## Mochizuki on ABC

[Update:  Lots of traffic coming in from Hacker News, much of it presumably from outside the usual pro number theory crowd that reads this blog.  If you're not already familiar with the ABC conjecture, I recommend Barry Mazur's beautiful expository paper "Questions about Number."]

[Re-update:  Minhyong Kim's discussion on Math Overflow is the most well-informed public discussion of Mochizuki's strategy.  (Of course, it is still very sketchy indeed, as Minhyong hastens to emphasize.)   Both Kim's writeup and discussions I've had with others suggest that the best place to start may be Mochizuki's 2000 paper "A Survey of the Hodge-Arakelov Theory of Elliptic Curves I."]

Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV:  log-volume computations and set-theoretic foundations.”

I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:

I hope it’s true:  my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.

Well, now the time has come.  I have not even begun to understand Shin’s approach to the conjecture.  But it’s clear that it involves ideas which are completely outside the mainstream of the subject.  Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

Let me highlight one point which is clearly important, which I draw from pp.3–6 of the linked paper.

WARNING LABEL:  Of course my attempt to paraphrase is based on the barest of acquaintance with a very small section of the work and is placed here just to get people to look at Mochizuki’s paper — I may have it all wrong!

Mochizuki argues that it is too limiting to think about “the category of schemes over Spec Z,” as we are accustomed to do.  He makes the inarguable point that when X is a kind of thing, it can happen that the category of Xes, qua category, may not tell us very much about what Xes are like — for instance, if there is only one X and it has only one automorphism. Mochizuki argues that the category of schemes over a base is — if not quite this uninformative — insufficiently rich to handle certain problems in Diophantine geometry.  He wants us instead to think about what he calls the “species” of schemes over Spec Z, where a scheme in this sense is not an abstract object in a category, but something cut out by a formula.  In some sense this view is more classical than the conventional one, in which we tend to feel good about ourselves if we can “remove coordinates” and think about objects and arrows without implicitly applying a forgetful functor and referring to the object as a space with a Zariski topology or — ptui! – a set of points.

But Mochizuki’s point of view is not actually classical at all — because the point he wants to make is that formulas can be intepreted in any model of set theory, and each interpretation gives you a different category.  What is “inter-universal” about inter-universal Teichmuller theory is that it is important to keep track of all these categories, or at least many different ones.  What he is doing, he says, is simply outside the theory of schemes over Spec Z, even though it has consequences within that theory — just as (this part is my gloss) the theory of schemes itself is outside the classical theory of varieties, but provides us information about varieties that the classical theory could not have generated internally.

It’s tremendously exciting.  I very much look forward to commentary from people with a deeper knowledge than mine of Mochizuki’s past and present work.

## 52 thoughts on “Mochizuki on ABC”

From page 31 of the first of the 4 papers, it almost looks as if the method doesn’t (as presently written) produce explicit constants, as one would use to apply ABC to bound heights of points on curves. Hopefully as the ideas become more widely understood, effective constants will be able to be extracted. Nonetheless, certainly very exciting.

3. Terence Tao says:

Well, I’m certainly no expert on any of Mochizuki’s work (having only heard about it through you), and can’t pass judgement on it, but I have always been fond of the idea that model-theoretic connections between objects (e.g. relating two objects by comparing the sentences that they satisfy) are at least as important in mathematics as the more traditional category-theoretic connections (where morphisms are the fundamental connective tissue between objects) or topological connections (where the objects are gathered into some common topological space or metric space in order to compare them). A good example is the Ax-Grothendieck theorem, in which a result that is easy to prove in the positive characteristic setting can then be transferred to the characteristic zero setting by a model-theoretic connection, even though there is no immediately obvious morphism, functor, or natural transformation from positive characteristic to zero characteristic, nor is there an immediately obvious topological sense in which the zero characteristic setting is a limit of the positive characteristic one. (This is not to say that there the connection between positive characteristic and
zero characteristic that is relevant for Ax-Grothendieck can’t be thought of in categorical or topological terms if one really wanted to view it that way – it probably can – but that the model-theoretic way of looking at it seems much more natural, in my opinion.)

