# Newton Polygons of p-Divisible Groups

I really wanted to move on from this topic, because the theory gets much more interesting when we move to ${p}$-divisible groups over some larger rings than just algebraically closed fields. Unfortunately, while looking over how Demazure builds the theory in Lectures on ${p}$-divisible Groups, I realized that it would be a crime to bring you this far and not concretely show you the power of thinking in terms of Newton polygons.

As usual, let’s fix an algebraically closed field of positive characteristic to work over. I was vague last time about the anti-equivalence of categories between ${p}$-divisible groups and ${F}$-crystals mostly because I was just going off of memory. When I looked it up, I found out I was slightly wrong. Let’s compute some examples of some slopes.

Recall that ${D(\mu_{p^\infty})\simeq W(k)}$ and ${F=p\sigma}$. In particular, ${F(1)=p\cdot 1}$, so in our ${F}$-crystal theory we get that the normalized ${p}$-adic valuation of the eigenvalue ${p}$ of ${F}$ is ${1}$. Recall that we called this the slope (it will become clear why in a moment).

Our other main example was ${D(\mathbb{Q}_p/\mathbb{Z}_p)\simeq W(k)}$ with ${F=\sigma}$. In this case we have ${1}$ is “the” eigenvalue which has ${p}$-adic valuation ${0}$. These slopes totally determine the ${F}$-crystal up to isomorphism, and the category of ${F}$-crystals (with slopes in the range ${0}$ to ${1}$) is anti-equivalent to the category of ${p}$-divisible groups.

The Dieudonné-Manin decomposition says that we can always decompose ${H=D(G)\otimes_W K}$ as a direct sum of vector spaces indexed by these slopes. For example, if I had a height three ${p}$-divisible group, ${H}$ would be three dimensional. If it decomposed as ${H_0\oplus H_1}$ where ${H_0}$ was ${2}$-dimensional (there is a repeated ${F}$-eigenvalue of slope ${0}$), then ${H_1}$ would be ${1}$-dimensional, and I could just read off that my ${p}$-divisible group must be isogenous to ${G\simeq \mu_{p^\infty}\oplus (\mathbb{Q}_p/\mathbb{Z}_p)^2}$.

In general, since we have a decomposition ${H=H_0\oplus H' \oplus H_1}$ where ${H'}$ is the part with slopes strictly in ${(0,1)}$ we get a decomposition ${G\simeq (\mu_{p^\infty})^{r_1}\oplus G' \oplus (\mathbb{Q}_p/\mathbb{Z}_p)^{r_0}}$ where ${r_j}$ is the dimension of ${H_j}$ and ${G'}$ does not have any factors of those forms.

This is where the Newton polygon comes in. We can visually arrange this information as follows. Put the slopes of ${F}$ in increasing order ${\lambda_1, \ldots, \lambda_r}$. Make a polygon in the first quadrant by plotting the points ${P_0=(0,0)}$, ${P_1=(\dim H_{\lambda_1}, \lambda_1 \dim H_{\lambda_1})}$, … , ${\displaystyle P_j=\left(\sum_{l=1}^j\dim H_{\lambda_l}, \sum_{l=1}^j \lambda_l\dim H_{\lambda_l}\right)}$.

This might look confusing, but all it says is to get from ${P_{j}}$ to ${P_{j+1}}$ make a line segment of slope ${\lambda_j}$ and make the segment go to the right for ${\dim H_{\lambda_j}}$. This way you visually encode the slope with the actual slope of the segment, and the longer the segment is the bigger the multiplicity of that eigenvalue.

But this way of encoding the information gives us something even better, because it turns out that all these ${P_i}$ must have integer coordinates (a highly non-obvious fact proved in the book by Demazure listed above). This greatly restricts our possibilities for Dieudonné ${F}$-crystals. Consider the height ${2}$ case. We have ${H}$ is two dimensional, so we have ${2}$ slopes (possibly the same). The maximal ${y}$ coordinate you could ever reach is if both slopes were maximal which is ${1}$. In that case you just get the line segment from ${(0,0)}$ to ${(2,2)}$. The lowest you could get is if the slopes were both ${0}$ in which case you get a line segment ${(0,0)}$ to ${(2,0)}$.

Every other possibility must be a polygon between these two with integer breaking points and increasing order of slopes. Draw it (or if you want to cheat look below). You will see that there are obviously only two other possibilities. The one that goes ${(0,0)}$ to ${(1,0)}$ to ${(2,1)}$ which is a slope ${0}$ and slope ${1}$ and corresponds to ${\mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$ and the one that goes ${(0,0)}$ to ${(2,1)}$. This corresponds to a slope ${1/2}$ with multiplicity ${2}$. This corresponds to the ${E[p^\infty]}$ for supersingular elliptic curves. That recovers our list from last time.

We now just have a bit of a game to determine all height ${3}$ ${p}$-divisible groups up to isogeny (and it turns out in this small height case that determines them up to isomorphism). You can just draw all the possibilities for Newton polygons as in the height ${2}$ case to see that the only ${6}$ possibilities are ${(\mu_{p^\infty})^3}$, ${(\mu_{p^\infty})^2\oplus \mathbb{Q}_p/\mathbb{Z}_p}$, ${\mu_{p^\infty}\oplus (\mathbb{Q}_p/\mathbb{Z}_p)^2}$, ${(\mathbb{Q}_p/\mathbb{Z}_p)^3}$, and then two others: ${G_{1/3}}$ which corresponds to the thing with a triple eigenvalue of slope ${1/3}$ and ${G_{2/3}}$ which corresponds to the thing with a triple eigenvalue of slope ${2/3}$.

