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The stable homology of Hurwitz modules and applications

Aaron Landesman, Ishan Levy

TL;DR

This work develops a comprehensive framework for the stable homology of Hurwitz spaces and their module/generalized variants (Hurwitz modules), achieving stability in all directions and computing the stable value via descent and scanning techniques. The authors connect topological stability to arithmetic statistics over function fields, deriving function-field analogues of Bhargava's conjecture, verifying Poonen–Rains (BKLPR) moments for Selmer groups, and establishing representation stability for single-component racks. The core strategy blends bar-construction morphisms, quotients by subracks, and Kan-fibration–assisted descent to pass from quotients to full Hurwitz-modules stability, with explicit identifications to configuration-space homology and the free $E_2$-algebra. Collectively, the results provide a robust toolkit for predicting and counting algebraic extensions over function fields, and set the stage for further arithmetic-statistics investigations in this topological-algebraic setting.

Abstract

We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function fields. As a second consequence, we deduce a version of Bhargava's conjecture, counting the number of $S_d$ degree $d$ extensions of $\mathbb F_q(t)$, for suitably large $q$. As a third consequence, we deduce that the homology of Hurwitz spaces associated to racks with a single component satisfy representation stability.

The stable homology of Hurwitz modules and applications

TL;DR

This work develops a comprehensive framework for the stable homology of Hurwitz spaces and their module/generalized variants (Hurwitz modules), achieving stability in all directions and computing the stable value via descent and scanning techniques. The authors connect topological stability to arithmetic statistics over function fields, deriving function-field analogues of Bhargava's conjecture, verifying Poonen–Rains (BKLPR) moments for Selmer groups, and establishing representation stability for single-component racks. The core strategy blends bar-construction morphisms, quotients by subracks, and Kan-fibration–assisted descent to pass from quotients to full Hurwitz-modules stability, with explicit identifications to configuration-space homology and the free -algebra. Collectively, the results provide a robust toolkit for predicting and counting algebraic extensions over function fields, and set the stage for further arithmetic-statistics investigations in this topological-algebraic setting.

Abstract

We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function fields. As a second consequence, we deduce a version of Bhargava's conjecture, counting the number of degree extensions of , for suitably large . As a third consequence, we deduce that the homology of Hurwitz spaces associated to racks with a single component satisfy representation stability.

Paper Structure

This paper contains 37 sections, 60 theorems, 114 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The moments of the distribution of Selmer groups in quadratic twist families
  3. Bhargava's conjecture
  4. Representation stability
  5. Homological stability results
  6. Summary of the proofs
  7. Outline
  8. Acknowledgements
  9. Hurwitz modules
  10. Definition of Hurwitz modules
  11. Subsets of Hurwitz modules
  12. Quotients of Hurwitz modules
  13. Scanning arguments
  14. Notation for scanning models
  15. A scanning model
  16. ...and 22 more sections

Key Result

Theorem 1.1.1

Choose $q$ with $\operatorname{char} \mathbb F_q > 3$ and $\nu$ an integer prime to $6q$. With notation as in notation:quadratic-twist-family, suppose $A$ is a nonconstant elliptic curve with squarefree discriminant. There is a constant $C_{\nu}$ depending on $\nu$ (but not on $A$) so that if $q^j

Figures (12)

  • Figure 1: This picture depicts the quotient $\mathcal{M}_{g,f,t}$ of the rectangle $\mathbf R - W$ in the case $g = 1, f = 2$. The boundary component of $\mathcal{M}$ consists of the union of the upper, left, and lower edges. The arrows indicate the orientations of the segments of the edges. The segments of the same color are glued to each other with the orientations indicated. The two black dots indicate the two punctures comprising $W$.
  • Figure 2: This depicts $\mathcal{M}_{g,f,1}^\epsilon$ in the case $g = 1, f = 2$. The surface $\mathcal{M}_{1,2,1}^\epsilon$ is a union of $2g + f = 4$ rectangles. There are eight rectangles pictured in four colors. Each pair of rectangles of the same color are glued along their right edge so that $\mathcal{M}_{1,2,1}^\epsilon$ consists of four rectangles.
  • Figure 3: This picture depicts a vector field on $\mathcal{M}_{g,f,1}$ as described in the proof of \ref{['lemma:deform-epsilon-distance']} in the case $g= 1, f = 2$.
  • Figure 4: This picture the allowable paths $\eta_1, \ldots, \eta_6$ in a particular configuration in $\mathcal{M}_{1,2,1}$ with $2$ points.
  • Figure 5: This picture shows the paths $\xi_1, \ldots, \xi_4$$\mathcal{M}_{1,2,1}$ which are the names we are using for the allowable paths $\eta_1, \ldots, \eta_4$ associated to the empty configuration.
  • ...and 7 more figures

Theorems & Definitions (167)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Remark 3
  • Remark 4
  • Remark 5
  • Conjecture 6: bhargava:mass-formulae-for-extensions-of-local-fields
  • Definition 8
  • Theorem 1.2.1
  • Remark 9
  • Remark 10
  • ...and 157 more