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I’ve created a page on the Artin-Mazur formal group.
Thanks for this nicely formatted new entry!
I have added in a few more hyperlinks, and linked to the entry from formal group.
When putting outside links in format [ somthing ] ( url ), the url has to have http part, otherwise it does not work. I have corrected this time. But I will change to show the permanent url of the numdam abstract page, rather than direct hard link to pdf so that people can choose between just reading abstract, and downloading either pdf or shorter djvu format and also see the links to some of the entries in bibliography.
Thanks! I definitely should have tried the link to make sure it worked.
added to the Idea-section at Artin-Mazur formal group the following, which I think is the actual Idea:
The Artin-Mazur formal group in dimension $n$ is a formal group version of the Picard n-group of flat/holomorphic circle n-bundles on the given variety. Therefore for $n = 1$ one also speaks of the formal Picard group and for $n = 2$ of the formal Brauer group.
I have expanded the Definition section. Just made it more verbose with more background information and more links.
Hope that’s okay with hilbertthm90. Please complain if you think I messed it up.
Looks good to me.
All right, thanks for the feedback.
Meanwhile I see that in their section III Artin-Mazur also discussed the formal groups of moduli of n-connections (Deligne complex). That’s neat. I should add something about this when I am not on the road, as now.
I put in a missing $S$ in one sentence, and changed the notation $X\otimes S$ to $X\times_{Spec(k)} S$, which to me is more explanatory.
There is also a “flat” version which I’ve been calling “the enlarged formal group.” For an elliptic curve, you get the full $E[p^\infty]$ which is always a height 2 formal group. The connected component is the formal Picard group which splits off a height 1 factor if it is ordinary and is the whole thing if it is supersingular.
That paper studies a bit the enlarged formal Brauer group of a K3 surface (of which the formal Brauer group is the connected component). The enlarged version is really useful when the K3 is ordinary. Nygaard used the structure of it to show that in that case the deformation functor is naturally isomorphic to the deformation functor of the enlarged formal group (this is “Serre-Tate theory” for $E[p^\infty]$ of an elliptic curve).
I spent a bit of time proving an analogous statement for Calabi-Yau threefolds. The hardest part in that case is just figuring out how to show that the enlarged formal group is pro-representable by a p-divisible group. I ended up assuming a bunch of hypotheses like ordinary, liftable, p>3, and torsion-free crystalline cohomology. This is written up in my thesis which I should be putting on the arxiv at some point in the next few weeks.
Thanks!
Where you say “that paper”: which paper? :-)
I just checked with Google (on my phone, alas..) and I find your post “Serre-Tate theory I”. (For one it is excellent that you have there - what nobody ever admits - the warning that it is morally wrong to think of the formal completion of the elliptic curve as being the formal group, which morally instead is the formal Picard group, otherwise nothing will make sense later on…)
I am very interested in whatever you have to say here. Whatever further details you are able to share, I’d be grateful for.
“That paper” was supposed to refer to Section IV of the Artin-Mazur paper. They build the enlarged formal Brauer group of a surface by switching to the fppf site. It is a bit more subtle to check various properties here, and already for surfaces (Proposition 1.8) they start needing a few more hypotheses than before.
I always find it strange when I have to convince a number theorist that you shouldn’t use the group structure on the elliptic curve if you don’t need to (the Galois representation is on the l-adic cohomology not the Tate module or the formal group is the formal Picard group). It is somehow accidental that you can think in terms of the group structure in that case. The generalizations to other varieties just work.
There is also something dangerous about constantly identifying $E$ with $PIc^0$ because it is naturally the dual elliptic curve which just happens to be isomorphic and choices of identification keep getting made. These choices have caused me headaches in the past that went away when I started thinking about things in “the right way.”
I took a look at this Szymik paper. I’m kind of amazed that people care about this stuff. Has anyone tried to look into “CY3 Spectra,” i.e. a triple (E, X, $\phi$) consisting of an even periodic ring spectrum $E$, a Calabi-Yau threefold $X$ over $\pi_0 E$ and an isomorphism, $\phi$, of the formal $H^3$ on $X$ with the formal group of $E$?
“That paper” was supposed to refer to Section IV of the Artin-Mazur paper.
Oh of course, sorry. I’ll have another look.
These choices have caused me headaches in the past that went away when I started thinking about things in “the right way.”
Interesting. The way I became aware of it was when “learning” equivariant elliptic cohomology. In that context one sees that the formal elliptic curve which is $spf E^\bullet(B U(1))$ for $E$ an elliptic spectrum “is not the elliptic curve one might think it is” because for any other compact Lie group $G$ one finds that $spf E^\bullet(B G)$ is the formal moduli of flat $G$-bundles on a given elliptic curve.
