This document is a lightly edited version of the mathematically substantive comments posted from April 6 - April 19, 2020 at this blog posting. The original
source should be consulted for comments that might be of interest posted before
and after the ones gathered here. For context, please consult the original blog
posting as well as this later one. This document was provided to the editorsof PRIMS who acknowledged its receipt but have decided that the Mochizuki
proof should be deemed correct and complete and published, despite the serious
objections raised here and elsewhere.
For a detailed background to this discussion, one should first consult the 2018
documents written by Scholze/Stix and Mochizuki (as well as Mochizuki/Hoshi)
available at Mochizuki’s website. David Roberts points out there is a more detailed version of what he has to say here.

Peter Scholze April 6, 2020 at 9:28 am
I may have not expressed this clearly enough in my manuscript with Stix, butthere is just no way that anything like what Mochizuki does can work. (I would
not make this claim as strong as I am making it if I had not discussed this with
Mochizuki in Kyoto for a whole week; the following point is extremely basic,
and Mochizuki could not convince me that one dot of it is misguided, duringthat whole week.) It strikes deep into my heart to think that in the name ofpure mathematics, an institute could be founded for research on such questions,and I sincerely hope that this will not come back to haunt pure mathematics.
The reason it cannot work is a theorem of Mochizuki himself. This statesthat a hyperbolic curve X over a p-adic field K (maybe with some assumptions,
all of which are always satisfied in all cases relevant to IUT) is determined up toisomorphism by its fundamental group π1 (X), and in fact automorphisms of X
are bijective with outer automorphisms of π1 (X). Thus, the data of X is completely equivalent to the data of π1 (X) as a profinite group up to conjugation.
In IUT, Mochizuki always considers the latter type of data, but of course up toequivalence of groupoids this makes no difference. (The passage back and forthis even constructive, by another result of Mochizuki.)
Mochizuki claims that by replacing X by π1 (X), things can happen thatcannot otherwise happen. Examples are given concerning the action of π1 (X)on certain associated monoids. We discussed this at very great length in Kyoto,
but none of these examples carried any actual content. Note that any potential
non-commutativity of some diagram that results from identifying π1 (X)’s via
isomorphisms of X’s could not possibly be resolved by using some other isomorphism of π1 (X)’s – all of them come from isomorphisms of X’s! Mochizuki
considers infinitely many distinct isomorphic copies of π1 (X)’s, but could nottell us what goes wrong if we simply identify all of them with one another,and with π1 (X0 ) for some fixed X0 – there is no diagram that commutes in
his situation but does not commute under this further identification. (In mymanuscript with Stix, we simply went through Mochizuki’s argument with thisfurther identification, pinpointing what goes wrong. If this further identification1

causes problems, just tell us which diagram it is whose commutativity is rescued
by not explicitly identifying π1 (X)’s.)
However, what I really want to do with this comment is to point out that
there seems to be significant confusion over just the above point on X’s vsπ1 (X)’s. Recently,arXiv:2003.01890v1 appeared, in which the author (Kirti
Joshi) gives some survey on results related to Mochizuki’s work. In the introduction, on page 7, he explicitly claims that one could find non-isomorphic X’s
giving rise to the same π1 (X), and even more, in Remark 2.1 on page 14 he
explains that my reading of the situation is a common misunderstanding. Even
more, in Corollary 21.2 on page 47, he states something “well-known to everyone at RIMS” giving an explicit example of this phenomenon of non-isomorphic
X’s giving rise to the same π1 (X).
With this appearing on arXiv, I was indeed quite confused – did I in factmisunderstand this basic point all this time? If the above claims would havebeen true, I would see how Mochizuki’s strategy might have a nonzero chanceof succeeding. But I was quite sure that in our discussions in Kyoto, Mochizuki
agreed with me on that basic point; and the proof of Theorem 21.1 in thatsurvey (of which Corollary 21.2 is indeed a corollary) was wrong. In any case,I emailed Joshi indicating my confusion, and he has since checked back with
Mochizuki and retracted all of these claims (he told me a new version will be onarXiv soon). In particular, the fact “well-known to everyone at RIMS” is wrong,
and in contradiction to this earlier correct anabelian theorem of Mochizuki.

Taylor Dupuy April 6, 2020 at 1:00 pm
Hi Peter!
First, hope your pandemic is going well. Mine is going ok. Hard to getthings done without daycare.
Second, let me say that for hyperbolic curves over a p-adic field K (withno extra hypotheses like strictly Belyi type or Belyi type or canonical lift) that
π1 (Z) “determines” Z is open. Also, I personally would advocate against using
words like “determines”, “reconstructs”, etc that have been causing sooo many
problems in discussing all of these things. For the uninitiated, let me say thatwhat Peter claims here is that outer isomorphisms of fundamental groups as
topological groups are in bijection with isomorphisms of curves over a field K.
The difference between what Mochizuki did in his relative case together with
his interpretation of GK and the absolute Grothendieck conjecture I claim is
subtle (maybe I am missing something though, I’m a little intimidated saying
this so publicly to be honest).
–to see this is a non-trivial topic consider for example the introduction tothis manuscript here:
http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1892.pdf
Mochizuki’s theorem states that for Z and W hyperbolic curves over a padic field K, outer isomorphisms π1 (Z) → π1 (W ) *over* GK — meaning theymorphisms in an overcategory where π1 (Z) → GK and π1 (W ) → GK — are in2

bijection with isomorphisms between Z and W.
To prove this, it suffices for example to show π1 (Z) interprets the field K.
I have written down an unreviewed proof which I don’t think it is so difficult
which I can share if you want.
Anyway, I think what you are thinking about is that π1 (Z) admits an interpretation GG(π1 (Z)) naturally isomorphic to GK and since we have both thefundamental group and the augmentation map we should be in the setting of
Mochizuki’s theorem and are done. This is not quite correct. What is at issue
is that given f : π1 (Z) → π1 (W ) one does not necessarily know that GG(f) is
inner (which would puts you into the hypotheses of the Mochizuki’s proof of the*relative* Grothendieck conjecture).
Also, Mochizuki has conjectured in print it is absolute Grothendieck overp-adic is not true—-See Remark 1.3.5.1 of this paper:
http://www.kurims.kyoto-u.ac.jp/~motizuki/Absolute%20Anabelian%
20Geometry.pdf
but seems to be nonspecific nowadays. I don’t know. We would need to askhim.
I personally believe if it is true then there is sort of a Zilber trichotemy/diochotemy
thing going on. That is “absolute Grothendieck over p-adic” iff π1 (Z) interpretsa field. Maybe I am using Zilber trichotemy/dichotemy wrong, but I stand by
the statement. The reason for this is because in all the special cases where absolute Grothendieck over a p-adic field holds, (canonical lifts, Belyi type, strictly
Belyi type) there is an interpretations of fields. Maybe this is wrong though anda reader can point out an example where we know the absolute Grothendieckconjecture over p-adic fields without some special hypothesis being imposed.
Third, your counterpoint is a heuristic and not a disproof. For example,
Dieudonné modules are equivalent to finite group schemes and we don’t callthem worthless. We could go blue in the face coming up with examples of two
equivalent objects one of which is useful and the other which is not. Maybe I
am missing something, but I’m not sure how productive these meta discussions
are. I think your point is that he needs to “use something”/“do something”. I
agree, but it is not a disproof.
Alternatively, we can argue who has the burden of proof here. . . which I
think is a more compelling argument for rejecting the whole thing. I think we
can all agree the notation in his manuscripts are a big dumpster fire.
Fourth, regarding the deformation theory, his entire theory is up to a genericisomorphism (=polyisomorphism) the base G. That is the fundamental objects
he considers are (like) a fundamental group Π with a map Π → GK where
GK is considered up to automorphism. In fact, any time he does one of thesepolyisomorphisms (which I think we should be calling generic isomorphisms),
we should probably be thinking of some sort of poor man’s universal family.
Next we can ask “does this *DO* anything”? Well, for one, it certainly
provides a formalism for talking about how things change under automorphism.
There are a lot of interesting representations of automorphisms of fundamentalgroups acting on various interpretations which seem to me to be vedry nontrivial. . . etale theta being a principal example. Here is another thing I haven’t
3been able to puzzle out but maybe you can help: Let Z be a hyperbolic curve overa p-adic field. Fix the augmentation map f : π1 (Z) → GK , let g : GK → GK
be an outer isomorphism. What is the base change of f by g?
π1 (X) ×GK ,g GK =???
It is like some sort of Frobenius twist looking thing—except not twisted by
the Frobenius but by some outer morphisms of the fundamental group of thebase. Is this the fundamental group of a curve? Anyway, it seems to me you
can apply many of his interpretations to this object and that constructions are
uniform in this sort of thing. This I am not 100% on and wish I didn’t have totalk about these things I don’t understand so well publicly. Maybe the readers
can tell us what breaks here.
So, that’s all I have to say about that for now.
Best, Taylor

W April 6, 2020 at 1:56 pm@ Taylor Dupuy
Your clarification is very helpful for arbitrary curves, but as you say at the
beginning, is relevant when there are no extra hypotheses like strictly Belyi
type. Since IUT focuses on étale fundamental groups of once-punctured elliptic
curves, and these are of strictly Belyi type, I don’t see why they are helpful.
In case anyone following is confused about what theorem of Mochizuki ismeant, it is Corollary 1.10 of his paper “Absolute Anabelian Cuspidalizationsof Proper Hyperbolic Curves”. (Not his earlier theorem about arbitrary curves.)[see next comment for correct reference]
Even if this “strictly Belyi type” condition were somehow avoided, then theexistence of extra isomorphisms of the abstract fundamental group would still
be an open problem. It seems hard to imagine how these isomorphisms could
ever be used in a proof of some concrete inequality between two real numbers
without proving that at least one exists.
It is probably possible to make rigorous a lot of what Peter is saying, thatany proof of XYZ form cannot possibly work, or maybe more precisely that
Mochizuki’s proof is equivalent to the wrong proof sketched by Scholze andStix.
But one also should include not-completely-rigorous evidence when deciding
on how much burden of proof to assign to the author of a paper. The fact thatevery plausible use of a particular mathematical construction would not help ina particular argument, but the construction adds unnecessary complexity to theargument, and thus would make it harder to see any mistake, and two smart
mathematicians have tried to remove the construction and found an incorrect
argument, is strong evidence against the paper, even though it is possible that(1) there is a different way to remove the construction which leads to a correct
argument or (2) there exists a clear explanation of how the construction is used
to solve a problem which is not possible without it.
4

Fix the augmentation map f : π1 (Z) → GK , let g : GK → GK
be an outer isomorphism. What is the base change of f by g?.... Is
this the fundamental group of a curve?
Isn’t the content of Mochizuki’s theorem that, if Z of strictly Belyi type,then this is not the fundamental group of a curve? Of course this may be a
curve in some other case.

Vesselin April 6, 2020 at 2:28 pm@W
Just to fix your precise reference for the convenience of others following along:
You mean either Corollary 2.3 on page 500 of that paper (Absolute anabeliancuspidalization of proper hyperbolic curves); or Corollary 1.10, part (iii) on page
43 of the subsequent paper (Topics in absolute anabelian geometry III).
Those disprove Corollary 21.2 of Joshi’s preprint.

Peter Scholze April 6, 2020 at 4:54 pm
Taylor, thanks a lot for your answer!
Regarding your first point: Yes, I’m doing fine in these strange times; I hope
you are too. Fortunately we are still allowed to enjoy the incoming spring.
Regarding your second point: As W observes, your objection only seems
relevant when the curves are not of strictly Belyi type, but all the relevant
ones for IUT are. So this objection is a red herring. (Meanwhile, it is clearlyinteresting to sort out whether strictly Belyi type is necessary, and I wish you
luck in improving Mochizuki’s anabelian results!)
Regarding your third point: Of course I am very well aware of the power ofcategory equivalences. But there must be something you can do on the otherside that you can’t do on the first. As I said, during one week Mochizuki was
not able to give a single relevant example. So to me it seems like a category
equivalence that simply obfuscates things.
Regarding your fourth point: I am at a complete loss what one wants to dowith full poly-isomorphisms. For the convenience of readers following along: A
full poly-isomorphism between two isomorphic objects A and B of a category
C is the set of all isomorphisms between A and B. Mochizuki often says that
he identifies two objects A and B along the full poly-isomorphism. To be clear,this is (up to equivalence of categories) no data at all, there is a unique fullpoly-isomorphism, so you can’t ”pick one” (or rather, you can always pick one,and only one). On the other hand, to identify to objects in a category, you needto pick a specific isomorphism! Of course, you can just pick any one of them,but you can’t pick all of them! If A and B are sets and a ∈ A is an element,
then what is the image of a in B under the full poly-isomorphism? It makes nosense. Mochizuki has repeatedly told us in Kyoto that because some diagram
does not commute when A and B are identified via the obvious isomorphism

5(that usually exists in his situation), he has to identify them only along thefull poly-isomorphism. But (in his situation) there is not a single isomorphism
between A and B that makes the diagram commute! So how does the full polyisomorphism help? You can’t make the logarithm map into a map of fields by
pre- and postcomposing with field automorphisms!
(To be clear, I am willing to accept that there is a nonzero chance that someof these things might make sense under certain circumstances. Again, let me
stress that we discussed these very matters for one week in Kyoto.)
From what I understand, the objection to my manuscript with Stix is that we
did some identifications that are not allowed. This is just the identification I was
talking about: Mochizuki considers infinitely many distinct copies of π1 (X) andis only allowing himself to identify them along the full poly-isomorphism. We
do not see any diagram that commutes with this choice, but does not commute
when we identify them along the identity map, once taking all π1 (X) to be equal
to the actual π1 (X0 ) for our fixed curve X0 . In any case, the passage from X’s to
π1 (X)’s is giving you no extra flexibility, as discussed in my previous comment.
Best wishes, Peter

