Davide Castelvecchi at Nature has the story this morning of a press conference held earlier today at Kyoto University to announce the publication by Publications of the Research Institute for Mathematical Sciences (RIMS) of Mochizuki’s purported proof of the abc conjecture.
This is very odd. As the Nature subheadline explains, “some experts say author Shinichi Mochizuki failed to fix fatal flaw”. It’s completely unheard of for a major journal to publish a proof of an important result when experts have publicly stated that the proof is flawed and are standing behind that statement. That Mochizuki is the chief editor of the journal and that the announcement was made by two of his RIMS colleagues doesn’t help at all with the situation.
For background on the problem with the proof, see an earlier blog entry here. In the Nature article Peter Scholze states:
My judgment has not changed in any way since I wrote that manuscript with Jakob Stix.
and there’s
“I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof.
I asked around this morning and no one I know who is well-informed about this has heard of any reason to change their opinion that Mochizuki does not have a proof.
Ivan Fesenko today has a long article entitled On Pioneering Mathematical Research, On the Occasion of Announcement of Forthcoming Publication of the IUT Papers by Shinichi Mochizuki. Much like earlier articles from him (I’d missed this one), it’s full of denunciations of anyone (including Scholze) who has expressed skepticism about the proof as an incompetent. There’s a lot about how Mochizuki’s work on the purported proof is an inspiration to the world, ending with:
In the UK, the recent new additional funding of mathematics, work on which was inspired by the pioneering research of Sh. Mochizuki, will address some of these issues.
which refers to the British government decision discussed here.
There is a really good inspirational story in recent years about successful pioneering mathematical research, but it’s the one about Scholze’s work, not the proof of abc that experts don’t believe, even if it gets published.
Update: See the comment posted here from Peter Scholze further explaining the underlying problem with the Mochizuki proof.
I now think a good reason to for not identifying $\pi_1$’s along log-links is as follows:
(It is very similar to [IUT-2, Rem 1.11.2 (ii)] for the theta-link; I am less precise here, taking the relevance of some reconstructions on faith, but it is a intuitive story, and precision can come later)
First, let us step back and look at the log-theta-lattice: It consists of Hodge-theaters and the log/theta-links as maps/functors between them. Now a formulation of (2) is: Why do we, say for the log-links, not just take the “identity” isomorphism between the relevant $\pi_1$’s instead of the full poly-isomorphisms between them?
Surly it is also a completely valid map between the theaters?
Yes, but: let us recall what we want to do with this log theta-lattice: we want at “suitable times(?)” glue the Hodge theaters along these links and then apply certain multiradial algorithms.
I think that if we would glue with the “identity” on the $\pi_1$’s (and all the other prescriptions of the log-link), then this gluing will be inconsistent/not well defined (when combined with some multiradial algorithms), in some vaguely similar sense as if you try to define a complex structure on a manifold by charts, but in some region the procedure leads to two different, incompatible complex structures, i.e. the transition map is not holomorphic.
Indeed, in one Hodge theater, the $\pi_1$ is not independent of the rest of the data; it acts via the quotient to the Galois group on the monoids. Thus $\pi_1$ is “related” to the monoids, and it is not apriori clear we are allowed to glue two Hodge theaters by gluing $\pi_1$ and the monoids separately in whatever way we wish (and then follow certain multiradial algorithms).
In the above analogy, a certain region of the Hodge theater is supposed to be “determined” jointly by the $\pi_1$ and the monoids.
The multiradial algorithm we consider is roughly as follows: The action of $\pi_1$ determines on the (from the Galois group reconstructed) local monoids (with 0) in addition a ring structure (a “holomorphic” structure).
The output is roughly some etale version of some of the data originally present in the Hodge theater, and the reconstruction algorithm yields a canonical isomorphism between the original data and the reconstructed one (see e.g. [Alien, §2.12] for essentially such constructions).
If one now glues two Hodge theaters along a log-link (with in addition “identity”-gluing for $\pi_1$’s), then this yields no consistent holomorphic/ring structure compatible with the above algorithm on some region:
Namely, consider the region of the ring (local units), which is glued via log (the actual log part of the log-link) to other Hodge theater; this yields one definition of “holomorphic” for this region. On the other hand, we can embed this region in the reconstructed monoids and use for the “holomorphic” structure on the reconstructed monoids the one induced by $\pi_1$ on both Hodge theaters and glue by the “identity”.
The above roughly describes two incompatible holomorpic/ring structures on some part of the glued Hodge theater; they are incompatible, “because the transition map, essentially the log, is not a ring map”. Such a situation is presumably avoided by taking the full polyisomorphism between $\pi_1$’s, where the log map on the units alone defines the holomorphic structure on the relevant region.
Dear UF,
thanks for your answer. (Taylor, I will answer in a separate comment later.)
Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is why you have to consider source and target up to some indeterminacy.
The only way I see to make this into a valid thing is if the indeterminacy somehow allows you to map the logarithm map into a map of rings — after all, this is what you critized in the first place. But any automorphism of $\pi_1$’s will only ever replace source and target by isomorphic rings. So the logarithm will still not be a map of rings. Note that I’m simply repeating what I already said in my second comment in this thread.
Dear Taylor,
addressing your last comment.
First half: I was refering to the current version on his webpage. I realize that this is dangerous, but it’s the only thing that at least currently makes it possible for everyone to follow. In any case, to me it sounds like you are saying that the Theta-pilot has nothing to do with Theta-values. Well, strictly speaking (modulo minor issues) what Mochizuki does is the following: He takes the Theta-value (some nonzero p-adic number, not a root of unity), and takes the (multiplicative) submonoid $M$ of the p-adic numbers generated by it, and then considers $M$ as an abstract monoid. As an abstract monoid, it is then of course isomorphic to $\mathbb N$, necessarily canonically so; its generator is referred to as a Theta-pilot. Does the abstract monoid $M$ know anything about the Theta-value? Of course not! But that is all that ever seems to enter the story of the Theta-link. At some point you do need to remember the interpretation of the Theta-pilot in terms of Theta-values — this is what Mochizuki calls the Theta-intertwining, and it plays a key role as I tried to reference. So I don’t think that Mochizuki simply cuts our diagram there.
Now David Roberts previously mentioned that Mochizuki seems to have a nonstandard point of view on what a mathematical object is, requiring separate copies when they are not actually needed; he often refers to some “history” of these objects. Now maybe one could try to argue that because $M$ was built out of Theta-values, it (the abstract monoid) still knows something about them, that can be even transported via an isomorphism of abstract monoids. (You made some similar remarks in your paragraph on $\mathbb N$ and Kummer theory, refering to “interpreted structures”.) But that’s evidently not the case. (That the abstract monoid $M$ knows something about Theta-values might be debatable if for you an abstract monoid is really encoded — as it is in ZFC — in terms of its actual set of elements. But even then, this structure (“the interpretation”) is clearly not transported by an isomorphism of abstract monoids.)
About the indeterminacies: I’m focusing on (Ind2); (especially) the part concerning automorphisms of local Galois groups (acting on local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity isomorphisms, but does commute when you conjugate by an automorphism of a local Galois group (acting on local units). Why do you need that, expect because “you need to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that? (There was actually a point where Mochizuki was surprised that by “forgetting the history of a group” one still has more than a group up to isomorphism: Namely, a group. That the datum of a group is strictly more than the datum of a group up to isomorphism seemed new to Mochizuki. I believe this is the (psychological) main reason he considers these full poly-isomorphisms. But really, forgetting “the history of the groups” you still have groups and they may still have natural commuting isomorphisms between them. Of course, for groups up to isomorphism you can’t ask for natural commuting isomorphisms between them, then you only have “full poly-isomorphisms”.)
Regarding the rest of your post, I don’t see how you’re really objecting to anything I said.
Finally a short answer to David Roberts’ last message: I highly doubt your sentiment that the possibility of doing mistakes is not correlated with how well your language is adapted to the mathematics at hand.
@Peter Schoze: I agree that the above should be surrounded by big caveats about the nature of the effects of passing to full poly-isomorphisms and I may interprete those incorrectly.
What my post above attempts to show, is: if passing to poly-isomorphism has the effect of doing no gluing/no identification of ring structures (arising from $\pi_1$, just the gluing from the actual log map), then the only gluing left is the actual log map, which gives one global chart, and no transition functions needed, essentially(?) since just one chart . I before never really seriously considered that full poly-isomorphism could have the effect of “no gluing arising from this part” (instead of “choose your favourite gluing”), but a similar thing is (I think) asserted for the case of the theta-link in [IUT II, 1.11.2(ii)], where the message is: full polyiso on $\pi_1$ means no sharing of ring structure arising from this.
Clearly this does not show by itself it really works like this, but it seems to reduce an argument for the log-link to some degree to an anologous one for the theta-link.
Dear UF (and Taylor and everyone),
I won’t comment any further here on statements of the form “well, maybe Mochizuki actually meant (vague statement)”.
A few comments up I summarized the situation with claims (1), (2) and (3). I have seen no valid objection to (1) and (2), and (2) alone would lead to a contradiction (as one gets too strong a form of ABC). To (3), Taylor indicated where to cut the diagram, but I really don’t think this is what happens, as this would isolate Theta-pilots from Theta-values and effectively remove the actual Theta-values from the proof; while Mochizuki does consider this “Theta-intertwining” which is the association of the Theta-pilot with the Theta-values.
I will only comment further here if either a valid objection to (1) or (2) is mentioned, or further clarification is given regarding (3). Any further technical discussions are probably best done via e-mail.
Best,
Peter
@Peter S
sorry, that’s not what I meant: from a purely mathematical point of view, my understanding of what M means by ‘copies’ is, divorced from all this context, mathematically consistent, if weird. I think things like working with functors up to isomorphism is a much bigger problem, especially when, eg, apparently more-rigid-than-usual “reconstruction functors” are at play. W asked about a very specific issue removed from the bigger context, which is the kind of counterfactual that is hard to answer sensibly. I agree than any language, mathematical or otherwise, that obfuscates what’s happening is going to increase the risk of error!
@David Roberts I’m happy to take responsibility for my understanding, which I think was caused by me formulating the question poorly.
My intent was to focus on what kinds of inferences you (or anyone) might be able to make about Mochizuki’s work, based on your knowledge of the response and your general knowledge of category theory, not based on reading through hundreds of pages of his papers as Peter Scholze did. I didn’t intend to single out one particular difference in notation that happened to be among the least concerning ones, but I guess I did.
Anyways the point of this is that if Mathematician A writes a paper and Mathematician B says a particular formula is off by a factor j^2 then, all else being equal, there is a good possibility that either side might be correct. If they go back and forth discussing it the probabilities change somewhat, but at every point, all else equal, both probabilities are pretty high.
But if we find that Mathematician A is using language that increases the risk of errors, and Mathematician B claims to have found the error by removing that language, and Mathematician A refuses to change this language, and Mathematician A claims that they are using this language to reduce the risk of errors, then we have found a bunch of pieces of evidence that A is the one who has made the error and not B, even if we don’t read the paper enough to follow either mathematician’s reasoning line-by-line.
