Given a rational variety V defined over 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 proven by William Duke [1].

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SynopsisCanonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.

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