1Departamento de Álgebra Universidad de Sevilla Avda. Reina Mercedes s/n, Apdo. 1160 41080 Sevilla, Spain 2Laboratoire de Mathématiques de Besançon UMR CNRS 6623 Université de Franche-Comté 16 route de Gray 25030 Besançon Cedex, France 3Mathematisch Instituut Universiteit Utrecht PO Box 80 010, 3508 TA Utrecht, The Netherlands 4Laboratoire de Mathématiques UMR 6620 CNRS Campus universitaire des Cézeaux 63171 Aubière, France 5Laboratoire de Mathématiques de Besançon, UMR CNRS 6623 Université de Franche-Comté 16 route de Gray 25030 Besançon cedex, France 6Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona, Spain
Acta Arithmetica, Tome 174 (2016) no. 4, pp. 339-366
Given a prime number $\ell \geq 5$, we construct an infinite family of three-dimensional
abelian varieties over $\mathbb{Q}$ such that, for any $A/\mathbb{Q}$ in the family, the
Galois representation $\overline{\rho}_{A,\ell} \colon G_{\mathbb{Q}} \to \mathrm{GSp}_6(\mathbb{F}_{\ell})$
attached to the $\ell$-torsion of $A$ is surjective. Any such variety $A$ will be the
Jacobian of a genus $3$ curve over $\mathbb{Q}$ whose respective reductions at two auxiliary
primes are prescribed to provide us with generators of $\mathrm{Sp}_6(\mathbb{F}_{\ell})$.
Keywords:
given prime number ell geq construct infinite family three dimensional abelian varieties mathbb mathbb family galois representation overline rho ell colon mathbb mathrm gsp mathbb ell attached ell torsion surjective variety jacobian genus curve mathbb whose respective reductions auxiliary primes prescribed provide generators mathrm mathbb ell
Affiliations des auteurs :
Sara Arias-de-Reyna
1
;
Cécile Armana
2
;
Valentijn Karemaker
3
;
Marusia Rebolledo
4
;
Lara Thomas
5
;
Núria Vila
6
1
Departamento de Álgebra Universidad de Sevilla Avda. Reina Mercedes s/n, Apdo. 1160 41080 Sevilla, Spain
2
Laboratoire de Mathématiques de Besançon UMR CNRS 6623 Université de Franche-Comté 16 route de Gray 25030 Besançon Cedex, France
3
Mathematisch Instituut Universiteit Utrecht PO Box 80 010, 3508 TA Utrecht, The Netherlands
4
Laboratoire de Mathématiques UMR 6620 CNRS Campus universitaire des Cézeaux 63171 Aubière, France
5
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623 Université de Franche-Comté 16 route de Gray 25030 Besançon cedex, France
6
Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona, Spain
@article{10_4064_aa8250_4_2016,
author = {Sara Arias-de-Reyna and C\'ecile Armana and Valentijn Karemaker and Marusia Rebolledo and Lara Thomas and N\'uria Vila},
title = {Large {Galois} images for {Jacobian} varieties of genus 3 curves},
journal = {Acta Arithmetica},
pages = {339--366},
year = {2016},
volume = {174},
number = {4},
doi = {10.4064/aa8250-4-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8250-4-2016/}
}
TY - JOUR
AU - Sara Arias-de-Reyna
AU - Cécile Armana
AU - Valentijn Karemaker
AU - Marusia Rebolledo
AU - Lara Thomas
AU - Núria Vila
TI - Large Galois images for Jacobian varieties of genus 3 curves
JO - Acta Arithmetica
PY - 2016
SP - 339
EP - 366
VL - 174
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4064/aa8250-4-2016/
DO - 10.4064/aa8250-4-2016
LA - en
ID - 10_4064_aa8250_4_2016
ER -
%0 Journal Article
%A Sara Arias-de-Reyna
%A Cécile Armana
%A Valentijn Karemaker
%A Marusia Rebolledo
%A Lara Thomas
%A Núria Vila
%T Large Galois images for Jacobian varieties of genus 3 curves
%J Acta Arithmetica
%D 2016
%P 339-366
%V 174
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4064/aa8250-4-2016/
%R 10.4064/aa8250-4-2016
%G en
%F 10_4064_aa8250_4_2016
Sara Arias-de-Reyna; Cécile Armana; Valentijn Karemaker; Marusia Rebolledo; Lara Thomas; Núria Vila. Large Galois images for Jacobian varieties of genus 3 curves. Acta Arithmetica, Tome 174 (2016) no. 4, pp. 339-366. doi: 10.4064/aa8250-4-2016
