  Large-scale polarization of the cosmic microwave background measured by the
WMAP satellite requires a mean optical depth to Thomson scattering, tau_e ~
0.17. The reionization of the universe must therefore have begun at relatively
high redshift. We have studied the reionization process using supercomputer
simulations of a large and representative region of a universe which has
cosmological parameters consistent with the WMAP results. Our simulations
follow both the radiative transfer of ionizing photons and the formation and
evolution of the galaxy population which produces them. A previously published
model with a standard stellar IMF, with a moderate photon escape fraction from
galaxies and with ionizing photon production as expected for zero metallicity
stars produces tau_e = 0.104, which is within 1.0 to 1.5-sigma of the ``best''
WMAP value. Values up to 0.16 can be produced by taking larger escape fractions
or a top heavy IMF. The data do not require a separate populations of
``miniquasars'' or of stars forming in objects with total masses below 10^9
M_sun. Reconciling such early reionization with the observed Gunn-Peterson
troughs in z>6 quasars may be challenging. Possible resolutions of this problem
are discussed.
