# Hurwitz space

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In [mathematics](/source/Mathematics), in particular [algebraic geometry](/source/Algebraic_geometry), **Hurwitz spaces** are [moduli spaces](/source/Moduli_space) of ramified [covers](/source/Covering_space) of the [projective line](/source/Projective_line), and they are related to the [moduli of curves](/source/Moduli_of_curves). Their [rational points](/source/Rational_point) are of interest for the study of the [inverse Galois problem](/source/Inverse_Galois_problem), and as such they have been extensively studied by [arithmetic geometers](/source/Arithmetic_geometry). More precisely, Hurwitz spaces classify isomorphism classes of [Galois covers](/source/Galois_covering) with a given [automorphism group](/source/Automorphism_group) G and a specified number of [branch points](/source/Branch_point). The [monodromy](/source/Monodromy) [conjugacy classes](/source/Conjugacy_class) at each branch point are also commonly fixed. These spaces have been introduced by [Adolf Hurwitz](/source/Adolf_Hurwitz)[1] which (with [Alfred Clebsch](/source/Alfred_Clebsch) and [Jacob Lüroth](/source/Jacob_L%C3%BCroth)) showed the connectedness of the Hurwitz spaces in the case of simply branched covers (i.e., the case where G is a [symmetric group](/source/Symmetric_group) and the monodromy classes are the conjugacy class of [transpositions](/source/Transposition_(mathematics))).

## Motivation

Let G be a finite group. The [inverse Galois problem](/source/Inverse_Galois_problem) for G asks whether there exists a finite [Galois extension](/source/Galois_extension) F \mid \Q whose [Galois group](/source/Galois_group) is isomorphic to G. By [Hilbert's irreducibility theorem](/source/Hilbert's_irreducibility_theorem), a positive answer to this question may be deduced from the existence, instead, of a finite Galois extension F \mid \Q(T) with Galois group G. In other words, one may try to find a connected ramified Galois cover of the projective line \mathbb{P}^1_{\Q} over \Q whose automorphism group is G. If one requires that this cover be geometrically connected, that is F \cap \bar\Q = \Q, then this stronger form of the inverse Galois problem is called the **regular inverse Galois problem**.

A motivation for constructing a moduli space of G-covers (i.e., geometrically connected Galois covers of \mathbb{P}^1 whose automorphism group is G) is to transform the regular inverse Galois problem into a problem of [Diophantine geometry](/source/Diophantine_geometry): if (geometric) points of the moduli spaces correspond to G-covers (or extensions of \bar\Q(T) with Galois group G) then it is expected that rational points are related to regular extensions of \Q(T) with Galois group G.

This geometric approach, pioneered by [John G. Thompson](/source/John_G._Thompson), [Michael D. Fried](/source/Michael_D._Fried), [Gunter Malle](/source/Gunter_Malle) and Wolfgang Matzat,[2] has been key to the realization of 25 of the 26 [sporadic groups](/source/Sporadic_group) as Galois groups over \Q — the only remaining sporadic group left to realize being the [Mathieu group M23](/source/Mathieu_group_M23).

## Definitions

### Configuration spaces

Let G be a finite group and n be a fixed integer. A [configuration](/source/Configuration_space_(mathematics)) is an unordered list of n distincts points of \mathbb{A}^1(\C). Configurations form a [topological space](/source/Topological_space): the [configuration space](/source/Configuration_space_(mathematics)) \operatorname{Conf}_n of n points. This space is the analytification (see [GAGA](/source/GAGA)) of an [algebraic scheme](/source/Algebraic_scheme) \mathcal{U}_n, which is the open subvariety of \mathbb{A}^n obtained by removing the closed subset corresponding to the vanishing of the [discriminant](/source/Discriminant).