Scheme theory, in my mind, does an excellent job of capturing the categorical and topological ways of connecting objects in algebraic or arithmetic geometry, but only engages in the model-theoretic connections in a rather restricted fashion. (An ideal in a commutative ring can be thought of model-theoretically as the set of all identities that can be deduced from a set of generator identities from the laws of commutative algebra (i.e. high school algebra), and so schemes capture the model theory of this algebra well; but there isn’t an obvious mechanism in place in scheme theory to capture the model theory of more sophisticated theories that might also be relevant in arithmetic geometry.)

4. Harald says:

What do you people think should go into a study plan leading into a state where one might be able to read the proof? Presumably model theory – and what else?

5. NDE says:

It would be shocking enough (and would fully justify JSE’s “holy crap” keyword) just to have a proof of ABC; effective (let alone explicit) constants may be too much to hope for, since we don’t have them yet even for Roth’s theorem, only an effective bound on the number of exceptions. But for Roth, and other ineffective estimates in number theory (ranging from Siegel on class numbers to Faltings), the source of ineffectivity is that one needs more than one counterexample to get a contradiction. Can anybody tell whether Mochizuki’s approach works this way too (assuming it works at all)?

Considering the extensive previous work of his upon which this all seems to depend (based on the Introductions to the papers), it seems to entail a whole lot more algebraic geometry than model theory (even though he is reinventing the theory of schemes to suit his purposes).

7. Terence Tao says:

In page 1 of the slides at http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf , Mochizuki has an interesting hierarchy of increasingly abstract and general theories, starting with schemes over Z, then geometry over F_1, then Arakelov geometry, and finally his inter-universal geometry. So I guess in particular that Mochizuki now has a rigorous interpretation of the field of one element?

8. [...] Conjecture: Did Shin Mochizuki solve the ABC Conjecture?Some preliminary details here: http://quomodocumque.wordpress.c…And some others here:http://en.wikipedia.org/wiki/Abc…Cannot add comment if you are logged out. [...]

9. [...] Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin [...]

10. Pasten says:

The ABC conjecture is just one consequence. If I understand correctly, he claims to have proved Vojta’s conjecture for curves and number fields of bounded degree (no, they are not equivalent, not even using Belyi maps unless you are actually assuming a very strong form of the ABC conjecture).

11. heres a fixed version of the link to the slides:

12. Tuck Fheman says:

http://is.gd/7ANqym

13. javier says:

I haven’t seen any rigorous F1 in Mochizuki’s notes, or any reference at all apart from that mention on the slides (although I have to say I only skimmed at them). I think in the slides he is just alluding at the ‘motivating principle’ of F1 geometry, i.e to find some category of schemes ‘beyond Spec Z’ which provides enough flexibility to prove some classical statements.

Sasha Smirnov found an interpretation of ABC conjecture in terms of F1 geometry which relied on studying the intersection theory of $\mathbb{P}^1_{\mathbb{F}_1}\times_{\mathbb{F}_1} Spec \mathbb{Z}$ and finding an analogue of the genus formula there. I cannot tell if this was Mochizuki’s original motivation, but Smirnov’s paper is not cited among his refernce, so I’d say his is a different angle from the F1-program.

14. [...] En este post de Quomodocumque nos hablan sobre todo este tema (con comentarios de, por ejemplo, Terence Tao). Y en éste de Not Even Wrong también tratan el asunto y además nos explican algunas cosas sobre lo que ha construido Mochizuki con su trabajo. La cuestión es que Shinichi Mochizuki ha desarrollado un conjunto de técnicas nuevas que generalizan la geometría algebraica al que ha llamado geometría inter-universal (reconozco que el nombre da que pensar, pero bueno, nada es perfecto). Según comenta el autor de Not Even Wrong, Peter Woit, In essence, he has created a new world of mathematical objects, and now claims that he understands them well enough to work with them consistently and show that their properties imply the abc conjecture. [...]

