Uniformization, Unipotent Flows and the Riemann Hypothesis

Abstract

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 ${\Gamma }_g\sim Sp(2g,\mathbb{Z})$-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 ${\Gamma }_{g-r}$ 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.

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• Error Estimates in Horocycle Averages Asymptotics: Challenges from String Theory