A NABELIAN A RITHMETIC G EOMETRY - A NEW G EOMETRY OF
F ORMS AND N UMBERS: Inter-universal Teichmüller theory or“beyond Grothendieck’s vision”
Benjamin Collas‡

This manuscript is a substantial revision of the paper accepted for publication in \Matematika i teoretiqeskie komp~ternye nauki" Tom 1, vypusk 1, 2023 – Mathematics and Theoretical Computer Science,
vol. 1, Iss. 1, 2023 – for researchers and students in sciences. This version replaces any previous versions; it
discusses no mathematical results obtained subsequent to October 2022.

This text presents an informal overview on how, in accordance with some deeply
rooted principles of the philosophy of Alexander Grothendieck concerning the practiceof mathematics, recent progress in anabelian arithmetic geometry has led to the interuniversal Teichmüller theory (IUT) of Mochizuki Shinichi. The new geometry of monoids
furnished by IUT may be understood as the result of a seminal encounter between
Grothendieck’s principle of resolving the tension between the discrete and continuous
realms, on the one hand, and 𝑝-adic Hodge theory and height theory, on the other. In
doing so, it opens a new research frontier that goes beyond the Grothendieck geometryof rings-schemes by providing a unifying framework for Diophantine and anabelianarithmetic geometry.
Keywords: Arithmetic homotopy and Galois theory, anabelian geometry, Diophantine geometry, Interuniversal Teichmüller geometry, Galois-Teichmüller theory, abc and Vojta Conjectures, Fermat’s Last Theorem.
MSC Classification (2020): 14F35, 11D75, 14-02 Primary; 14G32, 11G50, 14G40, 01A65 SecondaryC ONTENTS
A PANORAMA OF G ROTHENDIECK ’ S A NABELIAN P HILOSOPHY
Grothendieck’s Mathematics Philosophy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
“Esquisse” of a Fruitful International Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A “far from abelian” Galois Arithmetic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .
In Japan beyond the Grothendieckian Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T HE D ISCRETE - CONTINUOUS F RONTIER
2.1 - Number Theory, Diophantine Geometry & Arithmetization . . . . . . . .
2.1.1 - “abc” - The Shadow of a Network of Conjectures . . . . . . . . . . . . . . . . . . . . . .
2.1.2 - From Vojta to Mochizuki - Geometrization & Arithmetization. . . . . . . . . . . .

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2.2 - Anabelian Reconstructions - Schemes and Monoids . . . . . . . . . . . . . 9
2.2.1 - “Fukugen” - From Classical to Absolute Mono-anabelian . . . . . . . . . . . . . . . . 10
2.2.2 - “There exists a group-theoretic algorithm...” . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 - Mono-anabelian Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
I NTER - UNIVERSAL T EICHMÜLLER G EOMETRY
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3.1 - Categorification of a Diophantine Problem . . . . . . . . . . . . . . . . . . 14
3.1.1 - Hodge Theaters & Synchronization between Geometry and Arithmetic . . . . . 14
3.1.2 - Log-theta wandering - Algorithms to relate distinct Frobenius-like objects . . 16
3.1.3 - Universes, Species, and Logical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 - New Anabelian & Diophantine Frontiers... . . . . . . . . . . . . . . . . . . 18
3.2.1 - An Anabelian abc Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 - An Extended Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 - Anabelian progress in Grothendieck-Teichmüller theory . . . . . . . . . . . . . . . . . 20
R EFERENCES & N OTES‡

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan ● E-mail:bcollas@kurims.kyoto-u.ac.jp

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A PANORAMA OF GROTHENDIECK’S ANABELIAN PHILOSOPHY“[...] nothing seems to happen, and yet at the end a highly non-trivial theorem is there.”– P. Deligne on Grothendieck’s “Rising Sea” method, in [Del98].

A PANORAMA OF G ROTHENDIECK ’ S A NABELIAN P HILOSOPHY
At the most elementary level of human cognition, the essence of anabelian arithmetic geometry
concerns the articulation – and possible reconciliation – of two a priori complementary ways ofdeciphering the perceptible world, that is to say, the discrete and continuous realms. The discretenessof number theory – or its geometric variation, Diophantine geometry – imposes constraints on
the transcendental nature of the mind, while the continuous nature of geometry – or its discretevariation, arithmetic geometry – furnishes a comfortable receptacle for its various realizations.
§ Grothendieck’s Mathematics Philosophy. In response to the limitations of the human mind
to grasp the essence of the dichotomy between the discrete and the continuous, Grothendieck’s
approach is to propose a universal and structural vision that reconciles the two realms (the discreteand the continuous) – each as an avatar of a unique functorial and category-theoretic construction,
see Fig. 1 below and [R&S] more generally. This approach is both philosophical and practical:
languages are constructed that reveal pre-existing structures, which, in turn, stimulate the further
development of language1 . In Grothendieck’s twelve themes legacy, the unified treatment of Galois
symmetries of numbers and geometric forms – i.e., anabelian geometry and Galois-Teichmüller
theory – is presented as a “master theme” to the mathematical community, see Fig. 2 and ibid.
One can consider that the new geometry is, before anything else, a syn- The most profound (in my eyes)thesis between these two worlds, until then adjoining and closely interde- of these twelve themes [or “maspendent, but nevertheless separated: the “arithmetic” world, in which live ter themes” of my work], are thethe (so-called) “spaces” without principle of continuity, and the world of theme of motives, and the closely
the continuous magnitude, where live the “spaces” in the proper sense of related theme of anabelian algethe term, accessible to the means of the analyst [...]. In the new vision, braic geometry and the Galois[B]these two worlds, formerly separated, form only one [..., in the] vision of Teichmüller yoga.
an “arithmetic geometry” (as I propose to call this new geometry).[A]Fig. 2. A. Grothendieck in “RéFig. 1. A. Grothendieck in “Récoltes et semailles” (1986), § The new geometry
coltes et semailles” (1986), notes- or the marriage of number and grandeur [R&S].
22 & 23 [R&S].

The resolution of this discrete-continuous tension between number theory and geometry is what
constitutes the core of anabelian arithmetic geometry. This process not only involves multiple areasof mathematics – including class field theory, low-dimensional topology, topological group theory,complex algebraic geometry and analytic Teichmüller geometry, but also requires a dedicated effort
to acquire a specific way of mathematical thinking2 .
§ “Esquisse” of a Fruitful International Legacy. When Grothendieck retired from the mathematical community, he left his peers a stimulating research legacy that includes his anabelianvision: the structure is given, but the objects have yet to be defined. In his 1983 “Letter to Faltings”[Gro97], a first introduction to the anabelian yoga is sketched, that was later developed further as abroader “Galois-Teichmüller theory” in his 1.600 page private manuscript the “Long March through1

2

The philosophical aspects of Grothendieck’s practice of mathematics, i.e., more specifically, the creation of
unifying and revealing mathematical contexts are discussed in [R&S] § 2.8-“La vision - ou douze thèmes pour une
harmonie”. As discussed in a recent conference at Chapman University [Chap23], this approach could serve asa virtuous guide for the modern arithmetic geometer. Regarding Grothendieck’s approach, where “language isinvented and structures are discovered”, see Panza’s talk ibid.
This constant discrete-continuous tension – in addition to the use of a language-formalism-picture triangle and,following Grothendieck’s philosophy, the introduction of a rich and necessary terminology – is what makes
anabelian geometry of a total different nature – and thus difficult to grasp – even for specialists in closely related
fields, such as number theorists or non-anabelian algebraic geometers.

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Galois theory” [LMG]. This research finally culminated in his 1984 “Esquisse d’un programme”[Esq], that inspired the realization of an international research program3 (∼1990-2010).
What began as multiple isolated research directions in the international
community4 – for example, in France with the work of Lochak andSchneps on Grothendieck-Teichmüller theory, in Germany then in the
US with the work of Pop on anabelian birational geometry and the workof Fried on Regular Inverse Galois theory, and in Japan with the independent development of Ihara’s program – resulted in the 90’s in unified
international research efforts with multiple breakthroughs and long-term
collaborations. In Japan, the first generation – which centered aroundthe school of Ihara Yasutaka and Oda Takayuki – was followed by asecond generation – constituted by Matsumoto Makoto, Nakamura Hiroaki, Tamagawa Akio, and Mochizuki Shinichi. The school of Mochizuki
Shinichi, in turn, produced a third generation – namely, Hoshi Yuichiro– and, more recently, a fourth generation of junior researchers. All of Fig. 3. La “Longue Marche”
these developments in Japan centered around the “Research Institute à travers la théorie de Galois.
for Mathematical Sciences”, Kyoto University, Japan (RIMS), which is now the only international
mathematics institute that has preserved, vastly expanded and renewed this unique anabelian
culture5 .
§ A “far from abelian” Galois Arithmetic Geometry. The focus of anabelian arithmetic
geometry, which encompasses both anabelian and Galois-Teichmüller geometry, is to explore theabsolute Galois group Gal(Q̄⇑Q) of the rational number field – the seed of number theory that
controls all the Galois symmetries that arise from the rational numbers – in the framework ofGrothendieck’s theory of the étale fundamental group, the group 𝜋1et (𝑋) of paths or loops on a
space 𝑋, see Fig. 6. In this category-theoretic unifying context, the Galois group and the group ofpaths are two avatars of a unique construction that unifies the discreteness of the former and thecontinuity of the latter.
Fig. 4. Grothendieck-Teichmüller theory is agroup-theoretic, combinatorial approach to the absolute Galois group of the field of rational numbers,
which relies on an étale-analytic transport: on thesphere P1 ∖ {0, 1, ∞}, an analytic isomorphism transports the étale monodromy loop around 0 to the one
around 1 for an essential arithmetic invariant 𝑓𝜎 toappear.

Contrary to another theme of Grothendieck’s arithmetic geometry – linear Galois representation
theory, which was used by Grothendieck (1965), then Deligne (1974) to establish the Weil conjectures– and to the modularity program for elliptic curves – which leads, via Taniyama–Shimura–Weil and
Ribet’s Theorem, to Wiles’ proof of Fermat’s Last Theorem (1994) – anabelian arithmetic geometry
is far from being linear and abelian, i.e., based on a commutative fundamental group. The natural
domain of anabelian arithmetic geometry is that of hyperbolic curves of a given genus 𝑔 with 𝑚
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In the Russian mathematical community, where the “Esquisse” was distributed as an underground “Samizdat”, itshould be no surprise that George Shabat and Vladimir Voevodsky – first in terms of “Dessins d’enfants” [SV90],
then in terms of anabelian results [Voe91] – were the first to initiate a systemic study of Grothendieck’s legacy,see [Sha18]. It is worth remembering that the theory of “Dessins” indeed originates from a famous result of Belyı̆
– and a remark by Bogomolov – whose announcement made a strong impression on Grothendieck[C] . Anabeliangeometry over algebraic closed fields was later developed by Bogomolov and Tschinkel.
All these approaches share, in terms of their common Grothendieckian generating process and origins, somesimilarity with the anabelian geometry that is discussed in these notes.
It may be of interest to note that Japan was the first country in which “Récoltes et Semailles” was published, withthe authorization of Grothendieck, namely, by Gendai Sugakusha Ed., in 1989. It was translated into Japaneseby Tsuji Yuichi.

