# Rational pullbacks of Galois covers

Pierre Dèbes<EMAIL_ADDRESS>,  Joachim König
<EMAIL_ADDRESS>,  François Legrand<EMAIL_ADDRESS>and
Danny Neftin<EMAIL_ADDRESS>Laboratoire de Mathématiques Paul
Painlevé, Université de Lille, 59655 Villeneuve d’Ascq Cedex, France
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon
34141, South Korea  Institut für Algebra, Fachrichtung Mathematik, TU Dresden,
01062 Dresden, Germany  Department of Mathematics, Technion, Israel Institute
of Technology, Haifa 32000, Israel

###### Abstract.

The finite subgroups of ${\rm PGL}_{2}(\mathbb{C})$ are shown to be the only
finite groups $G$ with the following property: for some bound $r_{0}$, all
Galois covers $X\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ of group $G$ can be
obtained by pulling back those with at most $r_{0}$ branch points along all
non-constant rational maps
$\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$. For
$G\subset{\rm PGL}_{2}(\mathbb{C})$, it is in fact enough to pull back one
well-chosen cover with at most $3$ branch points. A worthwhile consequence of
the converse for inverse Galois theory is that, for $G\not\subset{\rm
PGL}_{2}(\mathbb{C})$, letting the branch point number grow always provides
truly new realizations $F/\mathbb{C}(T)$ of $G$. Our approach also leads to
some improvements of results of Buhler-Reichstein about generic polynomials
with one parameter. For example, if $G$ is neither cyclic nor odd dihedral, no
polynomial $P\in\mathbb{C}[T,Y]$ of Galois group $G$ can parametrize, via
specialization of $T$, all Galois extensions $E/k$ of group $G$; here $k$ is
some specific base field, not depending on $G$, viz. $k=\mathbb{C}((V))(U)$. A
final application, related to an old problem of Schinzel, provides, subject to
the Birch & Swinnerton-Dyer conjecture, a $1$-parameter family of affine
curves over some number field, all with a rational point, but with no rational
generic point.

###### Key words and phrases:

Galois extensions, covers, specialization, rational pullback, inverse Galois
theory, parametric and generic extensions

###### 2010 Mathematics Subject Classification:

Primary 12F12, 11R58, 14E20; Sec. 14E22, 12E30, 11Gxx

Acknowledgment. This work was supported in part by the Labex CEMPI
(ANR-11-LABX-0007-01) and ISF grants No. 577/15 and No. 696/13.

##  1\. Introduction

Specialization of field extensions $F/k(T)$ at points $T=t_{0}\in k$ is a
central tool in Galois theory, going back to Hilbert111The notion of
“specialization” or “specialized extension” is precisely recalled in §2.1..
Rational pullback is another one, in fact a variant.

Given a finite connected cover222For pullback, we rather use the cover
viewpoint rather than the function field equivalent one.
$f:X\rightarrow\mathbb{P}^{1}_{k}$ (over some field $k$), by a rational
pullback of $f$ we mean a cover
$f_{T_{0}}:X_{T_{0}}\rightarrow\mathbb{P}^{1}_{k}$ obtained by pulling back
$f$ along some non-constant rational map
$T_{0}:\mathbb{P}^{1}_{k}\rightarrow\mathbb{P}^{1}_{k}$. If $f$ is given by a
polynomial equation $P(t,y)=0$ (with $f$ corresponding to the $t$-coordinate
projection) and $T_{0}$ is viewed as a rational function $T_{0}(U)\in k(U)$,
then an equation for the pullback $f_{T_{0}}$ is merely $P(T_{0}(u),y)=0$.

As we recall below (§2.1), for “many” $T_{0}$, the cover $f_{T_{0}}$ remains
connected; hence if $f$ is Galois of group $G$, then so is $f_{T_{0}}$, which
in addition remains defined over $k$.

Rational pullback is thus a natural tool to create Galois covers of group $G$
over some field $k$ if one is known. Finding covers $g$ that are not rational
pullbacks of some given covers $f$ may be even more important for inverse
Galois theory: if none of these $f$ are defined over $k$, those covers $g$,
rather than the pullbacks $f_{T_{0}}$, are the best chance to find some and so
to realize $G$ as a regular Galois group over $k$ 333i.e., as the automorphism
group of a $k$-Galois cover $X\rightarrow\mathbb{P}^{1}$ with $X$
geometrically irreducible.. Theorems 1.1 and 1.2, presented in §1.1 below, fit
in this theme.

Another question relating rational pullbacks and specializations arises. The
set of $k$-specializations of $f$ (essentially the splitting fields of
polynomials $P(t_{0},Y)$, $t_{0}\in k$) clearly contains that of $f_{T_{0}}$
(same but with polynomials $P(T_{0}(u_{0}),Y)$, $u_{0}\in k$). The challenge
is the converse: are rational pullbacks $f_{T_{0}}$ of $f$ the only covers
with this property, in which case, being a rational pullback could be read off
from the specialization set containment? Our contribution to this question is
described at the end of §1.2.

A related theme, presented in §1.2, focuses on the set of $L$-specializations
of $f$ – same as above but for the polynomials $P(t_{0},Y)$ with $t_{0}$
ranging here over a given overfield $L$ of $k$. Can this set contain every
Galois extension of $L$ of group $G$? With $L=k(U)$, the question takes us
back to the rational pullback territory. With $L$ varying over all overfields
of $k$, it refers to the notion of generic extension (with one parameter).
Theorems 1.4 and 1.5 improve on results of Buhler-Reichstein [BR97] in this
context.

###  1.1. Regular parametricity

Assume first that $k=\mathbb{C}$. Consider the situation that all Galois
covers $X\rightarrow\mathbb{P}^{1}$ of given group $G$ can be obtained from a
proper subset of them by rational pullback; say then that the subset is
regularly parametric444The term “regularly” will be fully justified with the
general definition of “$k$-regular parametricity” for which the base field $k$
is not necessarily algebraically closed (see Definition 2.2). We also mean to
distinguish “regular parametricity” from the arithmetic notion of
“parametricity” (recalled later).. For some groups, a single cover $f$ may
suffice: for example, the degree $2$ cover
$\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ sending $z$ to $z^{2}$ is regularly
parametric. Such situations are however exceptional: for “general” groups, an
opposite conclusion holds.

######  Theorem 1.1.

The finite subgroups $G$ of ${\rm PGL}_{2}(\mathbb{C})$:
$\mathbb{Z}/n\mathbb{Z}$ for $n\geq 1$, $D_{n}$ for $n\geq 2$, $S_{4}$,
$A_{4}$, $A_{5}$, are exactly those finite groups which have a regularly
parametric cover $f:X\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$. Furthermore, if
$G\not\subset{\rm PGL}_{2}(\mathbb{C})$, not only one cover $f$ is not enough
but even the set of all Galois covers
$X\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ of group $G$ and with at most
$r_{0}$ branch points is not regularly parametric, for any $r_{0}\geq 0$.

Hence, for $G\not\subset{\rm PGL}_{2}(\mathbb{C})$, letting the branch point
number grow provides an endless source of “new” Galois covers of group $G$,
i.e., not mere rational pullbacks of some with a bounded branch point number,
and so truly new candidates to be defined over $\mathbb{Q}$.

Covers are understood here as algebraic branched covers, but Theorem 1.1 and
the following Theorem 1.2 may already be of interest at the topological level.

Both directions in the first statement of Theorem 1.1 are non-trivial. The one
showing that finite subgroups of ${\rm PGL}_{2}(\mathbb{C})$ have a regularly
parametric cover over $\mathbb{C}$ (with at most $3$ branch points) was proved
in [Dèb18, Corollary 2.5] as a consequence of the twisting lemma and Tsen’s
theorem. The part about “general” groups, those not contained in ${\rm
PGL}_{2}(\mathbb{C})$, is a new result of this paper. It solves in particular
Problem 2.14 from [Dèb18].

Here is a more precise version. Given an integer $r\geq 0$ and an $r$-tuple
${\bf C}$ of non-trivial conjugacy classes of $G$, denote the stack of all
Galois covers of group $G$ with $r$ branch points by ${\tensf H}_{G,r}$ and
the stack of those with ramification type $(r,{\bf C})$ by ${\tensf
H}_{G,r}({\bf C})$ (see §2.2); these are the so-called Hurwitz stacks. From
Theorem 1.1, if $G\not\subset{\rm PGL}_{2}(\mathbb{C})$,

${\tensf H}_{G,\leq r_{0}}=\bigcup_{r\leq r_{0}}{\tensf H}_{G,r}$

is never regularly parametric ($r_{0}\geq 1$). More precisely, we have the
following.

Theorem 1.1 (continued). Let $k$ be an algebraically closed field of
characteristic $0$ and let $G$ be a finite group.

(a) Assume $G\not\subset{\rm PGL}_{2}(\mathbb{C})$. Given an integer
$r_{0}\geq 0$, if $R$ is suitably large (depending on $r_{0}$) and $(R,{\bf
C})$ is a ramification type of $G$ such that ${\tensf H}_{G,R}({\bf
C})\not=\emptyset$, then not all $k$-covers in ${\tensf H}_{G,R}({\bf C})$ are
rational pullbacks of $k$-covers in ${\tensf H}_{G,\leq r_{0}}$.

(b) Assume $G$ has at least $5$ maximal non-conjugate cyclic subgroups (in
particular, $G\not\subset{\rm PGL}_{2}(\mathbb{C})$). Given $r_{0}\geq 0$, for
every suitably large even integer $R$, there is a non-empty Hurwitz stack
${\tensf H}_{G,R}({\bf C})$ such that no $k$-cover in ${\tensf H}_{G,R}({\bf
C})$ is a rational pullback of some $k$-cover in ${\tensf H}_{G,\leq r_{0}}$.

Subgroups of ${\rm PGL}_{2}(\mathbb{C})$ have at most $3$ maximal non-
conjugate cyclic subgroups. Other groups have exactly $3$: the quaternion
group $\mathbb{H}_{8}$ or more generally dicyclic groups ${\rm DC}_{n}$ of
order $4n$ ($n\geq 2$) (which include generalized quaternion groups ${\rm
DC}_{2^{k-1}}$ $k\geq 2$), groups $G={\rm SL}_{2}(\mathbb{F}_{q})$ with $q$ a
prime power, etc. For these groups, conclusion from (a) holds but that from
(b) is unclear. Replacing $5$ by $4$ in Theorem 1.1(b) seems feasible in light
of the techniques of §4, but hard and technical; for the sake of brevity, we
avoid this slight improvement. Groups with $4$ maximal non-conjugate cyclic
subgroups (for which conclusion from (b) might also hold) include
$\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$,
$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$, $A_{6}$, etc.

