Algebraic geometry III - selected topics

Summary

This course is aimed to give students an introduction to the theory of algebraic curves, with an emphasis on the interplay between the arithmetic and the geometry of global fields. One of the principle goals will be to understand the geometric formulation of global class field theory.

Content

Keywords

Algebraic geometry, algebraic curves, global fields, class field theory

Learning Prerequisites

Required courses

• Linear algebra
• Group Theory
• Rings and Modules
• Modern Algebraic geometry

Recommended courses

• Algebraic topology
• Differential geometry
• Complex Analysis

Learning Outcomes

By the end of the course, the student must be able to:

• Analyze basic problems in algebraic geometry of curves and solve them.
• Use the statements of basic theorems like Riemann-Roch, Serre duality, etc, and understand their proofs
• Reason intuitively about the geometry and topology of curves over the complex and finite fields.

Teaching methods

2h lectures+2h exercise sessions weekly.

Assessment methods

Oral Exam

Supervision

Resources

Bibliography

We will follow mainly

• Hartshorne, Algebraic Geometry
• R. Miranda, Algebraic Curves and Riemann Surfaces
• J. P. Serre, Algebraic Groups and Class Fields

Ressources en bibliothèque

• Algebraic Geometry / Hartshorne
• Algebraic Curves and Riemann Surfaces / Miranda
• Algebraic Groups and Class Fields / Serre

Moodle Link

• https://go.epfl.ch/MATH-535