## Philosophy behind Mochizuki's work on the ABC conjecture

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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?

15I'm afraid we do not permit the word "behooves." – Will Jagy – 2012-09-07T01:17:18.503

4There 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). – None – 2012-09-07T01:17:35.427

7The 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! – JSE – 2012-09-07T01:51:46.457

Correction: "an enthusiastic report". Sorry, Jordan! – James D. Taylor – 2012-09-07T01:55:39.737

24

@quid: the expositions I've seen (such as http://www.kurims.kyoto-u.ac.jp/~motizuki/2010-10-abstract.pdf) 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.

– James D. Taylor – 2012-09-07T02:00:17.460

5Well, 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 – None – 2012-09-07T02:13:42.620

6@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). – Kevin Ventullo – 2012-09-07T02:15:06.777

21

– Will Jagy – 2012-09-07T02:18:53.840

71@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! – James D. Taylor – 2012-09-07T02:21:01.920

2I reply on meta. – None – 2012-09-07T02:32:38.630

20@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). – David E Speyer – 2012-09-07T06:57:40.243

21The 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 E Speyer – 2012-09-07T07:00:34.173

4@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 – 2012-09-07T12:14:01.377

@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. – David E Speyer – 2012-09-07T12:37:16.620

15It 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. – Benjamin Steinberg – 2012-09-07T14:26:21.743

3David, your comments are precisely the type of answer I'm looking for. It is okay that it's 20 years old. – James D. Taylor – 2012-09-07T15:53:16.083

6I 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). – Todd Trimble – 2012-09-07T18:27:08.147

2James, 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 – Gerhard Paseman – 2012-09-07T20:21:58.407

16Not 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, ... – Emerton – 2012-09-07T22:24:46.530

8... 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! – Emerton – 2012-09-07T22:26:14.657

9Dear 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, – Emerton – 2012-09-07T22:31:41.290

8As discussed on the meta page, I just substantially edited the question to remove extraneous (and tendentious) material. – Andy Putman – 2012-09-08T18:01:00.857

12I 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. – David Corwin – 2012-09-09T07:33:32.850

6While 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. – David Corwin – 2012-09-09T07:33:42.787

2

what-is-the-insight-of-quillens-proof-that-all-projective-modules-over-a-polynom http://mathoverflow.net/questions/19584/what-is-the-insight-of-quillens-proof-that-all-projective-modules-over-a-polynom

– Alexander Chervov – 2012-09-12T11:02:40.177

1

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 Sidles – 2012-09-13T20:24:20.033

3John, 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. – Asaf Karagila – 2012-09-16T15:52:06.780

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hi all fyi ABC also has significant implications in CS theory see eg http://cstheory.stackexchange.com/questions/12504/implications-of-proof-of-abc-conjecture-for-cs-theory, & 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.

– vzn – 2012-09-28T15:41:37.033

"The proof must be correct for if it was not he wouldn't have the ideas to invent them" just joking – Koushik – 2013-10-02T13:33:56.650

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

– vzn – 2013-12-01T20:59:35.913

2

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: http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf )

– Trent – 2015-01-15T01:49:02.077

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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.

17

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 http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01.pdf

– David E Speyer – 2012-09-07T18:37:45.397

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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: http://front.math.ucdavis.edu/1209.0640

– Minhyong Kim – 2012-09-08T04:07:32.893

12By 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. – Minhyong Kim – 2012-09-08T04:09:10.193

Thank you for the very recent reference! I have voted to reopen, and if it reopens I will send an email. – Marty – 2012-09-08T04:18:39.733

1would you please elaborate on what you mean by abc like problems? can you tell problems of similiar flavour – Koushik – 2013-10-02T13:36:52.190

1

Worth sharing this link: https://mathoverflow.net/questions/852/what-is-inter-universal-geometry 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

– Samantha Y – 2017-09-19T05:32:18.917

205

I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle of ideas, regardless of the eventual status of the main result of interest. That is to say, the current sense of urgency to understand something seems generally a good thing. So I thought I'd give the flimsiest bit of introduction imaginable at this stage. On the other hand, as with many of my answers, there's the danger I'm just regurgitating common knowlege in a long-winded fashion, in which case, I apologize.

For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' [It has been noted here and there that the 'Survey of Hodge Arakelov Theory I,II' papers might be reasonable alternatives.][I've just examined them again, and they really might be the better way to begin.] These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is to decipher just a little bit those few paragraphs.

The beginning of the investigation is indeed the function field case (over $\mathbb{C}$, for simplicity), where one is given a family $$f:E \rightarrow B$$ of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial. There is an exact sequence $$0\rightarrow \omega_E \rightarrow H^1_{DR}(E) \rightarrow H^1(O_E)\rightarrow0,$$ which is moved by the logarithmic Gauss-Manin connection of the family. (I hope I will be forgiven for using standard and non-optimal notation without explanation in this note.) That is to say, if $S\subset B$ is the finite set of images of the bad fibers, there is a log connection $$H^1_{DR}(E) \rightarrow H^1_{DR}(E) \otimes \Omega_B(S),$$ which does not preserve $\omega_E$. This fact is crucial, since it leads to an $O_B$-linear Kodaira-Spencer map $$KS:\omega \rightarrow H^1(O_E)\otimes \Omega_B(S),$$ and thence to a non-trivial map $$\omega_E^2\rightarrow \Omega_B(S).$$ From this, one easily deduces Szpiro's inequality: $$\deg (\omega_E) \leq (1/2)( 2g_B-2+|S|).$$ At the most simple-minded level, one could say that Mochizuki's programme has been concerned with replicating this argument over a number field $F$. Since it has to do with differentiation on $B$, which eventually turns into $O_F$, some philosophical connection to $\mathbb{F}_1$-theory begins to appear. I will carry on using the same notation as above, except now $B=Spec(O_F)$.

A large part of HAT is exactly concerned with the set-up necessary to implement this idea, where, roughly speaking, the Galois action has to play the role of the GM connection. Obviously, $G_F$ doesn't act on $H^1_{DR}(E)$. But it does act on $H^1_{et}(\bar{E})$ with various coefficients. The comparison between these two structures is the subject of $p$-adic Hodge theory, which sadly works only over local fields rather than a global one. But Mochizuki noted long ago that something like $p$-adic Hodge theory should be a key ingredient because over $\mathbb{C}$, the comparison isomorphism $$H^1_{DR}(E)\simeq H^1(E(\mathbb{C}), \mathbb{Z})\otimes_{\mathbb{Z}} O_B$$ allows us to completely recover the GM connection by the condition that the topological cohomology generates the flat sections.

