Jacobian varieties of genus 3 and the inverse Galois problem

The inverse Galois problem, first addressed by D. Hilbert in 1892, asks which finite groups occur as the Galois group of a finite Galois extension K/Q$. This question is encompassed in the general problem of understanding the structure of the absolute Galois group G_Q of the rational numbers.

A deep fact in arithmetic geometry is that one can attach compatible systems of Galois representations of G_Q to certain arithmetic-geometric objects, (e.g. abelian varieties). These representations can be used to realise classical linear groups as Galois groups over Q.

In this talk we will discuss the case of Galois representations attached to Jacobian varieties of genus n curves. For n=3, we provide an explicit construction of curves C defined over Q such that the action of G_Q on the group of l-torsion points of the Jacobian of C provides a Galois realisation of GSp(6, l) for a prefixed prime l.
This construction is a joint work with Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas and Núria Vila, and was initiated as a working group in the Conference Women in Numbers Europe (CIRM, 2013).