`samuele.anni@gmail.com`

The inverse Galois problem is one of the greatest open problems in group theory and also one of the easiest to state: is every finite group a Galois group?

My interest around this problem is connected to the realization of linear and symplectic groups as Galois groups over $\mathbb{Q}$ and over number fields. In particular, I am interested in "uniform realizations": realizations of all elements in a family of groups (e.g. $GL_2(\mathbb{F}_\ell)$ for every prime $\ell$) simultaneously using only one "object". In this talk I will describe uniform realizations using elliptic curves, genus 2 and 3 curves.

After this introduction, I will explain how to extend these results via Jacobians of higher genus curves. This is joint work with Vladimir