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The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces

Aaron Landesman, Ishan Levy

TL;DR

This work establishes the Cohen–Lenstra moments for quadratic function fields over $\\mathbb{F}_q(t)$ in a large-$q$ regime by connecting arithmetic statistics to topology. The authors compute the stable rational homology of non-splitting Hurwitz spaces, then translate these topological invariants into exact moment formulas for odd-order abelian groups and certain non-abelian cases, including roots of unity. Their approach hinges on localization and completion via homological epimorphisms, a detailed rack/hurwitz-space analysis, and point-counting via the Grothendieck–Lefschetz trace formula. The results illuminate the function-field Cohen–Lenstra heuristics, providing the first nontrivial moments beyond the cubic-case and laying groundwork for further investigations into roots-of-unity phenomena and non-abelian analogues.

Abstract

We compute the average number of surjections from class groups of quadratic function fields over $\mathbb F_q(t)$ onto finite odd order groups $H$, once $q$ is sufficiently large. These yield the first known moments of these class groups, as predicted by the Cohen--Lenstra heuristics, apart from the case $H = \mathbb Z/3\mathbb Z$. The key input to this result is a topological one, where we compute the stable rational homology groups of Hurwitz spaces associated to non-splitting conjugacy classes.

The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces

TL;DR

This work establishes the Cohen–Lenstra moments for quadratic function fields over in a large- regime by connecting arithmetic statistics to topology. The authors compute the stable rational homology of non-splitting Hurwitz spaces, then translate these topological invariants into exact moment formulas for odd-order abelian groups and certain non-abelian cases, including roots of unity. Their approach hinges on localization and completion via homological epimorphisms, a detailed rack/hurwitz-space analysis, and point-counting via the Grothendieck–Lefschetz trace formula. The results illuminate the function-field Cohen–Lenstra heuristics, providing the first nontrivial moments beyond the cubic-case and laying groundwork for further investigations into roots-of-unity phenomena and non-abelian analogues.

Abstract

We compute the average number of surjections from class groups of quadratic function fields over onto finite odd order groups , once is sufficiently large. These yield the first known moments of these class groups, as predicted by the Cohen--Lenstra heuristics, apart from the case . The key input to this result is a topological one, where we compute the stable rational homology groups of Hurwitz spaces associated to non-splitting conjugacy classes.

Paper Structure

This paper contains 49 sections, 39 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results toward the Cohen--Lenstra heuristics
  3. Results toward the Cohen--Lenstra heuristics with roots of unity
  4. Results toward the non-abelian Cohen--Lenstra heuristics
  5. Stable homology of Hurwitz spaces
  6. In topology as covers of the disc
  7. In homotopy theory as an orbit space
  8. In homotopy theory as a free $\mathbb{E}_2$ algebra
  9. In algebraic geometry as a moduli space
  10. The stable homology for $S_3$ and degree $3$ covers
  11. Conjectures on stable homology of Hurwitz spaces
  12. Idea of the Proof
  13. Outline
  14. Acknowledgements
  15. Notation for Hurwitz spaces
  16. ...and 34 more sections

Key Result

Theorem 1.1.1

Suppose $H$ is a finite abelian group of odd order. Let $q$ be an odd prime power with $\gcd( |H|, q(q-1)) = 1$. There is an integer $C$, depending only on $H$, so that if $q > C$ and $i \in \{0,1\}$,

Figures (3)

  • Figure 1: This is a pictorial description of the homotopy from \ref{['proposition:tensorproducthomotopy']}. We take $b \in c'$ but $g \notin c'$. We pass the element $\xi(g)$ from the left to the right. When it meets the right boundary, it acts on the boundary by $b \triangleright (g\triangleright \xi(g)))$. By assumption $g\triangleright \xi(g) \notin c'$. Since $b \in c'$, $b \triangleright (g\triangleright \xi(g)))$ is also not in $c'$. Therefore, the right boundary becomes the base point, and this whole configuration is then identified with the base point.
  • Figure 2: This is a picture depicting the compatibility of the map $\tilde{g}$ in the proof of \ref{['proposition:modelquotdescription']} with relations $(i)$ and $(ii)$ from \ref{['definition:quotientmodel']}. The first two columns depict the paths $\gamma$ and $\gamma'$ the points take during the map $\tilde{g}$ under relation $(i)$, and the composite $\gamma' \gamma^{-1}$ is depicted underneath. The last two columns depict the paths $\gamma$ and $\gamma'$ which the points take under relation $(ii)$, and the composite $\gamma' \gamma^{-1}$ is again depicted underneath. In the diagrams, there are $4$ points, $p_1$ which is purple, $p_2$ which is red, $p_3$ which is green, and $p_4$ which is blue. So $r = 2$ and $k = 4$ in the notation of \ref{['definition:quotientmodel']}. The point $p_2$ is on the boundary. When $p_2$ is on the left boundary, the path is $\sigma_1 \cdots \sigma_{r-1} = \sigma_1$ since $r = 2$. When $p_2$ is on the right boundary, the path is $\sigma_r \cdots \sigma_{k-1} = \sigma_2 \sigma_3$.
  • Figure 3: This is a picture of the scanning argument of \ref{['lemma:scanningunpointed']}. In the diagrams, there vertical distance at least $\varepsilon$ between any two red points, and all red points have second coordinate lie in $[\epsilon, 1-\epsilon]$.

Theorems & Definitions (126)

  • Theorem 1.1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1.2.1
  • Theorem 1.3.1
  • Definition 7
  • Theorem 1.4.5
  • Remark 8
  • ...and 116 more