  We prove equidistribution of certain multidimensional unipotent flows in the
moduli space of genus $g$ principally polarized abelian varieties (ppav). This
is done by studying asymptotics of $\pmb{\Gamma}_{g} \sim
Sp(2g,\mathbb{Z})$-automorphic forms averaged along unipotent flows, toward the
codimension-one component of the boundary of the ppav moduli space. We prove a
link between the error estimate and the Riemann hypothesis. Further, we prove
$\pmb{\Gamma}_{g - r}$ modularity of the function obtained by iterating the
unipotent average process $r$ times. This shows uniformization of modular
integrals of automorphic functions via unipotent flows.
