Geodesic
Large Galois images for Jacobian varieties of genus 3 curves
Sara Arias-de-Reyna1 ; Cécile Armana2 ; Valentijn Karemaker3 ; Marusia Rebolledo4 ; Lara Thomas5 ; Núria Vila6
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
Acta Arithmetica, Tome 174 (2016) no. 4, pp. 339-366.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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 $\mathrm{Sp}_6(\mathbb{F}_{\ell})$.
DOI : 10.4064/aa8250-4-2016
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

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 
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     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},
     publisher = {mathdoc},
     volume = {174},
     number = {4},
     year = {2016},
     doi = {10.4064/aa8250-4-2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8250-4-2016/}
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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. http://geodesic.mathdoc.fr/articles/10.4064/aa8250-4-2016/

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