# Serre-Tate Theory 2

I guess this will be the last post on this topic. I’ll explain a tiny bit about what goes into the proof of this theorem and then why anyone would care that such canonical lifts exist. On the first point, there are tons of details that go into the proof. For example, Nick Katz’s article, Serre-Tate Local Moduli, is 65 pages. It is quite good if you want to learn more about this. Also, Messing’s book The Crystals Associated to Barsotti-Tate Groups is essentially building the machinery for this proof which is then knocked off in an appendix. So this isn’t quick or easy by any means.

On the other hand, I think the idea of the proof is fairly straightforward. Let’s briefly recall last time. The situation is that we have an ordinary elliptic curve ${E_0/k}$ over an algebraically closed field of characteristic ${p>2}$. We want to understand ${Def_{E_0}}$, but in particular whether or not there is some distinguished lift to characteristic ${0}$ (this will be an element of ${Def_{E_0}(W(k))}$.

To make the problem more manageable we consider the ${p}$-divisible group ${E_0[p^\infty]}$ attached to ${E_0}$. In the ordinary case this is the enlarged formal Picard group. It is of height ${2}$ whose connected component is ${\widehat{Pic}_{E_0}\simeq\mu_{p^\infty}}$. There is a natural map ${Def_{E_0}\rightarrow Def_{E_0[p^\infty]}}$ just by mapping ${E/R \mapsto E[p^\infty]}$. Last time we said the main theorem was that this map is an isomorphism. To tie this back to the flat topology stuff, ${E_0[p^\infty]}$ is the group representing the functor ${A\mapsto H^1_{fl}(E_0\otimes A, \mu_{p^\infty})}$.

The first step in proving the main theorem is to note two things. In the (split) connected-etale sequence

$\displaystyle 0\rightarrow \mu_{p^\infty}\rightarrow E_0[p^\infty]\rightarrow \mathbb{Q}_p/\mathbb{Z}_p\rightarrow 0$

we have that ${\mu_{p^\infty}}$ is height one and hence rigid. We have that ${\mathbb{Q}_p/\mathbb{Z}_p}$ is etale and hence rigid. Thus given any deformation ${G/R}$ of ${E_0[p^\infty]}$ we can take the connected-etale sequence of this and see that ${G^0}$ is the unique deformation of ${\mu_{p^\infty}}$ over ${R}$ and ${G^{et}=\mathbb{Q}_p/\mathbb{Z}_p}$. Thus the deformation functor can be redescribed in terms of extension classes of two rigid groups ${R\mapsto Ext_R^1(\mathbb{Q}_p/\mathbb{Z}_p, \mu_{p^\infty})}$.

Now we see what the canonical lift is. Supposing our isomorphism of deformation functors, it is the lift that corresponds to the split and hence trivial extension class. So how do we actually check that this is an isomorphism? Like I said, it is kind of long and tedious. Roughly speaking you note that both deformation functors are prorepresentable by formally smooth objects of the same dimension. So we need to check that the differential is an isomorphism on tangent spaces.

Here’s where some cleverness happens. You rewrite the differential as a composition of a whole bunch of maps that you know are isomorphisms. In particular, it is the following string of maps: The Kodaira-Spencer map ${T\stackrel{\sim}{\rightarrow} H^1(E_0, \mathcal{T})}$ followed by Serre duality (recall the canonical is trivial on an elliptic curve) ${H^1(E_0, \mathcal{T})\stackrel{\sim}{\rightarrow} Hom_k(H^1(E_0, \Omega^1), H^1(E_0, \mathcal{O}_{E_0}))}$. The hardest one was briefly mentioned a few posts ago and is the dlog map which gives an isomorphism ${H^2_{fl}(E_0, \mu_{p^\infty})\stackrel{\sim}{\rightarrow} H^1(E_0, \Omega^1)}$.

Now noting that ${H^2_{fl}(E_0, \mu_{p^\infty})=\mathbb{Q}_p/\mathbb{Z}_p}$ and that ${T_0\mu_{p^\infty}\simeq H^1(E_0, \mathcal{O}_{E_0})}$ gives us enough compositions and isomorphisms that we get from the tangent space of the versal deformation of ${E_0}$ to the tangent space of the versal deformation of ${E_0[p^\infty]}$. As you might guess, it is a pain to actually check that this is the differential of the natural map (and in fact involves further decomposing those maps into yet other ones). It turns out to be the case and hence ${Def_{E_0}\rightarrow Def_{E_0[p^\infty]}}$ is an isomorphism and the canonical lift corresponds to the trivial extension.

But why should we care? It turns out the geometry of the canonical lift is very special. This may not be that impressive for elliptic curves, but this theory all goes through for any ordinary abelian variety or K3 surface where it is much more interesting. It turns out that you can choose a nice set of coordinates (“canonical coordinates”) on the base of the versal deformation and a basis of the de Rham cohomology of the family that is adapted to the Hodge filtration such that in these coordinates the Gauss-Manin connection has an explicit and nice form.

