# Taming Genus 0 (or 1) components on
variables-separated equations

Michael D. Fried  Emeritus, UC Irvine
3547 Prestwick Rd, Billings MT 59101<EMAIL_ADDRESS>
###### Abstract.

To describe curves of form
${{\mathcal{C}}}_{f,g}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\\{(x,y)|f(x)-g(y)=0\\}$ and their number theory properties, you
must address ${{\mathcal{C}}}_{f,g}$ whose projective normalization has a
genus 0 (or 1) component. For $f$ and $g$ polynomials and $f$ indecomposable,
[Fr73a] distinguished ${{\mathcal{C}}}_{f,g}$ with $u=1$ versus $u>1$
components (Schinzel’s problem). For $u=1$, [Fr73b, (1.6) of Prop. 1] gave a
direct genus formula. To complete $u>1$ required an adhoc genus computation.

[Fr12] revisited later work. Pakovich [Pak18b], an example, dropped the
indecomposable and polynomial restrictions, but added ${{\mathcal{C}}}_{f,g}$
is irreducible ($u=1$). He showed – for fixed $f$ – unless the Galois closure
of the cover for $f$ has genus 0 (or 1), the genus grows linearly in
$\deg(g)$. Cor. 2.20 and Cor. 2.21 extend [Fr73b, Prop. 1] and use Nielsen
classes to generalize Pakovich’s formulation for $u>1$.

Using the solution to the Davenport and Schinzel problems, Hurwitz families
track the significance of these components, an approach motivated by Riemann’s
relating $\theta$ functions and half-canonical classes.

###### Key words and phrases:

Davenport’s Problem, Schinzel’s Problem, factorization of variables separated
polynomials, Riemann’s Existence Theorem, imprimitive groups, Pakovich’s
Theorem, Nielsen classes, genus 0 groups

###### 2020 Mathematics Subject Classification:

Primary 14D22, 20B15, 30F10, 12F10 Secondary 12D05, 12E30, 20E22

###### Contents

  1. 1 Fiber products of rational function covers
    1. 1.1 Characterizing what happens when $g\in{{\mathcal{R}}}_{f}$
      1. 1.1.1 Our approaches vs [Pak18b] and [Pak22]
      2. 1.1.2 Characterizing reducibility of $\tilde{{\mathcal{C}}}_{f,g}$
      3. 1.1.3 Relevant branch cycles
    2. 1.2 Results of the paper and role of the Classification
      1. 1.2.1 Results of the paper
      2. 1.2.2 Decomposable rational functions
      3. 1.2.3 Groups versus equations
      4. 1.2.4 Why consider rational function fiber products?
  2. 2 Branch cycles and Nielsen classes
    1. 2.1 Covers/branch cycles in a Nielsen class
      1. 2.1.1 Nielsen classes generalize curves of genus g
      2. 2.1.2 Components in families
      3. 2.1.3 Coelescing Nielsen classes
    2. 2.2 Intransitivity and genus formulas
      1. 2.2.1 Using conjugacy classes on a fiber product
      2. 2.2.2 Genus Corollaries of Prop. 2.17
  3. 3 Example genus component computations
    1. 3.1 Displaying examples of $g\in{{\mathcal{R}}}_{f}$
      1. 3.1.1 Example Nielsen classes
      2. 3.1.2 Essential differences between the ${\boldsymbol{\sigma}}\,$s and ${\boldsymbol{\tau}}\,$s
      3. 3.1.3 Degree 7 example components: Cors. 2.20 and 2.21
    2. 3.2 Context of §3.1.3
      1. 3.2.1 Coalescing and the Pakovich Goal
      2. 3.2.2 Coalescing and braids
      3. 3.2.3 §3.1.3 and the genus 0 problem
  4. 4 Dropping irreducibility hypothesis (1.1b)
    1. 4.1 $\tilde{{\mathcal{C}}}_{f,g}$ properties from $T_{1},T_{2}$
      1. 4.1.1 Using Prop. 1.8
      2. 4.1.2 Pieces to extend Thm. 1.1
      3. 4.1.3 Handling pairs of reps. $(T_{1},T_{2})$ of $G$
    2. 4.2 Treating ${{\text{\rm{\bf o}-char}}}>0$ when $f\in{{\mathbb{C}}}[x]$ has degree 7
      1. 4.2.1 Handling ${}_{j}{\boldsymbol{\sigma}}$, $j=1,2$, in (3.10)
      2. 4.2.2 Satellite Nielsen classes related to ${{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2)$
      3. 4.2.3 Significance of ${{\mathcal{H}}}_{7}$
    3. 4.3 The Main Theorem
      1. 4.3.1 Results of [Pak22]
      2. 4.3.2 The Main Theorem extending Thm. 1.1
  5. A Comments on fiber products
    1. A.1 Presenting the Galois closure cover
    2. A.2 When the orbifold characteristic is nonnegative
      1. A.2.1 Examples with ${{\text{\rm{\bf o}-char}}}=0$
      2. A.2.2 Examples with ${{\text{\rm{\bf o}-char}}}>0$
  6. B Expectations for $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ genus 0 components
    1. B.1 $\infty$-ly many $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ with $u=2$ components
    2. B.2 Series of genus 0 components of $\tilde{{\mathcal{C}}}_{f,g}$

##  1\. Fiber products of rational function covers

We use notation from [Fr12], quoting from it to relax the hypotheses (1.1a)
and (1.1b) below. We start with the same language as in [Pak18b] as if
referencing one rational function $f$ at a time. Yet, almost always, results
depend only on the Nielsen class (§2.1.1; §2.1 gives a quick survey) of the
cover given by $f$. That allows focusing on discrete data packets that help
find such low genus components.

For $z$ a complex variable, denote the projective line uniformized by the $z$
– including $z=\infty$ – by ${\mathbb{P}}^{1}_{z}$. Consider the set of
variables separated equations

${{\mathcal{C}}}_{f,g}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\\{(x,y)|f(x)-g(y)=0\\}\text{ with }f,g\in{{\mathbb{C}}}(x)\text{,
and $f$ fixed}.$

Regard ${{\mathcal{C}}}_{f,g}$ as an affine piece of the fiber product
${\mathbb{P}}^{1}_{x}\times_{{\mathbb{P}}^{1}_{z}}{\mathbb{P}}^{1}_{y}$ of the
rational function pair $(f,g)$ of respective degrees $m$ and $n$. Denote the
genus of a compact connected Riemann Surface $W$ by ${{\text{\bf g}}}_{W}$.

The projective normalization, $\tilde{{{\mathcal{C}}}}_{f,g}$, of
${{\mathcal{C}}}_{f,g}$ is a disjoint union of (say, $u$) compact connected
Riemann surfaces – components.111Most statements on compact Riemann surfaces
apply to projective nonsingular curves in any characteristic, except where
branch cycles enter. Despite Grothendieck’s theorem on characteristic $p$
fundamental groups, tame ramification is more complicated. Each has a genus,
given by notation like ${}_{i}{{\text{\bf g}}}_{f,g}$, $i=1,\dots,u$. Each
surface inherits canonical maps ${\text{\rm pr}}_{x}$ and ${\text{\rm
pr}}_{y}$, resp., to ${\mathbb{P}}^{1}_{x}$ and ${\mathbb{P}}^{1}_{y}$.
Excluding a finite set of points, $\tilde{{{\mathcal{C}}}}_{f,g}$ is the fiber
product without normalization. Prop. 1.18 reminds of the universal property of
fiber products of covers and a useful way to relate different fiber products.

For a (ramified) cover, $\varphi:X\to Z$, there is a Galois closure cover,
$\hat{\varphi}:\hat{X}\to Z$. Refer to the Galois group of that cover as the
monodromy group of $\varphi$. The Galois closure of an irreducible cover
$\varphi$ interprets as the (projective) normalization of a component of a
fiber product construction, compatible with our consideration of components of
$\tilde{{\mathcal{C}}}_{f,g}$ (§A.1). With $f\in{{\mathbb{C}}}(x)$ fixed,

$\text{consider }{{\mathcal{R}}}_{f}=\\{g\in{{\mathbb{C}}}(x)|f(x)-g(y)\text{
is reducible}\\}.$

######  Theorem 1.1 (Pakovich [Pak18b]).

Assume also:

  1. (1.1a)

the Galois closure of the cover ${\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$
has genus $>1$; and

  2. (1.1b)

we run over all $g$ not in ${{\mathcal{R}}}_{f}$.

Then, the genus of $\tilde{{\mathcal{C}}}_{f,g}$ goes to infinity with the
degree of $g$.

Much mathematics features equations of form $\tilde{{\mathcal{C}}}_{f,g}$. For
that, significant cases occur when $f$ and $g$ have an entanglement (as
discussed in §1.2.3), exposed through using their Galois closures as covers of
${\mathbb{P}}^{1}_{z}$. Foremost among those is the (nonobvious) reducibility
of $\tilde{{\mathcal{C}}}_{f,g}$, as occurred in considering Davenport’s and
Schinzel’s problems and now Pakovich’s result.

§1.1 starts by giving a precise characterization of the reducibility of these
fiber products, a characterization advantageous over dealing directly with
equations. At the end of the section, it separates this from finding
components of genus 0. It illustrates this with a running example that sets up
this reducibility property using these tools: Nielsen classes, the genus 0
problem and a formula to compute the genus of components of
$\tilde{{\mathcal{C}}}_{f,g}$. §1.2 then gives the results of the paper, which
are roughly these:

  1. (1.2a)

Illustrating the parameters for Nielsen classes for finding examples and
producing results generalizing those of Pakovich (§2).222Example: we would
never have discovered the complete description of Davenport’s problem if we
had relied on the ”look” of equations.

  2. (1.2b)

Using a formula – beyond [Fr73b] – for detecting and computing genuses of
components of reducible $\tilde{{{\mathcal{C}}}}_{f,g}$ in the (1a) families.

  3. (1.2c)

Finding/describing such components starting from (1b) based on coalescing
(§2.1.3) illustrated on running examples.

  4. (1.2d)

Generalizing Thm. 1.1 using the number of irreducible components as a
parameter. Then, characterizing when it changes in following a path of varying
$g\,$s.

The most important difference between [Pak18b] and this paper is that we start
with data about the fiber products $\tilde{{\mathcal{C}}}_{f,g}$. With this,
we can be specific about components, $W$, and then about the nature of the
projection ${\text{\rm pr}}_{y}:W\to{\mathbb{P}}^{1}_{y}$ appropriate for
considering extending Thm. 1.1. It was Pakovich’s idea to separate out covers
satisfying (1.1a), which we also adapted for ${\text{\rm pr}}_{y}$.

Cases with $\tilde{{\mathcal{C}}}_{f,g}$ having a genus 1 component are of
interest; §3.1.1 gives examples. There is one situation in which genus one
components must be considered separately, covered in Lem. 4.21, which
characterizes what must happen with ${\text{\rm pr}}_{y}$.

###  1.1. Characterizing what happens when $g\in{{\mathcal{R}}}_{f}$

In particular, we look at the $f\,$s failing the hypothesis (1.1a). §2.2 does
an exposition on the fiber product in terms of branch cycles.

Prop. 1.8 interprets reducibility of $\tilde{{\mathcal{C}}}_{f,g}$. The point
of [Fr73a] was to profitably relate pairs of covers
$\varphi_{X}:X\to{\mathbb{P}}^{1}_{z}$ and
$\varphi_{Y}:Y\to{\mathbb{P}}^{1}_{z}$ – for which even if given by rational
functions, by using relations between their monodromy (Galois closure) groups,
$G_{\varphi_{X}}$ and $G_{\varphi_{Y}}$. With that, we formulate dropping the
irreducibility assumption (1.1a). Using Prop. 1.8, we can start by fixing a
pair $(f^{*},g^{*})$ having the same Galois closures. Then, address the
following questions running over $g^{*}\circ g_{1}\in{{\mathbb{C}}}(x)$ with
$\deg(g_{1})\geq 1$.

  1. (1.3a)

When do component genuses on $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$
go to $\infty$ with $\deg(g_{1})$?

  2. (1.3b)

How to avoid $g_{1}\,$s for which $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ
g_{1}}$ has genus 0 (or 1) components for $\deg(g_{1})$ arbitrarily large,
contrary to (1.1a)?

We are heavy on examples illustrating using the tools. Thm. 4.23 uses this to
extend Thm. 1.1. Thm. 3.10 is a model for the best possible explicitness,
relating §3.1 to the genus 0 problem. Three subsections pave the way to
showing how we can be precise about when (1.1a) fails.

######  Definition 1.2.

A cover $\varphi:X\to Z$ is indecomposable if it does not decompose into
covers, $\varphi_{1}:X\to W$ and $\varphi_{2}:W\to Z$, with
$\deg(\varphi_{i})>1$, $i=1,2$. Equivalently, $\hat{\varphi}$ in its natural
$\deg(\varphi)$ permutation representation is primitive333Assume $\varphi$ is
irreducible. Then the permutation representation $T_{\varphi}$ is primitive if
there is no group properly between $G_{\varphi}(T_{\varphi},1)$ and
$G_{\varphi}$. [Fr70, Lem. 2]. [Fr12, Thm. 4.5] reviews characterizing this

(1.4) $\text{when }f,g\in{{\mathbb{C}}}[x]\text{ and }f\text{ is
indecomposable}.$

§1.1.1 compares Pakovich’s and our approaches/goals. Each considers a starting
place referencing pairs of rational functions $(f,g)$. Each considers
statements as $g$ changes. Pakovich considers ${{\mathcal{C}}}_{f,g}$ by
starting from particular pairs $(f,g)$ described by their coefficients, rather
than recognizing parameters for putting such pairs in natural families. Our
examples illustrate the following Principle.

######  Principle 1.3.

Results only depend on the Nielsen class, §2.1.1, in which $(f,g)$ falls.
Then, different Nielsen classes reveal distinct phenomena.555A general
phenomenon: Different Nielsen classes define different moduli problems.

§1.1.2 characterizes irreducible components of $\tilde{{\mathcal{C}}}_{f,g}$,
a characterization depending only on the Nielsen class of $(f,g)$. §1.1.3 uses
§1.1.2 to create, by example, notation for using branch cycles (and Nielsen
classes). This allows generalizing Thm. 1.1 with examples, without
coefficients for rational functions, consistent with his case. This includes
using systems of imprimitivity in decomposing rational functions.

Following these definitions and motivations, §1.2.1 lists general results,
illustrating using Ex. 1.6 as the start of a running example.

####  1.1.1. Our approaches vs [Pak18b] and [Pak22]

§2.1 gives an overview on using branch cycles (for covers of
$\varphi_{X}:X\to{\mathbb{P}}^{1}_{z}$). This effective computational tool
immediately gives the genus, ${{\text{\bf g}}}_{X}$, of $X$ from Riemann-
Hurwitz (RH, (2.3)), when $X$ is irreducible. We also discuss the orbifold
character, o-char, used by Pakovich.

§2.1.1 reminds of the (Hurwitz) space of covers in a given Nielsen class
(Princ. 2.3) generalizes the moduli space of curves of genus g. There are
several types. The most commonly used are absolute spaces. These come with a
group $G$, conjugacy classes C satisfying the properties in (2.1.1), and a
permutation representation, $T:G\to S_{n}$, attached to the degree $n$ covers,
$\varphi:X\to{\mathbb{P}}^{1}_{z}$, parametrized in the family. The notation
is ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$.

Assuming $\varphi$ is irreducible ($T$ is transitive), you immediately compute
the genus of the cover from a representing element
${\boldsymbol{\sigma}}\in{{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$ according to
Riemann-Hurwitz (RH) as in Ex. 1.6.666Another notation:
${\boldsymbol{\sigma}}\in{{\text{\bf C}}}$ – in some order the entries of
${\boldsymbol{\sigma}}$ fall in the classes C. The cycle type comes by
regarding the elements of C as in $S_{\deg(\varphi)}$.

Define ${{\mathcal{G}}}_{f,g}$ to be $(f^{*},g^{*})$ for which:

  1. (1.5a)

$f$ (resp. $g$) factors through
$f^{*}:{\mathbb{P}}^{1}_{x^{*}}\to{\mathbb{P}}^{1}_{z}$ (resp.
${\mathbb{P}}^{1}_{y^{*}}\to{\mathbb{P}}^{1}_{z}$) ; and

  2. (1.5b)

$\hat{f}^{*}:\hat{\mathbb{P}}^{1}_{x^{*}}\to{\mathbb{P}}^{1}_{z}$ and
$\hat{g}^{*}:\hat{\mathbb{P}}^{1}_{y^{*}}\to{\mathbb{P}}^{1}_{z}$, are
equivalent Galois covers.

Denote the common Galois group in (1.1.1b) by $G_{f^{*},g^{*}}$.

For Pakovich’s problem, we need to extend Nielsen classes to take account of
the involvement of two covers $(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$ as
entwined by Prop. 1.8. The notation we use here is ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$, with ${{\mathbb{T}}}=(T_{1},T_{2})$
two representations of $G$ as in (2.5). That uses inner Hurwitz spaces and the
monodromy group of the Galois closure of a cover.

For some groups, several conjugacy classes have the same cycle type.

######  Example 1.4 ($A_{n}$ example).

For $n\geq 3$ odd, $n$-cycles in $S_{n}$ are in $A_{n}$. There are two
conjugacy classes, ${\text{\rm C}}_{1},{\text{\rm C}}_{2}$, of such. They are
conjugate by an outer automorphism from $S_{n}$. That is, by the normalizer,
$N_{S_{n}}(G)$, of $G$ in $S_{n}$.

Use Def. 2.1: If C consists of ${\text{\rm C}}_{i}$ with multiplicity $m_{i}$,
$i=1,2$, then $N_{S_{n}}(G,{{\text{\bf C}}})=S_{n}$ if and only if
$m_{1}=m_{2}$. $\triangle$

######  Example 1.5 (Projective linear groups).

Take ${{\mathbb{F}}}_{q}$ the finite field of order a prime-power $q=p^{t}$.
The groups $G={{\text{\rm PGL}}}_{d}({{\mathbb{F}}}_{q})$, used in our
examples, have permutation representations of degree $n$ with several classes
of $n$-cycles. Here, though, what happens in Ex. 1.4 doesn’t hold.

For example, with $d=2$, take the permutation representation $T_{1}$ to be the
action on $n=\frac{p^{2{+}1}{-}1}{p{-}1}$ points and $T_{2}$ the action on
lines of projective $2$-space, ${\mathbb{P}}^{2}({{\mathbb{F}}}_{p})$. There
are pairs of $n$-cycles not conjugate by $N_{S_{n}}(G)$ [Fr73a, Lem. 5] and a
more general proof [Fr12, §4.2]. The cycle types for the permutation
representations $T_{1}$ and $T_{2}$ are the same, though the permutation
representations are different. $\triangle$

######  Example 1.6 (Genus of a cover from branch cycles).

RH computes the genus of covers given by the branch cycles called
${}_{1}{\boldsymbol{\sigma}}$ for the degree 7 example $f$ in §3.1.1 as
discussed in §1.1.2.