Hi hilbertthm90,
now I have some leisure to come back to your comment. I am really interested in understanding the Artin-Mazur formal groups for $n = 3$ and on CY 3-folds $X$.
Is it known which heights occur for $\Phi^3_{CY3}$ in this case? (Analogous to how it is known that $\Phi^2_{K3} \in \{1,2,3,4,5,6,7,8,9,10, \infty\}$, with all choices exhausted.)
What strikes me is that generally the Artin-Mazur group $\Phi^{n = 2k+1}_X$ is the classical perturbation theory of abelian self-dual higher gauge theory in (real) dimension $4k+2$. Moreover, by the discussion at equivariant elliptic cohomology – Interpretation in QFT we have that for $k = 0$, hence $n = 1$ that the elliptic cohomology associated to the formal elliptic curve $\Phi^1_{ell-curve}$ in a sense is the space of quantum states of the theory (namely of 3d $U(1)$-Chern-Simons)
Therefore analogy suggests for $k = 1$ hence $n = 3$ that $\Phi^3_{CY3}$ is the formal group of a multiplicative complex oriented cohomology theory which encodes the quantization of the $U(1)$-7d Chern-Simons theory.
I am aware that $\Phi^2_{K3}$ is used to discuss “K3-cohomology”. Onfortunately that sits in between these two cases and right now I am unsure how to think of this case in terms of QFT.
But it seems clear that from here one wants to proceed to defining a “CY3-spectrum” to be a complex oriented $E_\infty$-ring equipped with an equivalence between its formal group and the $\Phi^3_X$ of a given Calabi-Yau 3-fold.
Has this been considered? Or does any of this resonate with anything you have been thinking about?
To my knowledge it is not known which heights can occur. The really nice thing about K3 surfaces is that $b_2(X)=22$. The most naive estimate has to do with converting the height to the slope $[0,1)$ part of $H^2_{crys}(X/W)\otimes K$ and then thinking about which Newton polygons can occur on this F-isocrystal. By the Hard Lefschetz theorem there is a duality that implies that if the $[0,1)$ part has dimension $d$, then the slope $(1,2]$ part also has dimension $d$. Thus height $11$ is the highest. It turns out that this maximal height case really has all slopes equal to $1$ and hence is really the supersingular case and not height $11$. Thus the heights are in the range $1, ..., 10, \infty$.
That analysis is only possible because $b_2(X)=22$. For a Calabi-Yau threefold you want to look at $b_3(X)$. If we fix this number (e.g. we could just work on one component of the moduli space), then you can do a similar bounding trick. My guess is all possible heights are realized, since the height strata probably go down by $1$ dimension each height as in the K3 case and hence is non-empty (I’ve been meaning to look at Katsura and van der Geer’s paper on this to see if the same idea just works).
Summary: For a given $b_3(X)$ there are probably heights $1$ up to something weird like the ceiling of $3 b_3(X)/4$ and then - 1, and $\infty$. As far as I can tell, no one knows if there is a universal bound for $b_3(X)$, though.
I don’t really know anything about elliptic or K3 cohomology, but it seems like there should be a natural generalization in that direction. I just don’t know if it has an interpretation that people care about.
Anything you can say at height 3 would be interesting to homotopy theorists. Actually, Paul VanKoughnett and myself were planning on reading Artin-Mazur and trying to figure out if we could build some moduli of K3 surfaces + data such that each K3 surface has associated formal group of height 3.
Calabi-Yau would also be interesting… In particular, if any of these cohomology theories detected the gamma family in the homotopy groups of spheres, people would be really excited.
At the moment, the only geometry we have related to higher height phenomena in homotopy comes from PEL Shimura varieties- and you have to go all the way up to dimension n^2 to get information about height n, for various technical reasons. These moduli are really hard to compute with- no one really knows their cohomology, which is the necessary input to start computing with the associated cohomology theories. If we had some more manageable moduli, we might be able to compute something.
Thanks for the feedback! I have to dash away from my computer right now, but I just asked a question related to this here on MO and will love to come back to it later when I am more online again.
This paper should be useful on the moduli problem: http://arxiv.org/abs/math/9910061 They show the strata of height $h$ K3 surfaces is $20-h$ dimensional, so you should be able to produce 17 dimensional families of height 3 K3 surfaces (with a given polarization of fixed degree).
@Hilbertthm90,
thanks again, this is really useful information.