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dagger}\rightarrow E,$$ (the moduli space of line bundles with flat connection on $E$) whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the \'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form: $$\Xi: A_{DR}=\Gamma(E^{\dagger}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$ (I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by a calligraphic $H$. I can't seem to write calligraphic characters here.) Here, $L$ is a suitably chosen line bundle of degree $\ell$ on $E$, which can then be pulled back to $E^{\dagger}$. The inequality refers to the polynomial degree in the fiber direction of $E^{\dagger} \rightarrow E$. The isomorphism is effected via evaluation of sections at $$E^{\dagger}[\ell]\simeq E[\ell].$$ Finally, $$L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete \'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

Now for some corrections: As stated, the isomorphism is not true, and must be modified at the places of bad reduction, the places dividing $\ell$, and the infinite places. This correction takes up a substantial portion of the HAT paper. That is, the isomorphism is generically true over $B$, but to make it true everywhere, the integral structures must be modified in subtle and highly interesting ways, while one must consider also a comparison of metrics, since these will obviously figure in an arithmetic analogue of Szpiro's conjecture. The correction at the finite bad places can be interpreted via coordinates near infinity on the moduli stack of elliptic curves as the subtle phenomenon that Mochizuki refers to as 'Gaussian poles' (in the coordinate $q$). Since this is a superficial introduction, suffice it to say for now that these Gaussian poles end up being a major obstruction in this portion of Mochizuki's theory.

In spite of this, it is worthwhile giving at least a small flavor of Mochizuki's Galois-theoretic KS map. The point is that $A_{DR}$ has a Hodge filtration defined by

$F^rA_{DR}= \Gamma(E^{\dagger}, L)^{ < r}$

(the direction is unconventional), and this is moved around by the Galois action induced by the comparison isomorphism. So one gets thereby a map $$G_F\rightarrow Fil (A_{DR})$$ into some space of filtrations on $A_{DR}$. This is, in essence, the Galois-theoretic KS map. That, is if we consider the equivalence over $\mathbb{C}$ of $\pi_1$-actions and connections, the usual KS map measures the extent to which the GM connection moves around the Hodge filtration. Here, we are measuring the same kind of motion for the $G_F$-action.

This is already very nice, but now comes a very important variant, essential for understanding the motivation behind the IUTT papers. In the paper GTKS, Mochizuki modified this map, producing instead a 'Lagrangian' version. That is, he assumed the existence of a Lagrangian Galois-stable subspace $G^{\mu}\subset E[l]$ giving rise to another isomorphism $$\Xi^{Lag}:A_{DR}^{H}\simeq L\otimes O_{G^{\mu}},$$ where $H$ is a Lagrangian complement to $G^{\mu}$, which I believe does not itself need to be Galois stable. $H$ is acting on the space of sections, again via Mumford's theory. This can be used to get another KS morphism to filtrations on $A_{DR}^{H}$. But the key point is that

$\Xi^{Lag}$, in contrast to $\Xi$, is free of the Gaussian poles

via an argument I can't quite remember (If I ever knew).

At this point, it might be reasonable to see if $\Xi^{Lag}$ contributes towards a version of Szpiro's inequality (after much work and interpretation), except for one small problem. A subspace like $G^{\mu}$ has no reason to exist in general. This is why GTKS is mostly about the universal elliptic curve over a formal completion near $\infty$ on the moduli stack of elliptic curves, where such a space does exists. What Mochizuki explains on IUTT page 10 is exactly that the scheme-theoretic motivation for IUG was to enable the move to a single elliptic curve over $B=Spec(O_F)$, via the intermediate case of an elliptic curve 'in general position'.

To repeat:

A good 'nonsingular' theory of the KS map over number fields requires a global Galois invariant Lagrangian subspace $G^{\mu}\subset E[l]$.

One naive thought might just be to change base to the field generated by the $\ell$-torsion, except one would then lose the Galois action one was hoping to use. (Remember that Szpiro's inequality is supposed to come from moving the Hodge filtration inside De Rham cohomology.) On the other hand, such a subspace does often exist locally, for example, at a place of bad reduction. So one might ask if there is a way to globally extend such local subspaces.

It seems to me that this is one of the key things going on in the IUTT papers I-IV. As he say in loc. cit. he works with various categories of collections of local objects that simulate global objects. It is crucial in this process that many of the usual scheme-theoretic objects, local or global, are encoded as suitable categories with a rich and precise combinatorial structure. The details here get very complicated, the encoding of a scheme into an associated Galois category of finite \'etale covers being merely the trivial case. For example, when one would like to encode the Archimedean data coming from an arithmetic scheme (which again, will clearly be necessary for Szpiro's conjecture), the attempt to come up with a category of about the same order of complexity as a Galois category gives rise to the notion of a Frobenioid. Since these play quite a central role in Mochizuki's theory, I will quote briefly from his first Frobenioid paper:

'Frobenioids provide a single framework [cf. the notion of a "Galois category"; the role of monoids in log geometry] that allows one to capture the essential aspects of both the Galois and the divisor theory of number fields, on the one hand, and function fields, on the other, in such a way that one may continue to work with, for instance, global degrees of arithmetic line bundles on a number field, but which also exhibits the new phenomenon [not present in the classical theory of number fields] of a "Frobenius endomorphism" of the Frobenioid associated to a number field.'

I believe the Frobenioid associated to a number field is something close to the finite \'etale covers of $Spec(O_F)$ (equipped with some log structure) together with metrized line bundles on them, although it's probably more complicated. The Frobenious endomorphism for a prime $p$ is then something like the functor that just raises line bundles to the $p$-th power. This is a functor that would come from a map of schemes if we were working in characteristic $p$, but obviously not in characteristic zero. But this is part of the reason to start encoding in categories:

We get more morphisms and equivalences.

Some of you will notice at this point the analogy to developments in algebraic geometry where varieties are encoded in categories, such as the derived category of coherent sheaves. There as well, one has reconstruction theorems of the Orlov type, as well as the phenomenon of non-geometric morphisms of the categories (say actions of braid groups). Non-geometric morphisms appear to be very important in Mochizuki's theory, such as the Frobenius above, which allows us to simulate characteristic $p$ geometry in characteristic zero. Another important illustrative example is a non-geometric isomorphism between Galois groups of local fields (which can't exist for global fields because of the Neukirch-Uchida theorem). In fact, I think Mochizuki was rather fond of Ihara's comment that the positive proof of the anabelian conjecture was somewhat of a disappointment, since it destroys the possibility that encoding curves into their fundamental groups will give rise to a richer category. Anyways, I believe the importance of non-geometric maps of categories encoding rather conventional objects is that

they allow us to glue together several standard categories in nonstandard ways.

Obviously, to play this game well, some things need to be encoded in rigid ways, while others should have more flexible encodings.

For a very simple example that gives just a bare glimpse of the general theory, you might consider a category of pairs $$(G,F),$$ where $G$ is a profinite topological group of a certain type and $F$ is a filtration on $G$. It's possible to write down explicit conditions that ensure that $G$ is the Galois group of a local field and $F$ is its ramification filtration in the upper numbering (actually, now I think about it, I'm not sure about 'explicit conditions' for the filtration part, but anyways). Furthermore, it is a theorem of Mochizuki and Abrashkin that the functor that takes a local field to the corresponding pair is fully faithful. So now, you can consider triples $$(G,F_1, F_2),$$ where $G$ is a group and the $F_i$ are two filtrations of the right type. If $F_1=F_2$, then this 'is' just a local field. But now you can have objects with $F_1\neq F_2$, that correspond to strange amalgams of two local fields.

