SUMMARY OF LETTERS TO PARTICIPANTS OF THE WORKSHOP
ON THE THEORY OF SHINICHI MOCHIZUKI,
DECEMBER 7–11 2015, OXFORD
IVAN FESENKO
The webpage of the workshop on the inter-universal Teichmüller (IUT) theory, a.k.a. arithmetic
deformation theory, of Shinichi Mochizuki, Oxford, December 7–11 2015 is
https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html
1. There will be 24–26 hours of lectures, for the list of speakers at the workshop see the workshop
page. We will also arrange skype sessions of answers to them by Shinichi Mochizuki.
2. Questions from the participants of the workshop in relation to the theory: around 50 questions have
been received and answered prior to the workshop. Answers to questions on frobenioids are available
from http://www.kurims.kyoto-u.ac.jp/~motizuki/Responses%20to%20questions%20on%
20Frobenioids.pdf
3. Prerequisites for the workshop. They are mentioned or reviewed in sect. 1 of [17]: class field
theory, especially local class field theory; elliptic curves over finite, local and global fields, some
basic things about the set of Diophantine geometry conjectures the theory of Mochizuki deals with,
basics of (classical) anabelian geometry. For the latter Ch.4 of [21] is a very easy to read source. A
very short paper [1] elegantly proves a local version which involves higher ramification filtration, the
result plays a role in IUT theory.
4. Introductory texts, surveys, slides of talks. The participants of the workshop are recommended to
study the following materials, as well as the introductions of the main papers listed in section 7.
4i. Introductory texts by the author [13], [11], [12]. The first of these is the most recent text about
certain analogies with Bogomolov’s proof of the geometric Szpiro inequality.
4ii. Notes [17] and a recent survey [19] provide an external perspective.
4iii. These slides of a talk by Yuichiro Hoshi at RIMS workshop on IUT, March 2015 [18] is a good
introduction into mono-anabelian aspects of IUT. Slides of 3 talks by Yuichiro Hoshi at RIMS workshop in December 2015 are available from the top of this page http://www.kurims.kyoto-u.ac.
jp/~yuichiro/talks_e.html. Slides and notes of talks at the December workshop are available
from https://www.maths.nottingham.ac.uk/personal/ibf/files/iut-sch1.html.
5. Three motivating theories which can be read prior to the study of IUT, depending on your background and taste:
Date: Version of October 18, 2016.
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SUMMARY OF LETTERS TO PARTICIPANTS OF THE OXFORD WORKSHOP ON IUT
◦ the Bogomolov proof of the geometric version of the Szpiro inequality (sect. 5.3 of [14]
and [22]), which involves geometric considerations that are substantially reminiscent of the
geometry that underlies the Hodge theatres of [10]-I, for these analogies see 1.3, 2.10 of [17]
and a very recent paper [13] by Shinichi Mochizuki. Citing Rk 2.3.4 of [10]-III, "Bogomolov’s
proof may be thought of as a sort of useful elementary guide, or blueprint [perhaps even a sort
of Rosetta stone!], for understanding substantial portions of the theory".
◦ the classical theory of the functional equation of the theta-function which was one important
motivation for the development of the theory of [10]-II-III, see also [16] and [20] for more on
the archimedean theta-function.
◦ the classical theory of moduli of ordinary elliptic curves in positive characteristic and the
related structure of the Hecke correspondence (i.e. Tp ) in positive characteristic, which is also
substantially reminiscent of the geometry that underlies the Hodge theatres of [10]-I.
6. Other analogies between IUT and other theories. In addition to class field theory, inverse Galois theory and anabelian geometry and the theories of the previous section, there are many other
analogies.
There are analogies between IUT and p-adic Teichmüller theory and some analogies with complex
Teichmüller theory which are well described in [11], [12] and in §I4 of [10]-I.
There are certain analogies between IUT and p-adic Hodge theory (which is applied in the proofs of
mono-anabelian geometry). For example, the local and global functoriality of absolute anabelian algorithms corresponds to some degree to compatible local isomorphisms between Galois cohomology
modules in p-adic Hodge theory, see e.g. Fig.4.2 of [12].
Hodge–Arakelov theory [2] is not formally used in [10], but some of its ideas and expectations
motivate key concepts and objects of [10]; for more on this see [11], [12]. IUT can be viewed as
a mathematical justification and background for the realisation of a key idea from [2] concerning a
possible approach to establishing the hyperbolic Vojta conjecture.
A number of aspects of the theory of [15] can be viewed as abelian ancestors of certain aspects of
IUT, see Rk 2.3.3 of [10]-IV.
See the second part of sect. "Analogies and relations between IUT and other theories" of [17] for
analogies with higher adelic geometry and analysis.
See also Rk. 3.3.2 of [10]-IV.
7. Main texts. The 4 main papers are
[10]-I-IV.
To read them, the following papers are also needed, to varying degree:
[9]-III, [7], [8], [3], [4], [6].
Concerning the last papers on frobenioids, there are certain expectations that most of their material
could be omitted in future developments of the proper IUT theory. The theory of anabelioids is needed
to understand the proof of Th. B in [10]-I.
