  We review some recent developments in theoretical studies on the connection
between the progenitor systems of Type Ia supernovae (SNe Ia) and the explosion
mechanisms. (1) DD-subCh: In the merging of double C+O white dwarfs (DD
scenario), if the carbon detonation is induced near the white dwarf (WD)
surface in the early dynamical phase, it could result in the (effectively)
sub-Chandrasekhar mass explosion. (2) DD-Ch: If no surface C-detonation is
ignited, the WD could grow until the Chandrasekhar mass is reached, but the
outcome depends on whether the quiescent carbon shell burning is ignited and
burns C+O into O+Ne+Mg. (3) SD-subCh: In the single degenerate (SD) scenario,
if the He shell-flashes grow strong to induce a He detonation, it leads to the
sub-Chandra explosion. (4) SD-Ch: If the He-shell flashes are not strong
enough, they still produce interesting amount of Si and S near the surface of
C+O WD before the explosion. In the Chandra mass explosion, the central density
is high enough to produce electron capture elements, e.g., stable Ni-58.
Observations of the emission lines of Ni in the nebular spectra provides useful
diagnostics of the sub-Chandra vs. Chandra issue. The recent observations of
relatively low velocity carbon near the surface of SNe Ia provide also
interesting constraint on the explosion models.
