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Homological stability for Hurwitz spaces and applications

Aaron Landesman, Ishan Levy

TL;DR

This work advances the theory of Hurwitz spaces by proving integral homological stability for spaces attached to finite racks and computing the dominant stable homology after inverting finitely many primes. It leverages and extends the Holz–Ellenberg–Venkatesh–Westerland framework to accommodate general racks and non-splitting cases, enabling strong arithmetic applications. The authors establish function-field versions of Malle’s conjecture and the Cohen–Lenstra–Martinet moments, and prove an asymptotic Picard-rank conjecture for large branch data, all via a blend of derived algebraic techniques, Frobenius-equivariant stabilization, and log-geometry. The results illuminate how topological stability phenomena in Hurwitz spaces translate into precise asymptotics for counting Galois extensions and statistics of class groups in function fields, with potential further reach to higher-genus bases and integral invariants.

Abstract

We show the homology of the Hurwitz space associated to an arbitrary finite rack stabilizes integrally in a suitable sense. We also compute the dominant part of its stable homology after inverting finitely many primes. This proves a conjecture of Ellenberg--Venkatesh--Westerland and improves upon our previous results for non-splitting racks. We obtain applications to Malle's conjecture, the Picard rank conjecture, and the Cohen--Lenstra--Martinet heuristics.

Homological stability for Hurwitz spaces and applications

TL;DR

This work advances the theory of Hurwitz spaces by proving integral homological stability for spaces attached to finite racks and computing the dominant stable homology after inverting finitely many primes. It leverages and extends the Holz–Ellenberg–Venkatesh–Westerland framework to accommodate general racks and non-splitting cases, enabling strong arithmetic applications. The authors establish function-field versions of Malle’s conjecture and the Cohen–Lenstra–Martinet moments, and prove an asymptotic Picard-rank conjecture for large branch data, all via a blend of derived algebraic techniques, Frobenius-equivariant stabilization, and log-geometry. The results illuminate how topological stability phenomena in Hurwitz spaces translate into precise asymptotics for counting Galois extensions and statistics of class groups in function fields, with potential further reach to higher-genus bases and integral invariants.

Abstract

We show the homology of the Hurwitz space associated to an arbitrary finite rack stabilizes integrally in a suitable sense. We also compute the dominant part of its stable homology after inverting finitely many primes. This proves a conjecture of Ellenberg--Venkatesh--Westerland and improves upon our previous results for non-splitting racks. We obtain applications to Malle's conjecture, the Picard rank conjecture, and the Cohen--Lenstra--Martinet heuristics.

Paper Structure

This paper contains 54 sections, 53 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Application 1: Malle's conjecture
  3. Application 2: The asymptotic Picard rank conjecture
  4. Application 3: Cohen-Lenstra-Martinet heuristics
  5. Homological stability results
  6. Proof ideas
  7. Proof of the main homological stability results
  8. Proof idea for the Picard rank conjecture
  9. Proof idea for the Cohen-Lenstra-Martinet moments
  10. Proof idea for Malle's conjecture
  11. Outline
  12. Acknowledgements
  13. Background and notation for Hurwitz spaces
  14. Background on racks
  15. Hurwitz spaces over the complex numbers
  16. ...and 39 more sections

Key Result

Theorem 1.1.1

Fix a finite permutation group $G \subset S_d$. There are constants $a(G-\mathop{\mathrm{id}}\nolimits,\Delta), b_T(\mathbb F_q(t), (G - \mathop{\mathrm{id}}\nolimits)_\Delta)$, defined later in notation:malle, and a constant $C$, depending on $G$, with the following property: For $X$ sufficiently l

Figures (3)

  • Figure 1: This figure depicts known cases of the Picard rank conjecture, see \ref{['remark:known-picard-rank']} and \ref{['theorem:stable-picard-intro']}.
  • Figure 2: The top half of the diagram pictures the algebraic gluing map on the base of the cover, corresponding to the stabilization map. The map takes in a point of $\mathscr H^c_n$, corresponding to the blue line, and glues it to a fixed cover, corresponding to the green and pink lines, to obtain a point of $\overline{\mathscr H}^c_{n+Mdr}$. Here, there are $n$ additional branch points on the blue line, $Mdr$ additional marked point on the green line, and no additional marked points on the pink line. This is meant to be an algebraic incarnation of the topological structure pictured coming from the little discs operad in the bottom half of the diagram. The algebraic map replaces each disc with a copy of $\mathbb P^1$, with the white circles on the algebraic picture corresponding to the boundary of the disc.
  • Figure 3: A figure depicting the gluing construction described in \ref{['construction:gluing']}, inducing the Frobenius equivariant stabilization map.

Theorems & Definitions (153)

  • Theorem 1.1.1
  • Remark 1
  • Remark 2
  • Example 3
  • Remark 4: Comparison with Ellenberg--Tran--Westerland
  • Remark 5
  • Conjecture 6: Picard rank conjecture
  • Remark 7
  • Theorem 1.2.1
  • Remark 8
  • ...and 143 more