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Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635

In that preprint, Kirti Joshi claims that

  • he agrees with Scholze and Stix that Mochizuki's proof of ABC is incomplete,

  • Scholze and Stix's rigidity claim in Remark 9 of their paper "Why abc is still a conjecture" is wrong

  • "This paper provides the first proof of Mochizuki’s non-redundancy claim by establishing that the isomorphs are of distinct arithmetic-geometric provenance (and even continuous families of isomorphs exist) and therefore are non-redundant"

If these results are confirmed, what are the consequences of this preprint on the validity of IUT as a theory and Mochizuki's proof of the ABC conjecture?

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    $\begingroup$ I think "nonredundancy claim" is a really rubbery phrase. Mochizuki uses the word "redundant" in his expository documents to talk about his own formalism of diagrams he wishes to take colimits of, where there are multiple abstractly isomorphic objects (he claims you literally need an injective functor coding the diagram). Joshi is talking about existence of nontrivial "arithmetic" deformations. It may be that Mochizuki's expository documents are more like extended soft metaphors, but the examples he gives are so far from the actual problems at hand they are not so useful. $\endgroup$
    – David Roberts
    Nov 22 at 15:15
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    $\begingroup$ For those interested, Joshi has made some additional comments in this blog post: thehighergeometer.wordpress.com/2022/11/25/… (initially Joshi reached out to me in order to respond at this question—he isn't an MO user—but I thought that the intended purpose of MO made this not so amenable) $\endgroup$
    – David Roberts
    Nov 25 at 1:13
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    $\begingroup$ Can anybody explain to me the precise meaning of "family of isomorphs of ... parametrized by ..."? $\endgroup$ Nov 25 at 11:05
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    $\begingroup$ @PiotrAchinger The relevant data can also be described as "a family of spaces, each of whose fundamental groups is isomorphic to ..., parameterized by ...". $\endgroup$
    – Will Sawin
    Nov 25 at 12:31

2 Answers 2

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I should point out that Joshi's paper does not falsify Remark 9 of our note.

In Joshi's Theorem 4.8 (which he claims to falsify our Remark 9) the curve $X/E$ stays the same (and hence of course its tempered fundamental group stays the same). The only thing that changes is how $E$ is embedded into an untilt $K$ of an auxiliary characteristic $p$ perfectoid field $F$. But this extra data also doesn't have anything to do whatsoever with the situation -- of course one can't reconstruct it from the tempered fundamental group, as the latter doesn't even know about this extra data...

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    $\begingroup$ Forgive the query, but when you say "doesn't have anything to do whatsoever with the situation" (my emphasis), do you mean your Remark 9, the existence of deformations (as Joshi claims), or the whole ballgame (I guess Mochizuki's Cor 3.12)? I think, from my naive understanding, that Joshi would agree with you that $F$ and $K$ aren't coming from the anabelian data, but also that this is precisely the point. But this is very much a surface reading of the claims. $\endgroup$
    – David Roberts
    Nov 22 at 14:10
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    $\begingroup$ For some categories $C$ and $D$, there's a functor from $C$ to $D$, and we claim (and prove! -- by citing an old paper of Mochizuki) that this functor is fully faithful. Joshi defines a category $C'$ with a non-fully-faithful forgetful functor $C'\to C$ by endowing objects of $C$ with some extra data, and then notes that $C'\to C\to D$ is not fully faithful. Joshi makes the linguistic trick of calling the extra data he puts on objects of $C$ an "arithmetic holomorphic structure", but this is just linguistics... $\endgroup$ Nov 22 at 16:39
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    $\begingroup$ Thanks for the perspective. One might hope that the additional structure/data is doing some extra work over and above the anabelian geometry that so far hasn't managed to close the gap. But, I leave it to the experts to sort out :-) $\endgroup$
    – David Roberts
    Nov 22 at 23:59
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    $\begingroup$ @DavidRoberts I don't think there's too much more for experts to do until a proof which claims to use this extra structure to prove a Diophantine inequality appears. Before that, what can you say beyond "it doesn't seem to me like the extra data that's introduced will be helpful for the problem, because it's data about something that is apparently unrelated?" $\endgroup$
    – Will Sawin
    Nov 23 at 0:50
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    $\begingroup$ @Will Fair enough. We will see what Joshi's promised forthcoming paper does, when it lands. $\endgroup$
    – David Roberts
    Nov 23 at 1:08
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To give a simple answer: There would be no direct implications. The paper doesn't claim a proof of Corollary 3.12, the ABC conjecture, or any other Diophantine inequalities. I'm pretty sure that, if Joshi had a proof of one of these, he would say it.

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