The assumption in Theorem 1.1 that $k$ is algebraically closed can be weakened
to only assume that $k$ is ample, or even arbitrary (of characteristic $0$) in
some situations; see Theorem 3.7. Recall that a field $k$ is ample if for
every smooth $k$-curve $C$, either $C(k)=\emptyset$ or $C(k)$ is infinite.
Ample fields include separably closed fields, Henselian fields, fields
$\mathbb{Q}^{{\rm tot}\mathbb{R}}$, $\mathbb{Q}^{{\rm tot}p}$ of totally real,
totally $p$-adic algebraic numbers and their variants.

If instead of the sets ${\tensf H}_{G,\leq r}$, one considers the smaller
subsets ${\tensf H}_{G,r}({\bf C})$, one obtains even more striking
conclusions.

######  Theorem 1.2.

Let $G$ be a finite group not contained in
$\operatorname{PGL}_{2}(\mathbb{C})$ and let $(R,\bf C)$ be a ramification
type for $G$. Then there exists a ramification type $(R+1,\bf D)$ for $G$ such
that ${\tensf H}_{G,R+1}(\bf D)\not=\emptyset$ and no cover in ${\tensf
H}_{G,R+1}(\bf D)$ is a pullback from ${\tensf H}_{G,R}(\bf C)$.

These results have the following consequence. To ease notation, denote the set
of all Galois covers $X\rightarrow\mathbb{P}^{1}$ of group $G$ by ${\tensf
H}_{G}$ (over the algebraically closed field $k$). Say that a finite group $G$
has the Beckmann-Black regular lifting property if for any two Galois covers
$g_{1},g_{2}\in{\tensf H}_{G}$, there is a Galois cover
$f:X\rightarrow\mathbb{P}^{1}\in{\tensf H}_{G}$ and two non-constant rational
maps $T_{01},T_{02}:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ such that
$g_{i}=f_{T_{0i}}$ ($i=1,2$).

######  Corollary 1.3.

The finite subgroups of ${\rm PGL}_{2}(\mathbb{C})$ are exactly those finite
groups for which the Beckmann-Black regular lifting property holds.

The proof combines Theorem 1.1 above with Theorem 2.1 from [Dèb18]. It is
given in §2.3.2 where the former is recalled.

Our lifting property is a geometric variant of the Beckmann-Black arithmetic
lifting property for which $k$ is a number field and the $k$-Galois cover
$f:X\rightarrow\mathbb{P}^{1}\in{\tensf H}_{G}$ is requested to specialize to
some given Galois extensions $E_{i}/k$, $i=1,\ldots,N$ of group $G$ at some
points $t_{0i}\in\mathbb{P}^{1}(k)$. This property is already interesting with
$N=1$ as it supports Hilbert’s strategy to solve the Inverse Galois problem by
first realizing a $k$-Galois cover $f:X\rightarrow\mathbb{P}^{1}$ of given
group. It is only known to hold for some groups: abelian, odd dihedral555i.e.,
a dihedral group of order $2n$ with $n>1$ odd., $S_{n}$, $A_{n}$ and a few
others, and has no known counterexample. Our geometric variant is obvious for
$N=1$ (as $f_{T_{0}}=f$ for $T_{0}(U)=U$) and Corollary 1.3 shows that it
fails for $N\geq 2$ if $G\not\subset{\rm PGL}_{2}(\mathbb{C})$.

###  1.2. Regular parametricity and genericity

Before regular parametricity, a notion of mere parametricity was pre-existing:
given a field $k$ of characteristic zero and an overfield $L\supset k$, a
Galois extension $F/k(T)$ of group $G$ that is $k$-regular666that is, such
that $k$ is algebraically closed in $F$. is $L$-parametric if

($*$) every Galois extension $E/L$ of group $G$ is a specialization of the
extension $FL/L(T)$ at some point $t_{0}\in\mathbb{P}^{1}(L)$.

A rational pullback $f_{T_{0}}:X_{T_{0}}\rightarrow\mathbb{P}^{1}_{k}$ can in
fact be seen as a specialization and regular parametricity as some variant of
parametricity. Namely, view $f:X\rightarrow\mathbb{P}^{1}_{k}$ as its function
field extension $k(X)/k(T)$ and
$T_{0}:\mathbb{P}^{1}_{k}\rightarrow\mathbb{P}^{1}_{k}$ as an element
$T_{0}(U)$ of $k(U)$ (with $U$ some new indeterminate). Then, if $f_{T_{0}}$
is connected, the function field extension $k(X_{T_{0}})/k(U)$ of the pullback
$f_{T_{0}}:X_{T_{0}}\rightarrow\mathbb{P}^{1}_{k}$ is the specialization at
$T=T_{0}(U)\in k(U)$ of the function field extension $k(U)(X)/k(U)(T)$ of
$f\otimes_{k}k(U)$. Hence, the extension $k(X)/k(T)$ being $k(U)$-parametric
implies that the cover $f:X\rightarrow\mathbb{P}^{1}_{k}$ is $k$-regularly
parametric777The equivalence does not hold a priori as
$f:X\rightarrow\mathbb{P}^{1}_{k}$ being $k$-regularly parametric only implies
that the Galois extensions $E/k(U)$ (with the corresponding Galois group) that
are $k$-regular be specializations of $k(U)(X)/k(U)(T)$ (see Remark 2.3)..

In general, the $L$-parametricity property (condition ($*$)) is not preserved
by base change over bigger fields. A classical property – the genericity
property – in fact requests that the $L$-parametricity property holds for
every base change $L/k$. This property is much stronger. Finite groups
$G\not\subset{\rm PGL}_{2}(k)$ are automatically ruled out: they do not have
any generic extension $F/k(T)$ since the Noether invariant extension (of group
$G$)

$E/L=k({\bf T})/k({\bf T})^{G}$, with ${\bf T}=(T_{1},\ldots,T_{|G|})$,

can be a specialization of $FL/L(T)$ only if $F$ is of genus $0$, and so
$G\subset{\rm PGL}_{2}(k)$ (a Lüroth like consequence of $k({\bf T})$ being
purely transcendental; see [JLY02, Proposition 8.1.4]).

The Noether extension can no longer be used to disprove the weaker
$\overline{k}(U)$-parametricity property of a given $k$-regular Galois
extension $F/k(T)$ of group $G$ as it is then a Galois extension $E$ of
$L=\overline{k}(U)$ (instead of $L=k({\bf T})^{G}$) of group $G$, not being a
specialization of $FL/L(T)$, that should be produced. This is what Theorem 1.1
does for finite groups $G\not\subset{\rm PGL}_{2}(\mathbb{C})$: they have no
regularly parametric cover $f:X\rightarrow\mathbb{P}^{1}_{\overline{k}}$, a
fortiori no $\overline{k}(U)$-parametric extension $F/k(T)$.

Our approach also clarifies the question for finite subgroups of ${\rm
PGL}_{2}(\mathbb{C})$. Theorem 1.4 below recapitulates the whole situation,
over a non-necessarily algebraically closed field:

######  Theorem 1.4.

Let $G$ be a finite group, $k$ a field of characteristic zero, $F/k(T)$ a
$k$-regular Galois extension of group $G$, and $U,V$ two indeterminates.

(a) If $G\not\subset{\rm PGL}_{2}(\mathbb{C})$, then, $F/k(T)$ is neither
$\overline{k}(U)$-parametric nor $\overline{k}((V))(U)$-parametric.

(b) If $G$ is neither cyclic nor odd dihedral, then, $F/k(T)$ is not
$\overline{k}((V))(U)$-parametric.

We refer to Theorem 5.1 for a more general version which also covers most of
the $k$-regular Galois extensions of $k(T)$ with either cyclic or odd dihedral
Galois group.

Statement (b) applies in particular to these finite subgroups of ${\rm
PGL}_{2}(\mathbb{C})$: $S_{4}$, $A_{4}$, $A_{5}$, and even dihedral groups.
Theorem 1.4 implies that these groups, and, more generally, all neither cyclic
nor odd dihedral finite groups, do not have any generic extension $F/k(T)$.
This was known from the essential dimension theory of Buhler-Reichstein
[BR97]; see also [JLY02]888More precisely, it was already known that these
groups have no one parameter generic polynomial over $k$ (a polynomial version
of the notion of generic extension $F/k(T)$). We refer to Proposition B.2 for
a clarifying result about the equivalence between these two notions.. Our
alternate approach provides stronger conclusions:

(a) It explicitly produces a base change $L/k$ over which the
$L$-parametricity property fails: one can take $L=\overline{k}(U)$ or
$L=\overline{k}((V))(U)$. In comparison, the base change $k({\bf T})^{G}/k$
(coming from the Noether extension) is not purely transcendental in general,
is of high transcendence degree, and depends on $G$.

(b) Regarding generic extensions $F/k(T)$, our approach not only gives the
exact list of groups having one (as [BR97] did), but also the list of
corresponding extensions.

Namely, we have the following result, in which we take $k=\mathbb{Q}$ for
simplicity:

######  Theorem 1.5.

Let $G$ be a non-trivial finite group and $F/\mathbb{Q}(T)$ a
$\mathbb{Q}$-regular Galois extension of group $G$. The following three
conditions are equivalent:

(a) $F/\mathbb{Q}(T)$ is generic,

(b) $F/\mathbb{Q}(T)$ is $\overline{\mathbb{Q}}((V))(U)$-parametric and
$\mathbb{Q}(U)$-parametric,

(c) one of the following three conditions holds:

(i) $G=\mathbb{Z}/2\mathbb{Z}$ and $F/\mathbb{Q}(T)$ has two branch points,
which are $\mathbb{Q}$-rational,

(ii) $G=\mathbb{Z}/3\mathbb{Z}$ and $F/\mathbb{Q}(T)$ has two branch points,

(iii) $G=S_{3}$ and $F/\mathbb{Q}(T)$ has three branch points, which are
$\mathbb{Q}$-rational.

We refer to Corollary 5.4 for the case of an arbitrary field of characteristic
zero.

A final application relates to the following old problem of Schinzel,
presented in [Sch00, Chapter 5, §5.1, Problem 1]:

######  Question 1.6.

Let $k$ be a number field and $P\in\mathbb{C}[U,T,Y]$ a polynomial such that,
for all but finitely many $u_{0}\in\mathbb{Z}$, the polynomial $P(u_{0},T,Y)$
has a zero in $k^{2}$. Does P, viewed as a polynomial in T and Y, have a zero
in $k(U)^{2}$?

While the answer is “Yes” when $k=\mathbb{Q}$ and $P$ is of degree at most $2$
in $(T,Y)$ (see [DLS66, Theorem 2]), the answer seems to be negative in
general. However, only conjectural counter-examples, subject to a conjecture
of Selmer, are known. We refer to [Sch00, pp. 318-319] for more details and
references.

