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Large Galois images for Jacobian varieties of genus 3 curves

Volume 174 / 2016

Sara Arias-de-Reyna, Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas, Núria Vila Acta Arithmetica 174 (2016), 339-366 MSC: Primary 11F80, 11G30, 11G10; Secondary 12F12. DOI: 10.4064/aa8250-4-2016 Published online: 5 August 2016

Abstract

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

Authors

  • Sara Arias-de-ReynaDepartamento de Álgebra
    Universidad de Sevilla
    Avda. Reina Mercedes s/n, Apdo. 1160
    41080 Sevilla, Spain
    e-mail
  • Cécile ArmanaLaboratoire de Mathématiques
    de Besançon
    UMR CNRS 6623
    Université de Franche-Comté
    16 route de Gray
    25030 Besançon Cedex, France
    e-mail
  • Valentijn KaremakerMathematisch Instituut
    Universiteit Utrecht
    PO Box 80 010, 3508 TA
    Utrecht, The Netherlands
    e-mail
  • Marusia RebolledoLaboratoire de Mathématiques
    UMR 6620 CNRS
    Campus universitaire des Cézeaux
    63171 Aubière, France
    e-mail
  • Lara ThomasLaboratoire de Mathématiques de Besançon, UMR CNRS 6623
    Université de Franche-Comté
    16 route de Gray
    25030 Besançon cedex, France
    e-mail
  • Núria VilaDepartament de Matemàtiques i Informàtica
    Universitat de Barcelona
    Gran Via de les Corts Catalanes, 585
    08007 Barcelona, Spain
    e-mail

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