POINCAR ´ E'S REDUCIBILITY THEOREM WITH G-ACTION

A finite group G acting on an abelian variety A induces a decomposition of A up to isogeny. In this paper we prove an equivari- ant version of Poincare's reducibility theorem saying that up to isogeny A decomposes into a product of G-simple abelian subvarieties. This decom- position is unique up to isogeny. Let G be a finite group acting on an abelian variety A defined over an al- gebraically closed field k of arbitrary characteristic. The abelian variety A is called G-simple, if there is no G-equivariant isogeny A1 × A2 → A with non- trivial abelian varieties A1 and A2 with G-action. It is the aim of this note to show that any abelian variety A with G-action admits a G-equivariant isogeny ϕ : A1 × ... × Ar → A with G-simple abelian varieties Ai. Since the image of Ai under such an isogeny ϕ is a G-simple abelian subvariety of A, it suffices to decompose A into as um ofG-simple abelian subvarieties. Our main result is the following theorem which might be called Poincare's reducibility theorem with G-action.