Philosophy behind Mochizuki's work on the ABC conjecture

  • Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?

    I'm afraid we do not permit the word "behooves."

    There is some expository material on Mochizuki's website. Did you try to read it already? If not, please, do so first. For lack of any indication of having done so, vote close (as by FAQs 'homework' ought to be done before asking).

    The suggestion that what's on my blog constitutes even "an extremely vague glimpse" of what Mochizuki is trying to get at is false advertising of the most extravagant kind!

    Correction: "an enthusiastic report". Sorry, Jordan!

    @quid: the expositions I've seen (such as are mostly teasers to make people read more. My question is about the sketch underlying the proof of the ABC conjecture, which I don't see evident there. If you have an exposition that you would recommend, I suggest that you write it as an answer.

    Well, then, read more! And if you do not care enough or lack the appropriate background to do so, I do not see why you need to know this so urgently. If experts become optimistic and understand the thing well enough, expositions will be all around. Just wait

    @James: Have you looked at Remark 1.10.1 in IUTeich Theory IV? He actually states that the computations in the proof of the previous theorem, which seems to be the main theorem from which ABC is derived, were known to him as early as 2000, and actually compares this to the proof of the Weil Conjectures. So it might be worth trying to study the proof of Theorem 1.10 (obviously much easier said than done).

    @quid: you're being stubborn. Is it not legitimate to ask questions about mathematics that is available but difficult to read and understand? @Kevin: thanks!

    I reply on meta.

    @James Taylor I have not made any serious attempt to read the papers. However, I can point you to two things which I think are relevant, based on hints from the introductions. The first is the very easy proof of function field ABC, which turns into an analysis of the possible branching behavior of maps $\mathbb{CP}^1 \to \mathbb{CP}^1$. For number field ABC, the source $\mathbb{CP}^1$ should turn into $\mathrm{Spec}(\mathbb{Z})$ and the target should still be $\mathbb{P}^1$ (continued).

    The second is that I think the Mochizuki is thinking of the target $\mathbb{P}^1$ as the $j$-line, so that maps from $\mathrm{Spec} \mathbb{Z}$ to it correspond (roughly) to elliptic curves over $\mathbb{Q}$. This is very analogous to the way that introducing an elliptic curve made FLT provable. Are these things you already understand, or would it be useful for me to write them up in more detail? Again, this is all with the caveat that I haven't looked at anything beyond the introductions, and I understand only a little bit of them.

    @David: The link between ABC and Szpiro's Conjecture (which is the content of the application of the Frey-like construction) long predates Mochizuki's work, and the "function field case" of ABC seems to have nothing to do with the ideas relevant in Mochizuki's work in the number field case much as the "function field case" of FLT is totally irrelevant to the actual proof of FLT. So although each aspect is very interesting for someone who has never heard of the ABC Conjecture, neither of them sheds light on anything that has happened since the time Mochizuki began his work on these matters.

    @grp I absolutely agree that everything I am talking about is 20 years old, and it cannot be "what is new" in Mochizuki. I do not know what is new; someone else would need to write that and I hope they do. That's why I asked whether this sort of context is useful. I'll try to get together a reply re whether the function field analogy is relevant at some point later.

    It would seem that only Mochizuki could actually give a correct answer. Anything else would be speculation. Therefore, IMHO this should be a CW question since it cannot have a single correct answer (barring Mochizuki responding). That grp's popular answer was made CW by grp further substantiates this.

    David, your comments are precisely the type of answer I'm looking for. It is okay that it's 20 years old.

    I strongly agree with Benjamin's comment. I share the concerns of quid, grp, and others, that but for Mochizuki, no one is currently able to answer definitively, and therefore CW it should be (if it even remains open).

    James, I think your question is a reasonable one to ask. However "reasonable" and "appropriate for MathOverflow" are not the same thing. If you wish, we can discuss this further on meta.mathoverflow. Gerhard "Ask Me About Appropriate Asking" Paseman, 2012.09.07

    Not being active on MO anymore, I don't much care if this question survives or not, but I am very interested in understanding more about Mochizuki's argument, so to the extent that insight appear here, I'll happily take advantage of it, and I'm sure others will too. My own sense was that Mochizuki's program has been motivated by trying to get around Faltings's "no go" theorem on arithmetic KS, by constructing a new, non-linear (or perhaps anabelian) interpretation of Hodge theory (both classical and p-adic) and related ideas, leading to a construction of some sort of arithmetic KS map, ...

    ... which ultimately allows him to (in some vague sense, at least) mimic the function field argument. But perhaps this intuition is off, and in any case, it hasn't helped me much in penetrating what he is actually doing. That is going to take hard work!

    Dear James, Just to echo grp's answer, it is hard to overstate the extent to which most of the number theory community has not engaged with Mochizuki's work before now, and now people are desperately trying to catch up. It will take time before anyone can explain what is really going on. Regards,

    As discussed on the meta page, I just substantially edited the question to remove extraneous (and tendentious) material.

