In mathematics, inter-universal Teichmüller theory (IUT) is an arithmetic version of Teichmüller theory for number fields with an elliptic curve, introduced by Shinichi Mochizuki (2012a, 2012b, 2012c, 2012d).
Several previously developed and published theories in the previous 20 years by Shinichi Mochizuki are related in many ways to IUT. They include p-adic Teichmüller theory, Hodge-Arakelov theory, mono-anabelian geometry, frobenioids theory, anabelioids theory, and etale theta-functions theory.
Shinichi Mochizuki explains the name as follows: "in this sort of a situation, one must work with the Galois groups involved as abstract topological groups, which are not equipped with the 'labeling apparatus' . . . [defined as] the universe that gives rise to the model of set theory that underlies the codomain of the fiber functor determined by such a basepoint. It is for this reason that we refer to this aspect of the theory by the term 'inter-universal'." Alternative names for the theory are arithmetic deformation theory and Mochizuki theory.
Arithmetic deformation theory operates with full Galois and algebraic fundamental groups of various hyperbolic curves associated to an elliptic curve over a number field and related enhanced categorical structures called Hodge theaters. The key prerequisite for IUT is Mochizuki's mono-anabelian geometry and its powerful reconstruction results, which restore various one-dimensional and two-dimensional scheme-theoretic objects associated to the hyperbolic curves over the number field from the knowledge of its fundamental group or certain Galois groups. IUT applies algorithmic results of mono-anabelian geometry and three rigidities established in Mochizuki's etale-theta function theory to reconstruct relevant schemes after applying arithmetic deformations to them. Roughly speaking, arithmetic deformations change the multiplication of a given ring, and the task is to measure how much the addition is changed. Infrastructure for deformation procedures is decoded by certain links between theaters, such as a theta-link and a log-link. These links are not compatible with ring or scheme structures and are performed outside conventional arithmetic geometry. However, they are compatible with certain group structures, and absolute Galois groups as well as certain types of topological groups play a fundamental role in IUT. Considerations of multiradiality, a generalization of functoriality, imply that three mild indeterminacies have to be introduced. The effect of these indeterminacies eventually results in the epsilon term in the inequalities such as the strong Szpiro conjecture over any number field and part of the Vojta's conjecture for the case of hyperbolic curves over any number field, as well as other equivalent forms. Aside from the prime number theorem, IUT does not use analytic number theory.
The nearest geometric theory, over complex numbers, to IUT is Fedor Bogomolov's proof of the geometric Szpiro inequality.
IUT is a pioneering theory whose vision lies substantially outside the scope of arithmetic geometry. Rarely for mathematical work, this single theory opens a new branch of number theory. Shinichi Mochizuki, in section 4.4 of his most recent survey explains fundamental differences between IUT and some of the mainstream directions in number theory. Ivan Fesenko, in one of his surveys, lists twelve central new concepts of the theory. This partially explains the unusually difficult challenge for mathematicians to study the theory; some of other causes explaining why modern number theorists are essentially failing to study IUT are discussed in sect. 3.4 of.
Two texts Mochizuki (2013b) and Mochizuki (2014) gave a summary of progress in the study of the theory and its minor changes. Two surveys of IUT were produced by its author, two surveys by Ivan Fesenko and two surveys by Yuichiro Hoshi. Shinichi Mochizuki has invested a very substantial time to answer e-questions and to aid dissemination of his results in various seminars and meetings.
National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015. To assist mathematicians interested in the theory, two international workshops were organized. The first international workshop on Mochizuki's theory was held in Oxford in December 2015. Its report mentions, "The workshop helped its participants to go relatively fast through the prerequisites of the theory and to see many main new concepts of the theory in action." A further international workshop on IUT Summit was held at RIMS in July 2016. Among its files there is a document which includes, "As of July 2016, the four papers on IUT have been thoroughly studied and verified in their entirety by at least four mathematicians (other than the author), and various substantial portions of these papers have been thoroughly studied by quite a number of mathematicians (such as the speakers at the Oxford workshop in December 2015 and the RIMS workshop in July 2016). These papers are currently being refereed, and, although they have not yet been officially accepted for publication, the refereeing process is proceeding in an orderly, constructive, and positive manner."