$\begin{array}[]{c}{}_{1}\sigma_{1}=(x_{1}\,x_{3})(x_{4}\,x_{5}),{}_{1}\sigma_{2}=(x_{1}\,x_{4}\,x_{6}\,x_{7})(x_{2}\,x_{3}),{}_{1}\sigma_{3}=(x_{1}\,x_{2}\,x_{3}\,x_{4}\,x_{5}\,x_{6}\,x_{7})^{-1}\\\
2(7{+}{{\text{\bf g}}}_{f}{-}1)={{\text{\rm
ind}}}({}_{1}\sigma_{1}){+}{{\text{\rm ind}}}({}_{1}\sigma_{2}){+}{{\text{\rm
ind}}}({}_{1}\sigma_{3})=2{+}4{+}6\implies{{\text{\bf g}}}_{f}=0.\end{array}$

What you would expect from a cover given by a rational function. Notice:
${}_{1}\sigma_{1}\cdot{}_{1}\sigma_{2}\cdot{}_{1}\sigma_{3}=1$, the product-
one condition of (2.1.1). $\triangle$

A Nielsen class, with its accompanying Hurwitz monodromy action, gathers a
family of covers into an algebraic variety (Hurwitz spaces). The structure
gives tools and coordinates for collections of pairs $(f,g)$ for which the
properties – say, reducibility and number of components, genuses – are
constant on the families. So, specific solutions to the problems correspond to
points in these spaces.

If we stick to the $(f,g)$ formulation, then our central problem is to
consider pairs of $(f,g)$ for which $\tilde{{\mathcal{C}}}_{f,g}$ has a low
genus component. §1.1.2 starts with this pair notation but ends by noting all
properties are constant on covers in a given Nielsen class.

We give representative Nielsen class data illustrating how much things vary
when $(G,{{\text{\bf C}}},{{\mathbb{T}}})$ changes. Our main examples use when
$f$ is indecomposable. Since we must consider covers that are reducible, §2.2
extends Nielsen classes to drop the usual transitivity assumption, the main
point in generalizing the genus formula (for components) when
$\tilde{{\mathcal{C}}}_{f,g}$ is reducible.

####  1.1.2. Characterizing reducibility of $\tilde{{\mathcal{C}}}_{f,g}$

We start with data for constructing $(f,g)$ examples for which
$\tilde{{\mathcal{C}}}_{f,g}$ has more than one component.777Some examples,
mostly excluded by (1.1a), get repeatedly rediscovered. §A.2 revisits these
briefly to show why they are misleading about the nature of the problem.
Examples toward extending Thm. 1.1 will begin as a Nielsen class — lists of
branch cycles – defining pairs of covers, $(f^{*},g^{*})$, satisfying Prop.
1.8 properties.

The monodromy (Galois closure) group, $G_{f}$, of a cover with its (faithful)
representation $T_{f}:G_{f}\to S_{\deg(f)}$ acts on cosets of the subgroup,
$G_{f}(T_{f},i)$, stabilizing an integer, $1\leq i\leq\deg(f)=m$. Since $f$ is
irreducible, the $G_{f}(T_{f},i)\,$s are conjugate, giving equivalent
permutation representations. Then, the quotient on the Galois closure cover
$\hat{f}:\hat{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$ returns

$f:\hat{\mathbb{P}}^{1}_{x}/G_{f}(T_{f},i)\to{\mathbb{P}}^{1}_{z}:\text{covers
for different $i\,$s are equivalent, Def.~{}\ref{covequiv}}.$

Prop. 1.8 is about a group $G$ having two different permutation
representations for which their entanglement represents having a reducible
fiber product, much of which generally applies to the fiber produce
$\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$, for irreducible covers
$\varphi_{X}$ and $\varphi_{Y}$. With our intense preoccupation with genus
zero covers and our computations with branch cycles and Nielsen classes, it is
more efficient to use rational functions for most results. For the letters of
those representations –when we must distinguish them – we use
${x_{1},\ldots,x_{m}}$ (resp. ${y_{1},\ldots,y_{n}}$) representing the zeros
of $f(x){-}z$ (resp. $g(y){-}z$) on which $T_{1}$ (resp. $T_{2}$) act
transitively.

Prop. 1.8, [Fr73a, Prop. 2] or [Fr12, Lem. 4.2], transfers considering
reducibility of $\tilde{{\mathcal{C}}}_{f,g}$ to the case where $G_{f}=G_{g}$,
with two (faithful) permutation representations, $T_{1}=T_{f},T_{2}=T_{g}$.

######  Definition 1.7.

Given a cover of groups $\psi:G^{\dagger}\to G$, with respective transitive
permutation representations $T^{\dagger}$ and $T$ defined by the cosets of
$H^{\dagger}$ and $H$, we say $T^{\dagger}$ extends $T$, if
$\psi(H^{\dagger})$ is a conjugate of $H$.

Or, in Def. 1.7, we could say $T$ is a quotient of $T^{\dagger}$, with the
understanding there could be several $T^{\dagger}\,$s giving a specific $T$.
We follow Prop. 1.8 with clarifying comments on rational functions.

######  Proposition 1.8.

There exists $(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$ satisfying the following.

  1. (1.6a)

$\hat{f}^{*}:{\mathbb{P}}^{1}_{x^{*}}\to{\mathbb{P}}^{1}_{z}$ and
$\hat{g}^{*}:\hat{\mathbb{P}}^{1}_{y^{*}}\to{\mathbb{P}}^{1}_{z}$, are
equivalent as Galois covers.

  2. (1.6b)

Components of $\tilde{{\mathcal{C}}}_{f,g}$ and
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ correspond one-one.

For $(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$:

  1. (1.7a)

$f^{*}$ and $g^{*}$ have exactly the same branch points, with respective
branch cycles of $f^{*}$ and $g^{*}$ of the same order.

  2. (1.7b)

Components on
${\mathbb{P}}^{1}_{x^{*}}\times_{{\mathbb{P}}^{1}_{z}}{\mathbb{P}}^{1}_{y^{*}}\leftrightarrow$
orbits of $G_{f^{*}}(T_{f^{*}},1)$ under $T_{g^{*}}$
$\leftrightarrow$ orbits of $G_{g^{*}}(T_{g^{*}},1)$ under $T_{f^{*}}$ (switch
$T_{f^{*}}$ and $T_{g^{*}}$).

  3. (1.7c)

A component $W$ $\leftrightarrow$ $G_{f^{*}}(T_{f^{*}},1)$ orbit $I$ (resp.
$G_{g^{*}}(T_{g^{*}},1)$ orbit $J$) in (1.8b), has degree $|I|$ (resp. $|J|$)
over ${\mathbb{P}}^{1}_{x^{*}}$ (resp. ${\mathbb{P}}^{1}_{y^{*}}$).888Given an
irreducible factor $h(x_{1},y)$ of $f(x_{1}){-}g(y)$ over
${{\mathbb{C}}}(x_{1})$, if $h(x_{1},y_{1})=0$, the degree of $h$ in $y$ is
the number conjugates of $y_{1}$ under $G_{f^{*}}(T_{f^{*}},1)$.

(1.8) Comments: Proof and use of Prop. 1.8:

In (1.8b) using ${x_{1},\ldots,x_{m^{*}}}$ and ${y_{1},\ldots,y_{n^{*}}}$,
restate “orbits of $G_{f^{*}}(T_{f^{*}},1)$ under $T_{g^{*}}$” as “orbits of
the subgroup of $G_{f^{*}}$ fixing $x_{1}$ acting on
${y_{1},\ldots,y_{n^{*}}}$.”

If $I$ is such an orbit on the $y\,$s, then the degree of the component over
${\mathbb{P}}^{1}_{x^{*}}$ is $k_{I}=|I|$. The notation switching $x^{*}$ and
$y^{*}$ would have the degree of the component over ${\mathbb{P}}^{1}_{y^{*}}$
corresponding to an orbit $J$ on the $x\,$s as $\ell_{J}=|J|$.

[Fr73a, Prop. 2] uses the Theorem of natural irrationalities in a see-saw
argument. Here is the original paper:

$\text{http://www.math.uci.edu/\hbox{\~{}}mfried/paplist-ff/dav-red.pdf}.$

There is a natural partial ordering on ${{\mathcal{G}}}_{f,g}$ using Def. 1.7.
There is also a maximal simultaneous Galois cover of ${\mathbb{P}}^{1}_{z}$
through which both $\hat{f}$ and $\hat{g}$ factor (with group $G^{\scriptstyle
mx}$ in Rem. 1.12) which may not be represented in ${{\mathcal{G}}}_{f,g}$. If
it is, this would be a unique maximal element for the partial ordering.

(1.9) Comments: Our interest is in more than one component of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$:

  1. (1.10a)

Equivalence of $\hat{f}_{*}$ and $\hat{g}_{*}$ in (1.8a) implies
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ components are quotients of
$\hat{\mathbb{P}}^{1}_{x^{*}}$ as given in (2.9).

  2. (1.10b)

If $\tilde{{\mathcal{C}}}_{f,g}$ is irreducible, then
$(f^{*}(x)=g^{*}(y)=z)\in{{\mathcal{G}}}_{f,g}$. Our interest is (1.8b) beyond
Thm. 1.1: more than 1 component.

  3. (1.10c)

$(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$ can be a significant entanglement (the
same meaning as in §1.2.3), even if $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is
irreducible.

(1.11) Comments: Quotient representations with $u>1$ in (1.8):

Prop. 1.8 can give many several examples of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ having the same number of components as
does $(f,g)$.

  1. (1.12a)

From (1.1.1), there is has a natural partial ordering on
${{\mathcal{G}}}_{f,g}$: For
$(f^{*},g^{*}),(f^{\star},g^{\star})\in{{\mathcal{G}}}_{f,g}$ then
$(f^{*},g^{*})\geq(f^{\star},g^{\star})$ if the cover pair $(f^{*},g^{*})$
factors through $(f^{\star},g^{\star})$.

  2. (1.12b)

The count of components on $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is
nondecreasing (take $Z=Z^{\prime}$ in Prop. 1.18).

  3. (1.12c)

In our examples $(f^{\star},g^{\star})$ is minimal among quotients of
$(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$ with
$\tilde{{\mathcal{C}}}_{f^{\star},g^{\star}}$ having multiple components,
relying on naming such examples from the genus 0 problem.

Still, in (1.1.2c) you could start anywhere in a chain with a term with a
fiber having multiple components to consider a Pakovich-type result.

(1.13) Comments: Distinguished properties of rational function covers:

It is easy to write genus 0 covers of ${\mathbb{P}}^{1}_{z}$ – just list
coefficients of a rational function – the properties (1.1.2) may explain why
so many study them.

  1. (1.14a)

A general cover $\varphi:X\to Z$ can be reducible, but a curve cover from a
rational function $f$ is always irreducible.

  2. (1.14b)

Composites of rational functions are always defined, but such a composition
may not be unique.

Reminder of the proof of (1.1.2a): Assume $f=f_{1}/f_{2}$, with relatively
prime $f_{1},f_{2}\in{{\mathbb{C}}}[x]$. Then $f_{1}(x)-f_{2}(x)z$ is
irreducible, since any factors would have degree at least 1 in $z$. §1.2.4
reminds of motivations from (1.1.2b).

######  Definition 1.9 (Cover equivalence).

Two covers $\varphi_{i}:X_{i}\to Z$, $i=1,2$, are (absolute) equivalent if
there is $\psi:X_{1}\to X_{2}$ such that $\varphi_{2}\circ\psi=\varphi_{1}$.
When the $X_{i}\,$s are both ${\mathbb{P}}^{1}_{x}$ (and
$Z={\mathbb{P}}^{1}_{z}$), then $\varphi$ identifies with an element of
${{\text{\rm PSL}}}_{2}({{\mathbb{C}}})$ (Möbius transformation). Further,
call the $f_{i}\,$s, $i=1,2$, reduced equivalent if there are
$\psi,\psi^{\prime}\in{{\text{\rm PSL}}}_{2}({{\mathbb{C}}})$ for which
$\psi^{\prime}\circ f_{2}\circ\psi=f_{1}$.

See Rem. 1.15 for a warning on using reduced equivalence. Ex. 1.10 comments on
the most obvious cases when $\tilde{{\mathcal{C}}}_{f,g}$ is reducible.

######  Example 1.10.

Assume $f=f_{1}\circ f_{2}$, $\deg(f_{1})>1$, $g=g_{1}\circ g_{2}$, with
$f_{1}$ and $g_{1}$ equivalent covers of ${\mathbb{P}}^{1}_{z}$.

  1. (1.15a)

Then $\tilde{{\mathcal{C}}}_{f_{1},g_{1}}$ is reducible with a component
isomorphic to the diagonal in the fiber product of $f_{1}$ with itself.

  2. (1.15b)

Similarly, $\tilde{{\mathcal{C}}}_{f,g}$ is reducible with a component
isomorphic to the (projective normalization of) $\\{(x,y)\mid
f_{2}(x){-}g_{2}(y)=0\\}$.

Taking $f_{2}(y)=y$ in (1.10b) gives a genus 0 component of
$\tilde{{\mathcal{C}}}_{f,g}$. The components remaining complementary to the
diagonal could be significant. List the orbits in (1.8b) as
${I_{1},\ldots,I_{u}}$. The case (1.10b) that corresponds to $u=2$ is with
$T_{f^{*}}$ doubly transitive. As a case of the genus formula, Cor. 2.22 gives
the genus of the non-diagonal component. $\triangle$

The following lemma is automatic from the natural projection maps to
${\mathbb{P}}^{1}_{x}$ and ${\mathbb{P}}^{1}_{y}$ attached to a component of a
fiber product. Parallel to Prop. 1.8 for reducibility; it characterizes when a
component of the fiber product has genus 0. Neither Prop. 1.8 nor does it show
how to find such a component or compute its genus. Those are the main topics
of the paper.

######  Lemma 1.11 (Non-uniserial).

Suppose $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has a genus 0 component, $W$.
That gives rational functions $h_{x}\in{{\mathbb{C}}}(w)$ and
$h_{y}\in{{\mathbb{C}}}(w)$ that fit in a commutative diagram:

(1.16)
${W}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\tilde{{\mathcal{C}}}_{f^{*},g^{*}}}$${{\mathbb{P}}^{1}_{x}}$${{\mathbb{P}}^{1}_{y}}$${{\mathbb{P}}^{1}_{z}}$$\scriptstyle{h_{x}}$$\scriptstyle{h^{*}}$$\scriptstyle{h_{y}}$$\scriptstyle{f^{*}}$$\scriptstyle{g^{*}}$

  1. (1.17a)

Thus, (1.16) gives an $h^{*}\in{{\mathbb{C}}}(w)$ with two decompositions:

$h^{*}=f^{*}\circ h_{x}=g^{*}\circ h_{y}.$

  2. (1.17b)

Conversely, the rational function decomposition of (1.11a) produces a genus 0
component of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ as in (1.16).

  3. (1.17c)

For each $g_{2}$, $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ h_{y}\circ g_{2}}$
has a genus zero component.

Replacing $g^{*}$ by an equivalent cover $g^{*}\circ\alpha$,
$\alpha\in{{\text{\rm PSL}}}_{2}({{\mathbb{C}}})$, requires replacing $h_{y}$
by $\alpha^{-1}\circ h_{y}$ to keep the decomposition of $h^{*}$.

###### Proof.

The only item requiring further proof is (1.11c). Apply Ex. 1.10 to
$\tilde{{\mathcal{C}}}_{f^{*},f^{*}\circ h_{x}\circ g_{2}}$ using that
$g^{*}\circ h_{y}\circ g_{2}=f^{*}\circ h_{x}\circ g_{2}$. ∎

[Fr73b, Thm. 2] viewed searching for genus 0 components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ with $f^{*},g^{*}\in{{\mathbb{C}}}[x]$
(polynomials) as generalizing Ritt’s Theorem. That we regarded as isting all
cases of $h^{*}\in{{\mathbb{C}}}[x]$ (polynomial) in Lem. 1.11, as in

(1.18) $\text{{\sl Cyclic factor reduction Diagram}
\cite[cite]{[\@@bibref{}{Fr73b}{}{}, p.~{}46]}}.$

Prop. 4.7 has two distinct types of potential components that figure in our
generalization of Thm. 1.1. One from our main technique, the other – less
likely, called a decomposition variant – given by (1.16).

######  Remark 1.12.

Given covers $\hat{\varphi}_{X}:\hat{X}\to Z$ and
$\hat{\varphi}_{Y}:\hat{Y}\to Z$, there is a minimal simultaneous Galois
cover, $\hat{\varphi}_{X,Y}:\hat{V}_{X,Y}\to Z$ of them both. It has group
$G_{\varphi_{X},\varphi_{Y}}$ with fiber product description
$G_{\varphi_{X}}\times_{G^{{\scriptstyle mx}}}G_{\varphi_{Y}}$ where
$G^{{\scriptstyle mx}}$ the maximal group through which both
$\hat{\varphi}_{X}$ and $\hat{\varphi}_{Y}$ factor.

######  Remark 1.13.

We didn’t need an extra assumption on the covers from $f$ and $g$ being
irreducible in Prop. 1.8, because – (1.1.2a) – rational function covers are
always irreducible. Making that extra assumption, though, then extends Prop.
1.8 to any pair of covers.

######  Remark 1.14 (Subdegrees).

Orbit lengths, even without assuming $\varphi_{X}$ and $\varphi_{Y}$ are genus
zero cover, orbit lengths of, say, of $G(T_{1},x_{1})$ on
$\\{y_{1},\dots,y_{n}\\}$ are so-called subdegrees. For $f$ equivalent to
$x^{m}$, or resp. to Ex. A.4; (called Chebychev), then sub degrees are 1,
resp. 2. Other than these two examples, if $f\in{{\mathbb{C}}}[x]$ (a
polynomial) and $T_{1}$ is primitive, then the only sub degrees of
$\tilde{{\mathcal{C}}}_{f,f}$ are 1 and $m{-}1$ [Fr70, Thm. 1]: $T_{1}$ is
doubly transitive.

######  Remark 1.15.

I use reduced equivalence (Def. 2.5) often in my papers. It suits moduli
problems and produces reduced Hurwitz spaces whose distinct points correspond
to covers that truly seem distinct rather than related by an algebraic trick.
Here, though, it requires careful use; you must, when involving a pair
$(f^{*},g^{*})$, apply composite on the left by $\alpha\in{{\text{\rm
PSL}}}_{2}({{\mathbb{C}}})$ simultaneously to both rational functions.

