Finite Linear Groups, Lattices, and Products of Elliptic Curves

Let V be a finite dimensional complex linear space and let G be an irreducible finite subgroup of GL(V ). For a G-invariant lattice Λ in V of maximal rank, we give a description of structure of the complex torus V/Λ. In particular, we prove that for a wide class of groups, V/Λ is isogenous to a self-product of an elliptic curve, and that in many cases V/Λ is isomorphic to a product of mutually isogenous elliptic curves with complex multiplication. We show that there are G and Λ such that the complex torus V/Λ is not an abelian variety, but one can always replace Λ by another G-invariant lattice ∆ such that V/∆ is a product of mutually isogenous elliptic curves with complex multiplication. We amplify these results with a criterion, in terms of the character and the Schur Q-index of G-module V , of the existence of a nonzero G-invariant lattice in V .