  The aim of this work is the computation of the cosmic Type Ia supernova rates
at very high redshifts (z>2). We adopt various progenitor models in order to
predict the number of explosions in different scenarios for galaxy formation
and to check whether it is possible to select the best delay time distribution
model, on the basis of the available observations of Type Ia supernovae. We
also computed the Type Ia supernova rate in typical elliptical galaxies of
different initial luminous masses and the total amount of iron produced by Type
Ia supernovae in each case. It emerges that: it is not easy to select the best
delay time distribution scenario from the observational data and this is
because the cosmic star formation rate dominates over the distribution function
of the delay times; the monolithic collapse scenario predicts an increasing
trend of the SN Ia rate at high redshifts whereas the predicted rate in the
hierarchical scheme drops dramatically at high redshift; for the elliptical
galaxies we note that the predicted maximum of the Type Ia supernova rate
depends on the initial galactic mass. The maximum occurs earlier (at about 0.3
Gyr) in the most massive ellipticals, as a consequence of downsizing in star
formation. We find that different delay time distributions predict different
relations between the Type Ia supernova rate per unit mass at the present time
and the color of the parent galaxies and that bluer ellipticals present higher
supernova Type Ia rates at the present time.
