Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by *Publications of the Research Institute for Mathematical Sciences (RIMS)* of Mochizuki’s purported proof of the abc conjecture.

This is very odd. As the Nature subheadline explains, “some experts say author Shinichi Mochizuki failed to fix fatal flaw”. It’s completely unheard of for a major journal to publish a proof of an important result when experts have publicly stated that the proof is flawed and are standing behind that statement. That Mochizuki is the chief editor of the journal and that the announcement was made by two of his RIMS colleagues doesn’t help at all with the situation.

For background on the problem with the proof, see an earlier blog entry here. In the Nature article Peter Scholze states:

My judgment has not changed in any way since I wrote that manuscript with Jakob Stix.

and there’s

“I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof.

I asked around this morning and no one I know who is well-informed about this has heard of any reason to change their opinion that Mochizuki does not have a proof.

Ivan Fesenko today has a long article entitled On Pioneering Mathematical Research, On the Occasion of Announcement of Forthcoming Publication of the IUT Papers by Shinichi Mochizuki. Much like earlier articles from him (I’d missed this one), it’s full of denunciations of anyone (including Scholze) who has expressed skepticism about the proof as an incompetent. There’s a lot about how Mochizuki’s work on the purported proof is an inspiration to the world, ending with:

In the UK, the recent new additional funding of mathematics, work on which was inspired by the pioneering research of Sh. Mochizuki, will address some of these issues.

which refers to the British government decision discussed here.

There is a really good inspirational story in recent years about successful pioneering mathematical research, but it’s the one about Scholze’s work, not the proof of abc that experts don’t believe, even if it gets published.

**Update:** See the comment posted here from Peter Scholze further explaining the underlying problem with the Mochizuki proof.

@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 🙂

@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 is

> “given that and

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 that it doesn’t happen we can try a proof by contradiction.

We have a curve X over K and a map pi_1(X) => G_K. We fix K’ and an isomorphism G_K => G_{K’} which does not come from an isomorphism K => K’. You ask, does the composition pi_1(X) => G_K => G_{K’} arise from a curve X’ over K’?

Suppose it does. Then we have two hyperbolic curves X, X’ and an isomorphism between pi_1(X) and pi_1(X’). Then Mochizuki proves that, if X is of strictly Belyi type, this isomorphism arises from an isomorphism between X and X’. In particular, it implies the isomorphism on the Galois part arises from an isomorphism K to K’, contradicting our assumption.

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.

Dear Professor Woit,

This is to clarify the claims made about my paper in the context of your abc 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 initially appear so because I state a number of standard results without proofs) and contains 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 the paper 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 and myself) 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 my paper (below) will be to the forthcoming version.

To explain one of Mochizuki’s important ideas, let us begin with a classical result which says that there exist p-adic fields (i.e. finite extensions of the basic p-adic field Q_p) which are not isomorphic but which have topologically isomorphic 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 filtration (see the section: five fundamental theorems of … in my paper) for references to proofs 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 a dynamic variable while keeping its absolute Galois group fixed. Because of the above 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 group remains 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 and discriminants (these are standard measures of complexity of fields in number theoretic contexts). These examples can be easily verified by any one with a computer 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 E_K and E_L (i.e. E considered as curves over L, K respectively) do not have the same list of numerical invariants (in general). This is done by means of explicit examples computed using SAGEMATH (with no additional programming needed). Notably my examples establish quite clearly that the additive structure of the p-adic field controls many subtle invariants of 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 structure of 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 are affected 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 isomorphism class 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 the theory of Galois representations. In this theory L,K are two p-adic fields which have isomorphic absolute Galois groups, so one can pass from representations of G_K to G_L. 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 G_K representation as a G_L representation via any given isomorphism of these two groups does not preserve (p-adic) Hodge theoretic properties of a representation (for example an Hodge-Tate representation of G_K may not remain Hodge-Tate when considered as a G_L representation via an isomorphism of G_L with G_K). However I prove that an important subcategory, namely ordinary p-adic representations is preserved under this operation. This operation of viewing G_K representations as G_L representations is not the identity functor in general nor is it so on the subcategory of ordinary representations. One can say such things because I also demonstrate that 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 is of 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 since then). 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 theorem of my paper which I have admitted was incorrect (and the new version will appear 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 the incorrect 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 of p-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 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).

Dear Kirti Joshi,

thanks for chiming in here, and I’m sorry for concentrating on the parts of your 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 to me how this enters into the actual content of IUT. In particular, your example of 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, $\pi_1(E_K)$ and $\pi_1(E_L)$ are not usually isomorphic, although $\pi_1(E_K)$ and $\pi_1(E_L)$ both can be computed as pullbacks $\pi_1(E)\times_{G_F} G_K$ resp. $\pi_1(E)\times_{G_F} G_L$ where all terms are isomorphic — but not the fibre product, as the maps $G_K\to G_F$ and $G_K\cong G_L\to G_F$ are not the same. (This was the essential 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 (to pass to perfectoid spaces). Let me stress that perfectoid spaces have precisely this flexibility of changing geometry while preserving topology (like $\pi_1$), 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 (in fact, by $\pi_1$). 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 this fact 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 whole discussion seems tangential to IUT.

Best wishes!

