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Counting Components of Hurwitz Spaces

Béranger Seguin

TL;DR

This paper studies the asymptotics for the number of connected components of Hurwitz spaces of marked $G$-covers as the number of branch points grows, connecting to the distribution of $ ext{F}_q(T)$-extensions. It introduces the splitting number $igOmega(D_H)$ to quantify non-splitting among subgroups and demonstrates that the leading exponent in the growth is governed by this invariant, unifying and extending EVW’s homological stability results. The authors develop an intrinsic monoid of components, relate it to Hilbert functions of graded algebras, and, through the lifting invariant and Conway–Parker–Fried–Völklein–type theorems, compute leading coefficients in many cases. The work yields precise affine and projective counts, clarifies when the leading term factors through subgroups, and ties the asymptotics to the behavior of $ ext{F}_q(T)$-extensions with prescribed Galois groups, offering new tools for understanding arithmetic statistics of function field extensions.

Abstract

For a finite group $G$, we obtain asymptotics for the number of connected components of Hurwitz spaces of marked $G$-covers (of both the affine and projective lines) whose monodromy classes are constrained in a certain way, when the number of branch points grows to infinity. More precisely, we compute both the degree and (in many cases) the coefficient of the leading monomial in the count of components of marked $G$-covers whose monodromy group is a given subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, these asymptotics are related to the distribution of field extensions of $\mathbb{F}_q(T)$ with Galois group $G$.

Counting Components of Hurwitz Spaces

TL;DR

This paper studies the asymptotics for the number of connected components of Hurwitz spaces of marked -covers as the number of branch points grows, connecting to the distribution of -extensions. It introduces the splitting number to quantify non-splitting among subgroups and demonstrates that the leading exponent in the growth is governed by this invariant, unifying and extending EVW’s homological stability results. The authors develop an intrinsic monoid of components, relate it to Hilbert functions of graded algebras, and, through the lifting invariant and Conway–Parker–Fried–Völklein–type theorems, compute leading coefficients in many cases. The work yields precise affine and projective counts, clarifies when the leading term factors through subgroups, and ties the asymptotics to the behavior of -extensions with prescribed Galois groups, offering new tools for understanding arithmetic statistics of function field extensions.

Abstract

For a finite group , we obtain asymptotics for the number of connected components of Hurwitz spaces of marked -covers (of both the affine and projective lines) whose monodromy classes are constrained in a certain way, when the number of branch points grows to infinity. More precisely, we compute both the degree and (in many cases) the coefficient of the leading monomial in the count of components of marked -covers whose monodromy group is a given subgroup of . By the work of Ellenberg, Tran, Venkatesh and Westerland, these asymptotics are related to the distribution of field extensions of with Galois group .
Paper Structure (41 sections, 21 theorems, 52 equations)

This paper contains 41 sections, 21 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Motivation and previous work
  3. Approach of this work
  4. Disclaimer.
  5. Presentation of the results
  6. Terminology for subgroups.
  7. Relation with Hilbert functions.
  8. Main theorem.
  9. Number-theoretic meaning.
  10. Outline
  11. Organization of the proof.
  12. Acknowledgments.
  13. Presentation of the objects
  14. General conventions
  15. Tuples
  16. ...and 26 more sections

Key Result

Proposition 1.3

There is an integer $W \geq 1$ and integer-valued polynomials $Q_0, \ldots, Q_{W-1} \in \mathbb{Q}[X]$, with $\deg Q_0 \geq \deg Q_i$ for all $i \in \{1,\ldots,W-1\}$, such that, for all $n \in \mathbb{N}$ large enough, we have $\left | \pi_0 \mathrm{CHur}^*_X(H, D_H, n \xi_H) \right | = Q_{n \bmod

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 39 more