# P-Divisible Groups Mini Course: Talk 1

Over the next month, I will be giving a series of four introductory talks at UC Berkeley on $p$-divisible groups and related concepts. I plan to write a post for each talk and include a number of references for those wishing to see more details.

## Motivation

Let $A/K$ be an Abelian variety (a smooth proper group variety over some number field $K$). Associated with $A$ and a prime number $p$ is its Tate-module $T_pA := \varprojlim A[p^n]$ where $A[p^n]$ consists of the $p^n$-torsion elements of $A(K^{sep})$. We recall that in the present case, $T_pA$ is a free $\mathbb{Z}_p$-module of rank $2d$ (where $d$ is the dimension of $A$). The Tate-module is a very powerful invariant: for instance we have a theorem of Faltings that for Abelian varieties $A, B$

$\mathrm{Hom}_K(A,B) \otimes \mathbb{Z}_p \simeq \mathrm{Hom}_{\mathrm{Gal}(K^{sep}/K)}(T_pA, T_pB)$

The above construction is equally useful when $K$ is a finite field. However things break down when the characteristic of $K$ is $p$. To see this, lets focus in on the case of an elliptic curve $E$ over a field of characteristic $p$. Then the $E(K^{sep})[p^n]$ has either $p^n$ points in the ordinary case or just the identity in the supersingular case. The multiplication by $p$ map on $E$ is a degree $2p$-polynomial so somehow just considering geometric points is not capturing everything we’d like.

The missing information in the above picture corresponds to nilpotents so we can remedy the situation by considering the finite flat group scheme which is a kernel $A[p^n]$ of $[p^n]: A \to A$. For instance, if $E$ is an ordinary elliptic curve, then $E[p^n]= \mu_{p^n} \times \underline{\mathbb{Z}/p^n \mathbb{Z}}$ ( here $\mu_{p^n}$ is the scheme representing the functor on $K$-algebras given by $F(B)=\{ x \in B, x^{p^n}=1 \}$ and $\underline{\mathbb{Z} /p^n \mathbb{Z}}$ is the scheme representing the sheafification of the functor $F(B)= \mathbb{Z}/ p^n \mathbb{Z}$).

In this case the analog of the  Tate module that we want is $\varinjlim A[p^n]$ Unfortunately, direct limits need not exist in the category of finite flat group schemes. There are two (equivalent) ways of resolving this problem. Either we can consider directed systems of group schemes or we can enlarge our category to that of fppf sheaves. This brings us to the first key definition.

Definition: Let $S$ be a scheme and consider the site of fppf sheaves of groups on $S$. Such a sheaf $G$ is $p$-divisble (often Barsotti-Tate in the literature) if it satisfies each of the following conditions.

1. $G$ is $p$-torsion, i.e. $\varinjlim G[p^n]=G$.
2. $[p]: G \to G$ is an epimorphism
3. $G[p^n]$ is a finite, locally free group scheme

## Examples and Key Constructions

• If $A$ is an Abelian variety, then $\varinjlim A[p^n]$ is a $p$-divisible group.
• We write $\mu_{p^{\infty}} := \varinjlim \mu_{p^n}$ and $\underline{\mathbb{Q}_p / \mathbb{Z}_p} := \varinjlim \underline{ \mathbb{Z}/ p^n \mathbb{Z}}$.
• Cartier Duality: If $G$ is a finite flat group scheme then so is the functor $S \mapsto \mathrm{Hom}_{S Grps}( G_S, G_{m,S})$ and we denote this $G^*$. So for instance ${\mu_{p^n}}^*= \underline{ \mathbb{Z}/ p^n \mathbb{Z}}$ and ${G^*}^*=G$.
• We define the Cartier dual of a $p$_divisible group $G$ by $G^* := \varinjlim G[p^n]^*$. So ${\mu_{p^{\infty}}}^*=\underline{ \mathbb{Z}/ p^n \mathbb{Z}}$.
• We define the height of a $p$-divisible group $G$ to be $h$ so that $dim_{\mathcal{O}_S} G[p^n]=p^{nh}$.
• We call a $p$-divisible group $G$ etale/connected if it is a limit of etale/connected group schemes.  $\underline{ \mathbb{Z}/ p ^n \mathbb{Z}}$ is etale (as a scheme it is just $\mathrm{Spec} \coprod\limits_{i=1}^{p^n} S$), $\mu_{p^{\infty}}$ is connected when $S$ is in characteristic $p$ and etale otherwise.
• When $S$ is a complete noetherian local ring we get a short exact sequence $1 \to G^0 \to G \to G^{et} \to 1$ of group schemes and when $S$ is perfect  field $K$, the  sequence splits and gives a decomposition $G \simeq G^{et} \times G^0$. In this latter case a $p$-divisible group will likewise split into connected and etale parts.
• Finite flat group schemes over a base of positive characteristic have Frobenius and Verschiebung morphisms and these lift to morphisms of $p$-divisible groups. These morphisms satisfy $V \circ F=[p]$.