Also, the canonical lift admits a lift of the Frobenius which is also nice and compatible with how it acts on the above chosen basis on the de Rham cohomology. These coordinates are what give the base of the versal deformation the structure of a formal torus (product of ${\widehat{\mathbb{G}_m}}$‘s). One can then exploit all this nice structure to prove large open problems like the Tate conjecture in the special cases of the class of varieties that have these canonical lifts.

# Heights of Varieties

Now that we’ve defined the height of a ${p}$-divisible group we’ll define the height of a variety in positive characteristic. There are a few ways we can motivate this definition, but really it just works and turns out to be a very useful concept. We’ll mostly follow the paper of Artin and Mazur.

We could do this in more generality, but to keep things as simple as possible we’ll assume that we have a proper variety ${X}$ over a perfect field ${k}$ of characteristic ${p}$. The first motivation is that we can think about ${\mathrm{Pic}(X)}$. One way to get information about this group is to use deformation theory and look at the formal completion ${\widehat{\mathrm{Pic}}(X)}$.

The way to define this is to define the ${S}$-valued points (${S}$ an Artin local ${k}$-algebra with residue field ${k}$) to be the group fitting into the sequence ${0\rightarrow \widehat{\mathrm{Pic}}(X)(S)\rightarrow H^1(X\times S, \mathbb{G}_m)\rightarrow H^1(X, \mathbb{G}_m)}$.

So ${\widehat{\mathrm{Pic}}(X)}$ is a functor which by Schlessinger’s criterion is prorepresentable by a formal group over ${k}$. Notice that ${\widehat{\mathrm{Pic}}(X)(S)=\mathrm{ker}(\mathrm{Pic}(X\times S)\rightarrow \mathrm{Pic}(X))}$, so there is a pretty concrete way to think about what is going on. We take our scheme and consider some nilpotent thickening. The line bundles on this thickening that are just extensions from the trivial line bundle are what is in this formal Picard group.

There is no reason to stop with just ${H^1}$. We could define ${\Phi^r: Art_k\rightarrow Ab}$ by ${\Phi^r(S)}$ is the kernel of the restriction map ${H^r(X\times S, \mathbb{G}_m)\rightarrow H^r(X, \mathbb{G}_m)}$. In the cases we care about, modulo some technical details, we can apply Schlessinger type arguments to this to get that if the dimension of ${X}$ is ${n}$, then ${\Phi^n}$ is not only pro-representable, but by formal Lie group of dimension ${1}$. We’ll call this ${\Phi_X}$.

When ${n=2}$ this is just the well-known Brauer group, and so for instance the height of a K3 surface is the height of the Brauer group. We also have that if ${\Phi_X}$ is not ${\widehat{\mathbb{G}}_a}$ then it is a ${p}$-divisible group and amazingly the Dieudonne module of ${\Phi_X}$ is related to the Witt sheaf cohomology via ${D(\Phi_X)=H^n(X, \mathcal{W})}$. Recall that ${D(\Phi_X)}$ is a free ${W(k)}$-module of rank the height of ${\Phi_X}$, so in particular ${H^n(X, \mathcal{W})}$ is a finite ${W(k)}$-module!

Remember that we computed an example where that wasn’t finitely generated. So non-finite generatedness of ${H^n(X, \mathcal{W})}$ actually is related to the height in that if the variety is of finite height then ${H^n(X, \mathcal{W})}$ is finitely generated. Since we call a variety of infinite height supersingular, we can rephrase this as saying that ${H^n(X, \mathcal{W})}$ is not finitely generated if and only if ${X}$ is supersingular.

Just as an example of what heights can be, an elliptic curve must have height ${1}$ or ${2}$ and a K3 surface can have height between ${1}$ and ${10}$ (inclusive). As of right now it seems that the higher dimensional analogue of if the finite height range of a Calabi-Yau threefold is bounded is still open. People have proved certain bounds in terms of hodge numbers. For instance ${h(\Phi_X)\leq h^{1, 2}+1}$. For a general CY ${n}$-fold we have ${h\leq h^{1, n-1}+1}$.

This is pretty fascinating because my interpretation of this (which could be completely wrong) is that since for K3 surfaces the moduli space is ${20}$ dimensional, we get that (for non-supersingular) ${h^{1,1}=20}$ since this is just the dimension of the tangent space of the deformations, which for a smooth moduli should match the dimension of the moduli space. Thus we get a uniform bound (not the one I mentioned earlier).

But for CY threefolds the moduli space is much less uniform. They aren’t all deformation equivalent. They lie on different components that have different dimensions (this is a guess, I haven’t actually seen this written anywhere). So this doesn’t allow us to say ${h^{1,2}}$ is some number. It depends on the dimension of the component of the moduli that it is on (since ${h^{1,2}=\dim H^2(X, \Omega)=\dim H^1(X, \mathcal{T})}$ using the CY conditions and Serre duality). So I think it is still an open problem for how big that can be. If it can get unreasonably large, then maybe we can arbitrarily large heights of CY threefolds.

Next time maybe we’ll prove some equivalent ways of computing heights for CY varieties and talk about how height has been used by Van der Geer and Katsura and others in a useful way for K3 surfaces.