15. Pasten says:

I think that the meaning of “asymptotic behavior” and related comments in the page that you quote, are more of philosophical nature rather than hints on the structure of the proof. About effectivity (at least from a theoretical [non-practical] point of view) there are several reductions to bring the situation to a class of convenient elliptic curves where the main theory applies, and it looks to me that at least these reductions can be made effectively (ok… not a very informative remark but this is as far as I have gone in these few days since the proof was made public).

Above it would have been more accurate for me to have used the word “re-thinking” rather than “re-inventing”.

17. Tom Leinster says:

@Terry: Wilfrid Hodges begins his book A Shorter Model Theory with the slogan:

model theory is algebraic geometry minus fields.

18. Ulrik Buchholtz says:

It seems to me that Mochizuki defines species, mutation and morphism-of-mutation as being whatever defines a category, functor or natural transformation in any model of ZFC. Thus, by the completeness theorem, these are nothing but formalizable-in-ZFC categories, functors, and natural transformations.

I don’t understand from his examples (absolute anabelian geometry and the log-theta-lattice, of which I nothing), why this is useful, so I’d appreciate it if someone could provide a more down-to-earth example.

On another note, on p. 43 Mochizuki seems to claim that Grothendieck set theory is a conservative extension of ZFC, which it certainly is not. What the Feferman-paper he refers to shows, is that an extension of ZFC with a constant for a universe satisfying schematic reflection, ZFC/s, is a conservative extension of ZFC, and that’s obviously because every model of ZFC can be extended to a model of ZFC/s (using a Löwenheim-Skolem construction), but a model of ZFC/s need not be a model of Grothendieck set theory.

19. Terence Tao says:

On reading the Mochizuki paper, it seems to me (if I am not mistaken) that all the set theoretic and model theoretic stuff is really only used in Section 3, whereas the ABC conjecture is proven (at least allegedly) by Section 2. As far as I can tell, there are no forward references to Section 3 in previous sections other than side remarks. So perhaps all the set and model theory here is in fact something of a red herring as far as the application to ABC is concerned, and are primarily relevant for further development of Mochizuki’s inter-universal geometry instead? (Among other things, this would render the issue of the non-conservative nature of Grothendieck set theory somewhat moot.)

20. Ulrik Buchholtz says:

Certainly, and in any case I meant to add that the conservative extension claim has no bearing on the argumentation at all, and Mochizuki only added it to make the reader feel more comfortable with Grothendieck universes. I didn’t mean to make it sound like a potential issue.

21. Terence Tao says:

Minhyong Kim has described a zeroth order approximation to such a study plan at http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc/106658#106658

Short of Mochizuki himself making a suggestion, this is probably as good an answer to Harald’s question as one can hope for at this stage.

22. The ‘inter-universal’ aspect is used in previous papers in the series, I think mostly in paper III, dealing with the ‘log-theta’ lattice. This is used for paper IV (which is what I gather you mean by ‘the Mochizuki paper’, Terry). I think the complicated set-theoretical language is a red-herring, stemming from Mochizuki’s unfamiliarity with the relevant category-theoretical approaches in this area – and which perhaps are themselves waiting, like the statue in the stone, to be uncovered by the sculptor.

23. [...] If you’d like a taste of the math in Mochizuki’s work and a sense of the interest among research mathematicians in his work, check out the discussion on mathoverflow.  Also take a look at Jordan Ellenberg’s blog post. [...]

24. Terence Tao says:

Yes, I should have clarified that I was referring to the IUTT IV paper. (But on skimming through the previous papers, it appears to me that there is no significant amount of set theory explicitly used in any of the previous papers, so I am not sure how the inter-universal aspect of Mochizuki’s theory is actually being implemented in these papers.)

25. Thomas says:

My impression is that it is not so relevant what may or may not be expressable in the usual formalisms, but that Mochizuki thinks, both the “anabelian” and the “selfreferential” themes in algebraic geometry are acutally general and basic aspects of it and should be imaginated in a most basic and general way, therefore his “loops”. I find this idea nice and fitting to some other things, actually with a very “Grothendieckary” touch.

26. Vesselin says:

As far as I can see, unlike Roth’s theorem, Mochizuki’s logic seems to be, in this regard, completely straightforward: it does not even seem to proceed “by contradiction,” let alone by examining more than one counterexample. This would only make it all the more shocking, of course.