As another example, one might take a usual global object, such as $$(E, O_F, E[l], V)$$ (where $V$ denotes a collection of valuations of $F(E[l])$ that restrict bijectively to the valuations $V_0$ of $F$), and associate to it a collection of local categories indexed by $V_0$ (something like Frobenioids corresponding to the $E_v$ for $v\in V_0$). One can then try to glue them together in non-standard ways along sub-categories, after performing a number of non-standard transformations. My rough impression at the moment is that the 'Hodge theatres' arise in this fashion. [This is undoubtedly a gross oversimplification, which I will correct in later amendments.] You might further imagine that some construction of this sort will eventually retain the data necessary to get the height of $E$, but also have data corresponding to the $G^{\mu}$, necessary for the Lagrangian KS map. In any case, I hope you can appreciate that a good deal of 'dismantling' and 'reconstructing,' what Mochizuki calls surgery, will be necessary.

I can't emphasize enough times that much of what I write is based on faulty memory and guesswork. At best, it is superficial, while at worst, it is (not even) wrong. [In particular, I am no longer sure that the GTKS map is used in an entirely direct fashion.] I have not yet done anything with the current papers than give them a cursory glance. If I figure out more in the coming weeks, I will make corrections. But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

Allow me to make one remark about set theory, about which I know next to nothing. Even with more straightforward papers in arithmetic geometry, the question sometimes arises about Grothendieck's universe axiom, mostly because universes appear to be used in SGA4. Usually, number-theorists (like me) neither understand, nor care about such foundational matters, and questions about them are normally met with a shrug. The conventional wisdom of course is that any of the usual theorems and proofs involving Grothendieck cohomology theories or topoi do not actually rely on the existence of universes, except general laziness allows us to insert some reference that eventually follows a trail back to SGA4. However, this doesn't seem to be the case with Mochizuki's paper. That is, universes and interactions between them seem to be important actors rather than conveniences. How this is really brought about, and whether more than the universe axiom is necessary for the arguments, I really don't understand enough yet to say. In any case, for a number-theorist or an algebraic geometer, I would guess it's still prudent to acquire a reasonable feel for the 'usual' background and motivation (that is, HAT, GTKS, and anabelian things) before worrying too much about deeper issues of set theory.

58Such a great answer proves that the question was worth asking. – Olivier – 2012-09-08T12:47:06.353

41@Olivier: while I agree with your comment regarding the answer, I would like to add that your inference regarding this particular question to me comes close to saying: the great and heroic work of the fire-fighters prove that it was worth setting the house on fire. – None – 2012-09-08T13:18:08.747

155On MathOverflow, though, it is possible to edit the original house so that it is no longer inflammatory, while still recording the work of the firefighter. – Terry Tao – 2012-09-08T18:15:29.673

6Dear Minyong, it has been stated by experts, here on MO and elsewhere, that universes are not necessary for the uses to which EGA/SGA are put. Indeed Colin McLarty has shown that ZFC is far stronger than is necessary, with the results provable in some higher order version of arithmetic. As far as the interuniversal aspect of IUTT goes, I expect it to have a topos-theoretic interpretation. This to my mind makes switching between models of set theory cleaner i.e. functorial. – David Roberts – 2012-09-09T01:27:58.853

1Thanks David. I'll think about it. And thanks Tom for fixing my silly daggers. – Minhyong Kim – 2012-09-09T04:50:28.587

wrt @quids comment, what does any of this have to do with firefighting? sounds off topic to me.... & imho the proof attack, even if eventually proven to be erroneous or with a gap, is what can nevertheless be regarded as somewhat heroic.... – vzn – 2012-09-29T00:52:38.597

1@vzn: wouldn't asking about an off-topic comment in particular three weeks after it was made be off-topic, too? ;D Yet, what I mainly wanted to express (and this was a meta-commment, and you may call it off-topic if you like) is that just because a great text can be written as an answer to a question does not imply that the question was any good. I could also have expressed this with a different (or no) analogy, so there is nothing highly specific to firefighting. Yet, one could say that this and other nice texts (to some extent) helped to calm a heated discussion, so it seemed fitting. – None – 2012-10-16T12:58:30.757

85

Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )

Completely rewritten. (9/26)

It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.

Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 very badly.

Assumptions:

• A (Consequence of ABC) For all but finitely many elliptic curves over $\mathbb{Q}$, the conductor $N$ and the minimal discriminant $\Delta$ satisfy $\log{|\Delta|} < (\log{N})^2$.

• B (Uniform Serre Open Image conjecture) For each $d \in \mathbb{N}$, there is a constant $c(d) < \infty$ such that for every number field $F/\mathbb{Q}$ with $[F:\mathbb{Q}] \leq d$, and every non-CM elliptic curve $E$ over $F$, and every prime $\ell \geq c(d)$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/{\ell})$. (In fact, it is sufficient to take the weaker version in which $F$ is held fixed. )

Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking $F := F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5] \big)$ --- rather than simply $F := F_{\mathrm{tpd}}(\sqrt{-1})$ --- was to ensure that $E$ has semistable reduction over $F$. Since I will only work in what follows with semistable elliptic curves over $\mathbb{Q}$, I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over $F_{\mathrm{tpd}}$, we may actually take $F := F_{\mathrm{tpd}}(\sqrt{-1})$ in Theorem 1.10.

The infinite set of counterexamples. They come from Masser's paper [Masser: Note on a conjecture of Szpiro, Asterisque 1990], as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over $\mathbb{Q}$ whose conductor $N$ and minimal discriminant $\Delta$ satisfy $$(1) \hspace{3cm} \frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}.$$ (Thus, $N$ in these examples may be taken arbitrarily large. ) By (A) above, taking $N$ big enough will ensure that $$(2) \hspace{3cm} \log{|\Delta|} < (\log{N})^2.$$ Next, the sum of the logarithms of the primes in the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ is $2(\log{N})^2 + o((\log{N})^2)$, so it is certainly $> (\log{N})^2$ for $N \gg 0$ big enough. Thus, by (2), it is easy to see that the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ contains a prime $\ell$ which divides neither $|\Delta|$ nor any of the exponents $\alpha = \mathrm{ord}_p(\Delta)$ in the prime factorization $|\Delta| = \prod p^{\alpha}$ of $|\Delta|$.

Consider now the pair $(E,\ell)$: it has $F_{\mathrm{mod}} = \mathbb{Q}$, and since $E$ has rational $2$-torsion, $F_{\mathrm{tpd}} = \mathbb{Q}$ as well. Let $F := \mathbb{Q} \big( \sqrt{-1}\big)$. I claim that, upon taking $N$ big enough, the pair $(E_F,\ell)$ arises from an initial $\Theta$-datum as in IUTT-I, Definition 3.1. Indeed:

• Certainly (a), (e), (f) of IUTT-I, Def. 3.1 are satisfied (with appropriate $\underline{\mathbb{V}}, \, \underline{\epsilon}$);
• (b) of IUTT-I, Def. 3.1 is satisfied since by construction $E$ is semistable over $\mathbb{Q}$;
• (c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of $\ell$, as soon as $N \gg 0$ is big enough (recall that $\ell > (\log{N})^2$ by construction!), and by the observation that, for $v$ a place of $F = \mathbb{Q}(\sqrt{-1})$, the order of the $v$-adic $q$-parameter of $E$ equals $\mathrm{ord}_v (\Delta)$, which equals $\mathrm{ord}_p(\Delta)$ for $v \mid p > 2$, and $2\cdot\mathrm{ord}_2(\Delta)$ for $v \mid 2$;

while $\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}}$ consists of the primes dividing $\Delta$;

• Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser's examples $E$. (See page 37 of IUTT-IV).

