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Am making a start on trying to understand something of Mochizuki’s IUTT papers. I do not hold out any promises on how far I am going to get, or how long it is going to take me! Even this very first definition is going to me a long time, I think, as I intend to try to fill out all details. All help will be appreciated!
I’m actually already a bit confused (I am not an expert on these moduli spaces) by the terminology in 3.1 (b) in IUTTI, with regard to the distinction between a hyperbolic curve of type (1,1) and an elliptic curve. I’m sure I will be able to figure it out, but if someone is quicker, just go ahead and edit the page!
I’m also going to send an email to Mochizuki to ask if he can give or provide a reference a baby example of 3.1, which I can orient the exposition around.
Richard, I’ve noticed that when you want to start a new page you put a link to it in the Sandbox.
You can do the same thing in the nForum and just hit preview. This doesn’t bump up the Sandbox version.
Good idea, Rod, thanks for the tip!
If you remove the identity element, say, from an elliptic curve I think you get a hyperbolic curve (and if (1,1) means genus 1 with one puncture, you are done. But please email Mochizuki anyway, I don’t know the situation in arithmetic geometry
Yes, $(1,1)$ means genus $1$ with one marked point, but I thought (as I say, I do not really know these moduli spaces at all well) that these corresponded exactly to elliptic curves (with identity as the marked point). Whereas the following is written.
Write $E_{F}$ for the elliptic curve determined by $X_{F}$ [so $X_{F} \subseteq E_{F} ]$
I am sure that this particularly point is completely standard for those in the know.
But I think an explicit, and hopefully simple and elementary, example to illustrate the whole of 3.1 would be enormously helpful, and I have asked Mochizuki over email now if he could possible provide one. I have not found any explicit example, or even much discussion, concerning this definition in the various expository pieces by other authors.
I believe $(1,1)=(g,r)$ where $r$ is the number of points removed from $E_F$. So $X_F$ is $E_F$ with one point removed, hence the $X_F \subseteq E_F$. Indeed, a baby example would be most useful!
Thanks very much for joining the discussion, Richard! I agree with you. I had thought that Mochizuki was simply using “hyperbolic of type (1,1)” as a synonym for “stable curve of genus 1 with 1 marked point”, but in fact he means the complement at the marked point, as you say. See for example this article, at the bottom of page 1.
Thus I have now added the details for $X_{F}$ to the page, referring to newly added material at open subscheme. In the interest of simplicity, I have not mentioned moduli spaces or hyperbolic curves, as they are not needed just yet; though we will need them soon, I think.
Next up, we’re going to need some stuff on stable reduction for elliptic curves. I’m not aware of anything on the nLab for this, good reduction, etc. I’m not going to work more on this today; if anyone feels like creating pages for these concepts (which are reasonably elementary), please jump on in!
I’d just like to make one more quick remark about why I think it is reasonable to ask for a baby example. It is because, after all, we are just talking about an elliptic curve here, and some data concerning the field it is defined over! We should be able to write down a completely explicit equation and explore some things!
Re: #5, or, even better, start by making a link to it in an appropriate place from a real existing nLab page that should have a link to it. Pages that aren’t accessible from anywhere else aren’t nearly as useful anyway.
Indeed, the page is now linked to from inter universal Teichmüller theory, but I could have started with that.
Here is a basic question which I hope someone can help with. There is an explicit definition of good/stable reduction, multiplicative/semi-stable reduction, additive/unstable reduction for elliptic curves, by means of ’minimal Weierstrass equations’. Now, in the definition of initial Θ-data, Mochizuki refers to stable reduction not exactly of an elliptic curve, but of an elliptic curve with a point removed. Can the definition of the latter be taken to be the same as the one for the actual elliptic curve in terms of minimal Weierstrass models, except that one disregards the point one has removed, i.e. one does not need to check whether it is singular or non-singular, etc?
Richard Williamson, I hope you’re still continuing your pursuit of Mochizuki’s work. Despite not giving much of an appearance, I’ve been looking through his work in order to achieve some sort of “understanding” of IUT as well. As a slight aside to this thread, there appears to be reference to hyperbolic curves of type $(1, (\mathbb{Z} / l \mathbb{Z})^{\Theta})$ and hyperbolic curves of type $(1, l\text{-tors}^{\Theta})$. These make an appearance in IUT II and after much cat-and-mouse I was able to isolate their respective constructions in this paper. Filling out all the details is indeed difficult and might benefit from the wiki approach for note-taking (i.e., creating nLab pages for different definitions and constructions).