SUMMARY OF LETTERS TO PARTICIPANTS OF THE OXFORD WORKSHOP ON IUT
3
8. Main theories and concepts. Below is the list of main theories and concepts. They are of different
level of importance.
∗ mono-anabelian geometry, mono-anabelian reconstruction, [9], one mostly needs [9]-III; see
also sect. 2.4 of [17]
∗ frobenioids [6], e.g. [6]-I, §6; tempered frobenioids are discussed in §3–§5 of [7]
∗ nonarchimedean theta-functions and related line bundles on tempered coverings, [7] plus
comments to this paper on SM’s webpage, a review in §1 of [10]-II and a recently added Rk
2.3.4 of [10]-III; see also sect. 2.5 of [17]
∗ coric functions for and multiradical containers to transport elements of number fields, Rk
3.1.7 and Ex. 5.1 of [10]-I and Rks 1.5.2, 2.3.3, 3.12.2, Fig. 3.6, of [10]-III
∗ étale-like object and Frobenius-like objects, [9]-III
∗ generalised Kummer theory and multi-radial Kummer detachment [7], [10]-I-III; see also
sect. 2.6 of [17]
∗ noncritical Belyi maps and their applications, [3], [8]
∗ Belyi cuspidalisation and elliptic cuspidalisation and its applications, [9]
∗ principle of Galois evaluation, [10]-II, applications of absolute anabelian geometry (in an
extended form) to reconstruct a ring structure, [9]-III and [7]
∗ concept of mono-analiticity and arithmetic holomorphy immune to the logarithm, [9]-III
∗ concept of a global multiplicative subspace, [10]-I
∗ rigidities (discrete rigidity, constant multiplie rigidity, cyclotomic rigidity) and indeterminacies, [7], [10]-II-III
∗ multiradiality and indeterminacies, [10]-II-III
∗ Θ±ell NF theatres, [10]-I-II-III
∗ theta-link and two types of symmetry, [10]-I-III; see also sect. 2.7 of [17]
∗ log-link, log-shell, log-Kummer correspondence and upper semi-compatibility, [9]-III, [10]III
∗ log-theta-lattice, [10]-III
The key main theorems are Cor. 3.12 of [10]-III, Th. 1.10 and Cor. 2.2 of [10]-IV.
9. Animations for IUT are available from https://www.maths.nottingham.ac.uk/personal/
ibf/files/iut.anima.html.
10. Other workshops and conferences on IUT
RIMS workshop on IUT, March 2015. The programme of this two week workshop included a 90
minutes talk by Yu. Hoshi and 67 1/2 hours of talks of G. Yamashita (thus, setting the world record).
The slides for Hoshi’s talk, photos of formulas on blackboards from Yamashita’s lectures and a group
photo are available from http://www.kurims.kyoto-u.ac.jp/~motizuki/research-english.
html.
During the first week the following topics were covered:
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SUMMARY OF LETTERS TO PARTICIPANTS OF THE OXFORD WORKSHOP ON IUT
◦ various basic anabelian properties of absolute Galois groups of local fields that follow from
local class field theory, such as cyclotomic rigidity (this constituted a substantial portion of
Hoshi’s talk at the March workshop, as well as of the first few days of Yamashita’s talks, §1
of [5] and [1]
◦ noncritical Belyi maps and their application in [8]
◦ Belyi cuspidalisation and its application to the theory of §1 of [9]-III
◦ basic properties of theta functions on tempered coverings as discussed in §1–§2 of [7]
◦ basic properties and examples of frobenioids, as in §6 of [6]-I
◦ tempered frobenioids, i.e., a summary of §3–§5 of [7].
During the second week Yamashita’s talks essentially consisted of going through [10]-I-IV step by
step, although with substantial simplifications; for instance, he only discussed the local theory in the
case of bad reduction primes.
Bejing workshop on IUT, July 2015. This workshop on IUT, organised by Ch. P. Mok (Morningside
Center of Math.) and F. Tan (Michigan State Univ.), was held in Bejing, Morningside Centre of
Mathematics, July 20-25 2015: http://wiutt.csp.escience.cn/dct/page/1 . Its programme is
available from its pages. Four speakers K. Chen (Univ. Science and Technology of China), Ch. P.
Mok, F. Tan, J. Tong (Univ. Bordeaux) delivered 24 hours of lectures. Several files related to the
material of its lectures is available from the page of the workshop.
RIMS conference "IUT Summit" organised by Sh. Mochizuki (RIMS), Yu. Taguchi (Kyushu Univ.)
and I. Fesenko will take place at RIMS, July 18-27 2016.