To finish this post (and hopefully topic!) let’s bring this back to elliptic curves one more time. It turns out that ${D(E[p^\infty])\simeq H^1_{crys}(E/W)}$. Without reminding you of the technical mumbo-jumbo of crystalline cohomology, let’s think why this might be reasonable. We know ${E[p^\infty]}$ is always height ${2}$, so ${D(E[p^\infty])}$ is rank ${2}$. But if we consider that crystalline cohomology should be some sort of ${p}$-adic cohomology theory that “remembers topological information” (whatever that means), then we would guess that some topological ${H^1}$ of a “torus” should be rank ${2}$ as well.

Moreover, the crystalline cohomology comes with a natural Frobenius action. But if we believe there is some sort of Weil conjecture magic that also applies to crystalline cohomology (I mean, it is a Weil cohomology theory), then we would have to believe that the product of the eigenvalues of this Frobenius equals ${p}$. Recall in the “classical case” that the characteristic polynomial has the form ${x^2-a_px+p}$. So there are actually only two possibilities in this case, both slope ${1/2}$ or one of slope ${1}$ and the other of slope ${0}$. As we’ve noted, these are the two that occur.

In fact, this is a more general phenomenon. When thinking about ${p}$-divisible groups arising from algebraic varieties, because of these Weil conjecture type considerations, the Newton polygons must actually fit into much narrower regions and sometimes this totally forces the whole thing. For example, the enlarged formal Brauer group of an ordinary K3 surface has height ${22}$, but the whole Newton polygon is fully determined by having to fit into a certain region and knowing its connected component.

# More Classification of p-Divisible Groups

Today we’ll look a little more closely at ${A[p^\infty]}$ for abelian varieties and finish up a different sort of classification that I’ve found more useful than the one presented earlier as triples ${(M,F,V)}$. For safety we’ll assume ${k}$ is algebraically closed of characteristic ${p>0}$ for the remainder of this post.

First, let’s note that we can explicitly describe all ${p}$-divisible groups over ${k}$ up to isomorphism (of any dimension!) up to height ${2}$ now. This is basically because height puts a pretty tight constraint on dimension: ${ht(G)=\dim(G)+\dim(G^D)}$. If we want to make this convention, we’ll say ${ht(G)=0}$ if and only if ${G=0}$, but I’m not sure it is useful anywhere.

For ${ht(G)=1}$ we have two cases: If ${\dim(G)=0}$, then it’s dual must be the unique connected ${p}$-divisible group of height ${1}$, namely ${\mu_{p^\infty}}$ and hence ${G=\mathbb{Q}_p/\mathbb{Z}_p}$. The other case we just said was ${\mu_{p^\infty}}$.

For ${ht(G)=2}$ we finally get something a little more interesting, but not too much more. From the height ${1}$ case we know that we can make three such examples: ${(\mu_{p^\infty})^{\oplus 2}}$, ${\mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$, and ${(\mathbb{Q}_p/\mathbb{Z}_p)^{\oplus 2}}$. These are dimensions ${2}$, ${1}$, and ${0}$ respectively. The first and last are dual to each other and the middle one is self-dual. Last time we said there was at least one more: ${E[p^\infty]}$ for a supersingular elliptic curve. This was self-dual as well and the unique one-dimensional connected height ${2}$ ${p}$-divisible group. Now just playing around with the connected-étale decomposition, duals, and numerical constraints we get that this is the full list!

If we could get a bit better feel for the weird supersingular ${E[p^\infty]}$ case, then we would have a really good understanding of all ${p}$-divisible groups up through height ${2}$ (at least over algebraically closed fields).

There is an invariant called the ${a}$-number for abelian varieties defined by ${a(A)=\dim Hom(\alpha_p, A[p])}$. This essentially counts the number of copies of ${\alpha_p}$ sitting inside the truncated ${p}$-divisible group. Let’s consider the elliptic curve case again. If ${E/k}$ is ordinary, then we know ${E[p]}$ explicitly and hence can argue that ${a(E)=0}$. For the supersingular case we have that ${E[p]}$ is actually a non-split semi-direct product of ${\alpha_p}$ by itself and we get that ${a(E)=1}$. This shows that the ${a}$-number is an invariant that is equivalent to knowing ordinary/supersingular.

This is a phenomenon that generalizes. For an abelian variety ${A/k}$ we get that ${A}$ is ordinary if and only if ${a(A)=0}$ in which case the ${p}$-divisible group is a bunch of copies of ${E[p^\infty]}$ for an ordinary elliptic curve, i.e. ${A[p^\infty]\simeq E[p^\infty]^g}$. On the other hand, ${A}$ is supersingular if and only if ${A[p^\infty]\simeq E[p^\infty]^g}$ for ${E/k}$ supersingular (these two facts are pretty easy if you use the ${p}$-rank as the definition of ordinary and supersingular because it tells you the étale part and you mess around with duals and numerics again).

Now that we’ve beaten that dead horse beyond recognition, I’ll point out one more type of classification which is the one that comes up most often for me. In general, there is not redundant information in the triple ${(M, F, V)}$, but for special classes of ${p}$-divisible groups (for example the ones I always work with explained here) all you need to remember is the ${(M, F)}$ to recover ${G}$ up to isomorphism.

