instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
The inverse Galois problem, Jacobians and the Goldbach's conjecture
Seminario de Álgebra

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