In [Dèb18], a close variant of Question 1.6, called (WH), is introduced and
used as a Working Hypothesis; it is explicitly recalled in §5.4. By [Dèb18,
Proposition 2.17(a)], under (WH), we have a full positive answer to one of our
motivating questions:

######  Question 1.7.

Let $k$ be a number field and let $f_{1}:X_{1}\rightarrow\mathbb{P}^{1}_{k}$
and $f_{2}:X_{2}\rightarrow\mathbb{P}^{1}_{k}$ be two $k$-regular Galois
covers with the same Galois group such that the set of specializations of
$f_{2}$ is contained in that of $f_{1}$. Is $f_{2}$ a rational pullback of
$f_{1}$, after base change $\mathbb{C}/k$?

Here, subject to the Birch and Swinnerton-Dyer conjecture, we provide a
counterexample of genus $1$ over $k=\mathbb{Q}(\sqrt{-3},\sqrt[3]{11})$ both
to Question 1.6 and Question 1.7, and so to (WH) too; see Theorem 5.8. This
counterexample gives evidence that the genus $1$ case is exceptional. It is
plausible that the answer to Question 1.7 is positive for covers of genus at
least $2$.

The paper is organized as follows. §2 is a preliminary section providing the
basic notation and terminology together with some general prerequisites. §3
and §4 are concerned with our regular parametricity results, while §5 is
devoted to those around the connection between regular parametricity and
genericity. In particular, Theorem 1.1 is proved in §3, Theorem 1.2 in §4, and
Theorems 1.4 and 1.5 in §5.

##  2\. Notation, terminology, and prerequisites

###  2.1. Basic terminology (for more details, see [DD97b] and [DL13])

The base field $k$ is always assumed to be of characteristic $0$. We also fix
a big algebraically closed field containing the complex field $\mathbb{C}$ and
the indeterminates that will be used, and in which every field compositum
should be understood.

Given a field $k$, a finite extension $F/k(T)$ is said to be $k$-regular if
$F\cap\overline{k}=k$. We make no distinction between a $k$-regular extension
$F/k(T)$ and the associated $k$-regular cover
$f:X\rightarrow\mathbb{P}^{1}_{k}$: $f$ is the normalization of
$\mathbb{P}^{1}_{k}$ in $F$ and $F$ is the function field $k(X)$ of $X$. In
particular, we make no distinction between rational maps
$T_{0}:\mathbb{P}^{1}_{k}\rightarrow\mathbb{P}^{1}_{k}$ and the rational
functions $T_{0}\in k(U)$.

We also use affine equations: we mean the irreducible polynomial $P\in k[T,Y]$
of a primitive element of $F/k(T)$, integral over $k[T]$. We say defining
equation if the primitive element is not necessarily integral over $k[T]$;
then, $P\in k(T)[Y]$.

By group and branch point set of a $k$-regular extension $F/k(T)$, we mean
those of the extension $F\overline{k}/\overline{k}(T)$: the group of
$F\overline{k}/\overline{k}(T)$ is the Galois group of its Galois closure and
the branch point set of $F\overline{k}/\overline{k}(T)$ is the (finite) set of
points $t\in\mathbb{P}^{1}(\overline{k})$ such that the associated discrete
valuations are ramified in $F\overline{k}/\overline{k}(T)$.

The field $k$ being of characteristic $0$, we also have the inertia canonical
invariant ${\bf C}$ of the $k$-regular extension $F/k(T)$, defined as follows.
If ${\bf t}=\\{t_{1},\ldots,t_{r}\\}$ is the branch point set of $f$, then,
${\bf C}$ is an $r$-tuple $(C_{1},\ldots,C_{r})$ of conjugacy classes of the
group $G$ of $F/k(T)$: for $i=1,\ldots,r$, the class $C_{i}$ is the conjugacy
class of the distinguished999“distinguished” means that these generators
correspond to the $e_{i}$th root $e^{2i\pi/e_{i}}$ of $1$ in the canonical
isomorphism $I_{\mathfrak{P}}\rightarrow\mu_{e_{i}}=\langle
e^{2i\pi/e_{i}}\rangle$. generators of the inertia groups $I_{\mathfrak{P}}$
above $t_{i}$ in the Galois closure $\widehat{F}/k(T)$ of $F/k(T)$. The couple
$(r,{\bf C})$ is called the ramification type of $F/k(T)$. More generally,
given a finite group $G$, we say that a couple $(r,{\bf C})$ is a ramification
type for $G$ over $k$ if it is the ramification type of at least one
$k$-regular Galois extension $F/k(T)$.

We also use the notation ${\bf e}=(e_{1},\ldots,e_{r})$ for the $r$-tuple with
$i$th entry the ramification index $e_{i}=|I_{\mathfrak{P}}|$ of primes above
$t_{i}$; $e_{i}$ is also the order of elements of $C_{i}$, $i=1,\ldots,r$.

We say that two $k$-regular extensions $F/k(T)$ and $L/k(T)$ are isomorphic if
there is a field isomorphism $F\rightarrow L$ that restricts to an
automorphism $\chi:k(T)\rightarrow k(T)$ equal to the identity on $k$, and
that they are k(T)-isomorphic if, in addition, $\chi$ is the identity on
$k(T)$.

Given a Galois extension $F/k(T)$ and $t_{0}\in{\mathbb{P}}^{1}(k)$, the
specialized extension $F_{t_{0}}/k$ of $F/k(T)$ at $t_{0}$ is the Galois
extension defined as follows. For $t_{0}\in\mathbb{A}^{1}(k)$, consider the
integral closure $B$ of $A=k[T]$ in $F$. Then, $F_{t_{0}}/k$ is the residue
extension of an arbitrary prime ideal of $B$ above the ideal $\langle
T-t_{0}\rangle$. For $t_{0}=\infty$, do the same but with $A=k[1/T]$ replacing
$A=k[T]$ and $\langle 1/T\rangle$ replacing $\langle T-t_{0}\rangle$.

If $P\in k[T,Y]$ is an affine equation of $F/k(T)$ and $\Delta_{P}\in k[T]$ is
its discriminant w.r.t. $Y$, then, for every $t_{0}\in k$ such that
$\Delta_{P}(t_{0})\not=0$, $t_{0}$ is not a branch point of $F/k(T)$ and the
specialized field $F_{t_{0}}$ is the splitting field over $k$ of $P(t_{0},Y)$.

The following easy lemma will be used on several occasions in the sequel:

######  Lemma 2.1.

Let $L\supset k$ be an overfield and $t_{0}\in\mathbb{P}^{1}(k)$. Then, one
has $(FL)_{t_{0}}=F_{t_{0}}L$.

###### Proof.

Without loss of generality, we may assume $t_{0}\not=\infty$. Denote the
integral closure of $k[T]$ in $F$ by $B_{k}$. Pick $b_{1},\dots,b_{|G|}$ in
$B_{k}$ such that

$B_{k}=k[T]b_{1}\oplus\dots\oplus k[T]b_{|G|}.$

As $k$ has characteristic zero, the extension $L/k$ is separable (in the sense
of non-necessarily algebraic extensions; see, e.g., [Lan02, Chapter VIII,
§4]). Then, by, e.g., [FJ08, Proposition 3.4.2], the integral closure $B_{L}$
of $L[T]$ in $FL$ satisfies

$B_{L}=L[T]b_{1}\oplus\dots\oplus L[T]b_{{|G|}}.$

Let $\mathfrak{P}_{L}$ be a prime ideal of $B_{L}$ lying over that of $L[T]$
generated by $T-t_{0}$. Then, the restriction
$\mathfrak{P}_{k}=\mathfrak{P}_{L}\cap B_{k}$ of $\mathfrak{P}_{L}$ to $B_{k}$
lies over the prime ideal of $k[T]$ generated by $T-t_{0}$. One then has

$(FL)_{t_{0}}=B_{L}/\mathfrak{P}_{L}=L(\overline{b_{1}},\dots,\overline{b_{|G|}}),$

where $\overline{b_{1}},\dots,\overline{b_{|G|}}$ denote the reductions modulo
$\mathfrak{P}_{L}$ of ${b_{1}},\dots,{b_{|G|}}$, respectively. But these
reductions modulo $\mathfrak{P}_{L}$ are the reductions
$\underline{b_{1}},\dots,\underline{b_{|G|}}$ modulo $\mathfrak{P}_{k}$ of
$b_{1},\dots,b_{|G|}$, respectively. Hence, one has

$(FL)_{t_{0}}=k(\underline{b_{1}},\dots,\underline{b_{|G|}})L=F_{t_{0}}L,$

as needed for the lemma. ∎

If $f:X\rightarrow\mathbb{P}^{1}_{k}$ is a $k$-regular Galois cover of group
$G$ and $T_{0}\in k(U)\setminus k$, the pullback $f_{T_{0}}$ of $f$ along
$T_{0}$ is the $k$-cover $f_{T_{0}}:X_{T_{0}}\rightarrow\mathbb{P}^{1}_{k}$
obtained by pulling back $f$ along the rational map
$T_{0}:\mathbb{P}^{1}_{k}\rightarrow\mathbb{P}^{1}_{k}$. If $X_{T_{0}}$ is
connected, then, the pullback $f_{T_{0}}$ can be viewed as a specialization:

$k(X_{T_{0}})/k(U)=(k(U)(X))_{T_{0}}/k(U)$.

In this case, if $P\in k[T,Y]$ is an affine equation of $f$, then,
$\Delta_{P}(T_{0})\not=0$ and so $P(T_{0}(U),Y)$ is a defining equation of the
pullback $f_{T_{0}}$.

The subset $H_{P,k(U)}$ of $k(U)$ of all $T_{0}$ such that $P(T_{0}(U),Y)$ is
irreducible in $k(U)[Y]$ is a Hilbert subset of the field $k(U)$. For
$T_{0}\in H_{P,k(U)}$, the cover $f_{T_{0}}$ is connected, hence Galois of
group $G$. The field $k(U)$ being Hilbertian, the subset $H_{P,k(U)}$ is “big”
in various senses: it is infinite, it is dense for the Strong Approximation
Topology, etc.

###  2.2. More specific notation and terminology

Given a finite group $G$, an integer $r\geq 1$, an $r$-tuple ${\bf C}$ of non-
trivial conjugacy classes of $G$, and a field $k$ (of characteristic zero), we
use the following notation:

\- ${\tensf R}_{G}(k)$: set of all Galois extensions $E/k$ of group $G$,

\- ${\tensf H}_{G}(k)$: set of all $k$-regular Galois extensions $F/k(T)$ of
group $G$ or, equivalently, of all $k$-regular Galois covers
$f:X\rightarrow\mathbb{P}^{1}_{k}$ of group $G$,

\- ${\tensf H}_{G,r}(k)$: subset of ${\tensf H}_{G}(k)$ defined by the extra
condition that the branch point number is $r$,

\- ${\tensf H}_{G,r}({\bf C})(k)$: subset of ${\tensf H}_{G,r}(k)$ defined by
the extra condition that the inertia canonical invariant is ${\bf C}$.

