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\section*{p-divisible group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_cartier_dual}{The Cartier dual}\dotfill \pageref*{the_cartier_dual} \linebreak
\noindent\hyperlink{dieudonn_modules}{Dieudonn\'e{} modules}\dotfill \pageref*{dieudonn_modules} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{pdivisible_groups_and_crystals}{p-divisible groups and crystals}\dotfill \pageref*{pdivisible_groups_and_crystals} \linebreak
\noindent\hyperlink{relation_to_crystalline_cohomology}{Relation to crystalline cohomology}\dotfill \pageref*{relation_to_crystalline_cohomology} \linebreak
\noindent\hyperlink{in_derived_algebraic_geometry}{In derived algebraic geometry}\dotfill \pageref*{in_derived_algebraic_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{original_texts_and_classical_surveys}{Original texts and classical surveys}\dotfill \pageref*{original_texts_and_classical_surveys} \linebreak
\noindent\hyperlink{modern_surveys}{Modern surveys}\dotfill \pageref*{modern_surveys} \linebreak
\noindent\hyperlink{further_development_of_the_theory}{Further development of the theory}\dotfill \pageref*{further_development_of_the_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In great generality, for an [[integer]] $p$ a \emph{$p$-divisible group} is a [[inductive system|codirected diagram]] of [[abelian group|abelian]] [[group objects]] in a [[category]] $C$ where the abelian-group objects are (equivalently) the [[kernel|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 [[scheme|schemes]] over a commutative ring (mostly a field with prime characteristic) and the abelian [[group scheme|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 a group|order]] of the group schemes in the diagram.

Note that there is also a notion of [[divisible group]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
Fix a [[prime number]] $p$, a [[natural number|positive integer]] $h$, and a [[commutative ring]] $R$. Consider the group schemes $G$ over

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

\begin{displaymath}
0\to G_\nu \stackrel{i_\nu}{\to} G_{\nu +1}\stackrel{p^\nu}{\to} G_{\nu +1}
\end{displaymath}
is [[exact sequence|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.

\end{defn}
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]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
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 \href{http://math.stanford.edu/~malipnow/expository/pdiv.pdf}{pg.2, example (b)}

\end{example}
\begin{example}
\label{}\hypertarget{}{}
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 $2g$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A theorem of \hyperlink{Tate}{Serre and Tate} says that there is an [[equivalence of categories]] between divisible, commutative, formal [[Lie group]]s 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

\end{example}
\hypertarget{the_cartier_dual}{}\subsection*{{The Cartier dual}}\label{the_cartier_dual}

\begin{itemize}%
\item 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.


\item 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 $ht(G)=ht(G^D)=\dim G + \dim G^D$.



\end{itemize}
\hypertarget{dieudonn_modules}{}\subsection*{{Dieudonn\'e{} modules}}\label{dieudonn_modules}

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

\hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2}

\begin{itemize}%
\item The dual $\mu_{p^\infty}^D\simeq \mathbb{Q}_p/\mathbb{Z}_p$.


\item 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 $ht(X(p))=2g$.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The category of \'e{}tale $p$-divisible groups is equivalent to the category of $p$-adic representations of the fundamental group of the base scheme .

\hypertarget{pdivisible_groups_and_crystals}{}\subsection*{{p-divisible groups and crystals}}\label{pdivisible_groups_and_crystals}

(\ldots{})

References: \hyperlink{Weinstein}{Weinstein}

\hypertarget{relation_to_crystalline_cohomology}{}\subsection*{{Relation to crystalline cohomology}}\label{relation_to_crystalline_cohomology}

(\ldots{})

\hypertarget{in_derived_algebraic_geometry}{}\subsection*{{In derived algebraic geometry}}\label{in_derived_algebraic_geometry}

See \hyperlink{pLurie}{Lurie}. An important notion: [[nonstationary p-divisible group]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item Important tools in the study of $p$-divisible groups are [[Witt ring|Witt rings]], [[Dieudonné module|Dieudonné modules]] and more generally [[Dieudonné theory|Dieudonné theories]] assigning to a $p$-divisible group an object of [[linear algebra]] such as a [[display of a p-divisible group]].