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marked points – see Fig. 6 – or their moduli spaces ℳ𝑔,(︀𝑚⌋︀ , which classify the families, deformations,and internal symmetries of such curves.
In terms of mathematical objects – following the realization of “Esquisse” given by Ihara, Lochak,Matsumoto, Nakamura, and Schneps, and, as well as the realization given by Drinfel’d via quantum
group theory – anabelian geometry involves Teichmüller spaces (i.e., the analytic deformationof complex structures), mapping class groups and braid groups (where braid crossings are noncommutatively composed), topological group theory, and, more recently, the theory of operads. It
indeed thrives on seminal encounters – e.g., with Deligne’s theory of weights, Thurston’s progressin Teichmüller theory – from which it borrows seminal insight and techniques.
§ In Japan beyond the Grothendieckian Vision. The Japanese school achieved decisive progressin the anabelian program for curves, namely, with the work of Nakamura – via Deligne’s theory ofweights – then with the work of Tamagawa – via class field theory and the Lefschetz trace formula to detect rational points on covers – which also includes an anabelian Néron–Ogg–Shafarevich–Serre–Tategood reduction criterion. The Japanese school went further to produce higher dimensional anabelian
results for configuration spaces. The zero dimensional case, that is to say, the reconstruction of anumber field from its absolute Galois group, follows from previous work by Neukirch-Uchida (∼1977).
The next breakthrough, which includes several substantial strengthenings of Grothendieck’s original
anabelian conjecture for hyperbolic curves over number fields, came from a decisive shift of perspective
by Mochizuki Shinichi (1995), as the result of an encounter with Faltings’ 𝑝-adic Hodge theory, fromworking with spaces over number fields 𝐾⇑Q to working with spaces over 𝑝-adic local fields 𝐾⇑Q𝑝 6 ,i.e., over formal neighborhoods around a given prime 𝑝. This shift from Grothendieck’s original
global vision to working over local fields – which gives rise to stronger results that typically include
the global number field case – has since proven to be the most natural anabelian setting and has
yielded a plethora of progress (see [Hos22]), which also stimulated further work on other variants ofanabelian geometry7 .
Mochizuki’s anabelian breakthrough has, with the support of the Japanese anabelian community,
since ramified into some additional innovative absolute, mono-, and combinatorial anabelian variants,
each elucidating further the essential nature of Gal(Q̄⇑Q). The final reconciliation of the continuousand discrete realms – i.e., anabelian arithmetic geometry and the Diophantine geometry of estimates –however required another decisive step8 . The realization of this missing link – which may be regarded
as the final chapter of a 20-year long personal journey and indeed harks back to the mathematicaland philosophical legacy of Grothendieck – was provided by Mochizuki’s inter-universal Teichmüller
theory in 2012 – for a survey of the theory in its author’s own words, we recommend [Alien] andthe more recent [EssLgc]; for the general scientist, see [Fes16]. The new geometry furnished by
IUT opens new horizons that lie beyond Grothendieck’s original algebraic geometry and the realmof schemes, rings, and fields – structures with two operations – by working in the more flexible
realm of multiplicative and additive monoids – structures with only one operation – and has alreadyyielded some decisive and central results in number theory (see § An Anabelian abc Inequality), as
well as important breakthroughs related to the original framework of Galois-Teichmüller theory.

※ Note to the reader: This text is intended for mathematically oriented scientists or curious mathematics students.
For their convenience, special care has been given to include a reasonable use of mathematical notations. Such notations
should be considered as anchors for the mind and as concrete bridges to the original texts and cited manuscripts.
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For further details, we refer to the always stimulating survey of Nakamura-Tamagawa-Mochizuki [NTM98].
One could cite the “close-to-abelian” program of Nakamura and Tamagawa for curves, of Tamagawa and Saïdi forfields, and of Pop and Topaz for function fields (indeed originally Bogomolov’s 1990 program).
At this stage, it is interesting to note that, originally and for a long time, the bridge between anabelian andDiophantine geometry was expected to result from an argument to the effect that the Grothendieck Section
Conjecture implies Mordell’s Conjecture; we refer to “Letter from Deligne to Thakur” in [Sti11] and Remark 9
ibid. for a counter-argument.

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Acknowledgements. This manuscript should be understood as the outcome of multiple collaborative efforts
within the arithmetic anabelian community. It is the initial study of IUT by Yamashita Go, MohammedSaïdi, and Hoshi Yuichiro, together with the work of the speakers at the 2015 Oxford and 2016 Kyoto
conferences, that have led to what is today’s general understanding of IUT. At a more personal level, theauthor would like to express his gratitude to the speakers and participants of the “Promenade in Interuniversal Teichmüller theory” 2020-2021 RIMS-Lille seminar and the “Expanding Horizons of Inter-universal
Teichmüller Theory” 2021 workshop series, and especially to Mochizuki Shinichi, Ivan Fesenko, Pierre Dèbes,Hoshi Yuichiro, Emmanuel Lepage, Tsujimura Shota, and Minamide Arata. Regular discussions with thesemathematicians ensured the solidity of this text. This work was supported by the International Center for
Next-Generation Geometry, a center affiliated with the Research Institute for Mathematical Sciences locatedin Kyoto University.

T HE D ISCRETE - CONTINUOUS F RONTIER
Diophantine and anabelian arithmetic geometry, while both relying Fig. 5. Thue-Siegel-Roth Theoon Grothendieck’s geometry of schemes, are of an essentially different rem on bad rational approximationof algebraic numbers.
nature. The motivating force behind Diophantine geometry consists,
Given 𝛼 ∈ Q̄ an algebraic number,
for the most part, of questions concerning properties of numbers then for any 𝜀 > 0, there exist only
within geometry such as the existence of rational points in varieties finitely many 𝑝⇑𝑞 ∈ Q such thatand questions of estimates, see Fig. 5, where “the geometry governs⋃︀𝛼 − 𝑝⇑𝑞⋃︀ < 1⇑𝑞 2+𝜀 .
the arithmetic”. Anabelian arithmetic geometry, on the other hand,reflects a more symbiotic relationship between the symmetry properties of numbers and spaces interms of Galois groups and étale fundamental groups – or groups of paths, as in Fig. 6.
While the anabelian realm is dominated by a unique reconstruction goal, the Diophantine realm isspeckled with a vast intricate network of inter-related conjectures. Among them:• Oesterlé-Masser’s abc conjecture (1985) – or in its original geometric version, the Szpiro
conjecture for elliptic curves (1981) – is a seminal conjecture at the center of this network;• Mordell’s conjecture (1922) asserts the finiteness of the set of rational points on an algebraic
curve of any genus – proven by Faltings (1983) – and is the first decisive application of algebraic
geometry to height theory;• Vojta’s conjecture (1987) provides an innovative and fruitful geometric insight into Diophantine
problems.
Fig. 6. An anabelian hyperbolic curve.
Anabelian geometry deals
with the reconstruction of acurve from its group of paths,
as depicted in this geometricpicture of a genus 3 𝑚-pointed
curve, where each loop corresponds to, in the arithmetic
case, a “rotational” cyclotome⧹︂Z(1).
Abstract group-theoretic
synchronizations between the loops corresponding to cyclotomes and the disentanglement of their arithmetic andgeometric properties constitute essential steps in many anabelian reconstruction algorithms.

Anabelian arithmetic geometry, on the other hand, is already based on an essential unification of
number theory and geometry, since Grothendieck’s arithmetic fundamental group 𝜋1et (𝑋) may be
identified either with ⧹︂
𝜋1top (𝑋) – a profinite version of the group of paths on the underlying topologicalspace of 𝑋 when 𝑋 is a complex algebro-geometric space – or with the absolute Galois group¯Gal(𝑘⇑𝑘)when 𝑋 = Spec 𝑘. The étale fundamental group of a stack or scheme 𝑋 over a number field9¯
𝑘 appears furthermore as an extension of ⧹︂
𝜋1top (𝑋) by Gal(𝑘⇑𝑘)– see § Anabelian Reconstructions Schemes and Monoids. The theory has evolved into different variants – relative/absolute, bi/mono,combinatorial – each of which sheds light on certain aspects of the number-geometry intertwining.
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Strictly speaking, embedded in the field of complex numbers.

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Number Theory, Diophantine Geometry & Arithmetization
N UMBER T HEORY, D IOPHANTINE G EOMETRY & A RITHMETIZATION . The roots of modern Diophantine geometry may be traced back to the purely number-theoretic work of Thue-Siegel (seeFig. 5), as well as to the function field-theoretic insight of Hilbert, Kronecker, then Mordell andSiegel. On the other hand, the geometrization of Diophantine problems received substantial impetus
from the efforts of Lang in the 1960’s, who not only coined the term “Diophantine geometry”, butalso gave structure to the field by proposing numerous – now famous – conjectures that embodied
insights from analytic and complex geometry and, ultimately, Grothendieck’s algebraic geometry10 .
From these efforts of Lang resulted two of the most central conjectures of number theory andDiophantine geometry: Masser-Oesterlé’s abc conjecture – whose numerous variants include many
conjectures of Diophantine geometry, and which we present in its non-geometric (and non-Szpiro)
form for the reader’s convenience – and Vojta’s conjecture on heights of points of varieties11 , which,in the case of the abc Conjecture, constitutes a sort of quintessential example of the geometrizationof Diophantine estimates.
The shift from algebraic geometry to Galois-theoretic arithmetic geometry has its origins in
Mochizuki’s [GenEll] – which, at the time (2009), went largely unnoticed – where a link is established between Diophantine conjectures, on the one hand, and the introduction of powerful
methods from Faltings’ proof of the Mordell Conjecture12 , on the other. This shift may be seen in
the various Galois-theoretic aspects of the “dilation” of elliptic curves, which ultimately developed
into one of the core ideas underlying IUT geometry.

※ For an accessible, detailed, and up-to-date presentation of Diophantine geometry and height estimate theory, we
refer the reader to Bombieri-Gubler’s book [BG06].

§ “abc” - The Shadow of a Network of Conjectures. The abc Conjecture, proposed byMasser and Oesterlé, asserts, in its essence, a deep and subtle relationship between the addition andmultiplication of integers:
The abc Conjecture (1985). For every 𝜀 > 0, there exists a constant 𝐾𝜀 suchthat any triple of integers (𝑎, 𝑏, 𝑐) such that 𝑐 = 𝑎 + 𝑏 satisfies𝑐 ⩽ 𝐾𝜀 ⋅ rad(𝑎𝑏𝑐)1+𝜀 ,where rad(𝑛) denotes the product of prime divisors of 𝑛.
– or in Waldschmidt’s words: two integers divisible by large powers ofsmall prime numbers have a sum divisible by small powers of large prime
numbers. The abc Conjecture can be seen as a bound on the difference
between the additive and multiplicative structures of the rational numbers,
see also § An Anabelian abc Inequality.
Despite its elementary formulation, the abc Conjecture seems at first Fig. 7. An (𝑎, 𝑏) descripglance beyond reach. Various computational attempts to identify prop- tion of large ∼ 1012 abcerties of abc triples, including in terms of elliptic curves, did not provide triples.
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On the attitude of Lang toward the incorporation of Grothendieck’s developing and at that time already colossal
Éléments de géométrie algébrique, we refer to [Lan95]. This text, whose polemical tone does not diminish its
contemporary pertinence, indeed presents two interesting examples of social obstacles that may hinder thecirculation of new mathematical ideas – “If Algebraic Geometry really consists of (at least) 13 Chapters, 2.000
pages, all of commutative algebra, then why not just give up? The answer is obvious.” (ibid.).
A height function measures the size of rational points on varieties, both locally around primes and globally on
global number fields. An Arakelov-theoretic variant provides an even more functorial approach in terms ofarithmetic line bundles as in the context of Faltings’ and Mochizuki’s approaches, see § Inter-universal Teichmüller
Geometry.
While Faltings’ proof of the Mordell Conjecture constitutes a groundbreaking encounter between number theory
and geometry, the arithmetic of elliptic curves is however not yet fully revealed in Faltings’ work. We prefer tostress its seminal role in Mochizuki’s anabelian geometry breakthrough, see Fig. 10 and the corresponding section.

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any new insight into the conjecture – see for example Fig. 7, where colors are used to indicate the(1 + 𝜀)-quality of an abc-triple.
The abc Conjecture is deeply related, via its various variants, to some of the most central conjecturesof analytic number theory, Diophantine geometry and arithmetic geometry. For example, abc
implies (* denotes equivalences):• the Mordell Conjecture (Elkies 1991);• the Thue-Siegel-Roth Theorem of Fig. 5(Bombieri 1994);• no
Siegel zeros
for Dirichlet 𝐿-functions (Granville-Stark 1999);• the (strong) Hall Conjecture*;• the (generalized) Szpiro Conjecture*;

where the Szpiro Conjecture, via consideration of Frey-Hellegouarch curves 𝑦 2 = (𝑥 − 𝑎)(𝑥 − 𝑐) for(𝑎, 𝑏, 𝑐) such that 𝑎𝑛 + 𝑏𝑛 = 𝑐𝑛 , implies Fermat’s Last Theorem above a certain degree13 .
It was the brilliant insight of Vojta to reformulate abc as a geometric Diophantine inequality interms of heights and ramification, by considering an (𝑎, 𝑏, 𝑐) triple as a point (︀𝑎 ∶ 𝑏 ∶ 𝑐⌋︀ ∈ P1 ∖ {0, 1, ∞}on a genus 0 curve, or, alternatively, the projective line, see Fig. 8. While this new geometric insightmade precise a shadow area that had yet to be explored, no conceptual progress was made for20 years, thus illustrating the limits of traditional arithmetic thinking and suggesting the need for a
profound change of prism in arithmetic geometry.
§ From Vojta to Mochizuki - Geometrization & Arithmetization. As expressed by Dèbes,“the Vojta conjecture appears as a universal geometric Diophantine Approximation statement –rational points are rare because algebraic numbers are badly approximated by rational numbers inthe sense of [the] Thue-Siegel-Roth Theorem”.
The Vojta Conjecture, which was inspired by Nevanlinna
Fig. 8. The Vojta Conjecture (1987). For 𝑋 atheory in complex analysis, is a geometrization of Thue’svariety, 𝐾 a number field, 𝑆 a finite set of primes
inequality: 𝐷 can be seen as a subvariety of 𝑋, the functionof 𝐾, 𝐷 < 𝑋 a divisor with normal crossings, 𝐻 an
𝑚𝑆,𝐷 evaluates the proximity of the point to 𝐷 locally at theample divisor, and 𝜀 > 0:primes in 𝑆, see also (LHS) of Fig. 5; each height function ℎ●
𝑚𝑆,𝐷 (𝑃 ) + ℎ𝐾𝑋 (𝑃 ) ⩽ 𝜀 ⋅ ℎ𝐻 (𝑃 ) + 𝑂(1)evaluates the size of 𝑃 in terms of a certain referential systemfor every point 𝑃 of 𝑋 outside of some (proper)of coordinates describing 𝑋 that arises from the divisor “●” –subvariety of 𝑋.
see (RHS) of Fig. 5 with 𝑥𝑃 = 𝑞 as the coordinate of a point
𝑃 on the line. We refer to [BG06] Conj. 14.3.2 for further details and ibid. Conj. 14.3.11 for the ramification version.