# Heights of p-divisible Groups

Let’s try to define a few words I’ve thrown around for a few weeks. What is a ${p}$-divisible group, and how do we know what its height is? I’m going to do two things that will either make this easier to understand or way more confusing. We will be working with group schemes. To keep from repeating everything twice with the word “formal” in front of everything I will rarely specify whether I mean formal group scheme or group scheme. Obviously some things are different in the formal case, but not so much. The other thing I’ll do is assume our group schemes are affine to simplify notation, but if you want to do this more generally you can.

The point of these posts should be to give an overview of how these things fit together. When trying to learn about this stuff, there are so many hundreds of terms and papers and details in the papers that it is really easy to forget what is going on. For instance, basically any reference on ${p}$-divisible groups will get very caught up in all the technical details of Dieudonne theory, and I want to massively downplay this aspect for the purpose of defining this invariant called the height of a variety.

On to the definition. A ${p}$-divisible group is just an inductive system ${(G_\nu, i_\nu)}$ of group schemes satisfying two properties. First there must be an ${h}$ so that the order of ${G_\nu}$ is ${p^{\nu h}}$. Second, they must fit into an exact sequence ${0\rightarrow G_\nu \stackrel{i_\nu}{\rightarrow} G_{\nu +1 } \stackrel{ p^\nu}{\rightarrow} G_{\nu +1}}$. All this says is that if we look at the map that is multiplication by ${p^{\nu}}$ on ${G_{\nu +1}}$, the kernel of this is the copy of ${G_{\nu}}$ that sits inside ${G_{\nu +1}}$ via ${i_\nu}$.

This might seem like a strange set of conditions at first, but really the two most natural examples of forming inductive systems of group schemes already satisfy both of these. The first one is to take an abelian variety ${X}$ of dimension ${g}$. Then we have the isogenies multiplication by ${p}$, multiplication by ${p^2}$, etc. The kernels of these are all group schemes and it is well-known that they are isomorphic to ${(\mathbb{Z}/p)^{2g}}$, ${(\mathbb{Z}/p^2)^{2g}}$, etc. We just take the maps to be the inclusions and the ${h=2g}$.

The other main example is to do the same trick with ${\mathbb{G}_m}$ by taking successive kernels of multiplication by ${p}$. We just get the inductive system ${(\mu_{p^\nu}, i_{\nu})}$. In this case the orders are just ${p^\nu}$, so ${h=1}$. We’ve suggestively labelled this number ${h}$ which is the height of the ${p}$-divisible group. This one is usually denoted ${\mathbb{G}_m(p)}$.

There are lots of easy properties of ${p}$-divisible groups that can be verified mentally. For instance, you can put any two of ${G_\nu}$ and ${G_\alpha}$ into an exact sequence ${0\rightarrow G_\nu \rightarrow G_{\nu + \alpha} \rightarrow G_\alpha \rightarrow 0}$. A slightly harder property to check is that under mild base assumptions we have an equivalence of categories between ${p}$-divisible groups and divisible formal Lie groups. Under this equivalence we get that ${\mathbb{G}_m(p)}$ corresponds to the one-dimensional Lie group with group law ${F(X, Y)=X + Y + XY}$, so our earlier notation of calling this ${\widehat{\mathbb{G}_m}}$ makes sense because it comes from ${\mathbb{G}_m(p)}$.

Since ultimately we are concerned with computing heights, we should see if there is a way to figure out the height without computing orders. Let’s denote the Cartier dual of a group scheme by ${G^D=Hom(G, \mathbb{G}_m)}$. We can check that taking Cartier duals everywhere, we get another ${p}$-divisible group. I.e. ${(G_\nu^D, i_\nu^D)}$ is an inductive system satisfying the properties of a ${p}$-divisible group. This is often called the Serre dual. If we denote the whole inductive system by ${G}$, then it is customary to write ${G^t}$ for the Serre dual.

Note that duals can be quite different from the original group. In particular, the dimension can be different. In the case of ${\mathbb{G}_m(p)}$ we get a dimension ${1}$ etale group scheme, but it’s dual is ${(\mathbb{Z}/p^\nu)}$ and hence a dimension ${0}$ connected (with nilpotent structure) group scheme. If we add these two dimensions we get ${1}$, the height. This is true in general. We have the formula ${h=\textrm{dim}(G)+\textrm{dim}(G^t)}$. So we only need to know the dimensions of the group and its dual.

Another (much harder for calculating, but sometimes handy in theory) way to determine the height is to use the Dieudonne module we defined last time. If we take ${D(G)}$ it is a ${\mathbb{D}_k}$-module, but also a free of finite rank ${W(k)}$-module. The rank of this module turns out to be the height of ${G}$. In a similar fashion, you can form another module out of ${G}$ called the Tate module. By definition, multiplication by ${p}$ is a map ${G_{\nu +1}\rightarrow G_\nu}$, and hence we get an inverse system. We define ${T(G)=\lim G_{\nu}(\overline{k})}$ to be the Tate module. It is a free ${\mathbb{Z}_p}$-module. The rank of this is the height of ${G}$.

That is about the sketchiest crash course on ${p}$-divisible groups you can get, but I think it mentions enough to get to the next definition: the height of a variety in positive characteristic.