    I really don't get why anyone thought to close this question. Given that Mochizuki thought his methods might be able to prove the ABC conjecture years before he came up with his (supposed) proof, it seems reasonable to think he might have an intuitive idea of a proof in his mind, and then the years of development of IUTeich were a means of putting those intuitive ideas into rigorous mathematical reality.

    While this could hypothetically be a question that only Mochizuki can answer, it could instead be that there is a sketch of an "intuitive proof" known to experts for years before Mochizuki's work. In that scenario, no one knew how to put those intuitive (even wishful) ideas into a rigorous foundation, and Mochizuki developed his theory in part in order to create that foundation. But that proof sketch would be both obscure enough and well-known enough to put on MO.

    As a striking example of the increasing prevalence of the notion of *naturality* in contemporary mathematics, Mochizuki’s four preprints employ the word "natural" and its derivatives on more than six hundred separate occasions (for details and related mathematical quotations, see this post on *Gödel's Lost Letter and P=NP*).

    John, the word "natural" has a precise mathematical definition. It does not mean "natural" like the natural world, which mathematically (and perhaps philosophically) is completely not well-defined. Mathematics cannot be done without precision, and the world is completely imprecise.

    hi all fyi ABC also has significant implications in CS theory see eg, & just want to thank the math community & mathoverflow for keeping this question open to see extended engagement and analysis of the proof by professionals in the field, & hope mathoverflow will be open to further on-topic questions on the subj to facilitate further serious analysis.

    "The proof must be correct for if it was not he wouldn't have the ideas to invent them" just joking

    more on the behind-the-scenes efforts to verify the proof: paradox of proof by chen on mochizuki attack

    what is the status of this in 2015? (in december, mochizuki posted a progress report on the verification of universal teichmüller theory / his alleged proof: )

  • Marty

    Marty Correct answer

    8 years ago

    I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work, and I make no claims about the veracity of his recent work.

    First, some background which might satisfy the OP. For years, Mochizuki has been working on things related to Grothendieck's anabelian program. Here is why one might hope this is useful in attacking problems like ABC:

    Begin with the Neukirch-Uchida theorem. See "Über die absoluten Galoisgruppen algebraischer Zahlkörper," by J. Neukirch, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris, 1977. Also "Isomorphisms of Galois groups," by K. Uchida, J. Math. Soc. Japan 28 (1976), no. 4, 617–620.

    The main result of these papers is that a number field is determined by its absolute Galois group in the following sense: fix an algebraic closure $\bar Q / Q$, and two number fields $K$ and $L$ in $\bar Q$. Then if $\sigma: Gal(\bar Q / K) \rightarrow Gal(\bar Q / L)$ is a topological isomorphism of groups, then $\sigma$ extends to an inner automorphism $Int(\tau): g \mapsto \tau g \tau^{-1}$ of $Gal(\bar Q / Q)$. Thus $\tau$ conjugates the number field $K$ to the number field $L$, and they are isomorphic.

    So while class field theory guarantees that the absolute Galois group $Gal(\bar Q / K)$ determines (the profinite completion of) the multiplicative group $K^\times$, the Neukirch-Uchida theorem guarantees that the entire field structure is determined by the profinite group structure of the Galois group. Figuring out how to recover aspects of the field structure of $K$ from the profinite group structure of $Gal(\bar Q / K)$ is a difficult corner of number theory.

    Next, consider a (smooth) curve $X$ over $Q$; suppose that the fundamental group $\pi_1(X({\mathbb C}))$ is nonabelian. Let $\pi_1^{geo}(X)$ be the profinite completion of this nonabelian group. Basic properties of the etale fundamental group give a short exact sequence: $$1 \rightarrow \pi_1^{geo}(X) \rightarrow \pi_1^{et}(X) \rightarrow Gal(\bar Q / Q) \rightarrow 1.$$

    Now, just as one can ask about recovering a number field from its absolute Galois group ($Gal(\bar Q / K)$ is isomorphic to $\pi_1^{et}(K)$), one can ask how much one can recover about the curve $X$ from its etale fundamental group. Any $Q$-point $x$ of $X$, i.e. map of schemes from $Spec(Q)$ to $Spec(X)$ gives a section $s_x: Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$.

    One case of the famous "section conjecture" of Grothendieck states that this gives a bijection from $X(Q)$ to the set of homomorphisms $Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$ splitting the above exact sequence. One hopes, more generally, to recover the structure of $X$ as a curve over $Q$ from the induced outer action of $Gal(\bar Q / Q)$ on $\pi_1^{geo}(X)$. (take an element $\gamma \in Gal(\bar Q / Q)$, lift it to $\tilde \gamma \in \pi_1^{et}(X)$, and look at conjugation of the normal subgroup $\pi_1^{geo}(X)$ by $\tilde \gamma$, well-defined up to inner automorphism independently of the lift.)