####  1.1.3. Relevant branch cycles

We create the notation for a Nielsen class example. §3.1.1 uses it to
illustrate Prop. 1.8. This is for a natural family of pairs of polynomial
covers $(f^{*},g^{*})$ of degree 7, whose members show how to drop the
irreducibility hypothesis of Thm. 1.1.

Back to Prop. 1.8: Start, not with rational function pairs, but with a group
having two permutation representations (in this illustrating case of degree 7)
$G=G_{f^{*}}=G_{g^{*}}$, with two (faithful) permutation representations,
$T_{1}=T_{f^{*}}$, $T_{2}=T_{g^{*}}$; here $G={{\text{\rm
GL}}}_{3}({{\mathbb{Z}}}/2)={{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2)$.999§1.2
tells some of the story behind this example, and the Genus 0 Problem that
generalizes it. [Fr05b] gives more detail up to the early 2000s.

The actions of $T_{1}$ and $T_{2}$ are, respectively, on points and lines of
projective 2-space over ${{\mathbb{Z}}}/2$. Use ${x_{1},\ldots,x_{7}}$ and
${y_{1},\ldots,y_{7}}$ as the respective letters of the representations in
(3.1).

Now we want cover pairs $(f^{*},g^{*})$ satisfying Pakovich considerations.

  1. (1.19a)

Ensure $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is reducible;

  2. (1.19b)

Check that $(f^{*},g^{*})$ corresponds to a pair of genus 0 covers of
${\mathbb{P}}^{1}_{z}$ (so given by rational functions).

  3. (1.19c)

Compute if a component of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has genus 0
(or 1)?

  4. (1.19d)

If the desire is for $f^{*}$ to be a polynomial cover, guarantee a totally
ramified branch point.

This is how you get such. Choose a collection of $r$ conjugacy classes C to
form a nonempty Nielsen class with properties listed in (2.1.1):

(1.20) ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}})=\\{\boldsymbol{\mu}\mid{\langle\mu_{1},\mu_{2},\cdots,\mu_{r}\rangle}=G,\mu_{1}\cdot\mu_{2}\cdots\mu_{r}=1\text{
and }\boldsymbol{\mu}\in{{\text{\bf C}}}\\}.$

That gets us to satisfying (1.1.3a). This follows because for any $\sigma\in
G$, from equality of the traces of $T_{1}(\sigma)$ and $T_{2}(\sigma)$, a
special case of [Fr73a, Lem. 2]101010Not something that I would expect readers
interested in this problem to know. which also guessed – and later proved –
the generalization of where such triples $(G,T_{1},T_{2})$ came from. This,
and Prop. 1.16 – which we use to explain the goals of the genus 0 problem in
§1.2.2 – are documented in [Fr99].

Now, choose C judiciously to assure (1.1.3b) holds. Using RH, we find there is
a most general – Def. 2.13 on coalescing explains that phrase – set of
conjugacy classes C in $G$, so that all degree 7 rational function pairs
$(f^{*},g^{*})$ satisfying (1.1.3b) come from $\boldsymbol{\mu}\in{{\text{\rm
Ni}}}(G,{{\text{\bf C}}})$.

That most general ${{\text{\bf C}}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{{\text{\bf C}}}_{2^{6}}$ consists of six repetitions of the
involution class of $G$. We explain the last sentence of Prop. 1.16 by example
in our continuation below to discussing (1.1.3c) and (1.1.3d).

######  Proposition 1.16.

All degree 7 rational function pairs with six branch points and
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ reducible, come from coalescing in this
diagram:

(1.21) $\begin{array}[]{c}\boldsymbol{\mu}\in{{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2^{6}})\mapsto(T_{1}(\boldsymbol{\mu}),T_{2}(\boldsymbol{\mu}))\\\
\in{{\text{\rm Ni}}}({{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2^{6}},T_{1})\times{{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{6}},T_{2}).\end{array}$

Further, all other such Nielsen classes giving diagrams like that of (1.21)
come from coalescing elements in ${{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{6}})$.

The notation ${{\mathbb{T}}}$ on representations alluded to in §1.1.1
corresponds to the diagram (1.21). Statement (1.1.3a) is from the general
theory of Hurwitz spaces as in [Fr77]. Statement (1.1.3b), though, is a
consequence of $H_{6}$ transitivity on ${{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{3},{{\text{\bf C}}}_{2^{6}})$ as in (3.2.2). Denote by $U_{r}$, the
collection of $r$ distinct points on ${\mathbb{P}}^{1}_{z}$.

  1. (1.22a)

Any ${{\boldsymbol{z}}}\in U_{6}$ gives $|{{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{3}({{\mathbb{Z}}}/3),{{\text{\bf C}}}_{2^{6}})|$ degree 7 pairs
$(f^{*},g^{*})$ with branch points at ${{\boldsymbol{z}}}$ satisfying (1.1.3a)
and (1.1.3b).

  2. (1.22b)

Running over ${{\boldsymbol{z}}}\in U_{6}$, there is one connected Hurwitz
space component of the pairs in (1.1.3a).

In §3.1.2 we find that the components on $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$
given by (1.1.3) don’t have genus 0, and they certainly aren’t given by
polynomials. That is they satisfy neither of (1.1.3c) nor (1.1.3d).111111The
polynomials cases were for Davenport’s and Schinzel’s problems in classifying
when $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ was reducible.

To get the general cases satisfying polynomial condition (1.1.3d), take the
number of elements, $r$, in C to be 4; three are of the involution class,
${\text{\rm C}}_{2}$ and the 4th, called ${\text{\rm C}}_{\infty}$, is a
7-cycle under $T_{1}$ or $T_{2}$. See § 3.2.1. There are two 7-cycle conjugacy
classes in ${{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2)$. Denote the conjugacy
collection by ${{\text{\bf C}}}_{\infty\cdot 2^{3}}$. These appear again in
§3.2.1.

Cor. 4.14 has the two conjugacy classes sets for polynomials, ${{\text{\bf
C}}}_{2\cot 3\cdot 7}$ and ${{\text{\bf C}}}_{2\cot 4\cdot 7}$, that give
Nielsen classes whose fiber product representatives have (two each) components
of genus 0 or 1. We are getting close to Ex. 1.6, but those are 3-tuples, not
4-tuples. Further, again for ${{\text{\bf C}}}_{\infty\cdot 2^{3}}$, the
resulting $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ – formed analogous to (1.1.3);
to get polynomials take the branch point associated to ${\text{\rm
C}}_{\infty}$ to be $\infty$ – don’t have a genus 0 component. §3.1.3 uses
these to illustrate the general genus computation for components.

###  1.2. Results of the paper and role of the Classification

In lieu of Discreteness Princ. (1.3), the search is for Nielsen classes
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ defined by some pair
$(f^{*},g^{*})$ in the conclusion of Prop. 1.8. Especially they have the same
Galois closures with corresponding pairs of permutation representations for
which $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is reducible.

§1.2.1 describes the subsections using their emphases on Nielsen classes. A
Nielsen class gives an algebraic structure to the family of covers they
describe. Using Prop. 1.8 shows how crucial it is to understand that Galois
closure. It is a tool for describing Nielsen classes of covers $g$ for which
$\tilde{{\mathcal{C}}}_{f,g}$ is reducible. Also, it allows us to compute the
genuses of those components. The goal is to regulate existence of a rational
function series $\\{{}_{i}g_{1}\\}_{i=1}^{\infty}$ with:

  1. (1.23a)

$\lim_{t\mapsto\infty}\deg({}_{i}g_{1})=\infty$; and

  2. (1.23b)

$\tilde{{\mathcal{C}}}_{f,g\circ{}_{i}g_{1}}$ has, say, a genus 0 (or 1)
component.

Thereby, $\\{{}_{i}g_{1}\\}_{i=1}^{\infty}$ gives a failure for $(f,g)$ to
condition (1.1b).

§1.2.2 goes from our degree 7 examples into the central role of the
classification relevant to this paper: Resolution of the genus 0 problem. We
formulate that as the description of Nielsen classes, ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},T)$, of families of genus 0 indecomposable covers:
$T$ is a primitive representation of $G$.

It concludes with notation for going beyond that indecomposable case. Still,
Rem. 1.21 explains why we decided – since so few know the tools we discuss
here – to avoid, for now, creating detailed induction notation that would
start at decomposable rational functions. Pakovich’s topic includes dealing
with decomposable rational functions. We have generalized it with explicit
examples, and their tools, starting with indecomposable functions.

§1.2.3 discusses work on separated variable equation properties that benefit
from using groups but is entirely classification free. In particular, it
mentions examples where solutions started without reference to the
classification. Yet the solution was achieved as a collaboration combining the
work of adherents of arithmetic/geometry and group theory.

§1.2.4 touches on Pakovich’s motivation, places where separated variables
equations are significant but hidden as a topic of concentration. It ties to
the ubiquitous use of rational function fiber products in moduli problems,
specifically back to Riemann proving properties of $\theta$ functions.

####  1.2.1. Results of the paper

§2.1 and §2.1.1 do a 2-step introduction to Nielsen classes. They also
introduce Pakovich’s o-char (orbifold characteristic): apply Riemann-Hurwitz
(RH) to the genus ${{\text{\bf g}}}_{\hat{f}}$ of the Galois closure, then
divide by $|G_{f}|$. This gives a simplifying expression for quantities of
interest to us. We regard it as known to Riemann (despite its tie to real
3-manifolds).

§2.2 shows how to treat Nielsen classes of these pairs $(f,g)$. With a slight
abuse we also use $U_{f}$ (or $U_{{{\text{\rm Ni}}}(G,{{\text{\bf
C}}})^{\text{\rm abs}}}$) to refer to those Nielsen classes.

§2.2.2 gives the formula for the $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$
component genuses – Cors. 2.20 and 2.21 (Methods I and II) – when it is
reducible, extending the original formula of [Fr73b, (1.6) of Prop. 1]. As a
corollary, it gives the genus of the nondiagonal component of
$\tilde{{\mathcal{C}}}_{f^{*},f^{*}}$ when $T_{f^{*}}$ is doubly transitive
rather than just primitive. 121212Cor. 2.20 applies when components are on a
fiber product of nonsingular covers. [Fr73b, Prop. 1] was the case the fiber
product is irreducible. So, this precisely extends adhoc discussions in
[Fr73b] and [Fr74].131313Pakovich suggested a precise genus calculation is
computationally formidable. We think our argument is a counter, especially
including our Nielsen class formulation.

§1.1.2 has already taken this approach: 1st find $(f^{*},g^{*})$ (or rather
Nielsen classes; Prop. 1.8) that have at their core
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ reducible. Then focus on characterizing
when such components might have genus 0 (Lem 1.11) and how that defies a
simplistic version of Pakovich’s Theorem 1.1.

§3.1 ramps up our running example to illustrate treating Nielsen classes of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}\,$s that have genus 0 components.

§3.2 then applies coalescing to go from a large Nielsen class of reducible to
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}\,$s to those with genus 0 components. We
mostly refer to other literature for what has happened in those directions
that stem from the solution of Davenport’s and the genus 0 problem. My concern
here is how these characterizations can tackle generalizing Pakovich’s
Theorem.

§4.1 emphasizes the permutation pairs $(T_{1},T_{2})$ of a group $G$ that
appear in Prop. 1.8 and how using Nielsen classes allows direct access to the
results of the genus 0 problem. §4.2 completes our paper-long -example where
$f^{*}$ appears in Nielsen classes with $T_{1}$ of degree 7. In particular, we
use it to illustrate how precise we can be in generalizing Thm. 1.1.

R. Guralnick produced the final precise solution on what are the primitive
permutation representations $(G,T)$ represented by rational functions (genus 0
covers).141414He also directed the results for variants on covers of a fixed
higher genus. To keep the statement uncomplicated, we restrict to just those
primitive groups with their cores related to simple non-cyclic groups. Then,
there is a vast but finite list of such $(G,T)$ giving genus 0 covers.

Outside group theorists, I’ve found little knowledge of how simple and
primitive groups relate. Here are some comments. [AOS85] constructs a template
of five patterns of primitive groups. Into four of those, you insert almost
simple groups. Affine groups comprise the fifth. So, the solution of the Genus
0 Problem ran through two filters: [AOS85]; and the distinct series of finite
simple groups, together with affine. This lexigraphic procedure accounts for
the number and length of contributions to the genus 0 resolution (for covers
over ${{\mathbb{C}}}$).

Here are keywords that connect simple groups to the primitive group
classification. According to [GLS], a quasisimple group $G$ is a perfect
central cover $G\to S$ of a simple group $S$. Here: cover means onto
homomorphism; perfect means the commutators $g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}$
in $G$ generate $G$; and central means the kernel is in the center of $G$.
Such a cover is a special case – because we don’t assume $S$ is simple – of a
Frattini central cover: where the map, if restricted to a proper subgroup of
$G$, won’t be a cover. Then, if $S$ is perfect, so is $G$.

A component, $H\leq G$, of $G$, is a quasisimple subgroup with a composition
series, between $H$ and $G$, a sequence of groups each normal in the next. The
group generated by components and the maximal normal nilpotent subgroup of $G$
is called the generalized Fitting subgroup, $F^{*}(G)$, of $G$. [GLS] calls a
group $G$ almost simple if $F^{*}(G)$ is quasisimple.

The one significant exception to finiteness is where the core is an
alternating group (includes several different groups and representations,
including $G=S_{n}$). We give two examples that arose in actual applications
to show that, among series of such genus 0 pairs, the non-obvious conclusion –
when restricted to consider such pairs giving reducible fiber products – is
there can be both infinite series (§B.1) of such or just finitely many (§B.2).

§4.3 gives a Pakovich generalization using the characterization of what
$g_{1}\,$s to avoid, so that genus 0 components don’t appear on the fiber
products $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}\,$s in the
corresponding Nielsen class. Generalizing Thm. 1.1, the result characterizes
paths of $g_{1}\,$s for which the number of irreducible components doesn’t
change from that of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$. Then, for each such
path, it concludes a result akin to Thm. 1.1.

####  1.2.2. Decomposable rational functions

To create covers $f:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$, with
specific decompositions use groups constructed to have various systems of
imprimitivity, the topic of this subsection. This suits a Nielsen class given
by $G$, a permutation representation $T$, and conjugacy classes C, with the
stipulation that RH gives genus 0 for a cover associated with
${\boldsymbol{\sigma}}\in{{\text{\bf C}}}$.

Rem. 1.20 notes that the proof of Lem. 1.17 has taken advantage, notationally,
of the covers being given by rational functions.

######  Lemma 1.17 (The decomposition chain).

Maximal chains of decompositions of $f\in{{\mathbb{C}}}(x)$ ($\deg(f)=m$)
correspond to the following equivalent sets.

  1. (1.24a)

Maximal chains of subsets

$\begin{array}[]{rl}&\\{1\\}<I_{2}<\dots I_{u{-}1}<\\{1,\dots,m\\},\text{ with
$1\in I_{j}$; and for $\sigma\in G_{f}$},\\\ &\text{{\hskip 36.135pt} and
$1\leq j\leq u$, if $1\in(I_{j})T_{f}(\sigma)$, then
$(I_{j})T_{f}(\sigma)=I_{j}$.}\end{array}$

  2. (1.24b)

Maximal chains of (1.17a) correspond to maximal chains (by inclusion) of
groups between $G_{f}$ and $G_{f}(T_{f},1)$: overgroups of $G_{f}(T_{f},1)$.

###### Proof.

Take $H$ a group between $G(T_{f},1)$ and $G_{f}$. The Galois correspondence
produces a chain of fields
${{\mathbb{C}}}(x_{1})\geq{{\mathbb{C}}}(x_{H})\geq{{\mathbb{C}}}(z)$ with
$f_{H},f_{H}^{*}\in{{\mathbb{C}}}(x)$, $f_{H}^{*}(f_{H}(x))=z$ and
$x_{H}=f_{H}(x_{1})$. Then, $x_{1}$ is a zero of $f_{H}(x)=x_{H}$; the other
zeros are $\deg(f_{H})$ conjugates of $x_{1}$, all zeros of $f(x)=z$.

Given a fixed labeling of the zeros of $f(x)=z$, ${x_{1},\ldots,x_{m}}$, the
result is a set $I_{H}$ appearing in (1.17a). This is a system of
imprimitivity; $I_{H}$ and its conjugate sets by the action of $G_{f}$
partition $\\{1,\dots,m\\}$ into disjoint sets. ∎

Based on Prop. 1.8, given $f$, to find cases where Thm. 1.1 does not hold,
consider these steps.

  1. (1.25a)

List nontrivial composition factors $f^{*}$ of $f$.

  2. (1.25b)

For each $f^{*}$ in (1.2.2a) list (up to equivalence of covers) $g^{*}\,$s
(including $f^{*}$) where $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has more than
one component as in (1.8).

  3. (1.25c)

Compute the genuses of the components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

Locating proper subgroups $G_{f}\geq H\geq G_{f}(T_{f},1)$, with the Nielsen
class of $T_{H}$ of genus 0, is the 2nd key to finding genus 0 components of
reducible $\tilde{{\mathcal{C}}}_{f,g}$ as $g$ varies (Lem. 4.2). Take

$\varphi_{X,X^{\prime}}:X^{\prime}\to X,\varphi_{Y,Y^{\prime}}:Y^{\prime}\to
Y,\varphi_{Z,Z^{\prime}}:Z^{\prime}\to Z$

to be three (nonconstant) covers of irreducible compact Riemann surfaces.

######  Proposition 1.18 (Component image).

Suppose $W$ is an irreducible normal projective curve and $\varphi:W\to Z$ is
a morphism that factors through $\varphi_{Z,X}$ and $\varphi_{Z,Y}$ so that
$\varphi_{Z,X}\circ\varphi=\varphi_{Z,Y}\circ\varphi$. Then, $\varphi$ factors
through a (unique) component of
$\tilde{{\mathcal{C}}}_{\varphi_{Z,X},\varphi_{Z,Y}}$.

Assume given covers

$\varphi_{Z,X}:X\to Z\text{ and }\varphi_{Z,Y}:Y\to Z,$

and similar for $X^{\prime},Y^{\prime},Z^{\prime}$ compatible with the maps
above. These induce

$\varphi_{X,Y,Z;X^{\prime},Y^{\prime},Z^{\prime}}:\tilde{{\mathcal{C}}}_{\varphi_{Z^{\prime},X^{\prime}},\varphi_{Z^{\prime},Y^{\prime}}}\to\tilde{{\mathcal{C}}}_{\varphi_{Z,X},\varphi_{Z,Y}},$

an onto map on components that is genus nonincreasing.

If a component
$W^{\prime}\leq\tilde{{\mathcal{C}}}_{\varphi_{Z^{\prime},X^{\prime}},\varphi_{Z^{\prime},Y^{\prime}}}$
has ${{\text{\bf g}}}_{W^{\prime}}>1$, and the restriction map $W^{\prime}\to
W$ has degree exceeding 1, then ${{\text{\bf g}}}_{W^{\prime}}>{{\text{\bf
g}}}_{W}$. Also, ${{\text{\bf g}}}_{W^{\prime}}=1={{\text{\bf g}}}_{W}$ if and
only if the restriction map is unramified.

