### V5A4 - Selected Topics in Algebraic Geometry - p-adic Hodge theory

#### Winter Semester 2015/16, Wednesday 16.15-18.00, Seminar room 0.007

This course will provide an introduction to p-adic Hodge theory, a major area of arithmetic geometry, through p-divisible groups (these are also known as Barsotti-Tate groups, since the phrase "p-divisible group'' is so ambiguous). Historically, results and conjectures surrounding p-divisible groups were the main stimulus for the development of p-adic Hodge theory, and they continue to be highly relevant in modern research.

We will cover the main results in John Tate's 1967 paper "p-Divisible Groups'', though often with different proofs and providing many more details and examples; this will include affine group schemes, p-divisible groups, Tate-Sen theory, Hodge-Tate decomposition of a p-divisible group, applications to abelian varieties. Then we may do some Dieudonné theory to study p-divisible groups in characteristic p.

#### Prerequisites:

Standard commutative algebra (e.g., tensor products of k-algebras; local rings) will be sufficient for most of the course. Basic category theory (e.g., the notion of a functor) will be useful. The main applications of the theory concern abelian varieties, which require some knowledge of algebraic geometry, but these will not appear until later in the course; for most of the course algebraic geometry is not necessary.

### Some useful resources

• John Tate "p-Divisible groups". The original source, but rather terse. pdf.
• S. Shatz "Group Schemes, Formal Groups, and p-Divisible Groups" (a chapter of G. Cornell and J. Silverman's "Arithmetic Geometry"). A good exposition of most of Tate's paper. Can be found online.
• Thomas J. Haines "Notes on Tate's p-Divisible Groups". Some extra details on Tate's paper. pdf
• M. Demazure, "Lectures on p-Divisible Groups". Mainly on p-divisible group in characteristic p, which will not be our focus, but the first two chapters may be useful. link
• Richard Pink, Lecture notes of a course on finite group schemes. A good source for the general theory of finite group schemes. link
• Ildar Gaisin's masters dissertation "From the Hodge-Tate conjecture to p-adic Hodge theory". Provides a good summary of p-divisible groups in chapter 2. link
• Some notes by Brandon Levin: see Notes on Tate's article on p-divisible groups.