The Vojta Conjecture is an essential step in introducing geometric insight into the world of rational
numbers and estimates: the abc inequality now appears as a bound on the height of a point on a
curve by means of the ramification of the coordinates of the point14 ; in this form, the bound on
the height generalizes naturally to higher dimensions. In the case where 𝑋 = P𝑁 is 𝑁 -dimensionaland 𝐷 is a union of (linear) hyperplanes, the Vojta Conjecture translates into Schmidt’s Subspace
Theorem (1972), which for 𝑁 = 1 corresponds to the Thue-Siegel-Roth Theorem of Fig. 515 . In order
to illustrate the intricacy of this circle of ideas, it may be interesting to note that Vojta also provided
a new proof of the Mordell Conjecture – now simplified by Bombieri using the Thue-Siegel-RothTheorem and the theory of heights – that leads to further and rich generalizations. We refer to
Rémond’s survey [Rém03] for more details.
The decisive step that allows one to go beyond Vojta’s geometrization of the abc Conjecture – andwhich also paves the way for the introduction of anabelian geometry and the subsequent development
13

14

15

Interestingly, the polynomial analogue of abc – or Masson’s Theorem – is related to the additive group law of therational points of the circle and to Pythagorean triples. Masson’s Theorem can be established in an elementaryand beautiful way and also implies Fermat’s Last Theorem for function fields.
One passes from the abc inequality to the ramification version of the Vojta Conjecture by considering a suitablecovering of the projective line ramified at three points. This covering just happens to be the curve defined by the
Fermat equation.
It is indeed a rewarding and pleasant exercise to translate the special case of the Vojta Conjecture referred toabove into Schmidt’s Subspace Theorem.

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Number Theory, Diophantine Geometry & Arithmetizationof inter-universal Teichmüller theory – is given by the following seminal new insight due to Mochizuki,
see [GenEll] § 316 :under the assumption of a certain (in fact nonexistent) global multiplicative
subspace 𝜇ℓ < 𝐸, one may apply Faltings’ theory of isogeny invariance of heights
in the case of the ℓ-dilated quotient 𝐸⇑𝜇ℓ to obtain a certain bound on the height(roughly) of the form ht(𝐸) ⩽ log(ℓ)⇑(ℓ − 1) ⩽ 1.
We refer to [EssLgc] Example 3.2.1 for details. The simulation of this (in fact nonexistent) global
multiplicative subspace, which is implemented, in essence, by working with the collection of local
multiple subspaces (who do exist), is at the core of inter-universal Teichmüller and is what leads
to the categorification process (involving Frobenoids and Hodge theaters) in IUT geometry, see§ Categorification of a Diophantine Problem. Simulating a global multiplicative subspace requires
one, in effect, to “rearrange” the way in which the primes of a number field are distributed. Thisissue of “rearrangement of the distribution of primes” is precisely what led Mochizuki to introduce
mono-anabelian geometry 17 , see [EssLgc] Example 3.8.2 (iii) and (iv). Ultimately, mono-anabeliangeometry also plays an important role in IUT via its use in a certain transport process, see § Monoanabelian Transport, which allows one to derive the IUT version of the global height inequality (i.e.,ht(𝐸) ⩽ log(ℓ)⇑(ℓ − 1) ⩽ 1) discussed above, see § An Anabelian abc Inequality.
At a less essential level, the shift from the number field context of the Vojta Conjecture to the𝑝−adic local one of (absolute 𝑝-adic) mono-anabelian geometry, as in § “Fukugen” - From Classicalto Absolute Mono-anabelian, is prepared with a certain reformulation of the Vojta Conjecture. This
reformulation allows one to shift the focus of one’s attention – by working with certain local datasuch as a “compactly bounded set 𝒦”, which satisfies a certain Gal(Q̄𝑝 ⇑Q𝑝 )-invariance property,
for 𝑝 a prime number – from the classical notion of “size”, or “heights”, of numbers in the senseof the real or complex numbers (which plays a fundamental role in the traditional approach toDiophantine geometry) to the 𝑝-adic valuations of a number field. We refer to [GenEll] § 2 for moredetails. Moreover, this reformulation has the effect of recasting the Vojta Conjecture, which is botha centerpiece and a conceptual pinnacle in classical Diophantine geometry, as a seminal frontier ofdifferent nature.
It may be of interest to note at this point that the creative process of Mochizuki is quite similar
to Grothendieck’s principle of the “rising sea”, see below and also the quote of Deligne in the
introduction. Inter-universal Teichmüller geometry, the culmination of a 20-year journey initiatedby Mochizuki when he was a PhD student in Princeton with Faltings in 1991, crystallizes multiple
facets of his already innovative previous work in arithmetic geometry, see Fig. 11, and may be
regarded as the completion of a program that imparts new significance to the three main components– multiplicative subspaces, 𝑝-adicization of Vojta, and properties of general elliptic curves – ofMochizuki’s 2009 paper [GenEll].

“The unknown thing to be known appeared to me as some stretch of earth or hard marl,resisting penetration... the sea advances insensibly in silence, nothing seems to happen,
nothing moves, the water is so far off you hardly hear it... yet it finally surrounds theresistant substance.”– A. Grothendieck in [R&S] § 18.2.6.4 (translation by McLarty [McL07]).
16

17

Another component of [GenEll] involves establishing various arithmetic properties for “general” elliptic curves in
the moduli space ℳ1,1 of elliptic curves, some of which underlie the IUT notion of initial Θ-data. Such initialΘ-data must be explicitly constructed in order to apply the IUT algorithm of § Log-theta wandering - Algorithms
to relate distinct Frobenius-like objects.
We also refer to Fig. 10 for a more classical analogy between Diophantine and “anabelian” geometry.

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Anabelian Reconstructions - Schemes and Monoids
A NABELIAN R ECONSTRUCTIONS - S CHEMES AND M ONOIDS . Anabelian geometry deals with theissue of reconstructing – or “fukugen” in Japanese – a space 𝑋 from its group of étale paths, orthe étale fundamental group Π𝑋 of 𝑋, an object that functorially encodes certain symmetries
arising from (the étale coverings of) 𝑋. The theory of the étale fundamental group, which wasdeveloped at the beginning of Grothendieck’s “Séminaire de Géométrie Algébrique” ([SGA1], 1971),
naturally induces an intermingling between the discreteness of number theory (via the absolute¯
𝜋1top (𝑋) and the topological paths
Galois group Gal(𝑘⇑𝑘))and the continuity of geometry (via Δ𝑋 ≃ ⧹︂on 𝑋) in a sequence of fundamental groups, and indeed is (chronologically) the first main exampleof Grothendieck’s resolution of the tension between the discrete and the continuous realms:
𝑋 × Spec 𝑘¯𝑋Spec 𝑘functorially𝜋1 (−)?
∃
anabelian1¯
𝜋1et (𝑋 × 𝑘)

𝜋1et (𝑋)𝜋1et (𝑘)Δ𝑋Π𝑋¯
Gal(𝑘⇑𝑘)1.

Here 𝑋 × Spec 𝑘¯ can be be thought of as the space obtained by regarding the system of polynomial
equations with coefficients in 𝑘 that define the space 𝑋 as a system of polynomial equations withcoefficients in an algebraic closure 𝑘¯ of 𝑘. The goal of anabelian geometry is to obtain a reverse
process to this functorial construction. Figure 9 below describes the classical bi-anabelian formulationof this process and some of its variants.
Fig. 9. Anabelian Geometry reconstruction: Can
any morphism between the étale fundamental groups
Π𝑋 and Π𝑌 be reconstructed, essentially uniquely,
from a morphism between 𝑋 and 𝑌 ?
The mono-anabelian version involves a single object
𝑋, while the absolute version imposes the constraintof forgetting the relation between 𝑋 and 𝑘.

For 𝑋 and 𝑌 varieties over 𝑘, is the natural map
Isom𝑘 (𝑋, 𝑌 ) → IsomGal(𝑘⇑𝑘)(Π𝑋 , Π𝑌 )Δ𝑌¯
an isomorphism?

In the case where 𝑋 is of dimension 0, the anabelian reconstruction of 𝑋 corresponds to thereconstruction of a number field from its absolute Galois group and is known as the Neukirch–Uchida
Theorem (∼1977). In the case where 𝑋 is of dimension 1, that is to say, when 𝑋 is a curve,
Grothendieck conjectured that this anabelian property holds for a certain type of curves, namely,
the hyperbolic ones; this conjecture was solved in several steps by the Japanese school (Nakamura,
Tamagawa, Mochizuki, 1990-1995).
In Diophantine geometry and the abelian, i.e., commutative, realm, one cannot help but notice theanalogy between the above bi-anabelian formulation and Faltings’ Isogeny Theorem (i.e., the Tate
Conjecture) of Fig. 10 below.
For 𝐴 and 𝐵 abelian varieties over a number field
Fig. 10. Faltings’ Isogeny Theorem is a funda𝑘, the natural map
mental step in the proof of the Mordell Conjecture
and can also be understood as an “anabelian” result
for abelian varieties – where the fundamental groups
are replaced by their abelian counterparts, i.e., ℓ-adic
Tate modules 𝑇ℓ .

Hom𝑘 (𝐴, 𝐵) ⊗Z Zℓ → HomGal(𝑘⇑𝑘)(𝑇ℓ (𝐴), 𝑇ℓ (𝐵))¯is an isomorphism.

The étale fundamental group framework of [SGA1] is developed in the context of Grothendieck’s
arithmetic geometry of rings/schemes. Inter-universal Teichmüller theory, on the other hand, involves
an arithmetic geometry that goes beyond Grothendieck’s arithmetic geometry of ring and scheme
structures. In contrast to Grothendieck’s original anabelian context (as in [Gro97]), which is relativeand bi-anabelian, the IUT context relies on a general and thoroughly absolute approach, namely,
absolute mono-anabelian geometry 18 , where one is concerned with reconstruction techniques without
any reference to an alternative scheme for the sake of comparison19 or to a fixed base scheme.
18

19

Despite the many papers published in this field since its initiation in 2003, this aspect of anabelian geometry andits very distinctive flavor have been, over the last 20 years, largely ignored outside Japan.
One can also think in terms of the functor of points approach, which describes points of a space 𝑋 in terms ofarrows ● → 𝑋.

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Anabelian Reconstructions - Schemes and Monoids※ We refer the reader again to the general overview of anabelian principles in the classical [NTM98] and to the report[Hos22] on the latest advances of the field.

§ “Fukugen” - From Classical to Absolute Mono-anabelian. The approach of Mochizuki’s
early work in anabelian geometry, as in [pGC], builds on the pioneering group-theoretic insightand techniques of the earlier anabelian approaches of Nakamura and Tamagawa, but introduces
an important shift of perspective: exploiting Faltings’ 𝑝-adic Hodge theory leads to a decisive
breakthrough already in the classical non-mono-anabelian and non-absolute context. This approachprovides a canonical container P(𝐷𝑋 ) for the reconstruction of spaces, namely, the projective spaceassociated to the vector space 𝐷𝑋 of global differentials on the curve 𝑋, that allows one to construct
rational points à la Tamagawa, in this case as limits of 𝑝-adically convergent sequences of points
that appear in covers of the curve 𝑋, see [Fal98] for an overview.
This shift of perspective led, in effect, to a sort of regeometrization of anabelian geometry, froma discipline that relied primarily on classical arithmetic techniques over global number fields andfunction fields to a discipline that utilizes techniques of 𝑝-adic geometry such as line bundles andChern classes. This more geometric approach has since become the standard anabelian framework
and, moreover, may be regarded as a further step toward the sort of theory of heights in the style ofFaltings, which plays an important role in § Inter-universal Teichmüller Geometry.
Fig. 11. Anabelian Geometry – a Contemporary
Panorama. Inter-universal
Teichmüller geometry appears
as a bridge between Diophantine geometry and anabeliangeometry and as a new abutment of numerous tributaries
originating in diverse fields
of number theory and geometry; each ∗ indicates a major contribution by Mochizuki,
often based on 𝑝-adic Hodgetheoretic considerations.

