ON THE VERIFICATION OF INTER-UNIVERSALTEICHMÜLLER THEORY: A PROGRESSREPORT (AS OF DECEMBER 2013)

Shinichi Mochizuki (RIMS, Kyoto University, Professor)
ɹ Nearly a year and four months have elapsed since I released my series of papers
on inter-universal Teichmüller theory (IUTeich) in August 2012. In the following,
I would like to report on various developments that have come to pass during thistime with regard to the veriﬁcation of IUTeich.

(1) At the end of August 2012, I released my series of four papers concerning
IUTeich and submitted them to a mathematical journal. I have no intention of
disclosing the name of the journal or other information related to this journal
submission. The papers were released in order to make it possible for specialists
to study and assess the validity of IUTeich; they were never intended as an announcement to the general public. Moreover, I never envisaged that these papers
might elicit reactions consisting of non-mathematical content from non-specialists.
I have no intention of responding to such non-mathematical reactions.
(2) As of the time of writing, I have given two talks on IUTeich at Kyoto University— in
October 2010 (i.e., a one-hour “preview” talk) and
December 2012 (one hour)— and one talk on IUTeich at the University of Tokyo — inJune 2013 (one and a half hours).
The slides for these talks are available on my web site (on the page entitled “Traveland Lectures”). It is likely that I will again give at least one talk at some academic
institution in Japan in 2014. The various talks that I have given so far coveressentially the same content; this content is exposed in more detail in the survey[Pano]. This survey is available on the page of my web site entitled “Papers” andis scheduled to appear in the proceedings for the workshop at which my December
2012 talk was delivered.

2

ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY(3) I have been conducting a seminar devoted to the detailed examination of IUTeich with Go Yamashita (Research Fellow at Toyota Central Research and Development Laboratories; Project Lecturer at the Center for Research Interaction in
Mathematical Sciences of RIMS, Kyoto University)
once a month (for 2 days ≈ 12 hours), since October 2012.
More precisely, during the period
October ∼ December of 2012,
Yamashita studied the “preparatory papers” (i.e., [HASurI], [HASurII], [SemiAnbd], [FrdI], [FrdII], [EtTh], [AbsTopIII], [GenEll], etc.) and discussed theircontents with me in our seminar. During the periods
January ∼ March 2013 andApril ∼ September 2013,
he then proceeded to read through the entire “theory proper” constituted by thefour papers [IUTchI], [IUTchII], [IUTchIII], [IUTchIV] a total of two times. During
this period of roughly a year, I received from Yamashita, in writing (by e-mail),numerous detailed technical comments (roughly several hundred during this periodof a little over a year!) that exceed, by a substantial margin, the scope of a typicalreferee’s report for a mathematical journal. After discussing these comments in our
seminar, I proceeded to correct the relevant papers and post the corrected versions
on the page of my web site entitled “Papers”. Also, around April 2013, Yamashita
prepared an “FAQ”, which I posted on my web site, to answer questions posedby other researchers with whom he had occasion to interact at workshops and soon. As a result of these two readings of the four papers, Yamashita acquired athorough understanding of the theory and, in particular, veriﬁed the correctnessof the theory, although, in his words, this veriﬁcation constituted, for him, only a“single check” of the theory.
(4) Yamashita then proceeded to carry out, in his words, a “double check” of
IUTeich. This “double check” consisted of a seminar that was conducted
once a month (for 2 ∼ 3 days ≈ 16 ∼ 24 hours), May ∼ November 2013,
for the following three participants:
Akio Tamagawa (RIMS, Kyoto University, Professor)
Yuichiro Hoshi (RIMS, Kyoto University, Lecturer)
Makoto Matsumoto (Hiroshima University, Professor).
The seminar was conducted without my participation, in order to provide Yama-

ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY

3shita with an opportunity to explain the theory in his own words and thereby
verify his understanding of the theory. Yamashita’s presentation of the theory inthis seminar consisted of literally wading through — from the most elementary
portion of the “preparatory papers” to the ﬁnal portion of the “theory proper” —each Deﬁnition, Proposition, Theorem, etc. in the order written in these papers— an approach that required quite a substantial amount of eﬀort on the part ofboth the lecturer and the participants. When, either during the seminar or in thecourse of preparing for it, technical comments or questions arose (roughly 10 ∼ 30
a month, on average), Yamashita and I discussed and resolved them the following
month in our monthly seminar. Toward the end of the (“double check”) seminar,
one participant made a remark to the eﬀect that, although it seemed likely that
it would take some time for him to digest the theory, he felt that he had moreor less reached an understanding of the general outline of the theory. Moreover,the participants agreed that, from their present point of view, i.e., after having
seen Yamashita’s presentation of the theory, the content of the lectures and survey
discussed in (2) now seemed to be quite “appropriate”.
(5) Subsequent to the successful conclusion of the “double check” of (4), Yamashita
has started writing a detailed survey of IUTeich, i.e., which, in his words, will
constitute a “triple check” of the theory. He expects the survey to be about
200 ∼ 300 pages in length. Moreover, although his plans are not yet deﬁnite, heis considering, at the request of Associate Professor Yuichiro Taguchi of Kyushu
University, giving a (several week long) intensive course on IUTeich at Kyushu
University, at some date subsequent to April 2014.
(6) Mohamed Saı̈di (University of Exeter (UK), Associate Professor) visited RIMS,
Kyoto University, as a Visiting Professor, forthe three month period of July ∼ September 2013
during which I conducted a seminar with him devoted to the detailed examinationof IUTeich from various points of view. This seminar consisted ofroughly 10 meetings (≈ 24 hours in all).
He also met with Yamashita a few times to discuss IUTeich. During a periodof a little over half a year prior to his visit, Saı̈di apparently ﬁnished reading
the “preparatory papers”, as well as roughly half of the “theory proper”. Upon
arriving in Japan, he then ﬁnished reading the “theory proper”. During his visit, inorder to verify his understanding of the theory, he then proceeded to read throughthe entire four papers of the “theory proper” once again. These two readings of
the “theory proper” were supplemented by the seminars that I conducted with

4

ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY

him, roughly once a week during his visit, in which we discussed the content of
the papers in quite substantial detail. During these seminars, he provided me withnumerous detailed technical comments (roughly a hundred during the three monthsof his visit!) that exceed, by a substantial margin, the scope of a typical referee’s
report for a mathematical journal. Just as in the case of Yamashita, after discussing these comments in our seminar, I proceeded to correct the relevant papers
and post the corrected versions on the page of my web site entitled “Papers”. In
conclusion, Saı̈di stated to me, as well as, apparently, to at least one “third-party”
researcher outside Japan, that he believed the theory to be correct.
(7) Although both Yamashita and Saı̈di are researchers in the ﬁeld of arithmetic
geometry, they nonetheless come from substantially diﬀerent mathematical backgrounds, as may be veriﬁed by a brief look at their previous papers and otherwork. One important attribute that they share, however, is their proven trackrecord as referees, i.e., their wealth of experience in refereeing for mathematicaljournals and, in particular, being in a position to judge the suitability of submitted
manuscripts for publication in such journals. On the other hand, in the case of
IUTeich, in light of both the importance of the theory and the novelty of the ideasthat constitute the theory, it is desirable that the veriﬁcation process be conductedwith the utmost caution. In particular, the issue of whether or not one should regard the veriﬁcation of IUTeich as being, for all practical purposes, complete, i.e.,
as a result of the activities of Yamashita and Saı̈di, is by no means clear, and any
sort of “ﬁnal conclusion” on this topic must be regarded as a matter that lies beyond the scope of the present report. Nevertheless, such caveats notwithstanding,it seems to me that the degree of meticulousness and attention to detail exhibitedin the veriﬁcation activities summarized above in (3), (4), (6) — which, as noted
above, exceed, by a substantial margin, the scope of a typical referee’s report fora mathematical journal — together with the wealth of refereeing experience of
Yamashita and Saı̈di (i.e., as discussed above) should be regarded as lending quite
substantial weight to the extremely positive evaluation that I received from bothof them in the course of these activities.
(8) One further important “by-product” of the veriﬁcation activities of Yamashita
and Saı̈di concerning IUTeich is the fact that their experiences demonstrated, ineﬀect, in a very explicit fashion, that, despite the quite substantial diﬀerences intheir mathematical backgrounds,
it is quite possible to achieve a reasonably rigorous understanding of thetheory within a period of a little less than half a year.
Since there apparently exist researchers whose attempts to study IUTeich have notON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY

5

proceeded so smoothly, I asked Yamashita and Saı̈di on numerous occasions if they
had any “advice” for such researchers. Despite numerous discussions concerning
this topic, the only “advice” that I was able to elicit from them consisted of remarks
to the eﬀect that
if one proceeds to study carefully step by step, starting from the “preparatory papers”, there is no reason that one should encounter any insurmountable diﬃculties.
On the other hand, the potential reader should keep the following point in mind:
ɾAlthough various limited portions of IUTeich exhibit similarities to existing mathematical theories such as complex or p-adic Teichmüller theory
and the functional equation of the classical theta function, any attempt tostudy IUTeich under the expectation that the essential thrust of IUTeich
will proceed via a similar pattern of argument to existing mathematicaltheories is likely to end in failure.

Both Yamashita and Saı̈di voiced similar opinions concerning this point. Moreover,
over the past few years, I have frequently emphasized the following two points:
ɾ Although many individuals appear to be interested mainly in the numerical aspects of the inequality of the ABC Conjecture, in fact it ismore natural to regard the theory that constitutes IUTeich as the main
object of interest. That is to say, the inequality of the ABC Conjecture
is best understood as simply being one barometer of the depth of themathematical content of the theory.
ɾTypically, a one-hour lecture — or, indeed, even a week-long expositionof the theory — will be far from suﬃcient for most researchers to arrive
at a suﬃciently motivating appreciation of the essential framework that
underlies the theory. In particular, based on my past experiences, I
harbor substantial doubts that such short-term promotion activities canlead to meaningful results. On the other hand, although the precise
amount of time necessary to arrive at a reasonable understanding of thetheory will no doubt vary from individual to individual, at least in thecase of researchers familiar with conventional arithmetic geometry, theamount of time necessary should, as remarked above, be a matter ofmonths — i.e., roughly half a year or so — not a matter of years!

One aspect of my interaction with Saı̈di that left a particularly strong impression on me was the fact that he voiced opinions similar in content to these lasttwo points as “independent observations”, i.e., without even the slightest bit ofprompting on my part!

6

ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY(9) Finally, I would like to take this opportunity to express my deep gratitude to
Yamashita and Saı̈di, as well as the participants of (4), for their exceptional andzealous contributions to the goal of verifying IUTeich.

ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY

7

Bibliography[HASurI] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves I,
Arithmetic Fundamental Groups and Noncommutative Algebra, Proceedingsof Symposia in Pure Mathematics 70, American Mathematical Society (2002),
pp. 533-569.
[HASurII] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves II,
Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math. 36, Math. Soc.
Japan (2002), pp. 81-114.
[SemiAnbd] S. Mochizuki, Semi-graphs of Anabelioids, Publ. Res. Inst. Math. Sci. 42(2006), pp. 221-322.
[FrdI] S. Mochizuki, The Geometry of Frobenioids I: The General Theory, Kyushu
J. Math. 62 (2008), pp. 293-400.
[FrdII] S. Mochizuki, The Geometry of Frobenioids II: Poly-Frobenioids, Kyushu J.
Math. 62 (2008), pp. 401-460.
[EtTh] S. Mochizuki, The Étale Theta Function and its Frobenioid-theoretic Manifestations, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349.
[AbsTopIII] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms, RIMS Preprint 1626 (March 2008).
[GenEll] S. Mochizuki, Arithmetic Elliptic Curves in General Position, Math. J. Okayama
Univ. 52 (2010), pp. 1-28.
[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of Hodge
Theaters, RIMS Preprint 1756 (August 2012).
[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic
Evaluation, RIMS Preprint 1757 (August 2012).
[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of
the Log-theta-lattice, RIMS Preprint 1758 (August 2012).
[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume Computations and Set-theoretic Foundations, RIMS Preprint 1759 (August 2012).
[Pano] S. Mochizuki, A Panoramic Overview of Inter-universal Teichmüller Theory, RIMS Preprint 1774 (February 2013), to appear in RIMS Kōkyūroku
Bessatsu.