  The Gribov ambiguity problem is studied for compact U(1) lattice theory
within the Lorentz gauge. In the Coulomb phase, it is shown that apart from
double Dirac sheets all gauge (i.e. Gribov) copies originate mainly from the
zero-momentum modes of the gauge fields. The removal of the zero-momentum modes
turns out to be necessary for reaching the absolute maximum of the gauge
functional $F(\theta)$. A new gauge fixing procedure - zero-momentum Lorentz
gauge - is proposed.
