Consider ${{\text{\rm
Ni}}}(G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}}),{{\text{\bf
C}}}_{\infty\cdot 2^{3}},{{\mathbb{T}}})$, the Nielsen class where C consists
of three copies of the conjugacy class of $({}_{1}\sigma_{1},{}_{1}\tau_{1})$
in (3.1) and one copy of a class of a 7-cycle. Before coalescing, the
components of $\tilde{C}_{f,g}$ have genus 1 for the following reason. We can
take branch cycles for these covers to be

$((\sigma_{1}^{\prime},\tau_{1}^{\prime}),(\sigma_{2}^{\prime},\tau_{2}^{\prime}),(\sigma_{3}^{\prime},\tau_{3}^{\prime}),(\sigma_{\infty},\tau_{\infty}))$

with each $(\sigma_{i}^{\prime},\tau_{i}^{\prime})$, $i=1,2,3$, conjugate in
$G({\boldsymbol{\sigma}}\cdot{\boldsymbol{\tau}})$ to $(\sigma_{1},\tau_{1})$
in (3.1). So, for each of these indices in the representation computed, there
will be 8. The analog computation for (3.6) of this genus ${{\text{\bf
g}}}_{*}$ gives

$2(21+{{\text{\bf g}}}_{*}-1)=3\cdot 8+18,\text{ or }\text{\bf g}_{*}=1.$

######  Example 3.8 (A bigger coalescing).

Keeping $G={{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2)$, form Nielsen classes for
rational functions $(f^{*},g^{*})$ by replacing
$(\sigma_{\infty},\tau_{\infty})$ by a repetition of three copies of the
conjugacy class of $(\sigma_{1},\tau_{1})$. That is, ${{\text{\bf
C}}}={{\text{\bf C}}}_{2^{6}}$ consists of six repetitions of the involution
class in ${{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2)$. Here is the computation
of the genus, $g_{**}$, of the degree 3 component of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$:

$2(21+\text{\bf g}_{**}-1)=6\cdot 8,\text{ or }\text{\bf g}_{**}=4.$

####  3.2.2. Coalescing and braids

Here, we finish the context of these degree 7 coalescings, describing their
Nielsen classes as satellites cohering to one space attached to a fixed
Nielsen class (Thm. 4.14).

The braid action on an $r$-tuple
${\boldsymbol{\sigma}}=({\sigma_{1},\ldots,\sigma_{r}})$ satisfying the
branch-cycle conditions (2.1.1), is generated by two elements:

  1. (3.11a)

$q_{1}:{\boldsymbol{\sigma}}\mapsto(\sigma_{1}\sigma_{2}\sigma_{1}^{-1},\sigma_{1},\sigma_{3},\dots,\sigma_{r})$
the 1st (coordinate) twist, and

  2. (3.11b)

${{\text{\bf
sh}}}:{\boldsymbol{\sigma}}\mapsto(\sigma_{2},\sigma_{3},\dots,\sigma_{r},\sigma_{1})$,
the left shift.

Conjugating $q_{1}$ by sh, gives $q_{2}$: the twist moved to the right.
Repeating gives $q_{3},\dots,q_{r-1}$. Denote the group generated by the
braids by $H_{r}$ (more accurately described as the Hurwitz monodromy quotient
of the braid group). Here are uses of these braids as applied to a given
Nielsen class Ni. Denote the absolute Galois group of the number field $L$ by
$G_{L}$.

  1. (3.12a)

Each braid preserves generation, product-one and the conjugacy class
conditions. So it preserves branch cycles in Ni.

  2. (3.12b)

Given ${\boldsymbol{\sigma}}\in{{\text{\rm Ni}}}$, applying braids to
${\boldsymbol{\sigma}}$ allows forming ${\boldsymbol{\sigma}}^{\prime}$ whose
entries represent the elements of C in any desired order.

  3. (3.12c)

There is a minimal cyclotomic field $L_{{{\text{\rm Ni}}}}$ for which $G_{L}$
maps all covers over $\bar{L}$ in Ni into covers in Ni if and only if
$L_{{{\text{\rm Ni}}}}\subset L$.

The branch cycle argument ([Fr77, p. 62] or [Fr12, §5.1.3]) gives (3.2.2c).
Deeper points follow from this assumption [Fr77, Thm. 5.1]:

  1. (3.13a)

For $\varphi:X\to{\mathbb{P}}^{1}$ in Ni, $H_{r}$ transitive on Ni.

  2. (3.13b)

(3.2.2a) is equivalent to there being one connected component of covers in
Ni).

  1. (3.14a)

Then, for any ${\boldsymbol{\sigma}}\in{{\text{\rm Ni}}}$ (Def. 2.2), for some
classical generators (§2.1.1), ${\boldsymbol{\sigma}}$ is the branch cycle
description for $\varphi$.

  2. (3.14b)

For absolute Nielsen classes with $G(T_{\varphi},1)$ its own normalizer in
$G$, the intersection of all definition fields of covers in Ni is
$L_{{{\text{\rm Ni}}}}$.

Indeed, (3.2.2a) is equivalent to (3.2.2), and even without it,
$L_{{{\text{\rm Ni}}}}$ is the right definition field for the Hurwitz space.
Here, though, there are three such spaces,
${{\mathcal{H}}}_{7},{{\mathcal{H}}}_{13}$ and ${{\mathcal{H}}}_{15}$
corresponding to the most interesting of the degrees that appear in (3.18) and
referred to in the examples, Cor. 4.12 and LABEL:findeg7polyb. Hypothesis
(3.2.2) does hold for them.

Connectedness of ${{\mathcal{H}}}_{7}$ (as in §3.2.1) translates to transitive
braid action for $r=4$ on all elements of ${{\text{\rm Ni}}}({{\text{\rm
GL}}}_{3}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{{\text{\rm C}}_{\infty\cdot
2^{3}}},T_{1})$ ($T_{1}$ just taking the presentation corresponding to one
cover) as in [Fr05b, Prop. 4.1] (Princ. 2.3). This Nielsen class is of
4-branch point covers of ${\mathbb{P}}^{1}_{z}$. Lem. 3.9 effectively detects
orbits of $H_{3}$ on Nielsen classes of 3-branch point covers. There is no
such easy conclusion for $H_{r}$, $r\geq 4$ on Nielsen classes.

######  Lemma 3.9.

The group $H_{3}={\langle q_{1},q_{2}\rangle}$ acts on a Nielsen class
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$, with C consisting of 3 conjugacy
classes, as a quotient of the dihedral group $D_{3}=S_{3}$ of order 6, since
$q_{1}^{2}$ and $q_{2}^{2}$ act trivially. If all three classes in C are
distinct modulo $N_{S_{\deg(T)}}(G)$ then all $H_{3}$ orbits have length 6.

###### Proof.

This follows fairly straightforwardly from [BaFr02, §2.4.1], which notes that
for $\tau_{1}=q_{1}q_{2}$, $\tau_{1}^{3}=q_{1}^{2}=q_{2}^{2}$ from standard
braid group relations. Now consider the action of $q_{2}^{2}$ on a Nielsen
class element:

$(g_{1},g_{2},g_{3})q_{2}^{2}\mapsto(g_{1},\alpha(g_{2}\alpha^{-1},\alpha(g_{3}\alpha^{-1}),\text{
with }\alpha=(g_{2}g_{3})^{-1}.$

From the product-one condition, $\alpha=g_{1}^{-1}$. So, $q_{2}^{2}$ fixes the
absolute class of $(g_{1},g_{2},g_{3})$. Similarly for $q_{1}^{2}$. Therefore,
in its action on Nielsen classes, $H_{3}$ is generated by two involutions, and
therefore it is a dihedral group. It suffices to decide which dihedral group
it factors through.

Since $\tau_{1}$ has order 3, this is $D_{3}=S_{3}$, acting as permutations of
the conjugacy classes. As the classes are distinct modulo
$N_{S_{\deg(T)}}(G)$, the action on those will give all permutations since
their orderings are also distinct. ∎

We conclude with comments on the space attached to elements in a Nielsen
class. This is done for absolute classes in detail in [Fr77, §5]. It and
[BaFr02] emphasize that their properties come from the explicit Hurwitz
monodromy (braid) action on Nielsen classes. (3.2.2) is a special case.
Components, as in (2.9), correspond to orbits of $H_{r}$ on Nielsen classes.

[Fr12, §5] has exposition, using the branch cycle lemma for finding the
definition field of the whole Hurwitz space. [Fr12, §6.4] uses deg 7 again
([Fr12, Thm. 6.7]). That includes displaying the relation between inner and
absolute spaces ((2.7) does this for a given Nielsen class). [BaFr02, §4]
starts the more advanced topics on cusps, including effective computation of
the genus of reduced Hurwitz space components for 4 branch point covers. Then,
each component is an upper half-plane quotient and $j$-line cover, though it
is rarely a modular curve.

####  3.2.3. §3.1.3 and the genus 0 problem

This section expands on the available empirical data that can help transition
from using the results of the Genus 0 problem to more general cases, say in
testing computer programs (as suggested in §4.3.1). For developing insight, we
suggest testing the list from (3.18) for what Cors. 4.12 and
LABEL:findeg7polyb have done for degree 7. That would fulfill an
indecomposable polynomial extension of Pakovich.

The anomalous (decomposable) example $(f^{*},g^{*})=(T_{4},-T_{4})$ for which
$\tilde{{\mathcal{C}}}_{f,g}$ has two degree 2 components has been discovered
numerous times (Rem. A.5). For it, though, (1.1a) fails (the Galois closure
has genus 0). Also, if we keep $f^{*}$ a polynomial, but allow $g^{*}$ to be
any rational function, §B.2 has the significant case where
$\deg(f^{*})=5,\deg(g^{*})=10$.

Monodromy groups $(G,T)$ of indecomposable rational functions are called
(primitive) genus 0 groups. Guranicks’ version of the genus 0 problem (over
${{\mathbb{C}}}$) formulates what pairs $(G,T)$ – $\deg\ n$ permutation
representation $T$ – could be monodromy groups of $f:X\to{\mathbb{P}}^{1}_{z}$
satisfying:

(3.15) $X$ has genus 0 and $f$ is indecomposable ($T$ is primitive).

He extended this to genus 1 and any fixed genus $>1$. Our qualitative results
don’t require these higher genus generalizations.

The idea – akin to the classification, but not restricted to simple groups –
divided possible $(G,T)$ into two sets by appeal to the classification of
primitive groups – an offshoot primed by essentially indexing them using
simple groups [AOS85]. Primitive genus 0 monodromy groups have two types:

  1. (3.16a)

Genus 0 series: Infinite series of $(G,T)$ with $G$ having core either $A_{n}$
or $=({{\mathbb{Z}}}/p)^{a}\times^{s}\\!{{\mathbb{Z}}}/d$,
$d\in\\{1,2,3,4,6\\}$ appearing as genus 0 monodromy groups [Fr05b, p. 78] or
[Fr12, §7.1].

  2. (3.16b)

Genus 0 exceptional: excluding (3.2.3a) there are only finitely many
(primitive) genus zero $(G,T)\,$s.

Thm. 3.10 is a polynomial version of the main result, Thm. 4.23, with the
extra assumption $f$ is indecomposable. Further,

  1. (3.17a)

they produce polynomial pairs. So C has a conjugacy class

${\text{\rm C}}={\text{\rm C}}_{\infty}\text{ whose elements have order
}\deg(T_{1})=\deg(T_{2})\text{ and;}$

  2. (3.17b)

both $T_{1}$ and $T_{2}$ are primitive ($f^{*}$ and $g^{*}$ are
indecomposable).

Comments below give more on the other cases that arose in Davenport’s and
Schinzel’s problems.

######  Theorem 3.10.

Assume $f^{*},g^{*}\circ g_{1}\in{{\mathbb{C}}}[x]$, $f^{*}$ is indecomposable
and $f^{*}$ and $g^{*}$ give inequivalent covers. Then, (3.18) lists all
degrees of $T_{1}$ for the Nielsen classes ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ for which $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ
g_{1}}$ is reducible.

(3.18) $\deg(f^{*})=\deg(g^{*})$ is in $\\{7,11,13,15,21,31\\}$.

For each degree in (3.18), there is one main Nielsen class for which
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is reducible. That class is either
unique or in degrees 7, 13, and 15; or it is one of a finite number of Nielsen
classes obtained from coalescing the main class (as with the degree 7 case of
§3.1.3).

###### Comments.

All degrees in (3.18) can be analyzed with a rubric close to that of our
running deg 7 examples. Each gives several examples of Nielsen classes,
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$, of pairs of rational
functions $(f^{*},g^{*})$ all of whose representative fiber products
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ are reducible. For some of those Nielsen
classes, the representatives have components of genus 0 or 1. Further, the
rubric and Thm. 4.23 apply to produce examples of generalizing Thm. 1.1.

In each of these cases, $T_{1}$ and $T_{2}$ are distinct permutation reps,
equivalent as group representations. We have mostly considered the case where
$g^{*}$ has a totally ramified place, and composing with an element of
${{\text{\rm PSL}}}_{2}({{\mathbb{C}}})$ applied to ${\mathbb{P}}^{1}_{z}$
turns the cover into a polynomial. As however, with the degree 7 case, using
coalescing and regarding these cases as satellites of more general cases gives
rational function cases that I haven’t analyzed completely.

[Fr99, §9] wraps up classifying the cases (3.18), describing the spaces of
such pairs of polynomials. Like the degree 7 case, general covers in the
degrees 13 and 15 Nielsen classes have four branch points. [Fr99, §8.2] lists
elements of the degree 13 Nielsen class. Here $\\{1,2,4,10\\}$ is what
$\\{1,2,4\\}$ was in degree 7 (as in (3.2)): a difference set for the doubly-
transitive design. Now apply Cor. 2.21 to find the
$\tilde{{\mathcal{C}}}_{f,g}$ components genuses, as for degree 7.

[Fr99, §8.3] finds the braid group generator $q_{1}$ and $q_{2}$ actions. That
produces the genus of the reduced Hurwitz space ${{\mathcal{H}}}({1,2,4,10})$
of such covers. [Fr99, §8.4] finds the defining field, $K_{13}$, for
${{\mathcal{H}}}({1,2,4,10}$ as moduli of such pairs. it is the fixed field of
$M({1,2,4,10})$ in ${{\mathbb{Q}}}(\zeta_{13})$:
${{\mathbb{Q}}}(\zeta_{13}+\zeta_{13}^{3}+\zeta_{13}^{9})$. The Hurwitz space
for these Davenport pairs is a rational variety [Fr99, §8.5].

[Fr99, Thm. 9.1] bounds degrees of Davenport pairs, with detail for $n=31$,
the highest degree, in [Fr99, §9.2.2]. [Fr05b, §2.3.2 and §3] discuss
difference sets, and describes Nielsen classes for the maximal families of
Davenport pairs for $\deg(f)=7,13,15$. Here are two standout points.

