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Article overview
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Residual Representations of Semistable Principally Polarized Abelian Varieties | Samuele Anni
; Pedro Lemos
; Samir Siksek
; | Date: |
2 Aug 2015 | Abstract: | Let $A$ be a semistable principally polarized abelian variety of dimension
$d$ defined over the rationals. Let $ell$ be a prime and let
$overline{
ho}_{A,ell} : G_{mathbb{Q}}
ightarrow
mathrm{GSp}_{2d}(mathbb{F}_ell)$ be the representation giving the action of
$G_{mathrm{Q}} :=mathrm{Gal}(overline{mathrm{Q}}/mathrm{Q})$ on the
$ell$-torsion group $A[ell]$. We show that if $ell ge max(5,d+2)$, and if
image of $overline{
ho}_{A,ell}$ contains a transvection then
$overline{
ho}_{A,ell}$ is either reducible or surjective.
With the help of this we study surjectivity of $overline{
ho}_{A,ell}$ for
semistable principally polarized abelian threefolds, and give an example of a
genus $3$ hyperelliptic curve $C/mathbb{Q}$ such that
$overline{
ho}_{J,ell}$ is surjective for all primes $ell ge 3$, where $J$
is the Jacobian of $C$. | Source: | arXiv, 1508.0211 | Services: | Forum | Review | PDF | Favorites |
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