One can more generally define ${\tensf H}_{G,r}$ and ${\tensf H}_{G,r}({\bf
C})$ as stacks; ${\tensf H}_{G,r}(k)$ and ${\tensf H}_{G,r}({\bf C})(k)$ can
then be viewed as the set of $k$-rational points on these stacks.

\- For ${\tensf H}\subset{\tensf H}_{G}(k)$, we define

${\rm SP}({\tensf H})=\left\\{F_{t_{0}}/k\hskip 2.0pt\left|\hskip
2.84526pt\begin{matrix}F/k(T)\in{\tensf H}\hfill\cr
t_{0}\in\mathbb{P}^{1}(k)\hfill\cr\end{matrix}\right.\right\\}$

and

${\rm PB}({\tensf H})=\left\\{f_{T_{0}}\hskip 2.0pt\left|\hskip
2.84526pt\begin{matrix}f\in{\tensf H}\hfill\cr T_{0}\in k(U)\setminus
k\hfill\cr\end{matrix}\right.\right\\}.$

######  Definition 2.2.

Let ${\tensf H}$ be a subset of ${\tensf H}_{G}(k)$.

(a) ${\tensf H}$ is said to be $k$-parametric if ${\rm SP}({\tensf
H})\supset{\tensf R}_{G}(k)$.

(b) More generally, given a field extension $L/k$, ${\tensf H}$ is said to be
$L$-parametric if the set ${\tensf H}_{L}=\\{FL/L(T)\hskip 2.0pt|\hskip
2.0ptF/k(T)\in{\tensf H}\\}$ is $L$-parametric.

(c) ${\tensf H}$ is said to be generic if ${\tensf H}$ is $L$-parametric for
every field extension $L/k$.

(d) ${\tensf H}$ is said to be $k$-regularly parametric if ${\rm PB}({\tensf
H})\supset{\tensf H}_{G}(k)$.

(e) An extension $F/k(T)\in{\tensf H}_{G}(k)$ is said to be $L$-parametric for
a given extension $L/k$ (resp., is said to be generic) if
${\tensf{H}}=\\{F/k(T)\\}$ is $L$-parametric (resp., is generic).

######  Remark 2.3.

Given a subset ${\tensf H}$ of ${\tensf H}_{G}(k)$, we have the following
implications:

${\tensf H}$ generic $\Rightarrow$ ${\tensf H}$ $k(U)$-parametric
$\Rightarrow$ ${\tensf H}$ $k$-regularly parametric.

Indeed, only the latter requires explanations. Let $E/k(U)$ be a $k$-regular
Galois extension of group $G$ and assume without loss that $G$ is not trivial.
If $\tensf{H}$ is $k(U)$-parametric, there are $F/k(T)\in\tensf{H}$ and
$T_{0}\in\mathbb{P}^{1}(k(U))$ such that $E=(Fk(U))_{T_{0}}$. If $T_{0}$ is in
$\mathbb{P}^{1}(k)$, then, by Lemma 2.1, we get $E=F_{T_{0}}k(U).$ Hence, as
$E/k(U)$ is $k$-regular, the field $F_{T_{0}}$ is equal to $k$, that is,
$E=k(U)$, which cannot happen. Consequently, $T_{0}$ is not constant and
$E/k(U)$ is the pullback of $F/k(T)$ along $T_{0}$, thus showing that
$\tensf{H}$ is $k$-regularly parametric.

Compared to the right-hand side condition, the middle one also requires that
the Galois extensions of $k(U)$ of group $G$ which are not $k$-regular are
parametrized (in particular, the constant extensions $E(U)/k(U)$ with
$E/k\in{\tensf R}_{G}(k)$). However, these two conditions are equivalent if
$k$ is algebraically closed.

###  2.3. Prerequisites

The field $k$ is here algebraically closed and of characteristic $0$.

####  2.3.1. Riemann Existence Theorem

This fundamental tool of the theory of covers of $\mathbb{P}^{1}$ allows
turning questions about covers into combinatorics and group theory
considerations.

Riemann Existence Theorem (RET). Given a finite group $G$, an integer $r\geq
2$, a subset ${\bf t}$ of $\mathbb{P}^{1}(k)$ of $r$ points, and an $r$-tuple
${\bf C}=(C_{1},\ldots,C_{r})$ of non-trivial conjugacy classes of $G$, there
is a Galois extension $F/k(T)$ of group $G$, branch point set ${\bf t}$, and
inertia canonical invariant ${\bf C}$ iff there exists
$(g_{1},\ldots,g_{r})\in C_{1}\times\cdots\times C_{r}$ such that $g_{1}\cdots
g_{r}=1$ and $\langle g_{1},\ldots,g_{r}\rangle=G$. Furthermore, the number of
such extensions $F/k(T)$ (in a fixed algebraic closure of ${k(T)}$) equals the
number of $r$-tuples $(g_{1},\ldots,g_{r})$ as above, counted modulo
componentwise conjugation by an element of $G$.

The RET shows that a couple $(r,{\bf C})$ is a ramification type for $G$ over
$k$ if the set, traditionally called the Nielsen class, of all
$(g_{1},\ldots,g_{r})\in C_{1}\times\cdots\times C_{r}$ such that $g_{1}\cdots
g_{r}=1$ and $\langle g_{1},\ldots,g_{r}\rangle=G$ is non-empty. We shall use
the RET to construct Galois extensions of given group $G$ and with some
special ramification type.

####  2.3.2. Bounds for the branch point number and the genus of pulled-back
covers

######  Theorem 2.4.

Let $f:X\rightarrow\mathbb{P}^{1}$ be in ${\tensf H}_{G}(k)$ and
$T_{0}:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ a non-constant rational map.
Assume the pullback $f_{T_{0}}:X_{T_{0}}\rightarrow\mathbb{P}^{1}$ is
connected. Denote the branch point number of $f$ and the genus of $X$ (resp.,
the branch point number of $f_{T_{0}}$ and the genus of $X_{T_{0}}$) by $r$
and $g$ (resp., by $r_{T_{0}}$ and $g_{T_{0}}$). We have:

(a) $r\leq r_{T_{0}}$ and $g\leq g_{T_{0}}$,

(b) if $g>1$ and $T_{0}$ is not an isomorphism, then, $g<g_{T_{0}}$.

###### Comments on proof.

Regarding branch point numbers, it is a main result from [Dèb18]; see Theorem
2.1. Regarding genera, one may assume $g\not=0$ and $T_{0}$ is not an
isomorphism. The claim then follows from applying the Riemann-Hurwitz formula
to the cover $X_{T_{0}}\rightarrow X$. Namely, we obtain $2g_{T_{0}}-2\geq
N(2g-2)$ with $N=\deg(f)$, whence $g_{T_{0}}\geq 2(g-1)+1\geq g$ if $g\geq 1$,
with equality only if $g=1$. ∎

We can now explain how Corollary 1.3 is deduced from Theorem 1.1.

###### Proof of Corollary 1.3 assuming Theorem 1.1.

First, assume $G$ satisfies the Beckmann-Black regular lifting property. Pick
a Galois cover $g_{1}\in{\tensf H}_{G}$ (such a cover exists from the Riemann
Existence Theorem). Let $r_{1}$ be the branch point number of $g_{1}$. Then,
for any $g_{2}\in{\tensf H}_{G}$, there exists $f\in{\tensf H}_{G}$ such that
$g_{i}=f_{T_{0i}}$ ($i=1,2$). From Theorem 2.4, it follows from
$g_{1}=f_{T_{01}}$ that the branch point number of $f$ is $\leq r_{1}$. This
shows that ${\tensf H}_{G,\leq r_{1}}$ is regularly parametric. From Theorem
1.1, $G\subset{\rm PGL}_{2}(\mathbb{C})$.

Conversely, if $G\subset{\rm PGL}_{2}(\mathbb{C})$, then, $G$ has a regularly
parametric cover $f:X\rightarrow\mathbb{P}^{1}$ [Dèb18, Corollary 2.5], a
fortiori the Beckmann-Black regular lifting property holds. ∎

####  2.3.3. On varieties and their dimension

In the following, the term variety (over $k$) should always be understood to
mean “irreducible quasi-projective variety”. We recall some well-known facts
from algebraic geometry about the structure and dimension of images and
preimages under algebraic morphisms.

It is elementary that the image of an $n$-dimensional variety under an
algebraic morphism is always of dimension $\leq n$. A bound in the opposite
direction is given via the dimension of a fiber, cf., e.g., [Mum99, §1.8,
Theorems 2 and 3]:

######  Theorem 2.5.

Let $f:X\to Y$ be a dominant morphism between varieties $X$ and $Y$. Then,

$\dim(Y)\leq\dim(X)\leq\dim(Y)+\dim(f^{-1}(p)),$

where $p$ is any point in $f(X)$.

######  Theorem 2.6 (Chevalley).

Let $f:X\to Y$ be a morphism between varieties $X$ and $Y$. Then, the image of
any constructible subset of $X$ is constructible101010Here, a subset of a
topological space is called constructible if it is a finite union of locally
closed sets..

In particular, the image of any subvariety $X_{0}$ of $X$ is a finite union of
varieties $Y_{i}\subset Y$, each of dimension at most $\dim(X_{0})$.

For short, we shall say that a subset $S$ of a variety $X$ is of dimension
$\leq d$, if it is contained in a finite union of subvarieties of dimension
$\leq d$.

####  2.3.4. Defining equations for Galois covers and their pullbacks

To prove Theorem 1.1(a), we shall use affine equations $P(T,Y)=0$ to define
Galois covers of $\mathbb{P}^{1}_{k}$.

######  Lemma 2.7.

Let $G$ be a finite group and $r_{0}\in{\mathbb{N}}$. Then, every Galois cover
in ${\tensf H}_{G,\leq r_{0}}(k)$ can be defined by an affine equation
$P(T,Y)=0$, where $P$ is irreducible, monic, and integral in $Y$, of bounded
$T$-degree depending only on $r_{0}$ and $G$, and of degree $|G|$ in $Y$.

###### Proof.

This follows from [Sad99, Section 2.2], where it is shown that a Galois cover
of $\mathbb{P}^{1}_{k}$ with group $G$ and of prescribed genus $g$ can always
be defined by an affine equation of degree $\leq(2g+1)|G|\log|G|/\log(2)$ in
$T$ (and with the remaining assertions as in the statement above). See also
[Dèb17, Lemma 4.1]. It then suffices to note that the genus $g$ of a Galois
cover of $\mathbb{P}^{1}$ with $r_{0}$ branch points is bounded from above
only in terms of $G$; indeed, the Riemann-Hurwitz formula gives $2g\leq
2-2|G|+r_{0}(|G|\cdot(1-1/e_{{\rm{max}}}))$, where $e_{{\rm{max}}}$ is the
maximal element order in $G$. ∎

######  Definition 2.8.

Let $d,e\in{\mathbb{N}}$. We denote by $\mathcal{P}_{d,e}$ the space of
polynomials $P(T,Y)\in k[T,Y]$ in two variables, of degree exactly $d$ in $T$
and exactly $e$ in $Y$, viewed up to multiplicative constants. Furthermore,
denote by $\mathcal{R}_{d}$ the set of rational functions over $k$ in one
indeterminate $U$ of degree exactly $d$.