A pair ${(M,F)}$ of a free, finite rank ${W}$-module equipped with a ${\phi}$-linear endomorphism ${F}$ is sometimes called a Cartier module or ${F}$-crystal. Every Dieudonné module of a ${p}$-divisible group is an example of one of these. We could also consider ${H=M\otimes_W K}$ where ${K=Frac(W)}$ to get a finite dimensional vector space in characteristic ${0}$ with a ${\phi}$-linear endomorphism preserving the ${W}$-lattice ${M\subset H}$.

Passing to this vector space we would expect to lose some information and this is usually called the associated ${F}$-isocrystal. But doing this gives us a beautiful classification theorem which was originally proved by Diedonné and Manin. We have that ${H}$ is naturally an ${A}$-module where ${A=K[T]}$ is the noncommutative polynomial ring ${T\cdot a=\phi(a)\cdot T}$. The classification is to break up ${H\simeq \oplus H_\alpha}$ into a slope decomposition.

These ${\alpha}$ are just rational numbers corresponding to the slopes of the ${F}$ operator. The eigenvalues ${\lambda_1, \ldots, \lambda_n}$ of ${F}$ are not necessarily well-defined, but if we pick the normalized valuation ${ord(p)=1}$, then the valuations of the eigenvalues are well-defined. Knowing the slopes and multiplicities completely determines ${H}$ up to isomorphism, so we can completely capture the information of ${H}$ in a simple Newton polygon. Note that when ${H}$ is the ${F}$-isocrystal of some Dieudonné module, then the relation ${FV=VF=p}$ forces all slopes to be between 0 and 1.

Unfortunately, knowing ${H}$ up to isomorphism only determines ${M}$ up to equivalence. This equivalence is easily seen to be the same as an injective map ${M\rightarrow M'}$ whose cokernel is a torsion ${W}$-module (that way it becomes an isomorphism when tensoring with ${K}$). But then by the anti-equivalence of categories two ${p}$-divisible groups (in the special subcategory that allows us to drop the ${V}$) ${G}$ and ${G'}$ have equivalent Dieudonné modules if and only if there is a surjective map ${G' \rightarrow G}$ whose kernel is finite, i.e. ${G}$ and ${G'}$ are isogenous as ${p}$-divisible groups.

Despite the annoying subtlety in fully determining ${G}$ up to isomorphism, this is still really good. It says that just knowing the valuation of some eigenvalues of an operator on a finite dimensional characteristic ${0}$ vector space allows us to recover ${G}$ up to isogeny.

# A Quick User’s Guide to Dieudonné Modules of p-Divisible Groups

Last time we saw that if we consider a ${p}$-divisible group ${G}$ over a perfect field of characteristic ${p>0}$, that there wasn’t a whole lot of information that went into determining it up to isomorphism. Today we’ll make this precise. It turns out that up to isomorphism we can translate ${G}$ into a small amount of (semi-)linear algebra.

I’ve actually discussed this before here. But let’s not get bogged down in the details of the construction. The important thing is to see how to use this information to milk out some interesting theorems fairly effortlessly. Let’s recall a few things. The category of ${p}$-divisible groups is (anti-)equivalent to the category of Dieudonné modules. We’ll denote this functor ${G\mapsto D(G)}$.

Let ${W:=W(k)}$ be the ring of Witt vectors of ${k}$ and ${\sigma}$ be the natural Frobenius map on ${W}$. There are only a few important things that come out of the construction from which you can derive tons of facts. First, the data of a Dieudonné module is a free ${W}$-module, ${M}$, of finite rank with a Frobenius ${F: M\rightarrow M}$ which is ${\sigma}$-linear and a Verschiebung ${V: M\rightarrow M}$ which is ${\sigma^{-1}}$-linear satisfying ${FV=VF=p}$.

Fact 1: The rank of ${D(G)}$ is the height of ${G}$.

Fact 2: The dimension of ${G}$ is the dimension of ${D(G)/FD(G)}$ as a ${k}$-vector space (dually, the dimension of ${D(G)/VD(G)}$ is the dimension of ${G^D}$).

Fact 3: ${G}$ is connected if and only if ${F}$ is topologically nilpotent (i.e. ${F^nD(G)\subset pD(G)}$ for ${n>>0}$). Dually, ${G^D}$ is connected if and only if ${V}$ is topologically nilpotent.

Fact 4: ${G}$ is étale if and only if ${F}$ is bijective. Dually, ${G^D}$ is étale if and only if ${V}$ is bijective.

These facts alone allow us to really get our hands dirty with what these things look like and how to get facts back about ${G}$ using linear algebra. Let’s compute the Dieudonné modules of the two “standard” ${p}$-divisible groups: ${\mu_{p^\infty}}$ and ${\mathbb{Q}_p/\mathbb{Z}_p}$ over ${k=\mathbb{F}_p}$ (recall in this situation that ${W(k)=\mathbb{Z}_p}$).

Before starting, we know that the standard Frobenius ${F(a_0, a_1, \ldots, )=(a_0^p, a_1^p, \ldots)}$ and Verschiebung ${V(a_0, a_1, \ldots, )=(0, a_0, a_1, \ldots )}$ satisfy the relations to make a Dieudonné module (the relations are a little tricky to check because constant multiples ${c\cdot (a_0, a_1, \ldots )}$ for ${c\in W}$ involve Witt multiplication and should be done using universal properties).

