Chromotopy

Hypothetical abelian varieties from K-theory

I heard an idea tossed around recently that I’d like to share with you all. I worry that it might be a little half-baked, as I’ve only heard about it recently and so maybe haven’t spent enough time sketching out the edges of it. Maybe writing this will help. Throughout, $k$ is a perfect field of positive characteristic $p$.

Where Lie theorists study Lie algebras, formal algebraic geometers study (covariant) Dieudonné modules. The essential observation is that the sorts of formal Lie groups appearing in algebraic topology are commutative and one-dimensional, meaning that their associated Lie algebras are one-dimensional vector spaces with vanishing brackets, and so it is unreasonable to think that it would carry much interesting information about its parent formal Lie group. To correct for this, one takes the collection of all curves on the formal group (i.e., without reducing to their linear equivalence classes, as one does when building the Lie algebra) and remembers enough structure stemming from the group multiplication that this assignment

$$D: \left\{\text{formal groups}\right\} \longrightarrow \left\{\begin{array}{c}\text{collections of}\;p\text{-typical curves} \\ + \\ \text{structure} \end{array}\right\}$$

becomes an equivalence. Such collections of curves form modules, called “Dieudonné modules”, over a certain ground ring, called the Cartier ring, which is built upon the ambient field $k$. The three relevant pieces of structure are the actions of homotheties, of the Frobenius, and of the Verschiebung, which were all described way back in this post on Witt vectors. (In the notation of that post, we’re interested just in $F_p$ and $V_p$.) Altogether, this gives a formula for the Cartier ring:

$$\mathrm{Cart}(k) = \mathbb{W}(k)\langle F,V \rangle / \left(\begin{array}{c}VF = FV = p, \\ Fa = a^\sigma F, \\ aV = Va^\sigma\end{array}\right).$$

Say that a Dieudonné module is formal when it is: finite rank; free as a $\mathbb{W}(k)$-module; reduced, meaning that it’s $V$-adically complete; and uniform, meaning that the natural map $M / VM \to V^k M / V^{k+1} M$ is an isomorphism. Then, the Dieudonné functor on finite height formal groups lands in the subcategory of formal Dieudonné modules, and there it restricts to an equivalence. Actually, more is true: the Dieudonné module of a $p$-divisible group can be made sense of, and Dieudonné modules which are free of finite rank (but without reducedness or uniformity) are equivalent to $p$-divisible groups.

Dieudonné modules are thrilling because they are just modules, whereas $p$-divisible groups are these unwieldy ind-systems of finite group schemes. The relative simplicity of the data of a Dieudonné module allows one to compute basic invariants very quickly:

Theorem: Take $\mathbb{G}$ to be a $p$-divisible group and $M$ its Dieudonné module. There is a natural isomorphism of $k$-vector spaces $T_0 \mathbb{G} \cong M / VM$. In the case that $\mathbb{G}$ is one-dimensional, then the rank of $M$ as a $\mathbb{W}(k)$-module agrees with the height $d$ of $\mathbb{G}$. Moreover, if $\gamma$ is a coordinate on $\mathbb{G}$ (considered as a curve), then $\{\gamma, V\gamma, \ldots, V^{d-1} \gamma\}$ forms a basis for $M$.

There is also something like a classification of the simple Dieudonné modules over $\bar{\mathbb{F}}_p$ (where, among other things, the étale component of a p-divisible group carries little data):

Theorem (Dieudonné): For $m$ and $n$ coprime and positive, set

$$G_{m,n} = \mathrm{Cart}(\bar{\mathbb{F}}_p) / (V^m = F^n).$$

Additionally, allow the pairs $(m, n) = (0, 1)$ to get

$$G_{0,1} = D(\mathbb{G}_m[p^\infty])$$

and $(m, n) = (1, 0)$ to get

$$G_{1,0} = D(\mathbb{Q} / \mathbb{Z}).$$

(All these modules $G_{m,n}$ have formal dimension $n$ and height $(m+n)$.) For any simple Dieudonné module $M$, there is an isogeny $M \to G_{m,n}$, i.e., a map with finite kernel and cokernel. Moreover, up to isogeny every Dieudonné module is the direct sum of simple objects.