  1. (3.19a)

Given a particular conjugacy class of an $m$-cycle (resp. there are 2, 4, 2
for $m=7,13,15$ [Fr05b, p. 61]), then there is just one Nielsen class of
covers with 4 branch points (counting $\infty$).

  2. (3.19b)

Each Nielsen class of polynomials having these degrees comes from coalescing
from the Nielsen class in (3.2.3a).

[Mu95] (or [Fr05, App. C1]) describes all possible genus 0 monodromy groups of
polynomials. The polynomial map case is substantial – much given by the
results of the Davenport Problem – and sufficient for strong clues on the
genus 0 problem. Still, the latter is the poster child for distinguishing
between results on polynomials and rational functions. ∎

##  4\. Dropping irreducibility hypothesis (1.1b)

§3 is a primer on Nielsen classes. §4.1 sets up considering the pairs
$(f^{*},g^{*})$, (and their Nielsen classes) coming out of Prop. 1.8 with
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ reducible. Prop. 4.7 lists the
categories into which the components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ fall, and what $g_{1}\,$s to
avoid to get our particular generalization of Thm. 1.1..

§4.2 uses our running degree 7 examples to show those ingredients are
sufficient to produce Main Thm. 4.23. In §1.2.3 we referred to Cor. 4.12 and
Cor. 4.14 as the Degree 7 Corollaries. This is a model for how well such an
extension applies to Nielsen classes of rational functions $f$ of a given
degree, containing representative pairs $(f^{*},g^{*})$ with
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ reducible, and then those with genus 0
or 1 components.

Finally, §4.3 ties together the concepts and shows how to form the Nielsen
classes of excluded $g_{1}\,$s for our main theorem.

###  4.1. $\tilde{{\mathcal{C}}}_{f,g}$ properties from $T_{1},T_{2}$

§4.1.1 does some housekeeping on issues about computing with rational
functions. Then, it gets algorithmic on using Prop. 1.8 and on listing
components of representative $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ in a
Nielsen class. It also applies Pakovich’s o-char to some of our examples.
§4.1.2 precisely lists the technical obstructions, some akin to those Pakovich
gave to the generalization. §4.1.3 gets algorithmic about setting up Nielsen
classes.

Here are some preliminary reminders. If $\tilde{{\mathcal{C}}}_{f,g}$ is
reducible, then so is $\tilde{{\mathcal{C}}}_{f\circ f_{2},g\circ g_{2}}$ for
arbitrary (nonconstant) $f_{2},g_{2}\in{{\mathbb{C}}}(x)$. Prop. 1.8 points to
where we would find the heart of reducible $\tilde{{\mathcal{C}}}_{f,g}$. View
Thm. 1.1 as starting from a Nielsen class, ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{1})$, of genus-zero covers. Then, branch cycles for
$f:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$ in the Nielsen class are in
the classes C and $T_{1}=T_{f^{*}}$ as equivalent representations of $G$.

####  4.1.1. Using Prop. 1.8

We have already discussed aspects of $f^{*}$ and $g^{*}$ having a common left
composite (CLC).

  1. (4.1a)

Ex. 1.10 showed that gives reducibility of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

  2. (4.1b)

When $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has a genus 0 component, Lem. 1.11
showed, without amending the hypothesis, this violates the Thm. 1.1
conclusion.

Now consider how to recognize CLC; it can’t be eliminated trivially.

######  Lemma 4.1.

For $f,g\in{{\mathbb{C}}}[x]$, there is an effective check for whether a pair
$(f,g)$ have a CLC. This is equivalent to $f(x){-}g(y)$ has a variables
separated factor of form $u(x){-}v(y)$ with $f=f^{*}\circ u$ and
$g(y)=f^{*}\circ v(y)$ ($\deg(f^{*})>1$).

A Nielsen class equivalent without the polynomial assumption: The
representation $T_{f}\otimes T_{g}$ on the group $G_{f,g}$. has a component
isomorphic to the pullback of $T_{f^{*}}$ from $G_{f^{*}}$.

###### Comments.

The case $(f,g)$ are polynomials is the main theorem of [FrM69] with [FrM69,
§5] giving counterexamples when either $f$ is a rational function or the field
$K$ has characteristic dividing $\deg(f)$. These are also counter to the
extension of $K$ to its algebraic closure, preserving the lattice of fields.
Yet, for the general polynomial, maximal chains all have the same length and
relative degrees (in possibly different order).

The second paragraph, using Nielsen classes, is therefore stronger. ∎

Use notation like that of (1.21) . Assume $\boldsymbol{\mu}\in
G^{r}\cap{{\text{\bf C}}}$ generates $G$, and its entries have product 1.
Apply $T_{1}$ to $\boldsymbol{\tau}$, giving $T_{1}(\boldsymbol{\tau})$ and,
by RET, a representative $f^{*}$ of ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{1})$. Another faithful transitive representation $T_{2}$ of $G$
produces ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T_{2})$. Then,
$T_{2}(\boldsymbol{\tau})$ produces a cover
$X_{T_{2}(\boldsymbol{\tau})}\to{\mathbb{P}}^{1}_{z}$. A rational function
$g^{*}:{\mathbb{P}}^{1}_{y}\to{\mathbb{P}}^{1}_{z}$ represents this if covers
in ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T_{2})$ have genus zero.

Lem. 4.2 uses the Galois correspondence. Each cover
$\varphi_{W}:W\to{\mathbb{P}}^{1}_{z}$ through which $\hat{X}_{f^{*}}$ factors
has the form $\hat{X}_{f^{*}}/G(T_{\varphi_{W}},1)$. Its conclusion lists the
conjugacy classes of subgroups of $G$ defining components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

######  Lemma 4.2.

A representation $T$ of $G$ is faithful if and only if the intersection of all
conjugates of $G(T,1)$ in $G$ is trivial. If $G$ is a simple group, and $T$ is
a nontrivial permutation representation of $G$, then $T$ is faithful.

With $(f^{*},g^{*})$ as above, as in (1.8) assume a component $W$ of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ corresponds to an orbit, $I$, of
$G(T_{1},1)$ on $\\{{y_{1},\ldots,y_{n}}\\}$. Then $W\to{\mathbb{P}}^{1}_{z}$
is equivalent (as a cover) to $\hat{X}_{f}/H\to{\mathbb{P}}^{1}_{z}$ with
$H=G(T_{1},1)\cap G(T_{2},j)$ for any $j\in I$. The genus, ${{\text{\bf
g}}}_{W}$ of $W$, is the genus of the covers in ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},T_{W})$.

###### Proof.

The first statement is well-known, a consequence of characterizing $h\in G$
fixing all cosets $\\{G(T,1)g\\}_{g\in G}\leftrightarrow h\in\cap_{g\in
G}gG(T,1)g^{-1}$.

We show the second statement. The Galois correspondence says $H$ corresponds
to a cover $W^{*}=\hat{X}_{f}/H$ that is an image from
$\hat{X}_{f}=\hat{X}_{g}$.

Since $H\leq G(T_{1},1)$ and $\leq G(T_{2},j)$, then
$W^{*}\to{\mathbb{P}}^{1}_{z}$ factors through $\hat{X}_{f}/G(T_{1},1)$ (resp.
$\hat{X}_{g}/G(T_{2},j)$) a cover of ${\mathbb{P}}^{1}_{z}$ equivalent to
${\mathbb{P}}^{1}_{x}$ (resp. ${\mathbb{P}}^{1}_{y}$). From the universal
property of fiber products, this gives a map from $W^{*}$ to
$\tilde{{\mathcal{C}}}_{f,g}$. The image, $W$, of $W^{*}$ is a component of
$\tilde{{\mathcal{C}}}_{f,g}$, corresponding to a subgroup of $G(T_{1},1)\cap
G(T_{2},j)$. So $W^{*}=W$.

Now consider the converse: $W$ is a component of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$. Prop. 1.8 says $W$ corresponds to an
orbit, $I$, of $G(T_{f^{*}},1)$ in the representation $T_{g^{*}}$ (or an orbit
of $G(T_{g^{*}},1)$ in the representation of $T_{f^{*}}$). Suppose $j\in I$.
Conclude, in the Galois correspondence, that $W\to{\mathbb{P}}^{1}_{z}$ is
equivalent to the cover that corresponds to the subgroup $G(T_{f^{*}},1)\cap
G(T_{g^{*}},j)$ of $G(T_{f^{*}},1)$ leaving $j$ fixed.

The genus comment is a restatement of a cover’s genus being computed from the
Nielsen class of the cover by RH. This concludes the proof. ∎

We use Ex. 4.3 in Cor. 4.12. In each of the Nielsen classes pointed to by
(3.18), ${{\text{\rm{\bf o}-char}}}<0$. Yet, the condition ${{\text{\rm{\bf
o}-char}}}\geq 0$ arises in the complete description of Thm. 4.23, even in the
degree 7 cases if we allow $g^{*}$ to be a rational, rather than polynomial,
function.

######  Example 4.3 (Appearance of ${{\text{\rm{\bf o}-char}}}\geq 0$ for
degree 7).

Suppose in any of the degrees in (3.18) there is a(t least one) Nielsen class
${{\text{\rm Ni}}}(G_{f^{*}},{{\text{\bf C}}},{{\mathbb{T}}})$ so that
${{\mathcal{C}}}_{f^{*},g^{*}}$ has a genus 0 component $W$. In the notation
of Lem. 1.11, take $g=g^{*}\circ h_{y}$ with
$h_{y}:{\mathbb{P}}^{1}_{w}\to{\mathbb{P}}^{1}_{y}$ representing the cover
$W\to{\mathbb{P}}^{1}_{y}$.

Use the components of degrees 3 and 4 found in Lem. 3.3. We labeled the two
Nielsen classes with a pre-subscript $j=1$ or 2. Components are
correspondingly ${}_{j}W_{u^{\prime}}$: $u^{\prime}=1$ has degree 3, and
$u^{\prime}=2$ has degree 4.

Cor. 4.12 finishes handling the projections onto ${\mathbb{P}}^{1}_{y}$ of the
degree 3 (resp. degree 4) ${}_{u^{\prime}}W_{1}$ (resp.
${}_{u^{\prime}}W_{2}$) components of degree 7 polynomials. For example, with
$({}_{2}\sigma_{1},{}_{2}\tau_{1})$ and $({}_{2}\sigma_{2},{}_{2}\tau_{2})$ in
(3.9).

${{\text{\rm{\bf o}-char}}}=2+(1/3-1)-2(1/2)=1/3.$

Branch cycles for
${}_{2}h_{1}:{}_{2}W_{1}={\mathbb{P}}^{1}_{w}\to{\mathbb{P}}^{1}_{y}$ have a
3-cycle. Its Nielsen class is that of the degree 3 Chebychev polynomial
(§A.2.2).

From Rem. 3.5, the degree 4 complementary component ${}_{2}W_{2}$ also has
genus 0. Directly compute ${{\text{\rm{\bf
o}-char}}}=2+(1/3{-}1)+4(1/2{-}1)<0$. By inspection of the branch cycles of
the degree 4

${}_{2}h_{2}:{}_{2}W_{2}={\mathbb{P}}^{1}_{w}\to{\mathbb{P}}^{1}_{y}$

its monodromy is $S_{4}$; its branch cycles contain both a 2 and a 3-cycle.
$\triangle$

######  Remark 4.4.

Clearly, only if the Galois closure cover of $f$ has genus 0 or 1 is
${{\text{\rm{\bf o}-char}}}_{f}$ nonnegative: condition (1.1a) does not hold.
§A.2 presents, differently than the short exposition of [Pak18b, p. 2–3], on
how that produces examples violating the conclusion of Thm. 1.1.

[Da, p. 5-6] says orbifolds first appeared (as V-manifolds; see right below
(2.4)) in [Sa56]. [Th76, §13.3] changed the name V-manifold.

######  Remark 4.5 (Finding separated variable factors).

Factoring 2-variable polynomials is easier than finding composition factors of
a rational function. Then, checking if a two-variable polynomial has a
separated variables factor, as in Lem. 4.1 is easier still. Therefore,
checking for a given $g$, if $f$ and $g$ have a common composition factor, is
not so hard.

Handling genus 0 covers with imprimitive monody will require new ideas, going
beyond the genus 0 problem (for primitive groups), but Prop. 4.7 sets us in an
inductive direction to find new cases of it.

####  4.1.2. Pieces to extend Thm. 1.1

We use the (1.8) notation emphasizing the projections of components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ on ${\mathbb{P}}^{1}_{y}$.323232Despite
the symmetry in Prop. 1.8 between the $x$ and $y$ variables. So, the
components correspond to the orbits ${J_{1},\ldots,J_{u}}$ of $G(T_{2},y_{1})$
on ${x_{1},\ldots,x_{m}}$. The orbit $J_{j}\leftrightarrow W_{j}$ gives a
cover, ${\text{\rm pr}}_{y,j}:W_{j}\to{\mathbb{P}}^{1}_{y}$ of degree
$\ell_{j}=|J_{j}|$.

There are two Nielsen classes attached to the component $W_{j}$: That
represented by $W_{j}\to{\mathbb{P}}^{1}_{z}$ corresponding to the
representation $T_{H}$ in Lem. 4.2 and that from the natural projection map
from the fiber product, ${\text{\rm pr}}_{y,j}:W_{j}\to{\mathbb{P}}^{1}_{y}$.
For $1\leq j\leq t$ for which $W_{j}$ has genus 0, denote ${\text{\rm
pr}}_{y,j}$ by $h_{j}$, indicating representation by a rational
function.333333If you aren’t being careful about definition fields; the
rational function might not be over the definition field of ${\text{\rm
pr}}_{y,j}$.

Prop. 4.7 uses Prop. 1.8 to preclude growth of inappropriate components in
going from $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ to
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$. The lines (4.7a) and (4.7b)
inductively consider components of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ
g_{1}}$ that map to those of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$. Then.
(4.7c) considers additional components based on diagram (1.16), invoking
considerations of Ritt under the definition decomposition variant. Use Prop.
1.8 in the form of Cor. 1.19.

######  Definition 4.6 (Case $f^{*}$ indecomposable).

Call $g^{*}\circ g_{1}=g^{\star}\circ g_{1}^{\prime}$ a (nontrivial)
decomposition variant (dec-var in going) from
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ to
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ if $g^{\star}$ and $g^{*}$
are inequivalent covers and (1.8) holds by replacing $(f^{*},g^{*})$ with
$(f^{*},g^{\star})$.

Rem. 4.9 discusses generalizing Def. 4.6 without assuming $f^{*}$ is
indecomposable. Reference to $g_{1}$ in Prop. 4.7 (and elsewhere) assumes
$\deg(g_{1})>1$.

######  Proposition 4.7.

The possible relations between components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ and those of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is given by (4.7).

  1. (4.2a)

For $1\leq j\leq u$, components of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ
g_{1}}$ that map onto $W_{j}$ identify with components of
$\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$.

  2. (4.2b)

If for some $j$, $\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$, $1\leq
j\leq u$, has multiple components. Then, these give multiple components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ above $W_{j}$.

