FAQ on Inter-universal Teichmüller Theory
By Go Yamashita, RIMS, Kyoto University.
September 2018
Q1. Which papers do you recommend for people who wish to begin their study of inter-universal
Teichmüller theory?
A1. I recommend• [AbsTopIII] to learn the mono-anabelian reconstruction algorithms (which are important
ingredients in inter-universal Teichmüller theory) arising from the Belyi cuspidalisation
developed in [AbsTopII], and• [EtTh] to learn the theory of mono-theta environments (which is also an important
ingredient in inter-universal Teichmüller theory), especially the three important rigidities
that they satisfy. The algorithm underlying constant multiple rigidity uses the elliptic
cuspidalisation developed in [AbsTopII].
Both the Belyi cuspidalisation and the elliptic cuspidalisation mentioned above, which are
absolute anabelian results, are obtained as consequences of the relative anabelian result of[pGC], i.e., the relative Grothendieck Conjecture .
It seems to me worthwhile to remark that it was very important, from the point of view of
the application to inter-universal Teichmüller theory, that the relative Grothendieck Conjecture
was proved over local elds in [pGC], in spite of the implicit expectation at that time that, by
analogy with the Tate conjecture, the relative Grothendieck Conjecture should hold only overglobal elds.
Q2. In inter-universal Teichmüller theory, one often considers the full poly-isomorphism between distinct copies of mathematical objects such as, for example, copies † G and ‡ G of theabsolute Galois group GK of a local eld K . Considering distinct copies seems to be uselessand unnecessarily complicated. Why can't one simply replace † G and ‡ G by GK and replace
the full poly-isomorphism between † G and ‡ G by the identity morphism of GK ?
A2. Mathematical objects such as GK are a priori equipped with the histories of operationsthat appear in their constructions. Thus, for example, GK is equipped with its structure asa Galois group, i.e., its structure as a group of automorphisms of some eld. By taking an
isomorph! † G of GK as an abstract topological group , we forget the history of GK , i.e., † G
is no longer equipped with a Galois group structure (that is to say, structure as a group of
automorphisms of some eld). In particular, the only properties that † G shares with GK areproperties that arise from the abstract topological group structure of these topological groups.
It is only after forgetting such histories of operations that appear in the constructions of these objects and considering them as abstract topological groups, abstract topological monoids etc., that one may obtain a kind of symmetry (i.e., akind of switchability or exchangeability between two distinct copies of objects)

while maintaining the relationship between such switchable objects

More generally, he proved it over sub-p-adic elds.
i.e., the set of all isomorphisms.
!i.e., isomorphic object.
and a priori non-switchable objects ". This forms the essential content of
the multiradiality of the algorithms that play a central role in the main results ofinter-universal Teichmüller theory.

Q3. Suppose that, in the situation of (Q2), we proceed (as suggested in (Q2)) to replace †G and‡
G by GK and replace the full poly-isomorphism between † G and ‡ G by the identity morphismof GK , but that we also consent to treat GK as an abstract topological group when required.
Would inter-universal Teichmüller theory still function under these conditions?
A3. Once we forget their histories of operations, we should never recall back them. (In fact,
when we take an isomorph †G of GK as a topological group, †G can no longer recall back thehistory of operations.) The identity morphism from †G to ‡G does not make sense, since weneed their histories of operations as GK to dene the identity morphism, and †G and ‡G are
dierent objects! If we sometimes forget histories of operations, and sometimes recall themback, then it destroys the symmetry mentioned in (A3), especially, if we take the identity
morphism by recalling back their histories of operations, then the nal algorithm does not
work.
Q4. Suppose that, in the situation of (Q2), we consider the full poly-isomorphism between †Gand ‡G, but we replace (as suggested in (Q2)) †G and ‡G by GK , and we consent to treat GK
as an abstract topological group and to completely forget its history of operations. Wouldinter-universal Teichmüller theory still function under these conditions?
A4. In theory, the proof still works under these conditions, so long as one is careful not to
confuse oneself by the use of theGK(i.e., despite the fact thatthe notation  GK  suggests that any topological group denoted by  GK  is to be regarded as
the absolute Galois group of some eld K ). If one takes sucient care when using the notation†‡
 GK , thenGK
G
G
. In light of this state of aairs, it seems much
GKless confusing to work with such isomorphs †G and ‡G (treated as abstract topological groups)
from the beginning. Moreover, it is not clear that there is anything to be gained by using thesame notation  GK  to denote dierent topological groups, which are not,, supposed
to be regarded as Galois groups of elds. Thus, in summary, the use of the notation  GK 
only creates unnecessary confusion.
same notation

