Uniformization, Unipotent Flows and the Riemann Hypothesis

arXiv1102.1201MaRDI QIDQ6223355

S. L. Cacciatori, Matteo A. Cardella

Publication date: 6 February 2011

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 $pmb{Gamma}_{g}simSp(2g,mathbbZ)$-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.

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