By Federico Lo Bianco
By Bjorn Poonen
Appears in collection : Christophe Ritzenthaler: Geometry and arithmetic of curves of low genus
The first purpose is to introduce/recall basic properties of algebraic curves in order to be able to describe precisely these curves up to genus 5. Having this geometric description will be a first step to control their arithmetic. We will then study Weil’s conjectures for curves over finite fields and various corollary: these results encode deep control on the number of rational points. In particular, we will see that they give non-trivial bounds for the number of points. The course will largely follow the attached notes to which we refer for a more detailed description of the content. As the notes already exist, the videos won’t be a rewriting of the notes on the blackboard. The lecturer will motivate the study of each chapter, clarify some difficulties (for instance by giving details on a long example), and give further connections and illustrations.