Departamento de Matemáticas UAM

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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).

 

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