# Arithmetic of higher genus curves

**Project level:** PhD

Curves of genus 0 and 1 are relatively well understood. Starting in the 1990s explicit methods have been developed for the first time to deal with more complicated curves and determine all of their rational points. Open problems in this field include the role of an object called the Brauer-Manin obstruction and the question of determining whether the new explicit methods are guaranteed to determine all solutions.