  In multi-field reheating after modular $j$-inflation we investigate the
conditions under which baryogenesis via non-thermal leptogenesis can be
successfully realized. We introduce three heavy right-handed neutrinos to the
non-supersymmetric Standard Model of particle physics, assuming hierarchical
neutrino masses. Considering a typical mass for the first right-handed neutrino
of the order of $10^{11}~GeV$, suggested from the seesaw mechanism and also
from concrete $SO(10)$ grand unification models, we obtain the allowed
parameter space for viable baryogenesis. An upper bound for the inflaton mass
as well as a lower bound for its branching ratio to the pair of lightest
right-handed neutrinos are found and reported.