###### Proof.

The first paragraph is a restatement of the universality of fiber products
phrased to emphasize components.

The map $\varphi_{X,Y,Z;X^{\prime},Y^{\prime},Z^{\prime}}$ has the effect:
$(x^{\prime},y^{\prime})\mapsto(\varphi_{X,X^{\prime}}(x^{\prime}),\varphi_{Y,Y^{\prime}}(y^{\prime}))$
for
$\varphi_{Z^{\prime},X^{\prime}}(x^{\prime})=\varphi_{Z^{\prime},X^{\prime}}(y^{\prime})$.
Compatibility on maps means both coordinates lie above the same point of $Z$.
Extended to fiber product normalizations,

$\text{this follows from
}\varphi_{Z,X}\circ\varphi_{X,X^{\prime}}=\varphi_{Z,y}\circ\varphi_{Y,Y^{\prime}}.$

The genus nonincreasing statement follows from considering the map on any
component
$W^{\prime}\subset\tilde{{\mathcal{C}}}_{\varphi_{Z^{\prime},X^{\prime}},\varphi_{Z^{\prime},Y^{\prime}}}\to
W\subset\tilde{{\mathcal{C}}}_{\varphi_{Z,X},\varphi_{Z,Y}}$. The genus of the
image is the number of linearly independent holomorphic differentials on $W$.
The pullback of these differentials on $W^{\prime}$ remain linearly
independent. They give a subspace of the holomorphic differentials on
$W^{\prime}$. Thus, the genus of $W^{\prime}$ is at least that of the genus of
$W$.

Finally, the last paragraph is a well-known consequence of RH. ∎

Most of Cor. 1.19 is immediate from Prop. 1.8.151515It simplifies if $f^{*}$
is indecomposable. According to (1.8), such $g^{*}$ give covers
$Y_{g^{*}}\to{\mathbb{P}}^{1}_{z}$ with the same Galois closure as has
$f^{*}$. By the Galois correspondence, up to equivalence, covers correspond to
subgroups of $G_{f^{*}}$. There are only finitely many. In applying Prop. 1.8,
as in (1.1.2) our interest is in considering nontrivial
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ with $u$ components, $u>2$.

######  Corollary 1.19.

From (1.8a), given $f^{*}$, the complete collection of
$g\in{{\mathcal{R}}}_{f^{*}}$ have the form $g^{\star}\circ g_{1}$ where
$g^{\star}$, up to equivalence of covers, runs over a finite set,
${{\mathcal{R}}}_{f^{*}}^{\star}$, of ${\mathbb{P}}^{1}_{z}$ covers with the
following properties.

  1. (1.26a)

The Galois closure $\hat{f}^{*}:\hat{X}_{f^{*}}\to{\mathbb{P}}^{1}_{z}$ has
$\hat{g}^{\star}:\hat{Y}_{g^{\star}}\to{\mathbb{P}}^{1}_{z}$ as a quotient.

  2. (1.26b)

There is $f^{\star}\in{{\mathbb{C}}}(x)$ for which $f^{*}=f^{\star}\circ
f_{1}$ with $\hat{f}^{\star}$ and $\hat{g}^{\star}$ equivalent as Galois
covers.

  3. (1.26c)

With $(f^{\star},g^{\star})$ replacing $(f^{*},g^{*})$ and the permutation
representations $(T_{f^{\star}},T_{g^{\star}})$ replacing
$(T_{f^{*}},T_{g^{*}})$, properties (1.8) hold.

######  Remark 1.20 (Convenience of Luroth’s Theorem).

The proof of Lem. 1.17 took advantage of Luroth’s Thm. that the field
corresponding to $H$ between ${{\mathbb{C}}}(x_{1})$ and ${{\mathbb{C}}}(z)$
has a rational function generator $x_{H}$. That was a notational convenience,
but – thanks to the generality of Luroth – a version of (1.17) holds for any
cover.

######  Remark 1.21.

Using indecomposable $f^{*}\,$s The genus 0 problem, our genus formula,
Nielsen classes, and coalescing give a wealth of examples. I decided to leave
proceeding with the induction until someone writes a GAP or Maple or
Mathematica program for the genus computation using our examples, including
the comments we have labeled (4.5), (4.6) and (4.8). These will automatically
raise new starting points with decomposable rational functions. With more
empirical data, good notation for interesting cases, especially decomposition
variants, will fall in place.

####  1.2.3. Groups versus equations

[O15, p. 72] has a cocktail party description of the simple group
classification. Mathematicians could use a guide to using pieces of the
classification and its proof. A guide aimed at non-group theorists. [O15, p.
70] motivates the classification by reference to the Higgs boson. This has
problems.

  1. (1.27a)

Awe for the Higgs boson is more common than knowledge of how to define a boson
(hint: foremost it includes photons and gluons).

  2. (1.27b)

Practical molecular chemistry, unmentioned, as does much about elementary
particles seriously uses (simple et al) groups.

  3. (1.27c)

The genus 0 problem (§3.2.3) via [AOS85] – now commonly applied to monodromy
groups of those 9th grade rational functions – is the expertise of the four
mathematicians central to [O15].

Here’s a way to look at Davenport’s problem and its solution.161616Although
not stated as I have here, special forays into versions of Chebotarev Density
and Hilbert’s Irreducibility have taken up much literature, as noted in
[FrJ86]. Assume two covers, $\varphi_{X}:X\to{\mathbb{P}}^{1}_{z}$ and
$\varphi_{Y}:Y\to{\mathbb{P}}^{1}_{z}$ as above, is defined over a number
field $K$, with ring of integers ${{\mathcal{O}}}_{K}$. Consider a relation,
$R$, between the traces ${\text{\rm tr}}(T_{X})$ and ${\text{\rm tr}}(T_{Y})$
is interpreted as follows. Running over the Frobenius
$\text{Fr}_{z^{\prime},\varphi_{X}}$ attached to
$z^{\prime}\in{\mathbb{P}}^{1}_{z}({{\mathcal{O}}}/{{\boldsymbol{p}}})$ in the
cover $\varphi_{X}$, assume

(1.28) $\begin{array}[]{c}({\text{\rm
tr}}(T_{X})(\text{Fr}_{z^{\prime},\varphi_{X}}),{\text{\rm
tr}}(T_{Y})(\text{Fr}_{z^{\prime},\varphi_{Y}})\\\ \text{ satisfies $R$ for
almost all ${{\boldsymbol{p}}}$ and all
$z^{\prime}\in{\mathbb{P}}^{1}_{z}({{\mathcal{O}}}/{{\boldsymbol{p}}})$}\end{array}.$

We say $(\varphi_{X},\varphi_{Y})$ are $R$-entangled. Here are two examples.

(1.29) $\begin{array}[]{c}\text{Davenport-
entanglement}:R_{D}\implies{\text{\rm tr}}(T_{X})>0\equiv{\text{\rm
tr}}(T_{Y})>0.\\\ \text{Galois-entanglement}:R_{G}\implies{\text{\rm
tr}}(T_{X})={\text{\rm tr}}(T_{Y}).\end{array}$

I list steps to solving Davenport’s problem in this case: $\varphi_{X}$ and
$\varphi_{Y}$ are polynomials, $\varphi_{X}=f$ and $\varphi_{Y}=g$, $f$ is
indecomposable, $f$ and $g$ inequivalent.

  1. (1.30a)

$R_{D}$-entanglement ($(f,g)$ a Davenport pair) $\implies$
$R_{G}$-entanglement $\implies\tilde{{\mathcal{C}}}_{f,g}$ is reducible.

  2. (1.30b)

There are only finitely many Nielsen classes of Davenport pairs.

  3. (1.30c)

$K$ any number field, there is an explicit description of the Nielsen classes
of (1.2.3b) over $K$, and of Davenport pairs over $K$.

  4. (1.30d)

For $K={{\mathbb{Q}}}$ there are none.

There was no classification of simple groups in 1969 when I submitted [Fr73a,
Footnote p. 1] to The Journal of Algebra. It conjectured that primitive
permutation representations that satisfied Prop. 1.8 were on the groups
${{\text{\rm PSL}}}_{m}({{\mathbb{Z}}}_{q})\,$s – with one degree 11
exception.181818This was correct, so [Fr99, §9] gives tests of Cor. 2.21 on
remaining examples. A precise statement of (1.2.3b) – again, for polynomial
covers, based on genus 0 monodromy – has all degrees for Davenport pairs in
the list (3.18). As [Fr99, §8 and §9] documents, Hurwitz space properties gave
the main arithmetic results about indecomposable Davenport polynomial pairs.

Also, starting from [Fr73b], there were many applications to problems
involving covers over finite fields with wild ramification (otherwise from
Grothendieck’s theorem, the result would be the same). To contrast with
characteristic 0 phenomena, [Fr99, Thm. 5.7 and §7] shows the genus 0
conclusion limiting the monodromy groups won’t hold.

Though not an expert on the classification, I got the group theory to
work.191919I found that few group theorists are expert on the whole
classification. It hinged on technical discoveries [Fr73a, Lems. 1-5].

  1. (1.31a)

Properties in Prop. 1.8: $G_{f}=G$, supports a pair of permutation
representations $(T_{1},T_{2})$ attached to a doubly transitive design.

  2. (1.31b)

The $n$ cycles in these groups consist of more than one conjugacy class, even
modulo the normalizer in $S_{n}$ of $(G,T_{i})$, $i=1,2$.

  3. (1.31c)

-1 is not a multiplier in a doubly transitive design [Fr73a, Lem. 5].

  4. (1.31d)

The Davenport covers have no more than three finite (not including $\infty$)
branch points [Fr73a, Thm. 1].

We call Cor. 4.12 and Cor. 4.14 the Degree 7 Corollaries, is the core of the
case for (genus 0 cover) Nielsen classes of degree $n=7$ whose conjugacy class
set contains an $n=7$-cycle. The cases – as listed in Thm. 3.10 – for general
$n$, with an $n$-cycle where $T_{f}$ is a primitive representation of the
monodromy group – can be carried out as in our running example. We expect the
details will be illuminating.

[Fr12, Prop. 4.4] gives a modern proof of (1.2.3c); [Fr73a, Lem. 5] used a
classical idea hinted to the author by Tom Storer. The upshot was the same:
Davenport pairs weren’t defined over ${{\mathbb{Q}}}$. I eschewed writing
explicit equations. For the list generating (1.2.3b) [CoCa99] used Pari (and
[Fr73a]) to write such equations. [DLSc61], [Le64] and [Sc82] motivated
considering these problems.

The distinction between $K={{\mathbb{Q}}}$ in (1.2.3d) and general $K$
(1.2.3c) alludes to the two halves of the title of [Fr12]. That, with [FrGS93]
and [GMS03], shows how pieces of the classification – motivating the Genus 0
problem – appeared in the service of number theory. These papers worked by
detaching group theory’s role. This precluded having to manipulate algebraic
equations. Still, work goes on that centers on equation manipulation.

[AZ01], [AZ03], [B99], [BT00], [Haj97], [Haj98] and [BeShTi99] continued in
that vein, over ${{\mathbb{Q}}}$, but they added finding genus 1 curves with
infinitely many ${{\mathbb{Q}}}$ points. The last three papers took on
specific equations with complicated coefficients, [Fr99, §11]. Our general
results, however, reduced their problems to showing such particular
expressions as

$x(x+a)...(x+(k{-}1)a)=y(y+b)...(y+(m{-}1)b).$

were not compositions of polynomials linearly equivalent to Chebyshev
polynomials. They used Mazur’s famous result on modular curves – delineating
precisely the small torsion groups possible for elliptic curves over
${{\mathbb{Q}}}$ – by showing their equations had too many solutions to be
given by torsion. Again, Mazur’s theorem was only applicable if equations were
over ${{\mathbb{Q}}}$.

######  Remark 1.22.

[Fr12, §7.2.3] revisits [AZ03] and [BeShTi99]. Both run into showing
irreducibility of specific fiber products. They didn’t use much of [Fr73a]
which easily handles their examples. [AZ03] mistakenly thought [Fr73a] used
the simple group classification (see above; because they based their
connection to the problem through [Fr73b]?).

######  Remark 1.23.

[AZ03, Prop. 2.6] has six separated variable polynomial equations with
$\infty$-ly many ${{\mathbb{Q}}}$ solutions with $f\in{{\mathbb{Q}}}[x]$ and
$g(x)=cf(x)$, $c\not=0,\pm 1$. How do these examples fit into Prop. 1.8? Hint:
Use (1.8a) to see that $f$ and $cf$ don’t have equivalent Galois closure
covers.

####  1.2.4. Why consider rational function fiber products?

My early papers gave motivation for tools like Prop. 1.8 phrased as Davenport
and Hilbert-Siegel Problems. Their point was to understand how different
algebraic curve covers relate by separating their variables using Galois
Theory. It took advantage of the method by which Siegel proved his most famous
theorem: finiteness of quasi-integral points on any closed curve of genus
$>0$. He created separated variables equations giving an equivalent result, to
which he could apply the Thue-Siegel-Roth Theorem.

Riemann’s generalization of Abel’s Theorem – for which he created the
distinction between even and odd $\Theta$ functions – motivated Siegel. He
used odd functions for the generalization though he recognized that producing
all objects – including meromorphic differentials of various kinds – on
algebraic curves of genus g could use both. This he rephrased using the
technical device behind forming canonical $\Theta$ divisors: half-canonical
classes.

Denote the divisor class of one of these by $[D]$, then its degree is
${{\text{\bf g}}}{-}1$, and $[2D]$ is the canonical class.202020Half-canonical
classes form a homogeneous space for 2-division points on Jacobians of curves
of genus g. But they are not the same, except when ${{\text{\bf g}}}=1$. They
divide into two types, even and odd, according to the dimension of their
linear systems being even or odd. He needed that every curve had some of these
that were non-degenerate: dimension 0 or 1, respectively, for even or odd.

He knew those parities were deformation invariants. He found these desired
classes by going to the boundary of the space of genus g curves, the locus of
hyperelliptic curves. This locus has a description where the actual linear
system dimensions are constant on components, though only the parity is
constant on the moduli of genus ${{\text{\bf g}}}\ >2$ curves.

Only from these separated variable equations was he able to stratify their
loci to precisely describe the canonical classes with a given dimension to
their linear system. Based on details in [Fa73], [Fr10, §6.2 and §B] produced
a version of automorphic functions from even $\theta$-nulls on Hurwitz spaces
of odd-branching type, with one proviso. That the Hurwitz space covers
contained general genus g curves. A generalization would be to decide when
nondegenerate, even canonical classes could be found near the boundary of any
odd-branching212121That means branch cycles have odd-order. type Hurwitz
space. An approach would be to find the nature of hyperelliptics accessible
from curves on those Hurwitz spaces.

Prop. 2.8 uses the Hurwitz spaces ${{\mathcal{H}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ to display components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ – when it is reducible – in families.
This puts structure on those components – equations with interesting
properties – that would otherwise be hidden from view.222222There is a general
problem in identifying when a curve of genus g identifies as a component of a
separated variables equation.

##  2\. Branch cycles and Nielsen classes

There are many expositions on this 1st part of Riemann’s Existence Theorem
(RET) including [Fr12, §1-4], [Vo96, Chap. 5] and [FrRET, Chap. 4].

###  2.1. Covers/branch cycles in a Nielsen class

Consider a degree $n$ cover $\varphi:X\to{\mathbb{P}}^{1}_{z}$, ramified over
${{\boldsymbol{z}}}=\\{{z_{1},\ldots,z_{r}}\\}$. Associate to it classical
generators, ${{{\mathcal{P}}}_{1},\ldots,{{\mathcal{P}}}_{r}}$, of the
r-punctured sphere

$U_{{\boldsymbol{z}}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{\mathbb{P}}^{1}_{z}\setminus\\{{z_{1},\ldots,z_{r}}\\},\text{
based at a point $z_{0}$}.$

We have chosen an ordering of ${z_{1},\ldots,z_{r}}$ given by their
subscripts. We need a little extra notation on the branch points to avoid
ambiguity, so we use $B_{f}$ for $\\{{{\boldsymbol{z}}}\\}$. These generators
are disjoint (piecewise smooth) paths except for the base point, and they
issue from that base point in clockwise order according to the ordering on
${{\boldsymbol{z}}}$.

To the cover associate branch cycles
${\boldsymbol{\sigma}}=({\sigma_{1},\ldots,\sigma_{r}})$ relative to
${{{\mathcal{P}}}_{1},\ldots,{{\mathcal{P}}}_{r}}$: $\sigma_{i}\in S_{n}$
corresponds to a closed path from $z_{0}$ going (clockwise) around $z_{i}$.
§2.1.1 reminds us of the definition of the Nielsen class associated to
$(G,{{\text{\bf C}}},T)$, the data for a family of covers.

In dealing with Pakovich’s problem, §2.1.2 extends the Nielsen class notation
to consider the pairs of ${\mathbb{P}}^{1}_{z}$ covers – having a nontrivial
relation– giving fiber products with more than one component that arise from
Prop. 1.8. Then, assuming the self-normalizing condition on the pairs of
representations associated to the covers, an apt name for Prop. 2.8 would be
picturing $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ components in families.

Finally, §2.1.3 gives the coelescing lemma by which we go from the
reducibility hypothesis on fiber products to finding families (on the
boundary) where the components have genus 0 or 1.

####  2.1.1. Nielsen classes generalize curves of genus g

Basic properties of ${\boldsymbol{\sigma}}$ (giving branch cycles for
$\varphi$);

  1. (2.1a)

Generation: Its entries generate $G_{\varphi}\leq S_{n}$ (embedding by
$T_{\varphi}$);

  2. (2.1b)

Product-one: $\sigma_{1}\cdots\sigma_{r}=1$; and

  3. (2.1c)

Conjugacy classes: Independent of the classical generators,
${\boldsymbol{\sigma}}$ defines $r$ conjugacy classes (some possibly
repeated), C, in $G_{\varphi}$.

######  Definition 2.1.

For $T:G\to S_{n}$, and C a conjugacy class collection, denote the the
normalizer of $G$ in $S_{n}$ by $N_{S_{n}}(G)$, and the subgroup of
$N_{S_{n}}(G)$ that conjugates C into itself by $N_{S_{n}}(G,{{\text{\bf
C}}})$.

Further, $\varphi$ (for any ramified cover) produces a canonical permutation
representation, $T_{\varphi}:G_{\varphi}\to S_{n}$, up to conjugation by $G$.
It does this in 2 steps.

  1. (2.2a)

Produce the Galois closure
$\hat{\varphi}:\hat{X}_{\varphi}\to{\mathbb{P}}^{1}_{z}$ by the fiber product
construction (§A.1).

