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

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... 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 λϕ^4.