Throughout anabelian geometry – i.e., including classical non-absolute, non-mono-anabelian versionsof anabelian geometry, as well as more recent absolute, mono-anabelian developments – a fundamentalrole is played by the decomposition and inertia subgroups of the fundamental group, which areoften denoted 𝐷𝑥 and 𝐼𝑥 < Π𝑋 , respectively. In Mochizuki’s absolute anabelian geometry, these
decomposition and inertia subgroups of the fundamental group are reconstructed by means of thetechniques of elliptic and Belyi cuspidalization, which may be thought of as “hidden symmetries”
that relate a curve to covers of the curve with “missing” points.
For a survey that includes the most recent (at the time of writing of the present manuscript) progressin anabelian geometry, we refer to [Hos22]. For higher-dimensional anabelian results, see § 5 ibid.
and also Note 21.
§ “There exists a group-theoretic algorithm...”. While absolute mono-anabelian geometry
concerns the reconstruction of a space X from a single object Π𝑋 , it requires one to restrict one’s
attention to procedures and properties that are common to all objects considered20 . Mono-anabeliangeometry involves statements of the form:“There exists a functorial group-theoretic algorithm to reconstruct 𝑋 from Π𝑋 ”, or “Π𝑋 ↝ 𝑋”.
20

Regarding mono-anabelian geometry, Tamagawa speaks of a theory that deals with “one for all, all for one”
properties and proposes the terminology of “omni” or “pan” anabelian geometry.

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In this terminology, one works in an “absolute”, as opposed to relative Π𝑋 = 𝜋1et (𝑋) → Gal(𝑘⇑𝑘)context, in the sense that (the ring/scheme structure of) 𝑋 is detached from (the ring/scheme
structure of) its base field 𝑘.
In order to convey a sense of the flavor of a mono-anabelian (and absolute) reconstruction procedure,
let us present, in the case of a hyperbolic curve 𝑋 over a 𝑝-adic local base field 𝑘, a rough overview
of the mono-anabelian reconstruction of the base field 𝑘 (via the function field 𝐾𝑋 ):(i) Inertia and decomposition groups. Belyi cuspidalization allows one to reconstruct, first, thedecomposition group 𝐷𝑥 < 𝜋1et (𝑈 ) associated to a (closed) point 𝑥 of 𝑋, then the inertia group
𝐼𝑥 = 𝐷𝑥 ∩ Δ𝑈 < 𝜋1et (𝑈 ) associated to 𝑥, where 𝑈 is the complement in 𝑋 ∖ {𝑥} of a finite setof closed points.
(ii) Synchronization of geometric cyclotomes. One reconstructs the canonical isomorphisms between
geometric cyclotomes 𝐼𝑥 ≃ 𝐼𝑦 for 𝑥, 𝑦 ∈ 𝑋 ∖ 𝑈 that globally synchronize the (local) loops aroundvarious closed points.
×(iii) The multiplicative groups 𝐾𝑋
and 𝑘 × . One reconstructs the image of the Kummer map
×1𝜅∶ Γ(𝑈, 𝒪𝑈 ) ↪ 𝐻 (Π𝑈 , 𝜇Z⧹︂ (Π𝑋 )), where 𝑈 is allowed to vary among the non-empty open
subschemes of 𝑋. This yields a reconstruction of the multiplicative monoids (𝑘 × , ⊠) and××(𝐾𝑋, ⊠), where 𝑘 × < 𝐾𝑋
. Here 𝐻 1 (Π𝑈 , 𝜇Z⧹︂ (Π𝑋 )) plays the role of a common “container”,
and 𝜇Z⧹︂ (Π𝑋 ) is a synchronized cyclotome.
(iv) The field 𝑘. One reconstructs the additive structure on the multiplicative monoid (𝑘 × , ⊠)×
by considering the various quotients of the multiplicative monoid (𝐾𝑋, ⊠) corresponding to×valuations of 𝐾𝑋 . This yields a reconstruction of the field (𝑘 = 𝑘 ∪ {0}, ⊠, ⊞).
We refer to [AbsTopIII] § 1 for more details. One notes, in particular, the 2-step reconstruction ofthe ⊠/⊞-monoid structures in a compatible way, so that the (⊠, ⊞)-field structure of the base field 𝑘of the curve 𝑋 is recovered. For a completely different situation in the zero-dimensional case, i.e., ofa field that is not regarded as the base field of a hyperbolic curve, see § Mono-anabelian Transport
below.
While most anabelian theorems for curves admit corresponding 𝑝-adic absolute anabelian versions(which are much stronger), it is important to note that such adaptations require a deep expertise inanabelian geometry21 .
Mochizuki’s Belyi cuspidalization techniques lead in particular to a new anabelian class of geometricobjects, namely, the curves of strictly Belyi type, see [Hos22] § 6. The algorithmic reformulations ofabsolute mono-anabelian geometry allow reconstructions that are independent of a given fixed ring
structure, a feature that is exploited throughout inter-universal geometry.
§ Mono-anabelian Transport. Despite being the most natural context for anabelian geometry,the zero-dimensional case of a 𝑝-adic local field can already be quite subtle, since the isomorphismclass of such a field 𝑘 is not necessarily determined by the isomorphism class of its absolute Galois¯
group Gal(𝑘⇑𝑘).
One fundamental point concerns the issue of giving a compatible reconstruction of the two underlying⊠- and ⊞-monoid structures of 𝑘. Indeed, while
there exists a functorial group-theoretic algorithm to reconstruct from the group 𝐺 =¯¯Gal(𝑘⇑𝑘)
the additive (𝑘¯+ (𝐺), ⊞) and the multiplicative (𝒪× (𝐺), ⊠) Gal(𝑘⇑𝑘)-monoids¯𝑘¯ and 𝒪× (𝐺) < 𝑘¯× the multiplicative– where 𝑘¯+ (𝐺) denotes the underlying additive monoid of the field 𝑘,¯
𝑘monoid of units of the ring of integers, both of which are reconstructed from 𝐺 – it is important
to note that these two monoids cannot be reconstructed compatibly in such a way as to yield a¯ A similar conclusion also holds for 𝑘.
reconstruction of the ring or field structure of 𝑘.
21

One example may be seen in higher-dimensional anabelian geometry, namely, in the case of results concerning
the existence of an anabelian open basis — originally conjectured by Grothendieck — which may be naturally
derived from techniques of the Japanese anabelian school via 𝑝-adic absolute mono-anabelian methods, see [Hos20]
Cor. 3.4 & Rem. 3.4.1; see [SS16] for a result that holds only in the relative situation over number fields.

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As a result, this zero-dimensional anabelian context goes beyond the classical ring-scheme and Galois
theories and leads to a certain geometry of monoids. In this geometry, the two ⊠ and ⊞-monoids
structures are reconciliated by allowing additional indeterminacies such as (Ind1) and (Ind2). Some¯
new types of étale-like objects, e.g., the Galois group Gal(𝑘⇑𝑘),and some Frobenius-like objects,the ⊠ and ⊞-monoids, naturally appear. A new kind of mono-anabelian transport process22 defines
isomorphism classes of these more rigid objects, see Fig. 12 for an example.
⧹︂

𝐻 1 (𝐺𝑘 , 𝜇Z𝑘¯ (𝐺𝑘 ))

∼∗

⧹︂
Z×

↻

Fig. 12. Mono-anabelian transport for mulZ
tiplicative monoids of sub-𝑝-adic fields be- 𝐻 1 (𝐺𝑘 , 𝜇⧹︂¯ (𝐺𝑘 ))𝑘tween Frobenius-like objects via isomorphisms between étale-like objects. The isomorphism obtained𝜅
between Frobenius-like objects is subject to an ∗ =×⧹︂ Aut(𝐺𝑘 )-indeterminacy (Ind1) as well as a Z
𝒪𝑘×
indeterminacy (Ind2). Notation 𝜅 denotes the Kummer morphism.

étale-like

𝜅−1

𝒪𝑘×

Frobenius-like

In inter-universal Teichmüller geometry, mono-anabelian transport is a seminal process that usesétale containers to share Frobenius-like data across a certain non-schematic Θ-link, see § Log-theta
wandering - Algorithms to relate distinct Frobenius-like objects. In this context, the (Ind1) and(Ind2) indeterminacies, together with an additional (Ind3) indeterminacy that is discussed in loc.
cit., can be regarded as the mild deformations of the field structure necessary in order to rendercompatible the (a priori incompatible) ⊠ and ⊞-monoid structures on opposite sides of the Θ-link.
This construction bears some resemblance to the analytic transport that appears in classical
Grothendieck-Teichmüller theory between two isomorphic but distinct étale neighborhoods in the
projective line, see Fig. 4. In this case, the analytic link requires no-indeterminacy but produces
new arithmetic data, see also § Anabelian progress in Grothendieck-Teichmüller theory for a reverse
application of anabelian geometry to Grothendieck-Teichmüller theory.
On the path to IUT geometry, this shift to a mono-absolute framework and indeterminacies is a key
milestone in establishing a flexible geometry of monoids that allows one to deform, then estimate thediscrepancy between the additive and multiplicative structures on opposite sides of the deformation(i.e., the Θ-link), which yields the inequality of the abc Conjecture.

※ As we discuss later, one essential aspect of IUT involves the 𝑝-adic geometry of elliptic curves 𝐸𝑣 over𝑝-adic local fields 𝐾𝑣 ⇑Q𝑝 and a certain 𝑝-adic theta function. The corresponding fundamental group is in thiscase given by André’s tempered fundamental group, and the anabelian reconstructions involve combinatorial
anabelian geometry, see also § Anabelian progress in Grothendieck-Teichmüller theory. These tempered
reconstructions are indeed compatible with the profinite ones described above – we refer to [Lep10] § 1 for an
introduction to Mochizuki’s functorial results on this topic.

22

This elementary example indeed already involves the full techniques of poly-isomorphisms and synchronization of
cyclotomes later used in the IUT § Log-theta wandering - Algorithms to relate distinct Frobenius-like objects.
We refer to the introductory [Hos21] that was presented at the conference “Fundamental groups in arithmetic
geometry” at the Institut Henri Poincaré in Paris in 2016 for further details and comments.

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While Grothendieck’s Éléments de géométrie algébrique rewrote the foundations of algebraic geometry
using general ring theory and formal thickenings (the deformation of scheme structures within
ring/scheme theory), Mochizuki’s inter-universal Teichmüller theory23 studies deformations24 of
the ring structure by “untangling” the two ⊠/⊞-dimensions of a ring, using the theory of monoids,
and then estimating the degree to which the ring structure may be reconstructed by “juggling” the⊠/⊞-structures. Mono-anabelian geometry plays a central role in making this “juggling”, which
underlies the reconstruction of the ring structure, possible. In accordance with Grothendieck’s
philosophy, as discussed in § Grothendieck’s Mathematics Philosophy, it is based on the discovery of
new category-theoretic structures and the creation of new language, in a way that pushes arithmetic
geometry beyond the algebraic geometry of rings and of Galois groups. By doing so, it finalizes theinclusion, initiated by Vojta, of Diophantine discreteness into the already existing discrete-continuous
realm of anabelian geometry.
Fig. 13. Inter-universal Teichmüller theory. A triangle between language, formalism and geometric pictures
to resolve the tension between continuity and discreteness. The following excerpts are from the original IUT papers[IUTchI-IV]:“one may think of the fullness condition of multiradiality as the condition that there exist a sort of paralleltransport isomorphism between twocollections of radial data [i.e., corresponding to two “fibers”] that lifts
a given isomorphism between collections of underlying coric data [i.e.,corresponding to a path between the
points over which the two fibers lie].”