to denote dierent topological groups, which,moreover, are not to be regarded, a priori, as Galois groups of elds

working withis essentially equivalent to working with isomorphs

and

of

that are treated as abstract topological groupsa priori

It seems much more natural to use dierent notation fordierent objects, and to avoid notation for abstract topological groups that erroneously suggeststhat those groups are to be regarded as Galois groups of elds.

This sort of phenomenon may be understood by considering a, as follows#. Consider the following dialogue between aand aconcerning n-dimensional vector
spaces over R, for xed n > 0:
toy model

student

professor: People often speak of n-dimensional vector spaces V and W over R, but
this seems to be useless and unnecessarily complicated. Why can't one always
just work with Rn?

Student: Using the notation  Rn to denoten-dimensional vector
spaces over R would result in a substantial amount of confusion. For instance,
it might lead one to conclude (erroneously) that there exists a canonical isomorphism between V and its dual V ∗, or that it makes sense to consider composites"More precisely, Frobenius-like objects.
#This toy model arose in discussions with Akio Tamagawa.
Professor

arbitrary!

of the form  f ◦ f , where f is an R-linear map f : V → W between distinct V
and W .

Student: I still don't understand. It seems that there should be no problem withusing the notation  Rn  to denote arbitrary n-dimensional vector spaces over R,
so long as one is careful not to confuse dierent  Rn 's.

Professor: To regard some abstract n-dimensional vector space over R as  Rn 

is essentially equivalent to considering a basis of the vector space. The notionof a `vector space with a basis' is more complicated than the notion of `vector
space'. In particular, one can often simplify arguments and avoid confusion byomitting the specication of a particular basis and working with abstract vector
spaces  V  or  W .

Student: Working with such  V  and  W  seems to be useless and unnecessarily

complicated. They seem to be simply alternative names for the same vector
space Rn ! Moreover, it seems that we should be able to regard any isomorphism
between V and W as the identity morphism, simply by choosing suitable basesfor V and W .

Professor: If you take care not to get confused $ by the notation after choosing
bases for the various vector spaces that arise in an argument, then it is true that,in theory, one may always work with vector spaces equipped with a basis. On
the other hand, doing this can often result in confusion or in making arguments
unnecessarily complicated. Moreover, it is important to remember that the notion of the `identity morphism' is only dened when the domain and codomainof the morphism are equal, i.e., it does not make sense to speak of the `identity
morphism' between distinct vector spaces V and W .

There are many other toy models that one may consider. For example, a manifold is dened
by patching together open sets of Euclidean space. If one identies such open subsets with one
another, then the theory of manifolds collapses immediately %. Another example may be seen in
the theory of group representations, where, if, for instance, a representation of a group contains
multiple copies of an irreducible representation of the group, then it is of crucial importanceto distinguish the notion of equality of subrepresentations from the notion of isomorphism ofrepresentations. Yet another example may be seen in the elementary theory of eld extensions,where it is of crucial importance to distinguish the notion of equality of subextensions of a eldfrom the notion of isomorphism of elds .
Q5. Can you give some sort of toy model to illustrate the phenomenon of changing universes?
A5. Perhaps the most fundamental example of such a toy model is the phenomenon of changesof coordinates in Euclidean space. Please see (A4) of [FAQ] for other examples. The philosophy
of inter-universality also plays an important role in combinatorial anabelian geometry.
Q6. All of the constructions in the series of papers on inter-universal Teichmüller theory seemto be trivial and devoid of any nontrivial content. How can one deduce a nontrivial consequence$For instance, one must take care not to confuse the approach proposed in (Q4) with the approaches proposedin (Q2) and (Q3).
%This toy model is compatible with the point of view mentioned in (A4) of [FAQ] to the eect that changing
coordinates is a toy model of changing universes. See also (A3) of [FAQ].

"from such trivial constructions?