  2. (2.2b)

Take $G(T_{\varphi},1)$ to be the group giving $X\to{\mathbb{P}}^{1}_{z}$ as
the quotient of $\hat{X}_{\varphi}/G(T_{\varphi},1)\to{\mathbb{P}}^{1}_{z}$.

Riemann’s Existence Theorem (RET) uses that, given classical generators,
${{{\mathcal{P}}}_{1},\ldots,{{\mathcal{P}}}_{r}}$, elements
${\boldsymbol{\sigma}}\in G^{r}$ (replacing $G_{\varphi}$ by $G$), satisfying
(2.1.1), automatically define a cover $\varphi:X\to{\mathbb{P}}^{1}_{z}$ with
${\boldsymbol{\sigma}}$ as branch cycles, unique up to equivalence of covers
(Def. 2.5). We write ${\boldsymbol{\sigma}}\in G^{r}\cap{{\text{\bf C}}}$ to
indicate the conjugacy classes they define.

######  Definition 2.2 (Nielsen classes).

We say either a cover $\varphi$ or any branch cycle description of it is in
the Nielsen class ${{\text{\rm Ni}}}(G,{{\text{\bf C}}})$ defined by (2.1.1)
if $T_{\varphi}$ is equivalent $T$. We call ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}})/N_{S_{n}}(G,{{\text{\bf C}}})\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$ absolute classes. Inner
Nielsen classes account for using the Galois closure. Those with branch points
${{\boldsymbol{z}}}$ correspond one-one to ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}})/G$.232323Notice, a single element of a Nielsen class defines the Nielsen
class.

For a given set of branch points, ${{\boldsymbol{z}}}$, elements of
${{\text{\rm Ni}}}(G,{{\text{\bf C}}})/N_{S_{n}}(G)$ correspond one-one with
covers up to (absolute) equivalence.242424Without a choice of classical
generators based at the branch points, this correspondence is not canonical.
For any $\sigma\in S_{n}$, ${\text{\rm ord}}(\sigma)$ is the least common
multiple of its disjoint cycle orders. Fix the number $r$ of branch points for
the covers we take for our genus computations.

The genus stays the same if we move the branch points (keeping them separate)
to any location we desire. Therefore, we can take all branch cycles relative
to a fixed set of classical generators for any particular situation. Covers of
${\mathbb{P}}^{1}_{z}$ ramified over ${{\boldsymbol{z}}}$ correspond to branch
cycles computed from those classical generators.

######  Principle 2.3.

Deforming branch points – keeping them distinct – canonically pulls an initial
cover uniquely along a trail of covers over the branch point path. So, it
defines (moduli) spaces of cover equivalence classes in a Nielsen class,
${{\text{\rm Ni}}}(G({\boldsymbol{\sigma}}),{{\text{\bf C}}})$: C classes in
$G({\boldsymbol{\sigma}})$ defined by the initial cover.

Different equivalences between covers (of ${\mathbb{P}}^{1}_{z}$) in Princ.
2.3 define different moduli spaces. Here it is usually absolute equivalence
Def. 2.5: covers are isomorphic by a continuous252525Thus, automatically
analytic by Riemann’s removable singularities theorem. map commuting with the
maps to ${\mathbb{P}}^{1}_{z}$.

The index, ${{\text{\rm ind}}}(\sigma)$, of a cycle $\sigma\in S_{n}$ is
${\text{\rm ord}}(\sigma){-}1$. Extend ind to a product of disjoint cycles
additively. Then, the genus, $\text{\bf g}_{f}$, of an irreducible cover
$f:X\to{\mathbb{P}}^{1}_{z}$ is immediate from branch cycles defining the
cover, as is the genus, $\text{\bf g}_{\hat{f}}$, of the Galois closure
$\hat{f}:\hat{X}\to{\mathbb{P}}^{1}_{z}$. See Ex. 1.6.

(2.3) $\begin{array}[]{rl}2(\deg(f)+\text{\bf
g}_{f}{-}1)&=\sum_{i=1}^{r}{{\text{\rm ind}}}(\sigma_{i})\\\
2(|G_{f}|+\text{\bf g}_{\hat{f}}{-}1)&=\sum_{i=1}^{r}|G_{f}|/{\text{\rm
ord}}(\sigma_{i})({\text{\rm ord}}(\sigma_{i})-1).\end{array}$

The latter formula uses that $\sigma_{i}$ in the regular representation of
$G_{f}$ is the product of $|G_{f}|/{\text{\rm ord}}(\sigma_{i})$ disjoint
cycles of length ${\text{\rm ord}}(\sigma_{i})$.

Pakovich (Rem. 4.4) uses an orbifold characteristic for a cover. From branch
cycles ${\boldsymbol{\sigma}}$, its expression from RH for ${{\text{\bf
g}}}_{\hat{f}}$, appears essentially by dividing by $|G_{f}|$:

(2.4) ${{\text{\rm{\bf o}-char}}}_{f}=\frac{2(1{-}\text{\bf
g}_{\hat{f}})}{|G_{f}|}=2+\sum_{i=1}^{r}(1/{\text{\rm ord}}(\sigma_{i})-1).$

Riemann knew this. Still, Rem. 4.4 recounts its generalization and prestige
tied to (real) 3-manifolds.

Given branch cycles ${\boldsymbol{g}}$ for one (irreducible) cover
$\varphi:X\to{\mathbb{P}}^{1}_{z}$ with representation $T_{\varphi}$ as in
(2.1.1), interpret elements $\sigma_{i}$ in (2.3) as
$T_{\varphi}(\sigma_{i})$. Consider any other (faithful, transitive)
permutation representation $T_{\varphi^{\prime}}$ of $G_{\varphi}$.

Canonically produce a new cover
$\varphi^{\prime}:X^{\prime}\to{\mathbb{P}}^{1}_{z}$, quotient of the Galois
closure, $\hat{\varphi}:\hat{X}\to{\mathbb{P}}^{1}_{z}$ with branch cycles
$T_{\varphi^{\prime}}({\boldsymbol{\sigma}})$. The genus of this cover comes
from (2.3) by replacing $n_{\varphi}$ by $n_{\varphi^{\prime}}$ and each
$T_{\varphi}(\sigma_{i})$ by $T_{\varphi^{\prime}}(\sigma_{i})$.

####  2.1.2. Components in families

Ex. 2.6 extends Nielsen class notation for Pakovich’s problem. Ex. 2.6 uses
${x_{1},\ldots,x_{m}}$ (resp. ${y_{1},\ldots,y_{n}}$) for letters on which
$T_{1}$ (resp. $T_{2}$) act transitively, as suggested by Rem. 1.13.

######  Definition 2.4 (Self-normalizing).

Refer to (transitive) $T_{1}:G\to S_{n}$ as self-normalizing if the normalizer
of $G(T_{1},x_{i})$ in $G$ is itself. Transitivity implies this does not
depend on the choice of $i$.

We are after handling pairs of rational functions $(f,g)$ for which
$\tilde{{\mathcal{C}}}_{f,g}$ is reducible, indeed, for finding Nielsen
classes of such pairs by for which components have genus 0 (or 1), as in our
examples.

Therefore, use Prop. 1.8, and extend Nielsen classes to the notation
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ with
${{\mathbb{T}}}=(T_{1},T_{2})$ interpreted on the common Galois closures of
two covers with the same branch points. In earlier language,
$(f^{*},g^{*})\in{{\mathcal{G}}}_{f,g}$. Below $G$ is the common Galois
closure group of the covers for $f^{*}$ and $g^{*}$, with C their common
conjugacy classes. Then, $T_{1,i}$ and $T_{2,j}$ are the coset representations
defined from those of $f$ and $g$ by the images in $G=G_{f}=G_{g}$ of the
respective $G(T_{f},x_{i})$ and $G_{g}(T_{g},y_{j})$ given by Def.
1.7.262626Rem. 1.12 says this more generally using a simultaneous covering of
the Galois closure of $f$ and $g$, and the common quotient of these,
$G^{\scriptstyle mx}$. I see no gain in this.

Map $\boldsymbol{\mu}\in{{\text{\rm Ni}}}(G,{{\text{\bf C}}})$ into the
diagonal of ${{\text{\rm Ni}}}(G,{{\text{\bf C}}})$:

$\delta(\boldsymbol{\mu})\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}({\mu_{1},\ldots,\mu_{r}};{\mu_{1},\ldots,\mu_{r}})\in\Delta({{\text{\rm
Ni}}}(G,{{\text{\bf C}}})\times{{\text{\rm Ni}}}(G,{{\text{\bf C}}}))/G.$

There are still $r$ conjugacy classes in the diagonal of $G\times G$. For
${{\boldsymbol{z}}}\in U_{r}$ and $\delta(\boldsymbol{\mu})$ as above denote
by $N_{i,j}$ the collection

(2.5)
$\\{({\boldsymbol{\sigma}}=T_{1,i}(\boldsymbol{\mu});{\boldsymbol{\tau}}=T_{2,j}(\boldsymbol{\mu})).$

######  Lemma 2.5.

The Nielsen class of ${\boldsymbol{\sigma}}\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{1})$ and of ${\boldsymbol{\tau}}\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{2})$) gives a pair $(f^{*},g^{*})$ ramified over ${{\boldsymbol{z}}}$
satisfying the conditions of Prop. 1.8. Now we form
$\varphi_{{\boldsymbol{\sigma}},{\boldsymbol{\tau}}}:\tilde{{\mathcal{C}}}_{f^{*},g^{*}}\to{\mathbb{P}}^{1}_{z}$.
Then, $({\boldsymbol{\sigma}},{\boldsymbol{\tau}})$ gives a cover
$\varphi_{i,j}:W_{i,j}\to{\mathbb{P}}^{1}_{z}$ corresponding to a component of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

The cover equivalences on $\varphi_{i,j}$ commuting with the equivalences on
the covers given by $f^{*}$ and $g^{*}$ are elements of
$N_{S_{m}}(G,{{\text{\bf C}}})\cap N_{S_{n}}(G,{{\text{\bf C}}})$
corresponding to the representations $T_{1}$ and $T_{2}$.

###### Proof.

As noted in §2.3, fixing the cover corresponding to $\boldsymbol{\mu}$
requires classical generators of the fundamental group of
${\mathbb{P}}^{1}_{z}\setminus\\{{{\boldsymbol{z}}}\\}$. Now we figure what
are the equivalences of the cover given by $\boldsymbol{\mu}$ that give a
cover of ${\mathbb{P}}^{1}_{z}$ that is also a Galois closure of both $f^{*}$
and $g^{*}$.

Such an equivalence would induce an automorphism on $G$, that would stabilize
both sets of groups

(2.6) $\\{G(T_{f},x_{i})\mid 1\leq i\leq m\\}\text{ and }\\{G(T_{g},y_{j})\mid
1\leq j\leq n\\}.$

It would also conjugate the conjugacy classes of C. So, what we are allowed is
$\mu\mapsto h\mu h^{-1}$ with $h\in N_{S_{m}}(G,{{\text{\bf C}}})\cap
N_{S_{n}}(G,{{\text{\bf C}}})$. ∎

Now consider the implications of Lem. 2.5 at the level of Hurwitz spaces.
Assume ${{\mathcal{H}}}(G,{{\text{\bf C}}},T)={{\mathcal{H}}}^{\text{\rm
abs}}$ is an absolute Hurwitz space with fine moduli: self-normalizing holds
for $G(T,1)$ in $G$. Then, there is a unique total family, or fine moduli
structure, for both the absolute and inner spaces, with the inner space
constructed naturally by a fiber product construction from the absolute space
(§A.1).

(2.7) $\begin{array}[]{ccccc}{{\mathcal{T}}}^{\text{\rm
in}}&\smash{\mathop{\hbox to50.0pt{\rightarrowfill}}\limits^{\Psi_{\text{\rm
in}}}}&{{\mathcal{H}}}(G,{{\text{\bf C}}})^{\text{\rm
in}}\times{\mathbb{P}}^{1}_{z}&\smash{\mathop{\hbox
to50.0pt{\rightarrowfill}}\limits}&U_{r}\times{\mathbb{P}}^{1}_{z}\\\
\Big{\downarrow}\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle\Psi_{{\text{\rm
abs}},{\text{\rm in}}}$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\Phi_{{\text{\rm abs}},{\text{\rm
in}}}\times\text{\bf Id}_{z}$}}$\hss}&&\Big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\text{\bf Id}_{r}\times\text{\bf
Id}_{z}$}}$\hss}\\\ {{\mathcal{T}}}^{\text{\rm abs}}&\smash{\mathop{\hbox
to50.0pt{\rightarrowfill}}\limits^{\Psi_{\text{\rm
abs}}}}&{{\mathcal{H}}}(G,{{\text{\bf
C}}},T)\times{\mathbb{P}}^{1}_{z}&\smash{\mathop{\hbox
to50.0pt{\rightarrowfill}}\limits}&U_{r}\times{\mathbb{P}}^{1}_{z}.\end{array}$

  1. (2.8a)

The fiber of $\Psi_{\text{\rm abs}}$ over
${{\boldsymbol{p}}}\times{\mathbb{P}}^{1}_{z}$,
${{\boldsymbol{p}}}^{\prime}\in{{\mathcal{H}}}(G,{{\text{\bf C}}},T)$
represents the class of the covers associated to
${{\boldsymbol{p}}}^{\prime}\in{{\mathcal{H}}}(G,{{\text{\bf C}}},T)$.

  2. (2.8b)

Similarly, the fiber of $\Psi_{\text{\rm in}}$ defined by
$\hat{{\boldsymbol{p}}}\in{{\mathcal{H}}}(G,{{\text{\bf C}}})^{\text{\rm in}}$
lying over ${{\boldsymbol{p}}}$ gives a Galois cover that appears in Ex. 2.6.

######  Example 2.6 (Components).

Take
${\boldsymbol{\sigma}}=T_{1,i}(\boldsymbol{\mu}),{\boldsymbol{\tau}}=T_{2,j}(\boldsymbol{\mu})$
as in (2.5). At the level of Hurwitz spaces, suppose
$\hat{{\boldsymbol{p}}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\hat{{\boldsymbol{p}}}(\boldsymbol{\mu})\in{{\mathcal{H}}}(G,{{\text{\bf
C}}})^{\text{\rm in}}$ corresponds to the Galois cover,
$\hat{\varphi}:\hat{X}\to{\mathbb{P}}^{1}_{z}$ associated to
$\boldsymbol{\mu}$, with group identified with $G$. Consider
$W_{i,j}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}W_{i,j}(\boldsymbol{\mu})$,

(2.9) $\begin{array}[]{c}\text{the image of
}(\hat{\varphi}_{1}(i),\hat{\varphi}_{2}(j)):\hat{X}\to(\hat{X}/G(T_{1},x_{i}),\hat{X}/G(T_{2},y_{j})).\\\
\text{It factors through }\\\
\hat{X}/G(T_{1},x_{i})=X_{i}\to{\mathbb{P}}^{1}_{z}\text{ and
}\hat{X}/G(T_{2},y_{j})=Y_{j}\to{\mathbb{P}}^{1}_{z}.\end{array}$

That is, $W_{i,j}$ represents an irreducible component on
$\tilde{{\mathcal{C}}}_{\varphi_{1}(i),\hat{\varphi}_{2}(j)}$. $\triangle$

######  Definition 2.7.

With notation of Ex. 2.6, denote the set of
$\\{({\boldsymbol{\sigma}}=T_{1,i}(\boldsymbol{\mu}),{\boldsymbol{\tau}}=T_{2,j}(\boldsymbol{\mu}))\\}/N_{S_{m}}(G,{{\text{\bf
C}}})\cap N_{S_{n}}(G,{{\text{\bf C}}})\text{ by }{{\text{\rm
Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}}).$

Prop. 2.8 assumes self-normalizing holds for both $T_{1}$ and $T_{2}$.
Reminder: $r$ is the number of classes in C. Rem. 2.9 reminds of $H_{r}$
basics. Denote a collection of $u$ representatives of $G(T_{1},x_{1})$ orbits
on ${y_{1},\ldots,y_{n}}$ by $J$.

######  Proposition 2.8.

For $j_{0}\in J$ there is a space ${\mathcal{W}}_{1,j_{0}}$ that fits in a
diagram

(2.10) ${{\mathcal{W}}_{1,j_{0}}\subset{{\mathcal{H}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})}$${{{\mathcal{T}}}(G,{{\text{\bf
C}}},T_{1})}$${{{\mathcal{T}}}(G,{{\text{\bf
C}}},T_{2})}$${{{\mathcal{H}}}(G,{{\text{\bf
C}}},T_{1})\times{\mathbb{P}}^{1}_{z}}$${{{\mathcal{H}}}(G,{{\text{\bf
C}}},T_{2})\times{\mathbb{P}}^{1}_{z}}$${U_{r}\times{\mathbb{P}}^{1}_{z}}$$\scriptstyle{{\text{\rm
pr}}_{x}}$$\scriptstyle{{\text{\rm
pr}}_{y}}$$\scriptstyle{\Psi_{T_{1}}}$$\scriptstyle{\Psi_{T_{2}}}$

with the following properties.

  1. (2.11a)

For ${{\boldsymbol{z}}}\in U_{r}$, the pullback of
${{\boldsymbol{z}}}\times{\mathbb{P}}^{1}_{z}$ on ${\mathcal{W}}_{1,j_{0}}$ is
a cover given by branch cycles $(\boldsymbol{\sigma};\boldsymbol{\tau})$ as in
Ex. 2.6.

  2. (2.11b)

(2.8a) can be written as a quotient of the Galois cover with branch cycles
given by $\boldsymbol{\mu}$ as in (2.9).

  3. (2.11c)

Branch cycles ${\boldsymbol{\sigma}}$ (resp. ${\boldsymbol{\tau}}$) give the
genus zero cover $f^{*}$ (resp. $g^{*}$) by which we recognize the cover of
(2.8b) as a component of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

Running over $j_{0}\in J$, denote the disjoint union of the spaces
${\mathcal{W}}_{1,j_{0}}$ modulo the action of $N_{S_{m}}(G,{{\text{\bf
C}}})\cap N_{S_{n}}(G,{{\text{\bf C}}})$ by ${{\mathcal{H}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$.272727Thus, these representations in the fibers of
Nielsen class family don’t exceed that given by in Prop. 1.8.

  1. (2.12a)

$H_{r}$ acts on ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$
compatible with its action on ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T_{i})$,
$i=1,2$, thus producing ${{\mathcal{H}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$
as a cover of $U_{r}\times{\mathbb{P}}^{1}_{z}$.

  2. (2.12b)

Also, ${{\mathcal{H}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ fits in the diagram
(2.10) in place of ${\mathcal{W}}_{1,j_{0}}$.282828Similarly, if you only mod
out by $G$ (rather than $N_{S_{m}}(G,{{\text{\bf C}}})\cap
N_{S_{n}}(G,{{\text{\bf C}}})$) on the ${\mathcal{W}}_{1,j_{0}}\,$s there is a
diagram with the union of the results and everything else the same.