@W no worries 🙂
>Mathematician A refuses to change this language
this is the worrying part. And, IMHO, one of the biggest problems with the whole affair. The papers could have been rewritten in the past eight years to remove all the extraneous fluff, and using more standard terminology, based on feedback from the community. Forget about terminology and risk of error, it could have just simplified the papers.
>Mathematician A claims that they are using this language to reduce the risk of errors
I don’t know about this bit.
But all this is very meta-reasoning, and not mathematics, like a lot in this whole affair.
BTW, “language” here is perhaps not the best word to use, given that it has three potential meanings: just the general idiom, full of metaphors and not-incredibly-helpful “motivation; the mathematical language (what I prefer to call ‘terminology’); and the fact there are English- and Japanese-language documents on IUT (and unjustified questions about facility with one or the other out there).
@David Roberts
Sure, “terminology” is a better term. I don’t think anything to do with the English or Japanese languages is particularly relevant, and the general idiom is clearly relevant but only incidentally to the main issue.
By “Mathematician A claims they are using this terminology to reduce the risk of errors” I mean passages like these, from Mochizuki’s response:
> Omitting the labels leads to confusion concerning which copy
of the unity element 1 ∈ A is to be regarded as the unity element
for “Af ”. Such confusion may, of course, be misinterpreted as
an “internal contradiction” in the theory of localizations of
commutative rings with unity. In fact, however, there is no “internal contradiction” in the theory of such localizations; the
apparent “internal contradiction” is nothing more than a superficial consequence of the erroneous operation of omitting the
labels
> relying, in mathematical discussions, on declarations of “remembering”
that are not accompanied by precise, explicit documentation of the labeling apparatuses that are employed incurs the risk that different people
will “remember” different labeling apparatuses, which result in structurally non-equivalent mathematical structures.
I can think of a different way to interpret them, though (that Mochizuki is only justifying some of his terminological choices here, and this justification depends on other terminological choices, and is valid once you accept those choices) so maybe this gloss is not completely fair.
@Peter Scholze: Thank you for your comments.
Just to restate my view, in case I was unclear: I believe that the reasoning above for not including any specific identifications of the $\pi_1$’s (i.e. just full poly-isomorphisms) in the definition of the log-link is entirely parallel to Mochizukis reasoning in [IUT II, 1.11.2(ii)] for not including any specific identifications of the $\pi_1$’s in the definition of the theta-link.
@W
yes, I agree with your points. I just mentioned the other aspects to ‘language’, as there are readers here from a wide background, who might misconstrue what we are saying (there have been, I gather, accusations of racism thrown about that I want to head off here).
Those two quotes you have selected are indeed particularly weird, and reflect the psychological crutch I mentioned earlier. I even quoted the second one in my notes on Mochizuki’s report. If this is truly M’s thinking, then we are in “If a lion could speak” territory. This is not mathematics in the slightest.
I understand that it might not be the best use of everyone’s time but I have really enjoyed reading this. Seeing a well-mannered, real time discussion play out has been great (particularly in these times of lockdown). So while it might not be the most efficient way of trying to solve the problem / address the issues it is having some positive externalities (at least for me) .
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Hi Everyone,
I’m just going to make one last post to close out some loose ends for interested readers. I also want to point out some things that we haven’t covered that I think are important. I’ve omitted the discussions of asymptotics of Mochizuki’s formula but other than that, I think I covered my IOUs. If there are any analytic number theorists who were really looking forward to that please email me.
Sorry for the length.
**********************
Regarding (3): Is Mochizuki’s Proof Falsified?
**********************
There is no *proof* that Mochizuki’s method doesn’t work.
The following is what the Scholze-Stix manuscript proves:
Theorem. Assuming that one may identify Hodge Theaters in Mochizuki’s theory and simultaneously impose “concrete normalizations” of q-pilot and theta-pilot degrees then there is a contradiction.
Remarks.
1) The hypothesis of “simultaneously imposing concrete normalizations” is equivalent to $l(l+1)/12=1$ for a fixed prime $l$. In the manuscript no machinery from Mochizuki’s theory is used in this derivation. A protest to this identification is made in C12 of Mochizuki’s response. Despite this, Peter S. insists this is what Mochizuki means. See his comments above, or the quoted comments below.
2) The hypothesis that one can identify Hodge Theaters is also protested by Mochizuki and runs counter to his stated objective. It is prudent to observe that the thesis of the Gaussian integrals survey is that sometimes in Mathematics one can introduce “alien copies” of objects to prove something about your original object. This runs counter to the assumption of identifying Hodge Theaters in the above theorem.
I am not taking the position that Mochizuki’s proof is correct or will turn out to be correct but that, as it stands, the Stix-Scholze manuscript should not be held up as a reason to reject Mochizuki’s proof. On the contrary, I am just saying that a falsification of the proof requires a fluency in the definitions which seems hard or impossible to achieve. Many of the definitions use nonstandard terminology and come off as ambiguous to many readers (I’m not sure if I want to get into a definition of ambiguous). Case in point: questions about the “embeddedness” (or in Mochizuki’s terms “holomorphic structures”) of the monoids involved in the $\Vdash \blacktriangleright \times \mu$ prime strips; these came up in Peter’s previous post. Moreover, because of certain ambiguities in these definitions, readers are forced to search a large space of possible meanings. While these difficulties do not falsify the possibility of a proof, they make it so that what Mochizuki has written is not a proof in traditional terms. To speak plainly, flexing on the reader by omitting details is not something new in Arithmetic Geometry and IUT is a weird flex.
Finally, it is not outside the realm of reality that there could be 5 top notch international referees who have understood the proof as complete and correct. PRIMS has traditionally been a journal of very high quality and members of the editorial board have shown integrity in the past (look at how senior mathematicians are responding when asked by journalists to comment). Let’s see what the print version looks like (hopefully it has a table of contents and remove bolding and italics).
What can we do?: I think we should all think carefully about the damage omissions have next time we are tempted to do it. Additionally, we should praise the writers whose writing we find “easy” and useful. Because of these difficulties I personally hold Mochizuki’s document as a program for proving ABC and consider Mochizuki’s formula a conjecture (until I can see this with my own eyes). Then again, there are a lot of Theorems I know about and use but can’t prove. Also, be nice to everyone, we are all just people.
*****************
Regarding (1)
*****************
I see nothing wrong with (1). By this I understand replacing all constructions in IUT involving the fundamental group of an SBT curve with the corresponding construction in an actual curve. I think this is possible, but we would need to check that this doesn’t break somewhere. If the claim is that all fundamental groups in the theory need to be eliminated, I would be more reluctant. There are a lot of weird curves. I think I need to see what a more precise statement looks like in practice to say for sure.
*****************
Regarding (2)
*****************
Regarding (2), to be clear, we should repeat that this is not what Mochizuki does (only using one fundamental group). Also, the statement as Peter has stated is a moral one not to be taken literally — there are many more groups appearing in the construction that are not isomorphic to $\pi_1(X)$ (profinite etale) so, like, it doesn’t literally parse but I think I know what he means. Regarding its validity, I have reservations related to conjugacy synchronization and construction of “the diagonal” (points in Mochizuki’s theory are replaced by conjugacy classes of decomposition group and evaluation of functions is replaced by restriction of cohomology classes to decomposition groups — since we want to evaluate at many points simultaneously one needs them to be determined up to conjugation simultaneously — we need a diagonal action of a conjugation not an action by a product of conjugations. This process involves multiple fundamental groups.) — BUT if we are going to dispose of these things entirely as in (1) you can probably scrap a lot from section 2 of IUT2. I need to think about how this construction works in order to say something precise. I personally would ask Emmanuel Lepage or Jakob Stix about this.
A more middle ground statement is that one can replace all “base Hodge theaters” by a single “base Hodge theater”—which have more than just $\pi_1(X)$ and a bunch of maps between them. Yet another variant of this would involve contracting even more fundamental groups in the base Hodge Theater (and this is where I think you might run into conjugacy synchronization and diagonal problems).
Also, there are a lot of monoids built into the theory which are excluded from the statement and I don’t know if these are implicitly assumed to be identified as well. I think not.
Finally, to derive a contradiction, I think some things need to be said about *how* one proposes to reduce Mochizuki’s construction to “a single fundamental group”.
All in all to build this into an actual counter-example, as Peter claims, one needs to make the statement more precise.
Remark. In a post I read someone emphasized that Peter S. read “hundred of pages”, or that he met with Mochizuki “for a week”. Similarly, Fesenko has emphasized the number of hours people have studied the theory. These are personal reasons not to undertake something and not acceptable substitutes for proofs.
##################################
##################################
##################################
Now more comments…
**********************
Intertwining
**********************
>On the other hand, I read say on page 140 of IUT-3 that Mochizuki considers the Theta-intertwining, which I believe simply means this identification of abstract Theta-pilots with concrete Theta-pilots. He wants to be very careful with using this etc., but I do believe he wants to (and has to) use it somewhere. So I don’t think you can simply cut there. I think the closed loop that Mochizuki discusses on page 143 of IUT-3 is also relevant here. See also on page 144 the simultaneous Theta-intertwining and q-intertwining he wants to have (up to indeterminacies etc…).
There is more of this on page 188 too (April 16, 2020 copy)
In the quoted statement above the “up to indeterminacies” is the point.
Mochizuki asserts the existence of TWO theta monoids you need to be thinking about relating. One that identified with q-monoid and the other which has been interpreted alongside q in the same Hodge theater. In Mochizuki’s language one is in a “alien holomorphic structure” and the other is in interpreted from the same structure. Mochizuki’s game is then relate the auxillary theta pilot using anabelian techniques (amphoricity in the language of Joshi’s manuscript) in order to derive a relation between the two. This relation is a uniformity statement. All the thetas of the world can be blurred.
At one point I was confused about what is going on with the interpretations in absolute galois groups of p-adic fields. It is helpful to know that the measure spaces constructed from absolute galois groups of p-adic fields have measures which are well defined—in the initial construction in AAG3 (section 5?), the measures take values in an ordered one dimensional real vector space obtained by the process of ‘perfection’ ($\operatorname{colim} M$, nodes = $\mathbb{N}$, morphisms=$( a \mapsto ab)$ realized as multiplication by $b$) and completion. It turns out that because of initial theta data, the we know how to normalize the measures to give real numbers.
My point: the constructions of these measure spaces may look weird, but from the perspective of either interpreting structure these things give real numbers, and the values of these measures don’t lie in some abstract ordered one dimensional real vector space. Also, there exists a region $U \subset \mathbb{L}^{\vdash,et}$ (this is my notation for the monoanalytic etale like big log-shell) that is invariant under automorphisms of log-linked collections of Hodge Theaters. In Mochizuki’s terminology, “this region can be seen from both sides of the theta link”.