Hi Richard, thanks for your comment! I completely agree. Yes, I am definitely still pursuing this. I have not heard back from Mochizuki himself yet, but I also contacted Weronika Czerniawska at Nottingham, who was able to point me to someone who hopefully might be able to help. We’ll see! Thanks for following this thread!
Some developments. Firstly, I heard back from Wojciech Porowski at Nottingham regarding my question in #14. I have not fully digested the response yet (I am a bit rusty with algebraic geometry), which was the following.
the image of the chosen point in special fiber should lie in the smooth locus. That chosen point is to be considered as the origin of our elliptic curve.
Wojciech also wrote the following amongst other things, which looks helpful. Again, I have not fully digested it yet.
Basically, we are looking at elliptic curves such that at every bad reduction place the curve is a Tate curve (plus some other assumptions).
A few days later, Professor Mochizuki sent a very cordial and polite reply to my email, encouraging us in the creation of this wiki page. I appreciate this a lot, and respect him a lot for taking the time to write, despite being busy and having these kind of questions many times before. I do not think it appropriate to quote the entire email here, but he suggested a couple of small textual changes, which I have now made (see #17), and wrote the following concerning my question about an example. [Edit: I thought originally that the second paragraph is a quote from the IUTT papers, because it was indented in the email, but actually I’m now not sure; perhaps the indentation was simply for emphasis.]
in some suitable sense (cf. [IUTchIV], Corollary 2.2, (ii), for more details),
“most”/”suitably generic” elliptic curves over number fields arise from (i.e., as the “E” and “F” of) initial $\Theta$-data for some prime number “l” that is of a suitable size as to yield inequalities of interest concerning the height of the elliptic curve. Listing out explicitly/numerically those “E”/”F”/”l” that must be excluded (i.e., that are not “suitably generic”) would be technically very involved and not particularly enlightening from the point of view of understanding the logical structure of the theory.
Somewhat related issues are discussed in [IUTchI], Remark 3.1.3.
In short, from this, as well as Wojciech’s email, it seems that examples should actually be in abundance. Thus my feeling is that the best way to proceed is to just to explore some concrete examples. If we can create a few, I can write back to ask for further feedback, to check that things are correct.
[Edit: I’ve now looked at Corollary 2.2 (ii) in IUTT IV, and it does look extremely relevant. A second way to proceed could be to try to understand what happens here, where some initial $\Theta$-data is constructed explicitly. There are also several fascinating remarks just after this corollary, if you have not seen them before; in particular, there is a hint of the possibility that IUTT might be able to be adapted to say something about the Riemann hypothesis.]
The first thing we need is an example of an elliptic curve that has good reduction with respect to all non-archimedean valuations (over some extension of $\mathbb{Q}(i)$). Perhaps someone following can join in with trying to find one?
Sounds like a job for MathOverflow
Indeed, I’ve posted at Math StackExchange now here, as #19 is presumably fairly trivial for number theorists.
That is indeed a nice diagram, Tim! Deligne once gave a nice talk at the Grothendieck birthday conference in January 2009, where he expressed at the end the idea that the Langlands programme goes beyond motives.
I have now advertised the nLab page we are trying to fill out here on MathOverflow, and asked for help.
I am happy to work out details for myself, but for really basic stuff like the question I asked about in #21, it would surely be more efficient for some number theorist to just give an example! As I say, I presume it is fairly trivial.
Idiot questions coming up! I have been glancing at some of the work on Arithmetic Topology. How, if at all, does that stuff interact with the Mochizuki stuff? I am wondering for various reasons (apart from anything deep I was interested that, in his book, Moroshita was working with Alexander modules and what he called $\psi$-differential modules in an $\ell$-finite or $\ell$-adic context and I had worked on that for my profinite algebraic topology monograph… still being written very slowly!) It seemed to me that Richard’s knowledge of the knot theoretic stuff should interact with this stuff (or, Richard, is this one of your motivations?)
As a related question, Moroshita et al, seem only to be doing the analogues of knots in 3 spheres, but there is a lot of algebraic topology outside knot theory so what evidence is there of higher dimensional arithmetic topology? My ignorance is showing but might it be related to Grothendieck’s anabelian stuff and also to La Longue Marche à travers la théorie de Galois?
Small amendments suggested by Professor Mochizuki.
Way to go, Richard! And thanks a million for taking your notes on the $n$Lab. That’s the way science works. Hope you will have the energy to pursue this project for a bit.