Every abelian variety $A$ comes with a p-divisible group $A[p^\infty]$ arising from its system of $p$-power order torsion points. This connection is remarkably strong; for instance, there is the following theorem:

Theorem (Serre–Tate): Over a $p$-adic base, the infinitesimal deformation theories of an abelian variety $A$ and its $p$-divisible group $A[p^\infty]$ agree naturally.[1]

For this reason and others, the $p$-divisible group of an abelian variety carries fairly strong content about the parent variety. On the other hand, it is not immediately clear which $p$-divisible groups arise in this way. Toward this end, there is a symmetry condition:

Lemma (“Riemann–Manin symmetry condition”): As every abelian variety is isogenous to its Poincaré dual and the corresponding (Cartier) duality on $p$-divisible groups sends ($V$ to $F$ and hence) $G_{m,n}$ to $G_{n,m}$, these summands must appear in pairs in the isogeny type of the Dieudonné module of $A[p^\infty]$.

As a simple example, this gives the usual categorization of elliptic curves: an elliptic curve is $1$-dimensional, hence has $p$-divisible group of height $2$. One possibility, called a supersingular curve, is for the Dieudonné module to be isogenous to $G_{1,1}$; this is a $1$-dimensional formal group of height $2$ and it satisfies the symmetry condition. The only other possibility, called an ordinary curve, is for the Dieudonné module to be isogenous to $G_{1,0} \oplus G_{0,1}$; this is the sum of a $1$-dimensional formal group of height $1$ with an étale component of height $1$, and it too satisfies the symmetry condition.

A remarkable theorem is that the converse of the symmetry lemma hnews as well:

Theorem (Serre, Oort; conjectured by Manin): If a Dieudonné module $M$ satisfies the above symmetry condition, then there exists an abelian variety whose $p$-divisible group is isogenous to $M$.

Both proofs of this theorem are very constructive. Serre’s proof explicitly names abelian hypersurfaces whose $p$-divisible groups are of the form $G_{m,n} \oplus G_{n,m}$, for instance.

Now, finally, some input from algebraic topology. The Morava $K$-theories of Eilenberg–Mac Lane spaces give a collection of formal groups which can be interpreted in the following way:

Theorem (Ravenel–Wilson): For $1 < m \le n$, the $p$-divisible group associated to $K(n)^* K(\mathbb{Q}/\mathbb{Z}, m)$ is (in a suitable sense) the $m$th exterior power of the $p$-divisible group $K(n)^* K(\mathbb{Q}/\mathbb{Z}, 1)$. It is smooth, has formal dimension $\binom{n-1}{m-1}$, and has height $\binom{n}{m}$. (Additionally, it is zero for $m > n$.)

Theorem (Buchstaber–Lazarev): For $1 < m \le n$, the same $p$-divisible group has Dieudonné module isogenous to the product of $\frac{1}{n_0} \cdot \binom{n}{m}$ copies of $G_{n_0-m_0,m_0}$, where $m_0/n_0$ is the reduced fraction of $m/n$.

The conclusion of Buchstaber and Lazarev is that this means that these $p$-divisible groups almost never have realizations as abelian varieties, since they mostly don’t satisfy the symmetry condition. The only time that they do is something of an accident: when $n$ is even and $m = n/2$, then the corresponding Dieudonné module is isogenous to that of a large product of copies of a supersingular elliptic curve. However, Ravenel observed that Pascal’s triangle is symmetric:

Observation (I heard this from Ravenel, but surely Buchstaber and Lazarev knew of it): The sum of all of the Dieudonné modules

$$\bigoplus_{m=0}^{\text{$n$ (or $\infty$)}} D(\operatorname{Spf} K(n)^* K(\mathbb{Q}/\mathbb{Z}, m))$$

satisfies the Riemann-Manin symmetry condition.

