un 2 00 1 Abelian varieties with group action

Introduction. Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of G of JX up to isogeny. The most prominent example of such a situation is the case of the group G ≃ Z/2Z of two elements. Let π : X → Y = X/G denote the canonical quotient map. The first to notice that JX is isogenous to the product JY × P (X/Y) was probably Wirtinger (see [W]). The abelian variety P (X/Y) was later called by Mumford the Prym variety of the covering π, because of its relation to certain Prym differentials. The group action of some other special groups were studied by Recillas and Ries (who in [Re] and [Ri] decomposed the Jacobian of the Galois covering of a simple trigonal cover), There are also some general results: First of all there is the classical paper of Chevalley-Weil (see [CW]) which computes how many times an irreducible complex representation appears in the tangent space T 0 JX in terms of the fixed points under elements of G and its stabilizer subgroups. Finally Donagi in [D] and Mérindol in [M] found the decomposition of JX in the case of a finite group G all of whose irreducible Q-representations are absolutely irreducible.