Thanks Urs and Tim, I’ll reply later as I have to dash now! Just wanted to say thanks very much to David for helping out at the MathOverflow question! I am having some trouble posting there, so will just write here that I do not quite understand the situation described in the comments there. I mean, Mochizuki says that there are abundant examples, and the comments seem to suggest that there are no examples…! Maybe it has to do with the fact that we are actually asking for good reduction for the punctured elliptic curve, not the full elliptic curve? But I thought the puncture could be taken to be the origin (e.g. point at infinity), in which case I don’t quite see how it makes any difference…
There’s a difference between a marked curve and a punctured curve! The puncture is usually sent off to infinity.
Yes, what I was trying to say was that in the definition of initial $\Theta$-data it is actually required that the punctured curve (i.e. elliptic curve with one point removed) has ’stable reduction’, not that the original elliptic curve has stable reduction. I don’t know exactly what the definition of stable reduction is for a punctured curve, but I presume that it is basically the same as for the original elliptic curve (where it is also known as good reduction), except that maybe one allows the removed point to be discounted from consideration, i.e. maybe this point is allowed to become singular when one reduces. But if the removed point is the one at infinity, it seems that we have not really changed anything…, I mean one just has the same Weierstrass equation and one considers all points of it…? There must be something going for which the removed point is relevant, because otherwise the comments at MathOverflow say that there are no examples!
Ah, I see some kind soul has now made this point at MathOverflow.
Thank you very much for your thoughts, Tim! I find the idea of arithmetic topology fascinating; I’ve thought deeply about knot theory, and studied étale cohomology and the motivic world quite a lot as well, so in principle this should indeed be just the thing for me, as you suggest! Whilst I’ve come across some of the ideas many times, somehow I’ve just never tried to understand it properly. I should definitely do so!
I am not aware of a link between Mochizuki’s work and arithmetic topology, but don’t attach any weight to that, it could well just be my lack of knowledge! My motivation for looking into his work is mostly just for its own sake; I have tried to obtain some kind of feeling for what is happening, and it seems fascinating, but to appreciate it properly, I need to really get into the details. As an aside, it actually doesn’t bother me whether or not the work is completely correct, I think mathematicians often place too much emphasis on this. There can be valuable ideas in a flawed work. I am quite interested in and have thought a fair bit about algebraic geometry over $\mathbb{F}_{1}$; this is probably the closest link with Mochizuki’s work amongst things I know something about.
As a related question, Moroshita et al, seem only to be doing the analogues of knots in 3 spheres, but there is a lot of algebraic topology outside knot theory so what evidence is there of higher dimensional arithmetic topology?
I suppose you meant geometric topology rather than algebraic topology here? This is a good question, which I don’t know anything about! I suppose one issue is that I don’t think there is any known analogue for 2-knots, say, of the Lickorish-Wallace and Kirby theorems which allow 3-manifold theory to be thought of as knot theory. However, I do expect such an analogue to exist.
My ignorance is showing but might it be related to Grothendieck’s anabelian stuff and also to La Longue Marche à travers la théorie de Galois?
Even for ordinary arithmetic topology, I would guess that it should indeed be related to anabelian geometry. A bit like with the Riemann hypothesis, one needs to say something about $\mathsf{Spec}(\mathbb{Z})$, and probably one is going to need to work over a deeper base for that. Anabelian geometry seems loosely analogous to the question about when the fundamental group of a 3-manifold determines it (yes for knot complements, of course).
Thanks for the encouragement, Urs! I will certainly do my best to find the energy. Once/if we have an example or two to hand, things should be much easier. I am actually a bit baffled by the MathOverflow post so far; surely the first thing anybody would do to try to understand IUTT would be to ask this kind of question?!
Ah, I only now notice that there was a typo in my MathOverflow post, and that the comments were referring to elliptic curves defined exactly over $\mathbb{Q}(i)$. It doesn’t matter to me if they are defined over an extension of this, as long as that extension is completely explicit (as simple as possible is also good, of course!).
Thanks for the link, David! For me personally, as in #31, it wouldn’t make much difference whether or not it is flawed, I am mainly interested in trying to understand the approach and the ideas behind it. But it will be interesting to see what happens.
Do you know who the person is who writes that blog?
A number theory postdoc at Columbia. I saw someone else on Reddit say they’d already heard of this development, apparently with knowledge of who the alleged refuters are, and that Hansen shouldn’t have leaked the rumour.
Just in case anybody is wondering about the lack of activity here, I have contacted a couple more experts asking if they are able to provide an example.
As before, I cannot believe that nobody who has tried to understand IUTT has an example to hand! It may be trivial and a little tedious for an expert to give such an example, but surely it should not be so difficult. Again, I am happy to work out details, but I feel that it would be much more preferable to have an example outlined by an expert, and then fill in the details, than give one from scratch. Imagine how much one can learn from seeing the various parts of the definition of initial theta-data carefully discussed and illustrated…
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