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In this note we explore a connection between finite covers of surfaces andthe Teichm\"uller polynomial of a fibered face of a hyperbolic 3--manifold. Weconsider the action of a homological pseudo-Anosov homeomorphism $\psi$ on thehomology groups of a class of finite abelian covers of a surface$\Sigma_{g,n}$. Eigenspaces of the deck group actions on these covers arenaturally parametrized by rational points on a torus. We show that away fromthe trivial eigenspace, the spectrum of the action of $\psi$ on theseeigenspaces is bounded away from the dilatation of $\psi$. We show that theaction $\psi$ on these eigenspaces is governed by the Teichm\"uller polynomial.

Teichm\"uller polynomials, Alexander polynomials and finite covers of surfaces