Peter

Dear Kirti Joshi,

I had the impression that the theorem of Jannsen-Wingberg says that the absolute Galois group of p-adic field only knows the residue field, the degree over Qp, and the number of roots of unity in the field. Maybe I misunderstood the statement, but if what I just wrote is correct, it is quite clear that you will have p-adic fields with isomorphic Galois groups and different arithmetic invariants.

Also I am confused by “the additive structure of the field” as such a field is just a Qp-vector space of dimension the degree over Qp; so its additive structure is encoded in the Galois group. Maybe you meant the multiplicative structure?

Setting aside the matter of correctness, the referees have not fulfilled a crucial part of their assignment: ensuring that new ideas are explained much better, at least in the Introductions. If that was not part of their assignment then the editors (who have to be aware of the reasonable perception of the situation by the broader arithmetic geometry community) have been asleep at the switch.

If I understand correctly, after a lot of time and effort Peter Scholze has not identified any new insight in the papers that makes meaningful progress on the ABC Conjecture. The papers in their present form are therefore unsuitable for publication in a good journal, regardless of anything else (much as if Terry Tao were to regard a paper in harmonic analysis as devoid of new ideas then it is unsuitable for such publication too).

@PC: perhaps what is meant is that an identification of multiplicative groups (arising essentially from local class field theory) won’t also respect the additive structure upon appending {0} with its usual multiplicative property.

The last link doesn’t work.

Tomate,

Thanks. Fixed.

@PC: I think what you state is true if you consider the absolute Galois group only as a group, but if you consider it as a group together with its ramification filtration then it determines the p-adic field.

*********

W (=Will?) :

*********

I agree with your observation. Nice.

Here is my summary (slightly modified).

Lemma.

Let $X$ be an SBT curve over a $p$-adic field $K$.

Let $\Pi = \pi_1(X) \times_{G,g} G$ where $g:G \to G$ is an outer automorphism (=not inner).

The map $\operatorname{pr}_2: \Pi \to G$ is not isomorphic to the map between a fundamental group 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 $\Pi \to G$ can be viewed as $\Pi \to \pi_1(X) \to G$, or $\operatorname{pr}_2$ (indexing starting at 1 and not 0). Call this map $h$. In this case we have a diagram

$$\begin{CD}

\Pi @>\operatorname{pr}_1>> \pi_1(X) \\

@VhVV @VVV\\

G @>g>> G

\end{CD}$$

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 $\operatorname{pr}_1$ is geometric (the Grothendieck theorem is not just about isomorphisms but all morphisms). But the bottom map is outer. This gives a contradiction. (I’m assuming the $p$-adic relative Grothendieck for morphisms here… which I need to 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 $\operatorname{pr}_2: \Pi \to G$ is the same as $\Pi \to \mathbf{G}(\Pi)$ where $\mathbf{G}$ is the interpretation I referenced previously.

c) I think we only get that $\operatorname{pr}_2:\Pi \to G$ is not the natural map from and 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. $\operatorname{pr}_1:\Pi \cong \pi_1(X)$ is an isomorphism. In particular $\Pi$ is isomorphic to the fundamental group of an SBT curve.

Proof.

Let $f: \pi_1(X) \to G$ be the map from the fiber sequence.

Since $\Pi = \lbrace (a,b) : f(a) = g(b) \rbrace$ the kernel of the map $\operatorname{pr}_1$ 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 $\operatorname{pr}_1$ is an isomorphism.//

This means the composition $\pi_1(X) \to G \xrightarrow{g} G$ is something that Mochizuki considers but it not the isomorphic (as maps) to some $\pi_1(X) \to G_K$ of “geometric origin”.

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 be a 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 of Harris 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.)

*************

Daniel:

*************

More comments on $p$-adic Grothendieck:

The existence of some isomorphisms of fundamental groups of curves over p-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 $\sigma$ of $\pi_1(Z)$ such that $\mathbf{G}(\sigma)$ 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 be by 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 these counterexamples 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 structures in the setup for a later post (if I’m going to say anything at all) because I want to get it right for everyone.

*********

PC:

*********

What OP says is correct. It is a local class field theory thing. A classic theorem is that for $K$ a finite extension of $\mathbb{Q}_p$ we have $G_K^{ab} \cong \varprojlim K^{\times}/K^{\times n}=: \widehat{K^{\times}}$ where $G_K$ is the absolute galois group of $K$. There is a version of this “on crack” where you can describe the topological groups $\mathcal{O}_K^{\times}$ and $K^{\times}$ among other things ‘inside’ $G_K$. You only get multiplicative stuff though.

You can’t recover the full field structure (both binary operations satisfying the usual axioms) because there exists fields $K_1$ and $K_2$ with $G_{K_1} \cong G_{K_2}$ where $K_1$ and $K_2$ are not isomorphic. This is a theorem of Jarden and Ritter.

Anyway, at the end of the day you can have one binary operation or the other but not both. The equivalence of having multiplication or addition (but not both!) comes from $p$-adic logarithms (which can be defined ‘inside the group’). This is just the first isomorphism theorem of groups.