Now, take $\epsilon := \big( \log{N} \big)^{-2}$ in Theorem 1.10 of IUTT-IV; this is certainly permissible for $N \gg 0$ large enough. I claim that the conclusion of Theorem 1.10 contradicts (1) as soon as $N \gg 0$ is large enough.

For note that Mochizuki's quantity $\log(\mathfrak{q})$ is precisely $\log{|\Delta|}$ (reference: see e.g. Szpiro's article in the Grothendieck Festschrift, vol. 3); his $\log{(\mathfrak{d}^{\mathrm{tpd}})}$ is zero; his $d_{\mathrm{mod}}$ is $1$; and his $\log{(\mathfrak{f}^{\mathrm{tpd}})}$ is our $\log{N}$. By construction, our choice $\epsilon := \big( \log{N} \big)^{-2}$ then makes $1/\ell < \epsilon$ and $\ell < 3/\epsilon$, whence the finaly display of Theorem 1.10 would yield $$\frac{1}{6} \log{|\Delta|} \leq (1+29\epsilon) \cdot \log{N} + 2\log{(3\epsilon^{-8})} < \log{N} + 16\log{\log{N}} + 32,$$ where we have used $\epsilon \log{N} = (\log{N})^{-1} < 1$ for $N > 3$, and $2\log{3} < 3$.

The last display contradicts (1) as soon as $N \gg 0$ is big enough.

Thus Masser's examples yield infinitely many counterexamples to Theorem 1.10 of IUTT-IV (as presently written).

Added on 10/15, and revised 10/20. Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:

He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. (He estimates January 2013 to be a reasonable period). He confirms the following ["essentially"] anticipated revision of Theorem 1.10:

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$\frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + O\big( \log{(1/\epsilon)} \big)$$ $$< \log{N} + \Big( \epsilon \log{N} + \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big) + o\Big( \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big),$$ where $\omega(\cdot)$ denotes "number of prime factors." The second estimate comes from the prime number theorem in the form $\pi(t) = t/\log{t} + t/(\log{t})^2 + o\big( t/(\log{t})^2 \big)$, applied to $t := \epsilon^{-1}$, and is sharp if you restrict $\epsilon$ to the range $\epsilon^{-1} < (\log{N})^{\xi}$ with $\xi < 1$, as there nothing prevents $N$ from being divisible by all primes $p < (\log{N})^{\xi}$. In particular, as the Erdos-Stewart-Tijdeman-Masser construction is based on the pigeonhole principle, which cannot preclude that $N$ be divisible by all the primes $< (\log{N})^{2/3}$, the second estimate could very well be sharp in all the Masser examples. As it is easily seen that the bracketed term exceeds the range $\sqrt{\log{N}}/(\log{\log{N}})$ of Masser's examples, this has the implication that

the Erdos-Stewart-Tijdeman-Masser method cannot disprove Mochizuki's revised inequality,

which therefore seems reasonable.

On the other hand, if we take $\epsilon := (\log{N})^{-1}$ and assume $\omega(N_{\epsilon})$ bounded, this would yield $(1/6)\log{|\Delta|} < \log{N} + O(\log{\log{N}})$, just as before. (Thus, Mochizuki predicts that this last bound must hold for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded. I cannot see evidence neither for nor against this at the moment: again, the Masser and Erdos-Stewart-Tijdeman constructions are based on the pigeonhole principle, and do not seem to be able to exclude the small primes $< \log{N}$. So here we have an open problem by which one could probe Mochizuki's revised inequality. A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

A side remark: note that the inverse $1/\ell$ of the prime level from the de Rham-Etale correspondence $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$ in Mochizuki's "Hodge-Arakelov theory" ultimately figures as the $\epsilon$ in the ABC conjecture.

[I have deleted the remainder of the 10/15 Addendum, since it is now obsolete after Mochizuki's revised comments. ]

2Let $R=rad(abc)$. If (!?) is true, then take $\epsilon=B/log R$ and it becomes $c < A(e/B)^B\cdot R\cdot (\log R)^B$, so the $ABC$-conjecture would be true with the $R^\epsilon$ replaced by a power of $\log R$. The would indeed be very strong. I recall some people (Masser? others?) showed that $ABC$ is not true with exponent 1, but I don't remember if they handled the case with extra log terms. However, I have not looked at Mochizuki's papers, so I can't comment on whether the derivation of (!?) from his work is valid. – Joe Silverman – 2012-09-15T19:19:17.680

1Certainly Mochizuki does not claim that (!?) holds for all $(a,b,c)$; he claims instead that, for a given $\varepsilon$, he can effectively determine all counterexamples to (!?) using his (entirely elementary!) paper [GenEll]. This by itself is no contradiction; but it is very striking claim. See also his Remark 2.3.2 in IUTT-IV. – Vesselin Dimitrov – 2012-09-15T19:34:55.093

Stewart and Tijdeman showed (Monatsh. Math. 102 (1986), 251–257) that the upper bound cannot be $cR\exp ((4-\delta)\sqrt{\log R}/\log \log R)$, so (!?) cannot definitely hold for all $abc$ triples. – Felipe Voloch – 2012-09-15T19:38:31.010

Here is what, to my understanding, Mochizuki claims in his Theorem 1.10 on pp. 22-23 (see the final display before the proof on p. 23, and also see IUTT-I, p. 50 for the assumptions on the "initial $\Theta$-data" $(\mathbb{Q},E,\ell)$, where $E$ is the associated elliptic curve $y^2 = x(x-a)(x+b)$ needed to make the translation to (?!)):

For an arbitrary triple $(a,b,c)$, consider a prime $\ell$ which is generic for the associated elliptic curve $E$, in the sense that the assumptions of the third paragraph above are fulfilled. Then (?!) holds with $\varepsilon := 29/\ell$. – Vesselin Dimitrov – 2012-09-15T20:06:40.490

If the abc conjecture $c \leq K\varepsilon \hbox{rad}(abc)^{1+\varepsilon}$ holds for every $\varepsilon > 0$ (with no assumption on the dependence of $K\varepsilon$ on $\varepsilon$), then for any $\varepsilon > 0$, one has $c \leq \hbox{rad}(abc)^{1+\varepsilon}$ (i.e. the constant may be deleted) for all but finitely many triples a,b,c, as can be seen by applying the original version of the abc conjecture with $\varepsilon$ replaced by $\varepsilon/2$ to handle the case of large radical, and using the abc conjecture again to show that the number of small radical cases is finite. (cont) – Terry Tao – 2012-09-15T20:12:34.353

Basically, the constant $K_\varepsilon$ is a lower order term, and I wouldn't place much significance on exactly what value it is given if one is willing to lose a finite number of exceptions. – Terry Tao – 2012-09-15T20:14:45.767

... which certainly seems disturbing: assuming a uniform Serre's "open image theorem," it is then enough to take any big enough prime $\ell$ not dividing $abc$ as well as the prime exponents of $abc$.