A6. Let me cite the following passage from the Introduction of [Y]:... Indeed, once the reader admits the main results of the preparatory papers (especially [AbsTopIII], [EtTh]), the numerous constructions in the series of papers[IUTchI], [IUTchII], [IUTchIII], [IUTchIV] on inter-universal Teichmüller theory
are likely to strike the reader as being somewhat trivial.

On the other hand,

the way in which the main results of the preparatory papers are interpreted andcombined in order to perform these numerous constructions is highly nontrivial
and based on very delicate considerations (cf. Remark 9.6.2 and Remark 12.8.1)
concerning, for instance, the notions of multiradiality and uniradiality (cf. Section 11.1). Moreover, when taken together, these numerous trivial constructions,whose exposition occupies literally hundreds of pages, allow one to conclude ahighly nontrivial consequence (i.e., the desired Diophantine inequality) practically eortlessly! Again, from the point of view of the author, the point of theproof seems to lie in the establishment of a suitable framework in which one
may deform the structure of a number eld by abandoning the framework of
conventional scheme theory and working instead in the framework furnished byinter-universal Teichmüller theory (cf. also Remark 1.15.3).
In fact, the main results of the preparatory papers [AbsTopIII], [EtTh], etc.
are also obtained, to a substantial degree, as consequences of numerous constructions that are not so dicult.

On the other hand, the discovery of theideas and insights that underlie these constructions may be regarded as highly
nontrivial in content. Examples of such ideas and insights include the hidden
endomorphisms that play a central role in the mono-anabelian reconstruction
algorithms of Section 3.2, the notions of arithmetically holomorphic structure
and mono-analytic structure (cf. Section 3.5), and the distinction between étalelike and Frobenius-like objects (cf. Section 4.3).

Thus, in summary, it seems

to the author that, if one ignores the delicate considerations that occur in thecourse of interpreting and combining the main results of the preparatory papers,
together with the ideas and insights that underlie the theory of these preparatory papers, then, in some sense, the only nontrivial mathematical ingredient inpinter-universal Teichmüller theory is the classical result [ GC], which was alreadyknown in the last century!

Q7. Can you cite any nontrivial partial results that occur as consequences of intermediate steps
in the proofs of the main results of inter-universal Teichmüller theory?

A7. The constructions in the rst three papers of the series of papers on inter-universal Teichmüller theory, taken together, constitute a single algorithm , namely, the nal multiradialalgorithm of Theorem 3.11 in the third paper.

In particular, the structure of inter-universal

Teichmüller theory is such that it is only meaningful to search for nontrivial partial results inthe preparatory papers. For example, as mentioned before, the mono-anabelian reconstruction
algorithms arising from the Belyi cuspidalisation are very important results and have alreadyyielded further results (that are not directly related to inter-universal Teichmüller theory).

Q8. Can you give examples of further research or results that arose from inter-universal Teichmüller theory?

A8. I myself am interested in pursuing the possibility of applying various ideas that appear ininter-universal Teichmüller theory to the study of the Riemann zeta function. At the present#time, I have obtained some interesting observations, but no substantive results. Hoshi is studying an application of inter-universal Teichmüller theory to the birational section conjecture
in birational anabelian geometry, while Porowski and Minamide are studying numerical improvements of certain height inequalities in inter-universal Teichmüller theory. I also hear that
Dimitrov is studying the possibility of applying inter-universal Teichmüller theory to the studyof Siegel-zeroes.

References

[ GC]

p

S. Mochizuki,[EtTh]S. Mochizuki,

The Local Pro-p Anabelian Geometry of Curves.
!&The Étale Theta Function and its Frobenioid-theoretic Manifestations.
Invent. Math.

Inst. Math. Sci.
[AbsTopII]

S. Mochizuki,
Univ. Tokyo[AbsTopIII] S. Mochizuki,"# (2009), p.227349.

Topics in Absolute Anabelian Geometry II: Decomposition Groups.

 (2013), p.171269.

G. Yamashita,(2015), p.9391156.

FAQ on Inter-Universality, an informal note available at

http://www.kurims.kyoto-u.ac.jp/~motizuki/research-english.html[Y]

G. Yamashita,

Publ. Res.

J. Math. Sci.

Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms.

Math. Sci. Univ. Tokyo[FAQ](1999), p.319423.

A proof of the abc conjecture after Mochizuki, preprint available at

http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html

J.