Thanks to all commenters here for the remarkably informative discussion of the mathematics involved in the problem with Mochizuki’s claimed proof explained by Peter Scholze. Note an important aspect of this discussion: no one (including Joshi and Dupuy, two people who have been deeply involved in the study of IUT) has come forward to explain how Mochizuki can get around the problem pointed out by Scholze. The only place I know of publicly available that supposedly contains such an explanation is Mochizuki’s web-page

http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html

The only relevant materials there are absurd ad hominem arguments from Ivan Fesenko and Mochizuki’s own comments. Scholze and Stix are the only two who have had the experience of directly engaging in extensive discussion with Mochizuki of the problem, so their report that he has no answer to the problem must be taken as authoritative in the absence of some other strong evidence. The past two years of study of the problem do not seem to have led to anyone besides Mochizuki himself being willing or able to try to explain how Mochizuki’s claimed proof avoids the problem, and all experts I know find his explanation unconvincing.

Given this, the decision by PRIMS to hold a press conference announcing that the proof has been checked and will be published is completely outrageous. It may be good PR in Japan, but it is seriously damaging to the reputation of RIMS in the math community and those responsible for that institution need to come forward and address the issue.

Hi Peter Woit:

I feel surprised to see your comment, since Mochizuki did respond extensively in his personal blog:

https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/

I did not see any “absurd ad hominem argument” you are talking about. Scholze and Stix’s names did not even appear.

I am wondering – did anyone ever reached out directly to PRIMS’s editorial board on this? Isn’t it too quickly to reach a conclusion that “…is completely outrageous. It may be good PR in Japan, but it is seriously damaging to the reputation…” without some communication with the referees who have done respectable work on this paper? These people did spend eight years in going through the detail and understanding the paper. Should not these words be reserved for publishers that published Atiyah’s six sphere paper (https://www.springer.com/gp/book/9783319648125)?

I am not sure how you came to the conclusion that “…no one (including Joshi and Dupuy, two people who have been deeply involved in the study of IUT) has come forward to explain how Mochizuki can get around the problem…”. It is still possible that Mochizuki does not need to “get around the problem” because the problem does not exist in the first place; maybe it is caused by confusion and misunderstanding from both sides.

I feel confused reading through the blog. If someone with established expert status in a nearby field “give his/her nod” or “shake his/her head” regarding a paper, is this enough to replace the traditional journal referee process? If this is the case I would be more than happy to know Atiyah’s six sphere paper is complete, detailed, rigorous and correct – Atiyah claimed his proof was correct.

abc,

I read Mochizuki’s blog entry when it came out. To the extent I could make sense of it via Google Translate, it appears to be an argument that the people who see a problem with his proof (eg Scholze-Stix) are simply not understanding it because they are too dim-witted to understand the difference between an “and” and an “or” in a logical argument. This is just completely absurd.

I agree completely that it is the responsibility of the PRIMS editorial board to put forward publicly whatever mathematical report they received from referees showing that Scholze-Stix had misunderstood the argument and that the problem they pointed to does not exist.

As for the comparison to the sad story of Atiyah’s delusion in his declining years that he had a proof of the six-sphere question, it’s very telling that you see any relation at all between these two proofs.

As far as I understand the main objection brought up here by Peter Scholze is: Mochizuki considers infinitely many distinct isomorphic copies of

$\pi_(X) $’s, but can not tell us what goes wrong if we simply identify all of them with one another, and with $\pi_1(X_0)$ for some fixed $X_0$ – 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 $\pi_1(X)$’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 the $\pi_1(X)$’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: The ability to rigidify the relationship between the $\pi_1(X)$’s in one column when needed (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.

One way to prevent this absurd situation, i.e., the “proof” of abc being published by PRIMS, might be to organise an open letter/petition. The reason that I think that this might be useful is that even though many well-known mathematicians have expressed their views on Mochizuki’s papers, these have been scattered across several blog posts and comments. If there were a simple open letter, saying something to the effect that the undersigned have gone through the IUT papers and do not believe that they constitute a proof of abc, and this was signed by a relatively large number of experts, then this might have more of an effect and might even be picked up by the Japanese press.

Of course, for this to work some expert–I am not one, though I have read the papers–has to take the initiative.

naf,

The problems with such a petition are:

1. There’s a strong feeling among many that engaging in that kind of effort to get publicity for one side of a mathematics argument is inappropriate, that mathematical truth is not a topic for petitions.

2. Scholze-Stix are the only people who have spent the time directly engaging with Mochizuki needed to be completely sure, based purely on their own personal understanding of the mathematics, that he doesn’t have an answer to objections to his proof.

On the topic of 2., a huge part of the problem here is that, eight years after the proof came out, as far as I know no one except Mochizuki is willing to publicly claim that they fully understand the proof and can explain it to others, in particular that they can convincingly explain why Scholze and Stix are mistaken. Even in Mochizuki won’t travel or give talks explaining this point, why won’t anyone else?

There are mathematical organizations that arguably have some responsibility here and whose boards should consider some action. In particular the EMS is the publisher of PRIMS, and perhaps the IMU should consider the issue.

Peter Woit,

there is one aspect I find not really well represented in the current discussion. I think an author has also the right to make a mistake, and I do not mean deliberately, but in the process of scientific investigation. At the same time one has to take care not to inflate the meaning of scientific publishing and peer-review. It is a misconception to think the peer-review process should warrant absolute truth and correctness. Especially not in cases of relatively new and open fields.

In that sense one could take a lot of pressure out of the current debate, especially concerning the various close-to or ad-personam issues, if we would more agree on the fact that publication in an international journal is just another means of communication, if however one with a certain level of quality assurance.