###### Proof.

Much of the proof has already been established, so these are comments relating
the formulation of ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ and
its corresponding Hurwitz space, ${{\mathcal{H}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ as a cover of $U_{r}$, given by the action of $H_{r}$.

The construction of the total space ${{\mathcal{T}}}^{\text{\rm
in}}\smash{\mathop{\hbox to40.0pt{\rightarrowfill}}\limits^{\Psi_{\text{\rm
in}}}}{{\mathcal{H}}}(G,{{\text{\bf C}}})^{\text{\rm
in}}\times{\mathbb{P}}^{1}_{z}$ from the absolute space (as in §A.1) using the
self-normalizing condition has an inverse described by (2.6). The meaning of
(2.8a) is that the formation of ${{\mathcal{H}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ is a compatible analog.

Equivalences on Nielsen classes always include the action of $G$ (Rem. 2.9).
Running over the $(i,j)$, there is at most one representative for each
component because you can always take $x_{i}=x_{1}$ by conjugating by an
element of $G$. Since the stabilizer of that first position is then
$G(T_{1},x_{1})$, you have the leeway to conjugate by that group to set
$y_{j}$ to be any particular element in the orbit of $y_{j}$. ∎

######  Remark 2.9 ($H_{r}$ basics).

Braid orbits on the Nielsen classes correspond to components. In the action of
$H_{r}$ on any Nielsen class, braids always generate conjugation by $G$. That
is, for $g\in G$ there exists $q_{g}$ whose action on ${\boldsymbol{\sigma}}$
gives conjugation by $g$ [BFr82, Lem. 3.8]: we say $g$ is braided. As for
action of $N_{S_{m}}(G,{{\text{\bf C}}})\cap N_{S_{n}}(G,{{\text{\bf
C}}})=N_{m,n}$ on ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$, if
$N_{m,n}/G$ is not trivial, these elements may not be braided.

That means they may correspond to distinct inner Hurwitz space (Def. 2.2)
components if you don’t mod out by this group.

######  Example 2.10.

Examples – a la Rem. 2.9 – of nontrivial $N_{m,n}/G$.

  1. (2.13a)

When $G$ is $A_{n}$, the outer automorphism from $S_{n}$ usually permutes the
stabilising subgroups.

  2. (2.13b)

For $G={{\text{\rm PSL}}}_{n{+}1}({{\mathbb{F}}}_{q})$, with representations
on points and hyperplanes and $q=p^{e}$, $e>1$, the Frobenius acts
nontrivially in both.

For example in (2.10a), although §B.1 is with $G=S_{n}$, there are similar
examples with $G=A_{n}$. $\triangle$

######  Remark 2.11.

Davenport polynomial pair examples always had Hurwitz space components with
moduli definition field a proper extension of ${{\mathbb{Q}}}$, as revealed by
the Branch Cycle Lemma (exposition in the opening sections of [Fr10]). In the
analog of nonpolynomial cases (don’t include ${\text{\rm C}}_{\infty}$) as in
${{\text{\rm Ni}}}({{\text{\rm PSL}}}_{3}({{\mathbb{F}}}_{2}),{{\text{\bf
C}}}_{2^{6}},T_{i})$, $i=1,2$, this field can be ${{\mathbb{Q}}}$ (Prop.
1.16).

######  Remark 2.12.

The formulation of the components ${\mathcal{W}}_{1,j_{0}}$ in Prop. 2.8
doesn’t depend on having genus 0 covers in ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{i})$, $i=1,2$. For example, Rem. 2.11 applies to them, too, though it
might seem more striking that those fibers stratify inputs from rational
functions.

####  2.1.3. Coelescing Nielsen classes

Consider the coalescing operation.

######  Definition 2.13 (Restricted Coalescing).

Given an $r$-tuple $\boldsymbol{\sigma}\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T)$, with C consisting of $r$ classes, refer to
$\boldsymbol{\sigma}^{\prime}=(\sigma_{1},\dots,\sigma_{r{-}2},\sigma_{r{-}1}\cdot\sigma_{r})$
as a simple coalescing.292929It has an algebraic interpretation regarded as
going to the boundary of a Hurwitz space, but we don’t need that here. Call it
restricted if ${\langle\boldsymbol{\sigma}^{\prime}\rangle}=G$.

######  Definition 2.14 (Nielsen class coalescing).

Call ${{\text{\rm Ni}}}(G,{{\text{\bf C}}}^{\prime},T)$ a Nielsen class
coalescing of $\boldsymbol{\sigma}\in{{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$
if ${{\text{\bf C}}}^{\prime}$ is the end of a chain,

${{\text{\rm Ni}}}(G,{{\text{\bf C}}}_{j},T),j=0,\dots,v,\text{ with
}{{\text{\bf C}}}_{0}={{\text{\bf C}}}\text{ and }{{\text{\bf
C}}}^{\prime}={{\text{\bf C}}}_{v},$

for which there exists a representative $\boldsymbol{\sigma}_{j}\in{{\text{\rm
Ni}}}(G,{{\text{\bf C}}}_{j},T)$ that is a simple (restricted) coalescing of a
representative of $\boldsymbol{\sigma}_{j{-}1}\in{{\text{\rm
Ni}}}(G,{{\text{\bf C}}}_{j{-}1},T)$.

######  Lemma 2.15.

The genus of an irreducible cover represented by branch cycles
${\boldsymbol{\sigma}}^{\prime}$ given by a Nielsen class coalescing starting
from ${\boldsymbol{\sigma}}$ satisfies ${{\text{\bf
g}}}_{{\boldsymbol{\sigma}}^{\prime}}\leq{{\text{\bf
g}}}_{{\boldsymbol{\sigma}}}$.

###### Proof.

Use the induction set up by Def. 2.14. With no loss, show

${{\text{\rm ind}}}(\sigma_{r{-}1}\sigma_{r})\leq{{\text{\rm
ind}}}(\sigma_{r{-}1}){+}{{\text{\rm ind}}}(\sigma_{r}).$

Then, from the definition of ind, consider the restriction of
${\langle\sigma_{r{-}1},\sigma_{r}\rangle}$ to any orbit of it in the
representation $T$. That is, regarding the group as a transitive subgroup of
$S_{n}$ ($n$ the length of the orbit).

Now, apply RH to the triple
$(\sigma_{r{-}1},\sigma_{r},(\sigma_{r{-}1}\sigma_{r})^{{-}1})$:

$2(n{+}{{\text{\bf g}}}{-}1)={{\text{\rm ind}}}(\sigma_{r{-}1}){+}{{\text{\rm
ind}}}(\sigma_{r}){+}{{\text{\rm ind}}}(\sigma_{r{-}1}\sigma_{r})^{{-}1}).$

The lemma is proved unless ${{\text{\rm
ind}}}(\sigma_{r{-}1}\sigma_{r})>{{\text{\rm
ind}}}(\sigma_{r{-}1}){+}{{\text{\rm ind}}}(\sigma_{r})$.

Since ${{\text{\bf g}}}\geq 0$, this implies $2(n{-}1)<2\cdot{{\text{\rm
ind}}}(\sigma_{r{-}1}\cdot\sigma_{r})$. As the minimal index of an element is
$n{-}1$, this is a contradiction when it is an $n$-cycle. ∎

Use the notation of components on ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ of Ex. 2.6.

######  Corollary 2.16.

Coalescing $\boldsymbol{\mu}\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})\leftrightarrow\tilde{{\mathcal{C}}}_{T_{1}(\boldsymbol{\mu}),T_{2}(\boldsymbol{\mu})}$
to
$\boldsymbol{\mu}^{\prime}\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}}^{\prime},{{\mathbb{T}}})\leftrightarrow\tilde{{\mathcal{C}}}_{T_{1}(\boldsymbol{\mu}^{\prime}),T_{2}(\boldsymbol{\mu}^{\prime})}$
is non-increasing on component genuses.

Examples of why we need to consider Pakovich’s problem include the following
general idea. Items in (2.1.3) reference where illustrating them appear in our
examples.

Start with ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ for which
any representative gives a pair of covers
$(\varphi_{X}:X\to{\mathbb{P}}^{1}_{z},\varphi_{Y}:Y\to{\mathbb{P}}^{1}_{z})$
with $\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$ satisfying the equal
Galois closure and reducibility conditions of (1.8).

Then, use coalescing to move from a general Nielsen class ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ to Nielsen classes that have the
same $(G,{{\mathbb{T}}}),$ for which C has changed to give Nielsen class
representatives with the following properties.

  1. (2.14a)

Nielsen classes where $\varphi_{X}$ and $\varphi_{Y}$ are genus zero covers
represented by $(f^{*},g^{*})$ (list of groups and degrees (3.18); degree 7
§3.1.1).

  2. (2.14b)

From (2.1.3a) coalesce to where representative fiber products,
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$, have a genus 0 or 1 component (the
degree 7 case §3.1.3).

  3. (2.14c)

Compare (2.1.3b) results with Pakovich’s use of o-char (Ex. 3.4).

###  2.2. Intransitivity and genus formulas

§3.2 seriously uses Nielsen classes, but here we use it lightly. It has been
standard to assume the natural permutation representation on the group $G$ is
transitive. In this paper, we care about when it is not; the representation
breaks into a direct sum of transitive representations on the orbits.

§2.2.1 introduces the notation to calculate with the orbits of
$G(T_{2},y_{1})$ on ${x_{1},\ldots,x_{m}}$ that appear in Prop. 1.8. §2.2.2
gives the genus formulas that generalize [Fr73b, (1.6) of Prop. 1].

####  2.2.1. Using conjugacy classes on a fiber product

With notation of §2.1.3 suppose $\varphi_{X}$ (resp. $\varphi_{Y}$) –
irreducible covers; $T_{1}$ and $T_{2}$ are transitive – has branch cycles
${\sigma_{1},\ldots,\sigma_{r}}$ (resp. ${\tau_{1},\ldots,\tau_{r}}$). Denote
the subgroup of $G_{\varphi_{X}}\times G_{\varphi_{Y}}$ generated by

(2.15)
${\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}(\sigma_{1},\tau_{1}),\dots,(\sigma_{r},\tau_{r})\text{ by
}G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}}).$

Then, denote the subgroup of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ that stabilizes $y_{i}$ by
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{i})$.

  1. (2.16a)

Points on $X$ over $z_{i}$ correspond one-one with disjoint cycles of
$\sigma_{i}$ whose lengths are the ramification indices of those points.

  2. (2.16b)

$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ acts naturally on the
tensor product of the permutation representations $T_{\varphi_{X}}$ and
$T_{\varphi_{Y}}$.

  3. (2.16c)

Components of $\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$ identify with
orbits of $G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ on the symbols
$x_{i}\otimes y_{j}$, $i=1,\dots,m;j=1,\dots n$.

  4. (2.16d)

For $O$, a $G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ orbit in
(2.2.1c), branch cycles for the corresponding component $W_{O}$ are the
restriction of ${\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}}$ to $O$.

  5. (2.16e)

$G_{\varphi_{Y}}$ is transitive on $\\{{y_{1},\ldots,y_{n}}\\}$. So orbits of
(2.2.1c) correspond 1-1 to orbits of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{j})$ on
$\\{x_{1},\dots,x_{m}\\}\otimes y_{j}$.

Our examples usually have $X$ and $Y$ irreducible – as when those covers are
given by a pair $(f,g)$ of rational functions in one variable (1.1.2a) –
unless otherwise said. Denote points on $\tilde{{\mathcal{C}}}_{f,g}$
simultaneously over both $x^{\prime}\in X$ and $y^{\prime}\in Y$ by
$P_{x^{\prime},y^{\prime}}$. Let $\sigma_{i,x^{\prime}}$ (resp.
$\tau_{i,y^{\prime}}$) be the disjoint cycle in $\sigma_{i}$ (resp.
$\tau_{i}$) corresponding to $x^{\prime}$ (resp. $y^{\prime}$).

Then, $P_{x^{\prime},y^{\prime}}$ is nonempty if and only if
$f(x^{\prime})=g(y^{\prime})$, and it has only one element, unramified over
$z^{\prime}$ unless one of $s_{x^{\prime}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{\text{\rm ord}}(\sigma_{i,x^{\prime}})$ or
$t_{y^{\prime}}\stackrel{{\scriptstyle\text{\rm def}}}{{=}}{\text{\rm
ord}}(\tau_{i,y^{\prime}})$ exceeds 1. Consider ${{\boldsymbol{p}}}$ on
$\tilde{{\mathcal{C}}}_{f,g}$ with image
$x^{\prime}=x^{\prime}_{{\boldsymbol{p}}}\in{\mathbb{P}}^{1}_{x}$ (resp.
$y^{\prime}=y^{\prime}_{{\boldsymbol{p}}}\in{\mathbb{P}}^{1}_{y}$,
$z^{\prime}=z^{\prime}_{{\boldsymbol{p}}}\in{\mathbb{P}}^{1}_{z}$). We compute
the precise ramification, $e_{{{\boldsymbol{p}}}/y^{\prime}}$, of
${{\boldsymbol{p}}}$ over $y^{\prime}$. Use similar notation for other
ramification indices of a point over its image.

For $u,v\in{{\mathbb{Z}}}$, denote the greatest common divisor (resp. least
common multiple) of $u$ and $v$ by $(u,v)$ (resp. $[u,v]$). [Fr74, Proof of
Prop. 2]: Prop. 2.17 results from computing points and their ramification over
$y=0$ on the normalization of $\\{(x,y)\mid x^{u}=y^{v}\\}$.

######  Proposition 2.17.

Notation as above: Consider the points ${{\boldsymbol{p}}}\in
P_{x^{\prime},y^{\prime}}$ in
$\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$ corresponding to the
respective disjoint cycles $\sigma_{i,x^{\prime}}$ and $\tau_{i,y^{\prime}}$.

  1. (2.17a)

Then, there are $|P_{x^{\prime},y^{\prime}}|=(s_{x^{\prime}},t_{y^{\prime}})$
points of $\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$ each of
ramification index
$e_{{{\boldsymbol{p}}}/y^{\prime}}=\frac{[s_{x^{\prime}},t_{y^{\prime}}]}{t_{y^{\prime}}}$
over $y^{\prime}$.

  2. (2.17b)

${{\boldsymbol{p}}}\in P_{x^{\prime},y^{\prime}}$ correspond one-one to
disjoint cycles in $\sigma_{i,x^{\prime}}^{t_{y^{\prime}}}$.

######  Remark 2.18.

Assume $X$ and $Y$ (as usual) are irreducible, but
$\tilde{{\mathcal{C}}}_{f,g}$ has more than one component. Then,
${{\boldsymbol{p}}}\in P_{x^{\prime},y^{\prime}}$ as in (2.17b) corresponding
to the cycles in $\sigma_{i,x^{\prime}}^{t_{y^{\prime}}}$ may fall in
different components. This is precisely what happens to the two 2-cycles in
the 2nd line of (3.8).

####  2.2.2. Genus Corollaries of Prop. 2.17

Cor. 2.19 makes more precise statements toward getting branch cycles for
${\text{\rm pr}}_{y}:\tilde{{\mathcal{C}}}_{f,g}\to{\mathbb{P}}^{1}_{y}$, and
then for the restriction of ${\text{\rm pr}}_{y}$ to each component using the
notation of Prop. 2.17. Our applications to components of
$\tilde{{\mathcal{C}}}_{f,g}$ will start from Prop. 1.8, and then
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})=G({\boldsymbol{\sigma}})=G({\boldsymbol{\tau}})=G$.

We only need RH applied to (2.19a) and (2.19b) to get the Genus Corollary
statements that follow. Prop. 4.22 outlines a proof of the more precise
statement (2.19c) on branch cycles for ${\text{\rm
pr}}_{y}:W\to{\mathbb{P}}^{1}_{y}$ with $W$ a fiber product component. This is
based on RET and the references at the top of §2. We use that to interpret
generalizing Thm. 1.1 using Nielsen classes.

With ${\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}}$ branch cycles for
$\tilde{{\mathcal{C}}}_{f,g}$, Prop. 1.8 corresponds a component $W$ of the
fiber product with an orbit $J$ of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{1})$ on
${x_{1},\ldots,x_{m}}$. Prop. 2.17 runs over the disjoint cycles ,
$\tau_{i,y^{\prime}}$, of $\tau_{i}$, and so lists the points on $Y$ over the
$i$th branch point of $\varphi_{Y}$, with their ramification indices in the
cover ${\text{\rm
pr}}_{y}:\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}\to{\mathbb{P}}^{1}_{y}$.

With $W$ a component of $\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}$,
denote restriction of ${\text{\rm pr}}_{y}$ to $W$ by ${\text{\rm pr}}_{W}$.
Cor. 2.19 gives a representative of a branch cycle associated with every point
$\tau_{i,y^{\prime}}$, and then uses these to provide the Nielsen class of
${\text{\rm pr}}_{W}$. Denote the Nielsen class of ${\text{\rm pr}}_{y}$ by
${{\text{\rm Ni}}}_{{\text{\rm pr}}_{y}}$, and the Nielsen class of
${\text{\rm pr}}_{W}$ by ${{\text{\rm Ni}}}_{W}$. Use the Prop. 1.8
association $W\leftrightarrow J$, a $G(T_{2},y_{1})$ orbit on
$\\{{x_{1},\ldots,x_{m}}\\}$.

Cor. 2.19 is directly aimed at Method II, which hits a key point in applying a
generalization of Pakovich’s Thm. 1.1: Detecting the nature of the covers
${\text{\rm pr}}_{W}:W\to{\mathbb{P}}^{1}_{y}$ with $W$ – according to the
classification of Prop. 4.7 – a component of a fiber product
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

######  Corollary 2.19.

Denote $(\sigma_{i},\tau_{i})\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\gamma_{i}$. For each $(i,y^{\prime})$ choose $y_{b}$ in the
support of $\tau_{i,y^{\prime}}$. Then, choose $h_{y_{b},y_{1}}\in
G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ that maps $y_{b}$ to
$y_{1}$. Denote
$h_{y_{b},y_{1}}\gamma_{i}^{t_{y^{\prime}}}h_{y_{b},y_{1}}^{{-}1}$ by
$g_{i,y_{b}}$. This fixes $y_{1}$. Up to left action by $G(T_{2},y_{1})$,
$h_{y_{b},y_{1}}$ doesn’t depend on $y_{b}$.

Write $g_{i,y_{b}}$ as $(g_{i,y_{b}}^{\prime},g_{i,y_{b}}^{\prime\prime})$
with the first part a conjugate of $\sigma_{i}^{t_{y^{\prime}}}$ fixed on
$y_{1}$ and defined up to conjugation by elements of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ fixed on $y_{1}$.

