  Given a rational variety $V$ defined over $K$, we consider a principally
polarized abelian variety $A$ of dimension $g$ defined over $V$. For each prime
l we then consider the galois representation on the $l$-torsion of $A_t$, where
$t$ is a $K$-rational point of $V$. The largest possible image is
$GSp_{2g}(\mathbb{F}_l)$ and in the cases $g=1$ and 2, we are able to get
surjectivity for all $l$ and almost all $t$. In the case $g=1$ this recovers a
theorem originally proven by William Duke.