Taylor Dupuy April 6, 2020 at 6:39 pm
Hopefully this is productive...
second point (and W’s comment on the base change thing):
I’m not sure how much to engage here since this isn’t strictly logically necessary for a disproof. Since I think this is an important point in terms of ourunderstanding of what is going on I’m going to say a few words. First, forthe readers, Peter and W are focusing on the bad non-archimedeans cases, I
believe. In this situation, I’m not 100% sure that all the curves in what I would
call the “zoo of covers” are SBT but certainly most of them are as they are
analytifications of base changes of hyperbolic curves from number fields. Thisis Belyi’s Theorem. There are a number of other points that we should consider: there are log structures too which should make things even more rigid(my understanding is that you need this information for reconstruction of thespecial fiber), you have these punctured tempered universal covers (these arecertainly not algebraic), there are groups involved in mono-theta environments
going on, and stacks fundamental groups. My point is that it is still conceivable
to me that there is some sort of deformation theory going on and I don’t wantto confirm or deny this. But yeah, a lot of them are. The base fields certainly
are not.
In terms of uniformity of constructions in G or “strange base changes”;
Mochizuki performs many constructions with respect to (Π, G) where for a pair
we consider a generic isomorphism G(Π) → G – here G is the interpretation ofa structure isomorphic to GK in Π. This makes G “up to automorphism”. You
need Lemma 1.1.4.ii of “Absolute Anabelian Geometry of Hyperbolic Curves”
for the interpretation G.
W: “Isn’t the content of Mochizuki’s theorem that, if Z of strictly
6Belyi type, then this is not the fundamental group of a curve?”
I would say no, this is not the content of his theorem. But I don’t knowwhat you mean here. Mochizuki’s theorem is a statement about morphisms not
objects. The begining of the claim is“given that Π → G and Π0 → G are augmented fundamentalgroups of hyperbolic curves...”There may be statements that allow you to classify when a group is the fundamental group of something but you will have to ask Daniel Litt or EmmanuelLepage or Jakob Stix about this. They certainly will be much more knowledgeable than me. By the way, this is what Mochizuki means when he calls things“bianabelian”.
Let Π = π1 (Z) ×GK ,g GK . I think you can run the G(Π) interpretation on
this thing. I haven’t worked out what this does... it is unclear if pr2 : Π → GKis naturally the same as to pG : Π → G(Π).
Also, Peter, don’t hold your breath waiting for new results in this direction. What I said is pretty much the extent of what I “know” about absolute
Grothendieck conjecture over p-adic fields.
Also, I think we can all agree whether something can change or not due
to Mochizuki’s consideration of potentially lossy functors is one of the moreinteresting parts of this story and it is important that we as a community clarify
the situation. I think Kirti’s emphasis on this is a good idea.
fourth point:
https://imgflip.com/i/3vr2vv
Full polyisomorphisms should be considered as “generic isomorphisms”. I
completely agree with this viewpoint. There are some instances where you havepolyisomorphisms which not full and these keep track of finite indeterminacy
rather than an arbitrary choice.
For better or for worse Mochizuki decided this is the language he wanted touse to describe these things. When things are omitted without justification thestatements made are no longer Mochizuki’s. We can get mad at him all we wantfor over questionable style choices but it doesn’t change his assertions. This, I
believe, is grounds for rejecting a paper, but it doesn’t disqualify the proof.
I want to give a couple comments on polyisomorphisms for the readers.
Comment A) First, I want to point out a really really really bad style problem
that comes with this choice of “polyisomorphism language”: Many commutative
diagrams involving full polyisomorphisms are tautologically commutative whichmakes many claims vacuous.
example: Theorem 3.11.iii.a, the ` ×µ prime strip commutativity statements.
These are everywhere.
This style choice forces the readers to search unnecessarily for non-full polyisomorphisms which, frankly, is a big pain (an example of a non-full polyisomorphism can be found in the definitions of the bridges of the Hodge Theaters forexample). This doesn’t make him wrong though. Just not the best expositor.
7

Comment B) If A and B are isomorphic objects then Isom(A, B) = f Aut(A) =Aut(B)f for any fixed isomorphism f : A → B. In my head I always “push”
the generic isomorphism into an arbitary automorphisms of one object. This ispretty tautological and Peter already does this but I just wanted to say that
applying this systematically allows you to reduce a lot of things and perform
computations.
Comment C) I have spent a bit of time literally identifying objects if they hada polyisomorphisms between them. Although you can actually get some prettyinteresting “global objects” this is not what Mochizuki had in mind. One example comes from the so-called full polyisomorphism mono-theta environments
and the log-linked fields. If you do this you end up identifying a bunch of rootsof unity in a bunch of different fields – this is well defined because the fields are
all log-linked. Anyway, my point here is that this is not what Mochizuki had inmind so you maybe don’t want to do this.
(I can talk at length about other style issues. Another example is invocationof interpretations “in Hodge Theaters”. This is one I feel that just sends readers
on a wild goose chase reading page after page of definitions. Things are typicallydefined by much less and in an optimal presentation one shouldn’t consider
superfluous structure when a reduct will do.)Ok, so what do generic isomorphisms do besides confuse readers?
First, they are intended to keeps track of automorphisms. Mochizuki’s theory is really an investigation of the behavior of interpretations under automorphisms and permutations of the interpreting structure. I say permutations
because sometimes they don’t respect the category and are maps of sets. The
stupidest example I can think of: Let G → G0 be a map of groups with kernel N .
Set theoretic permutations of G which fix cosets gN setwise induce the identity
on G0 .
Here are some example questions Mochizuki addresses: –How does the Kummer class of (an l’th root of a pullback of) the Jacobi theta change when you
take an automorphims of π1temp (X + )? –What about the evaluation points (conjclasses of decomposition groups)? –What types of automorphisms/actions stabilize what construction? (I am thinking about mono-theta environments here
and the purpose they serve) –What happens to a the measure space we construct when we take automorphisms of the interpreting G? (this is what Ind2
is)Second, generic automorphisms can serve as a sort of “poor person’s deformation space”. I think Kirti discusses this well in his updated manuscripts
so I’m not going to talk about this so much. I think, as Kirti has suggested,
we should be asking ourselves which of these generic isomorphisms are actually
representable. Emmanuel Lepage, if he is reading, might be able to say more
about the usage of full and essentially surjective functors from the perspectiveof Gerbes. He told me something about this once.
Third, regarding your “what is the image of an element under a full polyisomorphism”. This goes back to what the point of all these things are. We
are really investigating interpretations under automorphisms. At the end of
the day we are looking to construct a “multiradial representation of the theta8

pilot object” which is a region obtained by a procedure involving certain orbits. This is supposed to be 1) independent of any choices and 2) relatableback to our original Pq . We partially address (1) in my first manuscript with
Anton which should be available soon. You can find some stuff on my vlog
about indeterminacy diagrams. Anyway, the point of these comments is thatthe polyisomorphisms are Mochizuki’s way of dealing with choices, it is also safe
to make a choice but you need to make things independent of this choice.
fifth point:–Regarding your manuscript with Jakob, the construction as you have statedimposes two normalizations that are alleged to be simultaneously enforced whichlead to a trivial contradiction of the form A=B and A!=B. I think we all agree
on this. Mochizuki says this is a straw man. I can’t find these assertions in the
manuscript.
I do however think that statements of the form “All proofs that use X musthave property Y” could be very useful provided 1) we can make X completely
rigorous and 2) Mochizuki’s language can be pinned down in a way that makes
X verifiable.
Aside: Actually, this was part of my motivation for looking at these interpretations. If you can show that two objects are equivalent (e.g. Frobenioids
and pairs (Π, M )) then if there exists a proof using Frobenioids then there exists
a proof using pairs (Π, M ). These sort of reductions allow you to make assumptions about the structures that are used in the proof. This is what safely allows
us to get rid of unappealing constructions.
I personally find a lot of the language very hard to falsify/parse.
–Regarding the sentences “This is just the identification I was talking about:
Mochizuki considers infinitely many distinct copies of π1 (X0 ) and is only allowing himself to identify them along the full poly-isomorphism. We do not see
any diagram that commutes with this choice, but does not commute when we
identify them along the identity map, once taking all π1 (X) to be equal to theactual π1 (X0 ) for our fixed curve X0 .”First, I don’t think this is a faithful presentation of Mochizuki’s setup but
one thing I can say that might make you feel better: in this particular example
of bad non-arch primes setting X0 = X v (this should be a double underline)
then automorphisms of the fundamental group do all sorts of things to the zooof covers. All of those dihedral symmetries act, all of the Z symmetries act etc.
Also, all of the stuff that is not interpreted from a fundamental group stays
fixed. This seems particularly relevant in the context of “monotheta cyclotomic
synchronization”, but again, if I could finish the proof we wouldn’t be having
this discussion at all.
Sorry if I made any mistakes anywhere...

DL April 7, 2020 at 8:09 pm
Taylor — you’ve written quite a lot, but frankly I don’t really see why anyone
should continue discussing this until someone can point to a specific spot in9the IUT papers which defeats Peter’s objection. For example, if I understandcorrectly, you suggest some juice might be obtained by looking at maps of π1
which are not over GK . If so, where are such maps used in IUT? And Peter gives
a specific challenge — point to some diagram whose commutativity is rescued
by not making the identifications Peter makes. If there is such a diagram, whereis it?

David Roberts April 7, 2020 at 10:19 pm
not making the identifications Peter makes
I’m convinced there is no such diagram, because that is not how category
theory works. The whole issue with identifications (as in: demanding objects are
distinct copies vs having them be the same object) is a red herring. Mochizuki
doesn’t want to do it for spurious technical reasons that one can ignore (oralternatively, humour him and agree to play along). I’m more suspicious thatpeople could be thinking about objects that are living in different categories.
But what do I know?

DL April 7, 2020 at 10:32 pm
David — indeed there are objects in different categories being identified here(eg a curve and its fundamental group). That’s because we’re speaking English,
not trying to make formal mathematical statements. Formally, one might (forexample) look for a diagram which is not commutative but whose image undersome functor becomes commutative, say. That is something that can happen in“category theory.” In any case, I agree it’s likely there’s no such diagram.

David Roberts April 7, 2020 at 11:21 pm@DL
There are more people reading here than just experts on arithmetic geometry,or even mathematicians. I’m also writing for their benefit. I’m just pointing
out that your interpretation of Peter’s challenge is not really the question one
should be asking—or at least, the way it comes across to me.
But oh well. I should keep my head down, perhaps. Much easier to discuss
this not via the internet, but we are all locked away after all.

W April 8, 2020 at 8:45 am@Taylor Dupuy,
I would say no, this is not the content of his theorem. But I
don’t know what you mean here. Mochizuki’s theorem is a statement
about morphisms not objects. The begining of the claim is10

“given that Π → G and Π0 → G are augmented fundamental groups of hyperbolic curves...”Despite what DL mentioned, I just want to respond to this bit. . .
OK but if we’re asked if something happens and we’re trying to prove thatit doesn’t happen we can try a proof by contradiction.
We have a curve X over K and a map π1 (X) → GK . We fix K 0 and an
isomorphism GK → GK 0 which does not come from an isomorphism K → K 0 .
You ask, does the composition π1 (X) → GK → GK 0 arise from a curve X 0 over
K 0?
Suppose it does. Then we have two hyperbolic curves X, X 0 and an isomorphism between π1 (X) and π1 (X 0 ). Then Mochizuki proves that, if X is ofstrictly Belyi type, this isomorphism arises from an isomorphism between X
and X 0 . In particular, it implies the isomorphism on the Galois part arises froman isomorphism K to K 0 , contradicting our assumption.
In fact, given Mochizuki’s previous proof of the p-adic Grothendieck conjecture theorem, the new theorem (except for the part about open injections,maybe) is precisely equivalent to the statement that the “base changes” of fundamental groups of Belyi curves are not fundamental groups of curves, plus thefact that the geometric part is identifiable as a normal subgroup.

kirti joshi April 8, 2020 at 10:15 am
Dear Professor Woit,
This is to clarify the claims made about my paper in the context of yourabc blog-post. First of all let me say that my paper (referred to by Scholze
in his comments) is not a survey of Mochizuki’s work (though it may initiallyappear so because I state a number of standard results without proofs) andcontains a number of new and original results (Scholze agrees with me on this).
I start with one of Mochizuki’s ideas and I build upon it in my paper. This is thepaper which is cited in Scholze’s comments and which can be found on the arxiv(though the update is not yet ready). In our correspondence Scholze has agreed(mostly) with all the changes in the new version and we (i.e. Scholze, Hoshi andmyself) continue to correspond to resolve any persisting issues (of which there
are very few). These issues and changes, at any rate, do not pertain to main
results of the paper but to how the contents relate to IUT. Since comments
about the retracted section of the paper may add to the confusion in a topic
which is already quite complicated for many reasons, so all references to mypaper (below) will be to the forthcoming version.
To explain one of Mochizuki’s important ideas, let us begin with a classicalresult which says that there exist p-adic fields (i.e. finite extensions of thebasic p-adic field Qp ) which are not isomorphic but which have topologicallyisomorphic absolute Galois groups. For examples of such fields see my paper.
(In the 1990s this was refined by Mochizuki: a p-adic field is determined by
its topological absolute Galois group equipped with its ramification filtration11

(see the section: five fundamental theorems of ... in my paper) for references toproofs of these results).
Two p-adic fields with isomorphic absolute Galois groups have distinct additive structures (the multiplicative groups of non-zero elements of such fields
are even topologically isomorphic). So the additive structure is the one which
is changing (even as the absolute Galois group remains fixed). Because of
Mochizuki’s Theorem one can view the (upper) ramification filtration as the
Galois theoretic manifestation of the additive structure of a p-adic field.
One of Mochizuki’s ideas, simply stated, is to treat the p-adic field as adynamic variable while keeping its absolute Galois group fixed. Because of theabove remarks, this should be seen as treating the additive structure of the field
as a (dynamic) variable. If readers are uncomfortable with this idea, they can
simply think of allowing the p-adic field to vary while its absolute Galois groupremains fixed. This makes complete sense and comes with highly non-trivial
consequences as my examples illustrate:(1) I show in my paper (with explicit numerical examples) that two p-adic
fields with isomorphic absolute galois groups do not have the same different anddiscriminants (these are standard measures of complexity of fields in number
theoretic contexts). These examples can be easily verified by any one with acomputer equipped with SAGEMATH and importantly more can be found by
my methods.
(2) I show that if E/F is an elliptic curve over a p-adic field F and if L,K are
two p-adic fields with isomorphic absolute Galois groups and both containing F,then the base changed curves EK and EL (i.e. E considered as curves over L, K
respectively) do not have the same list of numerical invariants (in general). Thisis done by means of explicit examples computed using SAGEMATH (with noadditional programming needed). Notably my examples establish quite clearly
that the additive structure of the p-adic field controls many subtle invariantsof elliptic curves over p-adic fields. At any rate computing these examples does
not require any of Mochizuki’s theory.
Important realization on which my paper is based is this: the upper numbering ramification filtration is a Galois theoretic stand-in for the additive structureof the field and through this stand-in, the additive structure leaves its imprint
on Galois representations.
(3) Notably invariants in (1) and (2) are also the sort of invariants which
are crucial in Szpiro’s conjecture and my work shows that these quantities areaffected by the changes in additive structure and so Mochizuki’s idea of using
the variation of the additive structure to understand Szpiro’s conjecture might
have significant merit (there are several other new ideas in Mochizuki’s paper as
he has reminded me on a number of occasions). Note that I am not claiming that
this is exactly what happens in the context of IUT, but I am simply reporting
my observations that these quantities are not determined by the isomorphismclass of the absolute Galois group of the relevant p-adic field. I do not know
how to use my examples to illustrate changes in the specific context of IUT.
(4) I also demonstrate that the idea of changing the additive structure (which
I have called anabelomorphy in my aforementioned paper) can be used in the12theory of Galois representations. In this theory L,K are two p-adic fields whichhave isomorphic absolute Galois groups, so one can pass from representationsof GK to GL . This does not affect many broad aspects of the representations
as these two groups have equivalent categories of finite dimensional representations. However this operation of considering a GK representation as a GL
representation via any given isomorphism of these two groups does not preserve(p-adic) Hodge theoretic properties of a representation (for example an HodgeTate representation of GK may not remain Hodge-Tate when considered as a
GL representation via an isomorphism of GL with GK ). However I prove thatan important subcategory, namely ordinary p-adic representations is preserved
under this operation. This operation of viewing GK representations as GL representations is not the identity functor in general nor is it so on the subcategoryof ordinary representations. One can say such things because I also demonstratethat important numerical invariants of a Galois representations, for example its
Swan conductor changes (again there are explicit numerical examples which I
provide). The fact that this operation does preserve ordinary representations isof importance not only in Mochizuki’s work (which uses two dimensional ordinary representations arising from Tate elliptic curves), but also in the broader
theory of Galois representations (Wiles, Taylor and most results in the area sincethen). This result opens up the possibility of wider applicability of Mochizuki’s
ideas to other areas of number theory. For additional results readers are referred
to my paper.
These ideas, proofs and examples have nothing to do with the one theoremof my paper which I have admitted was incorrect (and the new version willappear on the arxiv on thursday or friday this week) and my error should not
be viewed by the readers as an example of what is wrong in this business. [I sincerely apologize to my colleagues and friends in Kyoto and Japan for theincorrect statement in the old version of my paper and my assertion that this(incorrect statement) was ”well-known to everyone in Kyoto”.] Again let me
be clear that my errors (if any) should not in any case be attributed as issues
emanating from Mochizuki’s paper.
(5) I explain in my paper that even though K,L have topologically isomorphic
absolute Galois groups, it is possible to communicate meaningful (arithmetic)
information between them. For p-adic fields this idea is due to Mochizuki.
There is no direct interaction between the additive structures of these fields at
any point.
(6) In my paper I have also pointed out the analogy between Scholze’s work
(deeply extending earlier work of Fontaine) and Mochizuki’s idea (see the section
on perfectoid spaces in my paper) and in our personal correspondence Scholze
has said that he sees no issues with my claims in that section (modulo minor corrections). In particular I point out in that section that Scholze’s work is founded
on a similar idea of changing perfectoid fields, perfectoid varieties (instead ofp-adic fields and curves over p-adic fields) while keeping the absolute Galois
group (of the perfectoid field) fixed (resp. etale fundamental group fixed). In
the parlance of perfectoid geometry this corresponds to moving from one untilt
to another untilt–see my paper for details (or Scholze’s paper). Deepening of
13

