Karl Rökaeus: Curves over finite fields and their applications

Curves over finite fields is a topic at the intersection of number theory and algebraic geometry. It had its first heydays in the 1930s and 40s when Hasse and Weil proved fundamental theorems about the number of points possible on such a curve. This was later generalized to arbitrary varieties as the Weil conjectures, and was one of the driving motivations behind the development of algebraic geometry that took place in the late 1950s.

In the 1980s curves over finite fields came into fashion again when they turned out to have important practical applications: First as a tool to construct error correcting codes used in many forms of digital communications; and then also for security with the invention of elliptic curve cryptography.

In this talk, which is aimed at a general audience, I will give an overview of the theory of curves over finite fields, including some of my own contributions.