  3. (4.2c)

The possibility for a component $W^{\prime}$ of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ that does not lie above any
$W_{j}$ is that it appears as a dec-var from
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ to
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$.

If (4.7c) holds, then there is another decomposition of $g^{*}\circ g_{1}$ as
$g^{\star}\circ g_{1}^{\prime}$, and a correspondingly with
$f^{*}=f^{\star}\circ f_{1}$, for which some component on
$\tilde{{\mathcal{C}}}_{f^{\star},g^{\star}}$ contains the component
$W^{\prime}$. When $f^{*}$ is indecomposable, then $f^{\star}=f^{*}$.

###### Proof.

As above, fix a component $W_{j}$ of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$. We
leave the $\tilde{}$ decoration off the fiber products so that we can describe
and identify them before (projective) normalization.

The collection of components, $W^{\prime}_{j}$ of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ that map to $W_{j}$ identify
with the normalization of

(4.3) $W^{\dagger}_{j}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}\\{(x,y_{2})\in{\mathbb{P}}^{1}_{x}\times{\mathbb{P}}^{1}_{y_{2}}\mid
f^{*}(x)=g^{*}\circ g_{1}(y_{2}),(x,g_{1}(y_{2}))\in W_{j}\\}.$

We show normalization of $W^{\dagger}_{j}$ identifies with normalization of

(4.4) ${{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}=\\{(x,y;y_{2})\in
W_{j}^{\dagger}\times{\mathbb{P}}^{1}_{y_{2}}\text{ with }g_{1}(y_{2})=y\\}.$

To see that (4.3) and (4.4) define the same spaces, insert
$(g_{1}(y_{2}),y_{2})$ in place of the second coordinate of $(x,y_{2})$ from
(4.3).

That completes of the proof of (4.7a), and immediately gives (4.7b).

From Prop. 1.8 if there is another component $W^{\prime}$ not accounted,
(1.8b) fails. So, Prop. 1.8 implies there must another component on
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ accounted for by another
decomposition of $g^{*}\circ g_{1}$ and $f^{*}$ as given in the last paragraph
in the proposition.∎

Now for all the component types that arose in Prop. 4.7, we assume that none
have Galois closure of genus 0 or 1 (1.1a).

(4.5) The place of (4.7a) in Prop. 1.8.

If $g_{1}=h_{j}$, $1\leq j\leq t$, then for
$g_{2}\in{{\mathbb{C}}}(y)\setminus{{\mathbb{C}}}$,
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ h_{j}^{*}\circ g_{2}}$ has a genus 0
component. Use (1.11c) with $\tilde{{\mathcal{C}}}_{{\text{\rm
pr}}_{y,j},g_{1}\circ g_{2}}=\tilde{{\mathcal{C}}}_{h_{j},h_{j}\circ g_{2}}$.

In general, multiple components above $W_{j}$, implies that the pullback of of
$G(T_{1},x_{1})$ to $G_{g^{*}\circ g_{1}}$ is not transitive on the solutions
(in y) of $g_{1}(y)=y_{j}$.

(4.6) Possibilities related to (4.7b).

Here are the $g_{1}\,$s to avoid to assure the genus of all components of
$\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$ rises with the degree.

  1. (4.7a)

For $W_{j}$ of genus $>1$, assure $\tilde{{\mathcal{C}}}_{{\text{\rm
pr}}_{y,j},g_{1}}$ no degree 1 component.

  2. (4.7b)

For $W_{j}$ of genus $=1$, assure no component of
$\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$ is unramified over
$\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$.

  3. (4.7c)

For $W_{j}$ of genus 0, $1\leq j\leq t$, assure with
$(f^{*},g^{*})\mapsto(h_{j},g_{1})$ in Prop. 1.8, that $T_{h_{j}}$ and
$T_{g_{1}}$ are not entangled.

Lem. 4.8 is a step toward using Nielsen classes to avoid (4.1.2) $g_{1}\,$s.

######  Lemma 4.8.

In (4.1.2a) avoid all the genus 0 quotients of ${\text{\rm pr}}_{y,j}$. In
(4.1.2b) assure the branch points, ${{\boldsymbol{z}}}$, of ${\text{\rm
pr}}_{y,j}$ contain those of $g_{1}$ and for $z_{i}\in{{\boldsymbol{z}}}$,
each disjoint cycle of the branch cycle for ${\text{\rm pr}}_{y,j}$ at $z_{i}$
is a multiple of each disjoint cycle of the branch cycle for $g_{1}$ at
$z_{i}$.

In (4.1.2c) assure the stabilizer of a letter in the representation
$T_{h_{j}}$ is transitive on the letters of the representation of $T_{g_{1}}$.
As in §1.2.3, this is akin to avoiding Davenport-entanglement.

(4.8) Detecting the most difficult components: (4.7c)

The component denoted $W^{\prime}$ is a new genus zero component; not one on
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$, though it ends up on
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$. If $f^{*}$ is
indecomposable, then it corresponds to a new representation of $G_{f^{*}}$.

We may, though, luck out and find that
$\tilde{{\mathcal{C}}}_{f^{*},g^{\star}}$ is irreducible – there is no other
representation that could be $T_{g^{\star}}$ for which (1.8) holds – and
Pakovich’s Thm. applies to assure we need not worry about such a $g_{1}$. That
is the kind of data we get from the genus 0 problem.

For example, in our degree, 7 examples – and in all the other examples that
would come from the list of Thm. 3.10 in a similar style – we know that we do
get several Nielsen classes of degree 7 pairs, $(f^{*},g^{*})$, whose
representative $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ are reducible, and even
among those Nielsen classes many with genus 0 components. We also know there
is no distinct 3rd representation to worry about in (4.7c).

######  Remark 4.9.

To extend Def. 4.6 – decomposition variant – to the case $f^{*}$ is
decomposable, suppose in considering $\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ
g_{1}}$ that $g^{\star}\circ g_{1}^{\prime}=g^{*}\circ g_{1}$ and
$f^{\star}\circ f_{1}^{\prime}=f^{*}$ with the goal of an inductive argument
on the number of decomposition factors of $f^{*}$.

######  Remark 4.10.

[Pak22] raises iterated rational function applications. Some results try to
grab random rational functions, but “general” rational functions are not where
the action is.

The Nielsen approach gives names to the functions you deal with. Then, you add
to a particular Nielsen class further Nielsen class data for rational function
pairs to describe the $g_{1}\,$s to avoid, corresponding to each of the
comments above.

####  4.1.3. Handling pairs of reps. $(T_{1},T_{2})$ of $G$

Below, fix either $f$ (or its Nielsen class). The genus 0 problem (§1.2.2)
limits $G$ and the permutation representations, $T_{1}$, for genus 0 covers
and their Nielsen classes.

Start from a particular Nielsen class, ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{1})$ (say, for $f^{*}$), with $G_{f^{*}}=G$. Then, locate
corresponding $T_{2}$ for which one must deal in generalizing Pakovich,
without conditions (1.1). This shows the description of the $g^{*}\,$s with
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ having genus 0 (or 1) components has an
entirely Nielsen class formulation starting from $f^{*}$.

  1. (4.9a)

Find $T_{2}$ for which $T_{2}$ is intransitive on $G(T_{1},1)$ (as in (1.8b)).

  2. (4.9b)

(4.1.3a) $\implies$ $g\in{{\text{\rm Ni}}}(G,{{\text{\bf C}}},T_{2})$ fails
(1.1b): $\tilde{{\mathcal{C}}}_{f,g}$ is reducible.

  3. (4.9c)

Apply Cor. 2.21 to identify the genuses of the components of
$\tilde{{\mathcal{C}}}_{f,g}$.

§3.1 examples give Nielsen classes that label natural collections of pairs
$(f^{*},g^{*})$ that produce the reducibility phenomena. The Nielsen class of
$g^{*}$ is one of a finite number associated with the Nielsen class of
$f^{*}$. If, however, you change the Nielsen class of $f^{*}$, then you start
over again.

Comments on (4.1.3b): Branch cycles for $f^{*}$ consist of an $r$-tuple

$\text{$T_{1}(\boldsymbol{\tau})$ with
}({\boldsymbol{\tau}_{1},\ldots,\boldsymbol{\tau}_{r}})\in G^{r}\text{
satisfying \eqref{RET}}.$

Automatically create branch cycles for $g^{*}$ as $T_{2}(\boldsymbol{\tau})$.

The Prop. §B.1 example follows the steps in (4.1.3a) and (4.1.3b). Then, Cor.
B.2 does step (4.1.3c), finding genus 0 components of
$\tilde{{\mathcal{C}}}_{f,g}$.

We comment on what was essential about using the genus 0 condition and what
was not. Then, we add additional comments on $T_{1},T_{2}$ used in Thm. 4.23.
Start from Rem. 1.17.

  1. (4.10a)

Neither $f$ nor $g$ need be covers of genus 0 curves; components still
correspond to orbits of $G(T_{1},1)$ under $T_{2}$ in Prop. 1.8.

  2. (4.10b)

Orbits and component degrees apply even if $T_{1}=T_{2}$; one length 1 orbit
$\leftrightarrow$ a $\tilde{{\mathcal{C}}}_{f,g}$ component isomorphic to the
diagonal.

  3. (4.10c)

It is convenient, but not necessary, in Cor. 2.21 for the cover $f$ to have
genus 0 to find the genus of a component of $\tilde{{\mathcal{C}}}_{f,g}$.

In (4.1.3c), covers need only be nonsingular (irreducible) curves. As in §3.1,
singular points of ${{\mathcal{C}}}_{f,g}$ arise from coinciding images
$z^{\prime}$ of values of $x^{\prime}$ (resp. $y^{\prime}$) ramified of order
$e_{x^{\prime}}$ (resp. $e_{y^{\prime}}$) over $z^{\prime}$ with
$(e_{x^{\prime}}{-}1)(e_{y^{\prime}}{-}1)>0$.

Then, the $\gcd(e_{x^{\prime}},e_{y^{\prime}})$ (nonsingular) points of
$\tilde{{\mathcal{C}}}_{f,g}$ in a neighborhood over of $z^{\prime}$ are
locally isomorphic to
$\tilde{{\mathcal{C}}}_{x^{e_{x^{\prime}}},y^{e_{y^{\prime}}}}$ over
$z^{\prime}=0$. The notation for branch cycles as in, say, Prop. 2.17, is
still appropriate for a well-defined conjugacy class of elements in $G$ even
if $f$ is not a genus 0 cover.

######  Remark 4.11 (Comment on (4.1.3b)).

No need to limit $T_{1}$ and $T_{2}$ to faithful representations. For example,
assume $T_{1}$ is imprimitive, corresponding to a system of imprimitivity
$m_{1}$ (or $f=f_{1}\circ f_{2}$, as in §4.1.2). Then, the representation
${}_{m_{1}}T=T_{f_{1}}$ extends to a non-faithful representation of $G_{f}$ by
composing it with the natural cover $G_{f}\to{}_{m_{1}}G$.343434For
$\deg(f_{i})>1$, $i=1,2$, the wreath product (Rem. 4.15) says $G_{f_{1}}$ is a
proper quotient of $G_{f}$. A copy of the identity, from the diagonal on
$\tilde{{\mathcal{C}}}_{f,f}$, is in the kernel of $T_{1}\otimes
T_{1}\to{}_{m_{1}}T\otimes{}_{m_{1}}T$.

###  4.2. Treating ${{\text{\rm{\bf o}-char}}}>0$ when $f\in{{\mathbb{C}}}[x]$
has degree 7

This section uses our degree 7 example to show inputs to extend Thm. 1.1. Ex.
4.3 applies the o-char to the §3.1.1 examples of fiber products with genus 0
or 1 components. §3.10 (Cor. 4.12) finishes the analysis showing ${}_{2}h_{1}$
is equivalent to a Chebychev polynomial, and thus its Galois closure has genus
0 (A.5), while ${}_{2}h_{2}$ (degree 4) has group $S_{4}$. We see the value of
using the ${{\text{\rm{\bf o}-char}}}\geq 0$. Example: (A.4) describes those
Chebychev covers with

${{\text{\rm{\bf o}-char}}}=2+(1/3-1){+}2(1/2{-}1)>0.$

§4.2.2 discusses the many other Nielsen classes with $G={{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2)$ and the components of their fiber products (Cor.
LABEL:findeg7polyb). They are all satellites of one Nielsen class. These
examples cover most of the territory by example. Both the author and Schinzel
are/were number theorists. So, §4.2.3 includes a number theory discussion of
these Hurwitz spaces.353535There is a lesson here for those who think all
reasonable moduli spaces have their representing objects given only by
coordinates on their parameter spaces.

####  4.2.1. Handling ${}_{j}{\boldsymbol{\sigma}}$, $j=1,2$, in (3.10)

Use this Nielsen class ($j=1,2$):

(4.11) $\begin{array}[]{c}{{\text{\rm Ni}}}({{\text{\rm
PGL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2\cdot 3\cdot
7},{{\mathbb{T}}}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}(T_{1},T_{2}))\text{ of \eqref{bcfg} with representing}\\\
\text{branch cycles
}(({}_{j}\sigma_{1},{}_{j}\tau_{1}),({}_{j}\sigma_{2},{}_{j}\tau_{2}),(({}_{j}\sigma_{1}\cdot{}_{j}\sigma_{2})^{{-}1},({}_{j}\tau_{1}\cdot{}_{j}\tau_{2})^{{-}1})).\end{array}$

A fiber product representative would be labeled
$\tilde{{\mathcal{C}}}_{{}_{j}f^{*},{}_{j}g^{*}}$ corresponding to
$T_{{}_{j}f^{*}}=T_{1}$ and $T_{{}_{j}g^{*}}=T_{2}$, both of degree 7.

Prop. 4.7 classifies components on these fiber products. We figure what
restrictions (Lem. 4.8) on $g_{1}\in{{\mathbb{C}}}(y_{2})$ allow extending
Thm. 1.1 to $\tilde{{\mathcal{C}}}_{{}_{j}f^{*},{}_{j}g^{*}\circ g_{1}}$.

  1. (4.12a)

List restrictions on $g_{1}$ satisfying Nielsen class conditions; and

  2. (4.12b)

excepting (4.2.1a), show the two certain components on
$\tilde{{\mathcal{C}}}_{{}_{j}f^{*},{}_{j}g^{*}\circ g_{1}}$ are the only ones
and their genuses rise with $\deg(g_{1})$.

Use the notation of Rem. 4.3. For example, ${}_{2}W_{1}$ (resp. ${}_{2}W_{2}$)
is the degree 3 (resp. degree 4) component of
$\tilde{{\mathcal{C}}}_{{}_{2}f^{*},{}_{2}g^{*}}$ for the second Nielsen class
in §3.1.1: denoted there ${{\text{\bf C}}}_{\infty\cdot 2\cdot 3}$; here
${{\text{\bf C}}}_{2\cdot 3\cdot 7}$, ${\text{\rm C}}_{\infty}={\text{\rm
C}}_{7}$.

