The cohomology of the moduli space of curves at infinite level

2019/11/18 by Emanuel Reinecke, Reinecke, Emanuel
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1911.07392

openalex publication_date 2019/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Full level-n structures on smooth, complex curves are trivializations of the n-torsion points of their Jacobians. We give an algebraic proof that the etale cohomology of the moduli space of smooth, complex curves of genus at least 2 with "infinite level structure" vanishes in degrees above 4g-5. This yields a new perspective on a result of Harer who showed such vanishing already at finite level via topological methods. We obtain similar results for moduli spaces of stable curves and curves of compact type which are not covered by Harer's methods. The key ingredients in the proof are a vanishing statement for certain constructible sheaves on perfectoid spaces and a comparison of the etale cohomology of different towers of Deligne-Mumford stacks in the presence of ramification.

Citations

• Enhanced six operations and base change theorem for higher Artin stacks
• Relative p-adic Hodge theory: Foundations
• Good Reduction of Abelian Varieties
• Perfectoid covers of abelian varieties

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