Inter-universal Teichmüller theory is not an incremental enhancement of Grothendieckian geometry,but rather a fundamental conceptual advance based on new paradigms – see also Note 33 for the étale
fundamental group context discussed in § Anabelian Reconstructions - Schemes and Monoids and,even more generally, the discussion in § Universes, Species, and Logical Structure – that provides new
insight into (1) the interface between anabelian and Diophantine geometry, and (2) the relationshipbetween ring structures and underlying monoid structures in arithmetic geometry, with (3) new
techniques for defining a new research frontier, see § New Anabelian & Diophantine Frontiers...
These three aspects can be seen as part of a second regeometrization of arithmetic geometry – see§ Introduction of [MFO21] and the report [Moc23] – this time via anabelian geometry.
The category-theoretic approach of inter-universal Teichmüller geometry results in a rich andevocative language for guiding mathematical thinking25 : objects exists in étale-like and Frobeniuslike flavors – depending on whether one regards an object as anabelianly reconstructed, e.g., from
Galois groups, or, alternatively, as an object that is only defined relative to a particular ring or
monoid structure. Moreover, both étale-like and Frobenius-like objects come in two variants, namely,
holomorphic and mono-analytic, depending on whether they involve two or only one of the two
monoidal structures of a ring, see the discussion of [Alien] § 2.7 (vii) and § 3.6 (iv).
23

24

25

Mochizuki’s IUT was first officially presented to the international community during the 2010 “Developmentof Galois-Teichmüller Theory and Anabelian Geometry” conference [Moc10] – see also the 2004 “Arithmetic
Geometry” Tokyo international conference [Moc04] for a discussion of this work at a very preliminary stage.
The initial versions of the IUT manuscripts appeared in August 2012 on the official public RIMS preprints
server [RIMS1756, 1757, 1758, 1759] and were later updated on its author’s web page following comments of thecommunity and the referees (which resulted in 160 additional pages and 10 revisions of the submitted manuscript).
We refer to the discussion surrounding (RdVar) in [EssLgc] § 3.1 for the analogy between the classical deformationsof scheme theory and the deformations that appear in inter-universal Teichmüller theory.
Consider, for instance, the following typical IUT statement: “The (Ind2) indeterminacy arises from the passage
from mono-analytic Frobenius prime-strips to mono-analytic étale prime-stripes.” We refer to Marquis’ [Mar22]
on the cognitive value of a mathematical style (illustrated in loc. cit. in the case of Bourbaki’s structuralism).

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Categorification of a Diophantine Problem
C ATEGORIFICATION OF A D IOPHANTINE P ROBLEM . The natural geometric context of IUT is thatof elliptic curves that belong to certain subsets of (the set of rational points of) the associated moduli
space26 ℳ1,1 , i.e., over a global field 𝐾, where the goal, in order to establish a Vojta-like inequality,
is to construct a certain global multiplicative subspace 𝜇ℓ < 𝐸, see § From Vojta to Mochizuki Geometrization & Arithmetization.
To achieve this goal, one must overcome two issues, which in fact constitute essential guiding
principles in IUT theory:(i) While such a subspace exists locally – i.e., at certain places 𝑣, for elliptic curves 𝐸𝑣 ∈ ℳ1,1 (𝐾𝑣 )over local fields 𝐾𝑣 – such a subspace typically does not exist globally, i.e., for 𝐸 ∈ ℳ1,1 (𝐾).
(ii) The quotient morphism 𝐸 → 𝐸⇑𝜇ℓ – that gives rise to the height inequality, see ibid. –amounts, at the level of moduli spaces, to a sort of Frobenius morphism (−)ℓ that is not aring homomorphism – e.g., (𝑎 + 𝑏)ℓ ≠ 𝑎ℓ + 𝑏ℓ .
In order to overcome these obstructions, Mochizuki proposes (1) a categorification of the original
situation by constructing Hodge theaters27 ℋ𝒯 ● that simulate the existence of such a global
multiplicative subspace 𝜇ℓ < 𝐸 – see [EssLgc] Example 3.2.1 – and (2) an arithmetic geometry that
does not depend on ring structures.
Such a categorification of the arithmetic-geometric framework can be seen as a second exampleof Grothendieck’s principle of resolution of the tension between the discreteness of number theory
and the continuity of geometry, which this time, leads to the inclusion of the realm of Diophantine
estimates into the realm of anabelian geometry – for the first example, we refer to § Anabelian
Reconstructions - Schemes and Monoids.

※ Before going further, we recall that over a 𝑝-adic local field, an elliptic curve 𝐸𝑣 with bad reductionZ(1) may be identified with the Tate curve G𝑚 ⇑𝑞𝐸defined by its 𝑞𝐸𝑣 -parameter and (2) admits a canonical𝑣
function, called the Θ-function, that is determined by certain symmetry properties (studied by Mumford andTate). Indeed, the main results of Mochizuki’s [EtTh] establish some absolute mono-anabelian reconstruction
algorithms for this 𝑝-adic Θ context28 .

§ Hodge Theaters & Synchronization between Geometry and Arithmetic. Not only does
IUT fully exploit the category-theoretic setting of Grothendieck’s SGA1 theory of étale fundamentalgroups – at the interface of number and algebraic geometry theory in the most concrete way, seeFig. 14 and Note 30 – it also extends this geometry beyond Grothendieck’s framework of rings,
fields, and Galois groups by deforming the ⊠⇑⊞-monoid structures of a ring.
The category-theoretic simulation of a global multiplicative subspace 𝜇ℓ < 𝐸 is realized by means of
Θ±ell NF-Hodge theaters. These new category-theoretic objects appear as the result of an elegant
synchronization process involving:(i) the geometric symmetries of (a certain covering of) a 1-pointed genus 1 curve 𝑋 (i.e., a sortof hyperbolic version of the elliptic curve 𝐸), and(ii) the arithmetic symmetries of the number field 𝐾 which is the base field of 𝑋 – see Fig. 1429 .
26

27

28

29

The projective line minus three points P1 ∖ {0, 1, ∞}, which was mentioned earlier in § “abc” - The Shadow of a
Network of Conjectures, is a coarse version of this moduli space.
Formally, a Hodge theater is composed of Frobenioids – that category-theoretically encode the geometry ofline bundles – assembled into various kinds of prime-strips that gather local and local-to-global data related
to étale-like objects and Frobenius-like objects. We refer to Fig. I1.2 of Chap. I of [IUTchI-IV] for examples ofprime-strips.
... and a remark similar to the one of § From Vojta to Mochizuki - Geometrization & Arithmetization on therecasting aspect of Mochizuki’s work on Vojta in [GenEll] applies to the context-related significance of this work,i.e., it furnishes a new seminal perspective on classical mathematical objects.
The ⊠-symmetries result from the need to work at the level of (the places 𝑣𝑖 of) 𝐾, rather than the (perhaps
somewhat more natural) field of moduli 𝐹mod . The synchronization is realized by working with certain coverings
that support the ⊞-symmetries. We refer to [EssLgc] Example 3.8.2 for a detailed presentation.

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Categorification of a Diophantine Problem±ellSuch a Hodge theater ℋ𝒯 Θ NF is obtained by “gluing”30 together a ΘNF-part ℋ𝒯 ΘNF – NF for±ell“Number Field” – to a Θ±ell -part ℋ𝒯 Θ – where Θ refers to the Θ-function attached to the elliptic
curve under consideration. The realm of ℋ𝒯 ΘNF is Galois-theoretic/arithmetic and multiplicative,±ellwhile the realm of ℋ𝒯 Θ is geometric and additive, we refer to [Alien] § 3.3 (v) for further details
on these symmetries and the gluing construction.
In this category-theoretic context, the “ℓ-dilation” of the⊠-monoid structures31 , as in § From Vojta to Mochizuki- Geometrization & Arithmetization, is obtained via a Θlink that operates at the level of Frobenius-like objects,
see Fig. 12. This Θ-link is essentially defined, on the 𝑞𝐸 parameter of a Tate elliptic curve 𝐸, as the Frobenius𝑗2morphism 𝑞𝐸 ↦ 𝑞𝐸
for 𝑗 an integer – more precisely, a se-

Fig. 14. Local description of the two typesof symmetry that underlie the simulation of
the global multiplicative subspace.
⊞ − 𝑆𝑦𝑚.
𝜇1ℓ < 𝐸𝑣1...
𝜇𝑛ℓ < 𝐸𝑣𝑛
Δ𝑋

2

𝑗quence of such 𝑞𝐸 ↦ 𝑞𝐸
, where 𝑗 ∈ {1, . . . , (ℓ−1)⇑2}. Whilethe Θ-link is compatible with the ⊠-monoid structures anddetermines a gluing between two isomorphic copies of the
†

±ell

Θ ‡

±ell

𝐾𝑣1

...

𝐾𝑣𝑛
⊠ − 𝑆𝑦𝑚.

𝐹𝑚𝑜𝑑

Gal(𝐾⇑𝐹𝑚𝑜𝑑 )same Θ±ell NF-Hodge theater ℋ𝒯 Θ NF Ð
→ ℋ𝒯 Θ NF ,the Θ-link is not compatible with the ⊞-monoid structures. We thus denote the two distinct but
isomorphic ring-structures involved with distinct labels † (−) and ‡ (−) – or more generally with an(𝑛, 𝑚) indexing – so that one remembers that this framework is not compatible with a single fixed
field-ring structure.
In order to desynchronize-resynchronize the Θ-link deformation
Fig. 15. The log-link, the releof the multiplicative structure with respect to the (fixed) additive
vant ⊞/⊠-structures, and the log¯
structures of the rings on either side of the Θ-link, IUT relies
shell container ℐ(Π) ⊆ 𝑘(Π).
on the use of a certain sequence of log-links on either side oflog(𝒪†▷¯ , ⊠)(𝒪‡▷¯ , ⊞)the Θ-link. Each such log-link relates the ⊠-monoid structure𝑘𝑘▷in its source – denoted (𝒪● ; ⊠) – to the ⊞-monoid structure inℐ(Π)
its target – denoted (𝒪●▷ ; ⊞) – by embedding both in a common
Containerlog-shell container ℐ(−), whose introduction becomes unavoidable
as soon as one attempts to deal with the ⊠- and the ⊞-monoid structures together simultaneously, seeFig. 15. Here Π denotes a certain étale fundamental group that forms the input data in the absolutemono-anabelian reconstruction of the container objects in such a way that Π is detached from¯
the structure morphism Π → Gal(𝑘⇑𝑘)(which, a priori, depends on the ring structures involved),
so that the ring structure can be freely deformed during mono-anabelian reconstructions as in§ Mono-anabelian Transport.
Finally, we should emphasize that, in contrast to Grothendieck’s SGA1, which is essentially a theoryof the fundamental group attached to a single (geometric) base point, the anabelian reconstruction
algorithms of IUT geometry involve multiple base points 32 , as a result of the variation of the ring
structures involved. Indeed, this non-existence of a unique base point is a consequence of thefollowing three components of IUT: (a) the internal structure of Hodge theaters, which involves thesynchronizations of certain conjugates (see Note 30, above), (b) the Θ-link, which involves a dilationof the ring structure, and (c) the 𝑝-adic logarithm, which permutes, in a complicated fashion, the ⊠and ⊞-monoid structures, see ibid. Example 3.8.3 (i-d), (i-e), and (vi).
30

31

32

In particular, this construction makes possible the synchronization, for each valuation 𝑣, of the unavoidable
𝐺𝑣 -conjugate indeterminacies that originate from the functoriality of the SGA1 constructions, see [EssLgc]
Example 3.8.2 (iii) and 3.8.1.
This dilation of analytic structures can indeed be seen as the origin of the term “Teichmüller” in the terminology“Inter-universal Teichmüller Theory”.
In Grothendieck-Teichmüller theory already, see Fig. 4, the arithmetic importance of dealing with multiple basepoints is usually ignored by non-anabelian arithmetic geometers (and thus mistakenly described as “a gadget”
or “a trick”). In IUT theory, the use of multiple base points is an unavoidable requirement, as discussed in[EssLgc] Example 3.8.3.

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Categorification of a Diophantine Problem§ Log-theta wandering - Algorithms to relate distinct Frobenius-like objects. In thiscategory-theoretic setting, the above constructions33 are put together to form the log-theta lattice of
(𝑚,𝑛)

±ell

Hodge theatersℋ𝒯 Θ NF , for 𝑚, 𝑛 ∈ Z, see Fig. 16. The construction of structures, via a certain
multiradial algorithm, that are invariant with respect to (certain types of) highly non-commutative
wandering in the log-theta lattice forms the main theorem of IUT, see Fig. 17 and below. The
situation is qualitatively similar to the mono-anabelian transport situation of Fig. 12 and moregenerally of § Mono-anabelian Transport (in the case of wandering within a single column, we refer
to Fig. 15): Hodge theaters admit an étale-like variant, linked to the original Hodge theater via
Kummer morphisms, also up to certain indeterminacies.
Fig. 16. Log-theta latticeof Hodge Theaters.
In
the shadow of this lattice,anadditional
sub-layerofétale-like
containers“Kummer-connects” the Hodge
theaters in each verticalcolumn.
The multiradialalgorithm allows one tonavigate from this sub-layer
to a top-layer that consists of
the Frobenius-like componentsof the lattice.

