  We consider $4d$ compact lattice QED in the quenched approximation. First, we
briefly summarize the spectrum of the staggered Dirac operator and its
connection with random matrix theory. Afterwards we present results for the
low-lying eigenmodes of the Neuberger overlap-Dirac operator. In the strong
coupling phase we find exact zero-modes. Subsequently we discuss possibly
related topological excitations of the U(1) lattice gauge theory.
