Hurwitz space

In

${\displaystyle G}$ and a specified number of branch points. The monodromy conjugacy classes at each branch point are also commonly fixed. These spaces have been introduced by Adolf Hurwitz^[1] which (with Alfred Clebsch and Jacob Lüroth) showed the connectedness of the Hurwitz spaces in the case of simply branched covers (i.e., the case where ${\displaystyle G}$ is a ).

Motivation

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

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

This geometric approach, pioneered by John G. Thompson, Michael D. Fried, Gunter Malle and Wolfgang Matzat,^[2] has been key to the realization of 25 of the 26 sporadic groups as Galois groups over ${\displaystyle \mathbb {Q} }$ — the only remaining sporadic group left to realize being the Mathieu group M23.

Definitions

Configuration spaces

Let ${\displaystyle G}$ be a finite group and ${\displaystyle n}$ be a fixed integer. A configuration is an unordered list of ${\displaystyle n}$ distincts points of ${\displaystyle \mathbb {A} ^{1}(\mathbb {C} )}$. Configurations form a topological space: the configuration space ${\displaystyle \operatorname {Conf} _{n}}$ of ${\displaystyle n}$ points. This space is the analytification (see

${\displaystyle {\mathcal {U}}_{n}}$, which is the open subvariety of ${\displaystyle \mathbb {A} ^{n}}$ obtained by removing the closed subset corresponding to the vanishing of the discriminant.

The fundamental group of the (topological) configuration space ${\displaystyle \operatorname {Conf} _{n}}$ is the

${\displaystyle B_{n}}$, generated by elementary braids ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$ subject to the braid relations (${\displaystyle \sigma _{i}}$ and ${\displaystyle \sigma _{j}}$ commute if ${\displaystyle |i-j|>1}$, and ${\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}}$). The configuration space has the ${\displaystyle K(B_{n},1)}$.[3]^[4]

G-covers and monodromy conjugacy classes

A ${\displaystyle G}$-cover of ${\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}$ ramified at a configuration ${\displaystyle \mathbf {t} }$ is a triple ${\displaystyle (Y,p,f)}$ where ${\displaystyle Y}$ is a connected topological space, ${\displaystyle p:Y\to \mathbb {P} ^{1}\smallsetminus \mathbf {t} }$ is a

, and ${\displaystyle f}$ is an isomorphism ${\displaystyle \operatorname {Aut} (Y,p)\simeq G}$, satisfying the additional requirement that ${\displaystyle p:Y\to \mathbb {P} ^{1}\smallsetminus \mathbf {t} }$ does not factor through any ${\displaystyle Y\to \mathbb {P} ^{1}\smallsetminus \mathbf {t'} }$ where ${\displaystyle \mathbf {t'} }$ is a configuration with less than ${\displaystyle n}$ points. An isomorphism class of ${\displaystyle G}$-covers is determined by the monodromy morphism, which is an equivalence class of ${\displaystyle \pi _{1}(\mathbb {P} ^{1}\smallsetminus \mathbf {t} ,\,\infty )\to G}$ under the conjugacy action of ${\displaystyle G}$.

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

The conjugacy classes of ${\displaystyle G}$ containing the elements ${\displaystyle g_{1},\ldots ,g_{n}}$ do not depend on the choice of the generating loops. They are the monodromy conjugacy classes of a given ${\displaystyle G}$-cover. We denote by ${\displaystyle V_{n}}$ the set of ${\displaystyle n}$-tuples ${\displaystyle (g_{1},\ldots ,g_{n})}$ of elements of ${\displaystyle G}$ satisfying ${\displaystyle g_{1}\cdots g_{n}=1}$ and generating ${\displaystyle G}$. If ${\displaystyle \mathbf {c} =(c_{1},\ldots ,c_{n})}$ is a list of conjugacy classes of ${\displaystyle G}$, then ${\displaystyle V_{n}^{\mathbf {c} }}$ is the set of such tuples with the additional constraint ${\displaystyle g_{i}\in c_{i}}$.

Hurwitz spaces

Topologically, the Hurwitz space classifying ${\displaystyle G}$-covers with ${\displaystyle n}$ branch points is an unramified cover of the configuration space ${\displaystyle \operatorname {Conf} _{n}}$ whose fiber above a configuration ${\displaystyle \mathbf {t} }$ is in bijection, via the choice of a generating set of loops in ${\displaystyle \pi _{1}(\mathbb {P} ^{1}\smallsetminus \mathbf {t} ,\,\infty )}$, with the quotient ${\displaystyle V_{n}/G}$ of ${\displaystyle V_{n}}$ by the conjugacy action of ${\displaystyle G}$. 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 ${\displaystyle B_{n}}$ induced by the following formula:

This topological space may be constructed as the

${\displaystyle G\,\backslash \!\!\backslash \,V_{n}\,/\!\!/\,B_{n}}$:^[5][6] its homotopy type is given by ${\displaystyle {\widetilde {\operatorname {Conf} }}_{n}{\underset {B_{n}}{\times }}(V_{n}/G)}$, where ${\displaystyle {\widetilde {\operatorname {Conf} }}_{n}}$ is the ${\displaystyle EB_{n}}$ of the configuration space ${\displaystyle \operatorname {Conf} _{n}\cong BB_{n}}$, and the action of the braid group ${\displaystyle B_{n}}$ on ${\displaystyle V_{n}/G}$ is as above.

