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Residual representations of semistable principally polarized abelian varieties
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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 ISSN 2363-9555.
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Official URL: http://dx.doi.org/10.1007/s40993-015-0032-4
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 | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > 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: |
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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 (Creative Commons) | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
Grant number: | EP/K034383/1 (EPSRC) |
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