this analogy (as Taylor Dupuy and I hope to do in an ongoing project) should
provide further insights into this difficult topic.
(7) The following way of remembering Mochizuki’s idea may be useful:
A p-adic field wiggles and moves around in the isomorphism class of its
absolute Galois group. This wiggling is a (new) degree of freedom in number
theory and in algebraic geometry.
(This is illustrated in my paper with explicit examples and also see Mochizuki’s
paper and Hoshi’s work and other members of the Kyoto school).

Peter Scholze April 8, 2020 at 11:46 am
Dear Kirti Joshi,
thanks for chiming in here, and I’m sorry for concentrating on the parts ofyour paper/survey that were wrong.
The issue of non-isomorphic p-adic fields that have isomorphic absolute Galois groups is potentially interesting, and it is worthwhile to study which invariants are (un)changed under such an isomorphism. However, it is unclear tome how this enters into the actual content of IUT. In particular, your exampleof taking an elliptic curve E over a field p-adic field F and base-changing to
p-adic fields K/F and L/F with isomorphic absolute Galois groups is not relevant to IUT. Namely, π1 (EK ) and π1 (EL ) are not usually isomorphic, although
π1 (EK ) and π1 (EL ) both can be computed as pullbacks π1 (E) ×GF GK resp.
π1 (E) ×GF GL where all terms are isomorphic – but not the fibre product, as
the maps GK → GF and GK ∼= GL → GF are not the same. (This was theessential mistake in Joshi’s first version.)
Regarding (6), perfectoid geometry gives nontrivial examples of such relations, which however require a strong “softening” of algebraic varieties (topass to perfectoid spaces). Let me stress that perfectoid spaces have precisely
this flexibility of changing geometry while preserving topology (like π1 ), while
Mochizuki’s theorems I alluded to in the first comment prove that for the hyperbolic curves he considers, the geometry is determined by the topology (infact, by π1 ). It is clear that being able to change geometry while fixing topology
can be interesting – but Mochizuki is just not in a setup where this is possible!
Finally, regarding (7), while Mochizuki claims that his proof must use thisfact that p-adic fields are not determined by their Galois groups in some way, it
never actually enters – in particular, no construction of such exotic isomorphisms
is given or cited. In this sense, as you also say at the end of (3), this wholediscussion seems tangential to IUT.
Best wishes! Peter

Taylor Dupuy April 10, 2020 at 4:58 pm
W:
I agree with your observation. Nice.
Here is my summary (slightly modified).
14

Lemma. Let X be an SBT curve over a p-adic field K. Let Π = π1 (X) ×G,g G
where g : G → G is an outer automorphism (=not inner).
The map pr2 : Π → G is not isomorphic to the map between a fundamentalgroup of an SBT curve to its base field. (Maps between maps are taken to be
the obvious pair of maps satisfying the usual commutative diagram).
Proof. Suppose it is. First the map Π → G can be viewed as Π → π1 (X) → G,or pr2 (indexing starting at 1 and not 0). Call this map h. In this case we havea diagrampr1
Π −−−−→ π1 (X)




hy
y

g

G −−−−→ G
Where the bottom map is the outer g. If they were both SBT by Mochizuki’s
relative Grothendieck theorem + interpretability of the field we have that pr1
is geometric (the Grothendieck theorem is not just about isomorphisms but allmorphisms). But the bottom map is outer. This gives a contradiction. (I’massuming the p-adic relative Grothendieck for morphisms here... which I needto double check on).
I can’t parse your last remarks completely...
In fact, given Mochizuki’s previous p-adic Grothendieck conjecture theorem, the new theorem (except for the part about open
injections, maybe) is precisely equivalent to the statement that the“base changes” of fundamental groups of Belyi curves are not fundamental groups of curves, plus the fact that the geometric part is
identifiable as a normal subgroup.
Here are a couple more points (I’m not trying to be a jerk):a) What do you mean by “the new Theorem”?
b) We still don’t know if pr2 : Π → G is the same as Π → G(Π) where G is
the interpretation I referenced previously.
c) I think we only get that pr2 : Π → G is not the natural map from an SBT
curve to its base not that it is not the fundamental group of any hyperbolic
curve.
I believe we can resolve (b) and (c) with the Lemma below.
Lemma. pr1 : Π ∼= π1 (X) is an isomorphism. In particular Π is isomorphic to
the fundamental group of an SBT curve.
Proof. Let f : π1 (X) → G be the map from the fiber sequence. Since Π ={(a, b) : f (a) = g(b)} the kernel of the map pr1 contains elements of the form
(1, b). Since f (1) = g(b) this means b is in the kernel of g. Since g is an
isomorphism b = 1 and hence the map pr1 is an isomorphism.

15gThis means the composition π1 (X) → G −→ G is something that Mochizuki
considers but it not isomorphic (as maps) to some π1 (X) → GK of “geometricorigin”.
Also, I think you made an earlier comment about verifying the inequalities.
I personally think people should be looking at the Mochizuki’s inequalities after
3.12 but before Theorem 1.10 type inequalities. In my manuscript with Anton
we point to a couple places where improvements can be made; there seems to bea lot of room between the two inequalities. To do direct computations with Cor
3.12 it seems you need to work directly with Division Fields as in the work ofHarris Daniels, Alvaro Lozano-Robledo, and Drew Sutherland. Stuff like this:https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/
01/lozano-robledo_minimal_ramification_Rev1_v2.pdf(Maybe you can email me and we can talk about this more if you are interested.)
DL:
More comments on p-adic Grothendieck:
The existence of some isomorphisms of fundamental groups of curves overp-adic fields inducing isomorphisms of absolute galois groups of p-adic fields“not of geometric origin” seems to be related to the section conjecture but I
couldn’t figure out if there was an “obvious” implication. See the results here:
http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1892.pdf
I wanted to say something like “if the section conjecture is False then there
exists a hyperbolic curve Z over a p-adic field K and some automorphism σ ofπ1 (Z) such that G(σ) is not inner.” but I may be totally totally off here. Maybe
you can salvage this? I’m trying to get a sense of how difficult this should beby reducing this to a “really hard” problem.
Also, while the pro-p relative Grothendieck conjecture is true the pro-p section conjecture is False. This is a Theorem of Hoshi. I don’t understand thesecounterexamples but it is my understanding that Jakob views these as “accidents” or “lucky”. (pg 192 of his evidence for the section conjecture book.)
I’m going to postpone any more remarks about missing monoid structuresin the setup for a later post (if I’m going to say anything at all) because I wantto get it right for everyone.

UF April 11, 2020 at 9:02 pm
As far as I understand the main objection brought up here by Peter Scholze
is: Mochizuki considers infinitely many distinct isomorphic copies of π( X)’s,
but can not tell us what goes wrong if we simply identify all of them with one
another, and with π1 (X0 ) for some fixed X0 - there is no diagram that commutes
in his situation but does not commute under this further identification.
It seems to me that this is adressed in (C7) in Cmt2018-05. Mochizuki claims
there that if one (once and for all) identifies the various π1 (X)’s in one column(using the non-archimedean logarithms), then there is no switching symmetry
between the two neighboring columns. On the other hand, if one identifies

16the π1 (X)’s just as topological groups via poly-isomorphisms, then Mochizuki
claims that there is such a switching symmetry (i.e. some diagram commutes
at this weak level).
It seems to me that in the proposed proof both viewpoints are used: Theability to rigidify the relationship between the π1 (X)’s in one column whenneeded (e.g., I think, on both columns for the log-Kummer correspondence),
and then later the ability to forget about the rigid structure, to pass from the
LHS to the RHS.

Peter Scholze April 12, 2020 at 4:58 pm
As UF is trying to point to a specific objection to my manuscript with Stix,and it’s the first real try in this thread, let me try to answer this.
Actually, the objection is bizarre. If all your copies of π1 (X) are the same,then how can the situation be less symmetric? It’s totally symmetric. It’s
not symmetric if you also look at some other structures present in Mochizuki’s
setup, like the log-link: This just means that the logarithm map is a different
map from the one you obtain by switching source and target, which is selfevident (the logarithm is not its own inverse). It also cannot be salvaged byusing some other/indeterminate identification of π1 (X)’s – this would at mostchange source and target by some field automorphism. So it’s unclear to me how
the passage from X to π1 (X), and to π1 (X) up to indeterminate isomorphism,
is helping in this matter.

W April 13, 2020 at 9:49 am@Taylor Dupuy
I’m just saying that you can reverse the argument to an extent. Given
two curves X1 , X2 , with an isomorphism of their fundamental groups, if you
check that the geometric π1 of X1 is sent to the geometric π1 of X2 by thisisomorphism, then it follows that this isomorphism arises by “base change”
from an isomorphism of Galois groups of local fields. If this isomorphism of
Galois groups of local fields arises from an isomorphism of fields, then the factthat X1 is isomorphic to X2 comes from Mochizuki’s first p-adic Grothendieckconjecture theorem. If this isomorphism of Galois groups of local fields does
not arise from an isomorphism of local fields, then we can get a contradiction
if we know that base changes of strict Belyi type curves are never a curve. So
Mochizuki’s theorem that strict Belyi type curves are determined by their π1 s,
is basically equivalent, modulo these prior results (I think. . . ), to a special caseof your question.

Taylor Dupuy April 13, 2020 at 12:07 pm
W’s comment has insprired me to write a little bit more since he seems to17

be paying close attention (thank you W).
It seems to me that Peter S. has two claims:
1) One involving the diagram in his manuscript with Jakob in section 2.2.
That the setup here gives a contradiction is undisputed. The fact that the
setup is valid is disputed (see C8 of Mochizuki’s comments). It also seems very
dubious to me that the contradiction would be so tautological and I can’t findthese claims in the manuscript.
2) That one can identify all fundamental groups at bad non-archimedean
places using the identity and this will not change anything (maybe he wants toassert this for every structure in a “base Hodge theater”?). This includes acrossthe theta link and across log links. This is in say Footnote 8 of their manuscript.
Peter, how do you propose to show that (2) implies (1)? I also don’t understand the level of your assertion. Do you want to conjecture (2)? Do you want
to claim (2) as a theorem? Do you want to claim/conjecture that (2) is trueand implies some contradiction?
There is something in footnote 12 of your manuscript that I spent a little
time with and if it is supposed to claim this is the explanation of (2) implies (1)then it seems to be squeezing around 15 assertions into one footnote and omits
the explicit claim that (2) implies (1). In particular I don’t see how to recover
the normalization that needs to be simultaneously in force in (1) to derive theircontradiction.
Anyway, I still see Peter’s arguments as still being in the realm of a meta
argument.
Maybe it is obvious to him and everyone else how to apply the definitions
in IUT3 and convert these into proofs but this is not obvious to me. Again, forme 3.11 is made up of about 15 different assertions and uses a large number ofdefinitions that I don’t have at my fingertips...

Peter Scholze April 13, 2020 at 4:25 pm
Let me take Taylor’s comment as an opportunity to more clearly state several
related but distinct criticisms that are explicit or implicit in my manuscript with
Stix, and the previous discussion on this thread.
(1) The non-necessity of passing from X to π1 (X)(2) The non-necessity to replace π1 (X) with infinitely many distinct copies
of it(3) The inconsistency of the identification of various copies of ordered 1dimensional real vector spaces
Off these, only (3) points to (what seems to us) an actual mistake. (1)is about the question whether anabelian techniques – which are supposed to
be at the heart of the matter – are of relevance. (2) is about whether thehuge diagrams that Mochizuki considers are actually relevant to the argument.
Neither (1) nor (2) alone would really falsify anything; but if non-necessary, notmuch is left of these manuscripts.
As only (3) is about an actual mistake, we focused on this in our manuscript.

