crystalline Dieudonné theory

crystalline Dieudonné theory
Name	Crystalline Dieudonné theory
Field	Algebraic geometry, Arithmetic geometry
Introduced	1960s–1990s
Notable contributors	Jean Dieudonné, Alexander Grothendieck, Pierre Berthelot, Jean-Marc Fontaine, Christophe Breuil, Kazuya Kato, Luc Illusie, Jean-Pierre Serre, Gerd Faltings, Alexander Beilinson

crystalline Dieudonné theory is a framework in arithmetic geometry that relates p-adic Hodge theory objects to algebraic structures controlling p-divisible groups and deformation theory via crystalline cohomology. It refines classical Dieudonné module theory using the crystalline site and crystalline cohomology to classify and study isogeny classes, families, and integral structures in characteristic p-adic settings. The theory sits at the crossroads of work by Jean Dieudonné, Alexander Grothendieck, Pierre Berthelot, Jean-Marc Fontaine, and later contributors like Christophe Breuil and Kazuya Kato.

Introduction

Crystalline Dieudonné theory connects the study of p-divisible groups, formal group laws, and deformation problems to linear algebraic data defined on the crystalline site of a scheme over a perfect field of characteristic p-adic. Originating from the confluence of Dieudonné module methods and crystalline cohomology, it provides functorial equivalences between geometric objects and modules equipped with Frobenius and filtration structures, enabling interaction with p-adic Hodge theory and integral models of Shimura varietys. The subject builds on foundations laid by figures such as Alexander Grothendieck, Luc Illusie, and Pierre Berthelot.

Historical development and motivation

Development traces through the mid-20th century: Jean Dieudonné and John Tate formulated algebraic classifications of finite commutative group schemes and p-divisible groups, while Grothendieck proposed crystalline cohomology as a Weil cohomology affording a good theory in characteristic p-adic. Subsequent advances by Luc Illusie and Pierre Berthelot formalized crystals and the crystalline site; Jean-Marc Fontaine linked Galois representations to filtered modules with Frobenius, influencing crystalline Dieudonné perspectives. Later enhancements by Christophe Breuil, Gerd Faltings, and Kazuya Kato addressed integral structures, displays, and links with (phi,Gamma)-module techniques. Motivations include classification of abelian varietys over p-adic bases, integral comparison theorems, and constructing canonical models of Shimura varietys.

Foundations: crystals, displays, and Dieudonné modules

Core objects include crystals in quasi-coherent sheaves on the crystalline site as developed by Pierre Berthelot and Luc Illusie, classical Dieudonné modules from Jean Dieudonné and John Tate, and Zink's notion of displays influenced by Thomas Zink and Kazuya Kato. A typical crystalline Dieudonné module is a crystal equipped with a Frobenius endomorphism and a filtration reflecting Hodge structures studied by Jean-Marc Fontaine and Gerd Faltings. The category-theoretic framework invokes ideas from Alexander Grothendieck's topos theory and uses deformation methods also employed by Nicholas Katz and Pierre Deligne. Displays give integral models bridging Cartier theory and Breuil–Kisin modules investigated by Christophe Breuil and Mark Kisin.

Crystalline Dieudonné functor and classification theorems

The crystalline Dieudonné functor assigns to a p-divisible group over a scheme its Dieudonné crystal with Frobenius and Verschiebung operators; foundational equivalences were proved in stages by Jean Dieudonné, John Tate, Luc Illusie, and Jean-Marc Fontaine. Integral classification results, including equivalences between categories of nilpotent displays and p-divisible groups, were established by Thomas Zink and refined by Christophe Breuil and Mark Kisin for Breuil–Kisin modules. Comparison theorems relate crystalline Dieudonné modules to etale cohomology and de Rham cohomology via works of Pierre Berthelot, Gerd Faltings, and Jean-Marc Fontaine, yielding classification and rigidity statements used in studying isogeny classes and deformation rings associated with Galois representations.

Examples and computations

Concrete computations include Dieudonné crystals of formal group laws such as the additive and multiplicative groups, Lubin–Tate group laws studied by Jonathan Lubin and John Tate, and the Dieudonné modules of elliptic curves with supersingular or ordinary reduction analyzed by Jean-Pierre Serre and John Tate. Explicit Breuil modules for potentially semi-stable Galois representations and Breuil–Kisin modules for finite flat group schemes have been computed in works by Christophe Breuil, Mark Kisin, and Matthew Emerton. Computations often use techniques from local field theory associated to p-adic extensions and deformation theory as in the work of Barry Mazur.

Relations to p-divisible groups, displays, and prismatic theory

The theory gives equivalences between categories of p-divisible groups and suitably defined crystalline Dieudonné modules or displays, an approach expanded by Thomas Zink and Kazuya Kato. Recent developments connect crystalline Dieudonné theory to prismatic cohomology and prismatic Dieudonné theory initiated by Bhargav Bhatt and Peter Scholze, linking prisms to Breuil–Kisin frameworks developed by Christophe Breuil and Mark Kisin. These connections integrate methods from perfectoid spaces by Peter Scholze and integral p-adic Hodge theory by Bhargav Bhatt, Gerd Faltings, and Jean-Marc Fontaine, enabling new classifications for families over higher-dimensional bases and implications for Langlands program contexts studied by Michael Harris and Laurent Fargues.

Applications and further directions

Applications include classification of integral models of Shimura varietys, study of reduction types of abelian varietys, and contribution to the construction and analysis of Galois representations arising in the Langlands correspondence as pursued by Michael Harris, Richard Taylor, and Laurent Fargues. Ongoing directions involve deepening links with prismatic cohomology by Bhargav Bhatt and Peter Scholze, refining integral comparison theorems by Gerd Faltings and Jean-Marc Fontaine, and extending classification results for families over bases with singularities following research by Kazuya Kato, Mark Kisin, and Christophe Breuil. Future work aims to integrate crystalline Dieudonné perspectives into broader programs in arithmetic geometry and the study of moduli spaces tied to automorphic forms and the Langlands program.

Category:Arithmetic geometry