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In [[mathematics]], '''inter-universal Teichmüller theory''' ('''IUT''') is an arithmetic version of [[Teichmüller theory]] for [[number field]]s with an [[elliptic curve]], introduced by {{harvs|txt|authorlink=Shinichi Mochizuki|first=Shinichi|last=Mochizuki|year1=2012a|year2=2012b|year3=2012c|year4=2012d}}.<ref>{{citation|last1=Mochizuki| first1=Shinichi|url=http://mathsoc.jp/meeting/msjsi10/abstract/mochizuki.pdf|title=Development of Galois–Teichmüller Theory and Anabelian Geometry|chapter=Inter-universal Teichmüller Theory: A Progress Report|series=The 3rd Mathematical Society of Japan, Seasonal Institute|year=2011}}.</ref><ref>{{Citation|last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory I: Construction of Hodge Theaters | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf|year=2012a}}.</ref><ref>{{Citation|last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf | year=2012b}}.</ref><ref>{{Citation| last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice| url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf | year=2012c}}.</ref><ref>{{Citation| last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf | year=2012d}}.</ref>
In [[mathematics]], '''inter-universal Teichmüller theory''' ('''IUT''') is an arithmetic version of [[Teichmüller theory]] for [[number field]]s with an [[elliptic curve]], introduced by {{harvs|txt|authorlink=Shinichi Mochizuki|first=Shinichi|last=Mochizuki|year1=2012a|year2=2012b|year3=2012c|year4=2012d}}.<ref>{{citation|last1=Mochizuki| first1=Shinichi|url=http://mathsoc.jp/meeting/msjsi10/abstract/mochizuki.pdf|title=Development of Galois–Teichmüller Theory and Anabelian Geometry|chapter=Inter-universal Teichmüller Theory: A Progress Report|series=The 3rd Mathematical Society of Japan, Seasonal Institute|year=2011}}.</ref><ref>{{Citation|last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory I: Construction of Hodge Theaters | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf|year=2012a}}.</ref><ref>{{Citation|last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20II.pdf | year=2012b}}.</ref><ref>{{Citation| last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice| url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf | year=2012c}}.</ref><ref>{{Citation| last1=Mochizuki|first1=Shinichi|title=Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf | year=2012d}}.</ref>


Several previously developed and published theories by Mochizuki are related in various ways to IUT. They include [[p-adic Teichmüller theory]], [[Hodge–Arakelov theory|Hodge-Arakelov theory]], mono-anabelian geometry and etale theta-functions theory.
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|Hodge-Arakelov theory]], mono-anabelian geometry, frobenioids theory, anabelioids theory, and etale theta-functions theory.


Mochizuki explains the name as follows: "in this sort of a situation, one must work with the [[Galois group]]s 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 ''<nowiki>'inter-universal'</nowiki>''."<ref>{{Citation | last1=Mochizuki | first1=Shinichi|title=A Panoramic Overview of Inter-universal Teichmuller Theory|url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf | year=2013}}</ref> Alternative names for the theory are arithmetic deformation theory<ref name="Fesenko2015">{{Citation|first=Ivan|last=Fesenko|year=2015|title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015|url=https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf}}</ref> and Mochizuki theory.
Shinichi Mochizuki explains the name as follows: "in this sort of a situation, one must work with the [[Galois group]]s 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 ''<nowiki>'inter-universal'</nowiki>''."<ref>{{Citation | last1=Mochizuki | first1=Shinichi|title=A Panoramic Overview of Inter-universal Teichmuller Theory|url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf | year=2013}}</ref> Alternative names for the theory are arithmetic deformation theory<ref name="Fesenko2015">{{Citation|first=Ivan|last=Fesenko|year=2015|title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015|url=https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf}}</ref> 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, for a given ring arithmetic deformations change multiplication and the task is to measure how much addition is changed.<ref name="Fesenko2016">{{Citation|first=Ivan|last=Fesenko|year=2016|title=Fukugen, Inference: International Review of Science, 2016|url=http://inference-review.com/article/fukugen}}</ref> Infrastructure for deformation procedures is decoded by certain links between theaters, such as a theta-link and a log-link<ref name="Mochizuki2016">{{Citation|first=Shinichi|last=Mochizuki|year=2016|title=The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/}}</ref> These links are not compatible with ring or scheme structures and are performed outside conventional arithmetic geometry. However, they are complatible with certain group structures and absolute Galois groups as well as certain types of topological groups play fundamental role in IUT. Considerations of multiradiality, a generalization of functoriality, imply that certain three mild indeterminacies have to be introduced.<ref name="Mochizuki2016"/> The effect of these indeterminacies eventually results in the epsilon term in the inequalities such as the [[Szpiro's conjecture|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.<ref name="Fesenko2015"/><ref name="Fesenko2016"/>
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, for a given ring arithmetic deformations change multiplication and the task is to measure how much addition is changed.<ref name="Fesenko2016">{{Citation|first=Ivan|last=Fesenko|year=2016|title=Fukugen, Inference: International Review of Science, 2016|url=http://inference-review.com/article/fukugen}}</ref> Infrastructure for deformation procedures is decoded by certain links between theaters, such as a theta-link and a log-link<ref name="Mochizuki2016">{{Citation|first=Shinichi|last=Mochizuki|year=2016|title=The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf}}</ref> These links are not compatible with ring or scheme structures and are performed outside conventional arithmetic geometry. However, they are complatible with certain group structures and absolute Galois groups as well as certain types of topological groups play fundamental role in IUT. Considerations of multiradiality, a generalization of functoriality, imply that certain three mild indeterminacies have to be introduced.<ref name="Mochizuki2016"/> The effect of these indeterminacies eventually results in the epsilon term in the inequalities such as the [[Szpiro's conjecture|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.<ref name="Fesenko2015"/><ref name="Fesenko2016"/>