Of course, $\mathcal{P}_{d,e}$ is a variety in a natural way, via identifying
a polynomial $P(T,Y)=\sum_{i=0}^{d}\sum_{j=0}^{e}\alpha_{i,j}T^{i}Y^{j}$ with
the coordinate tuple $(\alpha_{i,j})_{i,j}$. In the same way,
$\mathcal{R}_{d}$ is a variety, by identifying a rational function
$T_{0}=T_{0}(U)=(\sum_{i=0}^{d}\beta_{i}U^{i})/(\sum_{j=0}^{d}\gamma_{j}U^{j})$
with the coordinate tuple
$(\beta_{0}:\cdots:\beta_{d}:\gamma_{0}:\cdots:\gamma_{d})\in\mathbb{P}^{2d+1}$.
Note that this identification is well-defined since either the numerator or
the denominator of $T_{0}(U)$ are of degree $d$.

We can now define pullback maps on the level of the above spaces of
polynomials and rational functions.

######  Lemma 2.9.

Let $d_{1}$, $d_{2}$, and $d_{3}$ be positive integers. Then, the map

${\rm
PB}:\mathcal{P}_{d_{1},d_{2}}\times\mathcal{R}_{d_{3}}\to\mathcal{P}_{d_{1}\cdot
d_{3},d_{2}},$

defined by $(P(T,Y),T_{0}(U))\mapsto$ “numerator of $P(T_{0}(U),Y)$” is a
morphism of algebraic varieties.

###### Proof.

Let $P(T,Y)=\sum_{i=0}^{d_{1}}\sum_{j=0}^{d_{2}}\alpha_{i,j}T^{i}Y^{j}$ and
$T_{0}=(\sum_{k=0}^{d_{3}}\beta_{k}U^{k})/(\sum_{k=0}^{d_{3}}\gamma_{k}U^{k})$.
Then,

${\rm
PB}(f,T_{0})=\sum_{i,j}\alpha_{i,j}(\sum_{k}\beta_{k}U^{k})^{i}(\sum_{k}\gamma_{k}U^{k})^{d_{1}-i}Y^{j}$

,

and identification with the spaces of coordinate tuples shows that ${\rm PB}$
is given by a polynomial map. This shows the assertion. ∎

##  3\. Regular Parametricity - Part One

This section and the following §4 articulate around our results on regular
parametricity: §§3.1, 3.2, §4 respectively correspond to Theorems 1.1(a),
1.1(b), 1.2 from §1.

Throughout this section, except in §3.3, the field $k$ is algebraically closed
of characteristic $0$. To simplify the notation, we then generally omit the
reference to the rationality field: for example, we write ${\tensf H}_{G}$ for
${\tensf H}_{G}(k)$.

Our proofs make use of the fact that $k$ is algebraically closed but certain
parts carry over to more general fields. We collect such considerations in
§3.3.

###  3.1. Proof of Theorem 1.1(a)

In this section, apart from the sets ${\tensf H}_{G,r}({\bf C})$ of covers
with ramification type $(r,\bf C)$, we also use (inner) Hurwitz spaces. These
are moduli spaces of equivalence classes of covers. We will denote them by
$\mathcal{H}^{in}_{G,r}({\bf C})$, and the equivalence class of $f\in{\tensf
H}_{G,r}({\bf C})$ by $[f]$. For a precise definition, cf., e.g., [FV91]. We
will use only the following well-known facts here:
Firstly, for any ramification type $(r,\bf C)$, the set
$\mathcal{H}^{in}_{G,r}({\bf C})$ is equipped with an algebraic structure
which makes it a finite union of quasi-projective varieties of dimension $r$.
Secondly, if $Z(G)=\\{1\\}$, then $f\in{\tensf H}_{G,r}({\bf C})$ is defined
over a field $K$ if and only if $[f]$ is a $K$-rational point of
$\mathcal{H}^{in}_{G,r}({\bf C})$.

####  3.1.1. Reduction to Theorem 3.1 and Lemma 3.2

The most challenging part in the proof of Theorem 1.1(a) is the following
result about covers of genus at least $2$.

######  Theorem 3.1.

Let $G$ be a finite group, let $r_{0}\in{\mathbb{N}}$, and let

${\tensf H}_{G,\leq r_{0},g\geq 2}(k):={\tensf H}_{G,\leq
r_{0}}(k)\setminus\\{\text{genus }\hskip-5.0pt\leq 1\text{ covers}\\}$

be the set of all $k$-Galois covers $f:X\rightarrow\mathbb{P}^{1}$ of genus at
least $2$ with Galois group $G$ and at most $r_{0}$ branch points. Then there
exists $R_{0}\in{\mathbb{N}}$ such that for every ramification type $(R,{\bf
C})$ for $G$ with $R\geq R_{0}$, one has

${\tensf H}_{G,R}({\bf C})(k)\not\subset{\rm PB}({\tensf H}_{G,\leq
r_{0},g\geq 2}(k))$.

In particular, ${\tensf H}_{G,\leq r_{0},g\geq 2}(k)$ is not $k$-regularly
parametric.

Theorem 3.1 is proved in §3.1.2-4 below. The following lemma shows non-
parametricity for sets of Galois covers of genus $1$.

######  Lemma 3.2.

Let ${\tensf H}_{G,g=1}(k)$ be the set of Galois covers
$f:X\rightarrow\mathbb{P}^{1}$ of genus $1$ with Galois group $G$. Then there
exists a ramification type $(r,{\bf C})$ for $G$ such that no cover in
${\tensf H}_{G,r}({\bf C})(k)$ is a pullback from ${\tensf H}_{G,g=1}(k)$. In
particular, ${\tensf H}_{G,g=1}(k)$ is not $k$-regularly parametric.

###### Proof.

Let $f:X\to\mathbb{P}^{1}$ be an element of ${\tensf H}_{G,g=1}$. As an
immediate consequence of the Riemann-Hurwitz formula, the tuple of element
orders in the inertia canonical invariant of $f$ is one of $(2,2,2,2)$,
$(3,3,3)$, $(2,4,4)$, or $(2,3,6)$. Furthermore, in each case $G$ has a normal
subgroup $N$ with cyclic quotient group $G/N$ of order $2$, $3$, $4$ and $6$
respectively; and such that $X\to X^{N}$ is an unramified cover of genus-$1$
curves over $k$. Assume first that $|N|=1$. Then $G$ is cyclic111111In fact,
$|G|=\\{2,3,4,6\\}$, since $G$ is then a group of automorphisms of some
elliptic curve. Cf. [Sil09, Chapter III, Theorem 10.1] and therefore possesses
coverings of genus zero. In particular, no set of coverings of genus $\geq 1$
can have those as pullbacks, by Theorem 2.4.

Assume therefore that $|N|>1$. Let $x\in N\setminus\\{1\\}$, and let $(r,{\bf
C})$ be any ramification type for $G$ involving the conjugacy class of $x$.
Since $X\to X^{N}$ is unramified, any rational pullback of it must also be
unramified. But of course for any cover $\widetilde{X}\to\mathbb{P}^{1}$ with
inertia canonical invariant ${\bf C}$, the subcover
$\widetilde{X}\to\widetilde{X}^{N}$ is ramified by definition. Therefore no
cover of inertia canonical invariant ${\bf C}$ can be a pullback of $f$. ∎

######  Remark 3.3.

In the case that $G$ is non-cyclic, the above proof in fact shows immediately
that, for $(r,{\bf C})$ a ramification type of genus $1$ with group $G$
($r\in\\{3,4\\}$), and for each $s\geq r+1$, there exists a ramification type
$(s,{\bf D})$ for $G$ such that no cover in ${\tensf H}_{G,s}({\bf D})$ is a
pullback from ${\tensf H}_{G,r}({\bf C})$. Indeed, for the only critical case
$s=r+1$, it suffices to replace $(x_{1},\ldots,x_{n})\in{\bf C}$, where
$x_{1}\notin N$ without loss, by $(x_{0},x_{0}^{-1}x_{1},\ldots,x_{n})$ with
$x_{0}\in N\setminus\\{1\\}$.

Assuming Theorem 3.1 and Lemma 3.2, we can now derive Theorem 1.1(a).

###### Proof of Theorem 1.1(a).

By assumption, $G$ does not possess any Galois covers of genus zero. Let ${\bf
C}:=(C_{1},\ldots,C_{R})$ be a ramification type for $G$. From Theorem 3.1, we
know that not all covers of inertia canonical invariant ${\bf C}$ can be
rational pullbacks of some element of ${\tensf H}_{G,\leq r_{0}}$ of genus
$\geq 2$, as soon as the length $R$ of ${\bf C}$ is sufficiently large
(depending on $r_{0}$). From Lemma 3.2 and its proof, we know that no cover of
inertia canonical invariant ${\bf C}$ can be a rational pullback of an element
of ${\tensf H}_{G,\leq r_{0}}$ of genus $1$, as soon as ${\bf C}$ contains
certain conjugacy classes. Altogether, if ${\bf C}$ contains all classes of
$G$ sufficiently often, then certainly not all covers of inertia canonical
invariant ${\bf C}$ can be reached via rational pullback from ${\tensf
H}_{G,\leq r_{0}}$. ∎

####  3.1.2. Proof of Theorem 3.1: Some dimension estimates

To prepare the proof of Theorem 3.1, we investigate the behaviour of rational
pullbacks of Galois covers.

Recall that we have introduced two different ways of associating algebraic
varieties to certain sets of Galois covers: the Hurwitz spaces, and the spaces
of defining equations. In Lemma 3.4 below, we relate the two concepts via a
dimension estimate, stating in particular that in order to obtain defining
equations for all covers in an $r$-dimensional Hurwitz space, we require at
least $r$-dimensional subvarieties in the space of defining equations.

To state the lemma, denote by $\mathcal{P}_{d,e}^{\textrm{sep}}$ the subset of
separable (in $Y$) polynomials in $\mathcal{P}_{d,e}$. Note that this is a
dense open subset of $\mathcal{P}_{d,e}$. Due to Lemma 2.7, when looking for
defining equations for covers in some ${\tensf H}_{G,\leq r_{0}}$, we can
restrict without loss to a suitable finite union of
$\mathcal{P}_{d,e}^{\textrm{sep}}$ (for $d$ smaller than some bound depending
only on $G$ and $r_{0}$, and in fact always with $e:=|G|$).

