Abstract
Let \(A/{\mathbb {Q}}\) be a semistable principally polarized abelian variety of dimension d≥1. Let ℓ be a prime and let \(\overline {\rho }_{A,\ell }\colon G_{\mathbb {Q}} \rightarrow \text {GSp}_{2d}({\mathbb {F}}_{\ell })\) be the representation giving the action of \(G_{\mathbb {Q}} :=\text {Gal}(\overline {{\mathbb {Q}}}/{\mathbb {Q}})\) on the ℓ-torsion group A[ℓ]. We show that if ℓ≥ max(5,d+2), and if image of \(\overline {\rho }_{A,\ell }\) contains a transvection then \(\overline {\rho }_{A,\ell }\) is either reducible or surjective.
With the help of this we study surjectivity of \(\overline {\rho }_{A,\ell }\) for semistable polarized abelian threefolds, and give an example of a genus 3 hyperelliptic curve \(C/{\mathbb {Q}}\) such that \(\overline {\rho }_{J,\ell }\) is surjective for all primes ℓ≥3, where J is the Jacobian of C.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arias-de Reyna, S, Armana, C, Karemaker, V, Rebolledo, M, Thomas, L, Vila, N: Galois representations and galois groups over \(\mathbb {Q}\) (2014). ArXiv e-prints.
Arias-de-Reyna, S, Dieulefait, L, Wiese, G: Classification of subgroups of symplectic groups over finite fields containing a transvection (2014). ArXiv e-prints.
Bosma, W, Cannon, J, Playoust, C: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997). Computational algebra and number theory (London, 1993).
Dieulefait, LV: Explicit determination of the images of the Galois representations attached to abelian surfaces with \({\text {End}}(A)=\mathbb {Z}\). Experiment. Math. 11(4), 503–512(2003) (2002).
Hall, C: An open-image theorem for a general class of abelian varieties. Bull. Lond. Math. Soc. 43(4), 703–711 (2011). With an appendix by Emmanuel Kowalski.
Khare, C, Wintenberger, J-P: Serre’s modularity conjecture. I. Invent. Math. 178(3), 485–504 (2009).
Lombardo, D: Explicit open image theorems for some abelian varieties with trivial endomorphism ring (2015). ArXiv e-prints.
Mazur, B: Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44(2), 129–162 (1978).
Mumford, D: A note of Shimura’s paper “Discontinuous groups and abelian varieties”. Math. Ann. 181(4), 345–351 (1969).
Raynaud, M: Schémas en groupes de type (p,…,p). Bull. Soc. Math. France. 102, 241–280 (1974).
Serre, J-P: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972).
Serre, J-P: Oeuvres/Collected papers. IV. 1985–1998. Springer Collected Works in Mathematics. Springer, Heidelberg (2013). Reprint of the 2000 edition [MR1730973].
Zywina, D: An explicit Jacobian of dimension 3 with maximal Galois action (2015). ArXiv e-prints.
Acknowledgements
The first-named and third-named authors are supported by EPSRC Programme Grant ‘LMF: L-Functions and Modular Forms’ EP/K034383/1.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Anni, S., Lemos, P. & Siksek, S. Residual representations of semistable principally polarized abelian varieties. Res. number theory 2, 1 (2016). https://doi.org/10.1007/s40993-015-0032-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-015-0032-4