  Using finite size scaling and histogram methods we obtain numerical results
from lattice simulations indicating the logarithmic triviality of scalar
quantum electrodynamics, even when the bare gauge coupling is chosen large.
Simulations of the non-compact formulation of the lattice abelian Higgs model
with fixed length scalar fields on $L^{4}$ lattices with $L$ ranging from $6$
through $20$ indicate a line of second order critical points.
Fluctuation-induced first order transitions are ruled out. Runs of over ten
million sweeps for each $L$ produce specific heat peaks which grow
logarithmically with $L$ and whose critical couplings shift with $L$ picking
out a correlation length exponent of $0.50(5)$ consistent with mean field
theory. This behavior is qualitatively similar to that found in pure
$\lambda\phi^{4}$.