(BTW, to produce hyperlinks here type
[linktext](url)
)
@Dylan,
also thanks again. Might you have a pointer to more information on the business related to PEL Shimura varieties that you are alluding to?
Ah, I have found Tyler Lawson’s “An overview of abelian varieties in homotopy theory”. That’s just what I needed here.
I need to get a feeling for this strategy in TAF to reduce to 1d formal groups, I understand that this allows to apply powerful tools, without which one would be stuck, but is there also some intrinsic motivation, or are we just looking for the keys where the streetlight is?
@hilbertthm90: Thanks! That looks great!
@Urs (Sorry for the long, probably mostly unnecessary, response): I think what’s going on is that we just have no way to import algebraic geometry into homotopy theory except through 1-dimensional formal groups- that’s sort of our only in. We built cohomology theories that tell us everything that we can get from formal groups (MU, BP, Morava E theories, and Morava K theories) but these are just a little too hard to compute with past height 2 or 3, because you end up having to compute some pretty gnarly profinite group cohomology with highly nontrivial coefficients. This profinite group is the Morava stabilizer group, which is the group of automorphisms of a height n formal group, and it acts on the space of functions on a universal deformation.
So how do we deal with this crazy group? Well, we look at big finite subgroups of it. A good way to cook up finite groups acting on a formal group is to realize that formal group as the germ of some honest gadget that’s rigid enough that it only has finitely many automorphisms. The simplest nontrivial example is the multiplicative group, which only has an action of Z/2, even though the associated formal group admits all sorts of love at each prime (Z_p^* much love). In homotopy theory this spits out K theory and KO, which are just a little easier to handle than the K(1)-local sphere (which is what you get when you take fixed points with respect to the whole thing).
The next example up is an elliptic curve, which has only finitely many automorphisms (so that’s good). Some of the associated formal groups have height 1, others have height 2. The ones with height 2 (supersingular curves) admit lots of automorphisms- in fact, they give maximal finite subgroups inside the automorphisms of the associated formal group, so the associated cohomology theory is like the most manageable approximation to the K(2)-local sphere we can get without facing our profinite fears. In this case we get the K(2)-localization of TMF. Understanding exactly how much distance there is between TMF and the K(2)-local sphere (the fixed points of Morava E theory wrt the whole stabilizer group) was figured out by Goerss, Henn, Mahowald, and Rezk and later by Behrens: it turns out that you can write down a small homotopy limit diagram realizing the K(2)-local sphere that involves TMF and TMF with some level structure.
Ok, so all this happened, and it’s been a huge success! But we’re in trouble: there’s a lot more 1-dimensional formal groups to study, but we’ve run out of geometric objects with obvious “germs” that are 1-dimensional formal groups… There’s no more 1-dimensional abelian varieties. So Behrens and Lawson decided to look at higher dimensional abelian varieties with some data that let them split off a 1-dimensional summand. There was also a technical consideration: it’s easiest to produce cohomology theories from gadgets whose deformations are controlled by deformations of their associated (one-dimensional) formal group. So they needed enough data to split off a 1-dimensional formal group and have that group completely control the deformations.
Now what I find really crazy is that all this stuff is highly motivated by the concerns of a homotopy theorist… and yet: Right there, in nature, number theorists had been studying such moduli of abelian varieties for many years. These are the PEL shimura varieties, and that’s the only motivation I know of from the point of view of homotopy theory.
The trouble with these shiny new objects is that they are hard to compute with: we struck gold with the moduli of elliptic curves for all sorts of reasons, but probably the biggest is that you can write it down. I don’t know if anyone knows a description of the types of varieties associated to height 3 that we can compute with. We hardly know anything about automorphic forms, from what I can gather, let alone the higher cohomology groups of these stacks… (Also, for some reason algebraic geometers and number theorists never seem to compute these cohomology groups, which just leaves more work for us to do!)
Anyway, I like this Artin-Mazur stuff because somehow these moduli (K3 surfaces, Calabi-Yau 3 folds, …) seem more approachable than the Shimura varieties… we’ll see if that’s true.
Dylan, this is interesting. This deformation issue is exactly the issue I’ve been studying. A “height 1” Calabi-Yau threefold has this height 1 one-dimensional AM formal group attached, but it doesn’t contain enough information to remember all of the deformation theory. This is why I switched to the enlarged version. Under enough hypotheses, the enlarged formal group of the Calabi-Yau is still one-dimensional because the connected component is just the AM formal group. This controls the deformation theory, and there is still this height 1 part that splits off (the enlargement pushes the height up though).
hilbertthm90- who knew our interests would end up intersecting so heavily!
