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 t
heory dual to the Villain action on toroidal lattices up to lattice sizes o
f 28(4) give results consistent with both, a vanishing and a nonvanishing d
iscontinuity in the thermodynamic limit. A decision on the order of the pha
se transition requires still larger lattice sizes.