18

(1) is Remark 9, and (2) is Footnote 8 in my manuscript with Stix. In my firstcomment here, I concentrated on (1), while point (2) was addresses in my second
comment in response to the fourth point raised by Taylor. [I should apologizethat while my current (2) corresponds to the (2) of Taylor’s last comment, it is
my (3) that corresponds to his (1).]The reason I brought up (1) and (2) here is that if one only stresses (3),as we did in our manuscript with Stix, then it may seem plausible that we
simply misunderstood something at this point of the argument, but some quite
powerful machinery had been built and one could plausibly finish the argument
differently. However, (1) and (2) mean that quite the opposite, all machinerythat is in place seems to have no power.
To me, (1), (2) and (3) seem logically independent. For (3), we explained
everything in Section 2.2 of the manuscript with Stix: What the various ordered 1-dimensional real vector spaces are, what identifications one wishes todo, and that these identifications lead to nontrivial monodromy, i.e. are inconsistent. The loop one has to take in order to get the monodromy is rather
large (6 identifications). I have seen no convincing argument that this nontrivial
monodromy does not lead to problems. Of course, one could just decide to cut
the loop at any point, but then one has to make sure that the argument neveruses that disallowed identification. In particular, one has to decide in advance
where to cut – if you know where, please let me know. (That in various parts of
the argument only small, locally consistent, parts of the diagram are relevant,does of course not help as the argument altogether must be consistent. So the
only way to achieve a consistent argument is to decide not to use one of theseidentifications.) I agree that this contradiction is very tautological. It seems
the more surprising to me that it can be altogether neglected. (Yes, Mochizuki
can’t possibly mean this. But regardless of what he means, this inconsistency
is simply there!)
About (1), Mochizuki’s anabelian theorem states that relevant X’s are equivalent to relevant π1 (X)’s, which is as much as can be asked for regarding a proof.
About (2), it’s hard for me to prove this (and it’s not required for themain point, (3)), as Mochizuki of course asserts that this is false, and there arethousands of diagrams one might have the occasion to look at, and I can’t look
into Mochizuki’s mind to find all of them. What I can say, and this is basicallyFootnote 12 in my manuscript with Stix, is that we checked that Theorem 3.11
holds up with (2) in place, and in fact is completely tautological, so if anything
Footnote 12 is meant to “prove” (2) to the extent this seems possible. Moregenerally, (2) seems extremely plausible to me, and with all diagrams I’ve looked
at in Mochizuki’s papers it was holding up; and if you doubt it, you can justanswer to my challenge of pointing to a single diagram whose commutativity is
rescued by allowing some indeterminate isomorphism.
I do not claim that (2) implies (3) or anyway leads to a contradiction, although my recollection is that Mochizuki said that (2) alone would contradict
his papers – he agreed that it is impossible to think that his argument mightwork if all of these copies are simply the same. (Maybe the idea is that oneof the identifications we use to get nontrivial monodromy in (3) is omitted by19having distinct copies of π1 (X)’s. But that’s not actually the case.)
As I said, it’s very easy to convince me that (2) is wrong: Just point to
one diagram whose commutativity is rescued by allowing this indeterminate
isomorphism of π1 (X)’s.

David Roberts April 13, 2020 at 8:31 pm
I’m sorry to keep banging on about it, but Peter S’s (2) is an artefact ofa weird approach to diagrams in categories that Mochizuki is using. Reading
M’s 2018 Report closely, he claims things like that you can’t define manifolds if
you build them out of colimits of diagrams where the objects are all the same
‘copy’ of Rn (this is LbEx5). Or that you incorrectly calculate the perfection ofa ring if in the usual sequential colimit you don’t create separate copies of thering in advance (this is LbEx3). This is patently absurd, but makes sense if one
assumes that diagrams *must* be injective functors, or rather, literal subcategories. Recall that M assumes that he is identifying isomorphic functors, so thatthe concept of a diagram qua functor is severely underdetermined. Working up
to isomorphism like this, and replacing a diagram with one that produces an isomorphic (co)limit, one can safely assume that diagrams are subcategories—but
it is super weird, and it took me ages to realise that was what he was thinking.
This is why he talks about things like “forgetting histories”, because he is thinking that you need to somehow create fresh, distinct copies of objects in order
to not collapse the subcategory down, and thereby give a different diagram. So
when someone versed in standard category-theoretic language says “let’s identify these objects”, he seems to hear it as “let’s collapse this subcategory tosomething trivial”. And when he says “I need distinct copies”, it seems totally
weird and unmotivated. So when I look at LbEx3 in the 2018 Report it looks
like the sort of mistake a student would make, when learning category theory
for the first time. The problem is his conceptions of basic notions seem to be so
idiosyncratic that without a serious translation filter, what he is saying seems
to be completely off the wall.
If one takes category theory seriously, and DL poked a bit of fun at me forthis (we talked privately afterwards), then one can *at the level of foundations*forget the equality predicate on the objects of categories, so that the equality
or otherwise of random objects of a category is not even something you canconsider. Alternatively, one can pass to a skeleton of the category at hand, inwhich case you just don’t have distinct copies at hand, but *nothing breaks*.
So there can be zero mathematical effect of the whole issue of distinct copies
or otherwise, it is merely a psychological crutch. Once one realises this, theactual mathematics can then be discussed, for instance something like creating
a formal colimit of a diagram all of whose objects are π1 (X) (or whatever), withappropriate gluing maps. But this is not what I wanted to address, and is outof my sphere of expertise.

20

UF April 13, 2020 at 10:16 pm@Peter Scholze
Thank you very much for addressing these comments of Mochizuki relevant
to your (2). Let us consider the simplest situation where we just consider twoneighboring columns of log-links and disregard for the moment the theta-link (Ithink we agree compatibility with theta link at most creates more trouble).
Then even in this simple situation (omitting theta-, but keeping log-links)
Mochizuki seems to claim (C7) in Cmt2018-05 that if one rigidifies the π1 (X)’sin the columns (by identifying them using identity maps, making the logarithms Galois-equivariant), there is no “switching-symmetry” permuting thetwo columns.
As you say, this seems quite bizarre, since one has two columns of isomorphic
data, so why should one not be allowed to switch?
I think a way out may be the following: It seems quite likely Mochizucki uses
“switching-symmetry” in a technical sense, synonymous with “multiradiality”
of some algorithm reconstructing the data (here a rigidified column of log-links)
at hand from some choric data, as he often does, compare e.g. [Alien, p.51].
His statement would then mean that if we rigidify the vertical columns, thenthere is (unlike in the non-rigidfied case) no multiradial algorithm to recover thiscolumn from certain choric data. This does not sound so absurd anymore (tome).
Now which multiradial algorithm does he mean here? I would suggest it
may be the multiradial algorithm in [IUT III, Cor 2.3 ], more specifically, thefirst part of 2.3 (ii) which concerns its compatibility with log-links. Note thatclose by, [IUT III, Rem 2.1.1 (ii)] the issue we are talking about “why π1 only
upto indeterminate iso?” is discussed. For further discussion see also in [IUT
II, Rem 3.6.4 (i)].
In any case, I agree this is an important issue to track down.

Taylor Dupuy April 14, 2020 at 5:30 pm(written before seeing next comment)
I think we are mostly on the same page now. No worries about the indexing.
I’m going to use your indexing in what follows.
–Regarding (3): The “cut” is supposed to occur at identification of the thetaside of the theta link; in that global realified frobenioid we don’t normalize that
Picard group according what you would want the degree of the theta pilot object
to be. For the interested reader, details are in my manuscript with Anton after
we introduce theta pilot divisors. I think Mochizuki includes this in one of his
responses too.
–Regarding (1) and (2) not implying a contradiction (directly): we agree. I
would consider (1) and (2) open as well. We agree on that too. Also, as I’m sureeveryone will agree, statements need to be pinned down a bit more and directlytied to what Mochizuki has written. So I think even *exact* statements of (1)and (2) are also open. I will say a little more about (1) below.
For reasers who want to look at this, the obvious thing to do is to isolate a21

single claim in IUT and just start running with it, then build out from there.
–Regarding not using automorphisms of fundamental groups: we shouldobserve that without indeterminacies, there are no indeterminacies. This is
tautological, yes, but indeterminacies appear in the statement of Cor 3.12 andwithout them we are really talking about something else entirely.
Also, this all goes back to how the indeterminacies allegedly afford us theability to compare the “volume” of the hull of the multiradial representation of
the theta pilot region with the degree of the q-pilot divisor (definitions can befound in my paper with Anton)—this is the infamous “Mochizuki switcharoo”.
I could talk more about this more but right now but I will be stating a bunchof (useful) isolated facts that readers would need to assemble for themselves (ifit is even possible to do so). I think I might be burned at the stake for this as
well as run the risk of making mistakes in public!
–Regarding (2): I think we need more language about “infinitely many
copies” but basically I’m of the same mindset as David on this. I think everyone agrees we shouldn’t think about ”different copies” so much in the same
way that you don’t need two copies of R to talk about R2 . I’m not sure this iswhat you or Mochizuki means though Peter. On the other hand there certainly
are cases where you need multiple monoids.
–Regarding (1): we basically agree here. I think groups may be much moreconvenient and I think the representations are definitely important. The statements Mochizuki makes involve etale theta and the reconstruction of evaluation
points and the representation of Aut(π1temp (X v )) on these interpretations (herewe need to take an analytification or formal scheme with log structure). Technically speaking, I suppose Aut(X v ) acts on the same objects but maybe it isn’t
as easy for me to see these actions.
–Getting your hands dirty in the definitions like UF has begun doing is
the way to proceed in these investigation. I can give some basic definitionsof “switching” but I believe they are not adequately developed for discussing
IUT3.3.11. Maybe we should take that offline? Whatever you guys want.
Mochizuki discusses the formalization of multiradiality in a remark following
IUT3.3.11 but that remark mostly says “you can do it” without any details as
I recall.
–As a side remark, and I know Mochizuki would hate me saying this, it does
feel like mind reading at times. I strongly agree with this sentiment. He tries
though. I believe attributions of malintent are misplaced. But still, lots of mind
reading.

Peter Scholze April 14, 2020 at 5:17 pm
David Roberts: Thanks for your thoughts on this!
Actually, something related happened in our Kyoto discussions: We realizedthat we could not get on common grounds regarding the issue of whether one
needs separate copies of a ring to form its perfection, so we decided that we
simply have different psychological crutches (as you call it) on this, and that22

we better focus on some actual mathematical statement where related issues
indisputably become important. However, no such focal point ever appeared,
despite us going through the essence of the IUT manuscripts. So it seems to us
that to the common mathematician, his whole big log-Theta-lattice essentiallycomes down to one Hodge theater – which is really just the elliptic curve you
started with (the category of Hodge theaters is equivalent to the category of
elliptic curves isomorphic to your given one) – together with the p-adic logarithmmap (“the log-link”) and some isomorphism (“the Theta-link”) of two copies
of a local Galois group acting (trivially) on a monoid isomorphic to N (I’mobviously simplifying, but not too much – I’m basically considering only one
bad place, while you have to consider all places, but I’m already telling you
about the most interesting place). The generator of this monoid N is one time
regarded as the value of the Tate parameter q, one times as the value of the
Theta-function (a collection of such values, really, but never mind). But ofcourse this isomorphism of abstract monoids is totally incompatible with these“interpretations”. Initially, one might think that Mochizuki claims that there
is some isomorphism of the pair of (local Galois group acting on local units)that takes the q-value to the Theta-value – this would obviously have great
consequences, and would probably require the use of exotic isomorphisms of local
Galois groups – but this is just not the case, one can easily give counterexamples;and it can’t even be true locally “up to blurring”, only a global statement,
averaging over all places, can be true. The isomorphism is just on the level of(local Galois group acting trivially on monoid isomorphic to N) and the relationof this monoid isomorphic to N to the local units is completely external.
So for all we can see we simply followed the procedure you suggested, andreinterpreted his distinct copies in the way usual mathematicians think. We
recorded the outcome of this in our manuscript: You can read there the detailsof what everything is, in particular that his log-Theta-lattice really boils down
to essentially the simple data above. How could one possibly go from here to
any nontrivial result? Of course, as also W suggests, all of this is much harder
to see through in his language.
@UF
The remarks from IUT that you cite make heavy reference to his paper on
etale theta-functions, which seems to play a key role in the IUT papers. This
paper gives some neat algorithm to start from the fundamental group of a oncepunctured elliptic curve with bad semistable reduction, and recover its Tateparameter q and some Theta function; I forget the details. While this is all
good and well, I don’t see the relevance: Mochizuki’s more general anabelian
theorems, discussed previously on this thread, tell you that from the fundamental group you can simply recover the whole curve. In these comments ofMochizuki that you reference, Mochizuki is discussing some nitty-gritty detailsof this algorithm, but this seems completely besides the point if you just remember that relevant π1 (X)’s are equivalent to relevant X’s, so of course you
can recover all invariants of X, and you can do so functorially in π1 (X).
Generally, a point seems to be made that Mochizuki’s algorithms have some23

magic power and that really the content of the algorithms is critical, so in thecontext of the previous paragraph, it would matter in some way how I invert thefunctor X 7→ π1 (X) using some explicit construction (or that I don’t actually
invert it but only read off Theta-values using some other roundabout algorithm).
This seems very surprising to me.
Regarding Mochizuki’s algorithms, let me add that I was surprised that thefollowing procedure counts as an algorithm for Mochizuki.
In IUT-3, Theorem 3.11 (i), an algorithm is discussed, that does the following. The input is data concerning only p-adic fields; it is basically a profinitegroup isomorphic to the absolute Galois group of your given p-adic field. The
output is something like the Theta-value of the elliptic curve you chose at the
beginning of the IUT papers (cf. part (b) of that data).
How is this possible? The input data doesn’t even know anything about theelliptic curve! This is completely magic!
The resolution is that the elliptic curve has indeed been fixed once and for
all in these papers, and so of course you can produce that Theta-value – simply
look at the elliptic curve you have fixed, and take its Theta-value.
Of course, there is some packaging done around this, but this is the essenceof this “algorithm”; Mochizuki confirmed this.

Taylor Dupuy April 14, 2020 at 6:07 pm
Quick comments:–Peter, I think the N’s should be regarded as embedded inside a monoid
with enough roots of unity and nth roots to do Kummer theory.
–You definitely need more than one part of the “Frobenius-like” objects ofthe Hodge Theater — the abstract monoids. I also think you should think about
overloading free variables; this thing is some sort of F’d up quotient of a freeconstruction. This of course isn’t precise.
–The term ”algorithm” is trash. I think it is a big part of the problem. Also,
we should note that has been going on in anabelian geometry for a very longtime and it isn’t a “Mochizuki thing”. For the most part I have found thatalgorithms = interpretations in infinitary first order model theory. And I very
very emphatically agree with you that it does *not* suffice to treat “functorial
algorithms” as functors alone, you need more. This is certainly omitted in
Mochizuki’s exposition.
Also, this interpretations stuff also goes off the rails a little bit...
1) you need to be able to consider structures up to automorphism as well.
2) There is also a backwards version of this where he considers “lifts” ofstructures that interpret lower structures. I talk about this a little in my firstmanuscript with Anton.
–Regarding 3.11.ii this is an independence results. He is saying it doesn’t
matter which lift of all these things that interpret the absolute galois groups “atthe bottom” you take. This thing you construct independent of the lift (by theway this points to the automorphisms of fundamental groups you were looking

24for).
Here is a funny observation: there exists a “shitty multiradial representation”
where you can take the union of all images of the theta pilot region in measurespaces cooked up from absolute galois groups of p-adic fields at v ∈ V . This is
ALSO independent of the lift. Proving that it is “multiradial” is tautological.
Yet, this one doesn’t use log-links and it isn’t claimed that this ”shitty
multiradial representation” can be related back to the minimal discriminant.