>In any case, to me it sounds like you are saying that the Theta-pilot has nothing to do with Theta-values.
On the contrary, all theta pilots are defined *in* structures which includes an evaluation map like (theta monoid)$\times$(evaluation points)$\to$(constant monoids). Look at the diagram in IUT Remark 3.10.2 and read it clockwise starting from gau and ending at etale $\mathfrak{lgp}$. Each of these $\Vdash \blacktriangleright \times \mu$ strips are parts of a larger structure where multiplication of the “value group portions” may act on the “unit group portions”. Eventually these “Kummer maps” lead to the construction of the region $U_{\Theta}$ described previously.
> Well, strictly speaking (modulo minor issues) what Mochizuki does is the following: He takes the Theta-value (some nonzero p-adic number, not a root of unity), and takes the (multiplicative) submonoid M of the p-adic numbers generated by it, and then considers M as an abstract monoid. As an abstract monoid, it is then of course isomorphic to N, necessarily canonically so; its generator is referred to as a Theta-pilot. Does the abstract monoid M know anything about the Theta-value? Of course not! But that is all that ever seems to enter the story of the Theta-link.
This goes back to my comments about his objects not having good black boxes (without some sort of language like the of interpretations). This also points to my point about his confusing writing style in the beginning. It is typically not sufficient to perform his constructions without knowing about the interpretation.
>At some point you do need to remember the interpretation of the Theta-pilot in terms of Theta-values — this is what Mochizuki calls the Theta-intertwining, and it plays a key role as I tried to reference. So I don’t think that Mochizuki simply cuts our diagram there.
Again, from the horses mouth, read C12 of his responses. He says he does not do this. Also, please look at the displayed diagram below. I think this may clarify something for you. Also, in this “interwining”, this is where the theorem about the amphoric/characteristic nature of the interpretation of the Jacobi theta function is supposed to be applied. I made some remarks before on this and refer to those.
>Now David Roberts previously mentioned that Mochizuki seems to have a nonstandard point of view on what a mathematical object is, requiring separate copies when they are not actually needed; he often refers to some “history” of these objects. Now maybe one could try to argue that because M was built out of Theta-values, it (the abstract monoid) still knows something about them, that can be even transported via an isomorphism of abstract monoids. (You made some similar remarks in your paragraph on N and Kummer theory, refering to “interpreted structures”.) But that’s evidently not the case. (That the abstract monoid M knows something about Theta-values might be debatable if for you an abstract monoid is really encoded — as it is in ZFC — in terms of its actual set of elements. But even then, this structure (“the interpretation”) is clearly not transported by an isomorphism of abstract monoids.)
I think this quoted text reinforces my points about “depth searches” and the ambiguities (percieved or not) in Mochizuki’s writing. Peter I am going to use the words “My understanding” so you may want to skip this response.
Regarding ZFC, while this is true, we don’t go this deep. That is not what is going on here. (for the uninitiated see https://stacks.math.columbia.edu/tag/0009)
My understanding of the “history of objects” is different, and I don’t think this saves the day. “Forgetting history” is Mochizuki-speak for forgetting about the interpretation as you have suggested. Automorphisms of objects are no longer coming from the interpreting structure and this structure is no longer “going along for the ride”.
In Mochizuki’s language or the language of naked categories Mochizuki’s theory does not have many good black boxes. This makes parsing the examples even harder because you need to carry around these large definitions in our small human brains (or use a large amount of paper). If you pick up a Hodges a “a Shorter Model Theory” or Olivia Caramello’s book, or MacLane’s Sheaves in Logic book, you will find perfectly fine definitions of interpretations that allow you to package all of that sloppy anabelian geometry/ “reconstruction algorithms” into nice little formulas.
Here are some of David Marker’s notes:
http://homepages.math.uic.edu/~marker/math512-F13/inf.pdf
Exercise: Look at the first page of Marker’s notes then show that the torsion of an abelian group is definable using infinitary formulas.
Exercise: People often complain about the non first orderizability of topological spaces. Read this post from stack exchange on how to define topological spaces: https://math.stackexchange.com/questions/46656/why-is-topology-nonfirstorderizable
>About the indeterminacies: I’m focusing on (Ind2); (especially) the part concerning automorphisms of local Galois groups (acting on local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity isomorphisms, but does commute when you conjugate by an automorphism of a local Galois group (acting on local units). Why do you need that, except because “you need to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that?
I’m not sure what “that” is. I’m going to assume you mean ind2. I need to break this down in order to give you a coherent response. For me, ind2 is a representations of automorphisms absolute galois group of a p-adic field $G \cong G_K$ on a $\mathbb{Q}_p$-vector space $\mathcal{O}^{\times \mu}(G)\otimes \mathbb{Q} \cong (K,+)$ induced functorially. There are interpretations given from local class field theory (I’m going to release a table of these soon in my manuscript with Anton, alternatively you can find these in some papers of Hoshi). These preserve the $\mathbb{Z}_p$-lattice $\mathcal{O}^{\times \mu}(G)$ inside this vector space.
I’m not sure what sort of diagrams you want but here is one:
Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$ where $\log(1+\pi) = \pi$ and $\log(1+\pi^2)=\pi^2$.
Let $\lbrace x\rbrace$ be a one point set.
Let $f,g: \lbrace x\rbrace \to K$ be given by $f(x) = \pi$ and $g(x) = \pi^2$.
$$\begin{CD}
@>>> \lbrace x \rbrace \\
@AAA @VVgV \\
\lbrace x \rbrace @>>f> K
\end{CD}$$
(I don’t know how to make the diagonal arrow)
Considering $K$ up to ind2 makes the diagram commute. Something tells me this is not what you had in mind. Ind2 has the property of “mixing valuations”.
In terms of automorphisms of $G$ these give you the freedom to move around the higher ramification groups. The reference for this is Joshi’s manuscript.
Here is the standard warning: this will not save the day and you will NEED ind3 to make the formulas work as this sort of doesn’t mix very much. The $\mathbb{Z}_p$-submodules $p^j \log(\mathcal{O}_K)$ will always be preserved under ind2, so $K = \coprod_{j \in \mathbb{Z}} p^j \log(\mathcal{O}_K) \setminus p^{j+1} \log(\mathcal{O}_K)$ and each annulus is preserves setwise under ind2.
This is saying, in some sense, that ind2 (and ind1) do not do too much. Moral: in order for Mochizuki’s inequality to be true, you *do* need ind3.
The purpose of ind2 (and the other indeterminacies) is to perform a construction that is independent of the structure that it was constructed from.
>(There was actually a point where Mochizuki was surprised that by “forgetting the history of a group” one still has more than a group up to isomorphism: Namely, a group. That the datum of a group is strictly more than the datum of a group up to isomorphism seemed new to Mochizuki. I believe this is the (psychological) main reason he considers these full poly-isomorphisms.
Classic Mochizuki.
>But really, forgetting “the history of the groups” you still have groups and they may still have natural commuting isomorphisms between them. Of course, for groups up to isomorphism you can’t ask for natural commuting isomorphisms between them, then you only have “full poly-isomorphisms”.)
To be honest, I’m not sure what you mean by “history of groups” and it actually hurts my head to think about this or to think about you thinking about this. If you want to email me a reference I might be able to say what is going on here but I would need to see it. I’m also don’t want to defend the usage of “history” in these documents.
>A few comments up I summarized the situation with claims (1), (2) and (3). I have seen no valid objection to (1) and (2), and (2) alone would lead to a contradiction (as one gets too strong a form of ABC).
I addressed this in the introduction.
>To (3), Taylor indicated where to cut the diagram, but I really don’t think this is what happens, as this would isolate Theta-pilots from Theta-values and effectively remove the actual Theta-values from the proof; while Mochizuki does consider this “Theta-intertwining” which is the association of the Theta-pilot with the Theta-values.
I addressed how Mochizuki proposes to bring theta values back into the proof; using the anabelian geometry. How this actually works is unclear to me. Also, I think Mochizuki would object to your characterization “theta intertwining” here. He would say something about the “strong anabelian properties of the etale theta function” which might be equally imprecise. He would also point to a number of other constructions which would include MOD vs $\frak{mod}$ constructions, and the theory of cyclotomic synchronizations. How the $q$ and $\Theta$ pilot objects are tied is the mystery and has been the subject of my conversations with everyone for the last several years (which is embarassing to admit outloud).
What is missing to me is the comparison between the $U_{\Theta}$ and $q$. Mochizuki and Hoshi have very patiently been trying to explain this to me for a long time now. Maybe you want to call this a gap. Maybe I’m
dense. Mochizuki will undoubtedly call this a “fundamental misunderstanding” and point to some aspect of his manuscript. Me (and Emmanuel Lepage and others) have been talking about this since 2017 and are stumped.
***********************
Some “Refereeing” of the log-links discussions between Peter and UF
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@UF April 11, 2020 at 9:02 pm
–C7 of the 05 comments actually says that you *can* switch.
–I don’t think this rigidifying/derigidifying should be emphasized and will just confuse people. Calling a pair $(\Pi, \overline{M}) \in [\pi_1^{temp}(Z^+),\mathcal{O}^{\vartriangleright}_K]$ a “rigidified fundamental group” is weird.
@Peter Scholze, April 12, 2020 at 4:58 pm
In this Peter chooses to address the log-links as an example of indeterminate copies doing nothing. Here he claims these can be replaced by logs.
My understanding of the situation for log-links is different. I’m going to give an abstract setup and I claim something like this occurs in IUT. Fix $\overline{M} \cong \mathcal{O}^{\vartriangleright}_{\overline{K}}$. This can be decomposed as $\overline{M} \cong \pi^{\mathbb{Q}_{\geq 0}} \cdot \overline{M}_{tors} \cdot F$ where $\overline{M}_{tors}$ is the torsion subgroup and $F$ is the free part (at finite level these are free $\mathbb{Z}_p$ modules). Furthermore for the subgroup of units we have $\overline{M}^{\times}\cong \overline{M}_{tors} \cdot F$. Suppose now that one only knows $\overline{M}$ up to automorphisms of $\overline{M}_{tors}$. In order to get a well-defined object, one thing that we can do is mod out by torsion $\overline{M}^{\times} \to \overline{M}^{\times \mu} = \overline{M}^{\times}/\overline{M}_{tors}$ this is now well defined.
Mochizuki would say the converse. That we want to manipulate $M^{\times\mu}$ (via ind1 and ind2) and this manipulation needs to be independent (lifted to?) from the $\overline{M}_{tors}$, which plays a role in the exterior cyclotome of the frobenius-like mono-theta environment, which is used for passing structures “down to the bottom” in IUT3 3.11.
@UF, April 13, 2020 at 10:16 pm
It seems that you are both talking past each other. Log links are about relationships between monoids *not* fundamental groups.
>It seems quite likely Mochizucki uses “switching-symmetry” in a technical sense, synonymous with “multiradiality” of some algorithm reconstructing the data (here a rigidified column of log-links) at hand from some choric data, as he often does, compare e.g. [Alien, p.51].