The nearest geometric theory, over complex numbers, to IUT is [[Fedor Bogomolov]]'s proof of the geometric Szpiro inequality<ref name="Fesenko2015"/><ref name="Mochizuki2016">{{Citation|first=Shinichi|last=Mochizuki|year=2016|title=
The nearest geometric theory, over complex numbers, to IUT is [[Fedor Bogomolov]]'s proof of the geometric Szpiro inequality<ref name="Fesenko2015"/><ref name="Mochizuki2016">{{Citation|first=Shinichi|last=Mochizuki|year=2016|title=
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}}</ref>.
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}}</ref>.


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<ref name="Mochizuki2016">{{Citation|first=Shinichi|last=Mochizuki|year=2016|title=The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf}}</ref> explains fundamental differences between IUT and the mainstream directions in arithmetic geometry. [[Ivan Fesenko]], in one of his surveys, lists twelve central new concepts of the theory.<ref name="Fesenko2015"/> 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 <ref name="Fesenko2015"/>.
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<ref name="Mochizuki2016"/>
<ref name="Mochizuki2016"> explains fundamental differences between IUT and the mainstream directions in arithmetic geometry. [[Ivan Fesenko]], in one of his surveys, lists twelve central new concepts of the theory.<ref name="Fesenko2015"/> 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 <ref name="Fesenko2015"/>.


Two texts {{harvs|txt|last=Mochizuki|year=2013b}} and {{harvs|txt|last=Mochizuki|year=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,<ref>{{Citation|first=Shinichi|last=Mochizuki|year=2014|title=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 | url=http://www.kurims.kyoto-u.ac.jp/%7Emotizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf}}</ref><ref name="Mochizuki2016"/> two surveys by [[Ivan Fesenko]]<ref name="Fesenko2015"/><ref name="Fesenko2016"/> and two surveys by Yuichiro Hoshi.<ref name="Hoshi">{{cite web|url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Questions%20and%20Comments%20on%20IUT.pdf|title=On questions and comments concerning Inter-universal Teichmüller Theory}}</ref> Shinichi Mochizuki has invested a very substantial time to answer e-questions and to aid dissemination of his results in various seminars and meetings.<ref>{{Citation|title=Seminars, meetings, lectures on IUT in Japan|url=https://www.maths.nottingham.ac.uk/personal/ibf/files/tab1.pdf}}</ref>
Two texts {{harvs|txt|last=Mochizuki|year=2013b}} and {{harvs|txt|last=Mochizuki|year=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,<ref>{{Citation|first=Shinichi|last=Mochizuki|year=2014|title=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 | url=http://www.kurims.kyoto-u.ac.jp/%7Emotizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf}}</ref><ref name="Mochizuki2016"/> two surveys by [[Ivan Fesenko]]<ref name="Fesenko2015"/><ref name="Fesenko2016"/> and two surveys by Yuichiro Hoshi.<ref name="Hoshi">{{cite web|url=http://www.kurims.kyoto-u.ac.jp/~motizuki/Questions%20and%20Comments%20on%20IUT.pdf|title=On questions and comments concerning Inter-universal Teichmüller Theory}}</ref> Shinichi Mochizuki has invested a very substantial time to answer e-questions and to aid dissemination of his results in various seminars and meetings.<ref>{{Citation|title=Seminars, meetings, lectures on IUT in Japan|url=https://www.maths.nottingham.ac.uk/personal/ibf/files/tab1.pdf}}</ref>

Revision as of 15:17, 18 December 2016

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 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'."[6] Alternative names for the theory are arithmetic deformation theory[7] 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, for a given ring arithmetic deformations change multiplication and the task is to measure how much 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 complatible with certain group structures and absolute Galois groups as well as certain types of topological groups play fundamental role in IUT. Considerations of multiradiality, a generalization of functoriality, imply that certain 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 theory, over complex numbers, to IUT is Fedor Bogomolov's proof of the geometric Szpiro inequality[7][9].

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[9] Cite error: A <ref> tag is missing the closing </ref> (see the help page).[9] two surveys by Ivan Fesenko[7][8] and two surveys by Yuichiro Hoshi.[10] Shinichi Mochizuki has invested a very substantial time to answer e-questions and to aid dissemination of his results in various seminars and meetings.[11]

National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015.[12] 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.[13] Its report[14] 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.[15][16] Among its files there is a document[10] 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. ^ a b c d 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. ^ a b c Fesenko, Ivan (2016), Fukugen, Inference: International Review of Science, 2016
  9. ^ a b c d e Mochizuki, Shinichi (2016), The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory (PDF) Cite error: The named reference "Mochizuki2016" was defined multiple times with different content (see the help page).
  10. ^ a b "On questions and comments concerning Inter-universal Teichmüller Theory" (PDF).
  11. ^ Seminars, meetings, lectures on IUT in Japan (PDF)
  12. ^ Future and past workshops on IUT theory of Shinichi Mochizuki
  13. ^ https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html
  14. ^ "Report on the Oxford workshop on the IUT theory of Shinichi Mochizuki, by Ivan Fesenko".
  15. ^ Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016)
  16. ^ Brief report on Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016)
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