### Hurwitz spaces

This is an Introduction: The subject really goes back to Riemann, and in special cases to Abel whose introduction of modular curves so motivated Riemann. Our historical comments here are the briefest possible. We hope they do not abet a mistaken notion that the topic classically was only about the moduli of curves of genus g. This introduction leaves a list of precise questions in the last section, and URLs to files that outline answers to them.

I. Why the abelian case is not a good model: Even today, mysteries about the subject lie in the nature of algebraic relations between pairs of functions on Riemann surfaces. For that study, it makes so much sense to consider compact Riemann surfaces X together with one nonconstant function f on them. That is exactly the topic of Hurwitz spaces.

The space of pairs (X,f)  has a natural complex structure, though what we mean be the word space depends on what equivalence relation we put on the pairs of (X,f). Particular applications (all those mentioned here) start by asking how to recognize connected components of that complex space relevant to the application. This file is an introduction to this.

Why you need X: Its first role is to allow comparing two functions f1 and f2 on X. That is, X is essentially equivalent, as noted by Riemann, to its field C(X) of rational functions over the complexes C. Then, as elements in C(X), the field operations always give an algebraic relation between f1 and f2. As introduced by Galois, – forerunners by Abel, Galois and Gauss, among famous others – that attaches a (finite) group G to the ordered pair (f1,f2). Here are two algebraic relation problems that don't require fancy material.

• Genus 0 problem: Given a positive integer g, and a (finite) group G, is there an X of genus no more than g supporting a function pair (f1,f2) for which the attached group is G?
• R(egular) I(nverse) G(alois) P(roblem): Given G, is there an X supporting a function pair (f1,f2) with the attached group G and the (minimal degree) algebraic having coefficients in Q, the rationals?

The simplest answer to the first is that for any given g, there is a limited set of G for which some such X exists. When g=0, the explicit list of these groups, and how they arise is the work of many papers, in this program initiated by Guralnick and Thompson [§ 7.2.3, thomp-genus0.pdf]. A basic use of R(iemann's) E(xistence) T(heorem) is that the Genus 0 problem has answer “Yes!,” if you don't bound g. The RIGP is even more involved because it includes so many generalizations of modern arithmetic problems.

The second role of X is for it to be the carrier of deformation information. This document is an introduction to that topic.

One case where it appears you don't need X is when G is abelian. This seems so because the functions come through Kummer theory, roots of rational functions in the variable z used below. Even, however, here, the ambiguity of the roots – branches of log (as in [§3, chapanal.pdf]) – cause immense problems in a 1st year graduate complex variables course. Also, here the deformation information is trivial, given just by looking at the location of the zeros of the rational functions.

A statement often made is that one can extend the abelian case to the solvable case. That is unfortunate! The first case this could apply is G a dihedral group of order 2p, with p an odd prime and the branching data given by involutions. Yet, the deformation information here is equivalent to describing the modular curve Y0(p), Abel's discovery. Not only isn't this first case elementary, few know what the extension would be to other G or that the extension applies without restriction (G need not be solvable). That is what Hurwitz spaces are about.

II. A gamut of connectedness applications: In a 1891 paper, Hurwitz explains how the set of degree d simple covers (all ﬁbers consist of at least d-1 points) P1 (the projective line – Riemann sphere) has a structure of complex manifold. In this he follows a much earlier (1867) paper of Clebsch who showed the connectedness of the space of simple covers. Hurwitz's paper thereby applies to show the connectedness of the moduli space of compact surfaces of genus g. Nowadays Hurwitz spaces refer more generally to moduli spaces of covers with speciﬁed Galois covering group and with precise constraints on the ramiﬁcation. The data for that is a Nielsen class with an equivalence relation.

Many arithmetic questions interpret as a property of moduli spaces of covers. Maybe that is not so obvious, so it is illuminating to see that this is so for the R(egular) I(nverse) G(alois) P(roblem).  The translation is to ﬁnding rational points on inner Hurwitz spaces. Hurwitz's example is an absolute Hurwitz space, though there is always a natural covering map between any absolute space and its inner version, and in Hurwitz's case that is an isomorphism.

III. The RIGP interpretation: This looks at the constraints a given question imposes on the collection of covers in question, and then it investigates whether there exist possible solutions on the associated moduli space, ﬁrst over C, and then over the ground ﬁeld – often taken as Q.

This approach translates the RIGP to this pure existance statement: Does there exist one Qpoint on any one of an infinite set of varieties? The discussion below is sufficient for understanding the explanation of the method and corollaries in CFPV-Thm.html. The effectiveness of the approach depends on how explicit one can be about the Hurwitz spaces that arise for a given group G. Hurwitz space components defined by Nielsen classes from r conjugacy classes in G are natural (unramified) covers of an the open subset of projective space of dimension r called the complement of the discriminant locus.