\item [[formal group]]


\item [[group scheme]]


\item [[Artin–Mazur formal group]]


\item [[height of a variety]]



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[lectures on p-divisible groups|Lectures on p-divisible groups]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For references concerning [[Witt rings]] and [[Dieudonné modules]] see there.

\hypertarget{original_texts_and_classical_surveys}{}\subsubsection*{{Original texts and classical surveys}}\label{original_texts_and_classical_surveys}

\begin{itemize}%
\item Barsotti, Iacopo (1962), ``Analytical methods for abelian varieties in positive characteristic'', Colloq. Th\'e{}orie des Groupes Alg\'e{}briques (Bruxelles, 1962), Librairie Universitaire, Louvain, pp. 77--85, MR 0155827


\item Demazure, Michel (1972), [[lectures on p-divisible groups|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, \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}


\item Grothendieck, Alexander (1971), ``Groupes de Barsotti-Tate et cristaux'', Actes du Congr\`e{}s International des Math\'e{}maticiens (Nice, 1970), 1, Gauthier-Villars, pp. 431--436, MR 0578496


\item 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


\item Serre, Jean-Pierre (1995) 1966, ``Groupes p-divisibles (d'apr\`e{}s J. Tate) \href{http://www.numdam.org/item?id=SB_1966-1968__10__73_0}{web}, Exp. 318'', S\'e{}minaire Bourbaki, 10, Paris: Soci\'e{}t\'e{} Math\'e{}matique de France, pp. 73--86, MR 1610452


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


\item Tate, John T. (1967), ``p-divisible groups.'' \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.295.7566&rep=rep1&type=pdf}{pdf}, in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827



\end{itemize}
\hypertarget{modern_surveys}{}\subsubsection*{{Modern surveys}}\label{modern_surveys}

\begin{itemize}%
\item 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


\item Dolgachev, I.V. (2001), ``P-divisible group'' \href{http://www.encyclopediaofmath.org/index.php/P-divisible_group}{web}, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4


\item Richard Pink, finite group schemes, 2004-2005, \href{http://www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf}{pdf}


\item Hoaran Wang, moduli spaces of p-divisible groups and period morphisms, Masters Thesis, 2009, \href{http://people.math.jussieu.fr/~dat/enseignement/M2HaoranWang.pdf}{pdf}


\item [[Jared Weinstein]], [[Weinstein, the geometry of Lubin-Tate spaces|the geometry of Lubi-Tate spaces]], Lecture 1: Formal groups and formal modules, \href{http://www.math.ias.edu/~jaredw/FRGLecture.pdf}{pdf}


\item Liang Xiao, notes on $p$-divisible groups, \href{http://math.uchicago.edu/~lxiao/files/notes/p-Divisble%20Groups.pdf}{pdf}



\end{itemize}
\hypertarget{further_development_of_the_theory}{}\subsubsection*{{Further development of the theory}}\label{further_development_of_the_theory}

\begin{itemize}%
\item Paul Goerss, p-divisible groups and Lurie's realization result, 2008, \href{http://www.math.ku.dk/~jg/homotopical2008/goerss.lec9.pdf}{pdf slides}


\item Jacob Lurie, [[A Survey of Elliptic Cohomology]], section 4.2, \href{http://www.math.harvard.edu/~lurie/papers/survey.pdf}{pdf}


\item Peter Scholze, Moduli of p-divisible groups, \href{http://arxiv.org/abs/1211.6357}{arxiv}


\item Thomas Zink, a dieudonn\'e{} theory for p-divisible groups, \href{http://www.math.uni-bielefeld.de/~zink/CFTpaper.pdf}{pdf}


\item Thomas Zink, list of publications and preprints, \href{http://www.math.uni-bielefeld.de/~zink/z_publ.html}{web}


\item T. Zink, On the slope filtration, Duke Math. Journal, Vol.109 (2001), No.1, 79-95, \href{http://www.math.uni-bielefeld.de/~zink/slopes.pdf}{pdf}


\item T. Zink, the [[display of a p-divisible group|display of a formal p-divisible group]], to appear in [[Astérisque]], \href{http://www.math.uni-bielefeld.de/~zink/display.pdf}{pdf}


\item T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkh\"a{}user 2001, \href{http://www.math.uni-bielefeld.de/~zink/Texel.pdf}{pdf}



\end{itemize}
[[!redirects p-divisible group]] [[!redirects p-divisible groups]]



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