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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?

YCor
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Madeleine Birchfield
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    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. – David Roberts Nov 22 '22 at 15:15
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    For those interested, Joshi has made some additional comments in this blog post: https://thehighergeometer.wordpress.com/2022/11/25/a-study-in-basepoints-guest-post-by-kirti-joshi/ (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) – David Roberts Nov 25 '22 at 01:13
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    Can anybody explain to me the precise meaning of "family of isomorphs of ... parametrized by ..."? – Piotr Achinger Nov 25 '22 at 11:05
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    @PiotrAchinger The relevant data can also be described as "a family of spaces, each of whose fundamental groups is isomorphic to ..., parameterized by ...". – Will Sawin Nov 25 '22 at 12:31

5 Answers5

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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...

Peter Scholze
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    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. – David Roberts Nov 22 '22 at 14:10
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    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... – Peter Scholze Nov 22 '22 at 16:39
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    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 :-) – David Roberts Nov 22 '22 at 23:59
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    @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?" – Will Sawin Nov 23 '22 at 00:50
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    @Will Fair enough. We will see what Joshi's promised forthcoming paper does, when it lands. – David Roberts Nov 23 '22 at 01:08
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    This looks related to a point Mochizuki seems to make at various places (e.g. panoramic overview, p.38-39): If one takes two copies of a curve $X/k$ and relates the base fields by $log:k\rightarrow k$, then even though the abstract groups $\pi_1(X)$ are (poly)isomorphic compatibly with their actions on $k$ and the log-map between the two copies of $k$, there is no scheme theoretic isomorphism between the two $X/k$ compatible with log (Mochizuki's equivalence of categories result is not applicable in this situation, as the two copies of $k$ are not related via a homomorphism of fields.) – user_1789 Nov 25 '22 at 12:30
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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.

Will Sawin
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    Now he says it : https://arxiv.org/abs/2401.13508 – Héhéhé Jan 28 '24 at 07:14
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    @Héhéhé It seems to me Joshi is currently claiming Corollary 3.12 but not the ABC conjecture or any other Diophantine inequalities. – Will Sawin Jan 29 '24 at 15:09
  • I'm not sure if Joshi's Corollary 3.12 is the same as Mochizuki's Corollary 3.12. He writes at the end of section 7.8 that "On the other hand, the above inequality suggests that the passage to the tensor product version $\hat{\hat{Θ}}_{\mathrm{Joshi}}^B$ should be expected to provide tighter upper bounds!" but he never proved the upper bounds, while Mochizuki's inequality specifically includes the upper bounds. – Madeleine Birchfield Feb 03 '24 at 17:48
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    OK, but now he really does say it: https://arxiv.org/abs/2403.10430 – Gro-Tsen Mar 18 '24 at 19:06
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    Here is a response from Mochizuki to the preprints https://www.kurims.kyoto-u.ac.jp/~motizuki/Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf (u/Valvino on reddit had posted this first with some discussion here: https://www.reddit.com/r/math/comments/1bmp1vk/very_salty_mochizukis_report_about_joshis/) – Sidharth Ghoshal Mar 25 '24 at 00:28
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My response to Mochizuki's Comments on my papers [Preprints: Construction of Arith. Holo. Strs I, II, II(1/2), III, IV. I will restrict myself to mathematics.

My work shows that Mochizuki's [IUT 1-3] requires modern p-adic Hodge Theory and not group theory of the fundamental group as many may have previously believed. The theory of [IUT 1-3] is about working with distinct arithmetic holomorphic structures and averaging over the many distinct p-adic periods (of a fixed elliptic curve) they give rise to.

My work [Preprints: Constr. of Arith. Holo. Strs I,II,II(1/2),III] provides a precise way for doing this and provides the correct mathematical tools to discuss and verify Mochizuki's [IUT 4]. My [Preprint: Constr. of Arith. Holo. Strs IV] is Mochizuki's proof of the abc-conjecture, and I think that up to any necessary modifications of my preprints [Preprints: Constr. of Arith. Holo. Strs III, IV], we are now in a position to robustly verify the validity of the main theorem of [IUT 4]. My approach to [IUT4], based on my theory, is in [Preprint: Constr. of Arith. Holo. Strs IV].