  1. (2.18a)

Running over $(i,y^{\prime},y_{b})$ as chosen above, Prop. 2.17 gives
$g_{i,y^{\prime},y_{b}}^{\prime}$ as a representive of the branch cycle for
${\text{\rm pr}}_{y}$ over the point $y^{\prime}$.

  2. (2.18b)

Restricting the $g_{i,y^{\prime},y_{b}}^{\prime}\,$s of (2.19a) to include
only disjoint cycles supported in $J$ gives conjugacy classes of branch cycles
for ${\text{\rm pr}}_{W}$.

  3. (2.18c)

Including reordering the $g_{i,y^{\prime},y_{b}}^{\prime}\,$s in (2.19b),
there are choices of $h_{y_{b},y_{1}}\,$s giving branch cycles for ${\text{\rm
pr}}_{W}:W\to{\mathbb{P}}^{1}_{y}$.303030We mean the result is actual branch
cycles, not just conjugacy classes, including generation and product-one of
(2.1.1).

###### Proof.

Denote $(\sigma_{i},\tau_{i})$ by $\beta$. Suppose we choose $y_{b^{\prime}}$
in the support of $\tau_{i,y^{\prime}}$ instead of $y_{b}$. Then, a power, say
$m$, of $\tau_{i}$ maps $y_{b^{\prime}}$ to $y_{b}$. Form
$h_{y_{b},y_{1}}\beta^{m}$. It maps $b^{\prime}$ to $y_{1}$, and conjugating
$\beta$ by it has exactly the same effect as conjugating by $h_{y_{b},y_{1}}$.

From Prop. 2.17, the result is an element of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ that fixes $y_{1}$, and in
its first slot gives a conjugate of $\sigma_{i}^{t_{y^{\prime}}}$ that is a
branch cycle for ${\text{\rm pr}}_{y}$ for the branch point $y^{\prime}$.
Thus, (2.19a) follows because we are running over the collection of branch
points in the cover ${\text{\rm pr}}_{y}$. Then, (2.19b) amounts to
restricting to disjoint cycles that correspond to points in $W$.

Finally, (2.19c) follows from RET: There exists conjugations by the stabilizer
of $y_{1}$ for each element listed in (2.19b) so you get an element of the
Nielsen class of the cover ${\text{\rm pr}}_{W}$. See Rem. 2.25.∎

######  Corollary 2.20 (Method I).

Take a component $W$ of $\tilde{{\mathcal{C}}}_{f,g}$, but use the permutation
representation $T_{W}$ of $G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$
for the cover $W\to{\mathbb{P}}^{1}_{z}$. Compute the permutations
$T_{W}(\sigma_{1},\tau_{1}),\dots,T_{W}(\sigma_{r},\tau_{r})$. Then, the genus
$\text{\bf g}_{W}$ of $W$ is computed from RH:

$2(|\deg(W/{\mathbb{P}}^{1}_{z})|{+}{{\text{\bf
g}}}_{W}{-}1)=\sum_{i=1}^{r}{{\text{\rm ind}}}(T_{W}(\sigma_{i},\tau_{i})).$

The first case of Cor. 2.21 is included in the general case $u\geq 1$; both
cases count ramification over a particular component of the fiber product
running over each point on $Y\to{\mathbb{P}}^{1}_{z}$ that contributes
nontrivially to the ramification. If $u=1$, running over all disjoint cycles
is equivalent to running over all ramified points in the unique component.

######  Corollary 2.21 (Method II; See Rem. 2.23).

If $\tilde{{\mathcal{C}}}_{f,g}$ is irreducible, then the genus, $\text{\bf
g}_{f,g}$, of its unique component is given is in [Fr73b, (1.6) of Prop. 1]:

(2.19) $\begin{array}[]{rl}&2(\deg(f)+\text{\bf
g}_{f,g}{-}1)=\sum_{{{\boldsymbol{p}}}\in{{\mathcal{C}}}_{f,g}}e_{{{\boldsymbol{p}}}/y^{\prime}}{-}1\\\
=&\sum_{\text{branch points z' of }f}\sum_{(x^{\prime},y^{\prime})\mapsto
z^{\prime}}(s_{x^{\prime}},t_{y^{\prime}})\Bigl{(}\frac{[s_{x^{\prime}},t_{y^{\prime}}]}{t_{y^{\prime}}}-1\Bigl{)}.\end{array}$

When $u\geq 1$, for $1\leq u^{\prime}\leq u$, using the correspondence
$W_{u^{\prime}}\leftrightarrow J_{u^{\prime}}$ to get the disjoint cycles,
$\beta^{*}$, of all the elements of (2.19b) (on $J_{u^{\prime}}$). That gives
the analog of (2.19) to compute the genus ${{\text{\bf g}}}_{W_{u^{\prime}}}$
of $W_{u^{\prime}}$.

  1. (2.20a)

Left side of RH: $2(\deg(W_{u^{\prime}}/{\mathbb{P}}^{1}_{y}){+}{{\text{\bf
g}}}_{W_{u^{\prime}}}{-}1)$.

  2. (2.20b)

Right side of RH: Sum over ${{\text{\rm ind}}}(\beta^{*})$.

Consider a Nielsen class ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ with

${{\mathbb{T}}}=(T_{1},T_{2}),\text{ both }\text{ transitive faithful
representations of }G.$

Assume ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T_{i})$, $i=1,2$, are both
Nielsen classes of genus 0 covers. Then, the genuses of the components in
(2.19) depend only on ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$;
not on branch cycles for the particular $(f,g)$.

Cor. 2.22 is a particular case of Cor. 2.21: The hypothesis implies that all
ramification must appear in $W_{2}$, and the contribution to the sum where
$s_{x^{\prime\prime}}=s_{x^{\prime}}$ is automatically trivial.

######  Corollary 2.22.

For, $f\in{{\mathbb{C}}}(x)$, $T_{f}:G_{f}\to S_{m}$ is doubly transitive if
and only if $\tilde{{\mathcal{C}}}_{f,f}$ has the diagonal and one other
component $W_{2}$ of degree $m{-}1$ over ${\mathbb{P}}^{1}_{y}$. If so, then
the genus, ${{\text{\bf g}}}_{W_{2}}$ of $W_{2}$, satisfies:

$\begin{array}[]{rl}&2(m{-}1{+}{{\text{\bf g}}}_{W_{2}}{-}1)\\\
=&\sum_{\text{branch points z' of
}f}\sum_{(x^{\prime},x^{\prime\prime},x^{\prime}\neq x^{\prime\prime})\mapsto
z^{\prime}}(s_{x^{\prime}},s_{x^{\prime\prime}})\Bigl{(}\frac{[s_{x^{\prime}},s_{x^{\prime\prime}}]}{s_{x^{\prime}}}-1\Bigl{)}.\end{array}$

######  Remark 2.23.

The last statement in Cor. 2.21 (dependency on only the Nielsen class) follows
because the inclusion of particular $\beta^{*}\,$s for counting a particular
component depends only on conjugacy classes in C.

######  Remark 2.24 ($T_{f}$ double transitive in Cor. 2.22).

Note: $T_{f}$ is automatically primitive if it is doubly transitive. Examples
of this include the case when $f\in{{\mathbb{C}}}[x]$ (polynomial) is
indecomposable, and neither a cyclic nor Chebychev polynomial [Fr70, Thm. 1].
The genus zero problem is easier but not trivial if we assume $f$ is doubly
transitive. An easy characterization (as in the polynomial case, Rem. 1.14)
for double transitivity is unlikely.

######  Remark 2.25 ($H_{r}$ transitivity).

Elements of (2.19c) define the Nielsen class associated to the correspondence
$W\leftrightarrow J$. If $r^{\prime}$ is the number of branch points of the
covers ${\text{\rm pr}}_{W}:W\to{\mathbb{P}}^{1}_{y}$, then transitivity of
$H_{r^{\prime}}$ allows listing all elements in the Nielsen class of
${\text{\rm pr}}_{W}$, from identifying one such element from Cor. 2.19
applied to the Nielsen class of
$\tilde{{\mathcal{C}}}_{\varphi_{X},\varphi_{Y}}\to{\mathbb{P}}^{1}_{z}$ as
noted in (LABEL:Schreier).

If, though, $H_{r^{\prime}}$ is not transitive, this is a subtler problem that
arises in each component type given by Prop. 4.7. While that includes those
for which ${\text{\rm pr}}_{W}$ has genus or 1 Galois closure, those Nielsen
classes usually are recognized and known to have $H_{r^{\prime}}$ transitivity
as in §A.2.

##  3\. Example genus component computations

Cors. 2.20 and 2.21 have our genus computing results for fiber products of
components of nonsingular covers. This section shows how one Nielsen class can
provide several example challenges to extending Pakovich’s Theorem.

We complement [Fr74, Ex. 5, p. 246] to produce examples of Nielsen classes
with elements giving $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ reducible, and then
having genus 0 or 1 components. §3.2.3 explains they are the first of a set of
six examples in (3.18), all from groups with their core a projective linear
group. Expository sections [Fr12, §1-4], stemming from [Fr73a], document the
literature.

These examples arise from coalescing branch cycles from one Nielsen class with
$G={{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2)$ and explicit classes C in $G$,
Ex. 3.7.

###  3.1. Displaying examples of $g\in{{\mathcal{R}}}_{f}$

Use the notation of Prop. 1.8 where $(f^{*},g^{*})$ have the same Galois
closures, or $(T_{1},T_{2})$ are faithful, transitive representations of the
same group $G$, and consider the Nielsen class ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$.

In keeping with Pakovich’s problem, §4 considers
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}=\cup_{j=1}^{u}W_{j}$ with $u>1$
components (so the irreducibility hypothesis of Thm. 1.1 does not hold), and
we can control comparing the components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ and
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$.

Subsections follow the progression in (2.1.3). §3.1.1 displays Nielsen classes
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ whose representative
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ are reducible, and $f^{*}$ has a totally
ramified place (so including polynomial covers). Review Comment (1.11) for why
among the possibilities offered by Prop. 1.8 we have chosen as our running
example the progenitor that shaped the genus 0 problem: Otherwise, we hardly
know how to name groups and their permutation representations.

§3.1.2 explains that both sets of Nielsen classes in §3.1.1 come by coalescing
from one more general Nielsen class with $r=4$ branch points. §3.1.3 computes
the genus of the components.

####  3.1.1. Example Nielsen classes

Use the notation $({}_{1}f^{*},{}_{1}g^{*})$ and $({}_{2}f^{*},{}_{2}g^{*})$
for respective pairs of degree 7 polynomials in these two Nielsen classes.
Each polynomial has three branch points (including $\infty$) which we can – if
we desire – take to be $0,1,\infty$ up to linear transformation (fixing
$\infty$) on ${\mathbb{P}}^{1}_{z}$.

The monodromy groups are ${{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2)=G_{{}_{j}f^{*}}=G_{{}_{j}g^{*}}$, $j=1,2$,
corresponding to the inequivalent representations on points ($T_{P}$; for
${}_{j}f^{*}$) and on lines ($T_{L}$; for ${}_{j}g^{*}$) in the dimension 2
projective plane ${\mathbb{P}}^{2}({{\mathbb{Z}}}/2)$.

Designate the permutation representations by $T_{{}_{j}f^{*}}$ (resp.
$T_{{}_{j}g^{*}}$) on the letters ${x_{1},\ldots,x_{7}}$ with the branch
cycles given by permutations ${}_{j}\sigma_{i}$ (resp. using letters
${y_{1},\ldots,y_{7}}$ in permutations ${}_{j}\tau_{i}$).

Example branch cycles in the Nielsen classes correspond to (right) subscripts
1 and 2 on the $\sigma\,$ and $\tau\,$s, as in [Fr74, (2.42)] and [Fr74,
(2.41)].

(3.1)
$\begin{array}[]{rlrl}T_{{}_{1}f^{*}}:{}_{1}\sigma_{1}=&(x_{1}\,x_{3})(x_{4}\,x_{5})&{}_{1}\sigma_{2}=&(x_{1}\,x_{4}\,x_{6}\,x_{7})(x_{2}\,x_{3})\\\
T_{{}_{1}g^{*}}:{}_{1}\tau_{1}=&(y_{1}\,y_{2})(y_{3}\,y_{5})&{}_{1}\tau_{2}=&(y_{1}\,y_{3}\,y_{6}\,y_{7})(y_{4}\,y_{5})\\\
T_{{}_{2}f^{*}}:{}_{2}\sigma_{1}=&(x_{1}\,x_{2}\,x_{3})(x_{4}\,x_{5}\,x_{7})&{}_{2}\sigma_{2}=&(x_{1}\,x_{4})(x_{6}\,x_{7})\\\
T_{{}_{2}g^{*}}:{}_{2}\tau_{1}=&(y_{1}\,y_{2}\,y_{7})(y_{3}\,y_{5}\,y_{6})&{}_{2}\tau_{2}=&(y_{3}\,y_{7})(y_{4}\,y_{5})\end{array}$

We indicate the cycle notation with integers when it is clear whether it is
$x$ or $y$ involved. For example, we have normalized each representation up to
one conjugation in $S_{7}$ so that the branch cycle at $\infty$ is always the
7-cycle $(1\,2\,3\,4\,5\,6\,7)^{-1}=\sigma_{\infty}=\tau_{\infty}$. §3.1.2
explains a potential confusion from doing this, despite its computational
advantage. Just remember that the ${\boldsymbol{\sigma}}$ entries act on
different letters than the ${\boldsymbol{\tau}}$ entries.

Now we name the Nielsen classes using the orders of their elements:

$\begin{array}[]{c}({}_{1}\sigma_{1},{}_{1}\sigma_{2},{}_{1}\sigma_{\infty};{}_{1}\tau_{1},{}_{2}\tau_{2},{}_{1}\tau_{\infty})\in{{\text{\rm
Ni}}}({{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{\infty\cdot
2\cdot 4},{{\mathbb{T}}})\\\
({}_{2}\sigma_{1},{}_{2}\sigma_{2},{}_{2}\sigma_{\infty};{}_{2}\tau_{1},{}_{2}\tau_{2},{}_{2}\tau_{\infty})\in{{\text{\rm
Ni}}}({{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{\infty\cdot
2\cdot 3},{{\mathbb{T}}})\end{array}$

The involution conjugacy class below will be referenced as ${\text{\rm
C}}_{2}$. The product, in order, of the branch cycles – as always – is 1,
where the 3rd cycle is $(1\,2\,3\,4\,5\,6\,7)^{-1}$. Permutations act on the
right of the letters: The result of $\sigma$ on $i\in\\{1,\dots,7\\}$ is
$(i)\sigma$, so

$T_{f^{*}}({}_{j}\sigma_{1})T_{f^{*}}({}_{j}\sigma_{2})=(1\,2\,\dots\,7),\
j=1,2,\text{ etc.}.$

Given these choices, the only leeway is conjugation by the branch cycle at
$\infty$. That amounts to a cycling, $(1\,2\,3\,4\,5\,6\,7)$ of
$\\{1,2,3,4,5,6,7\\}$.

####  3.1.2. Essential differences between the ${\boldsymbol{\sigma}}\,$s and
${\boldsymbol{\tau}}\,$s

Consider a pair $(f^{*},g^{*})$ of degree 7 rational functions that give
examples of simultaneous branch cycles for ${{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2)$ in the respective representations. In the
identification of $G_{f^{*}}$ and $G_{g^{*}}$ as subgroups of $S_{7}$, we have
simplified in two ways: Dropping the $x\,$s and $y\,$s in the notation, and
then identifying the 3rd terms of the ${\boldsymbol{\sigma}}\,$s and
${\boldsymbol{\tau}}\,$s in (3.1) with $(1\,2\,3\,4\,5\,6\,7)^{-1}$.

The actual relation between the two Galois closures is preserved by continuing
the embedding of $G_{f^{*}}\to S_{7}$ to ${{\text{\rm
GL}}}_{7}({{\mathbb{Q}}})$, and then conjugating by an incidence matrix
$I_{f^{*},g^{*}}$ ([Fr73a, Proof of Thm. 1] or [Fr12, §4.2]) that produces the
representation $T_{g^{*}}$ in ${{\text{\rm GL}}}_{7}({{\mathbb{Q}}})$. The
latter, in this case, happens to conjugate $S_{7}$ into itself, though
$I_{f^{*},g^{*}}$ is not a permutation matrix.

What is happening in these groups is that they have degree $n$ representations
and $n$-cycles $\sigma_{n}$. The conjugacy class of the $n$-cycle, ${\text{\rm
C}}_{n}$.

######  Definition 3.1 (multipliers).

The multipliers attached to ${\text{\rm C}}_{n}$ are integers $u$, with
$(u,n)=1$ for which $\sigma_{n}^{u}\in{\text{\rm C}}_{n}$. The quotient
$({{\mathbb{Z}}}/n)^{*}/{{\mathcal{M}}}$ with ${{\mathcal{M}}}$ the group of
multipliers of ${\text{\rm C}}_{n}$ is nontrivial if and only if there is more
than one conjugacy class of $n$-cycles.

Indeed, apply this to $\sigma_{\infty}$. The result is that the conjugacy
class of $\sigma_{\infty}^{-1}$ is different from that of $\sigma_{\infty}$.
That is, $-1$ is a non-multiplier of the design, or that $\sigma_{\infty}$ and
$\sigma_{\infty}^{-1}$ (albeit with the same cycle type) are not conjugate in
the image permutation group of $T_{f}$, but $I_{f,g}$ does conjugate them.

Similar statements apply to the other degrees $m$ in (3.18), though for those,
the quotient of $({{\mathbb{Z}}}/n)^{*}$ by the multipliers is a larger group.
For example, see Degree 13 in the comments following Thm. 3.10.

The most general Nielsen class that contains degree 7 polynomial covers, $f$ –
we denote it ${{\text{\rm Ni}}}({{\text{\rm PSL}}}_{2},{{\text{\bf
C}}}_{\infty\cdot 2^{3}},{{\mathbb{T}}})$ (note the ${{\mathbb{T}}}$ hasn’t
change) for which we get reducible $\tilde{{\mathcal{C}}}_{f,g}$ with branch
cycles of type

$((2)(2),(2)(2),(2)(2),(7));$

the integers here between ()’s are cycle lengths. Most general means the
largest number of branch points (elements in C).

As above, there are two Nielsen classes differing between the two conjugacy
classes of 7-cycles in this group. They come together in the tensor product of
the representation on points and lines as in §2.2. The design mentioned above
clarifies that the most possible fixed points for an element $M$ in this group
are 3. The fixed points in that extreme case correspond to 3 points on the
projective plane lying on a line $L$. RH shows that to get such branch cycles
for a genus 0 cover requires the three elements that aren’t 7-cycles have the
cycle type $(2)(2)$ (of index 2) above.