I don’t think I quite understand what the “enlarged formal group” is, but I lied a little bit in my post: we know how to produce spectra by a machine as soon as we have a formally étale map to the moduli of (1-dimensional) p-divisible groups. So if, as you say, this enlarged formal group ends up being a 1-dimensional p-divisible guy, then we’re good. Unless I’m misunderstanding, since you say “splits off” so now I’m afraid you went up a dimension, unless you mean take a connected component.
Out of curiosity, is there a bijection between deformations, or just an injection? That is: is the map from the moduli of height 1 Calabi-Yau threefolds to the moduli of p-divisible groups formally étale?
@Dylan,
thanks for the details. What I was trying to ask concerns right your first sentence:
we just have no way to import algebraic geometry into homotopy theory except through 1-dimensional formal groups- that’s sort of our only in
What I was wondering was: does this mean “we” concentrate on 1d formal groups because that just happens to be the only case “we” know how to handle in the given context (i.e.: “search under the streetlight”) or is there also an intrinsic reason?
I suppose the answer is actually “yes to both”: that the 1d case happens to be tractable is because abstractly it is… (and now fill in the blanks)
But before or while we are further discussing this, here something more boring, but to get the nLab entry straight (this may concern hilbertthm90 more than you, as he and me had been editing at Artin-Mazur formal group):
Charles Rezk was right to complain on MO that the CY-condition wasn’t really motivated (on the nLab/in my post). I went to the entry now and added more explicitly
some sufficient conditions that AM state, now in this proposition;
But as before, please check.
@Urs: In some ways the answer is “yes that’s all we know how to do.” But it’s not just because it’s “easier” it’s because we really have no idea how to do any other case! Complex cobordism is this natural thing and it’s magic that the universal 1-d formal group law describes its theory of Chern classes… We don’t know where to look for a higher dimensional formal group naturally occurring.
You might be able to artificially build some cohomology theories by building a derived enhancement of some algebro-geometric object. Often there’s some obvious way to do this, but the trouble is that this thing you built will probably have nothing to say about manifolds. For example, I’m pretty sure you could define a spectral Deligne-Mumford stack enhancing the moduli of elliptic curves pretty formally, but it would not have anything to do with tmf, or complex cobordism, etc. And good luck trying to compute anything about your answer.
And the other thing is that, by the classification of thick subcategories, and the nilpotence theorem, and the whole chromatic story- it’s hard to see where higher dimensional formal groups would come in! It sort of seems like 1-dimensional ones are telling us everything about stable homotopy theory already…
The only place I’ve heard of higher-dimensional phenomena occurring is this strange observation (maybe due to Ravenel, or Buchstaber-Lazarev) about the Dieudonne module associated to the Morava K-theories of Eilenberg-Maclane spectra, see here: http://chromotopy.org/hypothetical-abelian-varieties
@Dylan, thanks again for your thoughtful comments.
I had a related discussion with Aaron Mazel-Gee recently (in another thread here): while I am most pleased by the story of how “everything” in chromatic homotopy theory drops out canonically from the prime spectrum of the stable homotopy theory, I feel like I am still missing some sub-plots in this story.
Did anyone try to write up this story in as much detail as available? I very much liked how Aaron started out to do so in his seminar talk (pdf) but my impression was that after an initial “everything is god given” eventually the story drifted more into “and now we do this, because we can and it looks good”. I am not complaining, I am just wondering if one can do more.
Here is another question:
the Artin-Mazur formal groups are the deformation theory of ordinary (differential) cohomology. There is an obvious way in which one may ask the representability problem for deformation theory of generalized (differential) cohomology theories, such as for K-theory. Has this been considered? (It’s hard to google for this, because Google will always give me the formal group underlying K-theory and not formal groups (if any) formed by deformations of K-theory classes).
there is some motivation for this from string theory: while Witten argued/explained that the partition function of the M5-brane is computed by the phase space of 7d Chern-Simons theory whose perturbation theory is $\Phi^3_X$, there is next an argument that the RR-field of type II superstring theory is similarly given by an 11d Chern-Simons theory (see there for pointers) whose phase space is something like an “intermediate Jacobian” but not formed from ordinary cohomology, but from K-theory classes.
I have now briefly cross-linked (one way, and the other way) the discussion of the Deligne-cohomology version of Artin-Mazur formal groups with the Deligne-cohomology characterization of intermediate Jacobians.
It follows that the Deligne-version of Artin-Mazur formal groups are the formal theory intermediate Jacobians…
… or it would follow had Artin-Mazur discussed the case of the truncated Deligne complex, too (which I suppose they might just as well have, but didn’t think of. Or did they anywhere?)
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