There is a technical sense. Given a morphism of connected groupoids $F:\mathsf{A} \to \mathsf{B}$ one can form the category $\mathsf{A}\times_{\mathsf{B}}\mathsf{A}$ whose objects are $(A_1,A_2,f)$ where $f: F(A_1) \to F(A_2)$. When $F$ is full and essentially surjective the switching functor is $(A_1,A_2,f)\mapsto (A_2,A_2,f^{-1})$. In practice, as I have said before, this seems to be note enough for applications as objects in categories seem to be more. Also, I think there is a problem because we literally want a map of sets at the end of the day. I haven’t worked out how to use this formalism effectively.
>His statement would then mean that if we rigidify the vertical columns, then there is (unlike in the non-rigidfied case) no multiradial algorithm to recover this column from certain choric data. This does not sound so absurd anymore (to me).
Minor typo: “choric” should be “coric”
>Now which multiradial algorithm does he mean here? I would suggest it may be the multiradial algorithm in [IUT III, Cor 2.3 ], more specifically, the first part of 2.3 (ii) which concerns its compatibility with log-links. Note that close by, [IUT III, Rem 2.1.1 (ii)] the issue we are talking about “why $\pi_1$ only upto indeterminate iso?” is discussed. For further discussion see also in [IUT II, Rem 3.6.4 (i)]. In any case, I agree this is an important issue to track down.
This is addressed in the comments at the very very bottom.
@Peter Scholze, April 14, 2020 at 5:17 pm
>The remarks from IUT that you cite make heavy reference to his paper on etale theta-functions, which seems to play a key role in the IUT papers. This paper gives some neat algorithm to start from the fundamental group of a once-punctured elliptic curve with bad semistable reduction, and recover its Tate parameter q and some Theta function; I forget the details. While this is all good and well, I don’t see the relevance: Mochizuki’s more general anabelian theorems, discussed previously on this thread, tell you that from the fundamental group you can simply recover the whole curve. In these comments of Mochizuki that you reference, Mochizuki is discussing some nitty-gritty details of this algorithm, but this seems completely besides the point if you just remember that relevant pi1X’s are equivalent to relevant X’s, so of course you can recover all invariants of X, and you can do so functorially in pi1X.
If we are collection complaints about exposition I have another: As Peter has mentioned this and many other pieces of text in theorem environments punt you WAAAY back to topics in other papers. As Peter is mentioning this goes back to Etale Theta which I think is at least 5 papers back (IUT3>IUT2>IUT1>AAG3>AAG2>AAG1>EtTh) to be fair one could argue that AAGX and EtTh are independent. Either way, this is a big dependence, and we haven’t even touched the dependencies of the AAGX papers.
I responded to the second part of this remark already. You asked about 3.11.i. Also, this $\mathfrak{R}^{LGP}$ is actually not so complicated. The one in Cor 2.3 $\mathfrak{R}$ that is the bad one (no pun intended). I’m going to postpone this dicussion until the end.
@UF April 15, 2020 at 9:04 am
>I now think a good reason to for not identifying $\pi_1$’s along log-links is as follows:
(It is very similar to [IUT-2, Rem 1.11.2 (ii)] for the theta-link; I am less precise here, taking the relevance of some reconstructions on faith, but it is a intuitive story, and precision can come later)
>First, let us step back and look at the log-theta-lattice: It consists of Hodge-theaters and the log/theta-links as maps/functors between them. Now a formulation of (2) is: Why do we, say for the log-links, not just take the “identity” isomorphism between the relevant
$\pi_1$’s instead of the full poly-isomorphisms between them? Surely it is also a completely valid map between the theaters?
>Yes, but: let us recall what we want to do with this log theta-lattice: we want at “suitable times(?)” glue the Hodge theaters along these links and then apply certain multiradial algorithms.
>I think that if we would glue with the “identity” on the $\pi_1’s$ (and all the other prescriptions of the log-link), then this gluing will be inconsistent/not well defined (when combined with some multiradial algorithms), in some vaguely similar sense as if you try to define a complex structure on a manifold by charts, but in some region the procedure leads to two different, incompatible complex structures, i.e. the transition map is not holomorphic.
>Indeed, in one Hodge theater, the $\pi_1$’s not independent of the rest of the data; it acts via the quotient to the Galois group on the monoids. Thus $\pi_1$’s “related” to the monoids, and it is not apriori clear we are allowed to glue two Hodge theaters by gluing $\pi_1$ and the monoids separately in whatever way we wish (and then follow certain multiradial algorithms).
I’m going to refer you to my forthcoming manuscript with anton (the prequel to the one with the IUT4 style computaions). There is some stuff on my vlog that is a (poor) template for this stuff if you don’t want to wait.
>In the above analogy, a certain region of the Hodge theater is supposed to be “determined” jointly by the $\pi_1$ and the monoids.
The multiradial algorithm we consider is roughly as follows: The action of $\pi_1$ determines on the (from the Galois group reconstructed) local monoids (with 0) in addition a ring structure (a “holomorphic” structure).
>The output is roughly some etale version of some of the data originally present in the Hodge theater, and the reconstruction algorithm yields a canonical isomorphism between the original data and the reconstructed one (see e.g. [Alien, §2.12] for essentially such constructions).
This *is* a thing.
>If one now glues two Hodge theaters along a log-link (with in addition “identity”-gluing for $\pi_1$’s, then this yields no consistent holomorphic/ring structure compatible with the above algorithm on some region:
>Namely, consider the region of the ring (local units), which is glued via log (the actual log part of the log-link) to other Hodge theater; this yields one definition of “holomorphic” for this region.
I can’t parse this sentence.
>On the other hand, we can embed this region in the reconstructed monoids and use for the “holomorphic” structure on the reconstructed monoids the one induced by $\pi_1$ on both Hodge theaters and glue by the “identity”.
This is true. You definitely have one structure from the base and one structure from pulled back from log. The diagrams commute though.
>The above roughly describes two incompatible holomorpic/ring structures on some part of the glued Hodge theater; they are incompatible, “because the transition map, essentially the log, is not a ring map”. Such a situation is presumably avoided by taking the full polyisomorphism between $\pi_1$ where the log map on the units alone defines the holomorphic structure on the relevant region.
I’m not sure I parse this completely. I will say there is a way to get the a commutative diagrams of fields in the log-kummer correspondence. It depends on what maps you take. Any “log” has a backwards map which is an isomorphism of fields. I will refer to the forthcoming paper with Anton for details.
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@Peter Scholze April 15, 2020 at 2:05 pm
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>Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is why you have to consider source and target up to some indeterminacy.
The log-link is about the monoids and not the base (you use really only use the fundamental group to use Mochizuki’s interpretation of a field in order to pullback this structure so you can take the logarithm).
I don’t know if this helps but there is a “post-logarithm” that IS an isomorphism of rings and this IS used the definition of the log link. I think it is nice to factor the actual logarithm as $\log^{post}\circ \log^{pre}=\log$ where $\log^{pre}$ is the map in IUT. One can then use the map $\log^{post}:\mathcal{O}^{\times\mu}\otimes \mathbb{Q} \to K$ and pull back the field structure to get a new field $K_{\log}$ which is $\mathcal{O}^{\times \mu} \otimes \mathbb{Q}$ as a set but with this new field structure.
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@UF April 15, 2020 at 4:57 pm
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>What my post above attempts to show, is: if passing to poly-isomorphism has the effect of doing no gluing/no identification of ring structures (arising from $\pi_1$ just the gluing from the actual log map), then the only gluing left is the actual log map, which gives one global chart, and no transition functions needed, essentially(?) since just one chart.
Yes, but I think the point is what I was saying about killing indeterminacies. Doing log-links isn’t “for sport” as Mochizuki would say.
>I before never really seriously considered that full poly-isomorphism could have the effect of “no gluing arising from this part” (instead of “choose your favourite gluing”)
I don’t think I understand how you are thinking about this.
*******************
Remarks on IUT3 2.3.ii
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We are looking to see if this statement says something non-trivial about the bases of log-linked hodge theaters and multiradiality (I actually think this is barking up the wrong tree since, as I’ve said before, the bases in log-linked hodge theaters do nothing except impart ring structures to the monoids). Indeed this is the case because of assertion (2) below. Let me just remark though for Peter, that the maps between the $\times \mu$ prime strips are what are going to encode everything in Galois groups and what eventually gets the theta back to the other side.
Anyway, at the beginning of the proposition there is a certain amount of setup. BUT, notice in the setup, most of the structures are interpreted! This means that their morphisms are just functorially induced by the interpreting structures, in this case there is a prime strip of the $\times \mu$ variety, and a base hodge theater (actually, it is even simpler than this… see the note below). Looking at my notes… this looks tautological.
Also, just as a tip from as a person who has wasted his life doing this, the thing to look out for are maps that are NOT full polyisomorphisms.
In this item there are a total of three assertions.
1) log-links induce a full polyisomorphism of the $\mathfrak{R}$’s;
2) Theta links on log-links pairs of hodge theaters have $\mathfrak{D}_{\Delta}^{\vdash}$’s interpreted in $\mathfrak{D}_{\Delta}^{\vdash}$ which are isomorphic.
3) Using (1) and (2) the $\mathfrak{R}$ across is generically isomorphic across log-links; the $\mathfrak{D}^{\vdash}_{\Delta}$ data is generically isomorphic across Theta links.
Proof of 1: This only could be a problem because of the nature of the way the monoids are linked and the dependence of $\mathfrak{R}$’s on these monoids. In the data for $\mathfrak{R}$ there is exactly one place where the monoids interact. This is in the $\times \mu$ prime strips. Everything else is functorially induced. For example, in the definition he takes the monotheta environment interpreted from the base (this actually has well-definedness issues of its own!).
Proof idea of 2: As described somewhere in this note ind2 is isomorphisms of $\mathcal{O}^{\times\mu}(G)$’s induced by isomorphisms of $G$’s. If you look up the definition of $\mathfrak{D}^{\vdash}_{\Delta}$ you will see that it is just a bunch of $G$’s. This means the generic isomorphism of from the $G$’s induce an ind2 on the $\times \mu$ monoids they interpret; the $\times \mu$ monoid isomorphisms in the theta link are the same thing.
Expositional Note: This is an example of a proposition where too much structure was invoked for my taste. In the fine print you notice that the interpreters are $\mathfrak{D}_{>}$ and $\mathfrak{D}_{\Delta}^{\vdash}$ (=fundamental groups at each place and absolute galois groups at each place respectively). In the statement he references the full base hodge theater. That’s baggage. Also, you will notice that this could have been broken into many much simpler assertions each which can be individually checked.
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Remarks on IUT3 3.11.i
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Peter had asked (essentially) how can galois groups interpret theta values? How can they do anything?!
Well, I claim that Theorem 3.11.i and 3.11.ii are not the difficult parts of Theorem 3.11; Theorem 3.11.iii is the weird part. There are some subtle differences between the (abc)-module structure in both of these statements. In the first, the actors (b) and (c) are defined as subset of the $\mathbb{L}^{\vdash,et}$ (at the appropriate places) in part i. In part ii there is a clear module structure.