The [fundamental group](/source/Fundamental_group) of the (topological) configuration space \operatorname{Conf}_n is the [Artin braid group](/source/Artin_braid_group) B_n, generated by elementary braids \sigma_1, \ldots, \sigma_{n-1} subject to the braid relations (\sigma_i and \sigma_j commute if |i-j|>1, and \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}). The configuration space has the [homotopy type](/source/Homotopy_type) of an [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) K(B_n, 1).[3][4]

### *G*-covers and monodromy conjugacy classes

A G-cover of \mathbb{P}^1_{\C} ramified at a configuration \mathbf{t} \in \mathrm{Conf}_n is a triple (Y, p, f) where Y is a connected topological space, p : Y \to \mathbb{P}^1 \smallsetminus \mathbf{t} is a [covering map](/source/Covering_map) which does not extend into a cover of \mathbb{P}^1\smallsetminus\mathbf{t'} where \mathbf{t'} is a configuration with less than n points, and f is a group isomorphism \operatorname{Aut}(Y, p) \simeq G. Up to isomorphism, a G-cover is determined by its **monodromy morphism**, which is a conjugacy class of [group homomorphisms](/source/Group_homomorphism) \pi_1(\mathbb{P}^1 \smallsetminus \mathbf{t},\, \infty) \to G.

One may choose a generating set of the fundamental group \pi_1(\mathbb{P}^1 \smallsetminus \mathbf{t},\, \infty) consisting of homotopy classes of loops \gamma_1, \ldots, \gamma_n, each rotating once counterclockwise around each branch point, and satisfying the relation \gamma_1 \cdots \gamma_n = 1. Such a choice induces a correspondence between G-covers and conjugacy classes of tuples (g_1, \ldots, g_n) \in G^n satisfying g_1 \cdots g_n = 1 and such that g_1, \ldots, g_n generate G: here, g_i is the image of the loop \gamma_i under the monodromy morphism.

The conjugacy classes of G containing the elements g_1, \ldots, g_n do not depend on the choice of the loops \gamma_i. They are the **monodromy conjugacy classes** of a given G-cover. We denote by V_n the set of n-tuples (g_1, \ldots, g_n) of elements of G satisfying g_1 \cdots g_n = 1 and generating G. If \mathbf{c} = (c_1, \ldots, c_n) is a list of conjugacy classes of G, then V_n^{\mathbf{c}} is the set of such tuples with the additional constraint g_i \in c_i.

### Hurwitz spaces

Topologically, the Hurwitz space classifying G-covers with n branch points is an unramified cover of the configuration space \operatorname{Conf}_n whose fiber above a configuration \mathbf{t} is in bijection, via the choice of a generating set of loops in \pi_1(\mathbb{P}^1 \smallsetminus \mathbf{t},\, \infty), with the quotient V_n/G of V_n by the conjugacy action of G. Two points in the fiber are in the same connected component if they are represented by tuples which are in the same orbit for the action of the braid group B_n induced by the following formula:

\sigma_i.(g_1,\ldots,g_n) = (g_1,\ldots,g_{i+1}^{g_i}, g_i, \ldots,g_n).

This topological space may be constructed as the [Borel construction](/source/Borel_construction) G \,\backslash\!\!\backslash\, V_n \,/\!\!/\, B_n:[5][6] its homotopy type is given by \widetilde{\operatorname{Conf}}_n \underset{B_n}{\times} (V_n/G), where \widetilde{\operatorname{Conf}}_n is the [universal cover](/source/Universal_Cover) EB_n of the configuration space \operatorname{Conf}_n \cong BB_n, and the action of the braid group B_n on V_n/G is as above.

Using [GAGA](/source/GAGA) results, one shows that space is the analyfication of a complex scheme, and that scheme is shown to be obtained via [extension of scalars](/source/Extension_of_scalars) of a \Z\left[\frac{1}{|G|}\right]-scheme \mathcal{H}_{G, n} by a descent criterion of Weil.[7][8] The scheme \mathcal{H}_{G, n} is an étale cover of the algebraic configuration space \mathcal{U}_n. However, it is not a *fine* moduli space in general.

In what follows, we assume that G is centerless, in which case \mathcal{H}_{G, n} is a fine moduli space. Then, for any field K of characteristic relatively prime to |G|, K-points of \mathcal{H}_{G,n} correspond bijectively to geometrically connected G-covers of \mathbb{P}^1_K (i.e., regular Galois extensions of K(T) with Galois group G) which are unramified outside n points. The [absolute Galois group](/source/Absolute_Galois_group) of \Q acts on the \bar\Q-points of the scheme \mathcal{H}_{G,n}, and the fixed points of this action are precisely its \Q-points, which in this case correspond to regular extensions of \Q(T) with Galois group G, unramified outside n places.

