  We discuss the luminosity function of SNe Ia under the assumption that recent
evidence for dispersion in this standard candle is related to variations in the
white dwarf mass function (WDMF) in the host galaxies. We develop a simple
parameterization of the WDMF as a function of age of a stellar population and
apply this to galaxies of different morphological types. We show that this
simplified model is consistent with the observed WDMF of Bergeron et al. (1992)
for the solar neighborhood. Our simple models predict that WDMF variations can
produce a range of more than 1.8 mag in M$_B$(SN Ia), which is comparable to
the observed value using the data of Phillips (1993) and van den Bergh (1996).
We also predict a galaxy type dependence of M$_B$(SN Ia) under standard
assumptions of the star formation history in these galaxies and show that
M$_B$(SN Ia) can evolve with redshift. In principle both evolutionary and
galaxy type corrections should be applied to recover the intrinsic range of
M$_B$(SN Ia) from the observed values. Our current inadequate knowledge of the
star formation history of galaxies coupled with poor physical understanding of
the SN Ia mechanism makes the reliable estimation of these corrections both
difficult and controversial. The predictions of our models combined with the
observed galaxy and redshift correlations may have the power to discriminate
between the Chandrasekhar and the sub-Chandrasekhar progenitor scenarios for
SNe Ia.
