A study in basepoints: guest post by Kirti Joshi

[The text below the divider is a response by Kirti Joshi in response to some comments at MathOverflow regarding his recent preprint, “Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups — Arithmetic Holomorphic Structures“. Kirti reached out to me regarding making a response, and I suggested that a blog posting would be better than an answer at MathOverflow, since it is not in the format of an answer to the question, to which he agreed. Regular readers of the blog will know that I follow developments around IUT with close interest, but I am not an expert in that area. My long-stated hope is that some interesting mathematics comes out of the whole affair, regardless of specifics about the correctness of Mochizuki’s proof or otherwise. –tHG]

I wish to clarify my work in the context of the discussion here. For this purpose suppose that $X$ is a geometrically connected, smooth quasi-projective variety over a $p$ -adic field which I will take to be ${\mathbb{Q}}_p$ for simplicity. In Mochizuki’s context this $X$ will additionally required to be an hyperbolic curve.

First of all let me say this clearly: one cannot fix a basepoint for the tempered fundamental group of $X$ in Mochizuki’s Theory [IUT1–IUT4]. The central role of (arbitrary) basepoints play in Mochizuki’s theory is discussed in (print version) [IUT1, Page 24], and notably the key operations of the theory, namely the log-link and theta-link, change or require arbitrary basepoints on either side of these operations [IUT2, Page 324] (and similar discussion in [IUT3]).
This means one cannot naturally identify the tempered fundamental groups arising from distinct basepoints. [The groups arising from different basepoints are of course abstractly (and non-canonically) isomorphic. Mochizuki does not explicitly track basepoints while requiring them and so this makes his approach extremely complicated.]
In the context of tempered fundamental groups, a basepoint for the tempered fundamental group of $X$ is a morphism of Berkovich spaces $\mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p}$ , where $K$ is an algebraically closed complete valued field containing an isometrically embedded ${\mathbb{Q}}_p$ . [Such fields are perfectoid.] As I have detailed in my paper, arbitrary basepoints requires arbitrary perfectoid fields $K$ containing an isometrically embedded ${\mathbb{Q}}_p$ . [For experts on Scholze’s Theory of Diamonds, let me say that the datum $(X^{an}_{{\mathbb{Q}}_p}, \mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p})$ required to define tempered fundamental group with basepoint $\mathcal{M}(K)\to X^{an}_{{\mathbb{Q}}_p}$ is related (by Huber’s work) to a similar datum for the diamond $(X^{ad})^\diamond$ associated to the adic space for $X/{\mathbb{Q}}_p$ .]
In my approach I track basepoints explicitly (because of the (1) above) and I demonstrate how basepoints are affected by the key operations of the theory. [This is claimed in Mochizuki’s papers, but I think his proofs of this are quite difficult to discern (for me).]
Because basepoints have to be tracked, and tempered fundamental groups arising from distinct basepoints cannot all be naturally identified, assertions which involve arbitrarily identifying fundamental groups arising from distinct basepoints cannot be used to arrive at any conclusion about Mochizuki’s Theory.
In arithmetic geometry one typically works with isomorphism classes of Riemann surfaces i.e. with moduli of Riemann surfaces. Teichmuller space requires a different notion of equivalence and it is possible for distinct points of the classical Teichmuller space to have isomorphic moduli. This is also what happens in my $p$ -adic theory.
There is no linguistic trickery in my paper. I have developed my approach independently of Mochizuki’s group theoretic approach and my approach is geometric and completely parallels classical Teichmuller Theory. Nevertheless in its group theoretic aspect, my theory proceeds exactly as is described in [IUT1–IUT3] and arrives at all the principal landmarks with added precision because I bring to bear on the issues the formidable machinery of modern $p$ -adic Hodge Theory due to Fargues-Fontaine, Kedlaya, Scholze and others. This precision allows me to give clear, transparent and geometric proofs of many of the principal assertions claimed in [IUT1–IUT3] without using Mochizuki’s machinery. [Notably my view is that Mochizuki’s Corollary 3.12 should be viewed as consequent to the existence of Arithmetic Teichmuller Spaces (at all primes) as detailed in my papers. One version of Corollary 3.12 is detailed (with proof) in my Constructions II paper. The cited version works with the standard complete Fargues-Fontaine curve ${\mathscr{X}}_{{\mathbb{C}_p^\flat},{\mathbb{Q}}_p}$ , but there is a full version better adapted for Diophantine applications which works with the adic Fargues-Fontaine curve ${\mathscr{Y}}_{{\mathbb{C}_p^\flat},{\mathbb{Q}}_p}$ and its finite étale covers and which exhibits the tensor packet structure of [IUT3, Section 3] which will appear in Constructions III (the two proofs are similar).] No claims are presently being made about the main result of [IUT4]. That is a work in progress.
Apart from its intrinsic value, my original and independent work provides new evidence regarding Mochizuki’s work and I urge the mathematical community to reexamine his work using the emerging mathematical evidence.
I am happy to talk to any mathematician who is interested in my work and I have written my papers as transparently as possible. [If there are any concrete mathematical objections to my papers, I will be happy to address them.]
I thank David M. Roberts for a number of suggestions which have helped me improve this text.