## Applications

### The rigidity method

If conjugacy classes (c_1, \ldots, c_n) are given, the list (c_1, \ldots, c_n) is **rigid** when there is a tuple (g_1, \ldots, g_n) \in c_1 \times \cdots \times c_n *unique up to conjugacy* such that g_1 \cdots g_n = 1 and g_1, \ldots, g_n generate G — in other words, V_n^{\mathbf{c}}/G is a singleton (see also [rigid group](/source/Rigid_group)). The conjugacy classes c_1,\ldots,c_n are **rational** if for any element g_i \in c_i and any integer k relatively prime to the order of g_i, the element g_i^k belongs to c_i.

Assume G is a centerless group, and fix a rigid list of rational conjugacy classes \mathbf{c} = (c_1, \ldots, c_n). Since the classes c_1, \ldots, c_n are rational, the action of the absolute Galois group G_{\Q}=\operatorname{Gal}(\bar\Q\mid\Q) on a G-cover with monodromy conjugacy classes c_1,\ldots,c_n is (another) G-cover with monodromy conjugacy classes c_1,\ldots,c_n (this is an application of Fried's *branch cycle lemma*[9]). As a consequence, one may define a subscheme \mathcal{H}^{\mathbf{c}}_{G, n} of \mathcal{H}_{G, n} consisting of G-covers whose monodromy conjugacy classes are c_1, \ldots, c_n.

Take a configuration \mathbf{t}. If the points of this configuration are not globally rational, then the action of G_{\Q} on G-covers ramified at \mathbf{t} will not preserve the ramification locus. However, if \mathbf{t} \in \mathcal{U}_n(\Q) is a configuration defined over \Q (for example, all points of the configuration are in \mathbf{A}^1(\Q)), then a G-cover branched at \mathbf{t} is mapped by an element of G_{\Q} to another G-cover branched at \mathbf{t}, i.e. another element of the fiber. The fiber of \mathcal{H}^{\mathbf{c}}_{G, n} \to \mathcal{U}_n above \mathbf{t} is in bijection with V_n^{\mathbf{c}}/G, which is a singleton by the rigidity hypothesis. Hence, the single point in the fiber is necessarily invariant under the G_{\Q}-action, and it defines a G-cover defined over \Q.

This proves a theorem due to Thompson: if there exists a rigid list of rational conjugacy classes of G, and Z(G)=1, then G is a Galois group over \Q. This has been applied to the [Monster group](/source/Monster_group), for which a rigid triple of conjugacy classes (c_1, c_2, c_3) (with elements of respective orders 2, 3, and 29) exists.

Thompson's proof does not explicitly use Hurwitz spaces (this rereading is due to Fried), but more sophisticated variants of the rigidity method (used for other sporadic groups) are best understood using moduli spaces. These methods involve defining a curve inside a Hurwitz space — obtained by fixing all branch points except one — and then applying standard methods used to find rational points on algebraic curves, notably the computation of their [genus](/source/Genus_(mathematics)) using the [Riemann-Hurwitz formula](/source/Riemann-Hurwitz_formula).[2]

### Statistics of extensions of function fields over finite fields

Several conjectures concern the asymptotical distribution of field extensions of a given base field as the discriminant gets larger. Such conjectures include the Cohen-Lenstra heuristics and the Malle conjecture.

When the base field is a function field over a finite field \mathbb{F}_q(T), where q = p^r and p does not divide the order of the group G, the count of extensions of \mathbb{F}_q(T) with Galois group G is linked with the count of \mathbb{F}_q-points on Hurwitz spaces. This approach was highlighted by works of [Jordan Ellenberg](/source/Jordan_Ellenberg), [Akshay Venkatesh](/source/Akshay_Venkatesh), Craig Westerland and TriThang Tran.[10][6][11][12] Their strategy to count \mathbb{F}_q-points on Hurwitz spaces, for large values of q, is to compute the homology of the Hurwitz spaces, which reduces to purely topological questions (approached with combinatorial means), and to use the [Grothendieck trace formula](/source/Grothendieck_trace_formula) and [Deligne](/source/Pierre_Deligne)'s estimations of eigenvalues of Frobenius (as explained in the article about [Weil conjectures](/source/Weil_conjectures)).

