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 ΞgββΌSp(2g,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
Ξgβrβ 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