# Vertiefungsmodul: The Fargues-Fontaine Curve

**Lecture**

Monday, 4 - 6 pm, WSC-S-U-3.03

Tuesday, 4 - 6 pm, WSC-S-U-3.03

*first session on April 8, 2019*

**Problem session**

Thursday, 8:30 - 10 am, WSC-S-U-3.03

*first session on April 11, 2019*

**Prerequisites**

Algebra 1 & 2

Algebraic Number Theory 1 (basics of non-archimedean valued fields and their ramification theory; most of this will be recalled when we talk about Witt vectors)

Algebraic Geometry 1 (only for the second part of the lecture; essentially, it is enough if you know what Proj of a graded ring and what a vector bundle on a scheme is)

If you are unsure whether the course is suitable for you, please send me an email or come to the first lecture and talk to me. The first two problem sessions will be used for repetitions.

This lecture forms part of the ALGANT-program for master students and will be given in English. Of course, it is also open and suitable for PhD-students.

**Contents**

Starting from certain arithmetic data, Fargues and Fontaine have recently constructed a geometric object that they called the fundamental curve of p-adic Hodge theory. This is a one-dimensional regular noetherian scheme leading to geometric proofs of fundamental problems of number theory such as the p-adic monodromy theorem and local class field theory. The curve is also supposed to lead to a geometric formulation of the p-adic local Langlands correspondence. This is a highly active field of current research.

The lecture will focus on the construction of the Fargues-Fontaine curve and the classification of its vector bundles. It will also explain the link to p-adic Galois representations.

**Topics covered**

Ramified Witt vectors, Newton polygons, rings of p-adic periods, factorization theorems in period rings, construction of the curve, study of its basic geometric properties, construction and classification of vector bundles, vector bundles and p-adic Galois representations

**References**

[1] L. Fargues, J.-M. Fontaine: Courbes et fibrés vectoriels en théorie de Hodge p-adique, Astérisque, to appear

[2] L. Fargues: La Courbe, Proceedings of the ICM 2018 in Rio de Janeiro, to appear

[3] L. Fargues, J.-M. Fontaine: Vector bundles on curves and p-adic Hodge theory, in "Automorphic Forms and Galois Representations", London Mathematical Society Lecture Note Series **415** (2014)

[4] L. Fargues, J.-M. Fontaine: Vector bundles and p-adic Galois representations, AMS/IP Studies in Advanced Mathematics **51** (2011)

All references are available electronically through the webpage of Laurent Fargues.

**Credits**

In order to be given credit points for this course you will need to take an oral exam in the end. In order to be admitted to the exam it is necessary to obtain at least 45% of the points on the problem sheets and to actively participate in the problem sessions.

**Problem sheets**

The problem sheets will be collected on Thursdays during the problem sessions. You are encouraged to hand in your solutions in groups of up to three people.