In this case ${F}$ is bijective so the corresponding ${G}$ must be étale. Also, ${VW\subset pW}$ so ${V}$ is topologically nilpotent which means ${G^D}$ is connected. Thus we have a height one, étale ${p}$-divisible group with one-dimensional, connected dual which means that ${G=\mathbb{Q}_p/\mathbb{Z}_p}$.

Now we’ll do ${\mu_{p^\infty}}$. Fact 1 tells us that ${D(\mu_{p^\infty})\simeq \mathbb{Z}_p}$ because it has height ${1}$. We also know that ${F: \mathbb{Z}_p\rightarrow \mathbb{Z}_p}$ must have the property that ${\mathbb{Z}_p/F(\mathbb{Z}_p)=\mathbb{F}_p}$ since ${\mu_{p^\infty}}$ has dimension ${1}$. Thus ${F=p\sigma}$ and hence ${V=\sigma^{-1}}$.

The proof of the anti-equivalence proceeds by working at finite stages and taking limits. So it turns out that the theory encompasses a lot more at the finite stages because ${\alpha_{p^n}}$ are perfectly legitimate finite, ${p}$-power rank group schemes (note the system does not form a ${p}$-divisible group because multiplication by ${p}$ is the zero morphism). Of course taking the limit ${\alpha_{p^\infty}}$ is also a formal ${p}$-torsion group scheme. If we wanted to we could build the theory of Dieudonné modules to encompass these types of things, but in the limit process we would have finite ${W}$-module which are not necessarily free and we would get an extra “Fact 5” that ${D(G)}$ is free if and only if ${G}$ is ${p}$-divisible.

Let’s do two more things which are difficult to see without this machinery. For these two things we’ll assume ${k}$ is algebraically closed. There is a unique connected, ${1}$-dimensional ${p}$-divisible of height ${h}$ over ${k}$. I imagine without Dieudonné theory this would be quite difficult, but it just falls right out by playing with these facts.

Since ${D(G)/FD(G)\simeq k}$ we can choose a basis, ${D(G)=We_1\oplus \cdots \oplus We_h}$, so that ${F(e_j)=e_{j+1}}$ and ${F(e_h)=pe_1}$. Up to change of coordinates, this is the only way that eventually ${F^nD(G)\subset pD(G)}$ (in fact ${F^hD(G)\subset pD(G)}$ is the smallest ${n}$). This also determines ${V}$ (note these two things need to be justified, I’m just asserting it here). But all the phrase “up to change of coordinates” means is that any other such ${(D(G'),F',V')}$ will be isomorphic to this one and hence by the equivalence of categories ${G\simeq G'}$.

Suppose that ${E/k}$ is an elliptic curve. Now we can determine ${E[p^\infty]}$ up to isomorphism as a ${p}$-divisible group, a task that seemed out of reach last time. We know that ${E[p^\infty]}$ always has height ${2}$ and dimension ${1}$. In previous posts, we saw that for an ordinary ${E}$ we have ${E[p^\infty]^{et}\simeq \mathbb{Q}_p/\mathbb{Z}_p}$ (we calculated the reduced part by using flat cohomology, but I’ll point out why this step isn’t necessary in a second).

Thus for an ordinary ${E/k}$ we get that ${E[p^\infty]\simeq E[p^\infty]^0\oplus \mathbb{Q}_p/\mathbb{Z}_p}$ by the connected-étale decomposition. But height and dimension considerations tell us that ${E[p^\infty]^0}$ must be the unique height ${1}$, connected, ${1}$-dimensional ${p}$-divisible group, i.e. ${\mu_{p^\infty}}$. But of course we’ve been saying this all along: ${E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$.

If ${E/k}$ is supersingular, then we’ve also calculated previously that ${E[p^\infty]^{et}=0}$. Thus by the connected-étale decomposition we get that ${E[p^\infty]\simeq E[p^\infty]^0}$ and hence must be the unique, connected, ${1}$-dimensional ${p}$-divisible group of height ${2}$. For reference, since ${ht(G)=\dim(G)+\dim(G^D)}$ we see that ${G^D}$ is also of dimension ${1}$ and height ${2}$. If it had an étale part, then it would have to be ${\mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$ again, so ${G^D}$ must be connected as well and hence is the unique such group, i.e. ${G\simeq G^D}$. It is connected with connected dual. This gives us our first non-obvious ${p}$-divisible group since it is not just some split extension of ${\mu_{p^\infty}}$‘s and ${\mathbb{Q}_p/\mathbb{Z}_p}$‘s.

If we hadn’t done these previous calculations, then we could still have gotten these results by a slightly more general argument. Given an abelian variety ${A/k}$ we have that ${A[p^\infty]}$ is a ${p}$-divisible group of height ${2g}$ where ${g=\dim A}$. Using Dieudonné theory we can abstractly argue that ${A[p^\infty]^{et}}$ must have height less than or equal to ${g}$. So in the case of an elliptic curve it is ${1}$ or ${0}$ corresponding to the ordinary or supersingular case respectively, and the proof would be completed because ${\mathbb{Q}_p/\mathbb{Z}_p}$ is the unique étale, height ${1}$, ${p}$-divisible group.

# p-Divisible Groups Revisited 1

I’ve posted about ${p}$-divisible groups all over the place over the past few years (see: here, here, and here). I’ll just do a quick recap here on the “classical setting” to remind you of what we know so far. This will kick-start a series on some more subtle aspects I’d like to discuss which are kind of scary at first.

