Dieudonné theory: from classical to prismatic
Finite flat group schemes and $p$-divisible groups are fundamental objects in arithmetic geometry, as they arise in studying ($p$-power) torsion points of abelian varieties. Dieudonné theory provides an explicit way to understand finite flat group schemes / $p$-divisible groups via concrete linear algebraic objects, and thereby has wide-ranging applications in arithmetic geometry. This course will give an overview of Dieudonné theory, from classical results to modern developments. We will begin with foundations on finite flat group schemes / $p$-divisible groups, and study the classical results due to Dieudonné, Fontaine, Berthelot, and others. Then we will move on to discussing the result of Scholze-Weinstein, and give a quick overview of the prismatic Dieudonné theory due to Anschütz-Le Bras.
Lecturer
Date
16th September, 2026 ~ 6th January, 2027
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:05 | Qiuzhen | ZOOM 08 | 787 662 9899 | BIMSA |
Prerequisite
Basic graduate-level commutative algebra / algebraic geometry
Syllabus
1. Finite flat groups schemes and $p$-divisible groups (Oort, Tate)
2. Classical Dieudonné theory over perfect field of char. $p$
3. Crystalline sites and Dieudonné crystals of abelian schemes (Berthelot-Breen-Messing)
4. Dieudonné theory over perfect valuation ring of char. $p$ (Berthelot) and its generalization to perfect ring of char. $p$
5. Dieudonné theory over $\mathcal{O}_C$ (Scholze-Weinstein)
6. Overview of prisms and prismatic site
7. Prismatic Dieudonné theory (Anschütz-Le Bras)
2. Classical Dieudonné theory over perfect field of char. $p$
3. Crystalline sites and Dieudonné crystals of abelian schemes (Berthelot-Breen-Messing)
4. Dieudonné theory over perfect valuation ring of char. $p$ (Berthelot) and its generalization to perfect ring of char. $p$
5. Dieudonné theory over $\mathcal{O}_C$ (Scholze-Weinstein)
6. Overview of prisms and prismatic site
7. Prismatic Dieudonné theory (Anschütz-Le Bras)
Reference
1. Oort ``Commutative group schemes"
2. Tate ``$p$-divisible groups"
3. Fontaine ``Groupes $p$-divisibles sur les corps locaux"
4. Berthelot-Breen-Messing ``Théorie de Dieudonné cristalline II"
5. Berthelot ``Théorie de Dieudonné sur un anneau de valuation parfait"
6. Fargues ``Groupes analytiques rigides $p$-divisibles"
7. Scholze-Weinstein ``Moduli of $p$-divisible groups"
8. Anschütz-Le Bras ``Prismatic Dieudonné theory"
2. Tate ``$p$-divisible groups"
3. Fontaine ``Groupes $p$-divisibles sur les corps locaux"
4. Berthelot-Breen-Messing ``Théorie de Dieudonné cristalline II"
5. Berthelot ``Théorie de Dieudonné sur un anneau de valuation parfait"
6. Fargues ``Groupes analytiques rigides $p$-divisibles"
7. Scholze-Weinstein ``Moduli of $p$-divisible groups"
8. Anschütz-Le Bras ``Prismatic Dieudonné theory"
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
No
Notes Public
Yes
Language
English
Lecturer Intro
Yong Suk Moon joined BIMSA in 2022 fall as an assistant professor. His research area is number theory and arithmetic geometry. More specifically, his current research focuses on p-adic Hodge theory, Fontaine-Mazur conjecture, and p-adic Langlands program. He completed his Ph.D at Harvard University in 2016, and was a Golomb visiting assistant professor at Purdue University (2016-19) and a postdoctoral researcher at University of Arizona (2019 - 22).