In the situation of (i) you lift to an richer stronger and more powerful structure that allows you to define (b) and (c) then then you stick them inside the log shell. What the proposition is saying is that this is well-defined up to ind1,2 — so everything that was stuck inside these log shells now is considered up to a jumbling of the type described in a previous response. I will refer readers to my manuscripts with Anton Hilado for these formulas.
A subtle difference that I want to point out to everyone here which they may not have noticed is the difference between the usage of the MOD, LGP constructions and the $\frak{mod}$, $\frak{lgp}$ constructions in the later.
*************
Sorry if I made any errors anywhere. I didn’t mean to and tried to proofread this. Hopefully we learn as part of the corrections.
Best,
Taylor
Dear Taylor,
thanks for these final comments! I think I should answer to this. Let me first say that I agree with much of what you write, and for the sake of keeping this short, I only jump at the few places where I disagree.
> Regarding (3): Is Mochizuki’s Proof Falsified? […]
> Finally, it is not outside the realm of reality that there could be 5 top notch international referees who have understood the proof as complete and correct.
Really? I would have hoped that in that case at least one of them — not in their role as a referee, but simply as a mathematician who wants to share insight — would have come around and explain the key ideas in a way that is understandable.
> Because of these difficulties I personally hold Mochizuki’s document as a program for proving ABC and consider Mochizuki’s formula a conjecture (until I can see this with my own eyes). Then again, there are a lot of Theorems I know about and use but can’t prove.
I very strongly object to the implicit (I believe) assertion that Mochizuki’s result should now be considered just another of these really difficult theorems whose proof we never understood.
> *****************
> Regarding (2)
> *****************
> Also, there are a lot of monoids built into the theory which are excluded from the statement and I don’t know if these are implicitly assumed to be identified as well. I think not.
Well, I think I would want to identify some, too. By mode of thought is that starting from your elliptic curve, you can easily cook up all the data that forms a Hodge theater, including those monoids. Simply take that collection, and always the same one. Later, when you study log-links, try to understand what they do, which diagrams do not commute, and if necessary enlarge your diagram (of course you get a non-commutative diagram if you want that the logarithm equals the identity). My issue lies with possible non-identity isomorphisms of Hodge theaters being relevant at any point; they are not for all I can see, as all possible non-commutativity can’t be restored in terms of internal isomorphisms of Hodge theaters.
> Finally, to derive a contradiction, I think some things need to be said about *how* one proposes to reduce Mochizuki’s construction to “a single fundamental group”.
As the final step in Mochizuki (proof of Cor 3.12) is so unclear, it also seems impossible to fully justify where and how it breaks.
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Intertwining
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> Again, from the horses mouth, read C12 of his responses.
OK, I reread this. He talks about the q- and the Theta-holomorphic structure. This indeed sounds like there must be two distinct Hodge theaters around — Hodge theater is, I believe, the technical version of “ambient holomorphic structure”. But there are not! All of them are isomorphic. And if I identify them (using whatever isomorphism) I can see plainly that this does not make any sense.
> He says he does not do this. Also, please look at the displayed diagram below. I think this may clarify something for you.
Details seem to be off in the example, but in any case I totally see that some non-commutative diagrams become commutative up to these indeterminacies — otherwise they would not be indeterminacies. But I’m asking about a relevant one.
> Also, in this “interwining”, this is where the theorem about the amphoric/characteristic nature of the interpretation of the Jacobi theta function is supposed to be applied.
The “amphoric” nature here is just that you can recover $X$ from $\pi_1(X)$, so in particular the theta function (and all else). I’m still failing to see the importance of etale theta functions in all of this.
> Exercise: Look at the first page of Marker’s notes then show that the torsion of an abelian group is definable using infinitary formulas.
> Exercise: People often complain about the non first orderizability of topological spaces. Read this post from stack exchange on how to define topological spaces: https://math.stackexchange.com/questions/46656/why-is-topology-nonfirstorderizable
I’m totally lost about where you’re trying to go here.
>About the indeterminacies: I’m focusing on (Ind2); (especially) the part concerning automorphisms of local Galois groups (acting on local units if necessary). I don’t see any diagram that doesn’t commute when you take the identity isomorphisms, but does commute when you conjugate by an automorphism of a local Galois group (acting on local units). Why do you need that, except because “you need to forget the history of your objects in order to apply reconstruction algorithms” or some magic like that?
Thanks again for the example in reply to this, but I’m asking about a relevant diagram in the relevant abstract setting.
> The purpose of ind2 (and the other indeterminacies) is to perform a construction that is independent of the structure that it was constructed from.
This slogan I understand, but where does Ind2 help with anything? Why do you need to introduce it?
> I addressed how Mochizuki proposes to bring theta values back into the proof; using the anabelian geometry. How this actually works is unclear to me. Also, I think Mochizuki would object to your characterization “theta intertwining” here. He would say something about the “strong anabelian properties of the etale theta function” which might be equally imprecise. He would also point to a number of other constructions which would include MOD vs
constructions, and the theory of cyclotomic synchronizations. How the $q$ and $\Theta$ are tied is the mystery and has been the subject of my conversations with everyone for the last several years (which is embarassing to admit outloud).
> What is missing to me is the comparison between the $U_{\Theta}$ and $q$. Mochizuki and Hoshi have very patiently been trying to explain this to me for a long time now. Maybe you want to call this a gap. Maybe I’m
dense. Mochizuki will undoubtedly call this a “fundamental misunderstanding” and point to some aspect of his manuscript. Me (and Emmanuel Lepage and others) have been talking about this since 2017 and are stumped.
Well, I guess you are just pointing your finger to the same problem that I’m trying to point at. It is completely unclear how you get the actual Theta-values back in the game. You started this paragraph with “using the anabelian geometry”, but as I argued long before, it is completely unclear how anabelian geometry helps with anything in his setup (by (1)).
> *******************
> @Peter Scholze April 15, 2020 at 2:05 pm
> *******************
> >Removing all language from your message, I end up with the following message: The logarithm map is not a map of rings. This is why you have to consider source and target up to some indeterminacy.
> The log-link is about the monoids and not the base (you use really only use the fundamental group to use Mochizuki’s interpretation of a field in order to pullback this structure so you can take the logarithm).
> I don’t know if this helps but there is a “post-logarithm” that IS an isomorphism of rings and this IS used the definition of the log link. […]
My main question in the discussion with UF has always been the question of full poly-isomorphisms vs. identities between $\pi_1(X)$’s, i.e. issue (2). All you are writing is completely tangential to this. I think you actually agree that the log-link has so little to do with $\pi_1(X)$’s that it can’t possibly be the reason to consider $\pi_1(X)$ up to full poly-isomorphism. Yet Mochizuki was trying to make that exact point when we were discussing.
> *******************
> Remarks on IUT3 2.3.ii
> *******************
Why are you discussing this? I’m lost. Is it saying more than Theta-values being the same in all Hodge theaters?
> *******************
> Remarks on IUT3 3.11.i
> *******************
> Peter had asked (essentially) how can galois groups interpret theta values? How can they do anything?!
> Well, I claim that Theorem 3.11.i and 3.11.ii are not the difficult parts of Theorem 3.11;
You are simply ignoring my point!
Best wishes!
Peter
PS: I just realized that maybe the following information is worth sharing. Namely, as an outsider one may wonder that the questions being discussed at length in these comments (e.g., the issue of distinct copies etc.) are very far from the extremely intricate definitions in Mochizuki’s manuscripts (his notation is famously forbidding, some of it surfaced in Taylor’s comments), and feel almost philosophical, so one might wonder that one is not looking at the heart of the matter.
However, the discussions in Kyoto went along extremely similar lines, and these discussions were actually very much led, certainly initially, by Mochizuki. He first wanted to carefully explain the need for distinct copies, by way of perfections of rings, and then of the log-link, leading to discussions rather close to the one I was having with UF here. He agreed that one first has to understand these basic points before it makes sense to introduce all further layers of complexity. (I should add that we did also go through the substance of the papers, but kept getting back at how this reflects on the basic points, as we all agreed that this is the key of the matter.)
@Taylor Dupuy: Thanks for your comments.
The context for comment (C7) is Mochizukis interpretation of footnote 5 in [Scholze-Stix]. Mochizuki gives this interpretation in the first display of (C7): the rigidification mentioned there is (I think) not (some kind of) rigidification of $\pi_1$ in one Hodge theater (which you seem to mention, by considering pairs ($\pi_1$, monoid)), but the rigidification among the different $\pi_1$’s in one vertical column by choosing a specific isomorphism (“identity”) instead of full poly-isomorphism between the different $\pi_1$’s in one column.
Mochizuki then says that one can consistently identify the $\pi_1$’s as above, if one forgets about both log- and theta-links. (Mochizuki does not say so explicitly right here, but I think in this situation there would be a switching-symmetry. He says so in a closely related situation after the second display, which in my understanding amounts to the same situation).
In the second display in (C7), he seems to say (my ” ” are not meant here as quotes): In contrast, if one keeps log-links (or maybe similarly theta-links), then the rigidification of the $\pi_1$’s “depends” on the other data, say “0-column Frobenius-like data (at different vertical spots) with log-relations”, which do not admit a switching-symmetry between two vertical columns.
Here in my understanding “depends” indicates that the log-identifications between the 0-column Frobenius data cannot be made completely independently from the identity-identifications of the $\pi_1$’s in the 0-column, since at a fixed vertical position the $\pi_1$ and the monoids are “related/not independent” (via the action).
Implicit here (and relevant for the context of this C7) seems to be the assertion that because the rigidification of the $\pi_1$’s “depends” on the 0-column Frobenius data (which does not admit a switching-symmetry) , something has to go wrong with switching (when $\pi_1$’s are rigidified in a column), “because” it goes wrong for the related/dependent 0-column Frobenius-like data. (If this is *not* implicit here, then what in Mochizukis response in C7 is the objection to the vertical rigidification of $\pi_1$’s?)
After the second display he says: If one forgets about the 0-column Frobenius data (in particular the log-relationship between them at different vertical spots), then there is a switching-symmetry between two neighboring columns.
This is how I understand Mochizukis C7; I may certainly understand it wrong in parts, and I would be happy to understand it better. All the rest I wrote is an attempt make this (from my point of view) “implicit objection to rigidifying $\pi_1$’s” in the second display more explicit; but if my understanding concerning this is incorrect, then this is largely meaningless.
Hi Everyone,
I’m going to take a break today to attend WAGON:
https://sites.math.washington.edu/~jarod/wagon.html
Check it out!
After that I’ll take a look at the new comments and see if I have something to add.
Best,
Taylor
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Dr. Woit,
“I read Mochizuki’s blog entry when it came out. To the extent I could make sense of it via Google Translate …”
maybe DeepL
https://www.deepl.com/translator
would be worth a try; it’s got only few languages, but Japanese is one of them, and it gave me better results than Google Translate.
