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

Jordan S. Ellenberg; Akshay Venkatesh; Craig Westerland

doi:10.4007/annals.2016.183.3.1

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

Jordan S. Ellenberg, Akshay Venkatesh, Craig Westerland

School of Mathematics

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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, a positive fraction of quadratic extensions of F_q(t) have the ℓ-part of their class group isomorphic to A.

Original language	English (US)
Pages (from-to)	729-786
Number of pages	58
Journal	Annals of Mathematics
Volume	183
Issue number	3
DOIs	https://doi.org/10.4007/annals.2016.183.3.1
State	Published - May 1 2016

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© 2016 Department of Mathematics, Princeton University.

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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, a positive fraction of quadratic extensions of Fq(t) have the ℓ-part of their class group isomorphic to A.",

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

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