Also, for a fiber product component $W$, we are calculating branch cycles for
${\text{\rm pr}}_{y}:W\to{\mathbb{P}}^{1}_{y}$ as in Prop. 4.7. The opening
paragraph alludes to the representation given in Lem. 4.2.

######  Corollary 4.12.

The cover ${}_{2}W_{1}\to{\mathbb{P}}^{1}_{z}$ (resp.
${}_{2}W_{2}\to{\mathbb{P}}^{1}_{z}$) has associated representation on index
21 (resp. 28) cosets of a 2-Sylow (resp. $D_{3}$). These are respectively
defined by $G(T_{1},x_{j})\cap G(T_{2},y_{1})$ where $j\in J_{1}$ (resp.
$J_{2}$) the orbits of $G(T_{1},y_{1})$ on ${x_{1},\ldots,x_{7}}$, as in
(3.2).

Here are the results for the degree 4 $pr_{y}$ covers.

  1. (4.13a)

Degree 3 ${}_{2}{\text{\rm pr}}_{y}$: ${}_{2}W_{1}\to{\mathbb{P}}^{1}_{y}$
has group $D_{3}$ and genus 0 Galois closure.

  2. (4.13b)

Degree 4 ${}_{2}{\text{\rm pr}}_{y}$: ${}_{2}W_{2}\to{\mathbb{P}}^{1}_{y}$
has group $S_{4}$ and genus 3 Galois closure.

The Nielsen class of (4.12a) is of a degree 3 Chebychev polynomial.

Here are the comparable results for the degree 3 ${\text{\rm pr}}_{y}$ covers.

  1. (4.14a)

Degree 3 ${}_{1}{\text{\rm pr}}_{y}$: ${}_{1}W_{1}\to{\mathbb{P}}^{1}_{y}$
has group $D_{3}$ and genus 1 Galois closure.

  2. (4.14b)

Degree 4 ${}_{1}{\text{\rm pr}}_{y}$: ${}_{1}W_{2}\to{\mathbb{P}}^{1}_{y}$
has group $D_{3}$ and genus 0 Galois closure.

All, except the $S_{4}$ cover of (4.12b), are in the (1.1a) excluded cases.

###### Proof.

A well-known computation gives the order of any group of the form ${{\text{\rm
GL}}}_{m}(R)$ where $R$ is a finite ring. So,

$|{{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2)|=(2^{3}{-}1)(2^{3}{-}2)(2^{3}{-}2^{2}).$

Both $G(T_{1},x_{1})$ and $G(T_{2},y_{1})$ – by their orders – contain a
2-Sylow. By the Sylow Theorems, the 2-Sylows are conjugate. So, for some $j$,
$G(T_{1},x_{j})$ contains the same 2-Sylow as does $G(T_{2},y_{1})$. The two
groups define different permutation representations, so cannot be equal:
$G(T_{1},x_{j})\cap G(T_{2},y_{1})$ is exactly the 2-Sylow. The same argument
for the 3-Sylows, says there is a $j^{\prime}$ such that
$G(T_{1},x_{j^{\prime}})\cap G(T_{2},y_{1})$ is exactly a $D_{3}$.

The component ${}_{2}W_{v}$ defines a subgroup $H_{v}$, which we may assume is
contained in $G(T_{2},y_{1})$ and also in one of the conjugates of
$G(T_{1},x_{1})$, stabilized by conjugating by $G(T_{2},y_{1})$. These
containments correspond to the projection maps of ${}_{2}W_{v}$ on
${\mathbb{P}}^{1}_{x}$ and ${\mathbb{P}}^{1}_{y}$. The order of $H_{v}$ in
each of the two cases in the paragraph above characterizes whether $H_{v}$ is
a 2-Sylow or a $D_{3}$. That completes the first paragraph of the proposition.

Working on the Nielsen class with conjugacy classes ${}_{2}{{\text{\bf
C}}}={{\text{\bf C}}}_{2\cdot 3\cdot 7}$, to find conjugacy classes of branch
cycles for ${\text{\rm pr}}_{y}:{}_{2}W_{v}\to{\mathbb{P}}^{1}_{y}$, take
powers of conjugations of $({}_{2}\sigma_{i},{}_{2}\tau_{i})$, $i=1,2$, that
are fixed on $y_{1}$. Then, see what they do to the orbit corresponding to
$J_{v}$. We don’t need the subscript 7 (or ∞) because that only fixes $y_{1}$
if you take the trivial power. We have already done this just to find the
orbits $J_{1}$ and $J_{2}$.

The two separated expressions in (3.9) are the analogs of (3.7). For
${}_{2}W_{1}$, we look to the disjoint cycles with ∗ superscripts. The branch
cycle from one branch point $y^{\prime}_{1}$ is a 3-cycle corresponding to
$({}_{2}\sigma_{1},{}_{2}\tau_{1})$. There are two branch points,
$y^{\prime}_{2},y^{\prime}_{3}$; both have 2-cycle branch cycles corresponding
to $({}_{2}\sigma_{2},{}_{2}\tau_{2})$.

For the component ${}_{2}W_{2}$, we get a 3-cycle corresponding to an orbit of
$\sigma_{7}\cdot{}_{2}\sigma_{1}\cdot\sigma_{7}^{-1}$ for a branch point
$y_{1}^{\prime\prime}$ and three branch points
$y_{2}^{\prime\prime},y_{3}^{\prime\prime},y_{4}^{\prime\prime}$ with branch
cycles (respectively) of type $(2),(2)(2),2$. So, this degree 4 cover has
branch cycles with a 3-cycle and a 2-cycle, so it must have group $S_{4}$.
Apply (2.4) to compute the Galois closure has genus $3(>1)$.

We will be briefer on the examples ${}_{1}W_{1}$ and ${}_{1}W_{2}$ of (4.12).
For ${}_{1}W_{1}$, using (3.7)each of the branch cycles
$({}_{1}\sigma_{k},\tau_{k})$, $k=1,2$, gives two branch points on
${\mathbb{P}}^{1}_{y}$, each having a 2-cycle as branch cycle. The Galois
closure has genus 1. For ${}2W_{1}$, using (3.9), the branch cycle for
$({}_{2}\sigma_{1},\tau_{1})$ gives one branch point and corresponding branch
cycle a 3-cycle. For $({}_{2}\sigma_{2},\tau_{2})$ we get two branch points
and corresponding 2-cycles. Again, this is a Chebotarev case. ∎

####  4.2.2. Satellite Nielsen classes related to ${{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2)$

Cor. LABEL:findeg7polyb shows what we need to generalize Thm. 1.1 when $f$
falls among a reasonably limited set of Nielsen classes. Let ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}}_{b})$ and ${{\text{\rm Ni}}}(G,{{\text{\bf C}}}_{t})$
be two Nielsen classes with the same group $G$ ($b$ is for bottom, $t$ is for
top).

######  Definition 4.13 (Nielsen satellites).

We say ${{\text{\rm Ni}}}(G,{{\text{\bf C}}}_{b})$ is a satellite of
${{\text{\rm Ni}}}(G,{{\text{\bf C}}}_{t})$, if each
${\boldsymbol{\sigma}}\in{{\text{\rm Ni}}}(G,{{\text{\bf C}}}_{b})$ is a
coalescing from some ${\boldsymbol{\sigma}}^{*}\in{{\text{\rm
Ni}}}(G,{{\text{\bf C}}}_{t})$.

Notation is as in §3.1.2 where $T_{1}=T_{f}$ (resp. $T_{2}=T_{g}$), etc.

######  Corollary 4.14.

The branch cycles (3.1) – produced by the coalescings (3.10) – represent the
only Nielsen classes,

${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}{{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2\cdot 4\cdot
7},{{\mathbb{T}}})\text{ and }{{\text{\rm Ni}}}_{3\cdot 2\cdot
7}\stackrel{{\scriptstyle\text{\rm def}}}{{=}}{{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{3\cdot 2\cdot
7},{{\mathbb{T}}})$

with a pair of degree 7 representations $(T_{1},T_{2})$ and conjugacy classes
C with group $G$ containing ${\text{\rm C}}_{7}$ and for $f\in{{\text{\rm
Ni}}}(G,{{\text{\bf C}}},T_{1})$ there is $g\in{{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{2})$ with $\tilde{{\mathcal{C}}}_{f,g}$ reducible, having a genus 0
component.

Further, ${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}$ and ${{\text{\rm Ni}}}_{3\cdot
2\cdot 7}$ each consist of precisely six elements and

${{\text{\rm Ni}}}({{\text{\rm PSL}}}_{3}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2^{3}\cdot 7}),{{\mathbb{T}}})\text{ has both as satellites.}$

So, all degree 7 Davenport pairs, $(f,g)$, correspond to points on a
compactification of the irreducible space ${{\mathcal{H}}}_{7}$, with points
corresponding to the Nielsen class ${{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{3}\cdot 7},{{\mathbb{T}}})$
off the boundary.

###### Proof.

Expression (3.10) has the coalescings that give the two Nielsen classes above.
On the other hand, from ${}_{1}{\boldsymbol{\sigma}}$-coalesce, we get the
result of coalescing 4 other Nielsen class elements by the following device.

  1. (4.15a)

$\tau=(1\,4\,6\,7)(2\,3)$ is the product of the two entries of

$\mu=((1\,6)(2\,3),(6\,4)(1\,7)).$

  2. (4.15b)

Conjugate $\mu$ by $\tau$ to see that 4 elements in the Nielsen class (with
$\sigma_{\infty}$ in the 4th position) give exactly the same coalescings.

  3. (4.15c)

Achieve (4.2.2b) by the powers of the braid ${{\text{\bf
sh}}}q_{2}^{2}{{\text{\bf sh}}}$.

The references above (3.1) show that all possible branch cycles for Nielsen
classes of degree 7, containing $\sigma_{7}$, are contained in those listed in
(4.2.2), We now know these are coalescings from the single braid orbit
comprising ${{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{3}\cdot 7})^{{\text{\rm
abs}}_{1,2}}$. There are only two conjugacy classes of degree 7 elements in
${{\text{\rm PGL}}}_{2}({{\mathbb{Z}}}/2)$, completing all the Nielsen classes
with Davenport pair representatives. Therefore the Nielsen classes we have
listed are satellites of ${{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{3}\cdot 7},{{\mathbb{T}}})$.

See the beginning of §3.2, for initial discussion of ${{\mathcal{H}}}_{7}$,
and then §4.2.3 for the compactification we refer to. Finally, we document
that both ${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}$ and ${{\text{\rm Ni}}}_{3\cdot
2\cdot 7}$ consist of precisely six elements.

First, we have the natural map $\psi:H_{3}\to S_{3}$ given by the effect of
$q\in H_{3}$ on the order of the conjugacy classes. There are 3 distinct
conjugacy classes in C modulo $N_{S_{7}}(G)$ (because the classes have
distinct orders) and Lem. 3.9 implies $\psi$ is surjective. There are exactly
6 elements in each $H_{3}$ orbit in ${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}$ and
${{\text{\rm Ni}}}_{3\cdot 2\cdot 7}$.

Suppose each involution in a 3-tuple in one of our Nielsen classes, say
${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}$, conjugates to any other by some power
of $\sigma_{\infty}$. Assume further,

${\boldsymbol{g}}=(g_{1},g_{2},g_{3}),\
{\boldsymbol{g}}^{\prime}=(g_{1}^{\prime},g_{2}^{\prime},g_{3})\in{{\text{\rm
Ni}}}_{2\cdot 4\cdot 7}$

with say $g_{1}$ and $g_{1}^{\prime}$ involutions and
$g_{3}=\sigma_{\infty}^{-1}$. Again from product one, if $\sigma_{\infty}^{i}$
conjugates ${\boldsymbol{g}}$ to
$(g_{1}^{\prime},\sigma_{\infty}^{i}g_{2}\sigma_{\infty}^{-i},g_{3})$, then
$\sigma_{\infty}^{i}g_{2}\sigma_{\infty}^{-i}$ is $g_{2}^{\prime}$. Conclude:
${\boldsymbol{g}}$ and ${\boldsymbol{g}}^{\prime}$ are the same in
${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}^{\text{\rm abs}}$. So there are six
elements in this Nielsen class. Involutions, however, in this example (indeed
all examples in the groups of (3.18)), correspond to transvections (§3.1.2).
So, they are conjugate.

You can check this directly using the 7 Nielsen class representatives in
${{\text{\rm Ni}}}({{\text{\rm GL}}}_{3}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2^{3}\cdot 7},T_{1})$ listed, say, in [Fr05b, §3.3]. The same applies to
${{\text{\rm Ni}}}({{\text{\rm GL}}}_{3}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2\cdot 3\cdot 7},T_{1})$. This easily completes the theorem. ∎

######  Remark 4.15 (Wreath products).

We have typically labeled a Nielsen class with a group and a collection of
conjugacy classes, and that can be done for ${{\text{\rm
Ni}}}_{2,\deg{-}3,m}$, with $(m,3)=1$. Indeed, for any composite cover

$f_{1}\circ f_{2}:X_{2}\to X_{1}\to Z,$

we may describe $G_{f_{1}\circ f_{2}}$ as a subgroup of the wreath product of
the two covers, with their natural permutation reps. entwined [Fr70, §2]. This
is a fruitful way when, as in [BiFr86], $G$ is close to, even if not quite
equal, to the full wreath product.

Here it is not the full wreath product. The same ${{\mathbb{Z}}}/2$ quotient
of $D_{m}$ is mapped onto by ${{\text{\rm GL}}}_{3}({{\mathbb{Z}}}/2)$ in its
map to ${\mathbb{P}}^{1}_{y}$, for all $m$. We don’t make use of it here, so
give no details. More extreme even is when $\tilde{{\mathcal{C}}}_{f,g}$ has a
genus 0 component, producing the cover $h_{1}:C\to{\mathbb{P}}^{1}_{y}$ as in
§3.1. Then, the Galois closure of $g\circ h_{1}$ is still just $G$.

Also, we haven’t done any detail on the case when $(m,2)=2$, as in Ex. A.5.
The time for doing that would be when it plays a vital role in some example,
as it doesn’t work the same as the case $m$ is odd.

####  4.2.3. Significance of ${{\mathcal{H}}}_{7}$

The Hurwitz space ${{\mathcal{H}}}_{7}$ is one case of the varieties
constructed in [Fr77, §4]. For almost all Nielsen classes in this paper, such
as ${{\text{\rm Ni}}}_{2^{3}\cdot 7}$, each of those spaces,
${{\mathcal{H}}}$, is a fine moduli space, because the stabilizing group
$G(T,1)$ self-normalizes in $G$ [Fr77, §4, Prop. 3]. The remainder of this
subsection explains more on how the construction of [Fr77, §4] combined with
[Fr95b, Thm. 3.21] shows the cohering of those satellites.