######  Lemma 3.4.

Let $r,s,d,e\in{\mathbb{N}}$. Let $V\subset\mathcal{P}_{d,e}^{\rm{sep}}$ be a
subvariety of dimension $s$. Let $G$ be a finite group and let $(r,{\bf C})$
be a ramification type for $G$. Then the set of equivalence classes of covers
$[f]\in\mathcal{H}^{in}_{G,r}(\bf C)$ such that $f$ has a defining equation in
$V$ is of dimension $\leq s$.121212Note here that equivalent covers have the
same defining equations by definition. In particular, if $s<r$, then there are
infinitely many covers in ${\tensf H}_{G,r}({\bf C})$ which do not have a
defining equation in $V$.

###### Proof.

The discriminant map $\Delta:P(T,Y)\mapsto\Delta(P)\in k[T]$ (where $P$ is
viewed as a polynomial in $Y$) induces an algebraic morphism from
$\mathcal{P}_{d,e}$ into some space $\mathcal{P}_{\leq c}$ of polynomials in
$T$ of degree $\leq c$, viewed up to constant factors. Here we use the
definition of polynomial discriminant
$\Delta(\sum_{i=0}^{e}a_{i}Y^{i}):=a_{e}^{2e-2}\prod_{i<j}(r_{i}-r_{j})$,
where the $r_{i}$ are the roots of $\sum_{i=0}^{e}a_{i}Y^{i}$, counted with
multiplicities.

The fact that the degree of $\Delta(P)$ is bounded only in terms of $d$ and
$e$ follows easily from the fact that the discriminant is a polynomial
expression in the coefficients, viewed as transcendentals.

In particular, the image of $V$ inside $\mathcal{P}_{\leq c}$ is of dimension
at most $\dim(V)=s$. Note also that this image does not contain the zero
polynomial, since rational pullbacks of separable polynomials remain
separable.

Next, for any $r\leq t\leq c$ and any $r$-subset $R$ of $\\{1,\ldots,t\\}$,
consider the morphisms $u:(\mathbb{A}^{1})^{t}\to\mathcal{P}_{\leq c}$ given
by $(a_{1},\ldots,a_{t})\mapsto\prod_{i=1}^{t}(T-a_{i})$ and
$v:(\mathbb{A}^{1})^{t}\to(\mathbb{A}^{1})^{r}$ the projection on the
coordinates in $R$. For each of these finitely many possible maps $u,v$, the
map $u$ is finite and so $v(u^{-1}(W))$ is of dimension $\leq s$, where
$W:=\Delta(V)$. But since any branch point (assumed to be finite without loss)
of a cover is necessarily a root of the discriminant of a defining equation, a
cover can only have a defining equation in $V$ if its branch point set is in
$v(u^{-1}(W))$ for some $u,v$ as above. Now let $\mathcal{U}^{r}$ and
$\mathcal{U}_{r}$ denote the spaces of ordered, resp., of unordered $r$-sets
in $\mathbb{P}^{1}$. There is a well-defined finite morphism from
$\mathcal{H}^{in}_{G,r}(\bf C)$ to $\mathcal{U}_{r}$: the branch point
reference map. Now let ${\mathcal{H}}^{\prime}:=\mathcal{H}^{in}_{G,r}(\bf
C)\times_{\mathcal{U}_{r}}\mathcal{U}^{r}$ be the “ordered branch point set
version” of the Hurwitz space $\mathcal{H}^{in}_{G,r}(\bf C)$. Then the branch
point reference map induces a finite morphism
$\psi:\mathcal{H}^{\prime}\to\mathcal{U}^{r}(\subset(\mathbb{P}^{1})^{r})$.
Furthermore, $\mathcal{H}^{\prime}$ has a natural finite morphism $\pi$ to
$\mathcal{H}^{in}_{G,r}(\bf C)$ by definition. In particular, each set
$\pi(\psi^{-1}(v(u^{-1}(W))))$, and thus finally also the set of
$[f]\in\mathcal{H}^{in}_{G,r}(\bf C)$ such that $f$ has (only finite branch
points and) a defining equation in $V$, is of dimension $\leq s$. The
additional assertion in the case $s<r$ follows immediately, since
$\psi({\mathcal{H}}^{\prime})\cap(\mathbb{A}^{1})^{r}$ is of dimension $r$,
and in fact equal to the set of all ordered $r$-sets in $\mathbb{A}^{1}$, by
the Riemann Existence theorem. This concludes the proof. ∎

In the proof of Theorem 3.1, we will show as an intermediate result that a
rational function pulling a prescribed Galois cover $f$ back into a prescribed
${\tensf H}_{G,R}({\bf C})$ can only have a certain maximal number of branch
points outside of the branch point set of $f$. We then require the following
auxiliary result stating that varieties of rational functions with such
partially prescribed branch point sets cannot be too large.

######  Lemma 3.5.

Let $d,m,n,s\in\mathbb{N}$, and let $W$ be a subvariety of
$\mathcal{P}_{m,n}^{\textrm{sep}}$. Then the subset $W^{\prime}\subset
W\times\mathcal{R}_{d}$ of all $(P,T_{0})\in W\times\mathcal{R}_{d}$ such that
at most $s$ branch points of $T_{0}$ are not roots of $\Delta(P)$, is of
dimension at most $\dim W+s+3$.

###### Proof.

Denote the discriminant map on $\mathcal{P}_{m,n}^{\textrm{sep}}$ by
$\Delta_{1}$ and the one on $\mathcal{R}_{d}$ by $\Delta_{2}$. Here we define
the discriminant of a rational function $T_{0}(U):=T_{0,1}(U)/T_{0,2}(U)$ as
the discriminant of the polynomial $T_{0,1}(U)-T\cdot T_{0,2}(U)$ with respect
to $U$. Note that in the special case of rational functions, every root of the
discriminant is in fact a branch point; see e.g. [Mül02, Lemma 3.1] for a
stronger version of this statement. This means that, with
$u:(a_{1},\ldots,a_{t})\mapsto\prod_{i=1}^{t}(T-a_{i})$ as before, an element
of $u^{-1}(\Delta_{2}(T_{0}))$ is already the exact branch point set of
$T_{0}$, up to multiplicities.

Consider now the following chain of maps

$W^{\prime}\subset W\times\mathcal{R}_{d}\hskip 5.69054pt\smash{\mathop{\hbox
to34.1433pt{\rightarrowfill}}\limits^{{\rm id}\times\Delta_{2}}}\hskip
5.69054ptW\times\mathcal{P}_{\leq t}\hskip 5.69054pt\smash{\mathop{\hbox
to34.1433pt{\leftarrowfill}}\limits^{{\rm id}\times u}}\hskip
5.69054ptW\times(\mathbb{A}^{1})^{t}\hskip 5.69054pt\smash{\mathop{\hbox
to34.1433pt{\leftarrowfill}}\limits^{\alpha}}\hskip
5.69054ptW\times(\mathbb{A}^{1})^{t},$

where $\alpha$ is defined by
$\alpha(P,(a_{1},\ldots,a_{t})):=(P,(\Delta_{1}(P)(a_{1}),\ldots,\Delta_{1}(P)(a_{t})))$.
Clearly, all maps in this chain are morphisms, and except for the first map
${\rm id}\times\Delta_{2}$, they are all finite. Therefore $({\rm
id}\times\Delta_{2})(W^{\prime})$ is of the same dimension as $\alpha({\rm
id}\times(u^{-1}\circ\Delta_{2}))(W^{\prime})$, and by definition of
$W^{\prime}$, the latter is contained in one of finitely many varieties
isomorphic to $W\times(\mathbb{A}^{1})^{s}$; indeed, up to repetitions, all
except for at most $s$ roots of $\Delta_{2}(T_{0})$ are mapped to $0$ under
$\Delta_{1}(P)$, for $(P,T_{0})\in W^{\prime}$. Thus, $({\rm
id}\times\Delta_{2})(W^{\prime})$ is of dimension at most $\dim(W)+s$. Theorem
2.5 then yields that $W^{\prime}$ is of dimension at most
$\dim(W)+s+\dim(\Delta_{2}^{-1}(p))$ for any $p$ equal to the discriminant of
a rational function $T_{0}$ as above.

It remains to show that $\Delta_{2}^{-1}(p)$ is of dimension $\leq 3$. Now the
set of genus zero covers $\mathbb{P}^{1}\to\mathbb{P}^{1}$ (viewed up to
equivalence) of degree $d$ with prescribed branch point set is finite, and
each such cover is given by a degree-$d$ rational function, unique up to ${\rm
PGL}_{2}(k)$-equivalence. Since $\dim({\rm PGL}_{2}(k))=3$, the claim follows,
completing the proof. ∎

####  3.1.3. Proof of Theorem 3.1: reduction to Lemma 3.6

######  Lemma 3.6.

Let $G$ be a finite group and $f:X\to\mathbb{P}^{1}$ a Galois cover with group
$G$ and genus $\geq 2$. Then for every $j\in{\mathbb{N}}$, there exists a
constant $R_{0}\in{\mathbb{N}}$, depending only on $j$ and the branch point
number of $f$, such that for every class-$R$-tuple ${\bf C}$ of $G$ ($R\geq
R_{0}$) and for every rational function
$T_{0}:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ with more than $R-j$ branch
points outside the branch point set of $f$, the pullback of $f$ by $T_{0}$ is
not in ${\tensf H}_{G,R}({\bf C})$.

###### Proof of Theorem 3.1 assuming Lemma 3.6.

Let $f:X\to\mathbb{P}^{1}\in{\tensf H}_{G,\leq r_{0},g\geq 2}$. Let $g$ be the
genus of $X$. Let $F:=F(T,Y)$ be a separable defining equation for $f$, of
minimal degree in $T$. Using the pullback map ${\rm PB}$ as in Lemma 2.9,
denote by ${\rm PB}(f)$ the set of all pullbacks of $F$ by rational functions
of arbitrary degree, i.e., ${\rm PB}(f):=\cup_{d\in{\mathbb{N}}}{\rm
PB}(\\{F\\}\times\mathcal{R}_{d})$. Then ${\rm PB}(f)$ contains defining
equations for all rational pullbacks of the cover $f$.

Let $(R,{\bf C})$ be a ramification type for $G$. By the Riemann-Hurwitz
formula, the genus of a Galois cover with group $G$ arising as a degree-$d$
pullback of $f$ is at least $d(g-1)+1$. Since $g\geq 2$, this shows
immediately that there exists $d_{0}\in{\mathbb{N}}$, depending only on ${\bf
C}$, such that for all $d>d_{0}$, a degree-$d$ pullback of $f$ cannot have
inertia canonical invariant ${\bf C}$ (since the genus is the same for all
covers with invariant ${\bf C}$). In other words, to investigate the set of
polynomials in ${\rm PB}(f)$ which are defining equations for covers in
${\tensf H}_{G,R}({\bf C})$, it suffices to restrict to pullback functions
$T_{0}\in\mathcal{R}_{d}$, $d\leq d_{0}$, with some bound
$d_{0}\in{\mathbb{N}}$ depending only on ${\bf C}$.