Peter Scholze April 14, 2020 at 7:05 pm
Dear Taylor,first, thanks for explaining where to do the cut. In your comment, I don’t
actually understand where you want to cut, so I’ll have to look at your note with
Hilado. (In my manuscript with Stix, which direction does the isomorphism go?
Horizontal/slanted? In the lower or bottom half? In the left or right half?)
About (2) not leading to a contradiction, you actually made me realize thatprobably it does: I don’t see why part of Mochizuki’s indeterminacies, I believe
for example (Ind2), is necessary, but omitting it leads to a form of ABC that isprovably too strong. I agree it would be hard to “prove” a strong enough formof (2), but I think the burden of proof is on Mochizuki here, to show where theargument needs indeterminacies like (Ind2) – which is basically the issue (2)we’re discussing.
Reading your first comment above, I actually don’t really see where we
disagree.
Regarding your second comment, the relevant N’s are manifestly not part ofmonoids in which you can do Kummer theory! Yes, Mochizuki includes someextra factor, but that’s just along for the ride.
About the last bit, to be sure we’re on the same page: I was talking aboutpart (i), not (ii), of Theorem 3.11.

Peter Scholze April 14, 2020 at 7:37 pm
OK, I quickly looked at your manuscript. If I understand it right, that’s theupper left slanted arrow? In our terms, that’s the difference between the “abstract” Theta-pilot – a generator of an abstract monoid N – and the “concrete”
Theta-pilot – the actual Theta-values of your elliptic curve. (Mochizuki seemed
to conflate the two originally, or so it seemed to us.) If you cut there, thenthe Theta-link no longer links anything to actual Theta-values! This of course
removes any inconsistencies, but it also removes what’s supposed to be the key,namely the identification of q-values with Theta-values, in some form.
On the other hand, I read say on page 140 of IUT-3 that Mochizuki considers the Theta-intertwining, which I believe simply means this identification of
abstract Theta-pilots with concrete Theta-pilots. He wants to be very carefulwith using this etc., but I do believe he wants to (and has to) use it somewhere. So I don’t think you can simply cut there. I think the closed loop that25

Mochizuki discusses on page 143 of IUT-3 is also relevant here. See also on page
144 the simultaneous Theta-intertwining and q-intertwining he wants to have(up to indeterminacies etc...).

Taylor Dupuy April 14, 2020 at 8:37 pm
Yep, the theta pilot doesn’t map to the actual theta values *on the thetaside*. On the q-side it does.
so yeah, you don’t normalize the degree on the left hand side of your diagram.
It is like we are all saying you can’t something be equal to two different things
at the same time. . .
Also there is only *one* copy of the real numbers.

Taylor Dupuy April 15, 2020 at 5:17 am
Peter: Below is a more detailed response.
To recap, you made three posts. One in response to David and UF, andthen two smaller ones. I’m going to address the newer ones first and then go
back and address those older comments.
Responses to Newer Comments
Regarding the “cuts”: I think I addressed this. Let me know if you wantto talk about it more. I think if you are not taking two log-linked strips ofHodge Theaters and only have one theta monoid it is going to get rough. He
wants all the computations in on particular Hodge Theater in the log-linked
strip on the q-side to be the “usual normalization” (sorry for referencing suchlarge objects, I know this is super abusive... I’m sure there is a better way totalk about this than to make blanket references to Hodge Theaters — reallythe relevant “official constructions” here use the ≺ prime strips (I think) in the
Hodge Theater... that is not a fun reference chase.)
Regarding (2) [infinitely many fundamental groups]: So I think we need toclarify this what you mean by (2). I’m not sure which indeterminaciees you
want to get rid of. At some points I see that you are thinking about using ”onefundamental group” and at some points you are saying “use one Hodge theater”.
I think these have different consequences — one is about representations on
monoids and one is about the groups solo. Are you wanting to get rid of ind3
and this log-kummer correspondence? What do you want to kill exactly?
I agree with you that tracing the proof for simplifications which removes ormodifies inequalities to the point where they are false is a good strat for finding
flaws.
Aside: I will always agree with you that the burden of proof is on the writer
to explain things. In modern arithmetic geometry there is too much flexing on
the reader. IUT is a bit of a weird flex in some ways.
Regarding N’s and Kummer Theory in the I ×µ prime strips: As a recap:
You stated that they are isolated, I said they have a Kummer theory. I want to26

clarify: technically you are correct the monoids in a components at a bad place ofI ×µ prime strip are just N’s. BUT... they are interpreted structures. So theycome from other structures where it does make sense. Note also that in isolation,
there is no action of the N0 s on the corresponding copies of Ov×µ ⊗ Q. (Just
as a reading note: Mochizuki sometimes calls this his “holomorphic structure”.
I think he also uses the words “embedding” or “link between unit group andvalue group portions”). I can speculate a little about what is going on on eitherside of the theta link if you want.
*Subremark. This is sort of an ongoing theme I want to highlight: Givena structures A and B there may exist two ways of interpreting B in A (called
them B1 and B2 ) which are not equivalent. This for example could for example
be distinguished by the representations Aut(A) → Aut(Bi (A)) (I’m not saying
this is the case here, I’m just trying to give a concrete example).
*Subremark. Sometimes I find it useful to think about there being a single
I ×µ strip that lives in different “charts” (after applying the correct automorphisms). I am not sure how helpful this is, but it is kind of fun to think
about.
Regarding My Remarks on 3.11: A quick summary: You had stated how
ridiculous it is to construct anything involving Tate parameters from absolutegalois groups of p-adic fields. I said some words about lifting G’s to an interpretation so we view G as G(Π) for some fundamental group Π and said these
constructions are uniform in Π (let me say this is a cartoon picture right now,
we actually need a lot more structure on this. In particular what I had in mind
was a lift of the structures “at the bottom” of 3.11.i to log-linked collectionsof prime strips — I made need to modify the structures in this statement tomake it exactly correct). I had made some remarks about 3.11.ii being an independence result. Let me clarify, I was talking about UΘ ⊂ L`,et (the “coarse)
multiradial representation of the theta pilot region” inside the so-called “monoanalytic etale version of the log shells” all put together) being independent of
the lift. Let me think about 3.11.i. I need to refamiliarize myself with thisRLGP bullshit. If we are really going there right now then I think we need to
introduce “abc-modules” (the primary data discussed 3.11.i and 3.11.ii). These
are sort of pre UΘ structures (used to construct UΘ ). I need to think about
3.11.i and get back to you. Also, we are in territory where I am pretty shaky.
So... no promises...
Regarding “Theta Interwining”: Can you give me a reference with respect toa Theorem/Remark/Environment of some kind? I think we are reading different
versions of the manuscript. (Let me just make one side remark— there needs
to be a stable version of these documents somewhere in print; it has obviously
become an important historical document. I’m not saying where and how, it
just need to be immutable.)
Response to Response to David Roberts and UF
Let me just say, I don’t think your simplifications were an unreasonable guess(and I want to clarify to the readers that this is actually not what Mochizuki
meant in his setup. I’m just saying the whole thing is so crazy you need to start27

somewhere.) I think we all agree though if you take one Hodge theater it won’t
work as per your manuscript with Jakob.
Regarding Magic Powers: I really like this comment. It is a very interesting
thought regarding looking at the “conjugate constructions” from the perspectiveof curves. It does look funny. What is in this fundamental group sauce?! Hereare some possible explanations:1) There is more to what is going on than just fundamental group vs curves;
Mochizuki uses the monoids and cyclotomes extensively. In particular he applies
many ”orthogonality” results.
2) Fundamental groups are first order structure: they are a topological spacewith a binary operation. The formalism of interpretations applies here (maybe
by slightly increasing the signature), but you get to put everything in this nicebox where represenations become automatic and you get to see how everythingvaries with respect to automorphisms. (Full disclosure, Mochizuki objects to thelanguage of model theory for this stuff, he thinks it is unnecessarily complicating
things. This is a matter of taste maybe.)
I think we need to unpack the comment “The resolution is that the elliptic
curve has indeed been fixed once and for all in these papers, and so of course
you can produce that Theta-value — simply look at the elliptic curve you have
fixed, and take its Theta-value.” What Mochizuki means is ‘fixed in initial theta
data’ usually. I also want to repeat that I object to the usage of a single Hodge
Theater. If H is a log-linked collection of Hodge Theaters (which we view asa single massive infinitely sorted structure) the definable set we need is in like2
H (I said “like”).
Regarding the comment: “it can’t even be true locally ’up to blurring’, only
a global statement, averaging over all places, can be true....” This is correct.
His claim is a purely global statement. But... it is even WORSE than this! You
need to take the hull and then take the volume and only then does he claim you
see the comparison.
In this comment you also reminded me of some things concerning the shapeof the inequality. I’m not going to do it now, but I’m going to make a note totry to talk about:
2*Ind2 and moving around q j /2l – why you need Ind3
*Analytic number theory, what happens at large discriminants, and toy phenomenology for the inequality.
IOU: 3.11.i discussion.

UF April 15, 2020 at 9:04 am
I now think a good reason to for not identifying π1 ’s along log-links is asfollows: (It is very similar to [IUT-2, Rem 1.11.2 (ii)] for the theta-link; I am
less precise here, taking the relevance of some reconstructions on faith, but it isa intuitive story, and precision can come later)
First, let us step back and look at the log-theta-lattice: It consists of Hodgetheaters and the log/theta-links as maps/functors between them. Now a formu-28

lation of (2) is: Why do we, say for the log-links, not just take the “identity”
isomorphism between the relevant π1 ’s instead of the full poly-isomorphisms
between them? Surely it is also a completely valid map between the theaters?
Yes, but: let us recall what we want to do with this log theta-lattice: wewant at “suitable times(?)” glue the Hodge theaters along these links and thenapply certain multiradial algorithms.
I think that if we would glue with the “identity” on the π1 ’s (and all theother prescriptions of the log-link), then this gluing will be inconsistent/not
well defined (when combined with some multiradial algorithms), in some vaguely
similar sense as if you try to define a complex structure on a manifold by charts,
but in some region the procedure leads to two different, incompatible complex
structures, i.e. the transition map is not holomorphic.
Indeed, in one Hodge theater, the π1 is not independent of the rest of thedata; it acts via the quotient to the Galois group on the monoids. Thus π1 is“related” to the monoids, and it is not apriori clear we are allowed to glue two
Hodge theaters by gluing π1 and the monoids separately in whatever way wewish (and then follow certain multiradial algorithms). In the above analogy, acertain region of the Hodge theater is supposed to be “determined” jointly by theπ1 and the monoids. The multiradial algorithm we consider is roughly as follows:
The action of π1 determines on the (from the Galois group reconstructed) local
monoids (with 0) in addition a ring structure (a “holomorphic” structure). The
output is roughly some étale version of some of the data originally present in the
Hodge theater, and the reconstruction algorithm yields a canonical isomorphism
between the original data and the reconstructed one (see e.g. [Alien, §2.12] foressentially such constructions).
If one now glues two Hodge theaters along a log-link (with in addition“identity”-gluing for π1 ’s), then this yields no consistent holomorphic/ring structure compatible with the above algorithm on some region: Namely, consider theregion of the ring (local units), which is glued via log (the actual log part ofthe log-link) to the other Hodge theater; this yields one definition of “holomorphic” for this region. On the other hand, we can embed this region in thereconstructed monoids and use for the “holomorphic” structure on the reconstructed monoids the one induced by π1 on both Hodge theaters and glue by
the “identity”.
The above roughly describes two incompatible holomorpic/ring structures
on some part of the glued Hodge theater; they are incompatible, “because thetransition map, essentially the log, is not a ring map”. Such a situation ispresumably avoided by taking the full polyisomorphism between π1 ’s, where thelog map on the units alone defines the holomorphic structure on the relevant
region.

Peter Scholze April 15, 2020 at 2:05 pm
Dear UF,
thanks for your answer. (Taylor, I will answer in a separate comment later.)

29

Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is why you have to consider
source and target up to some indeterminacy.
The only way I see to make this into a valid thing is if the indeterminacy
somehow allows you to map the logarithm map into a map of rings – after all,this is what you critized in the first place. But any automorphism of π1 ’s will
only ever replace source and target by isomorphic rings. So the logarithm will
still not be a map of rings. Note that I’m simply repeating what I already said
in my second comment in this thread.

Peter Scholze April 15, 2020 at 4:31 pm
Dear Taylor,addressing your last comment.
First half: I was refering to the current version on his webpage. I realizethat this is dangerous, but it’s the only thing that at least currently makes
it possible for everyone to follow. In any case, to me it sounds like you aresaying that the Theta-pilot has nothing to do with Theta-values. Well, strictly
speaking (modulo minor issues) what Mochizuki does is the following: He takes
the Theta-value (some nonzero p-adic number, not a root of unity), and takesthe (multiplicative) submonoid M of the p-adic numbers generated by it, andthen considers M as an abstract monoid. As an abstract monoid, it is thenof course isomorphic to N, necessarily canonically so; its generator is referred
to as a Theta-pilot. Does the abstract monoid M know anything about the
Theta-value? Of course not! But that is all that ever seems to enter the storyof the Theta-link. At some point you do need to remember the interpretationof the Theta-pilot in terms of Theta-values – this is what Mochizuki calls the
Theta-intertwining, and it plays a key role as I tried to reference. So I don’t
think that Mochizuki simply cuts our diagram there.
Now David Roberts previously mentioned that Mochizuki seems to have anonstandard point of view on what a mathematical object is, requiring separate
copies when they are not actually needed; he often refers to some “history” ofthese objects. Now maybe one could try to argue that because M was built outof Theta-values, it (the abstract monoid) still knows something about them, thatcan be even transported via an isomorphism of abstract monoids. (You madesome similar remarks in your paragraph on N and Kummer theory, referring to“interpreted structures”.) But that’s evidently not the case. (That the abstract
monoid M knows something about Theta-values might be debatable if for you
an abstract monoid is really encoded – as it is in ZFC – in terms of its actual
set of elements. But even then, this structure (“the interpretation”) is clearly
not transported by an isomorphism of abstract monoids.)
About the indeterminacies: I’m focusing on (Ind2); (especially) the partconcerning automorphisms of local Galois groups (acting on local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity
isomorphisms, but does commute when you conjugate by an automorphism of30

a local Galois group (acting on local units). Why do you need that, except
because “you need to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that? (There was actually a point
where Mochizuki was surprised that by “forgetting the history of a group” onestill has more than a group up to isomorphism: Namely, a group. That thedatum of a group is strictly more than the datum of a group up to isomorphism
seemed new to Mochizuki. I believe this is the (psychological) main reason heconsiders these full poly-isomorphisms. But really, forgetting “the history of thegroups” you still have groups and they may still have natural commuting isomorphisms between them. Of course, for groups up to isomorphism you can’t
ask for natural commuting isomorphisms between them, then you only have“full poly-isomorphisms”.)
Regarding the rest of your post, I don’t see how you’re really objecting toanything I said.

UF April 15, 2020 at 4:57 pm@Peter Schoze
I agree that the above should be surrounded by big caveats about the natureof the effects of passing to full poly-isomorphisms and I may interpret thoseincorrectly.
What my post above attempts to show, is: if passing to poly-isomorphism
has the effect of doing no gluing/no identification of ring structures (arising from
π1 , just the gluing from the actual log map), then the only gluing left is theactual log map, which gives one global chart, and no transition functions needed,essentially(?) since just one chart. I before never really seriously considered thatfull poly-isomorphism could have the effect of “no gluing arising from this part”(instead of “choose your favourite gluing”), but a similar thing is (I think)asserted for the case of the theta-link in [IUT II, 1.11.2(ii)], where the message
is: full polyiso on π1 means no sharing of ring structure arising from this.
Clearly this does not show by itself it really works like this, but it seems to
reduce an argument for the log-link to some degree to an analogous one for thetheta-link.