These are just some thoughts that I have accumulated during 20 years of active scientific research.

RB,

I think you and many of my readers are not familiar with the culture of publication in mathematics, which by the nature of the field has a different aspect than that of other sciences. Assuring that the proofs of theorems given in a paper are correct is the main concern of the reviewing process at a math journal (secondary concerns are how interesting the theorems are and expository quality). What is going on here is completely unheard of.

Mathematics has a rigid, unyielding quality that differentiates it from other intellectual subjects. Either a proof is correct, with the claim logically following from the assumptions, or it isn’t. Unfortunately it’s not good enough for an argument to be fine at all steps except one, and this makes proving an important new theorem a high-pressure business. One can invest years in something which in the end simply doesn’t work. In order to make progress in mathematics, one needs to understand clearly when there is a correct and complete argument and when there isn’t. Mathematicians have the advantage of a much clearer boundary between what is understood and what isn’t than is typical in most fields.

Mostly this leads in my experience to a positive aspect of mathematical culture: people are used to finding that they are wrong and arguments about whether a mathematical argument is correct tend to not get personal. A colleague likes to explain that mathematics is the only subject he knows about where when two people go into a room disagreeing about whether something is true, almost always they come out with one of them agreeing he (or she) was wrong.

In this case, one side (Scholze-Stix) is making purely mathematical arguments, the other (Mochizuki-Fesenko) has engaged in unusual public argument about the competence of those who disagree with them. The very existence of this asymmetry is evidence for which side has the mathematics right.

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 $\pi_1(X)$ 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 self-evident (the logarithm is not its own inverse). It also cannot be salvaged by using some other/indeterminate identification of $\pi_1(X)$’s — this would at most change source and target by some field automorphism. So it’s unclear to me how the passage from $X$ to $\pi_1(X)$, and to $\pi_1(X)$ up to indeterminate isomorphism, is helping in this matter.

Peter Woit,

thank you for your explanations, you are fully right, to imply I am not a mathematician (also in our circles Mathematics is not considered a science but part of humanities (which is a bad translation for “Geisteswissenschaft”)). But for the one you don’t hit the nail fully on the head if you mean that in (other) sciences the peer reviews primarily “concerns are how interesting the theorems are and expository quality”. What I meant to address is that the peer review should mainly check technical and scientific consistency (call it quality). And here, definitely in the “sciences” is still plenty of space for technically and “scientifically” pretty flawless works to turn out to be “wrong” at a later stage of affairs.

It seems much less likely, but I am not convinced that all is that digital (either wrong or right) as you describe it. Let me attempt to construct an example: Lets take some theorem about a relatively involved concept like a perfectoid space with a technically fully correct proof. Let us assume there is some subtle issue in the definition of the perfectoid space which went undetected yet. Such a subtle issue might lead to a wrong theorem with a proof that relies on the definition, while the proof itself might still be technically fully correct.

I think even Maths is not immune against such problems. And such a problem could arise even if the proof is technically correct. In fields like arithmetic theory with very complex structures such cases seem not too unlikely.

Peter,

that it’s unheard of that a paper on mathematics gets published despite the referees not being completely sure of its correctness seems at odds with history; I guess Hales’ proof of the Kepler Conjecture would be the most famous counterexample. So what is unprecedented in this affair does not so much seem to be that Mochizuki has put forth a series of lengthy papers that the mathematical community is finding it hard to fully understand, but rather the complete breakdown in communication as is evident after the Scholze-Stix manuscript became public.

Any decision to (not) publish the articles will hardly change this highly unfortunate divergence of opinions that is going on now (and that I think is unprecedented).

BR,

The reference to concerns about interest of the theorems and the quality of the writing was just about mathematics, that these are important in evaluating a paper, but secondary to the issue of correctness of the proofs. In different fields of science I would think standards vary, depending on what characteristics of research are considered most important.

Yes, the example you give is relevant. Proofs can be wrong if they rely on problematic definitions or other results. When referees evaluate a math paper, normally they are starting from the same assumptions as the author about the correctness and unambiguity of the earlier literature. But this is exactly why it is considered so important that a new paper not contain an incorrect argument, since this can wreck the consistency and correctness of later published literature. What is going to happen if Mochizuki’s paper is published is essentially a fork in the mathematical literature. People will write papers that depend on Mochizuki’s results and may get these published in a new fork, but the majority of the community will have to reject such papers in order to preserve the consistency of the larger fork. This will not be a good situation.

Mahmoud,

The problem here is very different than the Hales case. In that case, referees were nearly certain the proof was correct, but could not be completely sure due to its complexity, making a computer check desirable. In this case there’s the bizarre situation that the referees seem to have agreed that the proof was correct, while the majority of experts believe the proof is simply incorrect.

If you read what Scholze has to say, he’s not saying that the Mochizuki proof is hard to check but probably correct. He’s saying that the proof is wrong, for fundamental reasons that he has explained and which the author has no answer to. The situation is in some sense relatively conventional: experts have looked at a claimed proof of a major result and identified a serious flaw, but the author has been unwilling to admit his proof has a flaw. The only unconventional thing is that given this situation a journal’s editors have decided to go ahead and publish the paper anyway.

