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.
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 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
: Ur → Ur. 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:
Xt →
P
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.
- (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 ∈
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): ΦH(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 h ∈
H(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.
- 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?
- To get this going you needed some pair (X, f).
How do you produce the cover f: X →
P1?
- 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