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Timeline for answer to Philosophy behind Mochizuki's work on the ABC conjecture by grp

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Apr 26, 2025 at 12:18 history edited kindasorta CC BY-SA 4.0
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Sep 30, 2012 at 19:01 comment added vzn 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: 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.
Sep 29, 2012 at 0:48 comment added vzn re 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.
Sep 8, 2012 at 23:27 history edited grp CC BY-SA 3.0
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Sep 8, 2012 at 18:03 comment added Andy Putman For later readers, I want to remark that this answer refers to an old version of the question.
Sep 8, 2012 at 14:22 comment added David E Speyer @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.
Sep 7, 2012 at 22:38 comment added Emerton Dear grp, Thanks very much for explaining what you meant; it's reassuring to know that my intuition isn't completely off! Best wishes, Matthew
Sep 7, 2012 at 22:35 comment added grp Dear 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.
Sep 7, 2012 at 22:21 comment added Emerton Dear 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
Sep 7, 2012 at 20:11 comment added Gerhard Paseman Disambiguating 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
Sep 7, 2012 at 16:51 comment added grp @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"... :)
Sep 7, 2012 at 15:48 comment added James D. Taylor You 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.
Sep 7, 2012 at 13:39 comment added Joël +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).
Sep 7, 2012 at 13:28 history answered grp CC BY-SA 3.0