Peter,

I am not convinced that there is any “strong feeling” against a petition, since such a situation has perhaps never occurred, at least in the last few decades. Also, the intended purpose would not be to “determine the truth”, but merely to express an opinion with the hope of convincing PRIMS .

It is clear that Mochizuki is unlikely to be ever convinced to withdraw his claims–no one outside his circle actually thinks the proof is correct and should be published in its current form–and nothing can be done about this. The unfortunate thing is that RIMS is now explicitly supporting him. I can only assume that there must be strong non-mathematical reasons–they cannot be unaware of the Scholze–Stix objections–for them doing so.

As you had said before, all doubts concerning the acceptance of the papers could be cleared if PRIMS released the referee reports, but this could only happen if they are acting in good faith and not under some strong external pressure.

naf,

I agree that something should be done about this, but there are significant problems with the petition idea, and I see no reason to believe that the PRIMS editorial board would agree to change its mind about publishing the papers in response to such a petition. There are a limited number of people who have responsibility for RIMS and this journal, and I suspect they’re hearing privately strong arguments that they should do something. I hope they live up to their responsibilities.

Peter,

I agree that the case of Hales is quite different from the IUT papers, but cf. the editorial note here for an approach one could take when publishing them. Furthermore, your contention that Scholze & Stix found a mistake in the proof and that Mochizuki should fix it or retract his claim isn’t unproblematic; I take the comment at the very end of his reply to S&S as saying that he completely agrees with their argument, but that this “absurd and meaningless” theory isn’t IUT but their own misguided simplification of it. So no essential progress at all has been made since he first made his papers public and the problem still remains that no one understands what he’s talking about. (I’m tempted to say that IUT appears to be not even wrong.)

To me the issue of whether PRIMS will publish the articles or not is minor since it seems most unlikely that any of the believers in IUT will change opinion if the editors decide to refuse. Having the work published would at least have the positive outcome of a final version and an end to the moving targets that Mochizuki has on his homepage (all the four IUT papers are updated this April and no changelog is provided).

mahmoud,

Yes, Mochizuki has claimed Scholze-Stix misunderstand him and are wrong, but virtually no experts in the subject believe this to be the case, based on hearing both sides of the argument.

The point of stopping PRIMS from publishing an incorrect proof is not to convince the true believers that it’s incorrect but to try and protect the integrity of the mathematics journal literature. If PRIMS does this, why shouldn’t other journals? Is it really all right if the standard for publication of a proof of a major conjecture becomes not whether it is right but whether the author and his allies have the political muscle to get it done? We increasingly live in Orwell’s world where truth no longer matters, but all need to resist being pushed farther down that path.

I am not sure that a petition is a right thing to do. However, a semester at IAS (or similar institutions) devoted to checking IUT would settle the matter. Experts who care about the general state of maths could step forward.

@Taylor Dupuy

I’m just saying that you can reverse the argument to an extent. Given two curves X_1, X_2, with an isomorphism of their fundamental groups, if you check that the geometrical pi_1 of X_1 is sent to the geometric pi_1 of X_2 by this isomorphism, 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 fact that X_1 is isomorphic to X_2 comes from Mochizuki’s first p-adic Grothendieck conjecture theorem. If this isomorphism of Galois groups of local fields does not arise from an isomorphism of local fields, then we can get a conjecture 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 pi_1s, is basically equivalent, modulo these prior results (I think…), to a special case of your question

@abc

> I feel confused reading through the blog. If someone with established expert status in a nearby field “give his/her nod” or “shake his/her head” regarding a paper, is this enough to replace the traditional journal referee process?

No one is suggesting that we decide this based on our prior knowledge of Scholze, Stix’s, and Mochizuki’s level of expertise. Instead you’re supposed to read Scholze-Stix’s critique and Mochizuki’s response and see – based on as much of the mathematics as you can understand, as well as the tone and such – who is more plausibly right.

One could also look at this comment thread. Specifically, note that everything Peter Scholze is saying is an elaboration of what’s in the original note. He’s not jumping to a new objection in every comment or anything strange like that. He’s just calmly explaining the key points of the argument. (Though much more clearly than I could explain them!)

@mahmoud

Even before the Scholze-Stix manuscript (but not too long), Scholze and Conrad (and maybe others) had publicly highlighted Corollary 3.12 as the key step in the proof that did not seem to make sense. There is no comparison to a long computational argument where case after case is checked but one can’t be completely sure there wasn’t a small but crucial mistake in one of the cases – here there are hundreds of pages of trivialities, followed by a single step which multiple mathematicians carefully read and independently observed was not properly justified. Of course many papers contain a step whose justification as given in the paper is not complete, but when these are pointed out, the purpose of the refereeing process is to fix such things.

@mahmoud again

What Peter Woit is attempting to do is not just report on some kind of they said – he said. He’s trying to look at what the two sides said and figure out if it is plausible or not. It’s not very plausible that someone would be able to look at their own theory and a radical simplification of it and not be able to explain why they are different, it’s also not very plausible that the explanations in Mochizuki’s note are valid.