Suppose ${G}$ is a ${p}$-divisible group over ${k}$, a perfect field of characteristic ${p>0}$. We can be extremely explicit in classifying all such objects. Recall that ${G}$ is just an injective limit of group schemes ${G=\varinjlim G_\nu}$ where we have an exact sequence ${0\rightarrow G_\nu \rightarrow G_{\nu+1}\stackrel{p^\nu}{\rightarrow} G_{\nu+1}}$ and there is a fixed integer ${h}$ such that group schemes ${G_{\nu}}$ are finite of rank ${p^{\nu h}}$.

As a corollary to the standard connected-étale sequence for group schemes we get a canonical decomposition called the connected-étale sequence:

$\displaystyle 0\rightarrow G^0 \rightarrow G \rightarrow G^{et} \rightarrow 0$

where ${G^0}$ is connected and ${G^{et}}$ is étale. Since ${k}$ was assumed to be perfect, this sequence actually splits. Thus ${G}$ is a semi-direct product of an étale ${p}$-divisible group and a connected ${p}$-divisible group. If you’ve seen the theory for finite, flat group schemes, then you’ll know that we usually decompose these two categories even further so that we get a piece that is connected with connected dual, connected with étale dual, étale with connected dual, and étale with étale dual.

The standard examples to keep in mind for these four categories are ${\alpha_p}$, ${\mu_p}$, ${\mathbb{Z}/p}$, and ${\mathbb{Z}/\ell}$ for ${\ell\neq p}$ respectively. When we restrict ourselves to ${p}$-divisible groups the last category can’t appear in the decomposition of ${G_\nu}$ (since étale things are dimension 0, if something and its dual are both étale, then it would have to have height 0). I think it is not a priori clear, but the four category decomposition is a direct sum decomposition, and hence in this case we get that ${G\simeq G^0\oplus G^{et}}$ giving us a really clear idea of what these things look like.

As usual we can describe étale group schemes in a nice way because they are just constant after base change. Thus the functor ${G^{et}\mapsto G^{et}(\overline{k})}$ is an equivalence of categories between étale ${p}$-divisible groups and the category of inverse systems of ${Gal(\overline{k}/k)}$-sets of order ${p^{\nu h}}$. Thus, after sufficient base change, we get an abstract isomorphism with the constant group scheme ${\prod \mathbb{Q}_p/\mathbb{Z}_p}$ for some product (for the ${p}$-divisible group case it will be a finite direct sum).

All we have left now is to describe the possibilities for ${G^0}$, but this is a classical result as well. There is an equivalence of categories between the category of divisible, commutative, formal Lie groups and connected ${p}$-divisible groups given simply by taking the colimit of the ${p^n}$-torsion ${A\mapsto \varinjlim A[p^n]}$. The canonical example to keep in mind is ${\varinjlim \mathbb{G}_m[p^n]=\mu_{p^\infty}}$. This is connected only because in characteristic ${p}$ we have ${(x^p-1)=(x-1)^p}$, so ${\mu_{p^n}=Spec(k[x]/(x-1)^{p^n})}$. In any other characteristic this group scheme would be étale and totally disconnected.

This brings us to the first subtlety which can cause a lot of confusion because of the abuse of notation. A few times ago we talked about the fact that ${E[p]}$ for an elliptic curve was either ${\mathbb{Z}/p}$ or ${0}$ depending on whether or not it was ordinary or supersingular (respectively). It is dangerous to write this, because here we mean ${E}$ as a group (really ${E(\overline{k})}$) and ${E[p]}$ the ${p}$-torsion in this group.

When talking about the ${p}$-divisible group ${E[p^\infty]=\varinjlim E[p^n]}$ we are referring to ${E/k}$ as a group scheme and ${E[p^n]}$ as the (always!) non-trivial, finite, flat group scheme which is the kernel of the isogeny ${p^n: E\rightarrow E}$. The first way kills off the infinitesimal part so that we are just left with some nice reduced thing, and that’s why we can get ${0}$, because for a supersingular elliptic curve the group scheme ${E[p^n]}$ is purely infinitesimal, i.e. has trivial étale part.

Recall also that we pointed out that ${E[p]\simeq \mathbb{Z}/p}$ for an ordinary elliptic curve by using some flat cohomology trick. But this trick is only telling us that the reduced group is cyclic of order ${p}$, but it does not tell us the scheme structure. In fact, in this case ${E[p^n]\simeq \mu_{p^n}\oplus \mathbb{Z}/p^n}$ giving us ${E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$. So this is a word of warning that when working these things out you need to be very careful that you understand whether or not you are figuring out the full group scheme structure or just reduced part. It can be hard to tell sometimes.

# 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.

# Serre-Tate Theory 1

Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and ${p}$-divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.

The idea is the following. Suppose you have an elliptic curve ${E/k}$ where ${k}$ is a perfect field of characteristic ${p>2}$. In most first courses on elliptic curves you learn how to attach a formal group to ${E}$ (chapter IV of Silverman). It is suggestively notated ${\widehat{E}}$, because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.

Since an elliptic curve is isomorphic to it’s Jacobian ${Pic_E^0}$ there is a conflation that happens. In general, if you have a variety ${X/k}$ you can make the same formal group by completing this group scheme and it is called the formal Picard group of ${X}$. Although, in general you’ll want to do this with the Brauer group or higher analogues to guarantee existence and smoothness. Then you prove a remarkable fact that the elliptic curve is ordinary if and only if the formal group has height ${1}$. In particular, since the ${p}$-divisible group is connected and ${1}$-dimensional it must be isomorphic to ${\mu_{p^\infty}}$.

