  In accreting WDs approaching the Chandrasekhar limit, hydrostatic carbon
burning precedes the dynamical breakout. During this \textit{simmering} phase,
$e-$captures are energetically favored in the central region of the star, while
$\beta-$decays are favored more outside, and the two zones are connected by a
growing convective instability. We analyze the interplay between weak
interactions and convection, the so-called convective URCA process, during the
simmering phase of SNe Ia progenitors and its effects on the physical and
chemical properties at the explosion epoch. At variance with previous studies,
we find that the convective core powered by the carbon burning remains confined
within the ${^{21}(Ne,F)}$ URCA shell. As a result, a much larger amount of
carbon has to be consumed before the explosion which eventually occurs at
larger density than previously estimated. In addition, we find that the
extension of the convective core and its average neutronization depend on the
the WD progenitor initial metallicity. For the average neutronization in the
convective core at the explosion epoch we obtain ${\overline{\eta}_{exp}} =
(1.094\pm 0.143)\times 10^{-3} + (9.168\pm 0.677)\times 10^{-2}\times Z$.
Outside the convective core, the neutronization is instead determined by the
initial amount of C+N+O in the progenitor star. Since S, Ca, Cr and Mn, the
elements usually exploited to evaluate the pre-explosive neutronization, are
mainly produced outside the heavily neutronized core, the problem of too high
metallicity estimated for the progenitors of the historical Tycho and Kepler
SNe Ia remains unsolved.