Such an element $M\in{{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2)$ is a
transvection: having the form

$M:{{\boldsymbol{v}}}\rightarrow{{\boldsymbol{v}}}{+}\varphi({{\boldsymbol{v}}}){{\boldsymbol{h}}}$

where $\varphi$ is the linear functional with the points of $L$ in its kernel,
and ${{\boldsymbol{h}}}$ is also in $L$. In, however, Ex. 3.7 we find that for
elements in these Nielsen classes the components of
$\tilde{{\mathcal{C}}}_{f,g}$ have genus exceeding 0 (actually 1).

The cycle-types of ${}_{1}{\boldsymbol{\sigma}}$ and
${}_{2}{\boldsymbol{\sigma}}$ are different. So it is no surprise in §3.1.3
their contributions generalizing Thm. 1.1 have a significant difference, as in
Cor. 4.12 and LABEL:findeg7polyb. §3.2.1 explains the following statements.

Reordering the conjugacy classes in $S_{7}$ of the entries of these branch
cycles is a minor change. Still, have we left out some significant 3-entry
branch cycles that come from coalescing of the main Nielsen class consisting
of 4-tuples? Cor. 4.14 shows we have not. §4.2.3 explains – under the rubric
that of all such coalescing giving Nielsen class satellites come one fixed
Nielsen class – the value of all these degree 7 examples cohering.

######  Remark 3.2 (Other degrees in (3.18)).

The case $n=7$ has several regularities that might be misleading. Example: In
Cor. LABEL:findeg7polyb we take advantage that three branch cycles for covers
in the main Nielsen class are transvections. That isn’t the case for $n=13$
and 15 [Fr05b, §3.4]. [Fr12, §4.3, Thm. 4.5] notes the missing difference set
for $n=15$ in [Fr73a, p. 134]. It also shows precisely how Fried (in 1969)
could write branch cycles for the Nielsen classes of the other examples.
Thereby, modulo the conjecture, verified later that we knew all the groups
with such doubly transitive designs, displaying the precise list of degrees
for Davenport pairs.

####  3.1.3. Degree 7 example components: Cors. 2.20 and 2.21

The distinction between Cor. 2.20 and Cor. 2.21: The former computes the genus
from an orbit of $x_{i}\otimes y_{1}$ under the group
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ (as in (2.15)). The latter
computes it from the orbit, under the smaller group
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{1})$ stabilizing $y_{1}$.

While the genus is of the same component (so, the same), the two methods
differ in what they reveal. Similarly, the two approaches play different roles
in Prop. 4.7.

Lem. 3.3 has two uses: It makes sense of how coalescing ties together many
examples. Also, it shows how conceptual ideas drove the simple group
classification and the genus zero problem. Out labeling of orbits is
consistent with Comments (1.8).

######  Lemma 3.3.

Orbits of $G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{1})$ for
$({}_{j}f^{*},{}_{j}g^{*})$, for either $j=1,2$, are:

(3.2) $J_{1}=\\{x_{1},x_{2},x_{4}\\}\otimes y_{1}\text{ and
}J_{2}=\\{x_{7},x_{3},x_{5},x_{6}\\}\otimes y_{1}.$

Indeed, you can figure the orbits from the conjugates of the 7-cycle on an
involution in the conjugacy class ${\text{\rm C}}_{2}$.

###### Proof.

The group is generated by any 2 of the 3 branch cycles. So, we only need
$\sigma_{\infty}$ and an element in ${\text{\rm C}}_{2}$. Use conjugation by
$\sigma_{\infty}^{t}$ on $\tau_{1}$ for various $t\,$s that fix $y_{1}$, and
check what orbits on the $x_{i}\,$s result.

(3.3)
$\sigma_{1}\mapsto(x_{2}\,x_{4})(x_{5}\,x_{6})(t=1);(x_{3}\,x_{5})(x_{6}\,x_{7})(t=2);(x_{5}\,x_{7})(x_{1}\,x_{2})(t=4).$

You can read the orbits $J_{1}$ and $J_{2}$ off the resulting three elements.
∎

In Lem. 3.3, $u=2$. Use $1\leq u^{\prime}\leq u$ as the index for referring to
a component. Denote the genus in each example computation by ${}_{j}\text{\bf
g}_{u^{\prime}}$. Example: ${}_{2}\text{\bf g}_{1}$ for the component
associated with $J_{1}$ and polynomials $({}_{2}f,{}_{2}g)$. To use Cor. 2.20,
compute the orbit of $x_{1}\otimes y_{1}$ under the action of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$.

Start with ${}_{1}{\boldsymbol{\sigma}}={\boldsymbol{\sigma}}$ and
${}_{1}{\boldsymbol{\tau}}={\boldsymbol{\tau}}$ in (3.1), dropping the pre-
subscript 1 to simplify. Apply $(\sigma_{1},\tau_{1})$ (resp.
$(\sigma_{2},\tau_{2})$) to $x_{1}\otimes y_{1}$ (resp. $x_{3}\otimes y_{3}$)
to get

(3.4) $x_{3}\otimes y_{2}\ (\text{resp. }x_{2}\otimes y_{6}).$

That alone gives us the expected orbit of length 21 by applying
$(\sigma_{\infty},\tau_{\infty})$ to the subscripts, equivalencing them
$\\!\\!\\!\mod 7$:

(3.5) $x_{1{+}i}\otimes y_{1+i},x_{3{+}i}\otimes y_{2{+}i},x_{2{+}i}\otimes
y_{6{+}i},i=0,\dots,6.$

In abbreviated notation the first two branch cycles give

$\begin{array}[]{rl}(\sigma_{1},\tau_{1}):&\\!\\!\\!\\!((1,1)\,(3,2))((2,2)\,(2,1))((3,3)\,(1,5))((4,4)\,(5,4))((5,5)\,(4,3))\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!((6,5)\,(6,3))((1,7)\,(3,7))((4,1)\,(5,2))\\\
(\sigma_{2},\tau_{2}):&\\!\\!\\!\\!((5,5)\,(5,4))((3,2)\,(2,2))((1,1)\,(4,3)\,(6,6)\,(7,7))\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!((7,6)\,(1,7)\,(4,1)\,(6,3))((1,5)\,(4,4)\,(6,5)\,(7,4))((2,6)\,(3,7)\,(2,1)\,(3,3)).\end{array}$

Indices of
$(\sigma_{1},\tau_{1}),(\sigma_{2},\tau_{2}),(\sigma_{\infty},\tau_{\infty})$
are the 3 numbers – in order – on the right of (3.6). RH applies to the degree
21 cover of ${\mathbb{P}}^{1}_{z}$ to compute ${}_{1}\text{\bf g}_{1}$:

(3.6) $2(21+{}_{1}\text{\bf g}_{1}-1)=8+14+18,\text{ or }{}_{1}\text{\bf
g}_{1}=0.$

Now we use Method II, Cor. 2.21. to compute ${}_{1}\text{\bf g}_{1}$ the genus
of the degree 3 component, ${}_{1}W_{1}$ of the fiber product by applying RH
to ${\text{\rm pr}}_{y}:{}_{1}W_{1}\to{\mathbb{P}}^{1}_{y}$.

The algorithm to produce the branch cycles of the cover
$({}_{1}\sigma_{1},{}_{1}\tau_{1})$ in Cor. 2.19 immediately shows we have
branch points (with nontrivial branch cycles) corresponding in
${}_{1}\tau_{1}$ with the length 1 cycles $y_{4},y_{6},y_{7}$. Powers of
$\sigma_{7}=\sigma_{\infty}$ translate the subgroups and give us the
respective branch cycles, as elements in $G(T_{2},y_{1})$ from (2.19b).

As in Lem. 3.3, use $\sigma_{\infty}$ to translate the subscripts to change
$y_{i}\,$ to $y_{1}$. Then check which cycles end up with their support in
$J_{1}$. Example:

$\begin{array}[]{rl}(x_{4}\otimes y_{4}\,x_{5}\otimes
y_{4})&\mapsto(x_{1}\otimes y_{1}\,x_{2}\otimes y_{1})\text{ \rm while }\\\
(x_{1}\otimes y_{4}\,x_{3}\otimes y_{4})&\mapsto(x_{5}\otimes
y_{1}\,x_{7}\otimes y_{1}).\end{array}$

In (3.7), cycles with superscript ∗ have support in $J_{1}$. The rest are in
$J_{2}$.

(3.7) $\begin{array}[]{rl}(x_{1}\otimes y_{4}\,x_{3}\otimes y_{4})^{\
}&(x_{4}\otimes y_{4}\,x_{5}\otimes y_{4})^{*}\\\ (x_{1}\otimes
y_{6}\,x_{3}\otimes y_{6})^{\ }&(x_{4}\otimes y_{6}\,x_{5}\otimes y_{6})\\\
(x_{1}\otimes y_{7}\,x_{3}\otimes y_{7})^{*}&(x_{4}\otimes y_{7}\,x_{5}\otimes
y_{7}).\end{array}$

Now do the computation where the branch cycle is
$({}_{1}\sigma_{2},{}_{1}\tau_{2})$. The 4-cycle in ${}_{1}\tau_{2}$
contributes nothing to ramification, but the fixed point ($y_{2}$) and the
2-cycle do. For the 2-cycle we may use either $v=4$ or 5 in its support in
(2.19b) after putting the 4-cycle in $\sigma_{2}$ to the power 2.

We choose $v=4$. As above, list those contributions with the ∗ superscript
from translating the subscripts to see which cycles end up in $J_{1}$.

(3.8) $\begin{array}[]{rl}(x_{1}\otimes y_{2}\,x_{4}\otimes
y_{2}\,x_{6}\otimes y_{2}\,x_{7}\otimes y_{2})&(x_{2}\otimes
y_{2}\,x_{3}\otimes y_{2})^{*}\\\ (x_{1}\otimes y_{4}\,x_{6}\otimes
y_{4})&(x_{4}\otimes y_{4}\,x_{7}\otimes y_{4})^{*}.\end{array}$

Both branch cycles over the 7-cycles are 7-cycles; they contribute nothing to
ramification. Denote cycles with ${}^{*}\,$ superscripts by $\mu^{*}$. Compute
${}_{1}\text{\bf g}_{1}$:

$2(3+{}_{1}\text{\bf g}_{{1}}-1)=\sum_{\mu^{*}}{{\text{\rm
ind}}}(\mu^{*})=4,\text{ or }{}_{1}\text{\bf g}_{{1}}=0.$

Finally, use Method II for orbit $J_{1}$ (in (3.2)) of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{1})$ for
$({}_{2}f,{}_{2}g)$. From the branch cycles in (3.1) the analogs of (3.7) and
(3.8) are

(3.9) $\begin{array}[]{rl}(x_{1}\otimes y_{4}\,x_{2}\otimes
y_{4}\,x_{3}\otimes y_{4})&(x_{4}\otimes y_{4}\,x_{5}\otimes
y_{4}\,x_{7}\otimes y_{4})^{*}\\\ &\text{ and }\\\ (x_{1}\otimes
y_{1}\,x_{4}\otimes y_{1})^{*}&(x_{6}\otimes y_{1}\,x_{7}\otimes y_{1})\\\
(x_{1}\otimes y_{2}\,x_{4}\otimes y_{2})^{\ }&(x_{6}\otimes
y_{2}\,x_{7}\otimes y_{2})\\\ (x_{1}\otimes y_{6}\,x_{4}\otimes y_{6})^{\
}&(x_{6}\otimes y_{6}\,x_{7}\otimes y_{6})^{*}.\end{array}$

We have already put the ${}^{*}\,$s on the translates in the right orbit when
all the $y_{i}\,$s are set back to $y_{1}$. As above we compute
${}_{2}\text{\bf g}_{1}$ by summing the indices of the $\mu^{*}\,$s (which is
4 again), to get ${}_{2}\text{\bf g}_{{1}}=0$.

Starting with Ex. 3.4, then going to Rem. 3.5 and Ex. 3.7, we illustrate how
Thm. 4.23 works.

######  Example 3.4 (Pakovich Goal).

The two conjugacy class sets ${{\text{\bf C}}}_{\infty\cdot 2\cdot 4}$ and
${{\text{\bf C}}}_{\infty\cdot 2\cdot 3}$, and the labeling for each
representative ${}_{j}{{\text{\bf C}}}$, $j=1,2$ corresponding to the two
permutation representations, appear in §3.1.1 (given by (3.1)). The difference
in the values of $j$ comes from the outer automorphism (§3.1.2) of
${{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2)$ that takes the letters of the
representation $T_{1}$ on points of the projective space to the letters of the
representation on lines. Consider the pairs $(f^{*},g^{*})$ in either Nielsen
class. Each $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has two components,
$W_{u^{\prime}}$, labeled by the orbits $J_{u}^{\prime}$, $u^{\prime}=1,2$ in
Lem. 3.3.

Use the notation of Prop. 1.8 (as in Comments (1.8)) to label the degrees of
the natural projections of the $W_{u^{\prime}}$, $u^{\prime}=1,2$, to
${\mathbb{P}}^{1}_{y}$: They are respectively $\ell_{1}=3$ and $\ell_{2}=4$
corresponding to the lengths of the orbits $J_{1}$ and $J_{2}$.

In going from $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ to
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ in each of these cases where
the components come out genus 0 or genus 1, Prop. 4.7 lists Nielsen class that
might contain $g_{1}$ for which the genus of components on
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ remains stable. Situations
(4.7b) and (4.7c) force recognizing them through either our genus formula, or
as belonging to a decomposition variant (Def. 4.6). $\triangle$

######  Remark 3.5.

For the complementary orbit of
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}},y_{1})$, $J_{2}$ in (3.2),
use the analogous formula for the genus g in either case. Then, the left side
of RH is $2(4+\text{\bf g}-1)$ and the right side is the sum of the indices of
the $\mu\,$s that don’t have a ∗ superscript.

In the resp. cases this sum is 4+4 and 6: genuses ${}_{1}\text{\bf g}_{2}$ and
${}_{2}\text{\bf g}_{2}$ of the complementary orbits are, resp. 1 and 0. That
gives three genus 0 components and one genus 1 component on reducible fiber
products of degree 7 polynomial covers.

###  3.2. Context of §3.1.3

§3.2.1 starts from a Nielsen class ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ whose representatives are reducible covers of
${\mathbb{P}}^{1}_{z}$ of the form $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ more
general than the Nielsen classes of §3.1. We refer to the latter as satellites
of it. The satellites come from coalescing, using Cor. 2.16 to assure that the
genus of components does not increase when coalesced. We follow the path of
(2.1.3) to where we end at the two Nielsen classes of §3.1.3.

The topic of §3.2.2 is that the Nielsen classes form a natural space with
their braid action. In each of our starting cases, with conjugacy classes
${{\text{\bf C}}}={{\text{\bf C}}}_{\infty,2^{3}}$ and ${{\text{\bf
C}}}_{2^{6}}$, but $G$ is still ${{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2)$,
there is just one component (of respectively four and six branch point
covers), making them natural to find target Nielsen classes as satellites.

Then, §3.2.3 returns to the genus 0 problem to expand from our running example
to situations that already have a place in the literature. The solution of the
genus 0 problem and its extensions shows how much it was influenced by using
aspects of Nielsen classes. Further, those are references for finding
permutation pairs that arise in applying Prop. 1.8.

####  3.2.1. Coalescing and the Pakovich Goal

Ex. 3.4 showing, by examples, how we would drop the irreducibility hypothesis
of Thm. 1.1.

Here is the source of 7-cycles in ${{\text{\rm
PGL}}}_{3}({{\mathbb{F}}}_{2})$. Regard projective three space over
${{\mathbb{F}}}_{2}$ as ${{\mathbb{F}}}_{2^{3}}$, the degree 3 extension of
${{\mathbb{F}}}_{2}$. Then, the multiplicative group of nonzero elements
${{\mathbb{F}}}_{2^{3}}^{*}$ is cyclic (Euler’s Theorem). Take a generator,
$\alpha$, of it, you get a 7-cycle.313131Of course, this works in general in
${{\text{\rm PGL}}}_{k}({{\mathbb{F}}}_{q})$ for $k\geq 3$.

Suppose $f^{*}$ (degree $m$) is indecomposable and the ranges of inequivalent
polynomials pairs $(f^{*},g^{*})$ over ${{\mathbb{Q}}}$ are the same for
almost all primes $p$. The first step in Davenport’s problem showed that Prop.
1.8 applies. Further:

######  Proposition 3.6.

Then, for ${\text{\rm C}}_{m}$ an $m$-cycle conjugacy class, there is $t$ with
$(t,m)=1$, for which ${\text{\rm C}}_{m}^{t}$ is not conjugate to ${\text{\rm
C}}_{m}$. From the Branch Cycle Lemma: $f^{*}$ and $g^{*}$ are conjugate over
a proper extension of ${{\mathbb{Q}}}$.

That is in [Fr73a], with the story of what it led to, including the Genus Zero
Problem, in [Fr12, §7.1].

We explain how §3.1.3 examples arise from degree 7 pairs $(f^{*},g^{*})$, each
branched at four (not three) points. A connected (Hurwitz) space
${{\mathcal{H}}}_{\infty\cdot 2^{3}}$ given by the Nielsen class ${{\text{\rm
Ni}}}_{\infty\cdot 2^{3}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{F}}}_{2}),{{\text{\bf C}}}_{\infty\cdot
2^{3}},{{\mathbb{T}}})$ parametrizes the pairs $(f^{*},g^{*})$ for which
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has two connected components, and
$(f^{*},g^{*})$ are represented by polynomials. We call this Hurwitz space
${{\mathcal{H}}}_{7}$ later.

We give two examples of these coalescings. Then, all examples naturally
related to the §3.1.3 examples of reducible $\tilde{{\mathcal{C}}}_{f,g}$,
with genus 0 components, come from coalescings in the Nielsen class of
$(f,g)$. Cor. LABEL:findeg7polyb finishes Rem. 3.1.2. Here are the
coalescings.

(3.10)
$\begin{array}[]{rl}\text{${}_{1}{\boldsymbol{\sigma}}$-coalesce:}&((1\,3)(4\,5),(1\,6)(2\,3),(6\,4)(1\,7),\sigma_{\infty})\\\
\rightarrow&((1\,3)(4\,5),(1\,4\,6\,7)(2\,3),\sigma_{\infty})\\\
\text{${}_{2}{\boldsymbol{\sigma}}$-coalesce:}&((1\,3)(4\,7),(2\,3)(5\,7),(1\,4)(6\,7),\sigma_{\infty})\\\
\rightarrow&((1\,2\,3)(4\,5\,7),(1\,4)(6\,7),\sigma_{\infty}).\end{array}$

The coalescing procedure: Multiply the 2nd and 3rd (resp. 1st and 2nd) entries
in the line for ${}_{1}{\boldsymbol{\sigma}}$-coalesce (resp.
${}_{2}{\boldsymbol{\sigma}}$-coalesce). This would be a 1-step coalescing,
but we can form many-step coalescings as in Ex. 3.7 and Ex. 3.8.

######  Example 3.7 (Before coalescing).