Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

• 成果类型: Article
• 署名作者: Bergstrom, Jonas; Faber, Carel; Payne, Sam
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.3.7
• 发表日期: 2024
• 页码: 1323-1365
• 关键词: stacks; number; covers

摘要

We compute the number of F-q-points on (M) over bar (4,n) for n <= 3 and show that it is a polynomial in q , using a sieve based on Hasse-Weil zeta functions. As an application, we prove that the rational singular cohomology group H-k((M) over bar (g,n)) vanishes for all odd k <= 9. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor 1, which are classified in low weight. Our vanishing result for odd cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.