Inter-universal Teichmüller theory

In mathematics, inter-universal Teichmüller theory (IUT) is a theory created by Shinichi Mochizuki (2012a, 2012b, 2012c, 2012d). It is an arithmetic version of Teichmüller theory for number fields with an elliptic curve, introduced by Shinichi Mochizuki (2012a, 2012b, 2012c, 2012d).[1][2][3][4][5]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.

Overview

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'."[6] Alternative names for the theory are arithmetic deformation theory[7] 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.

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 to reconstruct relevant schemes after applying arithmetic deformations to them; a key role is played by three rigidities established in Mochizuki's etale-theta function theory. Roughly speaking, arithmetic deformations change the multiplication of a given ring, and the task is to measure how much the addition is changed.[8] Infrastructure for deformation procedures is decoded by certain links between theaters, such as a theta-link and a log-link.[9] 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.[9] 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.[7][8]

The nearest geometric proof, over complex numbers, to IUT is a classical proof by Bogomolov and Zhang (in fact, its substantial part was already the content of one of Milnor's papers [10]) of the geometric Szpiro inequality.[7][11]

IUT is a theory whose vision lies substantially outside the scope of previous arithmetic geometry. In particular, its philosophy is very different from that of the Langlands Program and Galois representation theory. Uniquely in mathematics, IUT opens a new branch of number theory. Shinichi Mochizuki, in section 4.4 of his most recent survey[9] explains fundamental differences between IUT and some of the mainstream directions in number theory. Ivan Fesenko lists twelve central new concepts of the theory.[7] This partially explains the unusually difficult challenge for mathematicians to study the theory; some of the other causes explaining why modern number theorists are essentially failing to study IUT are discussed in sect. 3.4 of.[7]

Two texts Mochizuki (2013b) and Mochizuki (2014) gave a summary of progress in the study of the theory.[12] Changes in the texts of IUT have included hundreds of small corrections, some of which due to questions and comments of the first readers of the theory: Go Yamashita, Mohamed Saidi and Yuichiro Hoshi. Two surveys of IUT were produced by its author,[9][13] two surveys by Ivan Fesenko[7][8] and two surveys by Yuichiro Hoshi.[14] Shinichi Mochizuki has invested a very substantial time to answer e-questions and to aid dissemination of his results in various seminars and meetings.[15]

National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015.[16] 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.[17] Its report[18] mentions, "The workshop helped its participants to go relatively fast (sic) 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.[19][20] Among its files there is a document[14] 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."

References

  1. ↑ Mochizuki, Shinichi (2011), "Inter-universal Teichmüller Theory: A Progress Report", Development of Galois–Teichmüller Theory and Anabelian Geometry (PDF), The 3rd Mathematical Society of Japan, Seasonal Institute.
  2. ↑ Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF).
  3. ↑ Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF).
  4. ↑ Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF).
  5. ↑ Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF).
  6. ↑ Mochizuki, Shinichi (2013), A Panoramic Overview of Inter-universal Teichmuller Theory (PDF)
  7. 1 2 3 4 5 6 Fesenko, Ivan (2015), Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015 (PDF)
  8. 1 2 3 Fesenko, Ivan (2016), Fukugen, Inference: International Review of Science, 2016
  9. 1 2 3 4 Mochizuki, Shinichi (2016), The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory (PDF)
  10. ↑ Milnor, John (1958), On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215–223
  11. ↑ Mochizuki, Shinichi (2016), Bogomolov’s proof of the geometric version of the Szpiro conjecture from the point of view of inter-universal Teichmüller theory, Res. Math. Sci. 3(2016), 3:6
  12. ↑ Mochizuki, Shinichi (2017), News from Shinichi Mochizuki
  13. ↑ Mochizuki, Shinichi (2014), A panoramic overview of inter-universal Teichmüller theory, In Algebraic number theory and related topics 2012, RIMS Kôkyûroku Bessatsu B51, RIMS, Kyoto (2014), 301–345 (PDF)
  14. 1 2 "On questions and comments concerning Inter-universal Teichmüller Theory" (PDF).
  15. ↑ Seminars, meetings, lectures on IUT in Japan (PDF)
  16. ↑ Future and past workshops on IUT theory of Shinichi Mochizuki
  17. ↑ https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html
  18. ↑ "Report on the Oxford workshop on the IUT theory of Shinichi Mochizuki, by Ivan Fesenko".
  19. ↑ Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016)
  20. ↑ Brief report on Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016)

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