There has been some discussion about this comment thread that focused on the importance of the differentiating between fact and opinion. There has also been discussion that the long thread may obscure the points that Peter Scholze and Taylor Dupuy agree on by focusing on their areas of disagreement. I have written a (very long) comment to try to help with these issues.
The way this comment is intended to work is that the numbered paragraphs (1), (2), (3) give facts that are difficult or impossible to dispute and the paragraphs after them give opinions building on those facts.
(1) The first couple hundred pages of the IUT papers introduce a lot of difficult new terminology. Scholze and Stix demonstrated that none of this new terminology is necessary to prove any of the formal statements in these pages. Moreover, when the new terminology is ignored, the proofs all become simpler and easier to understand. They all follow quickly from Mochizuki’s previous work on the p-adic Grothendieck conjecture.
As a reader, this should already make you skeptical of the accuracy of the paper. How can dressing up simple statements in confusing and difficult terminology for hundreds of pages represent progress in turning the solution of one problem (the p-adic Grothendieck conjecture) into a completely different problem (ABC)? There are mathematical arguments that begin by introducing a lot of terminology (e.g. work of Grothendieck) but nothing like this ratio of terminology to content has ever been successful before. However, you certainly shouldn’t rule out Mochizuki’s work based only on this evidence.
As an editor or referee, you should also be very skeptical at this point. Setting aside concerns about correctness, you have some duty to ensure papers published in your journal are clear and get to the point. Having hundreds of pages that can be compressed to much less by the removal of new ideas rather than their introduction is not normally consistent with good mathematical writing, but one could imagine that the remainder of the paper justifies this sufficiently.
(2) Many people have tried to read the papers, read and understood the first few hundred pages, and then were unable to verify a key point – Corollary 3.12. This point is either the first or the second statement in the paper which doesn’t fit into the hundreds of pages mentioned in (1).
As a reader this should make you very worried! This is is not a pattern one typically sees when people try to understand a correct but difficult argument (regardless of how well or poorly it is written). Instead, different people almost always get stuck at different points. This is certainly true if the reason people are getting stuck is a lack of background in anabelian geometry, or a refusal to put enough work into it – both reasons suggested by defenders of IUT. You would then expect to see different confusions in different places from people with different levels of background, or from people who made careless errors at different points. Of course, what we see is exactly what you would expect to see if the proof of Corollary 3.12 is not a valid argument but the rest of the paper is valid.
As an editor or referee, this should be completely unacceptable. Even if the proof is technically correct in the sense that Mochizuki or someone else can give a clear and precise argument which the proof of Corollary 3.12 could be said to summarize, allowing the paper to continue in its present form would be a failure at their duty to ensure papers in their journal properly explain their arguments. A paper such that everyone who studies it seriously, except a small inner circle, gets stuck at the same point simply isn’t a proper explanation.
Some defenders of IUT like to point out that Scholze and Stix didn’t give their precise objection until 2018. But this phenomenon, given that it was noticed by most people who read the paper seriously, should have been turned up by the refereeing process before then. This is, I think, the starting point for ethical concerns about the refereeing process. (For instance, OP’s comment suggests that the editors could have asked a series of referees, ignoring those who have negative commentary, until they found someone willing to say it is good.)
Is it possible that the final version makes substantial changes to this argument and answers this objection? Everything is possible, but it seems unlikely. It is common practice in mathematics to edit the online version of the paper to the final submitted version, or something close to it, without the journal’s formatting. As far as I know the main reasons not to do this are because one doesn’t post online versions or doesn’t edit them often, neither of which apply to Mochizuki.
Even if the editors and Mochizuki want to avoid this for some reason (like they want to make people read the journal to see the complete proof, I guess…) they could easily post some comment online like “Corollary 3.12 has been replaced with a series of simpler statements, each with a detailed proof, culminating in the key inequality, totaling about 35 pages” and get much less criticism from mathematicians and mathematical physics bloggers. Does anyone like receiving criticism? Is there any reason not to do this?
(3) Two serious mathematicians have come up with a precise objection to the proof of Corollary 3.12. Despite the fact that it is not clear exactly what Mochizuki means, they have come up with a plausible interpretation, and shown that, in this interpretation, the inequality that Mochizuki proves is off from the inequality he states by a factor of j^2. This is important because the improvement of the stated inequality over a certain trivial inequality is exactly by a factor of j^2. They have come up with additional arguments, which are unsurprisingly not completely rigorous, that any plausible interpretation of what Mochizuki means will have the same problem.
(One could say that it is not plausible that Mochizuki made such a simple mistake, but I think it is plausible in light of the previously discussed information).
Neither Mochizuki nor his defenders have come up with a convincing rebuttal to this. Mochizuki’s response focused on arguing that it is possible for an argument like his to work, and that Scholze and Stix misunderstood it, rather than making basic clarifications to the argument that would aid in understanding whether the objection is valid, and that would certainly be possible if the argument is correct. For instance it is *provably* possible that any valid argument proving a concrete inequality using facts about affine spaces over the real numbers can be replaced with one that proves the concrete inequality using other concrete inequalities (or identities). Doing this would allow a more focused discussion because Scholze could either dispute one of the building-block concrete inequalities or dispute the implications from them to the desired statement.
For me this objection, combined with the other stuff, is devastating. Of course it is not the case that one must stop the presses of a journal every time a mathematician objects to an argument, and keep them stopped until that mathematician is satisfied. But combined with the other very worrying facts about the proof, this is an objection that must be answered for the paper to be published ethically. As mentioned earlier, if a better answer than what is in the currently online version has been delivered by Mochizuki to the journal, I would expect it, or at least an advertisement for its existence, to be posted online.
Despite the long comment thread, Peter Scholze and others have not exhausted all the reasons that a mathematician examining Mochizuki’s argument should be skeptical that it, or even any argument like it, could possibly work. I could list these additional reasons, but they are not so relevant when the objections outlined above already mean that it would be inappropriate to publish the article in its current form.
Kirti Joshi has now posted a revised manuscript ”On Mochizuki’s idea of Anabelomorphy and its applications” discussed earlier in this thread.
https://arxiv.org/abs/2003.01890
This paper of Joshi is remarkably unconvincing to me. If I may caricature it slightly, it seems to only contain the following types of results:
1. Statements of the form “(Thing X / Property Y) depends only on the absolute Galois group of a p-adic field.”
None of these are surprising or difficult: they all follow from basic class field theory or from the Jannsen-Wingberg theorem (which IS a difficult result, cf. here for a nice overview: http://www.numdam.org/article/AST_1982__94__153_0.pdf)
2. Statements of the form “(Thing X / Property Y) does not depend only on the absolute Galois group of a p-adic field.”
These are even less surprising, and they also follow from Jannsen-Wingberg, or from five seconds of thought.
3. Completely unmotivated results (e.g. Theorem 16.5, Theorem 22.6).
4. Vague suggestions that various things can be interpreted anabelomorphically.
What evidence is there here that this perspective of anabelomorphy is actually useful? What can you DO with it? The answer this paper seems to suggest is: nothing.
I am happy to be convinced otherwise.
For what it’s worth, the first two (of three) papers by Taylor and Anton Hilado are now on the arXiv:
“The Statement of Mochizuki’s Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies” https://arxiv.org/abs/2004.13228
“Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki’s Corollary 3.12” https://arxiv.org/abs/2004.13108
From the abstract of the first paper:
“This paper does not give a proof of Mochizuki’s Corollary 3.12. […] These manuscripts are designed to provide enough definitions and background to give readers the ability to apply Mochizuki’s statements in their own investigations. […] It is our hope that doing so will enable creative readers to derive interesting and perhaps unforeseen consequences Mochizuki’s inequality.”
It is my interpretation that the statement of Corollary 3.12 is reformulated in these papers. In the second paper, some consequences of the reformulation are derived. But aren’t these consequences of an unproven, highly debated corollary – which most likely “cannot work” (Scholze)? Which kind of “creative reader” should use these reformulation then?
Taylor was quite active here, maybe he can comment on how these papers fit into the story?
Martin,
Scholze’s argument is not that the inequality can’t be right, but that Mochizuki’s proof of the inequality can’t work. It’s a reasonable project to understand the implications of conjectured inequalities, better if they’re clearly labeled as conjectures…
In addition, another way to show Mochizuki’s proof doesn’t work and pinpoint exactly where it goes wrong would be to find a point where he derives something too strong, something that can be shown to be true by other methods.
Hi Everyone (and Peter),
In what follows I can give four proofs/reasons why a certain statement in Peter’s manuscript with Jakob about replacing Hodge Theaters with fundamental groups is false. I believe these to be correct. Please check for mistakes.
Best,
Taylor
====================================
I noticed the following claim embedded into the manuscript.
Claim.(Scholze-Stix manuscript page 6, paragraph 4)
There is an equivalence of categories between a connected groupoid of objects isomorphic to a big Hodge Theater and a connected groupoid of schemes isomorphic to a once punctured elliptic curve over a number field sitting in initial theta data. This allows us to replace Hodge Theaters by once punctured elliptic curves.
Here is the exact text:
“In other words, up to equivalence of categories, choosing a Hodge theater is equivalent to choosing a once-punctured elliptic curve abstractly isomorphic to $X$, and this equivalence of categories is constructive in the sense that one can give an explicit functor that takes a Hodge theater and produces a once-punctured elliptic curve. Of course, the category of elliptic curves abstractly isomorphic to $X$ (and isomorphisms of curves) is equivalent to the category whose only object is $X$ that we started with.”
This is stated in the manuscript without proof; they cite a number of propositions from IUT1 Corollary 6.12 (i), Proposition 6.6 (iii), Corollary 5.6 (ii), Proposition 4.8 (ii), Definition 6.13 mirroring Mochizuki’s style.
I just want to point out that this is a MUCH more extreme a claim than what I thought Peter was claiming previously. This is why I had softened it for him in my response warning readers not to take him literally. After looking at the manuscript again it appears that he *did* mean what he said, literally.
This, I can dispatch. It is actually False in four different ways. It is false the first time because of the equivalence of categories statement. It is then false the second time because of the assertion that “picking a once punctured elliptic curve is equivalent to choosing a Hodge theater” statement. It is false a third time because these two statements are themselves not equivalent. We then give another independent proof that this is false by means of interpretation of initial theta data (similar to Mochizuki’s original counter-argument in his “Report”).
Also, most importantly, this type of reduction used to simplify Mochizuki’s definitions is far from admissible. The notion of equivalence of categories and “equivalent” in a colloquial sense should not be confused. Equivalence of categories is too coarse of a notion to be useful in simplifying the definitions in IUT. The correct notion (for objects which are not considered up to automorphism) is the notion of bi-interpretability. This is a special type equivalence of categories which considers the types of formulas you are allowed to use. A reference to this notion in Olivia Carmello’s book at the bottom of this post and another to the wikipedia article.