It just simply is the case that any diagram in a category where all objects are naturally isomorphic is equivalent to a diagram where all vertices are labeled by the same object and edges are labeled by automorphisms or endomorphisms of that object. It is the case that, if the arrows are isomorphisms, the complexity and mathematical value of the diagram is controlled by the automorphism group of the object, and Mochizuki, in earlier work, bounded the automorphism groups of the relevant pi_1(X) (and even if he hadn’t, he isn’t able to point to any new lower bounds on the automorphism group or new isomorphisms more generally constructed in this work). It is the case that any argument for proving an inequality between two real numbers by combining a series of inequalities in various objects in the category of one-dimensional affine spaces over the real numbers can be converted into an argument for proving an inequality between two real numbers by combining a series of inequalities between two real numbers.

It is also the case that adding an ad hominem attack about a point of total irrelevance to your argument does not make it more convincing, but rather quite the reverse, and …

It is true that no progress has been made if you ignore these points and various similar ones. But if you accept any of these arguments then enormous progress has been made.

I think the sociological critiques of Mochizuki’s work are almost as damning as the specific mathematical critiques by Scholze-Stix et al. Namely, as Peter Woit has pointed out, there is no community of mathematicians who claim that they can understand and communicate Mochizuki’s work to the broader mathematical world; and furthermore the elaborate machinery developed by Mochizuki has so far apparently only been used to resolve exactly one famous conjecture in number theory, with no other applications. Consider by comparison the development of scheme theory. At the time they were developed (and still!) learning about schemes was considered very difficult; nevertheless, quickly several communities of experts across the world (in Paris, Boston, Moscow, etc.) emerged, and also almost immediately the theory was applied to many, many nontrivial problems. (I guess perfectoid spaces would be another similar example, although that might get too close to the personalities involved in the current dispute.)

anonymous,

Many people seem to remain under the misconception that the problem here is that Mochizuki’s proof is insufficiently well understood or checked. This was the problem a few years ago, but is not the problem now. The problem now is that a specific gap in the proof has been identified, but the author refuses to acknowledge this, while neither he nor anyone else can explain how to overcome the gap. Under these circumstances, no one competent is going to go spend a semester at the IAS discussing this (and, for his own reasons, Mochizuki himself is unlikely to do so either).

Peter, I largely agree with what you are saying, but I slightly disagree with this claim: “Assuring that the proofs of theorems given in a paper are correct is the main concern of the reviewing process at a math journal (secondary concerns are how interesting the theorems are and expository quality).” In our idealistic moments we might believe that this is the case, but I think that in most cases, the main concern of the refereeing process is to assess how interesting the theorems are. It is quite common for referees to

notcheck every detail of the proofs for correctness, but to trust the author.Having said that, I do agree that there are exceptions. If a paper claims to solve a well-known, difficult problem (as is the case here), then the referees are normally expected to scrutinize the correctness of the proof very carefully. Also, if a credible critic raises serious questions about the correctness of a proof (regardless of whether it’s a famous open problem), then everyone would expect a respectable journal not to publish the paper until those questions are satisfactorily addressed. I agree that the current situation is unprecedented. Even in the case of Hsiang’s proof of the Kepler conjecture, there was not this level of public criticism of the proof by other experts until

afterthe paper had been published.Will’s comment has insprired me to write a little bit more since he seems to be paying close attention (thank you Will). I kind of felt like Peter W.’s comments were a call to close the dialogue.

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 find these 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 to assert this for every structure in a “base hodge theater”?). This includes across the 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 true and 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 their contradiction.

Anyway, I still see Peter’s arguments as still being in the realm of a meta argument.

Maybe it is obvious to him an everyone else how to apply the definitions in IUT3 and convert these into proofs but this is not obvious to me. Again, for me 3.11 is made up of about 15 different assertions and uses a large number of definitions that I don’t have at my fingertips…

@w

I mostly agree with what you write, I only find it a sad sort of “progress” that it has become increasingly clear that Mochizuki can’t explain why he thinks the alleged gap isn’t really there. If one assumes that he is behaving honestly, he believes himself that he has a correct proof but is clearly failing completely to communicate his ideas to other mathematicians; and that issue has been apparent ever since he made his papers public.

My issue is mostly with the focus on publication in PRIMS; the “integrity of the mathematics journal literature” (to the extent there still is one) will survive I’m sure, even though the integrity of this particular journal might suffer. On the political side of this affair it’s more worrying that some people apparently are trying to obtain money from the UK government for research into a theory that increasingly seems like a mathematical case of being not even wrong.

This thread is somewhere between depressing and necessary – thank you PW for hosting it. But I would be far more interested in a comment thread about Peter’s Fontaine rings over Z. Wow. What is he about to do with them?

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 $\pi_1(X)$

(2) The non-necessity to replace $\pi_1(X)$ with infinitely many distinct copies of it

(3) The inconsistency of the identification of various copies of ordered $1$-dimensional 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 the huge diagrams that Mochizuki considers are actually relevant to the argument. Neither (1) nor (2) alone would really falsify anything; but if non-necessary, not much is left of these manuscripts.

As only (3) is about an actual mistake, we focused on this in our manuscript. (1) is Remark 9, and (2) is Footnote 8 in my manuscript with Stix. In my first comment 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 apologize that 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 machinery that 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 to do, 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 never uses 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 these identifications.) 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 $\pi_1(X)$’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 the main point, (3)), as Mochizuki of course asserts that this is false, and there are thousands 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 basically Footnote 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. More generally, (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 just answer 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 might work if all of these copies are simply the same. (Maybe the idea is that one of the identifications we use to get nontrivial monodromy in (3) is omitted by having distinct copies of $\pi_1(X)$’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 $\pi_1(X)$’s.