The Main Theorem of inter-universal Teichmüller theory states the existence of a multiradialalgorithm at the level of Hodge theaters, see Fig. 17 and Fig. 18, as follows:†

Fig. 17.
The Main Theorem of interuniversal Teichmüller theory (multiradial algorithm). Given a certain arithmetic-geometric context for elliptic curves, there exists an algorithm Ξ forconstructing a common container for the Frobeniuslike data associated to distinct Hodge theaters, labeled†(−) and ‡ (−), that are related to one another by thenon-schematic Θ-link.

Θ-link

ℋ𝒯

†

ℋ𝒯 𝒟

ℋ𝒯

‡

(−)

Kummer†‡●Ξ

Frobenius-like(−)

Kummer‡ℋ𝒯 𝒟●étale-like34

The arithmetic-geometric context for elliptic curves mentioned above is given by fixing the so-called
initial Θ-data; the common container corresponds to the étale-like data mentioned above; thealgorithm relies on the absolute mono-anabelian reconstructions discussed in § “There exists agroup-theoretic algorithm...”. We refer to [Alien] § 3.2 for additional examples of multiradiality.
The algorithm of Fig. 17 is defined up to certain indeterminacies (Ind1), (Ind2), and (Ind3) thatoriginate as follows: (Ind1) comes from comparing mono-analytic étale-like objects by monoanabelian transport; (Ind2) comes from a mono-analytic étale-like vs. Frobenius-like comparison;
finally, (Ind3) originates from the non-commutativity of the log-Kummer correspondence. Of these,(Ind 3) is the most essential indeterminacy of IUT and plays the most important role when thealgorithm of the Main Theorem of IUT is applied to the log-theta lattice of Hodge theaters forestablishing the height inequality, or Vojta-like inequalities, of the abc Conjecture – see the discussionof Θ-pilot/𝑞-pilot objects in § An Anabelian abc Inequality.
33

34

In this monoid-theoretic setting, one could ask what has become of the SGA1 “automorphism group of a fiber
functor over a geometric base point” ring-theoretic setting. We refer to the discussion on “a single unified basepoint”
and “single unified set-theoretic basepoint” of [EssLgc] Examples 3.8.3 and 3.8.4 – see also Example 3.8.1 ibid.
This figure is adapted from Minamide’s talk “Log-Theta Lattice: Symmetries and Indeterminacies” in [ExpHoriz1].

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Fig. 18. A Multiradial Algorithm allows one torecover objects Π ↠ 𝐺 that appear in the top row fromobjects constructed from the bottom row up to certain
indeterminacies; † (−) and ‡ (−) denote isomorphic but
distinct copies of isomorphic groups.

†Π ↠ †𝐺

full poly

≃

𝐺𝑘

‡

Π ↠ ‡𝐺

full poly

≃

𝐺𝑘

ℛ
Φ

†

‡

𝐺 ≃ 𝐺𝑘 ≃ 𝐺𝒞In Mochizuki’s own words, an essential inspiration for the notion of a multiradial algorithm isGrothendieck’s definition of the notion of a connection in the theory of crystals – see [Alien] § 3.1 (iv)(v) for details. Both notions may be regarded as a sort of a descent property – see [EssLgc] § 3.9.
§ Universes, Species, and Logical Structure. In the case of arithmetic geometry breakthroughs– such as, in the abelian cohomological context, the proof of Fermat’s Last Theorem by Wiles, andindeed already the proof of the Weil Conjectures by Deligne, see Mclarty’s discussion [McL10] andespecially § 7 ibid. – the issue of identifying their underlying set-theoretic foundational framework –e.g., Peano, Zermelo-Fraenkel, or ZFC + the existence of Grothendieck universes – is of specificimportance with respect to Grothendieck’s philosophy of mathematics since it is categories, notobjects, that provide a seminal context for the virtuous practice of mathematics – see also Note 36.
The foundational basis of IUT geometry can be approached at two distinct levels: an “external”
one that deals with the proper and logical articulation of sequences of statements, and an “internal”
one that ensures the correct interaction of objects, morphisms, categories, and functors that appear
in the theory. As presented in detail and in multiple contexts in [EssLgc], the former “externalone” boils down to a sequence of logical “OR” and “AND” relations35 – we refer to § 3. The logicalstructure of inter-universal Teichmüller theory ibid.
The “internal” level, which may also be described as category-theoretic, is related to Grothendieck’s
notion of “universe”, which provides a foundational framework in set theory and amounts to fixing
a ZFC model36 . In the Main Theorem of IUT of Fig. 17, it must be noted that each application of
the mono-anabelian reconstruction algorithms that appear involves a potential change of universeby successive enlargement (and thus a priori incompatible height or degree comparison, see thediscussion below concerning species and mutations, as well as § An Anabelian abc Inequality). As
examined in Chap. IV “Log-volume Computations and Set-theoretic Foundations” of [IUTchI-IV],“albeit from an extremely naive/non-expert point of view!” (dixit Mochizuki, ibid.)37 , this issue
in IUT is taken care of via the notions of species and mutations 38 – such as, respectively, theℛ (or 𝒞) and Φ in Fig. 18. Species and mutations are respective analogues of categories andfunctors, but defined in a sound set-theoretical framework to provide a foundational apparatus forthe various mono-anabelian and inter-universal Teichmüller reconstruction algorithms, which canthus eventually be expressed without any assumption on a fixed choice of universe – we refer to theabsolute anabelian and étale-Frobenius Examples 3.4-3.6 ibid. The following is a reformulation of
Remarks 3.1.4 and 3.6.3 ibid.:
35

36

37

38

While a modelization of IUT via proof assistants, such as Lean or Coq, would thus be of very limited interest, it
could reveal finer arithmetic-geometric structures in-between the discrete and continuous realms – we refer to the
Coq proof of the Four Colors Theorem by Gonthier et al (2005) and the introduction of combinatorial hypermaps
as presented in [Oli23].
In Grothendieck’s mathematical practice and philosophical view, the notion of universe may be considered as a“conceptual gadget” for resolving Bourbaki’s foundational category-theoretic problems, not as a virtuous groundfor subsequent developments, see Marquis’ contribution in [Chap23] and also [Krö06] § 6 – such a virtuous role isindeed explicitly attributed to topoi. By contrast to the category-theoretic prism, one notes that IUT is not ageometry of commutative diagrams – see Examples 3.6 (Syp2) and 3.10.2 in [EssLgc].
The reader will allow the author to take an even more naive position, whose goal is only to bring these considerations
to the eyes of experts in the hope of a potential formal and rigorous treatment.
While a priori distinct, both Bourbaki’s and Mochizuki’s notions of species can be seen as illustrating the same
issue of defining a set-theoretic foundation for the category-theoretic and functorial practice of mathematics, see[Krö06] § 6 and [Mar22].

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Although the mono-anabelian reconstruction algorithms can a priori give rise tonumerous enlargements of universe – and thus a priori numerous ZFC models –the foundation of IUT in terms of species and mutations ensures that the final
IUT algorithm only relies on their shared portion, i.e., on a certain “interuniversal” model.
Species and mutations further ensure that the non-scheme/ring-theoretic Θ-link and log-link are
well-defined in a sense that is independent of a fixed choice of ZFC model.
With regard to heights, or arithmetic degrees, because of the non-ring-theoretic anabelian reconstruction procedures involved at each local place 𝑣, it should be noted that each local height deg𝑣 takes
values, strictly speaking, in a copy R𝑣 of the real numbers that belongs to a different ZFC model.
In this anabelian context, the notion of universe is thus a substantive seminal concept which is of acompletely different nature from its auxiliary (“Helper”) role in establishing abelian cohomologicalarithmetic results – see [McL10] regarding Wiles’ FLT and Deligne’s Weil Conjectures, see § An
Anabelian abc Inequality regarding IUT.
Let us remark that if one follows Grothendieck’s viewpoint, as discussed in Note 36, even further, it
would be reasonable to expect for the theory of topoi – a theoretical framework that bridges category
theory, logic, and geometry – to provide an even finer and more revealing framework for thesechanges of ZFC model and universe – we refer to [Ble22] and references therein for an accessible
and introductory presentation of topoi from the perspective of logic and mathematical universes.
“[Mochizuki’s IUT] work [...] puts anabelian geometry on a higher level/perspective andopens new unprecedented horizons in the research surrounding arithmetic fundamentalgroups and their connection to diophantine geometry [...] The fact (established in these
papers) that Galois theory, and more precisely anabelian geometry, has a particular control on certain diophantine inequalities is an extraordinary fact which is unprecedented.”– M. Saïdi in [MR4225476]

N EW A NABELIAN & D IOPHANTINE F RONTIERS .... In inter-universal Teichmüller theory, the abc
inequality becomes an estimate on how far the additive and multiplicative structures of a ring can bedismantled from one another. Not only does the categorification approach of Mochizuki lead to ananabelian proof of the abc Conjecture and of Fermat’s Last Theorem, it also provides some seminal“reverse” insight into classical anabelian arithmetic geometry, or “Galois-Teichmüller theory”. The
latter is indeed a natural context in which to test new insights of anabelian arithmetic geometry.
§ An Anabelian abc Inequality. The abc inequality – also the first proof of the Vojta Conjecture“with ramification” in the case of curves – results from establishing the “Vojta-Mochizuki inequality”(i.e., [GenEll] Theorem 2.1), see also § From Vojta to Mochizuki - Geometrization & Arithmetization.
In IUT, this inequality follows from two inequalities between the arithmetic degrees – or “heights”,
or log-volumes – of two regions in containers arising from the log-theta lattice of Hodge theaters of
Fig. 16, that are described as a certain Θ-pilot object Θ𝐸 and a certain 𝑞-pilot object 𝑞𝐸 :{−⋃︀ deg 𝑞𝐸 ⋃︀ ⩽ −⋃︀ deg Θ𝐸 ⋃︀ < +∞−⋃︀ deg Θ𝐸 ⋃︀ ⩽ 𝑎ℓ − 𝑏ℓ ⋃︀ deg 𝑞𝐸 ⋃︀ with 𝑎ℓ , 𝑏ℓ ∈ R and 𝑏ℓ > 1

and thus deg 𝑞𝐸 ⩽ 𝑎ℓ ⇑(𝑏ℓ − 1),(HtIneq)

where the first inequality39 follows from the Main Theorem of IUT on the existence of a multiradialalgorithm as in Fig. 17 and Fig. 18, and the second one follows from some direct computations inclassical number theory, see Theorem 1.10 of Chap. IV of [IUTchI-IV]. It is interesting to observe,
once again, that in order to apply this argument, it is necessary to construct some explicit initialΘ-data, as in Corollary 2.2 ibid., that are related to concrete arithmetic and geometric properties of
elliptic curves. This sort of tight cohesion of IUT between the category-theoretic, geometric, andcomputational realms is a trademark of the soundest and most impactful mathematical theories.
39

For the original statement, we refer to Corollary 3.12 of Chap. III of [IUTchI-IV], where the notation “log(−)” isused instead of “deg”, as in the present discussion.

