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.
Fix a prime number $p$, a positive integer $h$, and a commutative ring $R$.
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
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 the $p$-divisible group as the colimit of the system $colim G_\nu$. Note that if everything is affine, $G_\nu=\mathrm{Spec}(A_\nu)$ and the limit $colim G_\nu = \mathrm{Spec}(\lim A_\nu)=\mathrm{Spf}(A)$.
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.
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 $1$.
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 $2g$.
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
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 $ht(G)=ht(G^D)=\dim G + \dim G^D$.
For the moment see display of a p-divisible group.
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 $ht(X(p))=2g$.
The category of étale $p$-divisible groups is equivalent to the category of $p$-adic representations of the fundamental group of the base scheme .
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References: Weinstein
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See Lurie.
Important tools in the study of $p$-divisible groups are Witt rings, Dieudonné modules and more generally Dieudonné theories? assigning to a $p$-divisible group an object of linear algebra such as a display of a p-divisible group.
For references concerning Witt rings? and Dieudonné modules see there.
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) [1966], “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.”, in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827
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
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