Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki. More than five years ago I wrote a posting with the same title, reporting on a talk by Lucien Szpiro claiming a proof of this conjecture (the proof soon was found to have a flaw). One change over the last five years is that now there are excellent Wikipedia articles about mathematically important questions like this conjecture, so you should consult the Wikipedia article for more details on the mathematics of the conjecture. To get some idea of the significance of this, that article quotes my colleague and next-door office neighbor Dorian Goldfeld describing the conjecture as “the most important unsolved problem in Diophantine analysis”, i.e. for a very significant part of number theory.

Jordan is an expert of this kind of thing, and he has some of the best mathematicians in the world (Terry Tao, Brian Conrad and Noam Elkies) commenting, so his blog is the place to get the best possible idea of what is going on here. After consulting a couple experts, it looks like this is a very interesting and possibly earth-shattering moment for this field of mathematics. In the case of the Szpiro proof, the techniques he was using were relatively straightforward and well-understood, so experts very quickly could read through his proof and identify places there might be a problem. This is a very different situation. What Mochizuki is claiming is that he has a new set of techniques, which he calls “inter-universal geometry”, generalizing the foundations of algebraic geometry in terms of schemes first envisioned by Grothendieck. 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.

What experts tell me is that, very much unlike the case of Szpiro’s proof, here it may take a very long time to see if this is really a proof. They can’t just rely on their familiarity with the usual scheme-theoretic world, but need to invest some serious time and effort into becoming familiar with Mochizuki’s new world. Only then can they hope to see how his proof is supposed to work, and be able to check carefully that a proof is really there, not just a mirage. It’s important to realize that this is being taken seriously because such experts have a high opinion of Mochizuki and his past work. If someone unknown were to write a similar paper, claiming to have solved one of the major open questions in mathematics, with an invention of a strange-sounding new world of mathematical objects, few if any experts would think it worth their time to figure out exactly what was going on, figuring instead this had to be a fantasy. Even with Mochizuki’s high reputation, few were willing in the past to try and understand what he was doing, but the abc conjecture proof will now provide a major motivation.

Mochizuki has been at this for quite a while. See this page for some notes from him about how he has been pursuing this project in recent years. This page has notes from lectures he has given on the topic, starting in 2004 with A Brief Introduction to Inter-universal Geometry. For the proof itself, see here, but this is the fourth in a sequence of papers, so one probably needs to understand parts of the other three too.

**Update**: Barry Mazur has recently made available his 1995 expository article on the abc conjecture, entitled Questions about Number.

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Safari doesn’t seem to like your link to the paper…

Not just Safari, my bad… Now fixed. Thanks Andrew!

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A pity he’s too old for the Fields Medal! If he were 36, there would be some pressure on the Fields committee to form an opinion of his work in the next two years…

The abc conjecture…surely it’s easy as 123, simple as doe rae mi?

There is some interesting discussion of this in response to a post by John Baez on Google+. See: http://tinyurl.com/cnh5gks

Apparently, Mochizuki declined to come to New York to discuss his work, which is interesting but there could be any number of reasons for it.

Marko,

With this kind of announcement, I suspect Mochizuki has all of a sudden received a very large number of invitations to give talks. From what I hear, he’s not someone who likes to travel, so he’s probably now turning down lots of such invitations (including the New York one).

You might be right that Mochizuki’s dislike of travelling could have something to do with his rejection of the invitation, but I suspect another reason may be that (as his friend Minhyong Kim says) Mochizuki is in all likelihood the only person in the world who is familiar with the concepts and ideas in the purported proof of the ABC Conjecture. A talk at this point would be premature because the audience could not understand what he was saying! In a new post, John Baez provides some comments by Minhyong Kim which I would recommend to anyone interested in this subject. Here is part of what Minhyong Kim said: “How long it will take for people to evaluate the work, it’s hard to say, possibly even a year or so. Among other difficulties, his work probes the very core of mathematical language such as what we might really mean by a number or a geometric figure, and how they might be interpreted in a manner quite different from usual conventions. In fact, it relies on deep relations of a geometric nature between such varying interpretations. Such questions have occupied philosophers for millennia, but are usually quite distant from the consciousness of modern mathematicians. But then, these seemingly philosophical questions have to be recast in the robust language of precise mathematics. You have to add to that some of the most sophisticated portions of 21st century arithmetic geometry. At the moment, I can fairly safely say that there is no one but the author who is familiar with all these things. Possibly his colleague Akio Tamagawa.” https://plus.google.com/117663015413546257905/posts/d1RsN4KnCUs#117663015413546257905/posts/d1RsN4KnCUs

Do we know whether Mochizuki is claiming the strong or the weak version of the conjecture?

In IUTT-IV Mochizuki writes, “In the present paper, estimates […] are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves.” The Szpiro Conjecture is, in fact, equivalent to the strong ABC Conjecture. [cf. Bombieri-Gubler, Heights in Diophantine Geometry p. 431]

In his analysis of Mochizuki’s articles on MathOverflow, Vesselin Dimitrov says that the key inequality stated in Section 2 of IUTT-IV is “asserted up to finitely many exceptions.” Dimitrov also says that “Mochizuki’s approach […] is entirely direct, and, consequently, effective.” http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc/106658#106658

But from what I can gather from other comments by experts, the question of which version of the ABC Conjecture Mochizuki claims to prove, and whether the claimed result is effective or not, are open questions at this point.

Mochizuki claims the strongest version of ABC that one could think of. (In particular, the effective one, and with the exponent 1 + epsilon). See Theorem A on p. 3 of his fourth paper (ABC with exponent 1+epsilon is a standard consequence of this).

As for an explicit effective statement, take a look at the inequality asserted on page 23. It concerns Szpiro’s inequality 1/6 log(D) < (1+epsilon) log(N) + Const. for the minimal discriminant D and conductor E of a (semistable) elliptic curve E. Here, log q on the left-hand side is precisely log(D). The f on the right-hand side is our conductor N, and the other term is a constant since we are concentrating on the single number field Q. In section 2, the full ABC conjecture is deduced, in an effective manner (by the paper [GenEll]), from this effective (Szpiro) inequality.

Vesselin,

Thanks very much for clarifying those points of your analysis. And sorry if the snippets I quoted from you may have misrepresented what you said. I found your analysis of Mochizuki’s articles on MathOverflow very interesting and helpful.

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If Mochizuki theory is proven right then will it also bring another proof of FLT? It is very interesting to see how long will take to confirm or infirm this new mathematics. Taking into account that in the past Mochizuki proved other deep theorems there is a great hope for a major math revolution.

@Nick Nazari: If Mochizuki’s work is correct (and this is a pretty big “If”…), it would certainly yield a new proof of FLT.

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