  We worked out in detail the three-Higgs-doublet extension of the standard
model when the $A_4$ symmetry, which is imposed to solve the flavor problem, is
extended to the scalar sector. The three doublets may be related to the fermion
mass generation and, in particular, they may be the unique responsible for the
generation of the neutrino masses. If this is the case, the respective VEVs
have to be quite smaller than the electroweak scale if no fine tuning in the
Yukawa couplings is assumed. We consider here the mass spectra in the scalar
sector in three different situations. In one of them there are no light scalars
at all, but in the other ones a light or two massless scalars, at the tree
level, may survive. The later fields are safe, from the phenomenological point
of view, since it couples mainly with neutrinos and/or becomes enough massive
at the tree level if there exist trilinear interactions. Quantum effects may be
important too.
