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Abelian varieties with group action 1 by

A finite group G acting on an abelian variety A induces a decomposition of A up to isogeny. In this paper we investigate several aspects of this decomposition. We apply the results to the decomposition of the Jacobian variety of a smooth projective curve with G-action. The aim is to express the decomposition in terms of the G-action of the curve. As examples of the results we work out the decompositions of Jacobians with an action of the symmetric group of degree 4, the alternating group of degree 5, the dihedral groups of order 2p and 4p, and the quaternion group.