Let $D\in{\mathbb{N}}$ be such that every $f\in{\tensf H}_{G,\leq r_{0}}$ has
a separable defining equation in some space $\mathcal{P}_{d_{1},|G|}$ with
$d_{1}\leq D$. Such $D$ exists by Lemma 2.7. Let $\delta$ be the dimension of
$\mathcal{P}(D,|G|)$ (to be explicit, $\delta=(D+1)(|G|+1)-1)$.

Fix an integer $j>\delta+3$, choose $R_{0}$ sufficiently large131313See e.g.
the proof of Lemma 3.6 for an explicit bound on $R_{0}$. and $R\geq R_{0}$,
and denote by $\mathcal{S}_{f}$ the set of all rational functions $T_{0}$
which pull a given $f\in{\tensf H}_{G,\leq r_{0},g\geq 2}$ back to a connected
$R$-branch-point cover. As seen above, the degree of such $T_{0}$ is
absolutely bounded from above (in terms of the genus, and thus the branch
point number of $f$), and by Lemma 3.6, all $T_{0}\in\mathcal{S}_{f}$ have at
most $R-j$ branch points outside the branch point set of $f$. A fortiori, they
are contained in the set $\mathcal{S}^{\prime}_{F}$ of rational functions (of
bounded degree as before and) with at most $R-j$ finite branch points outside
the set of roots of the discriminant of $F(T,Y)$, for a defining equation
$F(T,Y)=0$. The latter sets $\mathcal{S}^{\prime}_{F}$ can be defined for all
$F\in\mathcal{P}_{d_{1},d_{2}}^{\rm{sep}}$ (not just for those defining Galois
covers).

Now consider the set $\mathcal{S}:=\cup_{d_{1}\leq
D}\cup\\{F\\}\times\mathcal{S}^{\prime}_{F}$, where the inner union is over
all $F\in\mathcal{P}_{d_{1},|G|}^{\rm{sep}}$. From Lemma 3.5 (with
$W:=\mathcal{P}_{d_{1},|G|}^{\rm{sep}})$ it follows that $\mathcal{S}$ is
contained in a a finite union of varieties, of dimension at most
$\dim(\mathcal{P}_{D,|G|})+R-j+3\leq\delta+R-j+3<R$.

Therefore the image of $\mathcal{S}$ under ${\rm PB}$ is of dimension strictly
smaller than $R$ as well. On the other hand, Lemma 3.4 shows that no finite
union of varieties of dimension $<R$ can contain defining equations for all
elements of ${\tensf H}_{G,R}({\bf C})$. Hence ${\tensf H}_{G,\leq r_{0},g\geq
2}$ is not $k$-regularly parametric. This proves the assertion. ∎

####  3.1.4. Proof of Lemma 3.6

First step: Explicit choice of $R_{0}$.
Let $t_{1},\ldots,t_{s}$ be the branch points of $f$ and let
$e_{1},\ldots,e_{s}$ be the corresponding orders of inertia groups of $f$. By
the Riemann-Hurwitz formula, the genus $g$ of $f$ fulfills

$g-1=\frac{1}{2}|G|(s-2-\sum_{i=1}^{s}\frac{1}{e_{i}}),$

and since $g\geq 2$, Hurwitz’s automorphism theorem yields $|G|\leq 84(g-1)$,
yielding in total that

(1) $s-2-\sum_{i=1}^{s}\frac{1}{e_{i}}\geq 1/42.$

We may assume without loss that $j>3$, and we pick $R_{0}\in{\mathbb{N}}$ such
that $R_{0}>42s(j-3)$.

Let $R\geq R_{0}$, and let $(R,{\bf C})$ be a ramification type for $G$. Let
$T_{0}:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ be a rational function such
that the pullback of $f$ by $T_{0}$ defines a Galois cover with inertia
canonical invariant ${\bf C}$, and let $\mathcal{T}$ (resp., $d$) be the
ramification type (resp., the degree) of $T_{0}$.

Note that, since the branch point number of a degree-$d$ rational pullback of
$f$ is trivially bounded from above by $sd$, our choice $R_{0}>42s(j-3)$
implies $d>42(j-3)$, and thus, using (1), we obtain

(2) $d\sum_{i=1}^{s}{1/e_{i}}\leq d(s-2-1/42)<d(s-2)-(j-3).$

Second step: Translation into a combinatorial statement.
Let $\sigma_{1},\ldots,\sigma_{s}\in S_{d}$ be the inertia group generators of
$T_{0}$ at $t_{1},\ldots,t_{s}$, and $\sigma_{s+1},\ldots,\sigma_{s+m}$ the
non-trivial inertia group generators at further points. For $\sigma\in S_{d}$,
denote by $o(\sigma)$ the number of orbits of $\langle\sigma\rangle$, and set
${\rm{ind}}(\sigma)=d-o(\sigma)$. We claim that, assuming choice of $R_{0}$ as
above, the following holds:
Claim:

$\sum_{i=1}^{s}{\rm{ind}}(\sigma_{i})\geq 2d-2-R+j,$

or equivalently:

(3) $\sum_{i=1}^{s}o(\sigma_{i})\leq d(s-2)-j+2+R.$

The assertion then follows from (3), since $T_{0}$ defines a genus-zero cover,
whence the Riemann-Hurwitz genus formula yields
$\sum_{i=1}^{s+m}{\rm{ind}}(\sigma_{i})=2d-2$. Together with the claim, this
enforces $m\leq R-j$.

Third step: Transformation of cycle structures.
To prove the claim, consider the cycle structures of $\sigma_{i}$
($i=1,\ldots,s$). We will manipulate the cycle structures of the $\sigma_{i}$
in a controlled way, to make it easier to estimate the total number of orbits
of all $\sigma_{i}$ ($i=1,\ldots,s$).
By the definition of $T_{0}$ and by Abhyankar’s lemma, the cycle lengths of
the $\sigma_{i}$ are multiples of $e_{i}$, for $i=1,\ldots,s$, with a total of
exactly $R$ exceptions over all $i\in\\{1,\ldots,s\\}$.

Let $n_{i}\in{\mathbb{N}}$ be such that $n_{i}\cdot e_{i}$ is the sum of all
“non-exceptional” cycle lengths (i.e., of those which are multiples of
$e_{i}$) in $\sigma_{i}$. Now for each $i\in\\{1,\ldots,s\\}$, let $\tau_{i}$
be a permutation of cycle structure $[e_{i}^{n_{i}},r_{i}]$, where $r_{i}$ is
the sum of all exceptional cycle lengths in $\sigma_{i}$. We then have
$o(\tau_{i})\leq\lfloor d/e_{i}\rfloor+1$ for all $i=1,\ldots,s$, with strict
inequality if $r_{i}=0$ or $r_{i}\geq e_{i}$. Let $t\in{\mathbb{N}}$ be the
number of $i$ with $1\leq r_{i}<e_{i}$. It follows that
$\sum_{i=1}^{s}o(\tau_{i})\leq\sum_{i=1}^{s}\lfloor
d/e_{i}\rfloor+t\leq(d\sum_{i=1}^{s}{1/e_{i}})+t$. Using (2), this implies

$\sum_{i=1}^{s}o(\tau_{i})\leq(d(s-2)-(j-3)+t)-1=d(s-2)-j+2+t.$

Furthermore, it follows from the definition of the $\tau_{i}$ that

$\sum_{i=1}^{s}o(\tau_{i})\geq\sum_{i=1}^{s}o(\sigma_{i})-(R-t),$

since a total of $R$ cycles of length not divisible by $e_{i}$ in the
$\sigma_{i}$ gets replaced by a total of $\geq t$ cycles in the $\tau_{i}$,
and the number of non-exceptional cycles in $\tau_{i}$ is at least as large as
in $\sigma_{i}$. In total, $\sum_{i=1}^{s}o(\sigma_{i})\leq d(s-2)-j+2+R$,
showing the claim. This completes the proof.

###  3.2. Proof of Theorem 1.1(b)

We continue with proving the (b) part of Theorem 1.1, for which the assumption
on $G$ is stronger than not being a subgroup of ${\rm PGL}_{2}(\mathbb{C})$.

Fix a finite group $G$ with at least $5$ non-conjugate maximal cyclic
subgroups. At first assume that not all maximal conjugacy classes are of order
$2$. Let $\gamma_{1},\ldots,\gamma_{5}$ be generators of $5$ non conjugate
maximal cyclic subgroups of $G$ and let
${\mathcal{C}}_{1},\ldots,{\mathcal{C}}_{5}$ be their conjugacy classes.
Denote the order of $\gamma_{i}$ by $e_{i}$, $i=1,\ldots,5$ and without loss
of generality assume $e_{1}>2$. Consider then a tuple
$({\mathcal{C}}_{1},\ldots,{\mathcal{C}}_{5},{\mathcal{C}}_{6},\ldots,{\mathcal{C}}_{s})$
of conjugacy classes of $G$, not necessarily distinct, and satisfying the
following:

(A) all the non-trivial conjugacy classes of $G$, but the powers
${\mathcal{C}}_{i}^{j}$, $i=1,\ldots,4$, $j=1,\ldots,e_{i}-1$, appear in the
set $\\{{\mathcal{C}}_{5},\ldots,{\mathcal{C}}_{s}$}.

Consider the $(2s)$-tuple
$\underline{\mathcal{C}}=(\mathcal{C}_{1},\mathcal{C}_{1}^{-1},\ldots,\mathcal{C}_{s},\mathcal{C}_{s}^{-1})$.
Note that the integer $s$ can be taken to be any suitably large integer, for
example by repeating the conjugacy class ${\mathcal{C}}_{5}$. Picking an
element $g_{i}\in\mathcal{C}_{i}$, we form the tuple
$\underline{g}=(g_{1},g_{1}^{-1},\ldots,g_{s},g_{s}^{-1}$).

Since the elements of $\underline{g}$ and their powers contain at least one
element from each conjugacy class, a classical lemma of Jordan implies that
$\underline{g}$ forms a generating set of $G$. By construction the product in
$\underline{g}$ is $1$. From the Riemann Existence Theorem, the set ${\tensf
H}_{G,2s}(\underline{\mathcal{C}})$ of Galois covers
$X\rightarrow\mathbb{P}^{1}$ with group $G$, $2s$ branch points and inertia
canonical invariant $\underline{\mathcal{C}}$, is nonempty.

