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

@article{Ellenberg2012HomologicalSF, title={Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II}, author={Jordan S. Ellenberg and Akshay Venkatesh and Craig Westerland}, journal={arXiv: Number Theory}, year={2012} }

We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.

## 115 Citations

Moduli spaces of Riemann surfaces as Hurwitz spaces

- Mathematics
- 2021

This is the fourth article in a series about Hurwitz spaces. We consider the moduli space Mg,n of Riemann surfaces of genus g ≥ 0 with n ≥ 1 ordered and directed poles, and for d ≥ 2g + n − 1 we show…

Hurwitz monodromy and full number fields

- MathematicsAlgebra & Number Theory
- 2015

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting…

The little bundles operad

- Mathematics
- 2019

Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an…

Exposé Bourbaki 1164 : Homology of Hurwitz spaces and the Cohen-Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)

- MathematicsAstérisque
- 2020

Ellenberg, Venkatesh, and Westerland have established a weak form of the function field analogue of the Cohen--Lenstra heuristic, on the distribution of imaginary number fields with $\ell$-parts of…

$\mathrm{FI}_G$-modules and arithmetic statistics

- Mathematics
- 2017

This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of…

Partial Torelli groups and homological stability

- Mathematics
- 2019

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar…

Cohen-Lenstra heuristics and local conditions

- MathematicsResearch in Number Theory
- 2018

We prove function field theorems supporting the Cohen–Lenstra heuristics for real quadratic fields, and natural strengthenings of these analogs from the affine class group to the Picard group of the…

Motivic random variables and representation stability II: Hypersurface sections

- MathematicsAdvances in Mathematics
- 2019

Topological aspects of the dynamical moduli space of rational maps

- MathematicsAdvances in Mathematics
- 2022

Statistics of Number Fields and Function Fields

- Mathematics
- 2011

We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields…

## References

SHOWING 1-10 OF 88 REFERENCES

Hurwitz spaces

- Mathematics
- 2005

1.1 The classical Hurwitz space and the moduli of curves The classical Hurwitz space first appeared in the work of Clebsch [5] and Hurwitz [17] as an auxiliary object to study the moduli space of…

Homological stability for the mapping class groups of non-orientable surfaces

- Mathematics
- 2008

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that…

The stable moduli space of Riemann surfaces: Mumford's conjecture

- Mathematics
- 2002

D.Mumford conjectured in (30) that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes i of di- mension 2i. For the purpose of…

Results of Cohen-Lenstra type for quadratic function fields

- Mathematics
- 2008

Consider hyperelliptic curves C of fixed genus over a finite field F. Let L be a finite abelian group of exponent dividng N . We give an asymptotic formula in |F|, with explicit error term, for the…

Etale Cohomology and the Weil Conjecture

- Mathematics
- 1988

I. The Essentials of Etale Cohomology Theory.- II. Rationality of Weil ?-Functions.- III. The Monodromy Theory of Lefschetz Pencils.- IV. Deligne's Proof of the Weil Conjecture.- Appendices.- A I.…

Statistics of Number Fields and Function Fields

- Mathematics
- 2011

We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields…

Stabilization for mapping class groups of 3-manifolds

- Mathematics
- 2007

We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and…

The inverse Galois problem and rational points on moduli spaces

- Mathematics
- 1991

We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G…

Extending families of curves over log regular schemes

- Mathematics
- 1999

Abstract In this paper, we generalize to the “log regular case” a result of de Jong and Oort which states that any morphism satisfying certain conditions from the complement of a divisor with normal…

STABILITY THEOREMS FOR SPACES OF RATIONAL CURVES

- Mathematics
- 1999

on the space of smooth based maps Map(Σ, X), would encode a lot of the topology of the space in terms of the critical points of the functional. These critical points are harmonic maps; the absolute…