p-divisible group



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

In the classically studied case pp is a prime number, CC 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 pp-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 pp, a positive integer hh, and a commutative ring RR. Consider the group schemes GG over

A pp-divisible group of height hh over RR is a codirected diagram (G ν,i ν) ν(G_\nu, i_\nu)_{\nu \in \mathbb{N}} where each G νG_\nu is a finite commutative group scheme over RR of order p νhp^{\nu h} that also satisfies the property that

0G νi νG ν+1p νG ν+10\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 νG_\nu with the kernel of multiplication by p νp^\nu in G ν+1G_{\nu +1}.

Some authors refer to colim νG νcolim_\nu G_\nu (instead of the diagram itself) as the pp-divisible group.

It can be checked that a pp-divisible group over RR is a pp-torsion commutative formal group GG for which p:GGp\colon G \to G is an isogeny.



The kernel of raising to the p νp^\nu power on 𝔾 m\mathbb{G}_m (sometimes called p-torsion) is a group scheme μ p ν\mu_{p^\nu}. The limit lim μ p ν=μ p \lim_{\to} \mu_{p^\nu}=\mu_{p^\infty} is a pp-divisible group of height 11.

see Lipnowski pg.2, example (b)


The eponymous (pp-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 XX be an abelian variety over RR of dimension gg, then the multiplication map by p νp^\nu has kernel p νX_{p^\nu}X which is a finite group scheme over RR of order p 2gνp^{2g \nu}. The natural inclusions satisfy the conditions for the limit denoted X(p)X(p) to be a pp-divisible group of height 2g2g.


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

The Cartier dual

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

  • One of the important properties of the Cartier dual is that one can determine the height of a pp-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 pp-divisible group, GG, we have the formula that ht(G)=ht(G D)=dimG+dimG Dht(G)=ht(G^D)=\dim G + \dim G^D.

Dieudonné modules

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


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

  • For an abelian variety XX, the dual is X(p) D=X t(p)X(p)^D=X^t(p) where X tX^t denotes the dual abelian variety. Another proof that X(p)X(p) has height 2g2g is to note that XX and X tX^t have the same dimension gg, so using our formula for height we get ht(X(p))=2ght(X(p))=2g.


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

p-divisible groups and crystals


References: Weinstein

Relation to crystalline cohomology


In derived algebraic geometry

See Lurie.


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) [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 pp-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 pp-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

Revised on May 1, 2015 12:53:43 by Catherine Ray (