## The Associated Formal Group

Definition: A formal group of dimension $n$ over a complete Noetherian local ring $R$ with residue field $k$ of characteristic $p>0$ is the formal scheme $\mathrm{Spf} R[[x_1, ..., x_n]]$ with ideal of definition $(x_1, ..., x_n)$ and with a commutative group structure.

Explicitely, if $A$ is a formal group of dimension $n$, then the group structure is a family of $n$ power series $F=(F_i)$ in the $2n$ variables $X=(x_1, ..., x_n), Y=(y_1, ..., y_n)$ satisfying

1. $F(X,0)=F(0,X)=X$
2. $F(X, F(Y, Z))=F(F(X,Y),Z)$
3. $F(X,Y)=F(Y, X)$.

We say a formal group $A$ is $p$-divisible if multiplication by $p$ is an isogeny. In this case, $\varinjlim A[p^n]$ will be a $p$-divisible group. Conversely given a connected $p$-divisible group $G$, we can take the direct limit $\varinjlim G[p^n]$ in the category of locally ringed spaces and get a formal scheme  with a group structure inherited from the $G[p^n]$. The constructions are inverses and form an equivalence of categories. Note that even if this direct limit exists in the category of schemes, it may not be the same as that in the category of locally ringed spaces. For instance, $\mathrm{Spf} k[[x]]$ has one point while $\mathrm{Spec} k[[x]]$ has two.

With the above construction in hand, we can define the dimension of a $p$-divisible group $G$. It is the dimension of the connected component $G^0$. For example, we note that under the above assumptions, $\mu_{p^{\infty}}$ and $\underline{ \mathbb{Q}_p / \mathbb{Z}_p }$ both have height 1 and dimension 1 and 0 respectively.

In fact, we have a result that will often come in handy:

Proposition Let $G , G^*$ be dual height $h$ $p$-divisible groups over $R$ as above with respective dimensions $n, n'$.  Then $h=n+n'$.

Proof Sketch

We are only counting dimensions so we can tensor everything with the residue field of  $R$ and thus reduce to the case where we are working over a perfect field of characteristic $p$.

$[p]: G \to G, [p]: G^{(p)} \to G^{[p]}$ are both epimorphisms and since $V \circ F=[p]_G , F \circ V= [p]_{G^*}$, $F, V$ are both epimorphisms as well. It follows that we have an exact sequence $0 \to \mathrm{ker} F \to \mathrm{ker} [p] \to \mathrm{ker} V \to 0$.

Comparing dimensions, the middle is $p^h$ and the left and right are $p^n, p^{n'}$ respectively.

One way this result can prove useful is in analyzing $p$-divisible groups corresponding to Abelian varieties. For instance, the $p$-divisible group corresponding to an elliptic curve has height $2$. Moreover, if $A^t$ is the dual Abelian variety to $A$ with $G, G^t$ the corresponding $p$-divisible groups, one can show that $G^t=G^*$. Elliptic curves are self dual so this means the corresponding $p$-divisible group is as well. If $E$ is an elliptic curve over a finite field with $p$-divisible group $G$, then the equation $ht(G)= \mathrm{dim} G + \mathrm{dim} G^*$ yields $\mathrm{dim} G= 1$. Thus we can immediately tell that $G$ is not etale since then it would have dimension $0$. If $ht(G^0)=1$ then $G$ splits $G^0 \times G^{et}$ and each has height 1 (of course we’ve stated above explicitely what these groups are so this shouldn’t be surprising). If $ht(G^0)=2$. Then $G$ is a self-dual connected $p$-divisible group.

## A Glimpse at Further Motivations

For many questions in number theory, and in particular in global Langlands one considers moduli spaces of Abelian varieties with additional structure, for instance certain Shimura varieties. For local Langlands, the analog we want (in a way that I hope to make slightly clearer in a later talk) are deformation spaces of Abelian varieties.