First: ${{\mathcal{H}}}$ is an affine variety with a total space structure,
${{\mathcal{T}}}$, over it: ${{\mathcal{T}}}$ is a cover of
${{\mathcal{H}}}\times{\mathbb{P}}^{1}_{z}$ with this property. For
${{\boldsymbol{p}}}\in{{\mathcal{H}}}$, a fiber of ${{\mathcal{T}}}$ over
${{\boldsymbol{p}}}\times{\mathbb{P}}^{1}_{z}$ represents a cover in the
equivalence class of ${{\boldsymbol{p}}}$. This was shown by producing complex
analytic coordinates [Fr77, §4.B, pgs. 49–53], and then applying a famous
result of Grauert and Remmert [GraR57]: An analytic (unramified) cover $W$ of
a quasiprojective variety $V$ is quasi projective. So, you may complete $W$ to
a projective variety $\bar{W}$ by normalization of $\bar{{\mathcal{H}}}$ in
the function field of $W$.

[Fr77, Thm. 5.1] gives, from the Branch Cycle Lemma, the definition field of
${{\mathcal{H}}}$ as a moduli space. We can understand that nicely when fine
moduli holds to be the well-defined minimal definition of ${{\mathcal{T}}}$
with its structural maps. Particularly it says that the two connected families
of Davenport polynomials with respective Nielsen classes ${{\text{\rm
Ni}}}(GL_{3}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{3}\cdot 7})^{\text{\rm
abs}}$ and ${{\text{\rm Ni}}}(GL_{3}({{\mathbb{Z}}}/2),{{\text{\bf
C}}}_{2^{3}\cdot 7^{\prime}})^{\text{\rm abs}}$, are conjugate over
${{\mathbb{Q}}}(\sqrt{-7})$.

Continuing in generality, use the compactifications $\bar{{\mathcal{H}}}$ and
$\bar{{\mathcal{T}}}$ as projective varieties with extending maps
$\bar{{\mathcal{T}}}\to\bar{{\mathcal{H}}}\times{\mathbb{P}}^{1}_{z}$, through
normalization of (components $\bar{{\mathcal{H}}}^{\prime}$ of)
$\bar{{\mathcal{H}}}\times{\mathbb{P}}^{1}_{z}$ in the function field of
(components ${{\mathcal{T}}}^{\prime}$ of) ${{\mathcal{T}}}$. We recover the
families of covers attached to satellite Nielsen classes by inductively
coalescing, using this normalization stratification of $\bar{{\mathcal{H}}}$:

  1. (4.16a)

Restrict $\bar{{\mathcal{T}}}^{\prime}$ over the boundary
$\bar{{\mathcal{H}}}^{\prime}\times{\mathbb{P}}^{1}_{z}\setminus{{\mathcal{H}}}^{\prime}\times{\mathbb{P}}^{1}_{z}$.

  2. (4.16b)

Normalize (components of) that result and identify open unions of subsets of
them as spaces of covers attached to Nielsen classes.

  3. (4.16c)

Continue inductively on the dimension from (4.2.3a) applied to the Nielsen
classes in (4.2.3b).

[Fr95b, proof of Thm. 3.21 and Lem. 3.22] carried out these steps, under the
names specialization sequences and coalescing operators.

This means there is a path from a point on ${{\mathcal{H}}}$ to any point on
the space representing a satellite Nielsen class, a path that runs through a
family of nonsingular covers. This is the case, say, for a cover starting at
any element in the Nielsen class ${{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{3}\cdot 7},{{\mathbb{T}}})$
to a cover in the Nielsen class ${{\text{\rm Ni}}}_{2\cdot 4\cdot 7}$, or the
Nielsen class of the other satellite. A different, Deligne-Mumford style,
compactification was constructed by [DeEm99] and [We99]), motivated by the
application [Fr95b, Thm. 3.21].

###  4.3. The Main Theorem

§4.3.1, with reminders of [Pak22], uses our approach to mitigate Pakovich’s
skepticism of an effective genus formula for components of
$\tilde{{\mathcal{C}}}_{f,g}$ when it is reducible.

§4.3.2, based on §4.1.2, gives the Nielsen class formulation for avoiding
$g_{1}\,$s that might not give a generalization of Thm. 1.1. The genus of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ rises with $\deg(g_{1})$ for
$(f^{*},g^{*})$ in a Nielsen class for which
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is reducible, and even has components of
genus 0.

####  4.3.1. Results of [Pak22]

Pakovich switches the reference to the degrees of $f$ and $g$: His $m$ is my
$n$, et. cet. I keep mine, but I also use his (compatible to me) $k$ and
$\ell$ for the respective bi-degrees of projection of a component, $W$, of
$\tilde{{\mathcal{C}}}_{f,g}$ onto ${\mathbb{P}}^{1}_{x}$ and
${\mathbb{P}}^{1}_{y}$.363636The notation memorably extends to changing
${\mathbb{P}}^{1}_{x}$, ${\mathbb{P}}^{1}_{y}$, ${\mathbb{P}}^{1}_{z}$
respectively to $X$, $Y$, $Z$.

[Pak22, p. 2] is skeptical about an effective genus formula for components.
Since I have given such, l put it this way.

  1. (4.17a)

Using Nielsen classes, Prop. 1.8 gives a handle using the minimal left
composition factor of $f$.

  2. (4.17b)

As in (4.3.1a), divide what happens with that minimal left composition factor
according to properties listed for specific primitive groups by the solution
of the genus 0 problem.

  3. (4.17c)

The data, for example, Nielsen classes and the representation results of the
genus 0 problem, are programmable. 373737With experience, one easily detects
the likelihood of, say, Schur covering, Davenport pair, and Hilbert-Siegel
situations with rational functions – generalizing the known polynomial results
– directly from the extant results.

By programmable in (4.3.1c), we would include Mathematica, Maple, GAP, Cayley,
…. It would be part of applying programs to include the coelescings catching
the genus 0 or 1 Nielsen classes.

For example, start from ${{\text{\rm Ni}}}({{\text{\rm
PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf C}}}_{2^{6}},{{\mathbb{T}}})$, the
most general degree 7 case in Ex. 3.8, with components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ having genus 4. A program would find the
coalesced Nielsen classes of Rem. 3.5 for our running example, wherein each
has $k=3$ and $\ell=4$ and component genuses 0 or 1.

If $\tilde{{\mathcal{C}}}_{f,f}$ has (excluding the diagonal) no component of
genus 0 or 1, [Pak22, p. 3] refers to it as tame.383838For example, if the
natural representation for $f$ is double transitive, then Cor. 2.22 gives the
computation of the genus of the non-diagonal component of
$\tilde{{\mathcal{C}}}_{f,f}$. [Pak18a]: Up to reduced equivalence (Def. 2.5)
for a rational function $f$, $\hat{f}$ has genus 0 or 1 only for cyclic,
Chebychev and the Cheybchev Galois closure functions, plus a finite number of
other situations. [Pak22] notes the tame $f\,$s have Galois closures of genus
exceeding 1.

We note Pakovich’s other skeptical statements in Rem. 4.17. We should go
beyond the unrepresentative (and rare) anomalies of §A.2 or those with
extreme, but often insignificant, coefficient patterns. The goal would be to
progress in connecting the behavior of different algebraic functions to, say,
Galois/fundamental group theory and $\ell$-adic representations.

######  Theorem 4.16.

If $\tilde{{\mathcal{C}}}_{f,g}$ is irreducible, of genus $>1$, then its genus
exceeds $\frac{n{-}84m{+}168}{84}$, unless the genus of $\hat{f}$ is $\leq 1$
[Pak18b, Thm. 2].

[Pak22, p. 4] generalizes the fiber product diagram to specialize to where he
can drop the irreducibility hypothesis and replace
$\tilde{{\mathcal{C}}}_{f,g}$ by a component $W$. In his generalization, he
introduces the concept of having a right composition factor. In this case, the
projections of $W$ to ${\mathbb{P}}^{1}_{x}$ and ${\mathbb{P}}^{1}_{y}$ would
factor (nontrivially) through another cover $W^{\prime}$. The 1st paragraph of
Prop. 1.18, though, excludes this.

[Pak22, (10)] gives a formula using the bi-degree $(k,\ell)$ with

$k=\deg(\tilde{{\mathcal{C}}}_{f^{*},g^{*}}/{\mathbb{P}}^{1}_{x})>1,\ell=\deg(\tilde{{\mathcal{C}}}_{f^{*},g^{*}}/{\mathbb{P}}^{1}_{y})$

involving the use of the higher fiber products of $f$ as in §A.1 with the
fiber product description of the Galois closure. Here, though we can revert to
the framework of Prop. 1.8 and consider
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ with $f^{*}$ and $g^{*}$
having the same Galois closures. It says, if $f^{*}$ is tame, then

(4.18) ${{\text{\bf g}}}_{W}>2-m+n/m!\text{ unless }g^{*}\circ
g_{1}=f^{*}\circ f_{1}\text{ (and $W$ as in Ex.~{}\ref{diagcomp1})}.$

The genus bound from (4.18) is much smaller than 0. It wouldn’t point at the
results of Rem. 3.5 nor to the explicit Ex. 3.4 for the Pakovich formulation
in the 4 degree 7 cases where there is a genus 0 or 1 component.

######  Remark 4.17.

[Pak22] says obtaining a full classification of components on
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ of genus zero or one using the genus
formula seems to be hardly possible. In addition, such an analysis results
only in possible patterns of ramifications of $f^{*}$ and $g^{*}$. Pakovich
calls this the Hurwitz problem. [Pak09] provides Laurant polynomials,
$u(1/z)+v(z)$ with $u,v\in{{\mathbb{C}}}[x]$, examples.

This statement makes sense if your only input is from a combinatorial genus
formula. The Nielsen class approach, though, suggests information on examples
available for specific problems related to covers. There you would try to find
what groups $G$ offer possible solutions, as with the Schur, Davenport, etc.
problems. A more precise result on appropriate Nielsen classes would ask when
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},T)$ is non-empty.

######  Remark 4.18.

[Pak18b, p. 300] states Ritt’s second Theorem. Classifying curves
$\tilde{{\mathcal{C}}}_{f,g}$ of genus 0 when $f$ and $g$ are polynomials. He
also includes the result with at most two points at $\infty$ in [BT00].
Suppose $\\{{}_{i}g_{1}\\}_{i=0}^{\infty}$, rational functions, with
$\deg({}_{i}g_{1})\mapsto\infty$. He refers to an $f$ that has
$\tilde{{\mathcal{C}}}_{f,{}_{i}g_{i}}$ of genus 0 and irreducible for all i,
a basis of curves of genus 0 [Pak18b, p. 301]. [Pak18b, Thm. 1] says this
happens only if $\hat{f}$ has genus zero or one.

####  4.3.2. The Main Theorem extending Thm. 1.1

Prop. 4.7 gives a list of component types that can occur on a representative
fiber product $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ in a Nielsen class
${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ where these are
reducible. Following Pakovich, we have already excluded consideration of
$f^{*}$ for which the Galois closure of $f^{*}$ has genus 0 or 1: condition
(1.1a).393939As we take considerable space, especially in §A.3, to consider
this arising, there is reason to look further into this case, but not here.

######  Definition 4.19.

Property $P$ of fiber product representatives in ${{\text{\rm
Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ is a Nielsen (class) invariant if it
is constant on the Nielsen class. By contrast, $P$ is a braid invariant if it
is only constant on braid orbits.

All properties we consider are braid invariant, though not necessarily
obviously so. Def. 2.7 has the action of $N_{S_{m}}(G,{{\text{\bf C}}})\cap
N_{S_{n}}(G,{{\text{\bf C}}})$ used to define ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$ in (2.6) by quotienting out by this action. As we noted
in Rem. 2.9, these outer automorphisms are not always braided. When they
aren’t, they must be added to get the correct equivalence between covers on
those braid orbits. We know there is just one braid orbit in the cases we have
used as examples.404040We also know many Nielsen classes where there is more
than one braid orbit. Example: The Lift Invariant in the the Main Theorem of
[Fr10].

To generalize Thm. 1.1 we show the genus of representatives
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ goes up with the degree of
$g_{1}$ with four possible exclusions. All exceptions are made using
properties associated to fiber products denoted
$\tilde{{\mathcal{C}}}_{{\text{\rm pr}}_{y,j},g_{1}}$ in Prop. 4.7. Here
${\text{\rm pr}}_{y,j}={\text{\rm pr}}_{W_{j}}:W_{j}\to{\mathbb{P}}^{1}_{y}$
with $W_{j}$ a component of the fiber product.

There is a special situation, given by (4.3.2c) when $W_{j}$ has genus 1
requiring Def. 4.20 and Lem. 4.21.

######  Definition 4.20.

Given $\sigma\in S_{n}$ and $\tau\in S_{n^{\prime}}$, we say $\sigma$
dominates $\tau$ if the length of each disjoint cycle of $\sigma$ is a
multiple of the order of $\tau$.

We use the list of component types following the Prop. (and comments on them);
(4.3.2) is a reminder of what to avoid.

For a cover $\varphi_{X}:X\to{\mathbb{P}}^{1}_{y}$ denote its branch points by
${{\boldsymbol{z}}}_{\varphi_{X}}$ and a branch cycle for
$z^{\prime}\in{{\boldsymbol{z}}}_{\varphi_{X}}$ by
$g_{\varphi_{X},z^{\prime}}$.

  1. (4.19a)

Condition (1.1a): Galois closure of ${\text{\rm pr}}_{y,j}$ has genus $\leq
1$.

  2. (4.19b)

Condition (4.1.2a): $g_{1}$ giving any genus 0 quotients of ${\text{\rm
pr}}_{y,j}$.

  3. (4.19c)

Condition (4.1.2b): with ${{\text{\bf g}}}_{W_{j}}=1$, $g_{1}$ with
${{\boldsymbol{z}}}_{g_{1}}\subset{{\boldsymbol{z}}}_{{\text{\rm pr}}_{y,j}}$
and for $z^{\prime}\in{{\boldsymbol{z}}}_{g_{1}}$,
$g_{\varphi_{X},z^{\prime}}$ dominates $g_{g_{1},z^{\prime}}$.

  4. (4.19d)

Condition (4.1.2c): when ${\text{\rm pr}}_{y,j}=h_{j}$, $g_{1}$ entangled with
$h_{j}$, giving a new reducible $\tilde{{\mathcal{C}}}_{h_{j},g_{1}}$.

######  Lemma 4.21.

Suppose $W_{j}$ has genus 1, and there are $s$ branch cycles in
${\boldsymbol{g}}_{g_{1}}$ with respective orders ${d_{1},\ldots,d_{s}}$.
Then, (4.3.2b) implies

(4.20) $2\deg(W_{j})\geq\sum_{i=1}\deg(W_{j})(\frac{d_{i}{-}1}{d_{i}}).$

In particular, this situation can only occur with $s\geq 4$ if $s=4$ and all
$d_{i}\,$ are 2. In that case, $g_{1}$ is a cover with galois closure genus 1
(given in §A.4).

