Moduli of Curves at Infinite Level

Abstract

We present some new results for perfectoid rings and spaces and use them to study moduli of the following classes of complex algebraic curves: smooth, compact type, and stable. Full level-n structures on such curves are trivializations of the n-torsion points of their Jacobians. We give an algebraic proof that the étale cohomology groups of all three moduli spaces vanish in high degrees at "infinite level." For smooth curves, this yields a new perspective on a result of Harer who showed such vanishing already at finite level using topological methods. The statements for stable curves and curves of compact type are not covered by Harer's methods. Two of the main ingredients in the proofs are a vanishing statement for certain constructible sheaves on perfectoid spaces and a comparison of the étale cohomology groups of different towers of Deligne-Mumford stacks in the presence of ramification.