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Seminario de Álgebra y Combinatoria
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).
Location 13:00, Viernes, 11 de septiembre de 2015, Aula 520, Módulo 17, Departamento de Matemáticas