Barsotti–Tate group

Barsotti–Tate group
Name	Barsotti–Tate group
Field	Algebraic geometry, Number theory
Introduced	1960s

Barsotti–Tate group A Barsotti–Tate group is a type of p-divisible group arising in algebraic geometry and arithmetic, central to studying abelian varieties, formal groups, and deformation problems in characteristic p. It connects work of Carla Maria Barsotti, Jean-Marc Fontaine, and John Tate to foundations laid by Alexander Grothendieck, Jean-Pierre Serre, and Pierre Deligne, and plays a role in the Langlands program, the theory of Shimura varieties, and crystalline cohomology.

Definition and basic properties

A Barsotti–Tate group is a direct system of finite commutative group schemes whose transition maps are multiplication by p, defined over a base scheme such as Spec of a p-adic ring, a local field like Q_p, or a curve like the projective line. It generalizes the p-power torsion subgroup of an abelian variety studied by André Weil and is equipped with height and dimension invariants analogous to Hasse invariants used by Helmut Hasse and Ernst Witt. Basic properties tie to Tate modules studied by John Tate, to Néron models associated to André Néron, and to duality theories related to Alexander Grothendieck and Jean-Pierre Serre.

Examples and classification

Standard examples include the p-divisible group of an elliptic curve over a field treated in work by Gerd Faltings, Jean-Pierre Serre, and Pierre Deligne; multiplicative and étale p-divisible groups linked to cyclotomic characters studied by Ken Ribet and Barry Mazur; and Barsotti–Tate groups arising from Jacobians of curves considered by David Mumford and Friedrich Hirzebruch. Classification results over algebraically closed fields employ methods from Serre–Tate theory, crystalline Dieudonné theory by Jean-Michel Fontaine, and slope filtrations as in the work of Pierre Deligne and Jean-Loup Waldspurger. These classifications connect to moduli problems studied by Michael Artin, Alexander Grothendieck, and Michael Rapoport.

Relation to p-divisible groups and formal groups

Barsotti–Tate groups are synonymous with p-divisible groups in the literature of Grothendieck, Tate, and Barsotti, and relate intimately to formal groups introduced by John Tate and Michel Lazard. The connection to Lubin–Tate formal groups appears in local class field theory developed by Emil Artin and John Lubin, while Serre–Tate coordinates are used in deformation problems for abelian varieties studied by Jean-Pierre Serre, John Tate, and Nicholas Katz. These relations inform the study of reduction types of abelian varieties handled by Gerd Faltings and Jean-Pierre Serre, and influence the construction of integral models of Shimura varieties due to James Milne and Michael Harris.

Dieudonné modules and classification over perfect fields

Over a perfect field of characteristic p, the classification of Barsotti–Tate groups uses Dieudonné modules formulated by Jean Dieudonné and refined by Pierre Cartier, Grothendieck, and Jean-Marc Fontaine. The contravariant Dieudonné functor translates group schemes into modules with Frobenius and Verschiebung operators reminiscent of constructions in the work of Alexander Grothendieck, Jean-Pierre Serre, and Nicholas Katz. Slope decompositions parallel the Newton polygon techniques developed by Bernard Dwork and Pierre Deligne, and links to crystalline cohomology relate to theories by Jean-Michel Fontaine, Pierre Berthelot, and Arthur Ogus.

Deformation theory and moduli

Deformation theory of Barsotti–Tate groups uses methods from the study of moduli stacks by Alexander Grothendieck and Deligne–Mumford, with Serre–Tate theory providing local coordinates for deformations of abelian varieties as in work by Jean-Pierre Serre and John Tate. Moduli spaces of p-divisible groups and their integral models are central to the research by Michael Rapoport, Thomas Zink, and Robert Langlands, and connect to reduction theory studied by Gerd Faltings and Richard Taylor. These moduli problems inform the construction of Rapoport–Zink spaces and local Shimura varieties considered by Michael Harris and Laurent Fargues.

Applications in arithmetic geometry and Galois representations

Barsotti–Tate groups underpin the study of p-adic Galois representations via Tate modules and Fontaine's rings, influencing the proof of the Mordell conjecture by Gerd Faltings and the proofs of modularity theorems involving Andrew Wiles and Richard Taylor. They appear in the formulation of the Fontaine–Mazur conjecture and the p-adic local Langlands correspondence developed by Pierre Colmez, Laurent Clozel, and Christophe Breuil. Connections extend to Iwasawa theory studied by Kenkichi Iwasawa and Barry Mazur, to p-adic Hodge theory of Jean-Michel Fontaine and Kazuya Kato, and to the arithmetic of Shimura varieties studied by James Milne and Michael Harris.

Category:Algebraic geometry