Let $R$ be a ring killed by multiplication by $p^N$ for some $N \in \mathbb{N}$. Let $I$ be a nilpotent ideal in $R$ and let $R_0= R/I$. Consider $\mathcal{A}(R)$, the category of Abelian schemes over $R$ and $\mathrm{Def} (R, R_0)$ the category of triples $(A_0, G, \epsilon)$ where $G$ is a $p$-divisible group over $R$ and $\epsilon$ is an isomorphism from $G_0 := G \times \mathrm{Spec} R_0$ to the $p$-divisible group of $A_0$.

Theorem (Serre-Tate) The natural functor $\mathscr{A}(R) \to \mathrm{Def} (R, R_0)$ is an equivalence of categories.

Intuitively, this identitifies a space of Abelian schemes deformed from $R_0$ to $R$ with the deformation space of its $p$-divisible group. Thus if we are interested in local Langlands correspondence or simply want to study local moduli of Abelian varieties, we are naturally drawn to study moduli of $p$-divsible groups.

## Resources

Tate has an article $p$-Divisible Groups which was historically one of the first introductions to the topic and is very accessible.

Tate has also written an introductory paper Finite Flat Group Schemes which provides the basics of the theory as well as some concrete computations and more advanced topics.

If one wants a more complete treatment of the theory of group schemes and $p$-divisible groups, Demazure’s Lectures on $p$-Divisible Groups is good though a warning that some of the notation is a bit non-standard.

Katz’s  Serre-Tate Local Moduli has a nice proof of the Serre-Tate theorem at the beginning.

Messing’s The Crystals associated to Barsotti-Tate Groups is a classic on $p$-divisible groups and their deformation theory but is a harder read than the above.

## One thought on “P-Divisible Groups Mini Course: Talk 1”

1. Hey Bertie, just some comments.

There is a typo in your second paragraph. You either meant $E(K^\text{sep})[p^n]$ or $E[p^n](K^\text{sep})$.

The equality $E[p^n]=\underline{\mathbb{Z}/p^n\mathbb{Z}}\times\mu_{p^n}$ (as I know you know, but you didn’t make clear) only happens over $\overline{k}$. Also, over the algebraic closure there is also a nice description of, say, $E[p]$ when $E$ is supersingular. In this case it’s the non-trivial extension of $\alpha_p$ by itself. In fact, it’s evident that $\ker \text{Fr}\subseteq E[p]$ and since this can’t be $\mu_p$ or $\mathbb{Z}/p\mathbb{Z}$ for obvious reasons, it must be $\alpha_p$. Similarly, the quotient $Q:=E[p]/\ker(\text{Fr})$ is order $p$ and not $\mu_p$ or $\mathbb{Z}_p$. So, you get that it is such an extension. One can check that there are three groups which are extensions of $\alpha_p$ by itself—$\alpha_{p^2}$, $\alpha_p\times\alpha_p$, and something we’ll call $K$. We know that $E[p]$ can’t be the first two, and so it must be $K$. One can do all of this much easier with Dieudonne theory, but since that is the topic of your next talk, I assumed you’d want to avoid that.

You should note that Cartier duality is, in general, contravariant. So, when you write $G^\ast=\varinjlim G[p^n]^\ast$ your map $G[p^n]^\ast\to G[p^{n+1}]^\ast$ comes from the multiplication by $p$ map $G[p^{n+1}]\to G[p^n]$.

I guess it’s worth saying, although it might be obvious, that you’re not really considering $\varinjlim G[p^n]$ in the category of locally ringed spaces, but in the category of locally topologically ringed spaces. You should also mention why connectedness is important here.

Also, you should be careful when you say isogeny because it somehow calls to mind the wrong property of the morphism. The key point is that it’s locally free—that $A[[x_1,\ldots,x_n]]/([p]^ast(x_1),\ldots,[p]^\ast(x_n)]]$ is a projective $A$-module.

Also, although not strictly necessary, when it comes to thinking about things like $p$-divisible groups, I think that it really is helpful to recall Serre’s fact that one can think of $A^\vee$, for $A/S$ a projective abelian scheme, as being the $fppf$ sheaf given by $\mathcal{E}xt^1(\mathbf{G}_{m,S},A)$. This immediately proves the claim that for any $f:A\to B$ with dual map $f^\vee:B^\vee\to A^\vee$ that $\ker (f^\vee)=\ker(f)^\ast$. This can also be acheived by the generalized Weil pairing.

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