The starting point, as explained in Kim’s post, indeed seems to be the quest for an arithmetical interpretation of the “Kodaira-Spencer map” proof of Szpiro’s (geometric) inequality. But there is little indication in Mochizuki’s IV-th paper on the true origin – and the means of surmounting of – the quintessential “epsilon” of ABC. (Again very much unlike Roth’s theorem, where: (1) the epsion-threshold is transparently surmounted by using sufficiently many degrees of freedom, and sufficiently many counterexamples; and (2) the \epsilon.log(q) bound is transparently optimal: it may not be improved to a sublinear function of log q).

Could it really be that the “epsilon” in Mochizuki IV comes through, entirely, his use of noncritical Belyi maps in his previous paper “Arithmetic elliptic curves in general position,” used essentially in IUTT-IV? (cf. the bottom lines on p. 40 of IUTT-IV) http://www.math.okayama-u.ac.jp/mjou/mjou52/_01_mochizuki.pdf .

27. [...] “It is going to be a while before people have a clear idea of what Mochizuki has done,” Ellenberg told New Scientist. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” he added on his blog. [...]

28. Terence Tao says:

The polymath wiki happened to already have an existing page on the abc conjecture for a different reason (it was relevant to the Polymath4 project), but it now has a number of sections on the Mochizuki proof, in particular collecting links to various discussions about this proof (including this blog post).

30. [...] Ellenberg on Mochizuki [...]

31. [...] Times Science Tuesday section has an article on Mochizuki and his papers, which also links to a post by Jordan Ellenberg, in which Brian Conrad, Terence Tao, and Noam Elkies among others make [...]

32. I can’t tell whether http://mochizukidenial.wordpress.com/ (which is linked here, a “track back” I guess) is real or a parody.

33. Artie Prendergast-Smith says:

Yes, it’s very strange. It’s not at all clear to me what it should be a parody of; on the other hand, it’s full of (seemingly) sarcastic descriptions of the papers (“riveting”, “epic”, “masterpiece”), but not much mathematical content that I can discern.

34. [...] http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ This entry was posted in cstheory and tagged nt.number-theory by admin. Bookmark the permalink. [...]

35. [...] Even Wrong, Proof of the abc Conjecture, here.  Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin [...]

36. quasihumanist says:

Maybe it’s a thought experiment contrasting the philosophy of science and philosophy of mathematics?

37. [...] pronounce thing correctly. One is a little reassured when a professional mathematician blogs in Quomodocumque: I have not even begun to understand Shin’s approach to the conjecture. But it’s clear that it [...]

38. Artie Prendergast-Smith says:

(reply to quasihumanist: can replies only be nested three times?) Maybe, but that doesn’t seem to be what the author is saying in “About Mochizuki Denial”:

To be honest, I’m not sure it bears too much scrutiny, so maybe it’s best to wrap up the discussion at this point.

39. David Metzler says:

Is Mochizuki’s use of model theory at all related to, say, Hrushovski’s work? If so that would make it a little less out of the blue, correct? Or is this utterly different?

40. Terence Tao says:

I’m not sure which work of Hrushovski you are referring to, but in his work on approximate groups at least, the main use of model theory is to transfer a given problem (such as a finitary combinatorial problem) to a model with better model-theoretic properties (e.g. saturation with respect to a large ordinal, or a large group of automorphisms), in order to obtain a rich structural theory of definable sets and related objects. I don’t really understand how Mochizuki is using model theory, but it seems from a superficial reading that he is trying to capture the “functor-like” properties of various arithmetic geometry operations which, for one reason or another, aren’t functors in the classical sense, by instead viewing these operations as maps (he calls them “mutations”) connecting various models of set theory to each other. So it seems to be a rather different way to introduce model theory into the situation.

(But, as mentioned earlier, this model-theoretic formalism is only used at the end of the paper, after the applications to the abc conjecture and Diophantine geometry. So, as intriguing as this formalism may be, it might not be necessary to understand it if one is only interested in the abc application.)

