Contents

group theory

# Contents

## Idea

In great generality, for an integer $p$ a $p$-divisible group is a codirected diagram of abelian group objects in a category $C$ where the abelian-group objects are (equivalently) the kernels of the map given by multiplication with a power of $p$; these kernels are also called $p^n$-torsions.

In the classically studied case $p$ is a prime number, $C$ is the category of schemes over a commutative ring (mostly a field with prime characteristic) and the abelian group schemes occurring in the diagram are assumed to be finite. In this case the diagram defining the $p$-divisible group can be described in terms of the growth of the order of the group schemes in the diagram.

Note that there is also a notion of divisible group.

## Definition

###### Definition

Fix a prime number $p$, a positive integer $h$, and a commutative ring $R$. Consider the group schemes $G$ over

A $p$-divisible group of height $h$ over $R$ is a codirected diagram $(G_\nu, i_\nu)_{\nu \in \mathbb{N}}$ where each $G_\nu$ is a finite commutative group scheme over $R$ of order $p^{\nu h}$ that also satisfies the property that

$0\to G_\nu \stackrel{i_\nu}{\to} G_{\nu +1}\stackrel{p^\nu}{\to} G_{\nu +1}$

is exact. In other words, the maps of the system identify $G_\nu$ with the kernel of multiplication by $p^\nu$ in $G_{\nu +1}$.

Some authors refer to $colim_\nu G_\nu$ (instead of the diagram itself) as the $p$-divisible group.

It can be checked that a $p$-divisible group over $R$ is a $p$-torsion commutative formal group $G$ for which $p\colon G \to G$ is an isogeny.

## Examples

###### Example

The kernel of raising to the $p^\nu$ power on $\mathbb{G}_m$ (sometimes called p-torsion) is a group scheme $\mu_{p^\nu}$. The limit $\lim_{\to} \mu_{p^\nu}=\mu_{p^\infty}$ is a $p$-divisible group of height $1$.

see Lipnowski pg.2, example (b)

###### Example

The eponymous ($p$-divisible groups are sometimes called Barsotti-Tate groups) example is a special case of the previous one - namely the Barsotti-Tate group of an abelian variety. Let $X$ be an abelian variety over $R$ of dimension $g$, then the multiplication map by $p^\nu$ has kernel $_{p^\nu}X$ which is a finite group scheme over $R$ of order $p^{2g \nu}$. The natural inclusions satisfy the conditions for the limit denoted $X(p)$ to be a $p$-divisible group of height $2g$.

###### Example

A theorem of Serre and Tate says that there is an equivalence of categories between divisible, commutative, formal Lie groups over $R$ and the category of connected $p$-divisible groups over $R$ given by $\Gamma \mapsto \Gamma (p)$, where $\Gamma(p)=\lim_{\to} \mathrm{ker}(p^n)$. In particular, every connected $p$-divisible group is smooth

## The Cartier dual

• Given a $p$-divisible group $G$, each individual $G_\nu$ has a Cartier dual $G_\nu^D$ since they are all group schemes. There are also maps $j_\nu$ that make the composite $G_{\nu+1}\stackrel{j_\nu}{\to} G_\nu \stackrel{i_\nu}{\to} G_{\nu +1}$ the multiplication by $p$ on $G_{\nu +1}$. After taking duals, the composite is still the multiplication by $p$ map on $G_{\nu +1}^D$, so it is easily checked that $(G_{\nu}^D, j_{\nu}^D)$ forms a $p$-divisible group called the Cartier dual.

• One of the important properties of the Cartier dual is that one can determine the height of a $p$-divisible group (often a hard task when in the abstract) using the information of the dimension of the formal group and its dual. For any $p$-divisible group, $G$, we have the formula that $ht(G)=ht(G^D)=\dim G + \dim G^D$.

## Dieudonné modules

For the moment see display of a p-divisible group.

### Examples

• The dual $\mu_{p^\infty}^D\simeq \mathbb{Q}_p/\mathbb{Z}_p$.

• For an abelian variety $X$, the dual is $X(p)^D=X^t(p)$ where $X^t$ denotes the dual abelian variety. Another proof that $X(p)$ has height $2g$ is to note that $X$ and $X^t$ have the same dimension $g$, so using our formula for height we get $ht(X(p))=2g$.

## Properties

The category of étale $p$-divisible groups is equivalent to the category of $p$-adic representations of the fundamental group of the base scheme .

## p-divisible groups and crystals

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References: Weinstein

(…)

## In derived algebraic geometry

See Lurie. An important notion: nonstationary p-divisible group

For references concerning Witt rings and Dieudonné modules see there.

### Original texts and classical surveys

• Barsotti, Iacopo (1962), “Analytical methods for abelian varieties in positive characteristic”, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, pp. 77–85, MR 0155827

• Demazure, Michel (1972), Lectures on p-divisible groups, Lecture Notes in Mathematics, 302, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060741, ISBN 978-3-540-06092-5, MR 034426, web

• Grothendieck, Alexander (1971), “Groupes de Barsotti-Tate et cristaux”, Actes du Congrès International des Mathématiciens (Nice, 1970), 1, Gauthier-Villars, pp. 431–436, MR 0578496

• Messing, William (1972), The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, 264, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058301, MR 0347836

• Serre, Jean-Pierre (1995) , “Groupes p-divisibles (d’après J. Tate) web, Exp. 318”, Séminaire Bourbaki, 10, Paris: Société Mathématique de France, pp. 73–86, MR 1610452

• Stephen Shatz, Group Schemes, Formal Groups, and $p$-Divisible Groups in the book Arithmetic Geometry Ed. Gary Cornell and Joseph Silverman, 1986

• Tate, John T. (1967), “p-divisible groups.” pdf, in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827

### Modern surveys

• de Jong, A. J. (1998), Barsotti-Tate groups and crystals, “Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)”, Documenta Mathematica II: 259–265, ISSN 1431-0635, MR 1648076

• Dolgachev, I.V. (2001), “P-divisible group” web, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Richard Pink, finite group schemes, 2004-2005, pdf

• Hoaran Wang, moduli spaces of p-divisible groups and period morphisms, Masters Thesis, 2009, pdf

• Jared Weinstein, the geometry of Lubi-Tate spaces, Lecture 1: Formal groups and formal modules, pdf

• Liang Xiao, notes on $p$-divisible groups, pdf

### Further development of the theory

• Paul Goerss, p-divisible groups and Lurie’s realization result, 2008, pdf slides

• Jacob Lurie, A Survey of Elliptic Cohomology, section 4.2, pdf

• Peter Scholze, Moduli of p-divisible groups, arxiv

• Thomas Zink, a dieudonné theory for p-divisible groups, pdf

• Thomas Zink, list of publications and preprints, web

• T. Zink, On the slope filtration, Duke Math. Journal, Vol.109 (2001), No.1, 79-95, pdf

• T. Zink, the display of a formal p-divisible group, to appear in Astérisque, pdf

• T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, pdf