I stop here: I just wished to voice this as potential counterexample to Mochizuki's final estimate in section 1 of his fourth paper. My apology if I have got it wrong. – Vesselin Dimitrov – 2012-09-15T20:21:14.757

By the way, an uniform Serre open image theorem is widely believed and there is even a guess of how big $\ell$ has to be (a three digit number which I forget). – Felipe Voloch – 2012-09-15T20:36:32.603

Indeed, the $K{\varepsilon}$-constant is insignificant as far as the qualitative statement of ABC goes: one can just say that $c < \mathrm{rad}(abc)^{1+\varepsilon}$ has finitely many exceptions. But there is also interest in the form of $K{\varepsilon}$: for example, there is A. Baker's conjecture: $c \leq A \cdot \prod_{p \mid abc} (p/\varepsilon)^{1+\varepsilon}$ for all $0 < \varepsilon leq 1$, with an absolute constant $A$. In any case, Mochizuki has an extremely sharp version of the ABC-conjecture. It is claimed to be effective, too. – Vesselin Dimitrov – 2012-09-15T20:45:59.993

Here is a final post. Working backwards (taking big enough $\ell$ so that the conjectured "uniform open image theorem" holds for $\ell$, and considering only $abc$-triples which, together with all of their prime exponents, are not divisible by $\ell$), we can see that:

Assuming the uniform open image conjecture, Section 1 of IUTT-IV (see Theorem 1.10 and the backward-references there) implies, as written that there is an absolute $\epsilon_0 > 0$ such that, for any $0 < \varepsilon < \epsilon_0$, and any $abc$-triple such that (cont) – Vesselin Dimitrov – 2012-09-15T21:15:08.547

(cont) ... such that there is some prime in $(1/\varepsilon, 2/\varepsilon)$ not dividing neither $abc$ nor any of its prime exponents, then (!?) holds. (With absolute constants $A,B < \infty$).

This would be extremely disturbing. – Vesselin Dimitrov – 2012-09-15T21:17:39.423

Postscriptum: I commented upon two distinct parts of Mochizuki's IUTT-IV - my apology for the ambiguity. First of all I have in mind his section 1: this is completely independent of the paper [GenEll], and contains an unconditional inequality - in Thm, 1.10 - in which figures the auxiliary prime $\ell$ (then take $\varepsilon \sim 1/\ell$ and you get (?!)), but valid only under the assumption that $\ell$ is generic for $E$, in the sense mentioned above. It is here that I felt there might be a contradiction. [cont] – Vesselin Dimitrov – 2012-09-15T21:47:02.040

And Second, there is Mochizuki's Section 2. This reduces to the previous case with the help of the paper [GenEll], where in particular a valid choice of $\ell$ from Section 1 is ensured (among other reductions to "generic arithmetic elliptic curves"). There the inequality that figures is, literally (?!), but it is asserted up to finitely many exceptions covered by [GenEll] (in principle, their height is bounded above). So, by itself, this is no contradiction. – Vesselin Dimitrov – 2012-09-15T21:50:49.763

If you want to apply the theorem 1.10 with initial theta data having F=Q then you have a problem: F must contain i (square root of -1). If E was already semi-stable over Q then I guess that nothing happens, but otherwise the height of E gets smaller. Also, you need semi-stable reduction of E over F as part of the conditions. For example, for the Frey curve associated to an ABC triple to be semi-stable over Q, you need that the ABC triple (a,b,a+b) must be primitive (no common factor) and 16 must divide ab(a+b) (perhaps not 16...). This slightly reduces the list "too-good-to-be-true" examples. – Pasten – 2012-09-16T03:58:19.027

18This conversation is very interesting, but there seems to be a community consensus that we should avoid discussion of the validity of recently announced results. (Also, your post is not really an answer to the question that was asked.) – S. Carnahan – 2012-09-16T10:06:15.590

1Indeed, I was afraid this was probably not the place to write this, as it has nothing to do with the original question. – Vesselin Dimitrov – 2012-09-16T10:10:27.163

With $F=\mathbb{Q}$ and $E$ as above we have $F{\mathrm{mod}} = \mathbb{Q}$ and $F{\mathrm{tpd}}=F{\mathrm{mod}}(E[2])=Q$. Then note that the first displayed inequality before the proof on p. 23 involves only $F{\mathrm{tpd}}$, not $F$. That the ABC-triple is primitive is the essential assumption of the ABC-conjecture. As for semistability, it certainly holds away from 2; let's say we assume $16∣ab(a+b)$ as you say: it does not make much of a difference for the shocking outcome. – Vesselin Dimitrov – 2012-09-16T10:20:56.530

1On closer look, I see that the assumptions of Thm. 1.10 are indeed a bit more complicated: for the field $F$ obtained by adjoining $\sqrt{-1}$ and the $3 \cdot 5$-torsion points of $E$, it must be that $G_F \to \mathrm{GL}2(\mathbb{F}{\ell})$ is surjective. Still, a stronger version of uniform open image conjecture should make this automatic, as soon as $\ell >> 0$.

Anyway, I apologize if my post violates the rules of MO: if so, please feel free to remove it from this thread (or delete it). – Vesselin Dimitrov – 2012-09-16T11:09:57.330

6@VD: I think one should read more carefully the hypothesis. Also, I would not be surprised if the final Diophatine statement is not 100% correct as stated and needs to be refined - it is such a long and complicated work!. However, I think that the whole point is the technique: if it is correct$-\epsilon$ then people will make it work at some point. I do not remember the solution of a BIG problem that was 100% correct the first time it was released (perhaps I am exaggerating a little bit). – Pasten – 2012-09-16T13:58:31.077

14I agree. Let us keep on examining Mochizuki's papers, and hope that his daring but beautiful strategy (deforming, as it were, number fields!) really works. ABC is such a delicate problem: the slightest error in one part of the theory could easily ruin the whole proof (as with Miyaoka's earlier attempt).

As for the asymptotic discrepancy with ABC near-misses, Mochizuki himself seems to be a bit perplexed (see the remark on p. 40); hopefully this will be straightened out. – Vesselin Dimitrov – 2012-09-16T16:07:29.040

14If it is completely rewritten, then you should post it as a different answer, because people who upvoted/downvoted your post may change their mind and the relation with discussion in the comments are broken. – 36min – 2012-09-26T17:11:59.457

3

Furthermore, it doesn't really look like an answer to (any version of) the question to me. To be clear, it looks like great work! And might make a good blog post, or even a preprint. But this is in danger of becoming a detailed search for errors in Mochizuki's preprint, which is emphatically NOT what MO is about. (See here and the links therein: http://tea.mathoverflow.net/discussion/1422/discussing-recent-preprints-on-mo-again/#Item_23.)

– HJRW – 2012-09-27T07:59:57.930

134I think that people are working way, way too hard to define MathOverflow by what it isn't. Although I still like MO, too many babies have been thrown out with various bathwater. This posting by Dimitrov is as important as anything else that I have seen in MO lately. Yet a direct question about the validity of Mochizuki's work was closed down, and then this question was also closed down for a while. (Which may be a valid technical reason to revise an answer instead of posting another one.) What really is the point? It becomes rules just for the sake of rules. – Greg Kuperberg – 2012-09-27T16:09:17.547

2Greg - I suppose any further elements of this conversation should happen on meta, if anywhere. I'll just add that I agree about the 'importance' of this post. – HJRW – 2012-09-27T16:35:47.727

23Remark about mathematical communication: If Mochizuki's much-publicized papers need revision, then it would be much better for him to post to the arXiv for the sake of public commitment of versions. Preprints that announce important results shouldn't be a moving target. In fact, with his permission it could be done by proxy. – Greg Kuperberg – 2012-09-28T19:28:42.203

8

Mochizuki has commented that he will post revisions of IUTT-III and IUTT-IV "in the not so distant future": http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf . He acknowledges that there is an error tracing back to IUTT-III, which he believes to be completely fixable.

