Weyl groups and abelian varieties

Abstract

Let G be a finite group. For each integral representation rho of G we consider rho-decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X, H) with rho(G)-action, of dimension equal to the degree of rho, which admit a decomposition of the lattice for X into two G-invariant sublattices isotropic with respect to LH, with one of the sublattices ZG-isomor-phic to rho.

We give a construction for rho-decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves.

Conversely, if rho is absolutely irreducible, we show that each rho-decomposable p.p.a.v. is (isomorphic to) one of those constructed above, thereby characterizing them.

In the case of irreducible, reduced root systems, we consider the natural representation of its associated Weyl group, apply the preceding general construction, and characterize completely the associated families of principally polarized abelian varieties, which correspond to modular curves.