Seminar on adic spaces

Date: October - December 2012, each Monday at 14.00h (exceptions listed below); there are two 90-minutes talks.

Place: 4, place Jussieu (Paris, France); room 417, corridor 15-16 (4th floor)

Description of the seminar:

The goal of the seminar is two-fold. Firstly, we aim to report on the latest developments in the p-adic Hodge theory made by the use of adic spaces (and in particular perfectoid spaces). This includes some preliminary material on adic spaces in general [Hu], and then a discussion of perfectoid spaces [Sch1], Scholze's approach to the p-adic comparison theorem by the pro-etale site on adic spaces and the Fargues-Fontaine curve (from the perspective of perfectoid spaces) [FF].

Secondly, we hope to extend and apply those results. We will report on the work of Scholze-Weinstein ([SW], see also [We1] and [We2]) on Rapoport-Zink spaces at infinity which are proved to be perfectoid. This point of view permits to shorten and simplify a proof of Fargues-Faltings isomorphism between Lubin-Tate tower and Drinfeld tower ([Fal]). Then we hope to discuss p-divisible groups over (algebraically closed) perfectoid fields and use this to simplify (and extend to new cases) a proof of Harris conjecture (conjecture 5.2 from [Ha], [Mant]). We will also try to explore a link between Rapoport-Zink spaces at infinity and \phi-modules over adic spaces.

References:

[Fal] G. Faltings "Coverings of p-adic period domains"

[FF] L. Fargues, J.-M. Fontaine, "Courbes et fibrés vectoriels en théorie de Hodge p-adique"

[Far] L. Fargues, "Au dela de la courbe", notes from a talk

[Fo] J.M. Fontaine "Perfectoides, presque purete et monodromie-poids" n. 1057 Seminaire Bourbaki, Juin 2012

[GR] O. Gabber, L. Ramero "Almost ring theory", book

[Ha] M. Harris "Local Langlands correspondences and vanishing cycles on Shimura varieties", ECM 2000 talk

[Hu] R. Huber "Etale cohomology of Rigid Analytic Varieties and Adic Spaces", book

[Mant] E. Mantovan "On non-basic Rapoport-Zink spaces"

[Sch1] P. Scholze "Perfectoid spaces"

[Sch2] P. Scholze "p-adic Hodge theory for rigid-analytic varieties"

[SW] P. Scholze, J. Weinstein, "Moduli of p-divisible groups"

[We1] J. Weinstein "Formal vector spaces over a local field of positive characteristic"

[We2] J. Weinstein "Semistable models for modular curves of arbitrary level"

Timetable:

1st October

Lecture 1 (Przemyslaw Chojecki): Introduction

Contents: From perfectoid spaces via the Fargues-Fontaine curve to Rapoport-Zink spaces at infinity.

Lecture 2 (Florent Martin): Adic spaces I

Contents: Definition of adic spaces and basic properties. Morphisms between adic spaces: finite, quasi-finite, weakly finite, separated, proper, partially proper, unramified, smooth, etale ([chapter 1, Hu],[Sch1]).

8th October

Lecture 1 (Arthur-Cesar Le Bras): Adic spaces II

Contents: Etale site of adic spaces. Pseudo-adic spaces. Projective systems of adic spaces. Strict localisations and stalks of higher direct image ([chapter 2, Hu],[Sch1]).

Lecture 2 (Benjamin Schraen): Almost ring theory

Contents: Basic properties of almost modules: flatness, projectivity, finite generation ([chapter 2 and 3, GR], [Sch1]). Structure of almost finitely generated modules ([Sch2]).

15th October

Lecture 1 (Eugen Hellmann): Perfectoid spaces. Summary of results

Contents: Definition of perfectoid spaces. Tilting. Faltings' almost purity theorem ([Sch1], [Fo]).

Lecture 2 (Olivier Taibi): Pro-etale site

Contents: Perfectoid spaces as a basis for pro-etale topology on adic spaces. Structure sheaves and Fontaine's sheaves on a pro-etale site of adic spaces ([chapter 3, 4 and 6, Sch2]).

22th October

Lecture 1 (Wieslawa Niziol): Towards p-adic comparison theorem

Contents: Finiteness of p-adic etale cohomology ([chapter 5, Sch2]). Proof of the first comparison isomorphism ([Corollary 6.19, Sch2]).

Lecture 2 (Wieslawa Niziol): P-adic comparison theorem with coefficients

Contents: Filtered modules with integrable connections. Applications ([chapter 7 and 8, Sch2]).

29th October

All Saints (break)

5th November

Lecture 1 (Florent Martin): Formal vector spaces and p-divisible groups

Contents: Basic definitions. Reminder on p-divisible groups. Universal covers and formal vector spaces ([chapters 2 and 3, We1], [chapter 8, FF]).

Lecture 2 (Arthur-Cesar Le Bras): Lubin-Tate tower at infinity in positive characteristic

Contents: Definition of a deformation problem. Lubin-Tate tower at infinity as a perfectoid space ([chapters 4, 5 and lemma 6.0.1, We1]).

12th November

Lecture 1 (Arijit Sehanobish): The Fargues-Fontaine curve I

Contents: Definition of the curve. Basic properties ([Theoreme 10.2, FF]).

Lecture 2 (Przemyslaw Chojecki): The Fargues-Fontaine curve II

Contents: Classification of vector bundles. Kisin's theory on the curve and p-divisible groups ([Far]).

19th November

Lecture 1 (Przemyslaw Chojecki): Classification of p-divisible groups over O_C I

Contents: On Dieudonne module over semi-perfect rings ([chapter 4, [SW]).

Lecture 2 (Eugen Hellmann): Classification of p-divisible groups over O_C II

Contents: Universal covers of p-divisible groups. Vector bundles on the Fargues-Fontaine curve. Classification ([chapter 5, [SW]).

26th November

Lecture 1 (Arthur-Cesar Le Bras): Rapoport-Zink spaces at infinity and applications I

Contents: Basic defintions. The image of the period morphism ([chapter 6, [SW]).

Lecture 2 (Haoran Wang): Rapoport-Zink spaces at infinity and applications II

Contents: Duality of Rapoport-Zink spaces. Equivariant covers of Drinfeld's upper half-space ([chapter 7, [SW]).

3rd December

(no seminar)

10th December

Lecture (Jean-Marc Fontaine)

Contents: Generalizations of the curve.