This is an interesting observation. In light of the comments at the end of the Buchstaber–Lazarev paper, one wonders: why privilege $m = n/2$? But, even more honestly, why privilege $m = 1$ in our study of chromatic homotopy theory? A recurring obstacle to our understanding of higher-height cohomology theories has been the disconnection from the picture of globally defined abelian varieties. Could there be a naturally occurring abelian variety whose $p$-divisible group realizes the large ($2^{n-1}$-dimensional!) formal group associated to the $K$-theoretic Hopf ring of Eilenberg–Mac Lane spaces?

The explicit nature of the solutions to Manin’s conjecture show us that, yes, it is certainly possible to write down large products of hypersurfaces to give a positive answer to this question with the words “naturally occurring” deleted. This alone isn’t very helpful, however, and so to pin down what we might be even talking about there are a number of smaller observations that might help:

• If such an abelian variety existed, there would be a strange filtration imposed on its $p$-divisible group arising from the degrees of the Eilenberg–Mac Lane spaces. The Riemann–Manin symmetry condition tells us that these $G_{m,n}$ and $G_{n,m}$ factors must come in pairs, but in almost all situations, one pair appears on one side of the middle-dimension $n/2$ and the other appears on the other side. What could this mean in terms of the hypothetical abelian variety?
• Relatedly, this pairing arises from the “$\circ$-product” structure on the level of Hopf algebras, or as a sort of Hodge-star operation on the level of Dieudonné modules. What sort of structure on the abelian variety would induce such an operation, and in such an orderly fashion?
• There should be accessible examples of these hypothetical varieties at low heights. For instance, the one associated to height $1$ via mod-$p$ complex $K$-theory is (canonically, not merely isogenous to) a sum $G_{1,0} \oplus G_{0,1}$ — i.e., it comes from an ordinary elliptic curve. Can we identify which elliptic curve — and in what sense we can even ask this question? Is there a naturally occurring map from the forms of $\mathbb{G}_m$ to the ordinary locus on the (noncompactified?) moduli of elliptic curves? What about forms of $\hat{\mathbb{G}}_m$? What if we select a supersingular elliptic curve instead — is there an instructive assignment to abelian varieties whose Dieudonné module has isogeny type $G_{1,0} \oplus G_{1,1} \oplus G_{0,1}$? (On the face of it, this last bit doesn’t look so helpful, but maybe it is.)

[1] - Presumably this can be expressed by saying that the map from a moduli of abelian varieties to a moduli of $p$-divisible groups is formally étale, but no one seems to say this, so maybe I’m missing something.

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I’ve been making an effort to learn some arithmetic geometry recently. I started with local class field theory, which was mind-blowing. When I was a first year, someone sat me down and instructed me that I must take a course in complex geometry to become competent — and they were right, and it was a wonderful course, and I’m really glad I got that advice. I have no idea how (especially as I’ve been bumbling about with formal groups for so long!) it slipped past me that local class field theory is another one of these core competencies, and really one of the great achievements of twentieth century mathematics.

I gave a kind of measly talk about this a month ago, when I was trying to stir up interest in a reading group. The notes are a little batty, but they’re fun enough, and you can find them here.

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Speaking of mind-blowing things, last week there was a week-long workshop on perfectoid spaces at MSRI, which I attended between one-third and one-half of. There are video lectures available on the MSRI website; at the very least everyone should watch Scholze’s introduction, just to get a sense of what all the fuss is about, and then ideally both of Weinstein’s lectures, which were excellent and very much adjacent to the subject of this post — this blog, really.

And, as an uninspired parting remark, we topologists do have access to the pro-system

$$\left( \cdots \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty \xrightarrow{p} \mathbb{C}\mathrm{P}^\infty\right) \ldots$$

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