R EFERENCES
[1] S H . M OCHIZUKI , A version of the Grothendieck conjecture for p-adic local fields, The International Journal of Math. 8 (1997), 499–506, available with from http://www.kurims.kyoto-u.ac.jp/~motizuki/A%
20Version%20of%20the%20Grothendieck%20Conjecture%20for%20p-adic%20Local%20Fields.pdf
[2] S H . M OCHIZUKI , A survey of the Hodge-Arakelov theory of elliptic curves I, in Proc. of Symp. Pure Math. 70, AMS
(2002), 533–569; II, in Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math. 36, Math. Soc. Japan (2002), 81–
114, available with comment from http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[3] S H . M OCHIZUKI , Noncritical Belyi maps, Math. J. Okayama Univ. 46 (2004), 105–113, available with comments
from http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[4] S H . M OCHIZUKI , The geometry of anabelioids, Publ. Res. Inst. Math. Sci. 40 (2004), 819–881, available with
comments from http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[5] S H . M OCHIZUKI , The absolute anabelian geometry of hyperbolic curves, Galois Theory and Modular Forms,
Kluwer Academic Publishers (2004), 77–122, available with comments from http://www.kurims.kyoto-u.ac.
jp/~motizuki/papers-english.html
[6] S H . M OCHIZUKI , The geometry of frobenioids I: The General theory, Kyushu J. Math. 62 (2008), 293–400; II: PolyFrobenioids, Kyushu J. Math. 62 (2008), 401–460, available with comments from http://www.kurims.kyoto-u.
ac.jp/~motizuki/papers-english.html
[7] S H . M OCHIZUKI , The étale theta function and its frobenioid-theoretic manifestations, Publ. Res. Inst. Math.
Sci. 45 (2009), 227–349, available with comments from http://www.kurims.kyoto-u.ac.jp/~motizuki/
papers-english.html
SUMMARY OF LETTERS TO PARTICIPANTS OF THE OXFORD WORKSHOP ON IUT
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[8] S H . M OCHIZUKI , Arithmetic elliptic curves in general position, Math. J. Okayama Univ. 52 (2010),1–28, available
with comments from http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[9] S H . M OCHIZUKI , Topics in absolute anabelian geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19 (2012), 139–
242; II: Decomposition groups and endomorphisms, J. Math. Sci. Univ. Tokyo 20 (2013), 171–269; III: Global
reconstruction algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), 939–1156, available with comments from http:
//www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[10] S H . M OCHIZUKI , Inter-universal Teichmüller theory I: Constructions of Hodge theaters, preprint 2012–2015, II:
Hodge-Arakelov-theoretic evaluation, preprint 2012–2015; III: Canonical splittings of the log-theta-lattice, preprint
2012–2015; IV: Log-volume computations and set-theoretic foundations, preprint 2012–2015, available from http:
//www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html
[11] S H . M OCHIZUKI , Invitation to inter-universal Teichmüller theory (lecture note version), available
from
http://www.kurims.kyoto-u.ac.jp/~motizuki/Invitation%20to%20Inter-universal%
20Teichmuller%20Theory%20(Lecture%20Note%20Version).pdf
[12] S H . M OCHIZUKI , A panoramic overview of inter-universal Teichmüller theory, in Algebraic number theory
and related topics 2012, RIMS Kokyuroku Bessatsu B51, RIMS, Kyoto (2014), 301–345, available from
http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%
20Teichmuller%20Theory.pdf
[13] S H . M OCHIZUKI , Bogomolov’s proof of the geometric version of the Szpiro conjecture from the point of view
of inter-universal Teichmüller theory, available from http://www.kurims.kyoto-u.ac.jp/~motizuki/
Bogomolov%20from%20the%20Point%20of%20View%20of%20Inter-universal%20Teichmuller%
20Theory.pdf
[14] F. B OGOMOLOV, L. K ATZARKOV, T. PANTEV, Hyperelliptic Szpiro inequalities, J. Diff. Geometry 61(2002) 51–80,
available from http://arxiv.org/abs/math/0106212
[15] G. FALTINGS , Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Invent. Math. 73(1983), 349–366.
[16] J. FAY, Theta functions on Riemann surfaces, Lect. Notes Math., 352, Springer 1973.
[17] I. F ESENKO , Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions,
notes on the work of Shinichi Mochizuki, Europ. J. Math. (2015) 1: 405–440, available from https://www.maths.
nottingham.ac.uk/personal/ibf/notesoniut.pdf
[18] Y U . H OSHI , Mono-anabelian reconstruction of number fields, a talk at the RIMS workshop on IUT, March 2015,
available from http://www.kurims.kyoto-u.ac.jp/~yuichiro/talk20150309.pdf
[19] Y U . H OSHI , Introduction to inter-universal Teichmüller theory (in Japanese), a survey, November 2015, available
from the top of http://www.kurims.kyoto-u.ac.jp/~yuichiro/papers_e.html
[20] D. M UMFORD , Tata lectures on theta, Vols. I–II and III, Birkhäuser, 1983 and 1991.
[21] T. S ZAMUELY, Galois groups and fundamental groups, CUP 2009, corrections are available from http://www.
renyi.hu/~szamuely/erratafg.pdf
[22] S H . Z HANG , Geometry of algebraic points, In First Intern. Congr. Chinese Mathematicians (Beijing, 1998),
185–198, AMS/IP Stud.Adv.Math. 20 (2001), available from https://web.math.princeton.edu/~shouwu/
publications/iccm98.pdf