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つづき

The reason it cannot work is a theorem of Mochizuki himself. This states that a hyperbolic curve over a -adic field (maybe with some assumptions, all of which are always satisfied in all cases relevant to IUT) is determined up to isomorphism by its fundamental group
, and in fact automorphisms of are bijective with outer automorphisms of
. Thus, the data of is completely equivalent to the data of
as a profinite group up to conjugation. In IUT, Mochizuki always considers the latter type of data, but of course up to equivalence of groupoids this makes no difference. (The passage back and forth is even constructive, by another result of Mochizuki.)

Mochizuki claims that by replacing by
, things can happen that cannot otherwise happen. Examples are given concerning the action of
on certain associated monoids. We discussed this at very great length in Kyoto, but none of these examples carried any actual content. Note that any potential non-commutativity of some diagram that results from identifying
’s via isomorphisms of ’s could not possibly be resolved by using some other isomorphism of
’s ? all of them come from isomorphisms of ’s! Mochizuki considers infinitely many distinct isomorphic copies of
’s, but could not tell us what goes wrong if we simply identify all of them with one another, and with
for some fixed
? there is no diagram that commutes in his situation but does not commute under this further identification. (In my manuscript with Stix, we simply went through Mochizuki’s argument with this further identification, pinpointing what goes wrong. If this further identification causes problems, just tell us which diagram it is whose commutativity is rescued by not explicitly identifying
’s.)

つづく