41. [...] this is because they have to learn his new objects first. Here is part of what Jordan Ellenberg says about them: Mochizuki argues that it is too limiting to think about “the category of schemes [...]

42. Terence Tao says:

It looks like Vesselin has located a serious “red flag” in Mochizuki’s argument, in that the main Diophantine inequality claimed in IUTT-IV appears to have a robust family of counterexamples (assuming the truth some plausible conjectures, including abc): http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279

This doesn’t tell us exactly where the source of the error is coming from, though, or how fixable it would be. But it would be difficult to be optimistic about the proof until this issue is somehow resolved.

43. Vesselin says:

By the way, leaving aside history and motivation and dictionaries, I believe the two most essential papers logically preceding the IUTeich series to be actually, by and large:

- Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (Feb 2012);

and

- The Etale Theta Function and its Frobenioid-theoretic Manifestations (2008).

These are the two most prominent previous papers from each of the two main trends in Mochizuki’s proof: respectively, the “anabelian reconstruction” software [for number fields equipped with a hyperbolic curve related to a once-punctured elliptic curve]; and the “theta-evaluation on l-torsion points” in the spirit of HAT (developed, however, right from the beginning in the rather different framework of “geometry of categories”). They are certainly the two most heavily cited papers in IUTeich.

As for HAT and GTKS proper, it appears that as far as the actual proof is concerned, their results could be best read as guiding principles or heuristics. (While of course they are perfectly rigorous mathematics, and autonomous in their own right – only insufficient or ill-adapted for the actual diophantine applications.)

44. Is it just me or are the recent Mochizuki papers, notwithstanding their mathematical content, not polished to the extent one normally sees in a mathematics research paper. The statements of theorems – for example, Theorem 1.10 in IUTT-IV, do not seem as clear or self-contained as in other papers; and their grammar seems like its under considerable tension.

45. [...] that I have found two blog posts discussing it, one from Terrence Tao the other from the blog Quomodocumque, and just mentioning Terrence Tao, always seems to put some weight behind the validity. It even [...]

46. Vesselin says:

I am rather perplexed by what he writes in point (4.).

1. First, if “ramification index” and “moderately ramified” have the same meaning as in Def. 1.9 in IUTT-IV – rather than, say, the order of the q-parameter of the elliptic curve – then (by what he writes in the second sentence of (4.)!) there would be no change at all with the Masser examples at hand (which are over Q).

2. Second, at the beginning of page 3, he indicates an adjusted contribution on the order of \omega(N).log(log(N)) – log(N), which may be very negative. Taking \epsilon := (log N)^{-2} and an admissible prime l between log(N)^2 and 3.log(N)^2 (which surely exists, as argued for instance in the MO post), would yield an estimate of the kind

(1/6)log(D) < log(N) + [adjusted contribution] + O(log(log(N)))

(I write D for the minimal discriminant of the elliptic curve; log(D) is essentially Mochizuki's log(q), although technically speaking he is not counting the contribution at the place 2. This will not make any difference if we restrict to abc-triples with bounded 2-adic valuation).

Now, an inequality (1/6)log(D) < \omega.log(log(N)) + O(log(log(N))) is certainly wrong. (I could give a counterexample if it comes to this.)

On the other hand, Baker's conjectural form of ABC implies (upon taking \epsilon = (log(N))^{-2} again) that the bound

(1/6)log(D) < log(N) + \omega(n).log(log(n)) + O(log(log(N)))

is certainly plausible (it would be a terrific explicit form of the ABC conjecture!). But then the adjusted contribution ought to be simply \omega(N).log(log(N)), without the "-log(N)" term.

47. [...] most popular post, by a mile, was my post alerting the community to Mochizuki’s claimed proof of ABC, which was linked to by several big sites like Hacker News.  It’s been viewed over 50,000 [...]

48. Steve Silverman says:

So, what’s the deal? Is it correct or not?

49. [...] the attempted solution of the ABC conjecture by Shinichi Mochizuki. (See, e.g.,  here, here, here, and here.) I did not think that this has much to with me until I discovered yesterday in my room [...]

50. mentoR says:

Reblogged this on Rigorous Fantasy.