###### Proof.

The condition on the left of (4.20) is the sum of the indices of branch cycles
for ${\text{\rm pr}}_{W_{j}}$ assuming $W_{j}$ has genus 1, and the condition
on the right is the minimal value for the sum of the indices of branch cycles
corresponding to $z^{\prime}\in{{\boldsymbol{z}}}_{g_{1}}$. The rest of the
Lemma is evident from the characterization of §A.4 since the minimal value of
the index is $\frac{n}{2}$, and that only when the orders of the branch cycles
are 2. ∎

Prop. 4.22 gives the following general principle.

######  Proposition 4.22.

We can express branch cycles for the Nielsen class of ${\text{\rm pr}}_{y,j}$
in terms of those for $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$. From this, in
each case $v$ of (4.3.2) – possibly excluding (4.3.2c) – there is a (finite
collection of) Nielsen class(s) ${{\text{\rm Ni}}}_{v}$ containing those
$g_{1}\,$s to avoid to assure the genus rises with $\deg(g_{1})$.

In case (4.3.2c), using notation of Lem. 4.21, either $s=4$ and the Galois
closure of $g_{1}$ has genus 1, or $s=3$, and exceptional cases of (4.3.2c)
are given by classical triangle groups; the Galois closure of $g_{1}$ has
genus $\leq 1$.

###### Proof.

The algorithm for finding the Nielsen class of ${\text{\rm pr}}_{y,j}$ from
that of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ is given by Cor. 2.19, with the
illustrating examples of Cor. 4.12 the culmination of our degree 7 running
example. For each item in (4.3.2), we comment below on why it has a Nielsen
class description stemming from that of ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},{{\mathbb{T}}})$. Denote the Nielsen class of ${\text{\rm pr}}_{y,j}$ by
${{\text{\rm Ni}}}_{{\text{\rm pr}}_{y,j}}$.

Detecting condition (1.1a): Cor. 4.12 has given examples of using the ø-char
to find the Galois closure genus and also examples of detecting the main cases
of that condition happening from the list of §A.2.

Nielsen description of condition (4.3.2b): Given branch cycle representatives
${\boldsymbol{g}}$ of a Nielsen class ${{\text{\rm Ni}}}(G,{{\text{\bf
C}}},T_{W})$ of irreducible covers $\varphi_{W}:W\to{\mathbb{P}}^{1}_{y}$, we
consider how to find branch cycles for the genus 0 covers
$\varphi_{W^{\prime}}:W^{\prime}\to{\mathbb{P}}^{1}_{y}$ through which it
factors.

The subgroup $G(T_{W},1)$ defines $W$, and each cover through which
$\varphi_{W}$ factors is defined as a quotient of $\hat{\varphi}_{W}$ by a
group $G(T_{W},1)\leq H\leq G$ from the quotient $\hat{\varphi}_{W}/H$.
Compute the genus of this cover from its branch cycles, which are given by
${\boldsymbol{g}}\mod H$: The action of each entry on the distinct cosets of
$H$. Now apply this to the Nielsen class of ${\text{\rm pr}}_{W}$.

Nielsen description of condition (4.3.2c): As in the case above, given
${\boldsymbol{g}}$ an $s$-tuple of $S_{n}$ ($n=\deg(\varphi_{W})$), the
Nielsen classes to avoid contain elements
${\boldsymbol{g}}^{\prime}\in(S_{n^{\prime}})^{s}$, with $n^{\prime}$
arbitrary, subject to these conditions:

  1. (4.21a)

$g_{i}$ dominates $g_{i}^{\prime}$, $1\leq i\leq s$; and

  2. (4.21b)

${\langle{\boldsymbol{g}}^{\prime}\rangle}$ is a transitive subgroup of
$S_{n^{\prime}}$, satisfies product-one and RH gives the genus of a cover with
branch cycles ${\boldsymbol{g}}^{\prime}$ as 0.

Lem. 4.21 covers this situation by forcing $g_{1}$ to be in the excluded case
of having its Galois closure (as a cover) of genus $\leq 1$.

Nielsen description of condition (4.3.2d): We have already suggested this
situation is the complicated one, but that is because it calls for knowing if
there is another representation of $G$ that can be entangled with $T_{1}$.
Finding out about such examples would go into the solution of the genus 0
problem beyond the list of Thm.3.10 relating to the diagram (1.16).

Though this case is more complicated than the others, there still is a Nielsen
class description of the result. We need not exclude this situation except if
it requires avoiding what we have already considered above. ∎

######  Theorem 4.23.

Prop. 4.22 has gone through the list of types of components of
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$, and noted a Nielsen class description –
based on that of ${{\text{\rm Ni}}}(G,{{\text{\bf C}}},{{\mathbb{T}}})$ in
which $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ falls. It has given a Nielsen
class description of what to avoid with the covers given by $g_{1}$:

  1. (4.22a)

Anyplace where the Galois closure of a rational function cover has genus 0 or
1; and

  2. (4.22b)

(4.3.2b) where you must consider the genus 0 components of the starting
Nielsen class represented by $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$.

It also gives leeway to avoid (4.3.2c); that gets into an induction argument.
In what is allowed, the genus increases with the degree of $g_{1}$. But see
Rem. 4.24.

We conclude with final words on examples, beyond the many we have given based
on ${{\text{\rm Ni}}}({{\text{\rm PSL}}}_{2}({{\mathbb{Z}}}/2),{{\text{\bf
C}}},{{\mathbb{T}}})$. (3.2.3) has divided the genus 0 primitive monodromy
groups into two cases. In (3.2.3a) are those related to small semidirect
products and alternating groups.

Now consider the primitive genus 0 monodromy related to alternating groups.
§B.1 uses them to produce many examples of reducible fiber products with genus
0 components. That seems to illustrate alternating groups as the untamable
case of the (primitive) genus 0 problem. Yet, §B.2 reminds us of a (serious
diophantine motivated, really!) alternating group series (before [GSh07]) with
a surprising conclusion. Compatible with the genus 0 result, from infinitely
many possible examples, only one degree fulfilled the constraints on the
problem and produced rich diophantine examples.

Among the exceptional genus 0 groups of (3.2.3b), there are only finitely many
such $(G,T)$. Yet, the number is huge. A myriad of different authors
contributed, applying in each case their expertise on pieces of the (simple
group) classification through [AOS85]. We doubt any tractable classification
could come from these for a full, explicit, Pakovich extension. That is, with
the present state of the classification, that list might as well be infinite
even without dropping the primitive monodromy condition.

######  Remark 4.24.

With the Pakovich assumptions in Thm. 1.1, the genus of the irreducible
$\tilde{{\mathcal{C}}}_{f,g}$ rises with $\deg(g_{1})$. He doesn’t give the
formulation using Prop. 1.8 that considers testing for reducibility.
Nevertheless, the genus goes up, but we have been less careful about being
explicit about its rise.

##  Appendix A Comments on fiber products

§A.1 presents the Galois closure of a cover $f:X\to Z$ as defined by a fiber
product. [Fr10] and [Fr12] both do Hurwitz space versions of this for
conclusions about families of Galois closure covers. From the universality of
fiber products this gives a tight connection between the Galois closure of a
cover $f:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$ and its Galois closure
$\hat{f}$, especially considering the condition (1.8a) in Prop. 1.8 when
dealing with components of $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ for
$\hat{f}^{*}$ and $\hat{g}^{*}$ as equivalent covers.

§A.2 refers to the family of genus 0 or 1 covers of
$\tilde{{\mathcal{C}}}_{f,g_{1}}$ that arise from negating condition (1.1a) as
an o-char-fan. They fan into a web of $g_{1}$ values for which
$\tilde{{\mathcal{C}}}_{f,g\circ g_{1}}$ remains irreducible, but still has
bounded genus; unlike the other possible $g_{1}\,$s for which the genus rises
with $\deg(g_{1})$.

###  A.1. Presenting the Galois closure cover

To form the Galois closure, $\hat{f}:\hat{X}_{f}\to Z$, of a cover $f:X\to Z$,
start with the fiber product of $f$, $m=\deg(f)$ times. As usual, normalize,
then remove the fat diagonal (loci where two coordinates are equal) to get
$\hat{X}_{m}$.

Finally, take a connected component $\hat{X}(m)$, of $\hat{X}_{m}$, with its
natural projection (restriction of) $\hat{f}:\hat{X}(m)\to Z$. That’s the
Galois closure: By restricting the natural action of $S_{m}$ to it, the
stabilizer of the component has exactly the right order to be the Galois
closure (by the fundamental theorem of Galois theory). This works over any
field $F$ (of characteristic 0; even, with care about inseparability, in
characteristic $p$), but here take $F={{\mathbb{C}}}$.

Denote the decomposition group – subgroup of $S_{m}$, in its natural action on
$\hat{X}_{m}$, whose elements map $\hat{X}(m)$ into itself – by
$G(\hat{X}(m)/Z)$.

We can also construct $\hat{X}(k)$, based on the fiber product $k\leq m$
times. Given $k\leq m$, consider faithfulness condition (A.1) on
$G(\hat{X}(m)/Z)$:

(A.1) $F_{K}$: if $\sigma\in G_{f}$ fixes $k$ integers, then it $\sigma=1$.

######  Principle A.1.

There is an isomorphism $\hat{f}^{*}:G(\hat{X}(m)/Z)\to G_{f}$, uniquely
defined up to inner isomorphism of $G_{f}$. Given another connected component
$\hat{X}(m)^{\prime}$, then $\hat{X}(m)\to Z$ and $\hat{X}(m)^{\prime}\to Z$
are equivalent as covers by an isomorphism that induces an isomorphism of

$G(\hat{X}(m)/Z)\to G(\hat{X}(m)^{\prime}/Z),$

unique up to inner isomorphism of $G_{f}$.

Suppose $\hat{f}:\hat{X}(m)\to Z$ factors through a cover $\psi:V\to W$. Then,
$\hat{f}$ factors through $\hat{\psi}:\hat{V}\to W$, the Galois closure cover
of $\psi$.

Suppose $G_{f}$ satisfies $F_{k}$. Then, the Galois closure cover of $f$ is a
connected component of $\hat{X}(k)$.

###### Proof.

Since $|S_{m}|$ is the degree of $\hat{X}(m)\to Z$, the degree of $\hat{f}$ is
the same as the order of the decomposition group. The basic Galois principle
shows this gives the Galois closure cover. The restriction of $S_{m}$ to any
fiber $\hat{X}_{m,z}$ over $z\in Z$ is transitive. Therefore it is transitive
on components of $\hat{X}(m)$, and some element $\sigma\in S_{m}$ takes
$\hat{X}(m)$ to $\hat{X}(m)^{\prime}$. Conclude the result by tracing the
effect on the isomorphisms with $G_{f}$.

Now consider the morphism $\psi$. The morphism $\hat{X}(m)\to W$, factoring
through $\psi$ is a Galois cover factoring through $V$, while $\hat{\psi}$ is
the minimal Galois cover of $W$ factoring through $V$ by the Galois
correspondence.

Now assume $G_{f}$ satisfies $F_{k}$. Then, the action of $G_{f}$ on
$\hat{X}(k)$ is faithful. Take $Y$ to be an orbit of this action. Consider the
projection $\hat{X}_{m}\to\hat{X}_{k}$ onto the first $k$ coordinates. Then,
the pullback of $Y$ is the image of some connected component $\hat{X}(m)$ of
$\hat{X}_{m}$, and the action of $G(\hat{X}(m)/Z)$ commutes with the
projection. This identifies $Y\to Z$ as the Galois closure of $f$. ∎

The whole inner vs. absolute total space construction of (2.7) can be done in
the same fiber-product style based on assuming the self-normalizing condition
of Def. 2.4. The diagram is part of the [FrV91, Main Thm]; the fiber-product
construction is in [BaFr02, §3.1.1].

###  A.2. When the orbifold characteristic is nonnegative

We give Nielsen class formulations of two cases when the o-char is
nonnegative, with each appearing in examples in this paper’s body.

####  A.2.1. Examples with ${{\text{\rm{\bf o}-char}}}=0$

Pakovich notes cases where (1.1a) fails, go back to the late 1800s. True, but
the condition needs more discussion than [Pak18b, p. 2] gives. We give a
Nielsen class construction of families of towers of genus 0 components that
fail (1.1a). Those, for a given $f$, give irreducible
$\tilde{{\mathcal{C}}}_{f^{*},g^{*}\circ g_{1}}$ of genus 0, with
$\deg(g_{1})$ arbitrarily large.

Take $E$ a copy of the complexes ${{\mathbb{C}}}$. A 1-dimensional complex
torus has the form $E/L$ with $L$ isomorphic to ${{\mathbb{Z}}}^{2}$ viewed as
a rank 2 module of translations by complex numbers on $E$. For any integers
$n>1$ form $L\cdot\frac{1}{n}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}L_{n}$. An isogeny (group homomorphism) between complex torii has
the form

  1. (1.2a)

$\beta:E/L\to E/L^{\prime}$: $L\leq L^{\prime}\leq L_{n}$ for some integer
$n$.

  2. (1.2b)

For simplicity assume $|L^{\prime}/L|$ is odd.

To find a genus 0 cover $f:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$ whose
Galois closure has genus 1, note that $\beta$ commutes with modding out by
$\\{\pm 1\\}$ generated by multiplication by -1 on both $E/L$ and
$E/L^{\prime}$. From Weierstrass normal form:

$E/L^{\prime}\\!\\!\\!\mod\\{\pm 1\\}={\mathbb{P}}^{1}_{z}\text{ and
}E/L\\!\\!\\!\mod\\{\pm 1\\}={\mathbb{P}}^{1}_{x}.$

The induced cover $f:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$ has

(A.3) $G_{f}\equiv(L^{\prime}/L)\times^{s}\\!\\{\pm 1\\}.$

As $L^{\prime}/L$ is an abelian group of rank (minimal number of generators) 1
or 2. This produces two families of covers given by rational functions:
corresponding to $L^{\prime}/L={{\mathbb{Z}}}/n$ and
$L^{\prime}/L=({{\mathbb{Z}}}/n)^{2}$, rank 1 and rank 2 (degree $n$) cases.
Other covers from this method are cofinal in these [Fr74, §3].

######  Example A.2 ($L^{\prime}/L={{\mathbb{Z}}}/n$).

The Nielsen class is ${{\text{\rm Ni}}}(D_{n},{{\text{\bf C}}}_{2^{4}},T)$:
$D_{n}$ the dihedral group of order $2n$, C, the class of involutions, with
${{\text{\bf C}}}_{2^{4}}$ having C repeated 4 times; $T:D_{n}\to S_{n}$ with
the action on ${\langle\sigma\rangle}$ cosets, $\sigma\in{\text{\rm C}}$.

For $(n,n^{\prime})=1$, fiber products of covers, with the same branch points,
in ${{\text{\rm Ni}}}(D_{n},{{\text{\bf C}}}_{2^{4}},T)$ and ${{\text{\rm
Ni}}}(D_{n^{\prime}},{{\text{\bf C}}}_{2^{4}},T)$ are in ${{\text{\rm
Ni}}}(D_{n\cdot n^{\prime}},{{\text{\bf C}}}_{2^{4}},T)$. These are cases
where the Galois Closure has genus 1. Indeed, this follows immediately from
(2.4) and it having four branch cycles of order 2. So, the orbifold
characteristic is 0.

