a discussion ofthe 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 discussapplications of my theory to the theory of arithmetic loops and arithmeticknots.

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