    As in the case of the Neukirch-Uchida theorem, there is an active and difficult corner of number theory devoted to recovering properties of rational points of (hyperbolic) curves from etale fundamental groups. Here are two dramatically difficult problems in the same spirit:

    1. How can you describe the regulator of a number field $K$ from the structure of the profinite group $Gal(\bar Q / K)$?

    2. Given a section $s: Gal(\bar Q / Q) \rightarrow \pi_1^{et}(X)$, how can one describe the height of the corresponding point in $X(Q)$?

    I would place Mochizuki's work in this anabelian corner of number theory; I have always kept a safe and respectful distance from this corner.

    Now, to say something not quite as ancient that I gleaned from flipping through Mochizuki's recent work:

    Many people here on MO and elsewhere have been following research on the field with one element. It is a tempting object to seek, because analogies between number fields and function fields break down quickly when you realize there is no "base scheme" beneath $Spec(Z)$. But I see Mochizuki's work as an anabelian approach to this problem, and I'll try to describe my understanding of this below.

    Consider a smooth curve $X$ over a function field $F_p(T)$. The anabelian approach suggests looking at the short exact sequence $$1 \rightarrow \pi_1^{et}(X_{\overline{F_p(T)}}) \rightarrow \pi_1^{et}(X) \rightarrow Gal(\overline{F_p(T)} / F_p(T)) \rightarrow 1.$$ But much more profitable is to look instead at $X$ as a surface over $F_p$ which corresponds in the anabelian perspective to studying $$1 \rightarrow \pi_1^{et}(X_{\bar F_p}) \rightarrow \pi_1^{et}(X) \rightarrow Gal(\bar F_p / F_p) \rightarrow 1.$$ But this is pretty close to looking at $\pi_1^{et}(X)$ by itself; there's just a little profinite $\hat Z$ quotient floating around, but this can be characterized (I think) group theoretically within the study of $\pi_1^{et}(X)$ itself.

    I would understand (after reading Mochizuki) that looking at curves $X$ over function fields $F_p(T)$ as surfaces over $F_p$ is like looking at only the etale fundamental group $\pi_1^{et}(X)$ without worrying about the map to $Gal(\overline{F_p(T)} / F_p(T))$.

    So, the natural number field analogue would be the following. Consider a smooth curve $X$ over $Q$. In fact, let's make $X = E - \{ 0 \}$ be a once-punctured elliptic curve over $Q$. Then the absolute anabelian geometry suggests that to study $X$, it should be profitable to study the etale fundamental group $\pi_1^{et}(X)$ all by itself as a profinite group. This is the anabelian analogue of what others might call "studying (a $Z$-model of) $X$ as a surface over the field with one element".

    Without understanding any of the proofs in Mochizuki, I think that his work arises from this absolute anabelian perspective of understanding the arithmetic of once-punctured elliptic curves over $Q$ from their etale fundamental groups. The ABC conjecture is equivalent to Szpiro's conjecture which is a conjecture about the arithmetic of elliptic curves over $Q$.

    Now here is a suggestion for number theorists who, like myself, have unfortunately ignored this anabelian corner. Let's try to read the papers of Neukirch and/or Uchida to get a start, and let's try to understand Minhyong Kim's work on Siegel's Theorem ("The motivic fundamental group of $P^1 \backslash ( 0, 1, \infty )$ and the theorem of Siegel," Invent. Math. 161 (2005), no. 3, 629–656.)

    It would be wonderful if, while we're waiting for the experts to weight in on Mochizuki's work, we took some time to revisit some great results in the anabelian program. If anyone wants to start a reading group / discussion blog on these papers, I would enjoy attending and discussing.

    This is a really great answer, and dramatically subsumes anything I would write. A link that might help people: Some notes on the relation between Szpiro and ABC

    Dear Marty, Even though I'm grateful for the reference to my paper, it is a bit outdated. For a more recent (still incomplete) view of what my intentions really are, I would recommend the introduction to this paper:

    By the way, I was preparing some kind of an answer to this question when I noticed it was closed. If you folks end up opening it again, maybe someone can let me know by email. My primitive knowledge of the internet hasn't extended to automatic notifications and the like.

    Thank you for the very recent reference! I have voted to reopen, and if it reopens I will send an email.

    would you please elaborate on what you mean by abc like problems? can you tell problems of similiar flavour

    Worth sharing this link: in the slides linked in that question, there is background by mochizuki on 'interuniversal geometry' which he says is the correct context for viewing anabelian results of such a reconstructive nature. these slides are from 2009, so a few years before the IUT papers. might help in bridging the gap to the 2012 papers

License under CC-BY-SA with attribution

Content dated before 6/26/2020 9:53 AM