This applies to condition (1.1a) in Thm. 1.1. The rational function covers $f$
there have branch cycles that are coalescings (as in §3.2) in the Nielsen
class ${{\text{\rm Ni}}}(D_{n},{{\text{\bf C}}}_{2^{4}})^{\text{\rm abs}}$,
and their Galois closures (if they aren’t already Galois) are components
$\hat{f}_{2}:\hat{\mathbb{P}}^{1}_{x,2}:$ of the $k=2$-fold fiber product of
the cover $f$ minus the fat diagonal. $\triangle$

Especially the most important part is this. Over a given algebraic point
$j_{0}$ of ${\mathbb{P}}^{1}_{j}$, what the action of the absolute Galois
group $G_{{{\mathbb{Q}}}(j_{0})}$ does to the components of
$\hat{\mathbb{P}}^{1}_{x,2}$. Suppose we add the rank 2 case in a similar
style to this (with $G_{n}=({{\mathbb{Z}}}/n)^{2}\times^{s}\\!\\{\pm 1\\}$).
Then, this characterizes the covers that arise in three seemingly disparate
diophantine problems: Serre’s Open Image Theorem, the theory of complex
multiplication and deciding precisely when such $f$ represent exceptional
covers: Covers for which over their definition field $K$, for infinitely many
primes ${{\boldsymbol{p}}}$ of the ring of integers, ${{\mathcal{O}}}_{K}$,
provide one-one maps on
${{\mathcal{O}}}_{K}/{{\boldsymbol{p}}}\cup\\{\infty\\}$. [Fr05, §6.1] puts
that whole story together, referring back to the relevant points of [Fr78, §2]
and [GMS03].

In particular, only a handful of covers in the whole collection of Nielsen
classes with $G=D_{n}$ have definition field ${{\mathbb{Q}}}$. In contrast,
the ${{\mathbb{Q}}}$ covers, in the case $G=S_{n}$, are dense in the space of
covers because they correspond to elliptic curves over ${{\mathbb{Q}}}$ with
the degree $n^{2}$ isogenies given by multiplication by $n$. As [Fr74, §3]
notes, this is an enhancement of Ritt’s Second Theorem (which has no allusion
to number theory) and the very motivation for the generalization of that to
[Fr73b].

######  Example A.3 (o-char-fans for these Nielsen classes).

A natural Nielsen class, ${{\text{\rm
Ni}}}({{\mathbb{Z}}}\times^{s}\\!{{\mathbb{Z}}}/2,{{\text{\bf C}}}_{2^{4}})$
covers every Nielsen class in this subsection. It is one Nielsen class, but it
is special because to cover all these examples, the group $G$ is infinite. We
call this the ${}_{1}{{\text{\bf C}}}_{2^{4}}$ o-char-fan.

An even larger o-char-fan ${{\text{\rm
Ni}}}(({{\mathbb{Z}}})^{2}\times^{s}\\!{{\mathbb{Z}}}/2,{{\text{\bf
C}}}_{2^{4}})$ of genus 0, degree $n^{2}$, covers gives covers of all the
degree $n^{2}$ isogenies of elliptic curves $E\to E$ from multiplication by
$n$ followed by modding out on both sides by ${\langle\pm 1\rangle}$. Refer to
this as the ${}_{2}{{\text{\bf C}}}_{2^{4}}$ o-char-fan.

####  A.2.2. Examples with ${{\text{\rm{\bf o}-char}}}>0$

There is a natural polynomial case, by coalescing branch cycles just as was
done according to §3.2 to arrive at the genus 0 fiber product components of
§3.1.3. The corresponding Nielsen classes are ${{\text{\rm
Ni}}}(D_{n},{{\text{\bf C}}}_{2^{2}\cdot n})^{\text{\rm abs}}$ with
${{\text{\bf C}}}_{2^{2}\cdot n}$ indicating two repetitions of the class of
involutions together with the class of an $n$-cycle in $D_{n}$. Modulo
absolute (but not inner) equivalence there is just one $n$ cycle class,
because the outer automorphism group of $D_{n}$ is

$\\{{\big{(}\begin{array}[]{cc}\\!\lower 2.0pt\hbox{$\scriptstyle a$}&\lower
2.0pt\hbox{$\scriptstyle b$}\\!\\\ \\!\raise 2.0pt\hbox{$\scriptstyle
0$}&\raise 2.0pt\hbox{$\scriptstyle 1$}\\!\end{array}\big{)}}\mid
a\in({{\mathbb{Z}}}/n)^{*},b\in{{\mathbb{Z}}}/n\\}.$

These are cases where the Galois Closure has genus 0. This follows immediately
from (2.4) and two branch cycles of order 2, one of order $n$. So, the
orbifold characteristic is $>0$. The family of these is easy to understand as
they are just the Chebychev polynomials of degree $n$ modulo changes of
variable (as noted in [Fr70]). These are cases where the Galois closure of the
cover has genus 0.

######  Example A.4 (o-char-fan ${{\text{\rm
Ni}}}({{\mathbb{Z}}}\times^{s}\\!{{\mathbb{Z}}}/2,{{\text{\bf
C}}}_{2^{2}\cdot\infty})$).

Start with the o-char-fan that is the analog of the fans in Ex. A.3, for the
Nielsen classes in this subsection given by the projective limit of the
Nielsen classes ${{\text{\rm
Ni}}}({{\mathbb{Z}}}\times^{s}\\!{{\mathbb{Z}}}/2,{{\text{\bf C}}}_{2^{2}\cdot
m})^{\text{\rm abs}}$. Here the representation is on cosets of a group
$G(T,1)$ generated by an involution. For $m$ odd, the involutions form a
unique conjugacy class.

Covers in this Nielsen class form a connected family parametrized by

${{\mathcal{H}}}(D_{m},{{\text{\bf C}}}_{2^{2}\cdot
m})\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}({\mathbb{P}}^{1})^{3}\setminus\Delta_{3}/S_{2}\times\\{1\\}$

where: $\Delta_{3}$ is the fat diagonal; and $S_{2}\times\\{1\\}$ equivalences
$(y_{1},y_{2},y_{3})$ to $(y_{2},y_{1},y_{3})$. The cover over the equivalence
class of $(y_{1},y_{2},y_{3})$ is in (A.4).

Define the polynomial $T_{m}$ by

$T_{m}\Bigl{(}\frac{t+1/t}{2}\Bigr{)}=\frac{t^{m}{+}1/t^{m}}{2}=z.$

As a covering map it is in ${{\text{\rm Ni}}}(D_{m},{{\text{\bf
C}}}_{2^{2}\cdot m})$, with finite branch points $\pm 1$. For $(3,m)=1$, the
(normalized) fiber product $\tilde{{\mathcal{C}}}_{T_{3},T_{m}}$, as a cover
of ${\mathbb{P}}^{1}_{z}$, is
$T_{3m}:{\mathbb{P}}^{1}_{w}\to{\mathbb{P}}^{1}_{z}$, with
$w=\frac{t+1/t}{2}$. Take $L$ to be the linear fractional transformation on
${\mathbb{P}}^{1}$.Then, the fiber over $\ell\times{\mathbb{P}}^{1}_{y}$ of
the map

(A.4) $L\times{\mathbb{P}}^{1}_{w}\to L\times{\mathbb{P}}^{1}_{y}\text{ by
}\ell\times w\mapsto\ell\times l(T_{m}(\ell^{-1}(w)))$

has branch points $\ell(-1),\ell(+1)$; respective images of
$\ell(-1),\ell(+1)$.

This is, however, a case with a cover in ${{\text{\rm
Ni}}}({{\mathbb{Z}}}\times^{s}\\!{{\mathbb{Z}}}/2,{{\text{\bf C}}}_{2^{2}\cdot
m})^{\text{\rm abs}}$ having its Galois closure of genus 0:

(A.5) $2(2m{-}1{+}\textbf{g}_{m,{\text{\rm
in}}})=2(m{-}1)+2(2m/2)\implies\textbf{g}_{m,{\text{\rm in}}}=0.$

######  Remark A.5.

Take $n=4$, where (b) does not hold. Here

$G=D_{4}=({{\mathbb{Z}}}/4)\times^{s}\\!\\{\pm
1\\}=\bigl{\\{}{\big{(}\begin{array}[]{cc}\\!\lower 2.0pt\hbox{$\scriptstyle
a$}&\lower 2.0pt\hbox{$\scriptstyle b$}\\!\\\ \\!\raise
2.0pt\hbox{$\scriptstyle 0$}&\raise 2.0pt\hbox{$\scriptstyle
1$}\\!\end{array}\big{)}}\mid a\in\\{\pm 1\\},b\in{{\mathbb{Z}}}/4\bigr{\\}},$

has 2 classes of involutions ${\text{\rm C}}_{1}$ and ${\text{\rm C}}_{2}$,
resp. represented by $(a,b)=(-1,0)$ and $(-1,1)$. The Nielsen class analogous
to that for odd $n$ is ${{\text{\bf C}}}_{1^{2}2^{2}}$ indicating we take each
involution class twice. Then, we get a 2-component fiber product
$\tilde{{\mathcal{C}}}_{f,g}$ by taking the respective pairs of permutation
representations $T_{f}$ and $T_{g}$ as the resp. representations on the cosets
of ${\langle(-1,0)\rangle}$ and ${\langle(-1,1)\rangle}$.

There are two anomalies. First, $\tilde{{\mathcal{C}}}_{f,g}$ here fails both
of Pakovich’s hypotheses (1.1). The genus of the Galois closure of $f$ is 1,
and we get a reducible fiber product. Since the involutions fix two integers
in the representation, the 2-fold fiber product of $f$ will not give the
Galois closure (as in Princ. A.1).

Consider the polynomial version by coalescing in this case, where the Nielsen
class is ${{\text{\rm Ni}}}(D_{4},{{\text{\bf C}}}_{2^{2}\cdot 4})^{\text{\rm
abs}}$, with the repetition of an involution class twice, and a 4-cycle. The
orbifold characteristic is $2{+}2(\frac{1}{2}{-}1){+}(\frac{1}{4}{-}1)>0$.
Normalize the polynomials here to have branch points $1,-1,\infty$, and this
is the case of reducible degree 4, $T_{4}(x)+T_{4}(y)$ , or $g=-f$.

######  Example A.6 (Ubiquitous $T_{4}$ example).

Rem. A.5 gives group data about this example. Despite the rareness of
nontrivially reducible $f(x)-g(y)$ with $f,g\in{{\mathbb{C}}}[x]$, this
example pops up in many papers: $f(x)=T_{4}(x)$, $g=-T_{4}(y)$. Affine pieces
of the components of $\tilde{{\mathcal{C}}}_{f,g}$ appear in [Sc82, p. 57]:

$\bigl{(}x^{2}+\sqrt{2}xy+y^{2}-2\bigr{)}\bigl{(}x^{2}-\sqrt{2}xy+y^{2}-2\bigr{)}.$

Each component is the Galois closure of $X_{f}$ with corresponding groups both
$\\{1\\}$ and $(x,y)\mapsto(-x,y)$ displays the two components are isomorphic.

##  Appendix B Expectations for $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ genus 0
components

We use the notation of Prop. 1.8. §B.1 constructs $\infty$-ly many cases –
thanks to RET – where $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has $u=2$
components, with both $f^{*}$ and $g^{*}$ indecomposable, and having identical
Galois closures. It also shows that one of those components has genus 0. Yet,
§B.2 gives evidence that those are subtle exceptions to Genus zero problem
(§3.2) expectations.

###  B.1. $\infty$-ly many $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ with $u=2$
components

For each $m\geq 4$, we produce $m$ and $(f^{*},g^{*})$, with
$f^{*}\in{{\mathbb{C}}}[x]$ of $\deg(f^{*})=m$, and (nontrivially)
$\widetilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has two components. Use the notation
at the beginning of §1.1.2. Here is a branch cycle description for
$f^{*}:{\mathbb{P}}^{1}_{x}\to{\mathbb{P}}^{1}_{z}$, with $G_{f^{*}}=S_{m}$
and $T_{f^{*}}=T_{1}$ the standard representation of $S_{m}$:

${\boldsymbol{\sigma}}_{f}\stackrel{{\scriptstyle\text{\rm
def}}}{{=}}(\sigma_{1},\sigma_{2},\sigma_{3})=((1\,2),(1\,3\,4\,\dots\,m),(1\,2\,\dots\,m)^{-1})\in(S_{m})^{3}).$

Take $T_{g^{*}}=T_{2}$ the degree $\frac{m(m-1)}{2}=n$ rep. of $S_{m}$ on
unordered pairs

$\\{\\{i,j\\}\mid i\not=j,1\leq i,j\leq m\\}.$

The cover for $g^{*}$ has the branch cycles
${\boldsymbol{\tau}}=(T_{2}(\sigma_{1}),T_{2}(\sigma_{2}),T_{2}(\sigma_{3}))$.
A rational function represents the cover $g^{*}$ if ${{\text{\bf g}}}_{*}$ in
the following equation is 0.

(B.1) $2\Bigl{(}{\frac{m(m-1)}{2}}+{{\text{\bf
g}}}_{*}-1\Bigr{)}=\sum_{i=1}^{3}{{\text{\rm
ind}}}(T_{g}(\sigma_{i}))=\sum_{i=1}^{3}{{\text{\rm ind}}}(\tau_{i}).$

######  Proposition B.1.

Computing indices in (B.1) shows ${{\text{\bf g}}}_{*}=0$, producing
$g\in{{\mathbb{C}}}(y)$. Then, $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has
exactly two components ($g^{*}\in{{\mathcal{R}}}_{f}$, (1.1b) fails). For
$m>4$, the orbifold characteristic is negative ((1.1a) holds).

###### Proof.

The action of $(1\,2)$ on the pairs $\\{1,j\\}$, $j\neq 1$ or 2, moves them to
the pairs $\\{2,j\\}$. Conclude: ${{\text{\rm ind}}}(\tau_{1})=m-2$. Also,
every $\tau_{3}$ orbit has the form

$\\{\\{j,k{+}j{-}1\\}\mid 1\leq j\leq m\\},\ 2\leq k\leq m.$

This will be an orbit of length $m$, unless for a given $k$, there are two
distinct values of $j$ (say, $j^{\prime}$ and $j^{\prime\prime}$) for which

$j^{\prime}=j^{\prime\prime}{+}k{-}1\text{ and
}j^{\prime\prime}=j^{\prime}{+}k{-}1,\text{ or }2(k{-}1)\equiv 0\\!\\!\\!\mod
m.$

That is, $m$ is even and $k=\frac{m}{2}{+}1$. Also, the orbit under
translation of this set of two distinct elements is determined by the minimal
absolute difference between the elements in the set.

