Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

2009/12/02 by Jordan S. Ellenberg, Ellenberg, Jordan S., Akshay Venkatesh +3 · 8 citations
Mathematics · #11G20 #11S31 #14H10 #20J05 #55R80 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0912.0325

openalex publication_date 2009/12/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t) have the l-part of their class group isomorphic to A.

Cited by

• The leading constant in Malle's conjecture over function fields
• Homological stability and weak approximation
• Representation stability for ordered Hurwitz spaces
• Genus stabilization for the homology of moduli spaces of orbit-framed curves with symmetries-I
• The fluctuations of the mod p rank of triangular matrices
• The imaginary case of the nonabelian Cohen--Lenstra heuristics
• Burnside rings for racks and quandles
• Hecke reciprocity and class groups

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