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.

Table of
Contents:

I.
Why the abelian case is not a good model:

II.
A gamut of connectedness applications:

III.
The RIGP interpretation:

IV.
Cover Notation:

V.
Grabbing a cover by its Branch Points:

VI. What more do you need
to
know?:

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 f_{1} and f_{2} 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
f_{1} and f_{2}. As introduced by
Galois, – forerunners by Abel, Galois and Gauss, among famous others –
that
attaches a (finite) group G
to the ordered pair (f_{1},f_{2}). 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 (f
_{1},f_{2}) 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
(f
_{1},f_{2}) 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.

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 Y

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)

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 **Q**point 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
PGL_{2}(**C**)
(Möbius transformations). The result is an
r-3 dimensional space with a
natural map to an open subset of a space, J_{r},
generalizing the classical j-line.
Indeed, for r = 4,
J_{r} 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
PSL_{2}(**Z**). This already
resembles *modular curves*,
but the analogy goes deeper.

As such, f has a degree n, a Galois closure with some attached group G=G

Then, we might use

The space of r ordered distinct complex points is U

V. Grabbing a cover by its Branch Points: In each case, given any (continuous – piecewise analytic suffices) path P: [0,1] → U

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

Suppose t → P(t) is a closed path on U

- (
*X*,*f*)_{P }depends only on the homotopy class of P. - Only a
finite set of possible
covers Cov
_{(X, f)}{(X, f)_{i}}_{i ∈ I}could be equivalent to (*X*,*f*)_{P}. What those covers are – the indexing set I – depends on what equivalence we apply to covers.

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.

- The group G
_{f}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? - To get this going you needed some pair (
*X*,*f*). How do you produce the cover f: X →**P**^{1}? - What are the equivalences between covers that arise in practice?
- 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? - 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?)
- 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