transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
The generalization of Teichmüller theory to arithmetic geometry has been called inter-universal Teichmüller theory (often abbreviated IUTT) by Shinichi Mochizuki.
The term “inter-universal” apparently refers to the fact that the theory is meant to formulated explicitly in a way that respects universe enlargement, hence that it is universe polymorphic (Mochizuki 12d, remark 3.1.4, Yamashita 13).
It is claimed (Mochizuki 12d) but currently unchecked that a proof of the abc conjecture can be found from anabelian geometry in this context.
Shinichi Mochizuki, Inter-universal Teichmüller theory I, Construction of Hodge theaters (2012) (pdf)
Shinichi Mochizuki, Inter-universal Teichmüller theory II, Hodge-Arakelov-theoretic evaluation (2012) (pdf)
Shinichi Mochizuki, Inter-universal Teichmüller theory III, Canonical splittings of the Log-theta-lattice (2012) (pdf)
Shinichi Mochizuki, Inter-universal Teichmüller theory IV, Log-volume computations and set-theoretic foundations (2012) (pdf)
Surveys include
Shinichi Mochizuki, Panoramic overview of inter-universal Teichmuller theory, pdf
Yamashita, FAQ on ‘Inter-Universality’ (pdf)
Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, European Journal of Mathematics September 2015, Volume 1, Issue 3, pp 405-440 (publisher, pdf)
Minhyong Kim, Brief superficial remarks on Shinichi Mochizuki’s Interuniversal Teichmueller Theory (IUTT), version 1, 10/11/2015, (pdf).
Taylor Duypuy, Hodge Theaters: A First Look at the Big Hodge Theater, Confused Groups and Torsors
RIMS/Symmetries and Correspondences workshop: Inter-universal Teichmüller Theory Summit 2016
Jackson Morrow, Kummer classes and Anabelian geometry, notes from Super QVNTS: Kummer Classes and Anabelian Geometry 2017 (pdf)