Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps

We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detailed analysis of the unipotent terms give the relation between the eta invariant and the Selberg zeta function of odd type. This allows us to show the vanishing of the unipotent contribution. We show that the eta invariant and the Selberg zeta function of odd type satisfy certain functional equation. These results generalize earlier work of John Millson to hyperbolic manifolds with cusps. We also get the corresponding relation and functional equation for the regularized determinant and the Selberg zeta function of even type where the unipotent factor plays a non-trivial role.