Equations for abelian subvarieties

Abstract

Given a finite group $G$ and an abelian variety $A$ acted on by $G$, to any subgroup $H$ of $G$, we associate an abelian subvariety $A_H$ on which the associated Hecke algebra ${\mathcal{H}}_H$ for $H$ in $G$ acts. Any irreducible rational representation $\tilde{W}$ of ${\mathcal{H}}_H$ induces an abelian subvariety of $A_H$ in a natural way. In this paper we give equations for this abelian subvariety. In a special case these equations become much easier. We work out some examples.