– Vesselin Dimitrov – 2012-10-14T16:16:07.747

2No offence was meant by adding the preamble, but I see now that I have overstepped the mark. It was rude of me to place that above the answer. I am reminded of the maxim "Never put yourself above others". – user19172 – 2012-10-15T23:30:29.777

1

@Vesselin Dimitrov As this post is already CW and will have occasional revisions for months to come, I suggest updating the first line with the date of the latest revision. Meanwhile, the discussion on META about this question has entirely moved to http://tea.mathoverflow.net/discussion/1447/3/evaluation-of-potentially-credible-papers-should-be-allowed begun by Greg. It is now on the third "page," roughly 105 posts. If you wish to participate in that, there is an easy, but separate, registration. Also: http://tea.mathoverflow.net/discussion/1451/preamble-added-by-yahoo-anonymous-to-dimitro

– Will Jagy – 2012-10-19T19:30:29.747

1@Will Jagy: Thanks for the suggestion, as well as for the links! I just made the latest - and probably last - revision to write down the estimate that Mochizuki expects (which is corrected and confirmed by him), and to say that it can definitely not be disproved in a similar way as the current Thm 1.10 was (thus, it does now seems reasonable). I don't think I will make any more revisions to this post at least until Mochizuki uploads his corrections to IUTT-III and IUTT-IV. (Unless one could solve the open question that I outlined to probe the corrected inequality; but this seems unlikely). – Vesselin Dimitrov – 2012-10-20T17:46:26.500

4I'm sure you're aware of the revisions being released. Any comments? – David Roberts – 2012-12-18T23:47:29.607

I'd just like to point out that Mochizuki has released revised versions of IUTT I,III and IV. – Daniel Miller – 2013-04-14T23:57:26.707

1In fact, his webpage now features revisions of IUTT I,II, III, and V --- with a datestamp of 8-20-2013. – Suvrit – 2013-08-20T22:51:20.933

43

[The answer below is a response to an earlier version of the question that was rather different in certain respects. Minhyong Kim's answer gives excellent insight into ideas that Mochizuki had back in 2000 and that provide essential building blocks for the more recent work. But I still believe that it is too premature for a non-expert to seek insight into the new work, for reasons explained below, given that many top experts are presently trying to absorb the ideas Mochizuki developed back in 2000.]

This question appears to be inspired by an historical fallacy: the only "vision" of a proof of the Weil Conjectures that Grothendieck had when he began developing ideas related to his work on the problem (i.e., etale cohomology) was the one laid out in Weil's original paper. The yoga around the standard conjectures came much later.

That being said, although the new ABC developments are potentially very exciting, and it is understandable to want to "share in the excitement", for reasons specific to this situation it seems to be much too premature to ask for a sketch on MO or in a blog of Mochizuki's vision/proof with an expectation of insight into the new work. Let me try to indicate why this is the case.

As has been explained clearly by JSE elsewhere, there are plenty of top experts in arithmetic geometry who are presently struggling to get even a small handle on what is really going on in Mochizuki's papers (due entirely to the experts' lack of prior study of these ideas; Mochizuki's writing is extremely precise, detailed, thorough, and full of intuitive asides!). So the situation seems to be rather different from that of other tremendous advances in recent decades (by Perelman, Faltings, Wiles, etc.), for which the deep new work took place within a context that was already somewhat familiar to a good-sized community of experts in the field (who could then use their experience and expertise to quickly disseminate a "bird's eye view" to others of some of the key new ideas).

Because of the rather unique circumstances of this case, as just indicated, I believe that quid's initial urging of patience (if one isn't going to be directly engaged with the struggle to read the actual papers and the prior work upon which they depend) is appropriate.

But to end on a semi-positive note, let me explain why quid's mention of Mochizuki's survey papers is very apt. Some of those surveys are relatively short (e.g., less than 20 pages), and if you find them difficult to grok then you will get a real sense of the difficulties that a lot of top experts are current facing in their efforts to try to understand what Mochizuki has achieved. Please be patient! As quid has noted, in due time, as experts eventually come to acquire a genuine understanding of the overall structure of the arguments in these papers, plenty of expositions for wide dissemination of the ideas will emerge. Mochizuki has put a lot of effort into providing indications of his motivation and insights throughout his papers (which are a serious challenge even for top experts to absorb), and to respect his remarkable effort it seems best to engage with it directly (whether through reading the surveys or the main papers).

2+1. I am happy that you made us note the historical misconception at the beginning of the PO's question. The idea that a good cohomology theory would prove Weil's conjecture is due to Weil and much predates Grothendieck's work. Still, when Grothendieck began working in algebraic geometry (say around 1958), I believe he had the idea of how to define étale cohomology. Am I wrong ? But this illustrates the point you're trying to make. Who in 1958 could have anything interesting to say on the intuition of étale cohomology, Grothendieck aside ? (Only Serre, perhaps, but he doesn't write here). – Joël – 2012-09-07T13:39:23.020

1You are correct that I was inaccurate on that point, although I did know that it was Weil's idea. As for you argument for patience, I think you have misunderstood my question. I am not asking for a sketch of the methods, but only what those methods aim to achieve. An example of a good answer is David Speyer's comments. So saying "here is the rough argument in the function fields case, and so what we want is a number theoretic analogue of __" is precisely the answer I was looking for. – James D. Taylor – 2012-09-07T15:48:10.367

5@James: OK, good to hear the clarification. But one of Mochizuki's survey papers does address exactly what you suggest you'd like to hear about at the end of your comment, though using the sophisticated language of moduli stacks (which, if you pretend are schemes, can be inspiring even if you don't know about stacks): see 1.3.1 of "A survey of the Hodge-Arakelov Theory of Elliptic Curves I". Mochizuki is an extremely good writer!! If you elide unclear technical issues and try to just digest the flavor, you can get a lot of inspiration. I am reminded about "reading the masters"... :) – grp – 2012-09-07T16:51:23.957

11Disambiguating note: grp is a login name I use on meta.mathoverflow. Much as I might like to take credit for the answer above, I did not write it. Hopefully the entity will identify themselves further so as to clarify. I do not ask for a change of user name though, either on my part or on the entity's part. Gerhard "Really, That Is Not Me" Paseman, 2012.09.07 – Gerhard Paseman – 2012-09-07T20:11:01.720

3Dear grp, I'm curious as to why you wrote (in the comment thread above) that the function field case had nothing to do with, or provided not motivation for, the number field case. My impression was rather the opposite, that Mochizuki was/is trying to get around Faltings's "no go" theorem about an arithmetic KS map by radically reinterpreting the whole thing in very sophisticated non-linear, or anabelian, terms. I wonder if you can say more about what you meant? Best wishes, Matthew – Emerton – 2012-09-07T22:21:26.950