It might seem silly to think in these terms, but there is another “enlarged” ${p}$-divisible group attached to ${E}$ which always has height ${2}$. This is the ${p}$-divisible group you get by taking the inductive limit of the finite group schemes that are the kernel of multiplication by ${p^n}$. It is important to note that these are non-trivial group schemes even if they are “geometrically trivial” (and is the reason I didn’t just call it the “${p^n}$-torsion”). We’ll denote this in the usual way by ${E[p^\infty]}$.

I don’t really know anyone that studies elliptic curves that phrases it this way, but since this theory must be generalized in a certain way to work for other varieties like K3 surfaces I’ll point out why this should be thought of as an enlarged ${p}$-divisible group. It is another standard fact that ${E}$ is ordinary if and only if ${E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$. In fact, you can just read off the connected-etale decomposition:

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

We already noted that ${\widehat{E}\simeq \mu_{p^\infty}}$, so the ${p}$-divisible group ${E[p^\infty]}$ is a ${1}$-dimensional, height ${2}$ formal group whose connected component is the first one we talked about, i.e. ${E[p^\infty]}$ is an enlargement of ${\widehat{E}}$. For a general variety, this enlarged formal group can be defined, but it is a highly technical construction and would take a lot of work to check that it even exists and satisfies this property. Anyway, this enlarged group is the one we need to work with otherwise our deformation space will be too small to make the theory work.

Here’s what Serre-Tate theory is all about. If you take a deformation of your elliptic curve ${E}$ say to ${E'}$, then it turns out that ${E'[p^\infty]}$ is a deformation of the ${p}$-divisible group ${E[p^\infty]}$. Thus we have a natural map ${\gamma: Def_E \rightarrow Def_{E[p^\infty]}}$. The point of the theory is that it turns out that this map is an isomorphism (I’m still assuming ${E}$ is ordinary here). This is great news, because the deformation theory of ${p}$-divisible groups is well-understood. We know that the versal deformation of ${E[p^\infty]}$ is just ${Spf(W[[t]])}$. The deformation problem is unobstructed and everything lives in a ${1}$-dimensional family.

Of course, let’s not be silly. I’m pointing all this out because of the way in which it generalizes. We already knew this was true for elliptic curves because for any smooth, projective curve the deformations are unobstructed since the obstruction lives in ${H^2}$. Moreover, the dimension of the space of deformations is given by the dimension of ${H^1(E, \mathcal{T})}$. But for an elliptic curve ${\mathcal{T}\simeq \mathcal{O}_X}$, so by Serre duality this is one-dimensional.

On the other hand, we do get some actual information from the Serre-Tate theory isomorphism because ${Def_{E[p^\infty]}}$ carries a natural group structure. Thus an ordinary elliptic curve has a “canonical lift” to characteristic ${0}$ which comes from the deformation corresponding to the identity.

# Deformations of p-divisible Groups

I’ve made the official decision to not do a proof of anything with the deformation theory of ${p}$-divisible groups, but now that I’ve motivated it I’ll still state the results. The proof is incredibly long and tedious. It should be interesting to look at what this functor is, since it probably isn’t what you think it is. We’ll construct the moduli functor, but it isn’t just ${p}$-divisible groups up to isomorphism or isogeny, but involves the notion of a quasi-isogeny.

A few definitions are needed. We’ll work in great generality, so let ${S}$ be a base scheme with your favorite properties. Our ${p}$-divisible groups will only be required to be fppf sheaves on ${S}$ (with the ${p}$-divisible group property). An isogeny ${f:G\rightarrow G'}$ of ${p}$-divisible groups is a surjection of sheaves with finite locally free kernel. An example is multiplication by ${p}$ since the kernel is ${G_1}$ which is by definition of a ${p}$-divisible group a finite locally free group scheme (of order ${p^h}$).

The group ${\mathrm{Hom}_S(G,G')}$ is a torsion-free ${\mathbb{Z}_p}$-module. We can make the sheaf version by taking the Zariski sheaf of germs of functions ${\mathcal{H}om_S(G,G')}$. A quasi-isogeny ${G\rightarrow G'}$ is a section ${\rho\in\mathcal{H}om_S(G,G')\otimes_\mathbb{Z} \mathbb{Q}}$ with the property that ${p^n\rho}$ is an isogeny for some integer ${n}$.

Now we have enough to write down the moduli functor that we want. We have everything over an algebraically closed (I think a descent argument allows us to do this all over a perfect field) field, ${k}$, of characteristic ${p>0}$ and ${W}$ its ring of Witt vectors. Consider the category ${\mathrm{Nilp}_W}$ of locally Noetherian schemes, ${S}$, over ${W}$ such that ${p\mathcal{O}_S}$ is locally nilpotent.

Fix a ${p}$-divisible group ${G}$ over ${\mathrm{Spec}(k)}$. Our moduli functor ${\mathcal{M}}$ is a contravariant functor from ${\mathrm{Nilp}_W}$ to the category of of pairs ${(H, \rho)_S}$, where ${H}$ is a ${p}$-divisible group over ${S}$ and ${\rho}$ is a quasi-isogeny ${G_{\overline{S}}\rightarrow H_{\overline{S}}}$. Then we mod out by isomorphism where an isomorphism ${(H_1, \rho_1)\rightarrow (H_2, \rho_2)}$ in this category is given by a lift to an isomorphism of ${\rho_1\circ \rho_2^{-1}}$, i.e. an isomorphism that commutes with the quasi-isogenies.

