Seminario de Álgebra y Combinatoria


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Sara Arias de Reyna

Jacobian varieties of genus 3 and the inverse Galois problem

Abstract.-

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

$K/\mathbb{Q}$

. This question is encompassed in the general problem of understanding the structure of the absolute Galois group

$G_{\mathbb{Q}}$

of the rational numbers.

A deep fact in arithmetic geometry is that one can attach compatible systems of Galois representations of

$G_{\mathbb{Q}}$

to certain arithmetic-geometric objects, (e.g. abelian varieties). These representations can be used to realise classical linear groups as Galois groups over

$\mathbb{Q}$

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

$\mathbb{Q}$

such that the action of

$G_{\mathbb{Q}}$

on the group of

$\ell$

-torsion points of the Jacobian of

$C$

provides a Galois realisation of

$\mathrm{GSp}_6(\mathbb{F}_{\ell})$

for a prefixed prime

$\ell$

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).

Localización 13:00, Viernes, 11 de septiembre de 2015, Aula 520, Módulo 17, Departamento de Matemáticas