Peter Scholze April 15, 2020 at 5:10 pm
Dear UF (and Taylor and everyone),
I won’t comment any further here on statements of the form “well, maybe
Mochizuki actually meant (vague statement)”.
A few comments up I summarized the situation with claims (1), (2) and (3).
I have seen no valid objection to (1) and (2), and (2) alone would lead to acontradiction (as one gets too strong a form of ABC). To (3), Taylor indicatedwhere to cut the diagram, but I really don’t think this is what happens, as
this would isolate Theta-pilots from Theta-values and effectively remove theactual Theta-values from the proof; while Mochizuki does consider this ”Theta31

intertwining” which is the association of the Theta-pilot with the Theta-values.
I will only comment further here if either a valid objection to (1) or (2) is
mentioned, or further clarification is given regarding (3). Any further technicaldiscussions are probably best done via e-mail.
Best, Peter

Taylor Dupuy April 17, 2020 at 12:17 pm
Hi Everyone,
I’m just going to make one last post to close out some loose ends for interested
readers. I also want to point out some things that we haven’t covered that I
think are important. I’ve omitted the discussions of asymptotics of Mochizuki’s
formula but other than that, I think I covered my IOUs. If there are any analytic
number theorists who were really looking forward to that please email me.
Sorry for the length.
Regarding (3): Is Mochizuki’s Proof Falsified?
There is no *proof* that Mochizuki’s method doesn’t work.
The following is what the Scholze-Stix manuscript proves:
Theorem. Assuming that one may identify Hodge Theaters in Mochizuki’s
theory and simultaneously impose “concrete normalizations” of q-pilot andtheta-pilot degrees then there is a contradiction.
Remarks.
1) The hypothesis of “simultaneously imposing concrete normalizations” isequivalent to l(l +1)/12 = 1 for a fixed prime l. In the manuscript no machineryfrom Mochizuki’s theory is used in this derivation. A protest to this identification is made in C12 of Mochizuki’s response. Despite this, Peter S. insists thisis what Mochizuki means. See his comments above, or the quoted comments
below.
2) The hypothesis that one can identify Hodge Theaters is also protested by
Mochizuki and runs counter to his stated objective. It is prudent to observe thatthe thesis of the Gaussian integrals survey is that sometimes in Mathematics one
can introduce “alien copies” of objects to prove something about your original
object. This runs counter to the assumption of identifying Hodge Theaters in
the above theorem.
I am not taking the position that Mochizuki’s proof is correct or will turnout to be correct but that, as it stands, the Stix-Scholze manuscript should not
be held up as a reason to reject Mochizuki’s proof. On the contrary, I am justsaying that a falsification of the proof requires a fluency in the definitions which
seems hard or impossible to achieve. Many of the definitions use nonstandard
terminology and come off as ambiguous to many readers (I’m not sure if I
want to get into a definition of ambiguous). Case in point: questions about
the “embeddedness” (or in Mochizuki’s terms “holomorphic structures”) of themonoids involved in the I ×µ prime strips; these came up in Peter’s previous
post. Moreover, because of certain ambiguities in these definitions, readers are32

forced to search a large space of possible meanings. While these difficulties do
not falsify the possibility of a proof, they make it so that what Mochizuki has
written is not a proof in traditional terms. To speak plainly, flexing on thereader by omitting details is not something new in Arithmetic Geometry and
IUT is a weird flex.
Finally, it is not outside the realm of reality that there could be 5 top notch
international referees who have understood the proof as complete and correct.
PRIMS has traditionally been a journal of very high quality and members of theeditorial board have shown integrity in the past (look at how senior mathematicians are responding when asked by journalists to comment). Let’s see what theprint version looks like (hopefully it has a table of contents and remove bolding
and italics).
What can we do?: I think we should all think carefully about the damage
omissions have next time we are tempted to do it. Additionally, we should
praise the writers whose writing we find “easy” and useful. Because of these
difficulties I personally hold Mochizuki’s document as a program for proving
ABC and consider Mochizuki’s formula a conjecture (until I can see this with
my own eyes). Then again, there are a lot of Theorems I know about and use
but can’t prove. Also, be nice to everyone, we are all just people.
Regarding (1)I see nothing wrong with (1). By this I understand replacing all constructions
in IUT involving the fundamental group of an SBT curve with the corresponding
construction in an actual curve. I think this is possible, but we would need tocheck that this doesn’t break somewhere. If the claim is that all fundamentalgroups in the theory need to be eliminated, I would be more reluctant. There
are a lot of weird curves. I think I need to see what a more precise statement
looks like in practice to say for sure.
Regarding (2)Regarding (2), to be clear, we should repeat that this is not what Mochizuki
does (only using one fundamental group). Also, the statement as Peter has
stated is a moral one not to be taken literally – there are many more groups
appearing in the construction that are not isomorphic to π1 (X) (profinite etale)so, like, it doesn’t literally parse but I think I know what he means. Regarding
its validity, I have reservations related to conjugacy synchronization and construction of “the diagonal” (points in Mochizuki’s theory are replaced by conjugacy classes of decomposition group and evaluation of functions is replaced by
restriction of cohomology classes to decomposition groups – since we want toevaluate at many points simultaneously one needs them to be determined up to
conjugation simultaneously – we need a diagonal action of a conjugation not an
action by a product of conjugations. This process involves multiple fundamentalgroups.) — BUT if we are going to dispose of these things entirely as in (1)you can probably scrap a lot from section 2 of IUT2. I need to think about how
this construction works in order to say something precise. I personally would
ask Emmanuel Lepage or Jakob Stix about this.

33

A more middle ground statement is that one can replace all “base Hodge
theaters” by a single “base Hodge theater”—which have more than just π1 (X)and a bunch of maps between them. Yet another variant of this would involve
contracting even more fundamental groups in the base Hodge Theater (and thisis where I think you might run into conjugacy synchronization and diagonal
problems).
Also, there are a lot of monoids built into the theory which are excluded fromthe statement and I don’t know if these are implicitly assumed to be identified
as well. I think not.
Finally, to derive a contradiction, I think some things need to be said about*how* one proposes to reduce Mochizuki’s construction to “a single fundamentalgroup”.
All in all to build this into an actual counter-example, as Peter claims, one
needs to make the statement more precise.
Remark. In a post I read someone emphasized that Peter S. read “hundredof pages”, or that he met with Mochizuki “for a week”. Similarly, Fesenko hasemphasized the number of hours people have studied the theory. These are
personal reasons not to undertake something and not acceptable substitutes for
proofs.
****************************
Now more comments...
Intertwining
On the other hand, I read say on page 140 of IUT-3 that Mochizuki
considers the Theta-intertwining, which I believe simply means thisidentification of abstract Theta-pilots with concrete Theta-pilots.
He wants to be very careful with using this etc., but I do believe hewants to (and has to) use it somewhere. So I don’t think you cansimply cut there. I think the closed loop that Mochizuki discusses
on page 143 of IUT-3 is also relevant here. See also on page 144
the simultaneous Theta-intertwining and q-intertwining he wants tohave (up to indeterminacies etc. . . ).
There is more of this on page 188 too (April 16, 2020 copy)
In the quoted statement above the “up to indeterminacies” is the point.
Mochizuki asserts the existence of TWO theta monoids you need to be thinking about relating. One that identified with q-monoid and the other which has
been interpreted alongside q in the same Hodge theater. In Mochizuki’s language
one is in a “alien holomorphic structure” and the other is in interpreted fromthe same structure. Mochizuki’s game is then relate the auxillary theta pilot
using anabelian techniques (amphoricity in the language of Joshi’s manuscript)
in order to derive a relation between the two. This relation is a uniformity
statement. All the thetas of the world can be blurred.
At one point I was confused about what is going on with the interpretationsin absolute galois groups of p-adic fields. It is helpful to know that the measure34

spaces constructed from absolute galois groups of p-adic fields have measures
which are well defined—in the initial construction in AAG3 (section 5?), themeasures take values in an ordered one dimensional real vector space obtained by
the process of ’perfection’ (colim M , nodes = N, morphisms=(a 7→ ab) realized
as multiplication by b) and completion. It turns out that because of initial theta
data, the we know how to normalize the measures to give real numbers.
My point: the constructions of these measure spaces may look weird, butfrom the perspective of either interpreting structure these things give real numbers, and the values of these measures don’t lie in some abstract ordered one
dimensional real vector space. Also, there exists a region U ⊂ L`,et (this is
my notation for the monoanalytic étale like big log-shell) that is invariant under automorphisms of log-linked collections of Hodge Theaters. In Mochizuki’s
terminology, ”this region can be seen from both sides of the theta link”.
In any case, to me it sounds like you are saying that the Thetapilot has nothing to do with Theta-values.
On the contrary, all theta pilots are defined *in* structures which includes anevaluation map like (theta monoid)×(evaluation points)→(constant monoids).
Look at the diagram in IUT Remark 3.10.2 and read it clockwise starting from
gau and ending at etale lgp. Each of these I ×µ strips are parts of a larger
structure where multiplication of the “value group portions” may act on the“unit group portions”. Eventually these “Kummer maps” lead to the construction of the region UΘ described previously.
Well, strictly speaking (modulo minor issues) what Mochizuki
does is the following: He takes the Theta-value (some nonzero padic number, not a root of unity), and takes the (multiplicative)
submonoid M of the p-adic numbers generated by it, and then considers M as an abstract monoid. As an abstract monoid, it is thenof course isomorphic to N, necessarily canonically so; its generator
is referred to as a Theta-pilot. Does the abstract monoid M know
anything about the Theta-value? Of course not! But that is all thatever seems to enter the story of the Theta-link.
This goes back to my comments about his objects not having good blackboxes (without some sort of language like the of interpretations). This also
points to my point about his confusing writing style in the beginning. It is
typically not sufficient to perform his constructions without knowing about theinterpretation.
At some point you do need to remember the interpretation of the
Theta-pilot in terms of Theta-values — this is what Mochizuki calls
the Theta-intertwining, and it plays a key role as I tried to reference.
So I don’t think that Mochizuki simply cuts our diagram there.
Again, from the horse’s mouth, read C12 of his responses. He says he does
not do this. Also, please look at the displayed diagram below. I think this35

may clarify something for you. Also, in this “interwining”, this is where thetheorem about the amphoric/characteristic nature of the interpretation of theJacobi theta function is supposed to be applied. I made some remarks before
on this and refer to those.
Now David Roberts previously mentioned that Mochizuki seems
to have a nonstandard point of view on what a mathematical object
is, requiring separate copies when they are not actually needed; heoften refers to some “history” of these objects. Now maybe one
could try to argue that because M was built out of Theta-values, it(the abstract monoid) still knows something about them, that canbe even transported via an isomorphism of abstract monoids. (Youmade some similar remarks in your paragraph on N and Kummer
theory, refering to “interpreted structures”.) But that’s evidently
not the case. (That the abstract monoid M knows something about
Theta-values might be debatable if for you an abstract monoid isreally encoded — as it is in ZFC — in terms of its actual set ofelements. But even then, this structure (“the interpretation”) isclearly not transported by an isomorphism of abstract monoids.)
I think this quoted text reinforces my points about “depth searches” and theambiguities (percieved or not) in Mochizuki’s writing. Peter I am going to use
the words “My understanding” so you may want to skip this response.
Regarding ZFC, while this is true, we don’t go this deep. That is not what isgoing on here. (for the uninitiated see https://stacks.math.columbia.edu/
tag/0009)
My understanding of the “history of objects” is different, and I don’t think
this saves the day. “Forgetting history” is Mochizuki-speak for forgetting about
the interpretation as you have suggested. Automorphisms of objects are nolonger coming from the interpreting structure and this structure is no longer“going along for the ride”.
In Mochizuki’s language or the language of naked categories Mochizuki’s
theory does not have many good black boxes. This makes parsing the examples
even harder because you need to carry around these large definitions in oursmall human brains (or use a large amount of paper). If you pick up Hodge’s
a “a Shorter Model Theory” or Olivia Caramello’s book, or MacLane’s Sheaves
in Logic book, you will find perfectly fine definitions of interpretations thatallow you to package all of that sloppy anabelian geometry/ “reconstruction
algorithms” into nice little formulas.
Here are some of David Marker’s notes:
http://homepages.math.uic.edu/~marker/math512-F13/inf.pdf
Exercise: Look at the first page of Marker’s notes then show that the torsionof an abelian group is definable using infinitary formulas.
Exercise: People often complain about the non first orderizability of topological spaces. Read this post from stack exchange on how to define topologicalspaces: https://math.stackexchange.com/questions/46656/why-is-topology-nonfirstorderizable
36About the indeterminacies: I’m focusing on (Ind2); (especially)
the part concerning automorphisms of local Galois groups (acting on
local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity isomorphisms, but does commute
when you conjugate by an automorphism of a local Galois group(acting on local units). Why do you need that, except because “youneed to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that?
I’m not sure what “that” is. I’m going to assume you mean ind2. I needto break this down in order to give you a coherent response. For me, ind2 is a
representations of automorphisms absolute galois group of a p-adic field G ∼= GK
on a Qp -vector space O×µ (G) ⊗ Q ∼= (K, +) induced functorially. There areinterpretations given from local class field theory (I’m going to release a tableof these soon in my manuscript with Anton, alternatively you can find these insome papers of Hoshi). These preserve the Zp -lattice O×µ (G) inside this vector
space.
I’m not sure what sort of diagrams you want but here is one: Let K be a finite
extension of Qp with uniformizer π where log(1 + π) = π and log(1 + π 2 ) = π 2 .
Let {x} be a one point set. Let f, g : {x} → K be given by f (x) = π andg(x) = π 2 .

