Dieudonné theory

Dieudonné theory
Name	Dieudonné theory
Field	Algebraic geometry; Arithmetic geometry; Algebra
Introduced	1950s
Key contributors	Jean Dieudonné, Alexander Grothendieck, Pierre Deligne, John Tate, Michel Demazure, Jean-Pierre Serre, Nicholas Katz, Gerard Laumon, Luc Illusie, Christophe Breuil, Berthelot, Fontaine, Faltings

Dieudonné theory provides an algebraic classification of certain group schemes and p-adic-related structures via linear algebra over noncommutative rings. Developed in the mid‑20th century, it connects the work of Jean Dieudonné, Alexander Grothendieck, Pierre Deligne, and John Tate with later advances by Nicholas Katz, Luc Illusie, and Michel Demazure, and plays a central role in modern work of Gerd Faltings, Jean-Marc Fontaine, and Christophe Breuil on p-adic Hodge theory and Shimura varieties.

Introduction

Dieudonné theory began as an approach to classify finite commutative group schemes and p-divisible groups over fields of characteristic p. Early foundations were laid by Jean Dieudonné and Michel Demazure and extended by Alexander Grothendieck and Jean-Pierre Serre in the context of SGA seminars influenced by Pierre Deligne. The theory replaces geometric objects by modules over the Dieudonné ring, making contact with Galois representations, étale cohomology, crystalline cohomology, and the work of John Tate on Tate modules.

Dieudonné Modules and Dieudonné Rings

Dieudonné modules encode finite commutative group schemes and p-divisible groups as modules endowed with Frobenius and Verschiebung operators. The central algebraic object is the Dieudonné ring, a noncommutative ring generated by operators F and V with relations influenced by Frobenius endomorphisms studied by Évariste Galois-inspired arithmeticists and formalized by Alexander Grothendieck in SGA 3. Key contributors such as Luc Illusie and Nicholas Katz clarified connections with Witt vectors and the Witt ring used by Henri Cartan-era algebraists. The formalism interacts with structures studied by Michel Demazure, Jean-Pierre Serre, and Pierre Deligne, and with classification techniques in Tate's work on abelian varieties.

Classification of p-divisible Groups

Dieudonné theory yields a classification of p-divisible groups over perfect fields of characteristic p by isocrystals with Frobenius, linking to Newton polygons and Hasse invariants. This classification was refined by contributions from Nicholas Katz, Jean-Marc Fontaine, Gerard Laumon, and Gerd Faltings and features prominently in the study of reduction of abelian varieties and Shimura varieties by Richard Taylor and Michael Harris. The classification connects to the Honda–Tate theory as developed in the work of Taira Honda and John Tate, and to slope filtrations used by Ofer Gabber and Luc Illusie.

Crystalline Dieudonné Theory and Displays

Crystalline Dieudonné theory extends the classical theory to schemes over p-adic bases via crystalline cohomology and the theory of displays. Pioneering work by Berthelot, Grothendieck, and Luc Illusie established the crystalline perspective; later refinements and integral structures were introduced by Christophe Breuil, Jean-Marc Fontaine, Gerd Faltings, and Francesco Baldassarri. The notion of displays, as advanced by Th. Zink and proponents in p‑adic Hodge theory, created bridges to the integral models studied in research by Mark Kisin, Brian Conrad, and Keerthi Madapusi Pera for applications to moduli spaces and reduction theory of Shimura varieties.

Connections with Formal Groups and Abelian Varieties

Dieudonné theory tightly links formal groups, abelian varietys, and their p-divisible groups via the Serre–Tate theorem and deformation theory developed by Jean-Pierre Serre, John Tate, and Pierre Deligne. The theory informs the study of complex multiplication by connecting endomorphism algebras examined by Goro Shimura and Yutaka TANIYAMA-era results with integral invariants used by Nicholas Katz and Gerard Laumon. It underpins the analysis of ordinary locus and supersingular locus phenomena in reduction theory addressed by Eichler, Shimura, and Gross-style investigators, and helps describe isogeny classes as in Honda and Tate.

Applications and Further Developments

Dieudonné theory influences p-adic Hodge theory, the proof of cases of the Taniyama–Shimura conjecture by Andrew Wiles and Richard Taylor, the study of Galois representations in works by Pierre Deligne and Richard Taylor, and the integral models of Shimura varieties by M. Kisin and M. Rapoport. Recent activity involves refinements by Christophe Breuil, Mark Kisin, Tobias C. C. Liu, and Bhargav Bhatt linking derived methods of algebraic geometry and advances in perfectoid spaces by Peter Scholze with classical Dieudonné frameworks. Ongoing research explores interactions with motivic cohomology, automorphic forms studies by Michael Harris and Ramakrishnan, and arithmetic applications pursued by Gerd Faltings and collaborators.

Category:Algebraic geometry