The theorem is that ${\mathcal{M}}$ is representable by a formal scheme formally locally of finite type over ${\mathrm{Spf}(W)}$. The way to prove representability is the usual way of finding particularly nice open and closed subfunctors. In our case, part of the breakdown is already in this post. Since the definition of quasi-isogeny involves an integer ${n}$ for which ${p^n\rho}$ is an isogeny, you can break ${\mathcal{M}}$ up using this integer. See the book Period Spaces for p-divisible Groups by Rapoport and Zink for more details.

It comes out in the proof that for the nice cases we were considering in the past two motivational posts we get that the functor is representable by ${\mathrm{Spf}(W[[t_1, \ldots, t_d]])}$ where ${d=\dim G \dim G^t}$. One really interesting consequence of this is that if ${G}$ has height ${1}$, then since ${\mathrm{ht}(G)=\dim G + \dim G^t}$ we have that one of ${\dim G}$ or ${\dim G^t}$ is ${1}$ and the other is ${0}$, so in either case the product is ${0}$ and we get that the deformation functor of ${G}$ is representable by ${\mathrm{Spf}(W)}$. In general, it is always smooth and unobstructed.

I’m not sure if I should continue on with ${p}$-divisible groups now that I’ve done this. Maybe I’ll go back to crystalline stuff or move on to something else altogether.

# More Motivation for p-Divisible Groups

Perhaps the last post didn’t provide you with enough motivation to understand deformations of ${p}$-divisible groups. Today we’ll look at a much more general situation where it is again useful to know the deformation theory. Recall the general setup of trying to understand a class of objects. Say abelian varieties of dimension ${g}$. You probably want to form some sort of moduli space, so you usually say what the points of the space are and then try to prove that these points form some sort of scheme or stack that isn’t too horrible.

Often times it is horrible, though, so there is another trick to try to understand what is going on. You look at the deformations of some object. This will tell you what is happening in some formal local neighborhood of that object on the moduli space. This means that deformations actually are useful for understanding classification. For instance, all K3 surfaces can be deformed to eachother (analytically, but not algebraically), so this means that as manifolds they are all diffeomorphic, but they are not algebraically equivalent.

Here is the motivation. Recall that one of the easiest examples of a ${p}$-divisible group is that whenever you have an abelian variety ${A}$ of dimension ${g}$, you can take the ${p^\nu}$-torsion points ${A[p^\nu]=\mathrm{ker}(A\stackrel{p^\nu}{\rightarrow} A)}$. Set ${G_{\nu}=A[p^\nu]}$, then you have a height ${2g}$ ${p}$-divisible group ${\lim G_{\nu}=A[p^\infty]}$ (as long as the characteristic of ${k}$ is relatively prime to ${p}$ otherwise the height is different, but you still get a ${p}$-divisible group).

Here is an absolutely beautiful result of Serre and Tate: Fix some Artin local ${k}$-algebra ${R}$ and let AbSch(${R}$) be abelian schemes over ${R}$ and BT(${R}$) the category with objects ${(X,G,\epsilon)}$ where ${X}$ is an abelian variety over ${k}$, ${G}$ is a ${p}$-divisible group over ${R}$ and ${\epsilon}$ is a choice of isomorphism of the special fiber of ${G}$ with the ${p}$-divisible group associated to ${X}$, so ${\epsilon: G\otimes_R k\stackrel{\sim}{\rightarrow} X[p^\infty]}$. The result is that there is an equivalence of categories ${F:\mathrm{AbSch}(R)\rightarrow \mathrm{BT}(R)}$ given simply by ${F(A)=(A\otimes_R k, A[p^\infty], \epsilon)}$.

In particular, for an abelian variety ${X}$ we get an isomorphism of deformation functors ${Def_X\rightarrow Def_{X[p^\infty]}}$. So if you want to understand the deformation theory of abelian varieties you could try to understand the deformations of ${p}$-divisible groups. This is the topic of the great book by Messing The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes. Might I point out that whenever I find a book great and useful it seems to always have been moved to math storage as something no one ever looks at.

A similar thing happens with ordinary K3 surfaces (you can’t quite take the Artin-Mazur formal group, but instead an “enlarged” version of it). Many people are interested in knowing whether certain varieties lift to characteristic ${0}$, and not only that but in a very nice way (maybe you remember the Hodge filtration or you want it to satisfy the Tate conjecture). These are called “canonical lifts” and a really beautiful way to construct them is by showing that the deformation functor is isomorphic to the deformations of some particular ${p}$-divisible group of height ${1}$.

If you know that the deformations of a ${p}$-divisible group are unobstructed, then this produces a lift. If you know even more, like the deformation functor is represented by a smooth formal group scheme (which in the height ${1}$ case it is) you can take a canonical choice of lift, namely the one you get from the identity element. This is one way to get a canonical lift of ordinary abelian varieties and ordinary K3 surfaces.

Now hopefully everyone has at least a little interest in deformations of ${p}$-divisible groups. I have to say that I find this idea amazing and quite under-used. You want to know whether there is a lift to characteristic ${0}$, or maybe just in general you want to understand the deformations of some object. This in general is incredibly hard. Instead, you find a candidate ${p}$-divisible group associated to the object and show that the deformation functors are related or isomorphic. Then you can extrapolate information about the original deformation functor, since the deformations of ${p}$-divisible groups are well understood. I have to admit that this technique has not been very useful to me so far in my research, but I’m quite optimistic still…

# What are Crystals?

Since ${p}$-divisible groups are pretty awesome, I’m coming back to them for a little bit. Today will tie into the crystalline stuff we’ve been looking at, but after that we’ll move on to the deformation theory of them. First, you’ll need to recall the posts about the Dieudonne module and the heights of ${p}$-divisible groups (in particular, you should at the very least remind yourself what a ${p}$-divisible group is).

