Given a rational variety V defined over a number field K, we consider a principally polarized abelian variety A of dimension g defined over the function field K(V). For each prime l, we then consider the Galois representation on the l-torsion of At, where t is a K-rational point of V. The largest possible image is GSp_(2g)(l), and in the cases g = 1 and 2 we are able to attain this image for all l and almost all t. In the case g = 1, this recovers a theorem originally proved by Duke [C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 813–818].
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