Such Hurwitz spaces have reductions by a natural action of PGL2(C) (Möbius transformations). The result is an r-3 dimensional space with a natural map to an open subset of a space, Jr, generalizing the classical j-line. Indeed, for r = 4, Jr is the j-line \ {∞}. Much theory and application works by exploiting these coverings and their cusps  from their completions as projective covers. For example, when r = 4, reduced Hurwitz spaces are algebraic curves, and quotients of the upper-half complex plane H by a finite index subgroup of PSL2(Z). This already resembles modular curves, but the analogy goes deeper.

IV. Cover Notation: Suppose you start with any compact Riemann surface X, and a (nonconstant) complex analytic (rational) function f on it. Then, there is a fundamental way to create many more such surfaces with a function on them. First regard the function as an analytic map f: X → P1. This notation means that a complex coordinate chart on X comes from the chart on P1, uniformized by the (inhomogenous) variable z that appears in 1st year complex variables.

As such, f  has a degree n, a Galois closure with some attached group G=Gf, and (distinct) branch points z10, … ,zr0. The latter are the values z' for which the preimage on X, f -1(z') consists of fewer than n distinct points. An extremely valuable, and – are you surprised? – nontrivial case is when X itself is P1 (genus 0). For this and other reasons, it makes sense to label our reference copy as P1z.

Then, we might use P1w for a genus 0 X, and notate f as w f(w)=z. Applications demand we be able to handle discussions of branch points without putting an order on them, or assuming they locate at special places (like on the real line), where there is a natural order. Still, we often, temporarily, assume there is an order.

The space of r ordered distinct complex points is Ur and the space of r unordered distinct points – the configuration space of r-branched covers – is Ur. The natural map between them takes an ordered r-tuple to an unordered r-tuple: Ψr : UrUr. This cover is Galois with group the symmetric group Sr on r letters. So, we use {z10, … ,zr0}=z0 for the unordered set, with (z10, … ,zr0)=(z0) for the ordered set.

V. Grabbing a cover by its Branch Points:  In each case, given any (continuous – piecewise analytic suffices) path P: [0,1] → Ur starting at z0=P(0),  there is a unique and continuous assignment of ft: XtP 1z, t ∈[0,1], of compact surface covers, along the path, branched at P(t)=zt. This is a deeper use of RET, though its proof still concatenates material from standard 1st year complex variables.

Here is how it goes. Start with a set of classical generators (or in more detail [§ 1.4, chpret4-firsthalf.pdf]) of the fundamental group of P1z \ z0. Then, following the path P(t), deform the classical generators, applying the algorithm [§ 2.2, chpret4-firsthalf.pdf] for constructing the cover from the deformed classical generators. It is as if the original cover obediently followed your pulling it around by its branch points.

Suppose t → P(t) is a closed path on Ur resulting in a branched cover (X, f)P at the end point of P, with its branch points at z0. Then, the following considerations [§ 3, chpret4-firsthalf.pdf] produce a space H(X, f) we interpret as a (connected) component of a Hurwitz space.
1. (X, f)P depends only on the homotopy class of P.
2. Only a finite set of possible covers Cov(X, f){(X, f)i}iI could be equivalent to (X, f)P. What those covers are – the indexing set I – depends on what equivalence we apply to covers.
Conclusion: Consider any equivalence relation * – preserved by continuing along paths – on the set of covers (X, f)P as [P] runs over elements of the fundamental group π1(Ur, z0). This means * defines a permutation representation on Cov(X, f), producing H(X, f) as an unramified cover of Ur. The corresponding map Φ (X,f): &#934;H(X, f) Ur gives  H(X, f) a natural complex analytic structure. This all follows from the basic theory of fundamental groups [§7, chpfund.pdf]. Further, any point hH(X, f) corresponds to a cover (X, f) h . Fundamental group theory says it also corresponds to a homotopy class [P] where P has initial point z0 and end point the image in Ur of h by Φ (X, f)

VI. What more do you need to know?: We collect points that weren't answered above, but seem to need answering, with a clickable reference for where an outline answer appears below.

1. The group Gf wasn't defined by taking two functions on X, just by one. Is this an independent production of G from the definition given by two functions?
2. To get this going you needed some pair (X, f). How do you produce the cover f: X → P1?
3. What are the equivalences between covers that arise in practice?
4. Is there anything explicit in the production of the space H(X, f)? While you are at it, if this is just a component of a Hurwitz space, what is the actual space?
5. If you use Hurwitz spaces for algebraic problems, mustn't they have algebraic (quasi-projective) structure and well-defined definition fields? (Answer: Yes, but from where does it come?)
6. What is the relation between Hurwitz spaces and other spaces from algebraic geometry, like modular curves, spaces of Abelian varieties and the moduli of curves of genus g?

Items #3-#5 are discussed in Nielsen-ClassesCont.html. Items #6-#7 in Alg-Equations.html. Item #8 is the source of recent developments, under the name M(odular) T(owers. An overview in mt-overview.html uses the special cases of modular curves (and dihedral groups) to explain its topics.

Pierre Debes and Mike Fried, 03/13/08