Remark. In the Scholze-Stix manuscript, they never actually say *which* number field the elliptic curve is defined over. I’m going to assume it is F as defined in initial theta data in IUT. If you take it to be the field of moduli you can still find examples that give a contradiction.
******************
Lemma. $\operatorname{Out}(\pi_1(X_F))$ has cardinality strictly greater than 2 when $\operatorname{Aut}(F/\mathbb{Q})$ is non-trivial (which is the case for $F$ in initial theta data).
******************
We will use Tamagawa’s proof of the absolute Grothendieck conjecture for curves over fields finitely generated over $\mathbb{Q}$.
See Theorem 0.4 of Tamagawa’s paper “The Grothendieck conjecture for
affine curves” (Compositio 1997),
https://doi.org/10.1023/A:1000114400142
Here is a text version:
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1992BA14A2D63FA076DB39A34EC45E83/S0010437X97000614a.pdf/grothendieck_conjecture_for_affine_curves.pdf
Tamagawa’s Theorem:
If $F$ is finitely generated over $\mathbb{Q}$, and $Z/F$ is a hyperbolic curve and (geometrically connected over $F$) then $\operatorname{Out}(\pi_1(Z)) \cong \operatorname{Aut}(Z/\mathbb{Q})$.
Now for the application.
Let $E$ be an elliptic curve over $\mathbb{Q}$ which does not have CM geometrically (to be concrete you can take Cremona’s 11a1). Let $F = \mathbb{Q}(\sqrt{-1}, E[30])$ and let $X_F = E_F – o$ where $o$ is the origin so we have $X_F = \operatorname{Spec}(F[x,y]/(y^2-x^3-ax-b))$ for some $a,b\in F$.
Since $\operatorname{Aut}(X/\mathbb{Q}) \cong \operatorname{Aut}( F[x,y]/(y^2-f(x)))$ we see that $\operatorname{Aut}(X_F/\mathbb{Q})$ contains $G(F/\mathbb{Q})$ and it contains $\operatorname{Aut}(X_F/F)= \lbrace [-1]\rbrace$. These commute. Also note that $[F:Q]>1$. This means $\vert \operatorname{Out}(\pi_1(X_F)) \vert > 2$.//
Remark. I think the subtle point here is that $\operatorname{Out}(\pi_1(X_F)) = \operatorname{Aut}(X_F/\mathbb{Q})$ and not $\operatorname{Out}(\pi_1(X_F))= \operatorname{Aut}(X_F/F)$. Right? Either way, it is not so relevant because this isn’t the biggest point of contention.
******************
Lemma. Let $HT$ be a big Hodge Theater. $\operatorname{Aut}(HT)\cong C_2$
******************
Summary:
The automorphisms of the $\pm$ side of the Hodge theater is $\lbrace \pm \rbrace$.
There are only trivial automorphisms on the $\divideontimes$ side of the Hodge Theater.
These act independently because the way the two sides interact is through a fullpolyiso (this is described below).
This concludes the proof.//
Details According to Mochizuki:
Rem 3.5.2: “The morphisms of data are obvious.” (I can’t defend this remark. This is an omission.)
Prop 4.8(ii); Cor 5.6(ii): Auts of multiplicative base Hodge theaters are unique; these lift to morphisms of Hodge theaters.
Prop 6.6(iii); Cor 6.12(i): Auts of additive base Hodge theaters are unique; these lift to morphisms of Hodge theaters.
Rem 6.12.2(i)(ii): We regard the $\Theta$ bridge as being interpreted in the $\Theta\pm$ bridge. It talks about the $\Theta^{\pm \operatorname{ell}}\operatorname{NF}$ Hodge theater as being the additive and multiplicative part full polyisomorphismed along the interpretation in the previous item.
Defn 6.13(i): This is what I said above but I guess this is where he gives it a name.
Remark. For people who have no experience with Hodge Theaters, Figure 6.5 in IUT1 really contains all the information you need (with the technical definition of the “bridges”). There are 4 bridges and an “additive” and a “multiplicative side”. It turns out that the only “not full polyisomorphism” part of these bridges are the downward ones in that diagram which connect local and global information.
Remark. I would like to check this part again and give more the details, but I think this is all correct.
**************
Mochizuki’s Falsification
**************
Something like this was in Mochizuki’s original response. This is the most important thing to understand why these reductions are incorrect. I think it was overlooked because people find his writing hard to read. His argument was that “$\pi_1(X)$ doesn’t know about initial theta data”. This is true.
More concretely, the fundamental groups at (say) double underline covers can’t exist without the “global multiplicative subspace” (part of the initial theta data). They are defined in terms of the single underline cover which is the dual isogeny of the quotient determined by the finite group scheme $M \leq E$ (this $M$ is the “global multiplicative subspace”). This is not interpreted in $\pi_1(X)$ in Mochizuki’s papers or the Scholze-Stix manuscript (and I don’t see how to choose one in addition to the many with choices of $\underline{V} \subset V(K)$ in a definable way).
Reference: The global multiplicative subspace is explained in my “statement” manuscript with Hilado (https://arxiv.org/abs/2004.13228). See the initial theta data section. We spell out the required “compatibility” between $\underline{V}$ and $M$ there.
**************
Categorical Structure is Not What Matters
**************
I will explain why (naked) equivalence of categories is not the appropriate notion to be using to reduce structure in Mochizuki’s proof. The appropriate notion is bi-interpretability. The idea here is that one needs equivalence of *highly structured categories* and if this notion is not used we lose a lot of information. David Roberts actually makes this point as well in his Inference essay.
Example. Consider a non CM elliptic curve up to isomorphism (as a relative object). If we were just going to consider it by itself then the connected groupoid of curves determined by this object is equivalent to a category with a single object and two morphisms (because the automorphism group will be a group of order two)! A connected groupoid with two morphisms doesn’t allow you to do much.
Example. The structure $\mathbb{Z}$ (with its ring structure) has $\operatorname{Aut}(\mathbb{Z}) = \lbrace \pm 1 \rbrace$. Yet $Th(\mathbb{Z})$ is undecidable. This is a highly non-trivial but with a trivial automorphism group. The theory of a two morphism category with one object is certainly decidable.
The previous example shows the weird sort of things you can do if you admit replacing objects using equivalence of categories. In this case the connected groupoid of things isomorphic to a big hodge theater $HT$ is equivalent to a connected groupoid of things isomorphic to $\mathbb{Z}$ (as a ring). The sorts of things you can do with $\mathbb{Z}$ are quite different from the sorts of things you can do with $HT$.
For structures which don’t involve objects up to automorphism the correct notion I claim is bi-interpretability (btw, objects up to isomorphism and not up to isomorphism is something that is conflated in the SS manuscript). Two structures $A$ and $B$ are bi-interpretable if and only if there exists an interpretation of $A$ in $B$ and an interpretation of $B$ in $A$ such that when these interpretations are viewed as functors they provide an equivalence of categories. The functorial formulation can be found in Caramello’s book for example.
See
https://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf
Definition 6.12
See for a formulation in terms of sets:
https://en.wikipedia.org/wiki/Interpretation_(model_theory)
(As a warning, Mochizuki’s bridges and Caramello’s bridges have nothing to do with each other… as far as I know).
Dear Taylor,
thanks for your comment!
Unfortunately, I cannot follow what you are saying.
First, as is the case throughout IUT, the initial $\Theta$-data are fixed. In particular, we have fixed a number field $F$ etc. Our claim was:
Choosing a Hodge theater is equivalent to choosing a once-punctured elliptic curve over $F$, up to equivalence of categories.
Proof of Claim: As the relevant elliptic curve does not have CM, $\mathrm{Aut}(X/F)=C_2$ is the cyclic group with $2$ elements. Similarly, as you write, automorphisms of Hodge theaters are $C_2$. Moreover, both categories have precisely one object up to isomorphism. The induced map $C_2\to C_2$ can be checked to be nonzero, hence an isomorphism.
So you say that you gave 4 disproofs.
The first is a statement about $\pi_1(X)$ that does not seem relevant to the discussion — the question is about Hodge theaters vs. elliptic curves over $F$. I apologize that we did not explicitly say “over $F$” in the text.
The second is a statement about Hodge theaters that is a part of our proof. So actually, the correct version of the first (namely, automorphisms of $X$ over $F$ are $C_2$) and the second *prove* our claim.
The third is some vague statement. Regarding whether something “knows”, for example, the global multiplicative subspace: The initial $\Theta$-data are fixed throughout. Given an elliptic curve isomorphic to $X$, I can fix such an isomorphism, and import all structures that have been fixed for $X$, like the global multiplicative subspace. This may feel like cheating, but it is exactly what Mochizuki does say in Theorem 3.11 (i) as discussed previously.
The fourth is some vague statement. I don’t really understand what you mean by “bi-interpretability” or “highly structured category”; certainly the functors in both directions can be made explicit, which seems to go some way towards “bi-interpretability”. Note that Mochizuki goes to great pains to define Hodge theaters, including (the highly non-obvious) morphisms, so I naturally expect their category to be of relevance. (If you want to define some theory of diophantine equations over $\mathbb Z$, you should not spend your time defining the category of rings isomorphic to $\mathbb Z$! By the way, the ring $\mathbb Z$ does not have $-1$ as automorphism…) Of course, category equivalences can have power. But Mochizuki’s central claim is that he chooses *different* Hodge theaters and that something that looks like $5$ is more like $13$ in the other, except that they are also the same, up to some ambiguity… . So you have to choose Hodge theaters in his proof. But for this very step, the category equivalence does tell you that you might as well choose elliptic curves isomorphic to your given one (and take the corresponding Hodge theaters). I’m fine with going to the other side when I need to. But precisely because maps of Hodge theaters are a highly non-obvious notion, making it easy to get confused about all sorts of maps, I find it easier psychologically to first choose the elliptic curves and then pass to the other side.
Best wishes,
Peter
PS: You claim that objects and objects up to isomorphism are conflated in our manuscript. Where do we do this?
Sorry, I should have more carefully proofread. The claim is of course
Choosing a Hodge theater is equivalent to choosing a once-punctured elliptic curve over $F$ isomorphic to the given $X$, up to equivalence of categories.
(I omitted “isomorphic to the given $X$”.)
>By the way, the ring $\mathbb{Z}$ does not have $−1$ as automorphism.
https://images.app.goo.gl/zfRudP7opfTU8dVU8
That being said, you can just take some equally stupid thing like $\mathbb{Z}$ as an abelian group. You don’t get undecidability… but whatever. The point is you have something “nontrivial” equivalent to a category with one object and two morphisms.
I might give a more detailed response about the other stuff later. Right now, I hope that other people will start commenting about the Mathematics and that it leads to a productive discussion.
Dear Taylor,
I certainly understood your point there — you might also take the ring $\mathbb Z[\sqrt{-1}]$.
