Seminar on arithmetic geometry:
p-adic Hodge theory via Fargues-Fontaine curves

Date: each Tuesday, from 14.15 to around 16.30, room 3180 at the University of Warsaw.   
         For the information about upcoming talks, see bottom of the page.  

Classic Hodge theory furnishes an isomorphism between the Betti cohomology and the de Rham cohomology of complex manifolds. The p-adic Hodge theory arose by trying to obtain the same kind of result over p-adic fields.

The B_dR conjecture (now a theorem) of Fontaine says that the etale cohomology of a smooth, proper scheme over Spec Z_p is isomorphic after tensoring by a Fontaine's ring B_dR to the de Rham cohomology. As the etale cohomology is in particular a p-adic Galois representation (i.e. a continuous representation of a Galois group of Q_p over its algebraic closure), one is also interested in the study of p-adic Galois representations on their own.
This is the fundamental concern of the p-adic Hodge theory.

The B_dR and other Fontaine's rings allow a passage between p-adic Galois representations and much simpler, linear-algebraic objects (so-called (\phi,\Gamma)-modules). This kind of equivalence is used in the work of Fargues-Fontaine to compare semistable sheaves with representations (see below).

Description of the seminar:
Recently, Fargues and Fontaine in [1] has introduced a class of projective curves which "encodes" p-adic Hodge theory, that is, the category of semistable vector bundles (with some additional conditions) on those curves is equivalent to the category of p-adic Galois representations (general, de Rham, crystalline, etc. depending on the conditions put on the bundles). Using this interpretation, they were able to prove some classic results in the p-adic Hodge theory by using geometrical methods (e.g. semistable sheaves on curves, Harder-Narasimhan filtration, etc).

The objective of the seminar is to understand the results from [1] and then approach by classic geometrical methods some natural questions related to the Fargues-Fontaine curves and the p-adic Hodge theory. We will also discuss the theory around it. 

Knowledge of the p-adic Hodge theory is not neccessary to follow the seminar, it suffices to have a certain comprehension of algebraic geometry.

In [2], one can find a shorter survey of the results proven in [1].
For beautiful notes on p-adic Hodge theory, see [3]. Other useful sources are [4] and [5].

[1] L. Fargues, J.-M. Fontaine "Courbes et fibrés vectoriels en théorie de Hodge p-adique"
[2] L. Fargues, J.-M. Fontaine "Vector bundles and p-adic Galois representations"
[3] O. Brinon, B. Conrad "Notes on p-adic Hodge theory"
[4] L. Berger "Galois representations and (\phi, \Gamma)-modules"
[5] J.-M. Fontaine, Y. Ouyang "Theory of p-adic Galois representations"


28.02.2012  Overview
(Przemyslaw Chojecki): Introduction, from classical Hodge theory to p-adic, p-adic Galois representations, Fontaine's rings, geometric interpretation via the curve of Fargues-Fontaine. Explanation of objectives of the seminar.

(Przemyslaw Chojecki) Curves
Description: General definition of curves after chapter 1 in [1]. Example with B_cris.

(Piotr Pstrągowski) Vector bundles on curves
Description: Basic facts about vector bundles on curves after chapter 2 in [1].

(Agnieszka Bodzenta) Harder-Narasimhan filtration. Examples
Description: General formalism of semistable objects and HN filtration after chapter 3 in [1]. Examples.

(Adrian Langer) Classification of vector bundles
Description: Definition of Riemann spheres and generalized Riemann spheres, and classification of vector bundles on them after chapter 4 in [1].

(Maciej Zdanowicz) Fontaine's rings I
Description: Witt vectors. Definitions and properties of rings B^+ and R after chapter 5 in [1] and [3].

Easter break

Organisational meeting, break due to workshop on elliptic curves

(Maciek Zdanowicz) Continuation on adjoint functors W and R.

(Bartosz Naskręcki) Fontaine's rings II
Description: Period rings B_dR, B_cris, B_st and Galois representations.


(Piotr Pstrągowski)

(Sofia Tirabassi)

(Bartosz Naskręcki)  - moved to 24.05.2012; 

(Adrian Langer) - the last talk this semester

Institut Mathématique de Jussieu - Fondation Sciences Mathématiques de Paris - Fédération de recherche Mathématiques Paris Centre