It is my sincere hope that Mochizuki, who seems to be warming to my ideas, introduced in [Preprints: Constr. of Arith. Holo. Strs I,II,II(1/2),III], especially my use of perfectoids and untilts in this context, will help us arrive at the correct mathematical conclusions regarding [IUT 4], in spite of his current negativity about my papers themselves.

As my LaTeX source files of my preprints on arxiv.org will testify, I use the standard automated theorem numbering provided by AMSLaTeX (American Math. Society, LaTeX). Any numerical coincidences anyone finds in the theorem numbering in my papers must be considered entirely self-imagined.

Will Orrick
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Kirti Joshi
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    I do not know what to make of this answer: it is not clear whether the poster is truly Kirti Joshi himself, it is not clear what the references between square brackets point to, it is not clear what the last paragraph about theorem numbering alludes to, and in general it is not clear to me whether this post is trolling or not. – Alex M. Mar 27 '24 at 16:25
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    @AlexM. This is with high probability Joshi himself. In particular, the last paragraph is a reference to a rather strange (to put it mildly) comment in Mochizuki's recent response cited here, in the middle of page 7. Certainly nothing in this post suggests trolling. – Noah Schweber Mar 27 '24 at 16:29
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    The references appear to be I, II, II(1/2), III, IV. – Timothy Chow Mar 27 '24 at 18:19
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    It is very unfortunate that Mochizuki has written that paper you link in that fashion, it is extremely unprofessional the language he uses and some of the statements he makes. – Stiofán Fordham Mar 27 '24 at 21:55
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    What about Mochizukis argument "any essentially global inequality — i.e., such as the ABC/Szpiro inequalities or [IUTchIII], Corollary 3.12 — can never be obtained in this way, i.e., as a result of summing up local inequalities at each prime of a number field." ? – jjcale Mar 28 '24 at 16:18
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    @jjcale A MathOverflow question was asked about this a few days ago. – Will Orrick Mar 28 '24 at 16:50
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    @AlexM. It is Joshi – David Roberts Mar 29 '24 at 11:20
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For a detailed, mathematical and evidence based discussion of the Mochizuki-Scholze-Stix issues, let me point out my articles intended for a wider mathematical audience:

  1. Comments on Arithmetic Teichmuller Spaces (https://arxiv.org/pdf/2111.06771).

  2. Mochizuki’s Corollary 3.12 and my quest for its proof (https://www.math.arizona.edu/~kirti/joshi-teich-quest.pdf) [My mathematical conclusion is that Scholze-Stix claim is based on a flawed premise.]

  3. My guest blog post on David M. Roberts blog https://thehighergeometer.wordpress.com/2022/11/25/a-study-in-basepoints-guest-post-by-kirti-joshi/ is also recommended.

  4. Let me also add the following note which may be useful for many readers: https://www.math.arizona.edu/~kirti/Response-to-grouchy-expert.pdf.

Detailed proofs of my results can be found in my preprints related to Mochizuki’s work, the links to which can be found at (https://ncatlab.org/nlab/show/Kirti+Joshi) or in the preprints section of my webpage (https://www.math.arizona.edu/~kirti/).

Kirti Joshi
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@Madeleine Birchfield

My recently released paper (Constructions III) proves many versions of Corollary 3.12 including Mochizuki’s version and I provide a `Rosetta Stone’ which facilitates a detailed comparison of the two theories. My formulation and proof of the geometric case of Corollary 3.12 is given in Constructions II(1/2).

The abc-conjecture is an expected, but delicate consequence of Corollary 3.12 (and some other results) and my conclusions (with proofs) will appear in my forthcoming paper: Construction of Arithmetic Teichmuller Spaces IV.

To clarify the discussion about upper bounds, let me say that Mochizuki does not deal with upper bounds on this set in IUT III but takes this up in IUT IV. In my case too, upper bounds are taken up in my forthcoming paper, Constructions IV.

Timothy Chow
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Kirti Joshi
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