Hecke algebras acting on Abelian varieties

Abstract

The action of a finite group G on an abelian variety A induces a decomposition of A into factors related to the rational irreducible representations of G, the so called isotypical decomposition of A; when A=JZ is the Jacobian variety of a curve Z with G-action, for every subgroup H of G there is an induced canonical action of the corresponding Hecke algebra Q[H\\G/H] on the Jacobian of the quotient curve ZH=Z/H, and a corresponding isotypical decomposition of JZH. These results have provided geometric and analytic information on the factors appearing in the isotypical decomposition of JZ and JZH.In this paper we show that similar results hold for any abelian variety A with G-action: for every subgroup H of G there is a natural abelian subvariety AH of A fixed by H, such that the Hecke algebra Q[H\\G/H] acts on AH. We find the associated isotypical decomposition of AH, and the decomposition of the analytic and the rational representations of the action on AH.We also show that the notion of Prym variety for covers of curves may be extended to abelian varieties, and describe its isotypical decomposition with respect to the action of a natural induced subalgebra of its endomorphism ring. We apply the results to the decomposition of the Jacobian and Prym varieties of the intermediate cover given by H, in the case of smooth projective curves with G-action. We work out several examples that give rise to families of principally polarized abelian varieties, of Jacobian and Prym varieties, with large endomorphism rings.