3Dear Emerton: As you know, there is a purely algebraic proof of the "classical" (not "Szpiro") formulation for F(t), and that is what I was pretty sure David Speyer was considering to present. That argument (as one finds in elementary expositions) is what I meant isn't relevant, much like the F(t)-version of FLT proved by bare-hands algebra rather than by genus; sorry for being unclear. I agree that the F(t)-case proved via a classifying map from a curve to a moduli space of elliptic curves, thereby highlighting the role of Kodaira-Spencer maps, is a crucial perspective in Mochizuki's work. – grp – 2012-09-07T22:35:37.433

1Dear grp, Thanks very much for explaining what you meant; it's reassuring to know that my intuition isn't completely off! Best wishes, Matthew – Emerton – 2012-09-07T22:38:56.907

@grp For the record, I was somewhere between. I understood why you can't directly use the ramification of the map to the $\lambda$ line, as in the function field case, and that Mockiuzuki was using a study of the monodromy of torsion points to get around that issue. But I didn't understand how he was doing it in anything like the detail that the other answers now provide. – David E Speyer – 2012-09-08T14:22:44.397

10For later readers, I want to remark that this answer refers to an old version of the question. – Andy Putman – 2012-09-08T18:03:16.667

1re the example to wiles/FLT, didnt he work largely independently for 7yrs on his proof, with no intermediate papers or communications? my reading of history was that he was secretive of his master project. so imho maybe theres actually a similar pattern here. & anyway the conservativism on stackexchange related sites wrt various big/meaningful subjs is highly annoying at times. there are only a few experts in the entire world who can dissect the proof & gauge its real contribution, & many are on this site, & the software facilitates it, but the humans bicker about it. – vzn – 2012-09-29T00:48:15.670

further thought. it appears that it would be inaccurate or unfair to compare mochizuki with wiles or use the word "secretive" wrt mochizuki. he's been very communicative on his home page in near blog form: http://www.kurims.kyoto-u.ac.jp/~motizuki/thoughts-english.html so there is a symbiosis in research between independent work & communication, and perhaps he has been focused on the independent work, but he has been releasing intermittent papers over recent years, but apparently few outsiders have engaged with it. what can maybe be fairly said is that he hasnt promoted his results.

– vzn – 2012-09-30T19:01:59.637

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Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final estimate, already in 2000. (But let us recall, as a sane reminder of just how elusive the ABC conjecture has been, Miyaoka's flawed proof: did the idea of a Bogomolov-Miyaoka-Yau type bound involving arithmetic Chern numbers in Arakelov theory not seem equally beautiful, exciting, and promising?)

In brief, the main idea behind the IUTT-series is to construct, outside the rigid confines of algebraic geometry, a subtle object simulating a rank-1, Galois-stable quotient of $E[\ell]$. Here, $E/\mathbb{Q}$ is a (pretty much) arbitrary rational elliptic curve (and this is the main point: such a Galois-stable quotient will almost never exist!); and $\ell \geq 5$ is an auxiliary prime, generic for $E$ in a very mild sense, but otherwise free to optimize until the very final estimate. This is then applied to construct, in the non-linear discretized "Hodge-Arakelov theory," a comparison isomorphism between $(E^{\dagger}, <\ell)$ and $E[\ell]$, which is free of Gaussian poles at the bad places of $E$. For this then leads to a promising Galois-theoretic "Kodaira-Spencer map," as explained in Minhyong Kim's post, hopefully leading in the familiar way to the arithmetic Szpiro inequality for this very same elliptic curve: $\log{|\Delta_{\mathrm{min}}(E)|} \leq (6+\varepsilon) \log{N_E} + O_{\varepsilon}(1)$.

Let me, however, disagree with one point from M. Kim's post. My impression is that what Mochizuki calls an "initial $\Theta$-datum" - and which is, essentially, the pair of the rational elliptic curve $E$ (or equivalently, the $abc$-triple from the ABC-conjecture!) and, until the very final estimate in Ch. 2 of the fourth paper, the prime level $\ell$ - are fixed for good throughout the entire series of IUTT-papers. The deformation flavor of "Teichmuller theory" refers to dismantling the underlying number field, and not to the elliptic curve enhancement (indeed, in Mochizuki's dictionary with his own $p$-adic Teichmuller theory, it is the number field that corresponds to a hyperbolic curve; the elliptic curve enhancement corresponds to an "indigenous bundle" over the hyperbolic curve, and invites the anabelian philosophy via the \'etale fundamental group of the once-punctured elliptic curve). All the "Hodge theaters" associated to the initial $\Theta$-datum are isomorphic to one another, and form a vastly complicated $2$-dimensional non-commutative array - the "$\mathfrak{log}-\Theta$ lattice" - of non-ring theoretic translations between one another. What Mochizuki writes on p. 10 of IUTT-I is that the theory of $\Theta$-Hodge theaters "may be regarded as a sort of solution to the problem of constructing the global quotient $E[\ell] \twoheadrightarrow Q$" [needed for the application to arithmetic Kodaira-Spencer]. He does not seem to suggest that this is done by "moving the initial $E$ to a single elliptic curve via the intermediate case of an elliptic curve in general position," as M. Kim writes. (The term "elliptic curves in general position" indeed figures in Mochizuki's fourth paper, but it has a different, not-so-essential significance that comes through his entirely self-contained paper [GenEll], and whose purely technical purpose is to reduce the general ABC conjecture to the restricted version of Szpiro's inequality for $E$, in Thm. 1.10 of IUTT-IV, coming from the estimate in IUTT-III).

In particular, in sharp contrast to the Thue-Siegel-Roth tradition of Diophantine approximations, Mochizuki's program does not seem to compare different elliptic curves / $abc$-triples, all the way through to the key estimate $$(*) \hspace{3cm} \log{|\Delta_{\mathrm{min}}(E)|} \leq \big(6 + \varepsilon + 200/\ell\big)\log{N_E} + 12\log(\ell\varepsilon^{-7})$$ of IUTT-IV [asserted for all primes $\ell \geq 5$ that are generic for $E$ in a rather mild sense: essentially, $\ell$ has to be prime to the degenerate places and the $q$-parameters of $E$. Also, $\varepsilon \in (0,\epsilon_0)$ is arbitrary, with $\epsilon_0$ a numerical value. ] In this sense, Mochizuki's approach - nevermind the vast technical difficulties precipitated by the non-ring theoretic simulation of a global quotient $E[\ell] \twoheadrightarrow Q$ - is entirely direct and, consequently, effective.

So what does Mochizuki actually (claim to) prove?

Start with an $abc$-triple (co-prime rational integers with $a+b+c=0$). Since the discriminant $(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)$ of a cubic polynomial $x^3 + \cdots$ encapsulates exactly this equation, it is a profitable, traditional idea to interpret the $abc$-datum as the giving of the rational elliptic curve $E = E_{a,b,c}$ defined by the equation $y^2 = x(x-a)(x+b)$. The (apparently weaker, but virtually as powerful) ABC conjecture $abc < K_{\varepsilon}\cdot\mathrm{rad(abc)}^{3+\varepsilon}$ then translates into Szpiro's inequality: $\log{|\Delta_{\min}(E)|} \leq (6+\varepsilon)\log{N_E} + O_{\varepsilon}(1)$ between the minimal discriminant $\Delta_{\min}(E)$ and conductor $N_E$ of $E$ (which are, essentially, $(abc)^2$ and $\mathrm{rad}(abc)$). Pick the "auxiliary prime" $\ell \geq 5$ to be generic for $E$ in the sense that, essentially: (1) $\ell \nmid abc$; (2) $\ell$ does not divide the prime exponents in $abc$; (3) for $F := \mathbb{Q}( \sqrt{-1}, E[15] )$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2 (\mathbb{Z}/\ell)$. [Conjecturally, the last condition should only exclude a finite list of primes, independent of $E$!] Then Mochizuki [IUTT-IV, Thm. 1.10] claims that (*) should hold for any $\varepsilon < \epsilon_0$.

