Automorphisms

Abstract

An automorphism of a polarized abelian variety (X, L) is by definition an automorphism g of X satisfying g*L ≡ L. In Section 5.1 we saw that the group of automorphisms of (X, L) is always a finite group, and in Section 11.7 we showed that the group of automorphisms of some curves of genus two can be applied to compute period matrices of their Jacobians. More generally, automorphisms can be used to decompose an abelian variety into a product of smaller abelian varieties up to isogeny. As an example we saw in Chapter 12 that an action of the group of two elements on a Jacobian J decomposes J into the product of another Jacobian and a Prym variety. This chapter contains some of the main results on automorphism groups of abelian varieties. There are essentially two methods to employ a group G of automorphisms in order to decompose an abelian variety X: If G is cyclic, one can use the set of fixed points of G, and for an arbitrary group G the rational representation algebra ℚ[G], for decomposing X up to isogeny.