Therefore, $\tau_{3}$ is a product of $m$-cycles, $\frac{m-1}{2}$ of them if
$m$ is odd, but it has, besides $m$-cycles, one $\frac{m}{2}$-cycle if $m$ is
even. Thus:

${{\text{\rm ind}}}(\tau_{3})=\begin{cases}\frac{(m-1)(m-1)}{2}&\text{ if $m$
is odd; and }\\\ \frac{(m-1)(m-2)}{2}+\frac{m}{2}-1=\frac{m(m-2)}{2}&\text{ if
$m$ is even}.\end{cases}$

Now we compute ${{\text{\rm ind}}}(\tau_{2})$ based using that ${{\text{\bf
g}}}_{*}\geq 0$.

Case $m$ is even: From RH,

$\sum_{i=1}^{3}{{\text{\rm ind}}}(\tau_{i})\geq
2\Bigl{(}{\frac{m(m-1)}{2}}-1\Bigr{)}=m(m{-}1){-}2.$

Therefore, ${{\text{\rm ind}}}(\tau_{2})\geq
m(m{-}1){-}2{-}\frac{m(m{-}2)}{2}{-}(m{-}2)=\frac{m(m{-}2)}{2}$. Since
$\tau_{2}$ has order $m-1$, the maximal value of ${{\text{\rm
ind}}}(\tau_{2})$ occurs if $\tau_{2}$ is a product only of $(m{-}1)$-cycles,
$\frac{m}{2}$ of them. So, ${{\text{\rm ind}}}(\tau_{2})$ is this maximal
value and ${{\text{\bf g}}}_{*}=0$.

Case $m$ is odd: Same as in the Case $m$ is even, except that the maximal
possible value of $\tau_{2}$ if besides $(m{-}1)$-cycles, $\tau_{2}$ has one
$(\frac{m-1}{2})$-cycle.

Notice: $\tilde{{\mathcal{C}}}_{f^{*},g^{*}}$ has two components corresponding
to the two orbits of $G_{f}(1)$ on the unordered pairs: One orbit on all pairs
of the form $\\{1,k\\}$, $k\neq 1$; the other on the rest of the pairs.

This concludes the proof except to show that for $m>4$, the orbifold
characteristic is negative. Since it is given by

$2{+}(1/2-1){+}(1/m{-}1){+}(1/(m{-}1){-}1),$

a decreasing function of $m$. Check that it is -.05 for $m=5$. ∎

Cor. B.2 computes the genus of the two components discovered in the fiber
products of Prop. B.1.

######  Corollary B.2.

Let $m\geq 5$ in Prop. B.1 be odd. For $m=5$, both components of
$\tilde{{\mathcal{C}}}_{f,g}$ have genus 0.

For all $m\geq 5$, the component of degree $m{-}1$ over ${\mathbb{P}}^{1}_{z}$
has genus 0. The genus of the other component grows quadratically with $m$.

###### Proof.

We do the case where $m$ is odd in detail. Denote representatives of the
Nielsen class pairs $(f^{*},g^{*})$ corresponding to $m$ by
$(f^{*}_{m},g^{*}_{m})$. In applying Cor. 2.21, switch $f_{m}^{*}$ and
$g_{m}^{*}$; branch cycles for $f^{*}$ are simpler.

That is, we are dividing the points corresponding to cycles in the branch
cycles of $\tilde{{\mathcal{C}}}_{f^{*}_{m},g^{*}_{m}}$ over
${\mathbb{P}}^{1}_{x}$ between the two components. Start with $m=5$. The
method is completely analogous to that of §3.1.3. Since $\sigma_{3}$ is an
$m$-cycle, the corresponding point contributes nothing to RH in either
component. As $\sigma_{2}$ fixes only one point – corresponding to the integer
2 – and the cycles of $\tau_{2}$ all have order dividing $m{-}1$, we can take

(B.2)
$\begin{array}[]{rl}&((2,\\{2,1\\})\,(2,\\{2,3\\})\,(2,\\{2,4\\})\,(2,\\{2,5\\}))^{*}\\\
&((2,\\{1,3\\})\,(2,\\{3,4\\})\,(2,\\{4,5\\})\,(2,\\{5,1\\}))\\\
&((2,\\{3,5\\})\,(2,\\{1,4\\}))\end{array}$

as a rep. of the branch cycle conjugacy class in $G_{f^{*}}=G_{g^{*}}$ of
$\tau_{2}$. The ∗ superscript is used below.

Similarly, the branch cycle classes for the points of ${\mathbb{P}}^{1}_{x}$
corresponding to each of the three fixed points – $w=3,4,5$ – of $\sigma_{1}$
are represented by

(B.3)
$\\!\\!((w,\\{1,3\\})\,(w,\\{2,3\\}))((w,\\{1,4\\})\,(w,\\{2,4\\}))((w,\\{1,5\\})\,(w,\\{2,5\\})).$

We want conjugacy classes all in $G(T_{1},1)$ stabilizing 1 in the
representation $T_{1}$. Conveniently $\sigma_{3}$ is an $m$-cycle. As in
§3.1.3, translate subscripts uniformly to get in $G(T_{1},1)$. Translate by -1
in (B.2), and respectively by -2, +2, +1 in the expressions (B.3)
corresponding to $w=3,4,5$. Now we can drop the notation indicating 1 is
fixed, and thereby recognize the two components correspond to the orbits of
$G(T_{1},1)$ on

(B.4)
$\begin{array}[]{rl}\bar{S}_{1}=&\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{1,5\\}\\}\\\
\bar{S}_{2}=&\\{\\{2,3\\},\\{2,4\\},\\{2,5\\},\\{3,4\\},\\{3,5\\},\\{4,5\\}\\}.\end{array}$

We find one 4-cycle (denoted by ∗ in (B.2)) and one 2-tuple for each $w$ whose
support is in $\bar{S}_{1}$ in, resp., (B.2) and (B.3). Therefore the genus
${{\text{\bf g}}}_{5,1}$ for the component corresponding to $\bar{S}_{1}$, is
given by

$2(4{+}{{\text{\bf g}}}_{5,1}{-}1)=3+3\cdot 1=6,\text{ or }{{\text{\bf
g}}}_{5,1}=0.$

Similarly, for these cycles remaining from (B.2) and (B.3), the genus
${{\text{\bf g}}}_{5,2}$ for the component corresponding to $\bar{S}_{2}$, is
given by

$2(6{+}{{\text{\bf g}}}_{5,2}{-}1)=3+1+3\cdot 2=10,\text{ or }{{\text{\bf
g}}}_{5,2}=0.$

Induct on odd $m$. Assume, for $m^{\prime}=2(\ell{-}1){+}1$, we know the
correct result and notation. Then, tack on $\\{1,2\ell\\},\\{1,2\ell{+}1\\}$
to $\bar{S}_{1}$ in (B.4), and

$\\{w,k\\},k=2\ell,2\ell{+}1,w=2,\dots,2\ell{-}1,\text{ and
}\\{2\ell,2\ell{+}1\\}\text{ to $\bar{S}_{2}$}.$

Then, the ∗ term in (B.2) becomes the $(m{-}1$)-cycle by adding
$(2,\\{2,2\ell\\})$ and $(2,\\{2,2\ell{+}1\\})$ to the end. To (B.3) add

$((w,\\{1,2\ell\\})\,(w,\\{2,2\ell\\}))\text{ and
}((w,\\{1,2\ell{+}1\\})\,(w,\\{2,2\ell{+}1\\}))$

for $w=3,\dots,2\ell{+}1$ and for $w=2\ell,2\ell{+}1$, add

$((w,\\{1,k\\})\,(w,\\{2,k\\})),\ k=3,\dots,2\ell{-}1.$

Then, by writing $w=1{+}k$, translate all terms with a given value of $w$ by
$-k$, to conclude that only one 2-cycle – corresponding to the given value of
$w$ (again drop the $w$ slot) now occupied by 1 – will end up with support in
$\bar{S}_{1}$: $(\\{1{-}k,u{-}k\\}\,\\{2{-}k,u{-}k\\})$ where $u{-}k=1$.

Now compute the genus of the degree $m{-}1$ component as for $m=5$:

$2((m{-}1){+}{{\text{\bf g}}}_{m,1}{-}1)=m{-}2+(m{-}2)\cdot 1=2(m{-}2),\text{
or }{{\text{\bf g}}}_{m,1}=0.$

Finally, do the same for the degree $\frac{(m{-}1)(m{-}2)}{2}$ component by
adding indices of all the cycles not covered by the degree $m{-}1$ component:

$\begin{array}[]{rl}&2(\frac{(m{-}1)(m{-}2)}{2}+{{\text{\bf
g}}}_{m,2}{-}1)=m(m{-}3){+}2{{\text{\bf g}}}_{m,2}=\\\
&(m{-}4)(m{-}2){+}(\frac{m{-}1}{2}{-}1){+}(m{-}3)(m{-}2).\end{array}$

From the leading terms of the expression, $\textbf{g}_{m,2}/m^{2}$ has limit
$\frac{1}{2}$.

That completes the case for odd $m$. For even $m$, the only serious adjustment
comes in the case of arranging $\tau_{2}$, which now has $m/2$ disjoint cycles
of length $m{-}1$. The cycles, however, that appear in the degree $m{-}1$
component are entirely analogous. Thus, the genus computation has the same
terms on the right side of RHas a function of $m$. ∎

######  Remark B.3 (Values for $(f^{*},g^{*})$ in Cor. LABEL:[Sngenus0).

] In the case of Davenport pairs, the $(f^{*},g^{*})$ produced by them over a
field $K$ have the same values $\mod$ any residue class field of the ring of
integers of $K$. In the proof of Prop. B.1, we considered the index of
$\tau_{2}=T_{2}(\mu)$ and found that for $m$ even (resp. odd) in its action on
the unordered distinct pairs $\\{1\leq\neq j\leq m\\}$ it consisted of
distinct $m{-}1$-cycles (resp. one $\frac{m{-}1}{2}$-cycle and the rest
$m-1$-cycles). In either case it fixes no letter of the representation, though
$T_{1}(\mu)=\sigma_{2}$ is an $m{-}1$-cycle and it fixes an integer. The
Chebotarev density theorem implies that a positive density of primes $f^{*}$
and $g^{*}$ will have different ranges.

######  Remark B.4 (Summary of the Davenport case).

[Fr73a, Lem. 2]: Using that $T_{1}$ is a doubly transitive representation for
$f^{*}$ an indecomposable polynomial, this says that ${\text{\rm
tr}}(T_{1}(\sigma))>0$ if and only if ${\text{\rm tr}}(T_{2}(\sigma))>0$, then
the representations are equivalent. By the Chebotarev theorem, the images’
multiplicity is the same for both polynomials. The major work that went into
the rest of Davenport’s problem was in two parts:

  1. (2.5a)

The above implies these pairs are conjugate over ${{\mathbb{Q}}}$, so
Davenport’s problem was affirmative over ${{\mathbb{Q}}}$.

  2. (2.5b)

But over other numbers fields, there were only finitely many degrees giving
Davenport pairs. For each of the possible degrees in Thm. 3.10, we gave the
fields over which they occurred.

The explicitness of the result (B.4b) and the way it used the classification
motivated the full set of results in the genus 0 problem.

###  B.2. Series of genus 0 components of $\tilde{{\mathcal{C}}}_{f,g}$

What is the context for the infinite series of examples that appears in §B.1
for which genus 0 components appear on $\tilde{{\mathcal{C}}}_{f^{*},.g^{*}}$?
This section gives an historical example that suggests that such series
arising from alternating group-related series may be limited by a further
constraint.

[Fr12, §7.1] goes through many of the applications that fostered classifying
separated equations $f(x)-g(y)=0$, with $f,g\in{{\mathbb{Q}}}[x]$ that have
infinitely many quasi-integral points. As previously, a direct statement was
to generalize what we refer to as Ritt’s Theorem 2 as in §1.2. That used
Siegel’s famous result on quasi-integral points on affine curves over a number
field. The more general Hilbert-Siegel problems were expanding on that. 1st
Hilbert-Siegel Problem is Prop. B.5.

######  Proposition B.5.

Assumptions:

  1. (2.6a)

$f\in{{\mathbb{Q}}}[x]$ is indecomposable.

  2. (2.6b)

Quasi-integral reducibility: $\exists$ $a\in{{\mathbb{Z}}}$ and $\infty$-ly
many $y_{0}\in{{\mathbb{Z}}}[1/a]$

for which $f(x)-y_{0}$ is reducible, but it has no zero in ${{\mathbb{Q}}}$.

Conclusion: All but finitely many such $y_{0}$ are in the values of
$g\in{{\mathbb{Q}}}(x)$ where $\tilde{{\mathcal{C}}}_{f,g}$ has $u\geq 2$
components. There are two cases.

  1. (2.7a)

Polynomial: either $g\in{{\mathbb{Q}}}[x]$; or

  2. (2.7b)

Double-degree: with $\deg(f)=m$, $\deg(g)=2m$ and a branch cycle
$\sigma_{\infty}$ for $g$ over $\infty$ has the shape $(m)(m)$.

######  Corollary B.6.

That no $g$ works in the polynomial case of (B.5) comes from the branch cycle
argument alluded to in the title of [Fr12].414141As used in the solution of
Davenport’s problem in [Fr74]. If we didn’t restrict to ${{\mathbb{Q}}}$, we
would have to include the other cases of (3.18LABEL:genus0reductb).

Conclusions of the double-degree case of (B.5):

  1. (2.8a)

Only $m=5$ gives nontrivial examples.

  2. (2.8b)

There are several Nielsen classes, but – as in the examples of §3.1 – all are
coalescings of the main Nielsen class: $G=S_{5}$, $r=4$,${\text{\rm
C}}_{2}={\text{\rm C}}_{3}$ is the class of 2-cycles, ${\text{\rm C}}_{1}$ is
the class of products of two disjoint 2-cycles and ${\text{\rm C}}_{4}$ that
of 5-cycles.

  3. (2.8c)

[DeFr99, Thm. 1.2] concludes: Using the natural moduli space, for any
fractional ideal of ${{\mathbb{Z}}}$, the set of such $f$ produces solutions
dense in the points of the Hurwitz space.

This example surprised the authors of [DeFr99, Thm. 1.1 and 1.2] for two
reasons: Only one value of $m$ produced these. Yet, that one did in great
abundance. [Fr99, Exp. 6.3] has Guralnick’s conjecture ([Fr12, §7.1.4]; what
monodromy groups can arise often and the precise version of (3.2.3) for what
would be the exceptional genus 0 monodromy (over ${{\mathbb{C}}}$). Now it is
a theorem. In these lists, you see several related to $A_{n}$, including the
permutation representation of the cover acting on distinct, unordered pairs of
integers. That is the case above.

Thus, even Guralnick’s strong formulation of the genus 0 problem is
insufficient. [Fr86, §3] has the extra group theory for showing (B.5) holds
for the groups in Guralnick’s list for the Hilbert-Siegel problem when the
double-degree condition of (B.5) holds.

[Mu96] has results for more general versions of Hilbert Siegel Problems as in
[Fr86, §4]. He still relied on [Fr86, §2 and §3].

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