x−−−−→ {x}gy{x} −−−−→ Kf(I don’t know how to make the diagonal arrow) Considering K up to ind2 makes
the diagram commute. Something tells me this is not what you had in mind.
Ind2 has the property of ”mixing valuations”.
In terms of automorphisms of G these give you the freedom to move aroundthe higher ramification groups. The reference for this is Joshi’s manuscript.
Here is the standard warning: this will not save the day and you will NEED
ind3 to make the formulas work as this sort of doesn’t mix very much. The
Zp -submodules pj log(OK ) will always be preserved under ind2, soaK=
pj log(OK ) \ pj+1 log(OK )j∈Zand each annulus is preserves setwise under ind2. This is saying, in some sense,that ind2 (and ind1) do not do too much. Moral: in order for Mochizuki’s
inequality to be true, you *do* need ind3.
The purpose of ind2 (and the other indeterminacies) is to perform a construction that is independent of the structure that it was constructed from.
(There was actually a point where Mochizuki was surprised thatby “forgetting the history of a group” one still has more than a group37

up to isomorphism: Namely, a group. That the datum of a group isstrictly more than the datum of a group up to isomorphism seemed
new to Mochizuki. I believe this is the (psychological) main reason
he considers these full poly-isomorphisms.
Classic Mochizuki.
But really, forgetting “the history of the groups” you still havegroups and they may still have natural commuting isomorphisms
between them. Of course, for groups up to isomorphism you can’t
ask for natural commuting isomorphisms between them, then you
only have “full poly-isomorphisms”.)
To be honest, I’m not sure what you mean by “history of groups” and it
actually hurts my head to think about this or to think about you thinking
about this. If you want to email me a reference I might be able to say what
is going on here but I would need to see it. I’m also don’t want to defend theusage of “history” in these documents.
A few comments up I summarized the situation with claims (1),(2) and (3). I have seen no valid objection to (1) and (2), and (2)alone would lead to a contradiction (as one gets too strong a formof ABC).
I addressed this in the introduction.
To (3), Taylor indicated where to cut the diagram, but I really don’t think this is what happens, as this would isolate Thetapilots from Theta-values and effectively remove the actual Thetavalues from the proof; while Mochizuki does consider this “Thetaintertwining” which is the association of the Theta-pilot with the
Theta-values.
I addressed how Mochizuki proposes to bring theta values back into theproof; using the anabelian geometry. How this actually works is unclear to me.
Also, I think Mochizuki would object to your characterization “theta intertwining” here. He would say something about the “strong anabelian properties ofthe étale theta function” which might be equally imprecise. He would also point
to a number of other constructions which would include MOD vs mod constructions, and the theory of cyclotomic synchronizations. How the q and Θ pilot
objects are tied is the mystery and has been the subject of my conversations
with everyone for the last several years (which is embarassing to admit outloud).
What is missing to me is the comparison between the UΘ and q. Mochizuki
and Hoshi have very patiently been trying to explain this to me for a long time
now. Maybe you want to call this a gap. Maybe I’m dense. Mochizuki willundoubtedly call this a “fundamental misunderstanding” and point to some
aspect of his manuscript. Me (and Emmanuel Lepage and others) have beentalking about this since 2017 and are stumped.
38

Some “Refereeing” of the log-links discussions between Peter and UF@UF April 11, 2020 at 9:02 pm–C7 of the 05 comments actually says that you *can* switch. –I don’t think
this rigidifying/derigidifying should be emphasized and will just confuse people.
B] a “rigidified fundamental group” isCalling a pair (Π, M ) ∈ [π1temp (Z + ), OK
weird.
@Peter Scholze, April 12, 2020 at 4:58 pm
In this Peter chooses to address the log-links as an example of indeterminate
copies doing nothing. Here he claims these can be replaced by logs.
My understanding of the situation for log-links is different. I’m going to giveB.
an abstract setup and I claim something like this occurs in IUT. Fix M ∼= OK
QThis can be decomposed as M ∼= π ≥0 · M tors · F where M tors is the torsion
subgroup and F is the free part (at finite level these are free Zp modules).
×
Furthermore for the subgroup of units we have M ∼= M tors · F . Suppose now
that one only knows M up to automorphisms of M tors . In order to get a well××µ
defined object, one thing that we can do is mod out by torsion M → M=
×M /M tors this is now well defined.
Mochizuki would say the converse. That we want to manipulate M ×µ (viaind1 and ind2) and this manipulation needs to be independent (lifted to?) from
the M tors , which plays a role in the exterior cyclotome of the frobenius-likemono-theta environment, which is used for passing structures “down to thebottom” in IUT3 3.11.
@UF, April 13, 2020 at 10:16 pm
It seems that you are both talking past each other. Log links are about
relationships between monoids *not* fundamental groups.
It seems quite likely Mochizucki uses “switching-symmetry” in atechnical sense, synonymous with “multiradiality” of some algorithm
reconstructing the data (here a rigidified column of log-links) at handfrom some choric data, as he often does, compare e.g. [Alien, p.51].
There is a technical sense. Given a morphism of connected groupoids F :A → B one can form the category A ×B A whose objects are (A1 , A2 , f ) wheref : F (A1 ) → F (A2 ). When F is full and essentially surjective the switching
functor is (A1 , A2 , f ) 7→ (A2 , A2 , f −1 ). In practice, as I have said before, thisseems to be note enough for applications as objects in categories seem to be
more. Also, I think there is a problem because we literally want a map of sets
at the end of the day. I haven’t worked out how to use this formalism effectively.
His statement would then mean that if we rigidify the verticalcolumns, then there is (unlike in the non-rigidfied case) no multiradial algorithm to recover this column from certain choric data. This
does not sound so absurd anymore (to me).
39

Minor typo: “choric” should be “coric”
Now which multiradial algorithm does he mean here? I would
suggest it may be the multiradial algorithm in [IUT III, Cor 2.3 ],more specifically, the first part of 2.3 (ii) which concerns its compatibility with log-links. Note that close by, [IUT III, Rem 2.1.1 (ii)] theissue we are talking about “why π1 only up to indeterminate iso?”
is discussed. For further discussion see also in [IUT II, Rem 3.6.4(i)]. In any case, I agree this is an important issue to track down.
This is addressed in the comments at the very very bottom.
@Peter Scholze, April 14, 2020 at 5:17 pm
The remarks from IUT that you cite make heavy reference to his
paper on etale theta-functions, which seems to play a key role in the
IUT papers. This paper gives some neat algorithm to start fromthe fundamental group of a once-punctured elliptic curve with badsemistable reduction, and recover its Tate parameter q and some
Theta function; I forget the details. While this is all good andwell, I don’t see the relevance: Mochizuki’s more general anabelian
theorems, discussed previously on this thread, tell you that from thefundamental group you can simply recover the whole curve. In thesecomments of Mochizuki that you reference, Mochizuki is discussing
some nitty-gritty details of this algorithm, but this seems completely
besides the point if you just remember that relevant π1 (X)’s areequivalent to relevant X’s, so of course you can recover all invariantsof X, and you can do so functorially in π1 (X)’s.
If we are collecting complaints about exposition I have another: As Peter has mentioned this and many other pieces of text in theorem environments punt you WAAAY back to topics in other papers. As Peter is mentioning this goes back to Etale Theta which I think is at least 5 papers back
(IUT3>IUT2>IUT1>AAG3>AAG2>AAG1>EtTh) to be fair one could argue
that AAGX and EtTh are independent. Either way, this is a big dependence,
and we haven’t even touched the dependencies of the AAGX papers.
I responded to the second part of this remark already. You asked about
3.11.i. Also, this RLGP is actually not so complicated. The one in Cor 2.3 R
that is the bad one (no pun intended). I’m going to postpone this dicussion
until the end.
@UF April 15, 2020 at 9:04 am
I now think a good reason to for not identifying π1 ’s along loglinks is as follows:40

(It is very similar to [IUT-2, Rem 1.11.2 (ii)] for the theta-link; I am lessprecise here, taking the relevance of some reconstructions on faith, but it is a
intuitive story, and precision can come later)
First, let us step back and look at the log-theta-lattice: It consistsof Hodge-theaters and the log/theta-links as maps/functors between
them. Now a formulation of (2) is: Why do we, say for the log-links,
not just take the “identity” isomorphism between the relevant π1 ’sinstead of the full poly-isomorphisms between them? Surely it isalso a completely valid map between the theaters?
Yes, but: let us recall what we want to do with this log thetalattice: we want at “suitable times(?)” glue the Hodge theaters along
these links and then apply certain multiradial algorithms.
I think that if we would glue with the “identity” on the π10 s (andall the other prescriptions of the log-link), then this gluing will beinconsistent/not well defined (when combined with some multiradialalgorithms), in some vaguely similar sense as if you try to define acomplex structure on a manifold by charts, but in some region theprocedure leads to two different, incompatible complex structures,
i.e. the transition map is not holomorphic.
Indeed, in one Hodge theater, the π1 ’s not independent of therest of the data; it acts via the quotient to the Galois group on themonoids. Thus π1 ’s “related” to the monoids, and it is not apriori
clear we are allowed to glue two Hodge theaters by gluing π1 andthe monoids separately in whatever way we wish (and then follow
certain multiradial algorithms).
I’m going to refer you to my forthcoming manuscript with anton (the prequelto the one with the IUT4 style computaions). There is some stuff on my vlog
that is a (poor) template for this stuff if you don’t want to wait.
In the above analogy, a certain region of the Hodge theater is
supposed to be “determined” jointly by the π1 and the monoids.
The multiradial algorithm we consider is roughly as follows: The action of
π1 determines on the (from the Galois group reconstructed) local monoids (with0) in addition a ring structure (a “holomorphic” structure).
The output is roughly some etale version of some of the data
originally present in the Hodge theater, and the reconstruction algorithm yields a canonical isomorphism between the original data
and the reconstructed one (see e.g. [Alien, §2.12] for essentially such
constructions).
This *is* a thing.

41

If one now glues two Hodge theaters along a log-link (with inaddition “identity”-gluing for π1 ’s, then this yields no consistentholomorphic/ring structure compatible with the above algorithm onsome region: Namely, consider the region of the ring (local units),
which is glued via log (the actual log part of the log-link) to other
Hodge theater; this yields one definition of “holomorphic” for thisregion.
I can’t parse this sentence.
On the other hand, we can embed this region in the reconstructed monoids and use for the “holomorphic” structure on thereconstructed monoids the one induced by π1 on both Hodge theaters and glue by the “identity”.
This is true. You definitely have one structure from the base and one structure from pulled back from log. The diagrams commute though.
The above roughly describes two incompatible holomorpic/ring
structures on some part of the glued Hodge theater; they are incompatible, “because the transition map, essentially the log, is nota ring map”. Such a situation is presumably avoided by taking thefull polyisomorphism between π1 where the log map on the units
alone defines the holomorphic structure on the relevant region.
I’m not sure I parse this completely. I will say there is a way to get the a commutative diagrams of fields in the log-kummer correspondence. It depends on
what maps you take. Any ”log” has a backwards map which is an isomorphismof fields. I will refer to the forthcoming paper with Anton for details.
@Peter Scholze April 15, 2020 at 2:05 pm
Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is
why you have to consider source and target up to some indeterminacy.
The log-link is about the monoids and not the base (you use really only use
the fundamental group to use Mochizuki’s interpretation of a field in order topullback this structure so you can take the logarithm).
I don’t know if this helps but there is a ”post-logarithm” that IS an isomorphism of rings and this IS used the definition of the log link. I think it is nice
to factor the actual logarithm as logpost ◦ logpre = log where logpre is the map
in IUT. One can then use the map logpost : O×µ ⊗ Q → K and pull back thefield structure to get a new field Klog which is O×µ ⊗ Q as a set but with thisnew field structure.
@UF April 15, 2020 at 4:57 pm
42What my post above attempts to show, is: if passing to polyisomorphism has the effect of doing no gluing/no identification ofring structures (arising from π1 just the gluing from the actual logmap), then the only gluing left is the actual log map, which gives
one global chart, and no transition functions needed, essentially(?)
since just one chart.
Yes, but I think the point is what I was saying about killing indeterminacies.
Doing log-links isn’t “for sport” as Mochizuki would say.
I before never really seriously considered that full poly-isomorphism
could have the effect of “no gluing arising from this part” (insteadof “choose your favourite gluing”).
I don’t think I understand how you are thinking about this.
Remarks on IUT3 2.3.ii
We are looking to see if this statement says something non-trivial about
the bases of log-linked hodge theaters and multiradiality (I actually think thisis barking up the wrong tree since, as I’ve said before, the bases in log-linked
hodge theaters do nothing except impart ring structures to the monoids). Indeed
this is the case because of assertion (2) below. Let me just remark though forPeter, that the maps between the ×µ prime strips are what are going to encode
everything in Galois groups and what eventually gets the theta back to the otherside.
Anyway, at the beginning of the proposition there is a certain amount ofsetup. BUT, notice in the setup, most of the structures are interpreted! Thismeans that their morphisms are just functorially induced by the interpreting
structures, in this case there is a prime strip of the ×µ variety, and a basehodge theater (actually, it is even simpler than this... see the note below).
Looking at my notes... this looks tautological.
Also, just as a tip from as a person who has wasted his life doing this, thething to look out for are maps that are NOT full polyisomorphisms.
In this item there are a total of three assertions.
1) log-links induce a full polyisomorphism of the R’s;2) Theta links on log-links pairs of hodge theaters have D`∆ ’s interpreted inD`∆ which are isomorphic.
3) Using (1) and (2) the R across is generically isomorphic across log-links;
the D`∆ data is generically isomorphic across Theta links.
Proof of 1: This only could be a problem because of the nature of the way
the monoids are linked and the dependence of R’s on these monoids. In thedata for R there is exactly one place where the monoids interact. This is in the×µ prime strips. Everything else is functorially induced. For example, in thedefinition he takes the monotheta environment interpreted from the base (thisactually has well-definedness issues of its own!).
Proof idea of 2: As described somewhere in this note ind2 is isomorphisms ofO×µ (G)’s induced by isomorphisms of G’s. If you look up the definition of D`∆
43

you will see that it is just a bunch of G’s. This means the generic isomorphism of
from the G’s induce an ind2 on the ×µ monoids they interpret; the ×µ monoid
isomorphisms in the theta link are the same thing.
Expositional Note: This is an example of a proposition where too much
structure was invoked for my taste. In the fine print you notice that the interpreters are D> and D`∆ (=fundamental groups at each place and absolutegalois groups at each place respectively). In the statement he references the fullbase hodge theater. That’s baggage. Also, you will notice that this could havebeen broken into many much simpler assertions each which can be individuallychecked.
Remarks on IUT3 3.11.i
Peter had asked (essentially) how can galois groups interpret theta values?
How can they do anything?!
Well, I claim that Theorem 3.11.i and 3.11.ii are not the difficult parts ofTheorem 3.11; Theorem 3.11.iii is the weird part. There are some subtle differences between the (abc)-module structure in both of these statements. In thefirst, the actors (b) and (c) are defined as subset of the L`,et (at the appropriate
places) in part i. In part ii there is a clear module structure.
In the situation of (i) you lift to an richer stronger and more powerful structure that allows you to define (b) and (c) then then you stick them inside thelog shell. What the proposition is saying is that this is well-defined up to ind1,2— so everything that was stuck inside these log shells now is considered up toa jumbling of the type described in a previous response. I will refer readers to
my manuscripts with Anton Hilado for these formulas.
A subtle difference that I want to point out to everyone here which they
may not have noticed is the difference between the usage of the MOD, LGP
constructions and the mod, lgp constructions in the later.
*************
Sorry if I made any errors anywhere. I didn’t mean to and tried to proofread
this. Hopefully we learn as part of the corrections.
Best, Taylor