I very much do not want to close the remarkable and very valuable dialogue going on here about the mathematical issues with the Mochizuki proof. Taylor Dupuy however was correctly picking up on my wanting to help make sure the mathematical discussion stays focused on these issues. A broader discussion of the mathematics would quickly get beyond my already marginal abilities to moderate something like this.

Note that this particular comment thread is being moderated with an even heavier hand than usual: most submitted comments are getting deleted in an effort to keep the discussion focused and informative.

I’m sorry to keep banging on about it, but Peter S’s (2) is an artefact of a 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 R^n (this is LbEx5). Or that you incorrectly calculate the perfection of a ring if in the usual sequential colimit you don’t create separate copies of the ring 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 that the 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 to something 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 for this (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 can consider. Alternatively, one can pass to a skeleton of the category at hand, in which 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, the actual mathematics can then be discussed, for instance something like creating a formal colimit of a diagram all of whose objects are pi_1(X) (or whatever), with appropriate gluing maps. But this is not what I wanted to address, and is out of my sphere of expertise.

(on a separate note: thanks to Peter W for his patience and willingness to host this public discussion)

@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 two neighboring columns of log-links and disregard for the moment the theta-link (I think 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 $\pi_1(X)$’s in the columns (by identifying them using identity maps, making the logarithms Galois-equivariant), there is no “switching-symmetry” permuting the two 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, 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).

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)] the issue we are talking about “why $\pi_1$ 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.

@David Roberts

Please keep banging on about it! I think your interpretation of Mochizuki’s argument in the response was an important insight and underappreciated.

Let me ask you what I think is an important follow-up question. Suppose you take the same argument and present it in two different languages – one, the standard categorical language, and two, Mochizuki’s language where distinct copies of an isomorphic object are relevant for colimits and other categorical constructions. Assuming no other knowledge of what the argument actually is or how it is written, which language is more likely to conceal a subtle error in calculations or other mistake, and which language is more likely to make such mistakes easier to see?

I think this question is within your sphere of expertise, and the sphere of expertise of many other mathematicians – far more than have read carefully a portion of the document.

I also think its relevance to the broader topic of discussion is clear.

David Roberts: Thanks for your thoughts on this!

Actually, something related happened in our Kyoto discussions: We realized that 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 that we better focus on some actual mathematical statement where related issues undisputably 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 essentially comes 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 logarithm map (“the log-link”) and some isomorphism (“the Theta-link”) of two copies of a local Galois group acting (trivially) on a monoid isomorphic to $\mathbb{N}$ (I’m obviously 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 $\mathbb{N}$ 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 of course 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 $\mathbb{N}$) and the relation of this monoid isomorphic to $\mathbb{N}$ to the local units is completely external.

So for all we can see we simply followed the procedure you suggested, and reinterpreted his distinct copies in the way usual mathematicians think. We recorded the outcome of this in our manuscript: You can read there the details of 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 once-punctured elliptic curve with bad semistable reduction, and recover its Tate parameter 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 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 $\pi_1(X)$’s are equivalent to relevant $X$’s, so of course you can recover all invariants of $X$, and you can do so functorially in $\pi_1(X)$.

Generally, a point seems to be made that Mochizuki’s algorithms have some magic power and that really the content of the algorithms is critical, so in the context of the previous paragraph, it would matter in some way how I invert the functor $X\mapsto \pi_1(X)$ 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 the following 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 profinite group 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 the elliptic 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 essence of this “algorithm”; Mochizuki confirmed this.

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 theta side 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 sure everyone will agree, statements need to be pinned down a bit more and directly tied 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 take isolate single claim in IUT and just start running with it, then build out from there.

–Regarding not using automorphisms of fundamental groups: we should observe that without indeterminacies, there are no indeterminacies. This is tautological, yes, but indeterminacies appear in the statement of Cor 3.12 and without them we are really talking about something else entirely.

Also, this all goes back to how the indeterminacies allegedly afford us the ability 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 be found 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 bunch of (useful) isolated facts that readers would need to assemble for themselves (if it 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 $\mathbb{R}$ to talk about $\mathbb{R}^2$. I’m not sure this is what 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 more convenient 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 $\operatorname{Aut}( \pi_1^{\operatorname{temp}}(X”_{\underline{v}}))$ on these interpretations (last time I tried a double underline it didn’t work out so I’m using double prime this time, here we need to take an analytification or formal scheme with log structure). Technically speaking, I suppose $\operatorname{Aut}( (X”)_{\underline{v}})$ 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 definitions of “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. 😛

–I am intentionally omitting a discussion on “power”. If people want me to step into this world of speculation I can do it. I’m getting more of a “show me the money” vibe though.

Peter: I think we posted at the same time… I’m going to read your comment now.

Quick comments:

–Peter, I think the $\mathbb{N}$’s should be regarded as embedded inside a monoid with enough roots of unity and $n$th roots to do Kummer theory.

–You definitely need more than one part of the “Frobenius-like” objects of the 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 free construction. 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 long time and it isn’t a “Mochizuki thing”. For the most part I have found that

algorithms = 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” of structures that interpret lower structures.