There is of course a big difference between the ring $\mathbb Z$ and the “theory” it defines, i.e. roughly the class of all subsets of all finite powers $\mathbb Z^n$ that are definable by polynomial equations. The latter is indeed a highly nontrivial category (where morphisms are given by definable graphs); it is of course not equivalent to a category with one object and two morphisms. If a category like this is in place in Mochizuki’s work, I’m happy to hear about it!
Reading the IUT papers, however, you are presented with some extremely difficult notion of a Hodge theater, together with a highly non-obvious notion of isomorphisms of such: Isomorphisms do not preserve nearly as much structure as you would expect them to, and this is by design as Mochizuki points out. So I find it very hard to “guess” what something like a surrounding “theory” might be. For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry https://en.wikipedia.org/wiki/Interpretation_(model_theory) you linked to, nor the topos-theoretic framework of Caramello. (Regarding the first one: A “structure” in the sense of model theory has first of all an underlying set. I find it hard to take a Hodge theater and produce some interesting set that is functorial in isomorphisms of Hodge theaters, the problem being the very lax notion of isomorphisms of Hodge theaters.)
However, these long discussions are all about interpretations. Regarding the mathematics proper: I stand by the claim made in our manuscript, and have indicated the proof above.
Best wishes!
Peter
Hi Peter,
I think this discussion kills (1). More comments are below.
(Sorry if there are any mistakes or typos)
Best,
Taylor
#################
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>The first is a statement about $\pi_1(X)$ that does not seem relevant to the discussion. The question is about Hodge theaters vs. elliptic curves over $F$. I apologize that we did not explicitly say “over F” in the text.
So maybe I am misinterpreting you (math joke) but I thought your point was that because of absolute Grothendieck, $\pi_1(X)$ and $X$ are “the same” and that you could just replace one with the other willy-nilly in any statement. I think you are saying that any proof with fundamental groups could be replaced by a proof with $X$’s (whenever absolute Grothendieck holds).
Look at your comments April 6, 2020 at 9:28 am.
Look at the text following your claims on April 13, 2020 at 4:25 pm (this is the (1),(2),(3) post).
This seems to contradict your position now that $X_F$ should be viewed as relative. I have a feeling say you want $X_F$ to be absolute in one case and relative in the other (which seems to violate your own heuristic).
Remark. Before getting into the equivalence of categories stuff below let me stress again that I think this is a red herring as it is not what we should be talking about.
In either case if you want to $X_F$ as an absolute scheme, you get to apply absolute grothendieck but don’t have the equivalence. In the case that you view $X_F$ as a relative scheme you get the equivalence of categories. The equivalence of categories is a red herring since it isn’t the appropriate notion to use for reduction. In the relative case I agree that the associated groupoids of isomorphic Hodge Theaters, Isomorphic $F$-schemes, and isomorphic rings are all equivalent to a category with one object and two morphisms. We both agree this is a ridiculous category that contain little information (if all we are allowed to talk about are objects and morphisms). We can interpret nothing from this category. There are no q-pilots, no theta-pilots, no bogus diagrams, nothing. Hence, this must not be what is going on.
####################
####################
>For all I can see, Hodge theaters fit neither into the framework of “structures” as used in the wikipedia entry
We want sets. (as you said)
You might notice that I made the caveat of bi-interpretability being correct for things in IUT not considered up to automorphism. The way I currently think about it is in two phases.
In phase one you have your structures (I replace all the Frobenioids with pairs $(\Pi, \overline{M})$, fundamental groups by $\Pi$’s etc), in this phase we do all this interpreting.
In phase two whenever we add the polyisomorphisms, I exchange that for an automorphisms and let those act on whatever it is that was interpreted.
Remark. Mochizuki’s Species/Mutation stuff in IUT4 is his way of explaining this (poorly). I have not made it though that stuff and at this point I don’t think I need it. My point is, I think one can write this all down carefully, but I haven’t done so and I don’t think Mochizuki has done so. There is an intereting discussion on ncatlab that discusses this stuff. It seems that Skoda alluded to what I am talking about now back in 2012!
https://nforum.ncatlab.org/discussion/4260/abc-conjecture/ read posts 13, 16-20, 37.
Remark. Hodge Theaters seem to be used more of a catch-all for all the stuff he will ever need. Every individual construction I have encountered always requires much less than the full Hodge Theater. I certainly haven’t worked out what the signature of a Hodge Theater is but I do think this should be worked out. You will also see from the big tables
First orderizability is explained in the stack exchange post I mentioned earlier. I think you were a little confused about the context of that post but hopefully this is clear now.
>The third is some vague statement. Regarding whether something “knows”, for example, the global multiplicative subspace: The initial theta-data are fixed throughout. Given an elliptic curve isomorphic to $X$, I can fix such an isomorphism, and import all structures that have been fixed for $X$, like the global multiplicative subspace. This may feel like cheating, but it is exactly what Mochizuki does say in Theorem 3.11 (i) as discussed previously.
I claim this is can be made precise. I view his claims as statements about “functorial algorithms” (which I view as interpretations). For whatever definition of a “phase 1 Hodge Theater” you take as described above, there exists a prime strip as a reduct (I regard coming up with a minimal signature for a “working” Hodge Theater as an open problem). From this prime strip you get a group isomorphic to $\pi_1^{temp}(\underline{\underline{X}}_{\underline{v}})$ as a reduct: view the Frobenioids their as pairs $(\Pi,\overline{M})$ and then forget the monoid. In your formulation there is just no “functorial algorithm” so it is not comparing apples to apples. You are asking for something entirely different from what Mochizuki is doing.
I said this about 3.11.i before but I will repeat because I think you will be more receptive now that we are on the same page with this equivalence of categories vs bi-interpretability thing. What Mochizuki does in 3.11.i is a “lift/construct/push”.
You lift to a structure to a structure with more interpretive power that interprets it, you perform the desired interpretation, then you push it into a set that interpreted in your original structure. This construction is equivariant with respect to automorphisms of the lifted structure. You take the orbit of these (or in the case of ind3, you do something like “consider all possible lifts”) and consider the interpretation only as existing only up to this indeterminacy. The result is the type of crap you get in 3.11.i where the interpretation up to automorphism is independent of the choice of lift.
Remark. David Roberts recently pointed me to this 5 year old note of Kim where he picked up on the significance of the global multiplicative subspace:
http://people.maths.ox.ac.uk/kimm/papers/pre-iutt.pdf
I was impressed by how early he picked it up.
>You claim that objects and objects up to isomorphism are conflated in our manuscript. Where do we do this?
So I think by allowing yourself to replace polyisomorphisms by fixed isomorphisms and then at the same time cite Propositions from Mochizuki this could get you in trouble. I was worried about this at least.
(1) The issue is not whether switching between X and pi_1(X) could ever under any circumstances induce an error, but whether switching between X and pi_1(X) could ever be the deciding factor in whether Mochizuki’s proof is valid. The idea that the discrepancy between the automorphism group of X as an absolute scheme and the automorphism group of X as a relative scheme could be responsible for confusion as to whether the proof is correct (either someone thinking it’s correct when it’s not, or thinking it’s incorrect when it is) is absurd.
For one, there are many fields F with no automorphisms that one could run the argument over, and the discrepancy vanishes there. In any case, Mochizuki’s argument is not an argument which is based on using the automorphisms of a field F to some kind of strategic effect.
(2) It is not, I think, in dispute whether it is possible to define an object on a structure that has few automorphisms, then “push” to a structure with more automorphisms by taking the orbit under its automorphism group. If you have two constructions which are each naturally defined in a different level of structure, then doing this might be helpful to compare them. But the issue here is that everything in sight is most naturally defined on the same structure, that being the structure of a (punctured) elliptic curve.
So this raises the question of what could be the benefit of artificially forcing some natural object to be invariant under some random unrelated group of automorphisms. Of course it is possible to imagine benefits that might possibly occur in some conceivable argument, but one has to explain what the benefit actually is in Mochizuki’s argument and where it occurs (e.g. commutativity of some diagram).
One fundamental issue is that invariance under a group action is very useful for proving identities, but not useful for proving inequalities, unless one proves an inequality by first deriving an identity and using a separate argument to convert that identity into an inequality. But there don’t seem to be any useful identities that can be proved using the invariance under automorphisms of the Galois group, because automorphisms of the Galois group never show up in an essential way at any other point in the argument.
Will, I completely 100% totally agree with (1). I think (2) is a misunderstanding of what I am talking about. We are not killing automorphisms but considering completely different structures. Also, I don’t think you understand the definition of a structure. It isn’t a hand-wavy thing. It is a collection of functions, relations, constants, and sorts. I need to think about your last two paragraphs before I can comment more.
Here are some relevant nLab entries that define the model-theoretic terms that Taylor has used in his recent comments. I hope this can clear up some confusion.
Structure: https://ncatlab.org/nlab/show/structure+in+model+theory
Interpretation: https://ncatlab.org/nlab/show/interpretation
Dear Taylor,
thanks for your further comments. I think W said it all.
Let me just make the following clarification regarding (1). You claim that you disproved some claim that I made in my manuscript with Stix, or some claim that I made in the current thread. This is wrong. In this thread, I looked at certain curves $X$ (of strictly Belyi type and defined over a $p$-adic field) and claimed
The abstract curve $X$ is equivalent to the topological group up to inner automorphism $\pi_1(X)$.
In the manuscript with Stix, this claim is also made, along with a different claim. Namely, for some other curve $X$ defined over a number field $F$ (appearing in the initial $\Theta$-data of Mochizuki)
The relative curve $X/F$ is equivalent to a Hodge theater.
In both cases, “equivalent” is meant in the sense of equivalences of categories of the relevant objects. (To be precise: a) The category of schemes that are hyperbolic curves of strictly Belyi type over a $p$-adic field, with morphisms being isomorphisms of abstract schemes, is equivalent to the category of topological groups that are isomorphic to $\pi_1(X)$ for some such $X$, and morphisms being outer isomorphisms of topological groups. b) The category of curves over $F$ isomorphic to the given $X$, with morphisms isomorphisms of curves over $F$, is equivalent to the category of Hodge theaters as defined by Mochizuki, with morphisms the ones defined by Mochizuki.) We agree that both claims are true; in fact, both are theorems of Mochizuki.
You seem to argue that there is some tension between these claims, as one talks about absolute vs. the other about relative curves. Well, also the right side is different, so no wonder the left side is!
Actually, in the context of the current thread, it would have been a good question whether, given that a Hodge theater is a collection of certain topological groups isomorphic to $\pi_1(X)$’s for certain different $X$’s, plus other data, with lots of relations, can you actually interpret all those things as some convenient structure on the side of schemes again? This is not a priori clear, and Mochizuki’s central claim is that by passing to $\pi_1(X)$’s, he can do interesting new things. So he cooks up a Hodge theater. And … you can beautifully interpret it in good old schemes, it “is” just a curve over your original number field $F$.
The same happens for everything else I’ve seen in IUT or your comments.
I’m happy to continue any further discussions by e-mail.
Best wishes!
Peter