p-divisible group

p-divisible group
Name	p-divisible group
Field	Algebraic geometry; Number theory; Representation theory
Related	p-adic Hodge theory; Dieudonné module; Formal group

p-divisible group

A p-divisible group is a central object in modern algebraic geometry and number theory that encodes infinitesimal and torsion phenomena attached to abelian varieties, formal groups, and Galois representations. Originating in the work of Alexander Grothendieck and further developed by Jean-Pierre Serre, John Tate, and Pierre Deligne, p-divisible groups bridge arithmetic geometry with p-adic Hodge theory and the theory of moduli of abelian varietys. They play a crucial role in the study of reduction of elliptic curves, the classification of finite flat group schemes, and the construction of integral models for Shimura varieties such as those studied by George Pappas and Michael Rapoport.

Definition and basic properties

A p-divisible group over a base scheme S is an inductive system of finite flat commutative group schemes {G_n, i_n} where each G_n is of p-power order p^n and the transition maps i_n: G_n → G_{n+1} identify G_n with the p^n-torsion subgroup of G_{n+1}; this structure was formalized in the foundational work of Grothendieck and Tate. For a scheme S over a prime p, p-divisible groups admit invariants such as height and dimension introduced in the study of abelian schemes and Barsotti–Tate groups; these invariants behave functorially under base change and interact with notions from crystalline cohomology and Hodge theory. Over perfect fields of characteristic p, p-divisible groups decompose into étale and connected components, a phenomenon closely analyzed by Jean-Marc Fontaine and Gerd Faltings in the context of p-adic representations and the study of Tate modules.

Examples and constructions

Basic examples include the p-power torsion of an abelian variety A → S, giving A[p^∞], and the formal multiplicative group μ_{p^∞} and the formal additive group Q_p/Z_p arising in the theory of local fields and Galois cohomology. The p-divisible group attached to an elliptic curve E encodes reduction types studied by Serre and Tate and connects to the Kodaira–Néron classification of reduction. One constructs further examples via extension and Cartier duality: Cartier duality exchanges multiplicative and étale factors as in the duality between μ_{p^∞} and Q_p/Z_p, a duality exploited by Pierre Cartier and used extensively in the work of Mazur on deformation rings. p-divisible groups also arise from Drinfeld modules and Lubin–Tate formal groups in the context of local Langlands correspondences studied by Drinfeld and Harris–Taylor.

Classification and structure theory

Over algebraically closed fields of characteristic p, classification results reduce p-divisible groups to Dieudonné modules and Newton polygons, paralleling the theory of isocrystals developed by Katz and Dieudonné. The slope decomposition yields isoclinic components indexed by rational slopes, connected to the classification of isocrystals with Frobenius by Dieudonné and refined by Manin and Grothendieck. Over complete discrete valuation rings, classification interacts with the theory of displays and Breuil–Kisin modules introduced by Laurent Fargues, Clausen–Scholze contexts, and Christophe Breuil, providing integral structures for p-adic Galois representations investigated by Breuil and Kisin. The study of endomorphism algebras of p-divisible groups involves results of Honda–Tate and connections to automorphism groups considered by Shimura and Taniyama.

Connections to formal groups and Dieudonné theory

p-divisible groups generalize formal group laws and are classified by covariant and contravariant Dieudonné modules over perfect fields, a theory initiated by Jean Dieudonné and expanded by Grothendieck and Demazure. The Dieudonné equivalence relates p-divisible groups to modules over the Dieudonné ring with Frobenius and Verschiebung operators, connecting to crystalline cohomology developed by Berthelot and Ogus. Formal group laws arising from complex multiplication by imaginary quadratic fields studied by Shimura and Tate produce specific p-divisible groups whose Tate modules realize Galois representations central to Iwasawa theory and the study of L-functions. The interaction with the theory of Lubin–Tate deformation spaces also links to the work of Gross and Hopkins on stable homotopy theory.

Cohomological and arithmetic applications

p-divisible groups are fundamental in the study of integral p-adic Hodge theory, crystalline comparison theorems of Faltings and Tsuji, and the formulation of the Fontaine–Mazur conjecture in the representation theory of Gal( Qbar / Q ). They provide the local p-adic data for the study of reduction of Shimura varieties by Kisin and Rapoport–Zink spaces and appear in the proof of modularity results by Wiles and Taylor–Wiles via deformation rings and Hecke algebras studied by Mazur and Diamond. Cohomological invariants from p-divisible groups feed into the theory of special cycles on moduli spaces investigated by Kudla and Rapoport and inform arithmetic intersection theory developed by Gillet–Soulé.

Deformations and moduli spaces

Deformation theory of p-divisible groups yields formal moduli spaces such as the Rapoport–Zink spaces and Lubin–Tate towers studied by Rapoport and Zink, which play a central role in local Langlands correspondence work by Harris–Taylor and in the construction of local models by Pappas and Kottwitz. These moduli spaces parameterize isogeny classes and stratifications by Newton polygons and Ekedahl–Oort types analyzed by Oort, Wedhorn, and Vasiu, linking to global moduli of abelian varietys and integral models of Shimura varieties explored by Deligne and Milne. Deformation rings arising from p-divisible groups are also pivotal in the study of Galois deformation problems by Mazur and in modularity lifting techniques developed by Skinner–Wiles and Calegari–Geraghty.

Category:Algebraic groups