  The early reionization of the intergalactic medium, which is favored from the
WMAP temperature-polarization cross-correlations, contests the validity of the
standard scenario of structure formation in the cold dark matter cosmogony. It
is difficult to achieve early enough star formation without rather extreme
assumptions such as very high escape fraction of ionizing photons from
proto-galaxies or a top-heavy initial mass function. Here we propose an
alternative scenario that is additional fluctuations on small scales induced by
primordial magnetic fields trigger the early structure formation. We found that
ionizing photons from Population III stars formed in dark haloes can easily
reionize the universe by $z \simeq 15$ if the strength of primordial magnetic
fields is larger than $0.6 \times 10^{-9}$Gauss.
