Inter-universal Teichmüller theory

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).This theory is considered to be the most fundamental development in pure mathematics in several decades. Other names for the theory are arithmetic deformation theory and Mochizuki theory.

Several previously developed and published theories in the previous 20 years by Shinichi Mochizuki are related and used in many ways to IUT. They include his fundamental pioneering work in anabelian geometry including its new areas of absolute anabelian geometry and mono-anabelian geometry, as well as p-adic Teichmüller theory, Hodge-Arakelov theory, 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.

On the one hand, IUT is a very novel theory which offers a large number of novel concepts and points of view on numbers and their properties. On the other hand, this theory is applied to prove some key problems in Diophantine geometry, such as the abc conjecture. The latter can be viewed as a concentrated expression of fundamental issues in Diophantine geometry. Solving the abc conjecture implies or is likely to imply solutions of many famous problems in number theory. For readers familiar with algebraic number theory, the place of IUT can be designated as closely related to anabelian geometry which is one of the three most important generalizations of class field theory. Two other generalizations are the Langlands program and higher class field theory. A very distinctive feature of anabelian geometry and IUT is its non-linearity: unlike the other two generalizations of class field theory, it operates with full groups of symmetries such as the absolute Galois group of a number field and its completions and the arithmetic fundamental group of a hyperbolic curve over a number fields.

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