Seminar on arithmetic geometry:

p-adic Hodge theory via Fargues-Fontaine curves

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.

Introduction:

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].

References:

[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"

Timetable:

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.

06.03.2012

(Przemyslaw Chojecki) Curves

Description: General definition of curves after chapter 1 in [1]. Example with B_cris.

13.03.2012

(Piotr Pstrągowski) Vector bundles on curves

Description: Basic facts about vector bundles on curves after chapter 2 in [1].

20.03.2012

(Agnieszka Bodzenta) Harder-Narasimhan filtration. Examples

Description: General formalism of semistable objects and HN filtration after chapter 3 in [1]. Examples.

27.03.2012

(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].

03.04.2012

(Maciej Zdanowicz) Fontaine's rings I

Description: Witt vectors. Definitions and properties of rings B^+ and R after chapter 5 in [1] and [3].

10.04.2012

Easter break

17.04.2012

Organisational meeting, break due to workshop on elliptic curves

24.04.2012

(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.

01.05.2012

Holidays

08.05.2012

(Piotr Pstrągowski)

15.05.2012

(Sofia Tirabassi)

22.05.2012

(Bartosz Naskręcki) - moved to 24.05.2012;

29.05.2012

(Adrian Langer) - the last talk this semester