Study Group on the Fargues–Fontaine Curve

Overview

The aim of this study group is to define and study the basic properties of the Fargues–Fontaine curve, a geometric object central to many recent developments in arithmetic geometry. The program is structured to provide both foundational knowledge and insights into the more advanced aspects of the theory.

We will start with an introduction to (or refresher about) perfectoid fields. Then, we will define the schematic Fargues–Fontaine curve \(X\), study the structure of its divisors, its Picard group, and show that \(X\) is, in fact, a Dedekind scheme. This will require some technical ideas, such as, for example, the study of Newton polygons for \(\mathbf{A}_\mathrm{inf}\), which we will cover in the first few talks.

Later talks will address the Harder–Narasimhan formalism and the classification of vector bundles. This includes the relation of semistable vector bundles with the Dieudonné–Manin classification of isocrystals. We will also discuss the étale fundamental group of the Fargues–Fontaine curve, showing in particular that \(X\) is geometrically simply connected.

Key topics include:

• Introduction to perfectoid fields, tilts, untilts, and relation to period rings.
• Schematic (and possibly adic) descriptions of the Fargues–Fontaine curve.
• Newton polygons, divisors, and factorization results on the curve.
• Harder–Narasimhan filtration for vector bundles and its applications.
• Connections to \(p\)-adic Hodge theory and vector bundles via isocrystals.

Organisers

• L. La Porta
• A. Marrama
• N. Mazzari

Program

For more details on the contents of each talk, click here. We plan to update the program week by week.
For dates, time and location, click here (and look for "GDL La Curva FF").

1. Overview (A. Marrama)
2. Background on perfectoid fields (F. Bambozzi)
3. Definition of the curve \(X\) (F. Baldassarri)
4. \(B_{\text{dR}}\) and divisors (N. Mazzari)
5. Newton polygons and factorisations (A. Bertapelle)
6. Structure of the curve (M. Longo)
7. Line bundles on the curve - Part 1, Part 2 (I. Vanni)
8. Harder–Narasimhan filtration (L. Fiorot)
9. Covers and semistable vector bundles (N. Coppola)
10. The étale fundamental group of \(X\) (B. Chiarellotto?)
11. Adic and relative curves
12. What else?

References

1. Johannes Anschütz, Lectures on the Fargues–Fontaine Curve.
2. Pierre Colmez, Préface par Pierre Colmez: la courbe de Fargues et Fontaine, Astérisque (2018), no. 406, 1–50. MR 3948097
3. Laurent Fargues, Some new geometric structures in the Langlands program, 2023, Eilenberg Lectures.
4. Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-adique, with a preface by Pierre Colmez, Astérisque 406: xiii-382 (2018).
5. Jacob Lurie, The Fargues–Fontaine Curve - Lecture notes.

Notes

We plan (hope) to share notes for each talk. They should be accessible by clicking on each talk's title above.