  The phase diagram and critical behavior of scalar quantum electrodynamics are
investigated using lattice gauge theory techniques. The lattice action fixes
the length of the scalar (``Higgs'') field and treats the gauge field as
non-compact. The phase diagram is two dimensional. No fine tuning or
extrapolations are needed to study the theory's critical behovior. Two lines of
second order phase transitions are discovered and the scaling laws for each are
studied by finite size scaling methods on lattices ranging from $6^4$ through
$24^4$. One line corresponds to monopole percolation and the other to a
transition between a ``Higgs'' and a ``Coulomb'' phase, labelled by divergent
specific heats. The lines of transitions cross in the interior of the phase
diagram and appear to be unrelated. The monopole percolation transition has
critical indices which are compatible with ordinary four dimensional
percolation uneffected by interactions. Finite size scaling and histogram
methods reveal that the specific heats on the ``Higgs-Coulomb'' transition line
are well-fit by the hypothesis that scalar quantum electrodynamics is
logarithmically trivial. The logarithms are measured in both finite size
scaling of the specific heat peaks as a function of volume as well as in the
coupling constant dependence of the specific heats measured on fixed but large
lattices. The theory is seen to be qualitatively similar to $\lambda\phi^{4}$.
The standard CRAY random number generator RANF proved to be inadequate