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The Θ-pilot and 𝑞-pilot objects may be thought of as regions inside some
intricate collection of systems of tensor products of log-shells that arise
from local portions 40 of the various Hodge theaters involved; log-shells
serve as common containers for the ⊞/⊠-monoid structures. There is acertain analogy between the log-theta lattice wandering that gives rise to
the comparison of Θ-pilot and 𝑞-pilot arithmetic degrees, up to certain
indeterminacies, and the classical theory of analytic continuation, which
involves a collection of multiple neighborhoods taken in various layers of the Riemann surface definedby the multi-valued complex logarithm – see the figure on the right.
Each Hodge theater contains various global realified Frobenioids, each of which may be understood asa collection of local copies {R𝑣 }𝑣∈V(𝐾) of R at each of the valuations 𝑣 of the number field 𝐾 underconsideration that are related to one another via a relationship that is analogous to the productformula in elementary algebraic number theory. In particular, such a global realified Frobenioid
gives rise to a global copy of R, whose elements may be thought of as global heights deg obtained assums of various local heights deg𝑣 . One Frobenius-like copy of such a global realified Frobenioid
plays a central role in the definition of the non-ring-theoretic Θ-link and hence in the proof of themain IUT inequality between the heights of the 𝑞-pilot and Θ-pilot objects as in Eq. (HtIneq). With
regard to the foundational aspects of the situation, as discussed in § Universes, Species, and Logical
Structure, we observe that:
Because of the non-ring-theoretic constructions involved, the various Hodge the(𝑚,𝑛)±ellℋ𝒯 Θ NF , for 𝑚, 𝑛 ∈ Z, in the log-theta lattice involve distinct andaters
incompatible ring structures — hence, in particular, distinct and incompatible
copies of R, i.e., that arise from distinct and incompatible global realified Frobenioids — which exist a priori in distinct universes (hence distinct ZFC models).
It is only after taking into account certain indeterminacies that the multiradialalgorithm of Fig. 17 is able to produce structures, in a common universe (hence
a common ZFC model), that are compatible with these distinct and incompatible
ring structures and hence allow one to compare in a meaningful way — and,in particular, to verify the height inequality between — the 𝑞-pilot and Θ-pilot
objects41 as in Eq. (HtIneq).
Regarding the arithmetic significance of considering isomorphic but non-identical objects in anabeliangeometry, we refer to the classical example of Grothendieck-Teichmüller theory in Fig. 4. In thecase of IUT geometry, the non-identification of isomorphic objects allows one to distinguish andthen compare various Frobenius-like objects42 . The seminal role of such non-identifications may
be understood in more detail by considering an alternative “RCS-IUT theory” – i.e., a theory
where isomorphic objects are identified, and which may be proven to be logically unrelated to theoriginal IUT theory – where the identification of isomorphic objects indeed produces an essential
contradiction. Here, “RCS” stands for the “Redundant Copies School” of thought, see [EssLgc].
The multiradial construction of the Θ-pilot object results from a certain wandering within the
log-theta lattice that allows one to reconcile the additive structures associated, via the respective
40

41

42

We remind the reader that these local portions are independent of one another and not constrained to any fixed
relationship with a global number field or indeed any other fixed ring structure.
From the point of view of the maturation of mathematical ideas, it is interesting to remember (1) that an
animated debate, led by Weierstrass, caused a stir in the mathematical community of the mid-19th century on thequestion of “how isomorphic (non-labeled) neighborhoods in a Riemann surface could produce any new geometric
information?” (the “geometric fantasies”, ∼1864), and (2) that the dispute between Leibniz and Bernoulli on thelogarithm of non-positive real numbers was resolved by the introduction of the argument-indeterminacy (∼1751),
see the discussion in [EssLgc] § 1.5. We also refer to [Moc18] (C17) for a discussion of what happens if one ignoresthe (𝑚, 𝑛) labeling issue and does not apply the whole IUT algorithm.
We also refer to [EssLgc] Example 2.4.8, which explains how this issue can be illustrated via the elementary
notions of rings and monoids.

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(distinct and incompatible) ring structures involved, to the dilated and undilated multiplicative
monoid structures; the three IUT indeterminacies (Ind1), (Ind2), and (Ind3) are just flexible enoughfor this to happen.
§ An Extended Fermat’s Last Theorem. As recalled in § “abc”
Fig. 19. Fermat’s Last Theorem.
- The Shadow of a Network of Conjectures, while it is natural to
For 𝑛 > 2 the equation
expect for a theory establishing abc to lead to a proof of Fermat’s
𝑥𝑛 + 𝑦 𝑛 = 𝑧 𝑛
has no positive integer solution.
Last Theorem (FLT), in order to obtain such an application to
FLT, it is necessary to establish an “effective” version of the abc Conjecture, i.e., where the constant𝐾𝜀 is explicitly given, see [GT02] – a condition that requires an essential enhancement of the
non-effective anabelian-Diophantine result of the original IUT.
The main obstruction to establishing such an effective version lies in the initial use of a compactly
bounded subset 𝒦 < P1 ∖ {0, 1, ∞} – see the final portion of § From Vojta to Mochizuki - Geometrization & Arithmetization – that supports the prime 2, and which, in the original version of IUT, is latereliminated by applying the (non-effective!) theory of noncritical Belyi maps. In [ExpEst], the original
version of IUT is refined by applying a construction that originates in [Por20], which constructs
an étale theta function – and establishes associated properties of the corresponding monoids – via
evaluation at 6-torsion points instead of at 2-torsion points as in the original Mumford construction
and thus eliminates the requirement of avoiding the prime 2 via the use of the compactly bounded
set 𝒦.
The result is an effective version of abc, which in turns implies an anabelian proof of Fermat’s Last
Theorem, see [ExpEst], Theorem B and Corollary C:⋃︀𝑎𝑏𝑐⋃︀ ⩽ 24 ⋅ exp(1.7 ⋅ 1030 ⋅ 𝜀−166⇑81 ) ⋅ rad(𝑎𝑏𝑐)3(1+𝜀) , and Fermat has no solution for 𝑝 > 1.615 ⋅ 1014
This lower bound – combined with a numerical result of Coppersmith (1990) and some new cyclotomic
estimates of Mihailescu and Rassias (2022) – is indeed lowered to 𝑝 > 257, hence yields, whencombined further with a classical result of Vandiver (1929), a new proof of FLT, as well as a certain
generalized version of FLT – see Cor. 5.9 ibid.
§ Anabelian progress in Grothendieck-Teichmüller
Fig. 20. GT theory - an arithmeticcombinatoric-geometric triangle.
theory. Let us mention another noteworthy example in
Mochizuki’s work of Grothendieck’s principle of the categoriGal(Q̄⇑Q)
Aut(︀𝜋1 (ℳ0,(︀𝑚⌋︀ )⌋︀
fication of arithmetic-geometric contexts – as discussed in§ Grothendieck’s Mathematics Philosophy, as well as in § Cate⇓𝐺𝑇
gorification of a Diophantine Problem – namely, combinatorial
anabelian geometry 43 , a theory developed by Mochizuki jointly with Hoshi (and later with Minamide
and Tsujimura). We refer to [Hos22] § 7-8 and the references therein for a survey.
Following Grothendieck’s “Esquisse d’un programme” [Esq], Grothendieck-Teichmüller theory (GT)concerns the search for a combinatorial description of the absolute Galois group Gal(Q̄⇑Q) in terms⇓ that arises from the (geometric) étale fundamental group of the moduli stack ℳ0,(︀𝑚⌋︀
of a group 𝐺𝑇of genus 0 curves with 𝑚 marked points – see Fig. 20, where the horizontal map is arithmetic, thevertical map is geometric, and the diagonal map is Galois-combinatoric. The leading question here
is to determine “how close are these three maps to being isomorphisms?”
While Galois-Teichmüller theory and Grothendieck-Teichmüller theory have typically, in the past,
provided some input for anabelian geometry, Mochizuki’s school recently obtained two unexpected
applications in the reverse direction, i.e., of combinatorial anabelian geometry to GT theory: (1) a⇓ is larger than was previously expected – see [CbGT]; and, most essentially,
result to the effect that 𝐺𝑇43

To avoid any confusion or potential misunderstanding, let us state clearly that combinatorial anabelian geometry
is a theory in its own right which does not involve any logical dependence on inter-universal Teichmüller theory.

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(2) a combinatorial description Q̄𝐵𝐺𝑇 of the algebraic closure Q̄ of the field of rational numbers andhence of Gal(Q̄⇑Q) itself, see [CbGal].
“En guise de conclusion”
Inter-universal Teichmüller theory opens a new chapter in arithmetic geometry in a way thatfaithfully follows and incorporates the most fundamental aspects of the philosophy of Alexander
Grothendieck concerning the practice of mathematics. Given the multi-layered coherency of its
geometric, category-theoretic, and explicit constructions – as well as how deeply rooted it is in themost essential techniques and progress of classical algebraic geometry and number theory – one
may say that IUT comes across as a stimulating and virtuous theory for the mind of the arithmetic
geometer.
IUT already opens new horizons “internally” – for example, in the form of refinements of theoriginal version of IUT involving elliptic curves over number fields, as well as degenerations of elliptic
curves over number fields, that are expected to lead to new number-theoretic applications that were
not within the range of applicability of the original version of IUT – but also more broadly, as
discussed in § Anabelian progress in Grothendieck-Teichmüller theory, with ramifications for otherrelated topics in arithmetic and homotopic Galois theory. As was already reported in [Hos21b], a“Galois-orbit version of inter-universal Teichmüller theory” that is currently under development for
hyperbolic curves of arbitrary genus is expected to yield important progress on the local-to-global
Grothendieck Section Conjecture – for an introduction to this conjecture, see [Saï12] and [Hos14].
We further refer to [Moc23] § 4 for a report on work in progress and connections with other theories
such as algebraic geometry (via “resolution of nonsingularities” in the sense of Tamagawa, see also
for example [Lep13]) or analytic number theory.
One can thus only expect Mochizuki’s inter-universal Teichmüller theory, as well as the generalphilosophy surrounding this theory, to act as one of the long-term beacons in the ongoing general
harmonization process between arithmetic, on the one hand, and homotopic Galois theory, on theother.

※ For additional introductory notes on inter-universal Teichmüller theory, we refer to the 2021 notes, videos andLeitfaden of [ExpHoriz1; ExpHoriz2]; for a broader introduction and additional references, we refer to the Program
and list of references of [Prom20].

R EFERENCES & N OTES
[Ble22] I. Blechschmidt, “Exploring mathematical objects from custom-tailored mathematical universes,” in Objects, structures, and logics—FilMat
studies in the philosophy of mathematics, ser. Boston
Stud. Philos. Hist. Sci. Vol. 339, Springer, Cham,
2022, pp. 63–95, MR4404380. (cit. on p. 18).

[MFO21] B. Collas, P. Dèbes, H. Nakamura,and J. Stix, Eds., Proceedings of the Homotopic
and Geometric Galois Theory Oberwolfach Workshop, Germany, vol. 18 no. 1, Oberwolfach Reports,
EMS - Publishing House, 2021, pp. 663–744. (cit. on
p. 13).

[BG06] E. Bombieri and W. Gubler, Heights in
Diophantine geometry (New Mathematical Monographs). Cambridge University Press, Cambridge,
2006, vol. 4, pp. xvi+652, MR2216774. (cit. on pp. 6,
7).

[Del98] P. Deligne, “Quelques idées maîtresses del’œuvre de A. Grothendieck,” in Matériaux pourl’histoire des mathématiques au XX𝑒 siècle (Nice,
1996), ser. Sémin. Congr. Vol. 3, Soc. Math. France,
Paris, 1998, pp. 11–19, MR1640253. (cit. on p. 2).

[Prom20] B. Collas, P. Dèbes, and B. Fresse,“RIMS-Lille seminar - Promenade in Inter-universal
Teichmüller theory,” https://www.kurims.kyotou.ac.jp/~bcollas/Promenade- IUT/index.html,[Online; accessed 15-Sept-2022], 2020. (cit. on p. 21).

[Fal98] G. Faltings, “Curves and their fundamental groups (following Grothendieck, Tamagawa andMochizuki),” in 252, Séminaire Bourbaki. Vol.
1997/98, 1998, Séminaire Bourbaki. Vol. 1997/98,
Exp. No. 840, 4, 131–150, MR1685577. (cit. on
p. 10).

Version 02/07/2024

21/24

[Fes16] I. Fesenko, ““Fukugen” - On Shinichi
Mochizuki’s Inter-universal Teichmüller Theory,” Inference, vol. 2, no. 3, Sep. 2016. (cit. on p. 4).

translated in “Sugaku Expositions” AMS). (cit. on
pp. 4, 10, 11, 20).

[GT02] A. Granville and T. J. Tucker, “It’s aseasy as 𝑎𝑏𝑐,” Notices Amer. Math. Soc., vol. 49,
no. 10, pp. 1224–1231, 2002, MR1930670. (cit. on
p. 20).

[CbGT] Y. Hoshi, A. Minamide, and S.
Mochizuki, “Group-theoreticity of numerical invariants and distinguished subgroups of configuration
space groups,” Kodai Mathematical Journal, vol. 45,
pp. 295–348, 2022. (cit. on p. 20).

[SGA1] A. Grothendieck, Revêtements étales etgroupe fondamental (SGA 1) (Lecture notes in mathematics). Springer-Verlag, 1971, vol. 224. (cit. onp. 9).

[CbGal] Y. Hoshi, S. Tsujimura, and S.
Mochizuki, “Combinatorial construction of theabsolute galois group of the field of rational numbers,” 97 pp. 2020, RIMS-1935. (cit. on p. 21).

[Gro97] ——, “Brief an G. Faltings,” in GeometricGalois actions, 1, ser. London Math. Soc. Lecture
Note Ser. Vol. 242, With an English translation on
pp. 285–293, Cambridge Univ. Press, Cambridge,
1997, With an English translation on pp. 285–293,
pp. 49–58, MR1483108. (cit. on pp. 2, 9).

Krömer,“La«machinede[Krö06] R.
Grothendieck» se fonde-t-elle seulement sur desvocables métamathématiques ? Bourbaki et lescatégories au cours des années cinquante,” Revued’histoire des mathématiques, vol. 12, no. 1, pp. 155–
198, 2006. (cit. on p. 17).

[LMG] ——, “La “longue marche” à travers lathéorie de Galois,” Institut Montpelliérain
Alexander Grothendieck (Univ. Montpellier), Ed., Cote n. 140-2 https://grothendieck.
umontpellier.fr, 1978-1982. (cit. on p. 3).