Using

of a ${\displaystyle \mathbb {Z} \left[{\frac {1}{|G|}}\right]}$-scheme ${\displaystyle {\mathcal {H}}_{G,n}}$ by a descent criterion of Weil.^[7][8] The scheme ${\displaystyle {\mathcal {H}}_{G,n}}$ is an étale cover of the algebraic configuration space ${\displaystyle {\mathcal {U}}_{n}}$. However, it is not a fine moduli space in general.

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

Applications

The rigidity method

If conjugacy classes ${\displaystyle (c_{1},\ldots ,c_{n})}$ are given, the list ${\displaystyle (c_{1},\ldots ,c_{n})}$ is rigid when there is a tuple ${\displaystyle (g_{1},\ldots ,g_{n})\in c_{1}\times \cdots \times c_{n}}$ unique up to conjugacy such that ${\displaystyle g_{1}\cdots g_{n}=1}$ and ${\displaystyle g_{1},\ldots ,g_{n}}$ generate ${\displaystyle G}$ — in other words, ${\displaystyle V_{n}^{\mathbf {c} }/G}$ is a singleton (see also

). The conjugacy classes ${\displaystyle c_{1},\ldots ,c_{n}}$ are rational if for any element ${\displaystyle g_{i}\in c_{i}}$ and any integer ${\displaystyle k}$ relatively prime to the order of ${\displaystyle g_{i}}$, the element ${\displaystyle g_{i}^{k}}$ belongs to ${\displaystyle c_{i}}$.

Assume ${\displaystyle G}$ is a centerless group, and fix a rigid list of rational conjugacy classes ${\displaystyle \mathbf {c} =(c_{1},\ldots ,c_{n})}$. Since the classes ${\displaystyle c_{1},\ldots ,c_{n}}$ are rational, the action of the absolute Galois group ${\displaystyle G_{\mathbb {Q} }=\operatorname {Gal} ({\bar {\mathbb {Q} }}\mid \mathbb {Q} )}$ on a ${\displaystyle G}$-cover with monodromy conjugacy classes ${\displaystyle c_{1},\ldots ,c_{n}}$ is (another) ${\displaystyle G}$-cover with monodromy conjugacy classes ${\displaystyle c_{1},\ldots ,c_{n}}$ (this is an application of Fried's branch cycle lemma^[9]). As a consequence, one may define a subscheme ${\displaystyle {\mathcal {H}}_{G,n}^{\mathbf {c} }}$ of ${\displaystyle {\mathcal {H}}_{G,n}}$ consisting of ${\displaystyle G}$-covers whose monodromy conjugacy classes are ${\displaystyle c_{1},\ldots ,c_{n}}$.

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

This proves a theorem due to Thompson: if there exists a rigid list of rational conjugacy classes of ${\displaystyle G}$, and ${\displaystyle Z(G)=1}$, then ${\displaystyle G}$ is a Galois group over ${\displaystyle \mathbb {Q} }$. This has been applied to the Monster group, for which a rigid triple of conjugacy classes ${\displaystyle (c_{1},c_{2},c_{3})}$ (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

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 ${\displaystyle \mathbb {F} _{q}(T)}$, where ${\displaystyle q=p^{r}}$ and ${\displaystyle p}$ does not divide the order of the group ${\displaystyle G}$, the count of extensions of ${\displaystyle \mathbb {F} _{q}(T)}$ with Galois group ${\displaystyle G}$ is linked with the count of ${\displaystyle \mathbb {F} _{q}}$-points on Hurwitz spaces. This approach was highlighted by works of Jordan Ellenberg, Akshay Venkatesh, Craig Westerland and TriThang Tran.^{[10][6][11][12]} Their strategy to count ${\displaystyle \mathbb {F} _{q}}$-points on Hurwitz spaces, for large values of ${\displaystyle q}$, 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 and Deligne's estimations of eigenvalues of Frobenius (as explained in the article about Weil conjectures).

See also

• Deformation theory
• Moduli space of curves
• Configuration space
• Inverse Galois theory
• Hilbert's irreducibility theorem
• Dessin d'enfant
• Grothendieck–Teichmüller group

References