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    $\begingroup$ @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"... :) $\endgroup$ Commented Sep 7, 2012 at 16:51
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    $\begingroup$ 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 $\endgroup$ Commented Sep 7, 2012 at 20:11
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    $\begingroup$ 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 $\endgroup$ Commented Sep 7, 2012 at 22:21
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    $\begingroup$ 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. $\endgroup$ Commented Sep 7, 2012 at 22:35
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    $\begingroup$ For later readers, I want to remark that this answer refers to an old version of the question. $\endgroup$ Commented Sep 8, 2012 at 18:03