## See also

- [Deformation theory](/source/Deformation_Theory)
- [Moduli space](/source/Moduli_space) and [Moduli space of curves](/source/Moduli_space_of_curves)
- [Configuration space](/source/Configuration_space_(mathematics))
- [Inverse Galois theory](/source/Inverse_Galois_theory)
- [Hilbert's irreducibility theorem](/source/Hilbert's_irreducibility_theorem)
- [Dessin d'enfant](/source/Dessin_d'enfant)
- [Grothendieck–Teichmüller group](/source/Grothendieck%E2%80%93Teichm%C3%BCller_group)

## References

1. Hurwitz, A. (1891-03-01). ["Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten"](https://doi.org/10.1007/BF01199469) (in German). *Mathematische Annalen*. **39** (1): 1–60. [doi:10.1007/BF01199469](https://doi.org/10.1007/BF01199469). [ISSN 1432-1807](https://www.worldcat.org/issn/1432-1807). [S2CID 123053696](https://api.semanticscholar.org/CorpusID:123053696)

1. Malle, Gunter & Matzat, B. Heinrich (1999). [*Inverse Galois Theory*](https://link.springer.com/book/10.1007/978-3-662-12123-8). Springer Monographs in Mathematics. [doi:10.1007/978-3-662-12123-8](https://doi.org/10.1007/978-3-662-12123-8). ISBN 978-3-642-08311-2.

1. Fadell, Edward & Neuwirth, Lee (1962-06-01). ["Configuration Spaces."](http://www.mscand.dk/article/view/10517). *Mathematica Scandinavica*. **10**: 111. [doi:10.7146/math.scand.a-10517](https://doi.org/10.7146/math.scand.a-10517). [ISSN 1903-1807](https://www.worldcat.org/issn/1903-1807)

1. Fox, R. & Neuwirth, L. (1962-06-01). ["The Braid Groups."](http://www.mscand.dk/article/view/10518). *Mathematica Scandinavica*. **10**: 119. [doi:10.7146/math.scand.a-10518](https://doi.org/10.7146/math.scand.a-10518). [ISSN 1903-1807](https://www.worldcat.org/issn/1903-1807)

1. Randal-Williams, Oscar (2019-06-18). "Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)". [arXiv:1906.07447](https://arxiv.org/abs/1906.07447)

1. Ellenberg, Jordan S.; Venkatesh, Akshay; Westerland, Craig (2015-12-01). "Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields". [arXiv:0912.0325](https://arxiv.org/abs/0912.0325)

1. ["Hurwitz spaces | Société Mathématique de France"](https://smf.emath.fr/publications/espaces-de-hurwitz). *smf.emath.fr*. Retrieved 2023-04-25.

1. Dèbes, Pierre. ["Arithmétique et espaces de modules de revêtements"](https://www.math.uci.edu/~mfried/othlist-cov/Debes-HurwSpaces-SchinCf.pdf)

1. Fried, Michael. ["The Branch Cycle Lemma"](https://www.math.uci.edu/~mfried/deflist-cov/Branch-Cycle-Lem.html)

1. Ellenberg, Jordan S. & Venkatesh, Akshay (2005), ["Counting extensions of function fields with bounded discriminant and specified Galois group"](https://doi.org/10.1007/0-8176-4417-2_7), *Geometric Methods in Algebra and Number Theory*, Bogomolov, Fedor (ed.), Boston, MA: Birkhäuser, pp. 151–168, [doi:10.1007/0-8176-4417-2_7](https://doi.org/10.1007/0-8176-4417-2_7). ISBN 978-0-8176-4417-8, retrieved 2023-04-25

1. Ellenberg, Jordan S.; Venkatesh, Akshay; Westerland, Craig (2013-11-19). "Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II". [arXiv:1212.0923](https://arxiv.org/abs/1212.0923)

1. Ellenberg, Jordan S.; Tran, TriThang; Westerland, Craig (2023-03-05). "Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields". [arXiv:1701.04541](https://arxiv.org/abs/1701.04541)

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