Anni, Samuele, Lemos, Pedro and Siksek, Samir (2016) Residual representations of semistable principally polarized abelian varieties. Research in Number Theory, 2 (1). pp. 1-12. doi:10.1007/s40993-015-0032-4

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Abstract

et A/QA/Q be a semistable principally polarized abelian variety of dimension d≥1. Let ℓ be a prime and let ρ¯¯¯A,ℓ:GQ→GSp2d(Fℓ)ρ¯A,ℓ:GQ→GSp2d(Fℓ) be the representation giving the action of GQ:=Gal(Q¯¯¯¯/Q)GQ:=Gal(Q¯/Q) on the ℓ-torsion group A[ℓ]. We show that if ℓ≥ max(5,d+2), and if image of ρ¯¯¯A,ℓρ¯A,ℓ contains a transvection then ρ¯¯¯A,ℓρ¯A,ℓ is either reducible or surjective.

With the help of this we study surjectivity of ρ¯¯¯A,ℓρ¯A,ℓ for semistable polarized abelian threefolds, and give an example of a genus 3 hyperelliptic curve C/QC/Q such that ρ¯¯¯J,ℓρ¯J,ℓ is surjective for all primes ℓ≥3, where J is the Jacobian of C.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Galois theory, Abelian varieties
Journal or Publication Title:	Research in Number Theory
Publisher:	SpringerOpen
ISSN:	2363-9555
Official Date:	15 February 2016
Dates:	Date Event 15 February 2016 Published 23 November 2015 Accepted 10 August 2015 Submitted
Volume:	2
Number:	1
Number of Pages:	12
Page Range:	pp. 1-12
DOI:	10.1007/s40993-015-0032-4
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/K034383/1 (EPSRC)

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