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07 February 2025
 
  » arxiv » 1508.0211

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Residual Representations of Semistable Principally Polarized Abelian Varieties
Samuele Anni ; Pedro Lemos ; Samir Siksek ;
Date 2 Aug 2015
AbstractLet $A$ 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
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