This is the essential Diophantine estimate. Anything further than that [i.e., the deduction of the full ABC conjecture in IUTT-IV, Section 2] consists of standard, and relatively straightforward reductions [such as, e.g., the use of non-critical Belyi maps] elaborated in Mochizuki's self-contained paper [GenEll]: "Arithmetic elliptic curves in general position." Mochizuki indeed writes, in his first paper, that the auxiliary prime level $\ell \geq 5$ from the Hodge-Arakelov discretized non-linear comparison isomorphisms/correspondences $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$, will be chosen in the Diophantine application to be large, roughly on the order of the height of $E$. But this comes entirely through Theorem 3.8 in [GenEll]: there, the various non-divisibility properties are ensured by simply taking $\ell$ to exceed all the primes of bad reduction / all the $q$-parameters (also, the full Galois action is ensured unconditionally). In (*), $\ell$ could be any prime satisfying the mentioned non-divisibility conditions. (This, by the way, is what I considered highly disturbing).

My apology if I have misunderstood - and misrepresented - the points from Mochizuki's papers that I have alluded to.

5For people wanting to known more details without having to read all the 500 pages: the answer provided by VD is a nice survey of the first 16 pages of IUTT-I (avoiding technicalities of how the several types of Hodge theaters are actually constructed, or what are the prime-strips). For the interested reader, the first 27 pages of IUTT-I indeed give a very good introduction. However, it is better to get used with the language of Frobenioids FIRST, otherwise the exposition can be intimidating. Unfortunately, it does not hint on the actual "source of inequality" (I mean, not beyond analogies). – Pasten – 2012-09-17T21:28:11.267

1(cont.) The table in page 27 of IUTT-I gives an idea of what are the roles played by some of the main objects introduced by Mochizuki (and as VD pointed out, the hyperbolic curve "is" the number field, not the elliptic curve). Anybody can read this directly from the paper, but the only reason why I am mentioning it is the following: I was very curious about the papers (as everybody else), but the first couple pages seemed very intimidating. However, after spending some time with the papers on Frobenioids then the introduction of IUTT-I became readable after all. I hope this suggestion helps! – Pasten – 2012-09-17T21:42:20.440

5This sounds like sensible mathematics, far from the cries of 'no one has any idea what he's doing!' reported in the press. Creating new mathematical objects that are formal (co)limits of existing objects without the required (co)limits is a long-established tradition (e.g. negative numbers! but more recently stacks etc). – David Roberts – 2012-09-18T01:01:12.617

6Deane Yang mentioned this post on Facebook, remarking that Vesselin Dimitrov 'dared to disagree' with me. Of course Deane meant this in a good humored way. But still, I copy here my reply, in case there's any misunderstanding: Hi Deane, 'Dare to disagree' is a strong term! I still haven't had any time to really look at Mochizuki, so it's quite likely I said something stupid that was noted by Dimitrov. I'll try to look at it more carefully soon, but I hope it doesn't take much daring for people to disagree with me (in general). – Minhyong Kim – 2012-09-18T11:05:01.897

1My apology for sounding argumentative in relating my impression from Mochizuki's papers: it was not my intention, and I should not have used the word "disagree." – Vesselin Dimitrov – 2012-09-18T20:10:27.033

2@Vesselin: I'm sorry I voted to close. Your answers (and the others too) are quite interesting, but it's a matter of principle. – Felipe Voloch – 2012-09-19T00:02:02.697

@DavidRoberts just wanted to say that your optimistic comment was very uplifting, and on a personal level, encouraged me to go back and not only take another stab at trying to understand IUT, but also led me to realize my understanding of colimits was woefully inadequate!

exactly the point of Mathoverflow comments, and MO in general, in my humble opinion – Samantha Y – 2017-11-16T22:15:34.167

34

I want to point out a bibliographical information that perhaps is not very well-known and can be taken as "evidence" for the possibility of applying anabelian geometry to the ABC conjecture successfully. However, I am not claiming that this is related in any sort of way to Mochizuki's work.

Here is the fact: There is a $\pi_1$ proof of the function field Szpiro conjecture (over the complex numbers, as far as I know). The proof is indeed easy and conceptually clear, you can find a nice exposition of it in some (expository) paper of Zhang, whose title is lost somewhere in my memories. (EDIT: the paper is "Geometry of algebraic points").

Anyway, I can tell you what is the key point of the argument. Let E be an elliptic fibration over the projective line L over the complex numbers. Assume that E has only multiplicative bad reduction. You can read the order of the discriminant at a point of L from the Kodaira type of the fibre, which in turn can be recovered in terms of monodromy representations of the fundamental group of L minus the points with bad fibres: smooth fibres have trivial monodromy, and the monodromy of singular fibres is determined by Dehn twists (assuming multiplicative reduction). You can look at all these local representations at once, after choosing loops to link bad points to some generic point p of L and then study the image of the global monodromy representation on the homology of the fibre above p. Choosing loops appropriately gives the usual commutator relation which in the image of the global representation gives a relation R=1 among generators of the local reps (and they "know" what the discriminant is). Everything here is inside $SL_2(Z)=Aut(Z^2)=Aut(H_1(E_p,Z))$ which acts on the real plane, and up to scalars it acts on the projective real line whose universal covering you already know (yes, the real line). One can lift the relation R=1 to a relation among automorphisms of the real line to get a relation R'=1' where now 1' knows the number of terms appearing on R, that is to say the number of singular fibres, which is the conductor of E in this setting. Then the Szpiro bound can be recovered from the relation R'=1'.

And there you have, a derivative-free proof of the Szpiro conjecture for function fields (a bit shocking at least for me the first time I saw it). All the diophantine information being supplied by fundamental groups.

5Are you thinking of "Geometry of algebraic points", Zhang's address to the 1998 International Congress of Chinese Mathematicians, www.math.columbia.edu/~szhang/papers/iccm98.ps ? – David E Speyer – 2012-09-08T14:13:28.923

1yes, you found it! – Pasten – 2012-09-08T14:26:21.437

3

Apparently, this answer written in September 2012 turned out to be not-so-unrelated to the question. See http://www.kurims.kyoto-u.ac.jp/~motizuki/Bogomolov%20from%20the%20Point%20of%20View%20of%20Inter-universal%20Teichmuller%20Theory.pdf

– Pasten – 2015-08-24T19:48:47.900

15

NEW !! (2013-02-21)

A Panoramic Overview of Inter-universal Teichmüller Theory By Shinichi Mochizuki

http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf

...as well as a brand new survey thereof: http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf

– Sylvain JULIEN – 2016-07-03T13:54:48.580