7 thoughts on “A study in basepoints: guest post by Kirti Joshi”

yonz says:

November 25, 2022 at 3:01 pm

I would like to hear what Mr. Scholze thinks about Mr. Joshi’s response. Have you notified Mr. Scholze about Joshi’s response? Maybe it’s a little bit too harsh to say that Mr. Joshi uses “linguistic trickery” in his paper. I hope Mr. Scholze responds to clarify (or possibly apologizes if he misunderstood Mr. Joshi’s paper) what he meant by “linguistic trickery” in Mr. Joshi’s paper

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1. David Roberts says:
  
  November 25, 2022 at 3:05 pm
  
  I haven’t notified Peter Scholze, but I don’t think he is unaware of Joshi’s work, and can understand it at a deeper level than I can. Should a discussion happen here in the comments, I’m happy to facilitate. But we will see.
  
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voloch says:

November 25, 2022 at 5:03 pm

If Kirti is claiming that Cor 3.12 is correct but is not claiming ABC, is there a (potential) problem somewhere else? Where?

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1. David Roberts says:
  
  November 25, 2022 at 5:28 pm
  
  He has explicitly said he’s not claiming any estimates (which I take to include Cor 3.12) at present, merely that the machinery really does do something non-trivial.
  
  My own personal take is that with such heavy reworking and radical strip-down of the Mochzuki tech, the gaps in it (or at least in the exposition of it) stand out more easily.
  
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  1. voloch says:
    
    November 25, 2022 at 5:36 pm
    
    In his point 7 above, he is explicitly saying that he proved Cor 3.12.
    
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  2. David Roberts says:
    
    November 25, 2022 at 5:59 pm
    
    I take his “One version of Corollary 3.12” to not mean the actual statement as given by Mochizuki, but something analogous but weaker. And “there is a full version better adapted for Diophantine applications” I think means including all the relevant arithmetic information. In the papers Kirti has written in the past couple of years he has been cautious and included disclaimers that he isn’t proving the inequality in Mochizuki’s Cor 3.12, but giving proofs of concepts.
    
    In an earlier draft Kirti wrote to me “For Diophantine applications a more elaborate version [than the version referred to above in 7] is needed.” Also, he mentioned that his work is local in nature, which he says should be cleared up first, before establishing global results as Mochizuki has claimed.
    
    I think the “work in progress” is just that: more of a “no comment” than hinting at a concrete resolution in any particular direction.
    
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  3. David Roberts says:
    
    November 25, 2022 at 6:24 pm
    
    In particular, in the paper referenced in point 7, Joshi writes, in §1.7:
    
    “…to be absolutely and perfectly clear and as is clarified below, Theorem 10.1.1 is structurally similar but not the same as [Mochizuki, 2021c, Corollary 3.12]. Importantly, Theorem 10.1.1 is still local (i.e. takes place at a single prime), but it can be globalized, i.e. adelized, quite easily in the presence of a number field using the adelic arithmetic Teichmuller spaces of [Joshi, 2021a]–but this adelic version is not treated here.”
    
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