Peter Scholze April 17, 2020 at 6:25 pm
Dear Taylor,
thanks for these final comments! I think I should answer to this. Let me
first say that I agree with much of what you write, and for the sake of keeping
this short, I only jump at the few places where I disagree.
Regarding (3): Is Mochizuki’s Proof Falsified? [...]
Finally, it is not outside the realm of reality that there could be
5 top notch international referees who have understood the proof ascomplete and correct.
Really? I would have hoped that in that case at least one of them – notin their role as a referee, but simply as a mathematician who wants to share44

insight – would have come around and explain the key ideas in a way that isunderstandable.
Because of these difficulties I personally hold Mochizuki’s document as a program for proving ABC and consider Mochizuki’s formula a conjecture (until I can see this with my own eyes). Thenagain, there are a lot of Theorems I know about and use but can’t
prove.
I very strongly object to the implicit (I believe) assertion that Mochizuki’s
result should now be considered just another of these really difficult theorems
whose proof we never understood.
Regarding (2)Also, there are a lot of monoids built into the theory which are
excluded from the statement and I don’t know if these are implicitly
assumed to be identified as well. I think not.
Well, I think I would want to identify some, too. My mode of thought is thatstarting from your elliptic curve, you can easily cook up all the data that formsa Hodge theater, including those monoids. Simply take that collection, andalways the same one. Later, when you study log-links, try to understand what
they do, which diagrams do not commute, and if necessary enlarge your diagram(of course you get a non-commutative diagram if you want that the logarithm
equals the identity). My issue lies with possible non-identity isomorphisms of
Hodge theaters being relevant at any point; they are not for all I can see, as all
possible non-commutativity can’t be restored in terms of internal isomorphismsof Hodge theaters.
Finally, to derive a contradiction, I think some things need to besaid about *how* one proposes to reduce Mochizuki’s construction
to “a single fundamental group”.
As the final step in Mochizuki (proof of Cor 3.12) is so unclear, it also seems
impossible to fully justify where and how it breaks.
Intertwining
Again, from the horses mouth, read C12 of his responses.
OK, I reread this. He talks about the q- and the Theta-holomorphic structure. This indeed sounds like there must be two distinct Hodge theaters around– Hodge theater is, I believe, the technical version of “ambient holomorphic
structure”. But there are not! All of them are isomorphic. And if I identify
them (using whatever isomorphism) I can see plainly that this does not makeany sense.
He says he does not do this. Also, please look at the displayed
diagram below. I think this may clarify something for you.
45

Details seem to be off in the example, but in any case I totally see that some
non-commutative diagrams become commutative up to these indeterminacies –otherwise they would not be indeterminacies. But I’m asking about a relevantone.
Also, in this “interwining”, this is where the theorem about theamphoric/characteristic nature of the interpretation of the Jacobi
theta function is supposed to be applied.
The “amphoric” nature here is just that you can recover X from π1 (X),
so in particular the theta function (and all else). I’m still failing to see theimportance of étale theta functions in all of this.
Exercise: Look at the first page of Marker’s notes then show thatthe torsion of an abelian group is definable using infinitary formulas.
Exercise: People often complain about the non first orderizabilityof topological spaces. Read this post from stack exchange on how
to define topological spaces: https://math.stackexchange.com/
questions/46656/why-is-topology-nonfirstorderizable
I’m totally lost about where you’re trying to go here.
Thanks again for the example in reply to this [from earlier comment by
Scholze]
About the indeterminacies: I’m focusing on (Ind2); (especially)
the part concerning automorphisms of local Galois groups (acting on
local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity isomorphisms, but does commute
when you conjugate by an automorphism of a local Galois group(acting on local units). Why do you need that, except because “youneed to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that?
but I’m asking about a relevant diagram in the relevant abstract setting.
The purpose of ind2 (and the other indeterminacies) is to perform a construction that is independent of the structure that it wasconstructed from.
This slogan I understand, but where does Ind2 help with anything? Why
do you need to introduce it?
I addressed how Mochizuki proposes to bring theta values backinto the proof; using the anabelian geometry. How this actuallyworks is unclear to me. Also, I think Mochizuki would object to yourcharacterization “theta intertwining” here. He would say something
about the “strong anabelian properties of the etale theta function”
which might be equally imprecise. He would also point to a number46

of other constructions which would include MOD vs mod constructions, and the theory of cyclotomic synchronizations. How the q andΘ are tied is the mystery and has been the subject of my conversations with everyone for the last several years (which is embarassing
to admit outloud).
What is missing to me is the comparison between the UΘ and q.
Mochizuki and Hoshi have very patiently been trying to explain thisto me for a long time now. Maybe you want to call this a gap. Maybe
I’m dense. Mochizuki will undoubtedly call this a “fundamental
misunderstanding” and point to some aspect of his manuscript. Me(and Emmanuel Lepage and others) have been talking about thissince 2017 and are stumped.
Well, I guess you are just pointing your finger to the same problem thatI’m trying to point at. It is completely unclear how you get the actual Thetavalues back in the game. You started this paragraph with “using the anabeliangeometry”, but as I argued long before, it is completely unclear how anabeliangeometry helps with anything in his setup (by (1)).
@Peter Scholze April 15, 2020 at 2:05 pm
Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is
why you have to consider source and target up to some indeterminacy.
The log-link is about the monoids and not the base (you use really
only use the fundamental group to use Mochizuki’s interpretationof a field in order to pullback this structure so you can take thelogarithm).
I don’t know if this helps but there is a “post-logarithm” that
IS an isomorphism of rings and this IS used the definition of the log
link. [...]
My main question in the discussion with UF has always been the questionof full poly-isomorphisms vs. identities between π1 (X)’s, i.e. issue (2). All you
are writing is completely tangential to this. I think you actually agree that the
log-link has so little to do with π1 (X)’s that it can’t possibly be the reason toconsider π1 (X) up to full poly-isomorphism. Yet Mochizuki was trying to makethat exact point when we were discussing.
Remarks on IUT3 2.3.ii
Why are you discussing this? I’m lost. Is it saying more than Theta-values
being the same in all Hodge theaters?
Remarks on IUT3 3.11.i
Peter had asked (essentially) how can galois groups interprettheta values? How can they do anything?!
Well, I claim that Theorem 3.11.i and 3.11.ii are not the difficult
parts of Theorem 3.11;
47

You are simply ignoring my point!
Best wishes! PeterPeter Scholze April 17, 2020 at 7:15 pm
PS: I just realized that maybe the following information is worth sharing.
Namely, as an outsider one may wonder that the questions being discussed at
length in these comments (e.g., the issue of distinct copies etc.) are very far fromthe extremely intricate definitions in Mochizuki’s manuscripts (his notation isfamously forbidding, some of it surfaced in Taylor’s comments), and feel almost
philosophical, so one might wonder that one is not looking at the heart of thematter.
However, the discussions in Kyoto went along extremely similar lines, andthese discussions were actually very much led, certainly initially, by Mochizuki.
He first wanted to carefully explain the need for distinct copies, by way of perfections of rings, and then of the log-link, leading to discussions rather close to
the one I was having with UF here. He agreed that one first has to understandthese basic points before it makes sense to introduce all further layers of complexity. (I should add that we did also go through the substance of the papers,
but kept getting back at how this reflects on the basic points, as we all agreed
that this is the key of the matter.)

UF April 17, 2020 at 8:59 pm@Taylor Dupuy
Thanks for your comments. The context for comment (C7) is Mochizuki’s
interpretation of footnote 5 in [Scholze-Stix]. Mochizuki gives this interpretation in the first display of (C7): the rigidification mentioned there is (I think)not (some kind of) rigidification of π1 in one Hodge theater (which you seemto mention, by considering pairs (π1 , monoid)), but the rigidification among
the different π1 ’s in one vertical column by choosing a specific isomorphism(“identity”) instead of full poly-isomorphism between the different π1 ’s in one
column.
Mochizuki then says that one can consistently identify the π1 ’s as above,
if one forgets about both log- and theta-links. (Mochizuki does not say so
explicitly right here, but I think in this situation there would be a switchingsymmetry. He says so in a closely related situation after the second display,which in my understanding amounts to the same situation).
In the second display in (C7), he seems to say (my “” are not meant hereas quotes): In contrast, if one keeps log-links (or maybe similarly theta-links),
then the rigidification of the π1 ’s “depends” on the other data, say “0-column
Frobenius-like data (at different vertical spots) with log-relations”, which do
not admit a switching-symmetry between two vertical columns.
Here in my understanding “depends” indicates that the log-identifications
between the 0-column Frobenius data cannot be made completely independently48

from the identity-identifications of the π1 ’s in the 0-column, since at a fixedvertical position the π1 and the monoids are “related/not independent” (via theaction). Implicit here (and relevant for the context of this C7) seems to be theassertion that because the rigidification of the π1 ’s “depends” on the 0-column
Frobenius data (which does not admit a switching-symmetry), something has
to go wrong with switching (when π1 ’s are rigidified in a column), “because” itgoes wrong for the related/dependent 0-column Frobenius-like data. (If this is*not* implicit here, then what in Mochizuki’s response in C7 is the objection
to the vertical rigidification of π1 ’s?)
After the second display he says: If one forgets about the 0-column Frobenius
data (in particular the log-relationship between them at different vertical spots),then there is a switching-symmetry between two neighboring columns.
This is how I understand Mochizuki’s C7; I may certainly understand it
wrong in parts, and I would be happy to understand it better. All the rest I
wrote is an attempt make this (from my point of view) “implicit objection torigidifying π1 ’s” in the second display more explicit; but if my understanding
concerning this is incorrect, then this is largely meaningless.

W April 19, 2020 at 9:53 am
There has been some discussion about this comment thread that focused on
the importance of the differentiating between fact and opinion. There has also
been discussion that the long thread may obscure the points that Peter Scholze
and Taylor Dupuy agree on by focusing on their areas of disagreement. I havewritten a (very long) comment to try to help with these issues.
The way this comment is intended to work is that the numbered paragraphs(1), (2), (3) give facts that are difficult or impossible to dispute and the paragraphs after them give opinions building on those facts.
(1) The first couple hundred pages of the IUT papers introduce a lot ofdifficult new terminology. Scholze and Stix demonstrated that none of this new
terminology is necessary to prove any of the formal statements in these pages.
Moreover, when the new terminology is ignored, the proofs all become simpler
and easier to understand. They all follow quickly from Mochizuki’s previous
work on the p-adic Grothendieck conjecture.
As a reader, this should already make you skeptical of the accuracy of thepaper. How can dressing up simple statements in confusing and difficult terminology for hundreds of pages represent progress in turning the solution of
one problem (the p-adic Grothendieck conjecture) into a completely different
problem (ABC)? There are mathematical arguments that begin by introducing
a lot of terminology (e.g. work of Grothendieck) but nothing like this ratio of
terminology to content has ever been successful before. However, you certainly
shouldn’t rule out Mochizuki’s work based only on this evidence.
As an editor or referee, you should also be very skeptical at this point. Setting aside concerns about correctness, you have some duty to ensure papers published in your journal are clear and get to the point. Having hundreds of pages49

that can be compressed to much less by the removal of new ideas rather thantheir introduction is not normally consistent with good mathematical writing,but one could imagine that the remainder of the paper justifies this sufficiently.
(2) Many people have tried to read the papers, read and understood the first
few hundred pages, and then were unable to verify a key point – Corollary 3.12.
This point is either the first or the second statement in the paper which doesn’t
fit into the hundreds of pages mentioned in (1).
As a reader this should make you very worried! This is is not a pattern one
typically sees when people try to understand a correct but difficult argument(regardless of how well or poorly it is written). Instead, different people almost
always get stuck at different points. This is certainly true if the reason people
are getting stuck is a lack of background in anabelian geometry, or a refusal
to put enough work into it – both reasons suggested by defenders of IUT. You
would then expect to see different confusions in different places from people
with different levels of background, or from people who made careless errors at
different points. Of course, what we see is exactly what you would expect to see
if the proof of Corollary 3.12 is not a valid argument but the rest of the paper
is valid.
As an editor or referee, this should be completely unacceptable. Even if theproof is technically correct in the sense that Mochizuki or someone else can givea clear and precise argument which the proof of Corollary 3.12 could be said tosummarize, allowing the paper to continue in its present form would be a failure
at their duty to ensure papers in their journal properly explain their arguments.
A paper such that everyone who studies it seriously, except a small inner circle,
gets stuck at the same point simply isn’t a proper explanation.
Some defenders of IUT like to point out that Scholze and Stix didn’t give
their precise objection until 2018. But this phenomenon, given that it wasnoticed by most people who read the paper seriously, should have been turned
up by the refereeing process before then. This is, I think, the starting point
for ethical concerns about the refereeing process. (For instance, OP’s commentsuggests that the editors could have asked a series of referees, ignoring those
who have negative commentary, until they found someone willing to say it isgood.)Is it possible that the final version makes substantial changes to this argument and answers this objection? Everything is possible, but it seems unlikely.
It is common practice in mathematics to edit the online version of the paper
to the final submitted version, or something close to it, without the journal’s
formatting. As far as I know the main reasons not to do this are because one
doesn’t post online versions or doesn’t edit them often, neither of which apply
to Mochizuki.
Even if the editors and Mochizuki want to avoid this for some reason (likethey want to make people read the journal to see the complete proof, I guess. . . )they could easily post some comment online like “Corollary 3.12 has been replaced with a series of simpler statements, each with a detailed proof, culminating in the key inequality, totaling about 35 pages” and get much less criticism
from mathematicians and mathematical physics bloggers. Does anyone like re50

ceiving criticism? Is there any reason not to do this?
(3) Two serious mathematicians have come up with a precise objection to
the proof of Corollary 3.12. Despite the fact that it is not clear exactly what
Mochizuki means, they have come up with a plausible interpretation, and shown
that, in this interpretation, the inequality that Mochizuki proves is off from theinequality he states by a factor of j 2 . This is important because the improvementof the stated inequality over a certain trivial inequality is exactly by a factorof j 2 . They have come up with additional arguments, which are unsurprisingly
not completely rigorous, that any plausible interpretation of what Mochizuki
means will have the same problem.
(One could say that it is not plausible that Mochizuki made such a simple
mistake, but I think it is plausible in light of the previously discussed information).
Neither Mochizuki nor his defenders have come up with a convincing rebuttal
to this. Mochizuki’s response focused on arguing that it is possible for an argument like his to work, and that Scholze and Stix misunderstood it, rather thanmaking basic clarifications to the argument that would aid in understanding
whether the objection is valid, and that would certainly be possible if the argument is correct. For instance it is *provably* possible that any valid argument
proving a concrete inequality using facts about affine spaces over the real numbers can be replaced with one that proves the concrete inequality using otherconcrete inequalities (or identities). Doing this would allow a more focused discussion because Scholze could either dispute one of the building-block concrete
inequalities or dispute the implications from them to the desired statement.
For me this objection, combined with the other stuff, is devastating. Of
course it is not the case that one must stop the presses of a journal every timea mathematician objects to an argument, and keep them stopped until thatmathematician is satisfied. But combined with the other very worrying facts
about the proof, this is an objection that must be answered for the paper to be
published ethically. As mentioned earlier, if a better answer than what is in thecurrently online version has been delivered by Mochizuki to the journal, I would
expect it, or at least an advertisement for its existence, to be posted online.
Despite the long comment thread, Peter Scholze and others have not exhausted all the reasons that a mathematician examining Mochizuki’s argument
should be skeptical that it, or even any argument like it, could possibly work.
I could list these additional reasons, but they are not so relevant when the objections outlined above already mean that it would be inappropriate to publish
the article in its current form.

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