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 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, 2015 (pdf)