ar X iv : m at h / 05 05 57 1 v 2 [ m at h . A G ] 2 4 Ju l 2 00 5 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 if 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 .