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).[1][2][3][4][5]

Several previously developed and published theories by Mochizuki are related in various ways to IUT. They include his p-adic Teichmüller theory, his Hodge-Arakelov theory, his categorical geometry theories of Frobenioids and anabelioids, his mono-anabelian geometry and his etale-theta functions theory.

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]

This theory deals with full Galois and fundamental groups of various hyperbolic curves associated to the elliptic curve and related enhanced categorical structures (systems of Frobenioids). It applies deep algorithmic results of mono-anabelian geometry to reconstruct the groups and schemes after applying various links which are not compatible with ring or scheme structure. Resulting synchronizations, rigidities and mild inderterminacies lead to applications to the strong Szpiro conjecture and its equivalent forms.

The main theorems (Mochizuki 2012d) include two inequalities on the log-volume change associated to appropriately chosen deformations. The theorems imply a proof of several equivalent fundamental conjectures in Diophantine geometry, including the strong Szpiro conjecture over any number field, the abc conjecture over any number field, and part of the Vojta's conjecture for the case of hyperbolic curves over any number field. IUT extends substantially the scope of arithmetic geometry.

The theory is complex, includes many new concepts in mathematics, and requires substantial efforts to understand. During different stages of study and verification of the theory, many mathematicians asked and made hundreds of questions and comments, all of which have been addressed by the author. Mochizuki (2013b) and Mochizuki (2014) gave a summary of progress in verifying his work. Surveys[7][8] provide external perspectives.

A workshop on IUT was held at RIMS in March 2015 and in Beijing in July 2015. The first international workshop on Mochizuki's theory was organized by Ivan Fesenko and held in Oxford at in December 2015.[9] The next workshop on IUT Summit will be held at the Research Institute for Mathematical Sciences in Kyoto in July 2016.[10]

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. ↑ 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. ↑ Hoshi, Yuichiro (2015), Introduction to inter-universal Teichmüller theory (PDF)
  9. ↑ https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html
  10. ↑ https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html

External links

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