[Lan95] S. Lang, “Mordell’s review, Siegel’s letterto Mordell, Diophantine geometry, and 20th century
mathematics,” Notices Amer. Math. Soc., vol. 42,
no. 3, pp. 339–350, 1995, MR1316133. (cit. on p. 6).

[Esq] A. Grothendieck, “Esquisse d’un programme,” in Geometric Galois actions, 1, ser. London Math. Soc. Lecture Note Ser. Vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 5–48,
MR1483107. (cit. on pp. 3, 20).
[R&S] ——, Récoltes et semailles, Réflexions et témoignage sur un passé de mathématicien. Gallimard,
Jan. 2022, pp. 717+1926. (cit. on pp. 2, 8).
[Hos14] Y. Hoshi, “Conditional results on the birational section conjecture over small number fields,”
in Automorphic forms and Galois representations.
Vol. 2, ser. London Math. Soc. Lecture Note Ser.
Vol. 415, Cambridge Univ. Press, Cambridge, 2014,
pp. 187–230, MR3444234. (cit. on p. 21).
[Hos20] ——, “A note on an anabelian open basis
for a smooth variety,” Tohoku Math. J. (2), vol. 72,
no. 4, pp. 537–550, 2020, MR4194185. (cit. on p. 11).
[Hos21] ——, “Introduction to Mono-anabelian Geometry,” Publications mathématiques de Besançon.
Algèbre et théorie des nombres, pp. 5–44, 2021. (cit.
on p. 12).
[Hos21b] ——, “On the Galois orbit version of Interuniversal Teichmüller theory I & II: A progress
report,” https : / / www . kurims . kyoto - u . ac .
jp/~motizuki/RIMS-workshop-homepages-20162021 / w4 / iut2 . html, Inter-universal Teichmüller
Theory Summit [Online; accessed 15-Sept-2022],
2021. (cit. on p. 21).
[Hos22] ——, “Progress in anabelian geometry (inJapanese),” Sūgaku, no. 1, pp. 1–30, 2022, (To be

22/24[Lep10] E. Lepage, “Tempered fundamental groupand metric graph of a Mumford curve,” Publ. Res.
Inst. Math. Sci., vol. 46, no. 4, pp. 849–897, 2010,
MR2791009. (cit. on p. 12).
[Lep13] ——, “Resolution of nonsingularities for
Mumford curves,” Publ. Res. Inst. Math. Sci.,
vol. 49, no. 4, pp. 861–891, 2013, MR3141726. (cit.
on p. 21).
[Mar22] J. Marquis, “The structuralist mathematical style: Bourbaki as a case study,” in Boston
Studies in the Philosophy and the History of Science,
S. B. C. T. G. Oliveri, Ed., 2022, pp. 199–231.
(cit. on pp. 13, 17).
[McL07] C. McLarty,“The rising sea:
Grothendieck on simplicity and generality,” in
Episodes in the history of modern algebra (1800–
1950), ser. Hist. Math. Vol. 32, Amer. Math. Soc.,
Providence, RI, 2007, pp. 301–325, MR2353501.
(cit. on p. 8).
[McL10] ——, “What does it take to prove Fermat’s
last theorem? Grothendieck and the logic of number
theory,” Bull. Symbolic Logic, vol. 16, no. 3, pp. 359–
377, 2010, MR2731249, Eprint available on-line. (cit.
on pp. 17, 18).
[pGC] S. Mochizuki, “The local pro-𝑝 anabeliangeometry of curves,” Invent. Math., vol. 138, no. 2,
pp. 319–423, 1999, MR1720187. (cit. on p. 10).
[Moc04] ——, “Categorical representation of arithmetic log schemes, with applications to thearithmetic of elliptic curves,” https : / / www .
ms . u - tokyo . ac . jp / video / conference /2004arithmeticgeometry/, “Arithmetic Geometry”
Version 02/07/2024

Tokyo Conference [Online; accessed 15-Sept-2022],
2004. (cit. on p. 13).

wolfach Reports, EMS - Publishing House, 2023, p. 6.
(cit. on pp. 13, 21).

[EtTh] ——, “The étale theta function and its
Frobenioid-theoretic manifestations,” Publ. Res.
Inst. Math. Sci., vol. 45, no. 1, pp. 227–349, 2009,
MR2512782. (cit. on p. 14).

[ExpEst] S. Mochizuki, I. Fesenko, Y. Hoshi,
A. Minamide, et al., “Explicit estimates in interuniversal Teichmüller theory,” Kodai Math. J.,
vol. 45, no. 2, pp. 175–236, 2022, MR4447653. (cit.
on p. 20).

[GenEll] ——, “Arithmetic elliptic curves in general
position,” Math. J. Okayama Univ., vol. 52, pp. 1–
28, 2010, MR2589843. (cit. on pp. 6, 8, 14, 18).
[Moc10] ——, “Inter-universal Teichmüller Theory: A Progress Report,” https : / / www .
mathsoc . jp / meeting / msjsi10 / JointMSJ RIMSprogram20101027 . pdf, “Development of Galois -Teichmüller Theory and Anabelian Geometry”
Joint MSJ-RIMS Conference [Online; accessed 15Sept-2022], 2010. (cit. on p. 13).
[AbsTopIII] ——, “Topics in absolute anabelian geometry III: Global reconstruction algorithms,” J.
Math. Sci. Univ. Tokyo, vol. 22, no. 4, pp. 939–
1156, 2015, MR3445958, Eprint available on-line.
(cit. on p. 11).
[Moc18] ——, “Comments on the manuscript by
Scholze-Stix concerning Inter-universal Teichmüller
theory (IUTCH),” https://www.kurims.kyotou.ac.jp/~motizuki/IUTch- discussions- 201803.html, Jul. 2018. (cit. on p. 19).
[IUTchI-IV] ——, “Inter-universal Teichmüller theory: (I) Construction of Hodge theaters, (II) HodgeArakelov-theoretic evaluation, (III) Canonical splittings of the log-theta-lattice, (IV) Log-volume computations and set-theoretic foundations,” Publ. Res.
Inst. Math. Sci., vol. 57, no. 1, pp. 3–723, 2021,
MR4225473. (cit. on pp. 13, 14, 17, 18).
[Alien] ——, “The mathematics of mutually alien
copies: From Gaussian integrals to inter-universal
Teichmüller theory,” in Inter-universal Teichmüller
Theory Summit 2016, ser. RIMS Kôkyûroku Bessatsu,
B84, Res. Inst. Math. Sci. (RIMS), Kyoto, 2021,
pp. 23–192, MR4279800, Eprint available on-line.
(cit. on pp. 4, 13, 15–17).
[EssLgc] ——, “On the Essential Logical Structureof Inter-universal Teichmüller Theory in terms of logical and “∧”/logical or “∨” relations: Report on theoccasion of the publication of the four main papers
on Inter-universal Teichmüller theory,” https : / /www.kurims.kyoto-u.ac.jp/~motizuki/papersenglish.html, [Online; accessed 22-Sept-2022], Sep.
2022. (cit. on pp. 4, 8, 13–17, 19).
[Moc23] ——, “Inter-universal teichmüller theory as
an anabelian gateway to diophantine geometry andanalytic number theory,” in Proceedings of the “Arithmetic Homotopy and Galois Theory” MFO-RIMS
Tandem workshop, Germany & Kyoto, B. Collas, P.
Dèbes, Y. Hoshi, and A. Mézard, Eds., ser. OberVersion 02/07/2024[ExpHoriz2] S. Mochizuki, Y. Hoshi, I. Fesenko, Y. Taguchi, et al., “Workshop - Interuniversal Teichmüller Theory Summit 2021,” https://www.kurims.kyoto-u.ac.jp/~motizuki/RIMSworkshop- homepages- 2016- 2021/w4/iut2.html,[Online; accessed 15-Sept-2022], 2021. (cit. on p. 21).
[ExpHoriz1] ——, “Workshop - Invitation to Interuniversal Teichmüller Theory,” https : / / www .
kurims . kyoto - u . ac . jp / ~motizuki / RIMS workshop- homepages- 2016- 2021/w3/iut1.html,[Online; accessed 15-Sept-2022], 2021. (cit. on pp. 16,
21).
[NTM98] H. Nakamura, A. Tamagawa, and S.
Mochizuki, “Grothendieck’s conjectures concerning fundamental groups of algebraic curves,” Sūgaku,
vol. 50, no. 2, pp. 113–129, 1998, (Eng. “Sugaku
Expositions” 14, AMS (2001), no. 1), MR1648427.
(cit. on pp. 4, 10).
[Oli23] F. Oliva, “What does a computer-assisted
proof prove?” Talk in the “French Colloquium on the
Philosophy of Mathematics : French Philosophy ofMathematics Workshop (FPMW 15)”, Paris, 2023.
(cit. on p. 17).
[Por20] W. Porowski, “Anabelian geometry ofpunctured elliptic curves,” University of Nottingham,
2020. (cit. on p. 20).
[Chap23] “Proceedingsoftheconference“Grothendieck, a multifarious giant: Mathematics,
Logic and Philosophy” (to be published),” https:/ / www . chapman . edu / scst / conferences - and events / grothendieck - conference . aspx, 2023.
(cit. on pp. 2, 17).
[Rém03] G. Rémond, “Vojta’s method in diophantine geometry, applications and related topics,” inProc. of the Diophantine Problems and Analytic
Number Theory Conference, ser. Kôkyûroku, RIMS,
2003, pp. 202–2010. (cit. on p. 7).
[Saï12] M. Saïdi, “Around the Grothendieck anabelian section conjecture,” in Non-abelian fundamental groups and Iwasawa theory, ser. London Math.
Soc. Lecture Note Ser. Vol. 393, Cambridge Univ.
Press, Cambridge, 2012, pp. 72–106, MR2905530.
(cit. on p. 21).
[MR4225476] ——, “Review of the four Interuniversal Teichmüller theory papers of S.

23/24

Mochizuki,” Mathematical Reviews, vol. 4225476,
2022. (cit. on p. 18).
[SS16] A. Schmidt and J. Stix, “Anabelian geometry with étale homotopy types,” Ann. of Math. (2),vol. 184, no. 3, pp. 817–868, 2016, MR3549624. (cit.
on p. 11).
[SV90] G. B. Shabat and V. A. Voevodsky,“Drawing curves over number fields,” in The
Grothendieck Festschrift, Vol. III, ser. Progr. Math.
Vol. 88, Birkhäuser Boston, Boston, MA, 1990,
pp. 199–227, MR1106916. (cit. on p. 3).

[Sha18] G. Shabbat, “The divine spark, To thememory of Vladimir Voevodsky,” Troitsky variant,no. 249, Mar. 2018, An interview by Olga Orlova,
English version: https : / / www . math . ias . edu /Voevodsky/other/Orlova.pdf. (cit. on p. 3).
[Sti11] J. Stix, “The Brauer-Manin obstruction
for sections of the fundamental group,” J. Pure
Appl. Algebra, vol. 215, no. 6, pp. 1371–1397, 2011,
MR2769238. (cit. on p. 4).
[Voe91] V. A. Voevodsky, “Galois representations
connected with hyperbolic curves,” Izv. Akad. NaukSSSR Ser. Mat., vol. 55, no. 6, pp. 1331–1342, 1991,
MR1152216. (cit. on p. 3).

[A] “On peut considérer que la géométrie nouvelle est avant toute autre chose, une synthèse entre ces deux mondes,
jusque là mitoyens et étroitement solidaires, mais pourtant séparés : le monde "arithmétique", dans lequel vivent les(soi-disants) "espaces" sans principe de continuité, et le monde de la grandeur continue, ou vivent les "espaces" au sens
propre du terme, accessibles aux moyens de l’analyste [...]. Dans la vision nouvelle, ces deux mondes jadis séparés,n’en forment plus qu’un seul [...,] vision d’une "géométrie arithmétique" (comme je propose d’appeler cette géométrie
nouvelle).”[B] “Les plus profonds (à mes yeux) parmi ces douze thèmes, sont celui des motifs, et celui étroitement lié de géométrie
algébrique anabélienne et du yoga de Galois-Teichmüller.”[C] “ Une telle supposition avait l’air à tel point dingue que j’étais presque gêné de la soumettre aux compétences en lamatière. Deligne consulté trouvait la supposition dingue en effet, mais sans avoir un contre-exemple dans ses manches.
Moins d’un an après, au Congrès International de Helsinki, le mathématicien soviétique Bielyi annonce justement cerésultat, avec une démonstration d’une simplicité déconcertante tenant en deux petites pages d’une lettre de Deligne –jamais sans doute un résultat profond et déroutant ne fut démontré en si peu de lignes !”24/24

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