I talk about this a little in my first manuscript 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 “at the bottom” you take. This thing you construct independent of the lift (by the way this points to the automorphisms of fundamental groups you were looking 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 measure spaces cooked up from absolute galois groups of p-adic fields at $\underline{v} \in \underline{V}$. 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.

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 that probably 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 is provably too strong. I agree it would be hard to “prove” a strong enough form of (2), but I think the burden of proof is on Mochizuki here, to show where the argument 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 $\mathbb{N}$’s are manifestly not part of monoids in which you can do Kummer theory! Yes, Mochizuki includes some extra 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 about part (i), not (ii), of Theorem 3.11.

OK, I quickly looked at your manuscript. If I understand it right, that’s the upper left slanted arrow? In our terms, that’s the difference between the “abstract” Theta-pilot — a generator of an abstract monoid $\mathbb{N}$ — 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, then the 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 careful with 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 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 to have (up to indeterminacies etc…).

Yep, the theta pilot doesn’t map to the actual theta values *on the theta side*. On the q-side it does.

I will check your manuscript again in a bit. Dinner then bedtime (I am barbecuing).

I want to check again to make sure I didn’t miss something.

I realize I have my phone… 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.

@W

gosh, thanks!

This is tricky: it depends who’s reading it. Who are you envisaging seeing mistakes? I can’t imagine (ignoring the fact this is IUT and tremendously baroque) that someone who’s had a decade of practice with their own idiosyncratic style of working would make mistakes more frequently that someone using the language of the majority, all things being equal, apart from the fact the latter person has more potential external checks and balances. This latter point I think can’t be overemphasised. Andrew Wiles was still speaking the language of his community by the time he emerged with his (first attempt at a) proof of FLT, and even engaged the help of someone else to try to check the subtle parts of the argument before that. This hasn’t happened here…

A bigger problem is the rigid commitment to definitions/structures that are explicitly admitted as being far more general than necessary (*cough* Frobenioids *cough*). This increases the friction for potential eyes on the IUT papers, if you’ll permit me a worrying metaphor mix.

Peter: Below is a more detailed response.

To recap, you made three posts. One in response to David and UF, and then two smaller ones. I’m going to address the newer ones first and then go back and address those older comments.

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Responses to Newer Comments

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Regarding the “cuts”: I think I addressed this. Let me know if you want to talk about it more. I think if you are not taking two log-linked strips of Hodge 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 such large objects, I know this is super abusive… I’m sure there is a better way to talk about this than to make blanket references to Hodge Theaters — really the relevant “official constructions” here use the $\prec$ 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 to clarify 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 “one fundamental 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 or modifies 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 $\mathbb{N}$’s and Kummer Theory in the $\Vdash \blacktriangleleft \times \mu$ prime strips:

As a recap: You stated that they are isolated, I said the have a Kummer theory. I want to clarify: technically you are correct the monoids in a components at a bad place of $\Vdash \blacktriangle\times \mu$ prime strip are just $\mathbb{N}$’s. BUT… they are interpreted structures. So they come from other structures where it does make sense. Note also that in isolation, there is no action of the $\mathbb{N}’s$ on the corresponding copies of $\mathcal{O}_{\underline{v}}^{\times \mu} \otimes \QQ$. (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 and value group portions”). I can speculate a little about what is going on on either side of the theta link if you want.

*Subremark. This is sort of an ongoing theme I want to highlight: Given a structures $A$ and $B$ there may exist two ways of interpreting $B$ in $A$ (called them $\mathbf{B}_1$ and $\mathbf{B}_2$) which are not equivalent. This for example could for example be distinguished by the representations $\operatorname{Aut}(A) \to \operatorname{Aut}(\mathbf{B}_i(A))$ (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 $\Vdash \blacktriangleleft \times \mu$ 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 absolute galois groups of p-adic fields. I said some words about lifting $G$’s to an interpretation so we view $G$ as $\mathbf{G}(\Pi)$ for some fundamental group $\Pi$ and said these constructions are uniform in $\Pi$ (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 collections of prime strips — I made need to modify the structures in this statement to make it exactly correct). I had made some remarks about 3.11.ii being an independence result. Let me clarify, I was talking about $U_{\Theta} \subset \mathbb{L}^{\vdash,et}$ (the “(coarse) multiradial representation of the theta pilot region” inside the so-called “mono-analytic 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 this $\mathfrak{R}^{LGP}$ 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 $U_{\Theta}$ structures (used to construct $U_{\Theta}$). 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 to a 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.)

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Response to Response to David Roberts and UF

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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 start 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 perspective of curves. It does look funny. What is in this fundamental group sauce?! Here are 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 space with a binary operation. The formalism of interpretations applies here (maybe by slightly increasing the signature), but you get to put everything in this nice box where represenations become automatic and you get to see how everything varies with respect to automorphisms. (Full disclosure, Mochizuki objects to the language 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 HTbar is a log-linked collection of Hodge Theaters (which we view as a single massive infinitely sorted structure) the definable set we need is in like HTbar^2 (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 shape of the inequality. I’m not going to do it now, but I’m going to make a note to try to talk about:

*Ind2 and moving around $q^{j^2/2l}$ — why you need Ind3

*Analytic number theory, what happens at large discriminants, and toy phenomenology for the inequality.

IOU: 3.11.i discussion.

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Oh dear, all my triangles are off. The pointy bit should be to the right…