Unless otherwise stated we will be working over a perfect field ${k}$ of positive characteristic. Almost everything done in the next several posts could actually be done in more generality by working on a general base and taking the ${p}$-divisible group to be a sheaf of groups on the fppf site of Sch/S. Recall that a ${p}$-divisible group is a formal group ${G=\lim_{\rightarrow}G_\nu}$ with the property that ${|G_\nu|=p^{\nu h}}$ and we call ${h}$ the height of ${G}$ (this is NOT the definition, but merely a fact about them).

For any ${p}$-divisible group we can form the Dieudonne module ${D(G)}$ which turns out to be a free ${W(k)}$-module of rank ${h}$. For our first new concept, ${D(G)}$ is an example of something called an ${F}$-crystal, or just crystal sometimes. For a definition, an ${F}$-crystal is a free ${W(k)}$-module ${M}$ together with an injective endomorphism ${\phi: M\rightarrow M}$ that is Frobenius semi-linear, i.e. ${\phi(\lambda m)=F(\lambda)\phi(m)}$, where ${F}$ is the lift of Frobenius to ${W(k)}$. The ${F}$-crystals over ${k}$ form a category in the obvious way (maps have to respect the endomorphism) and more importantly this is the category for which we get an equivalence with ${p}$-divisible groups just by the functor ${G\mapsto D(G)}$ (note ${D(G)}$ is an ${F}$-crystal by using the standard Frobenius as the endomorphism).

Suppose ${K}$ is the fraction field of ${W(k)}$. An ${F}$-isocrystal is a (finite dimensional) vector space over ${K}$ equipped with a Frobenius semi-linear automorphism. We always get one of these from a ${p}$-divisible group as well by tensoring with ${K}$. So define ${E(G)=D(G)\otimes K}$ and the automorphism is obtained by extending Frobenius linearly. It is a major theorem that every ${F}$-isocrystal has a direct sum decomposition called the slope decomposition ${\bigoplus E_\lambda}$. To see what this is, I typed it up here. The ${\lambda}$ are called the slopes and they are a finite set of rational numbers. If ${E}$ is an isocrystal we also write ${E_{[a,b]}}$ to mean ${\bigoplus_{\lambda\in [a,b]}E_\lambda}$.

This brings us to an incredibly fascinating tie in to the heights of varieties. Recall that we have a ${p}$-divisible group representing ${\Phi (S)=\mathrm{ker}(H_{et}^n(X\otimes S, \mathbb{G}_m)\rightarrow H_{et}^n(X, \mathbb{G}_m))}$ and the height of this is the height of the variety. But if we look at the crystalline cohomology ${H^n_{crys}(X/W)\otimes K}$ we get an ${F}$-isocrystal, and the part ${(H^n_{crys}(X/W)\otimes K)_{[0,1)}}$ with slopes strictly less than ${1}$ is some finite dimensional vector space, and the dimension of this is the height of ${X}$! Maybe I’ll explain this more some other time, but it is very far off topic for now.

Grothendieck saw how nice the theory of crystals helped analyze ${p}$-divisible groups and other things, so he tried to generalize it to crystalline cohomology (note, I have no idea about the accuracy of this historical tidbit). Suppose ${\mathcal{F}}$ is a Zariski stack on Sch. An ${\mathcal{F}}$-crystal is a Cartesian section of the fibered category ${\mathcal{F}\times_{Sch} Crys(X)}$ where the map from the crystalline site ${Crys(X)\rightarrow Sch}$ is given by ${(U\hookrightarrow T, \gamma)\mapsto T}$ (a functor we’ve talked about already). If you unravel all this, then you find that an ${\mathcal{F}}$-crystal is always a sheaf ${\mathcal{G}}$ on ${Crys(X)}$ that satisfies another condition. We’ll always think of ${\mathcal{O}_X}$-crystals in which case the extra condition is just that for any map ${u}$ in ${Crys(X)}$ we get ${u^*\mathcal{G}\rightarrow \mathcal{G}}$ is an isomorphism.

We see that this is some sort of “rigidity” condition. In Berthelot and Ogus, they claim that Grothendieck coined the term crystal because it has two properties, it is both “rigid” as we saw and it “grows” over PD thickenings (this is automatically satisfied because it is a sheaf). A trivial example of a crystal is ${\mathcal{O}_X}$ itself. A nontrivial, but incredibly useful example of a crystal is given any closed immersion (over some fixed PD scheme) ${i: Y\hookrightarrow X}$ the sheaf ${i_{crys *}(\mathcal{O}_Y)}$ is an ${\mathcal{O}_X}$-crystal.

Suppose now we have some fixed (PD) base ${S}$. In order to extend Dieudonne theory to this new sense of crystals, we might want a functor ${\mathbb{D}}$ that sends a ${p}$-divisible group to an ${\mathcal{O}_S}$-crystal. There is such a functor, but its construction requires us to know that the deformations of any ${p}$-divisible group are unobstructed. This brings me to the point of this post. We want to understand the deformation theory of ${p}$-divisible groups.

# 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.