  We address the question of the order of the deconfinement phase transition of
four dimensional U(1) lattice gauge theory. Simulations of the Z-gauge theory
dual to the Villain action on toroidal lattices up to lattice sizes of 28^4
give results consistent with both, a vanishing and a nonvanishing discontinuity
in the thermodynamic limit. A decision on the order of the phase transition
requires still larger lattice sizes.
