  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.