Let $h$ be a cover in the set ${\tensf H}_{G,2s}(\underline{\mathcal{C}})$.
Assume there exist $r_{0}\in\mathbb{N}$, a Galois cover $f\in{\tensf
H}_{G,\leq r_{0}}$ and $T_{0}\in\mathbb{C}(U)\setminus\mathbb{C}$ of degree
$N$, such that $h$ and $f_{T_{0}}$ are isomorphic. Denote the branch point
number of $f$ by $r$ ($r\leq r_{0}$) and its inertia canonical invariant by
${\bf C}=(C_{f,1},\ldots,C_{f,r}$). By [Dèb18, §3.1], the inertia canonical
invariant of $f_{T_{0}}$ is a tuple ${\bf C}_{f,T_{0}}$ obtained by
concatenating tuples of the form ${\bf
C}_{f,T_{0},j}=(C_{f,j}^{e_{j1}},\ldots,C_{f,j}^{e_{jr_{j}}}),\,j=1,\ldots,r,$
where $r_{j},e_{j,\ell}$ are integers with $r_{j}\geq 0$ and $e_{j,\ell}\geq
1$ for all $\ell=1,\ldots,r_{j}$, and $j=1,\ldots,r$. Note that some of the
classes in ${\bf C}_{f,T_{0}}$ might be trivial.

Denote by $p_{j}$ (resp., $q_{j}$) the number of $e_{j,\ell}$’s,
$\ell=1,\ldots,r_{j},$ equal to $1$ (resp., $>1$), for $j=1,\ldots,r$. For
$j=1$, denote further by $u_{j}$ (resp., $v_{j}$) the number of
$e_{1,\ell}$’s, $\ell=1,\ldots,r_{1},$ equal to $2$ (resp., $>2$). Recall
further from [Dèb18, §3.1] that since $e_{j1},\ldots,e_{jr_{j}}$ are the
ramification indices of $T_{0}$ over some point, we have
$\sum_{\ell=1}^{r_{j}}e_{j,\ell}=N$, for $j=1,\ldots,r$. Hence,
$p_{j}+2q_{j}\leq N$, and $p_{1}+2u_{1}+3v_{1}\leq N$, or equivalently:

(4) $q_{j}\leq\frac{N-p_{j}}{2}\leq\frac{N}{2}\text{ for }j=2,\ldots,r\text{
and }v_{1}\leq\frac{N-p_{1}-2u_{1}}{3}\leq\frac{N}{3}.$

By the Riemann–Hurwitz formula for $T_{0}$, we also have

(5)
$2N-2\geq\sum_{\ell=1}^{r_{1}}(e_{1,\ell}-1)+\sum_{j=2}^{r}\sum_{\ell=1}^{r_{j}}(e_{j,\ell}-1)=N-p_{1}-u_{1}-v_{1}+\sum_{j=2}^{r}(N-p_{j}-q_{j}).$

As $f_{T_{0}}$ and $h$ are isomorphic, the tuples ${\bf C}_{f,T_{0}}$ and
$\underline{\mathcal{C}}$ are equal, up to order. Without loss of generality
we may assume $\mathcal{C}_{1}\in{\bf C}_{f,T_{0},1}$. For each index
$i=1,\ldots,4$, if $j_{i}\in\\{1,\ldots,r\\}$ is an index such that
${\mathcal{C}}_{i}\in{\bf C}_{f,T_{0},j_{i}}$ 141414There may be a priori two
different indices $j_{i}\in\\{1,\ldots,r\\}$ such that
${\mathcal{C}}_{i}\in{\bf C}_{f,T_{0},j}$., then, as
$\langle\gamma_{i}\rangle$ is a maximal cyclic subgroup of $G$, one has
${\mathcal{C}}_{i}={C}_{f,j_{i}}^{w_{i}}$ for some integer $w_{i}$ relatively
prime to $e_{i}$. This together with the assumption that
$\mathcal{C}_{1},\ldots,\mathcal{C}_{4}$ are not powers of each other, implies
that the correspondence $i\mapsto j_{i}$ is injective. Thus (5) and (4) give

$\displaystyle 2N-2$ $\displaystyle\geq
N-p_{1}-u_{1}-v_{1}+\sum_{i=2}^{4}(N-p_{j_{i}}-q_{j_{i}})>\frac{13}{6}N-(p_{1}+u_{1})-\sum_{i=2}^{4}p_{j_{i}},$

and hence

(6) $N<6(p_{1}+u_{1}+\sum_{i=2}^{4}p_{j_{i}}).$

Since each $\mathcal{C}_{i}$, $i=1,\ldots,4$, appears at most twice in ${\bf
C}_{f,T_{0}}$ by (A), it follows that $p_{j_{i}}\leq 2$ for $i=1,\ldots,4$.
Furthermore, since the powers of $\mathcal{C}_{1}$ appear at most twice and
$e_{1}>2$, we also have $p_{1}+u_{1}\leq 2$. Thus (6) gives $N<48$. However,
by a priori choosing $s$ to be large, the degree $N$ of $T_{0}$ is forced to
be at least $48$, contradiction.

It remains to consider the case where all maximal conjugacy classes are of
order $2$. In this case $G$ is an elementary abelian $2$-group of rank at
least $3$. It follows that the number of maximal conjugacy classes of cyclic
groups is at least $7$. Repeating the above argument with seven classes
$\mathcal{C}_{1},\ldots,\mathcal{C}_{7}$, replacing the previous
$\mathcal{C}_{1},\ldots,\mathcal{C}_{5}$, (5) and (4) take the form:

$\displaystyle 2N-2$ $\displaystyle\geq\sum_{j=1}^{r}(N-p_{j}-q_{j})\text{ and
}q_{j_{i}}\leq\frac{N-p_{j_{i}}}{2}\leq\frac{N}{2}\text{ for }i=1,\ldots,6.$

Thus, their combination gives:

$2N-2\geq\sum_{i=1}^{6}(N-p_{j_{i}}-q_{j_{i}})>3N-\sum_{i=1}^{6}p_{j_{i}},$

and hence $N<\sum_{i=1}^{6}p_{j_{i}}$. Once again choosing $s$ and hence $N$
large enough we obtain a contradiction.

###  3.3. Extension to more general fields

We assume that the field $k$ is algebraically closed in our proof of Theorem
1.1 because we use the Riemann Existence Theorem. However, as we explain
below, this assumption can be relaxed in some situations.

######  Theorem 3.7.

Let $G$ be a finite group and let $k$ be a field of characteristic $0$.

(a) Theorem 1.1(a) holds in each of these situations:

(a-1) $k$ is ample151515Definition of “ample field” is recalled in §1.1.,

(a-2) $G$ is abelian of even order,

(a-3) $G$ is the direct product $A\times H$ of an abelian group $A$ of even
order and

of a non-solvable group $H$ occuring as a regular Galois group over $k$
161616In fact, the assumption on $H$ to be non-solvable can be removed with a
bit of extra effort.,

(a-4) $G=S_{5}$.

(b) Theorem 1.1(b) holds if either $k$ is ample or $G$ is abelian.

In particular, the full Theorem 1.1 holds if $k$ is ample.

####  3.3.1. Proof of Theorem 3.7(b)

In the proof of Theorem 1.1(b) (§3.2), the assumption $k=\overline{k}$ was
only used to guarantee that there is at least one $k$-cover in the Hurwitz
stack ${\tensf H}_{G,2s}(\underline{\mathcal{C}})$. When $k$ is no longer
algebraically closed, we first slightly modify the tuple
$\underline{\mathcal{C}}$ to make it $k$-rational, i.e., such that the action
of $\hbox{\rm Gal}(\overline{k}/k)$ on $\underline{\mathcal{C}}$ (taking the
power of the classes by the cyclotomic character) preserves
$\underline{\mathcal{C}}$, up to the order. This can be done by replacing, for
each index $i=1,\ldots,4$,

each pair $({\mathcal{C}}_{i},{\mathcal{C}}_{i}^{-1})$ by the tuple
$({\mathcal{C}}_{i},{\mathcal{C}}_{i}^{-1},,\ldots,{\mathcal{C}}_{i}^{e_{i}-1},{\mathcal{C}}_{i}^{-(e_{i}-1)}$).

The modified tuple is indeed $k$-rational and the proof of Theorem 1.1(b)
still holds after some slight adjustments: inequalities $p_{j_{i}}\leq 2$
become $p_{j_{i}}\leq 2(e_{i}-1)$, $i=1,\ldots,4$, which only changes the
constants in the final estimates.

With the tuple $\underline{\mathcal{C}}$ now $k$-rational, it is still true
that the Hurwitz stack ${\tensf H}_{G,2s}(\underline{\mathcal{C}})$ has a
$k$-cover, and so that Theorem 1.1(b) holds, in the following situations:

\- $G$ abelian and $k$ arbitrary (as a consequence of the classical rigidity
theory, see e.g., [MM99, Chapter I, §4] or [Völ96, §3.2]), and

\- $k$ ample and $G$ arbitrary. Namely, recall that over a complete discretely
valued field $k$, the so-called $1/2$-Riemann Existence Theorem of Pop [Pop96]
can be used to ensure that ${\tensf H}_{G,2s}(\underline{\mathcal{C}})(k)$ is
non-empty. Furthermore this last conclusion extends to ample fields, via some
classical specialization argument [Pop96, Proposition 1.1] [DD97a, §4.2].

####  3.3.2. Proof of Theorem 3.7(a)

We will deduce Theorem 3.7(a) from Theorem 3.9 below. Start with an arbitrary
field $k$ and consider the following condition, for a finite group $G$.

######  Condition 3.8.

  * (a)

There exist a constant $m\geq 0$, infinitely many integers $R$, and for each
$R$ a ramification type $(R,{\bf C})$ for $G$ such that the set of all
equivalence classes $[f]\in\mathcal{H}^{in}_{G,R}(\bf C)$ such that $f$ is a
$k$-Galois cover cannot be covered by finitely many varieties of dimension
$<R-m$ (in $\mathcal{H}^{in}_{G,R}({\bf C})(\overline{k})$).

  * (b)

Each ${\bf C}$ in (a) contains every conjugacy class of $G$ at least once.

We then have the following analog of Theorem 3.1 and Lemma 3.2:

######  Theorem 3.9.

Assume that $G$ fulfills Condition 3.8(a) (resp., Condition 3.8(a) and (b))
over $k$. Let $r_{0}\in{\mathbb{N}}$, and let ${\tensf H}:={\tensf H}_{G,\leq
r_{0},g\geq 2}(k)$ (resp., ${\tensf H}:={\tensf H}_{G,\leq r_{0},g\geq 1}(k)$)
be the set of $k$-Galois covers with group $G$, branch point number $\leq
r_{0}$ and genus $\geq 2$ (resp., $\geq 1$). Then there are infinitely
ramification types $(R,\bf C)$ for $G$ over $k$ such that ${\tensf
H}_{G,R}({\bf C})(k)\not\subset{\rm PB}({\tensf H})$. In particular, ${\tensf
H}$ is not $k$-regularly parametric.

###### Proof.

Observe that the crucial Lemma 3.6 in the proof of Theorem 3.1 (§3.1.3)
guarantees the following: there exists $R_{0}\in{\mathbb{N}}$, such that for
every ramification type $(R,{\bf C})$ for $G$ ($R\geq R_{0}$), the set of
(defining equations of) $G$-covers with inertia canonical invariant ${\bf C}$
which arise as rational pullbacks of some cover with $\leq r_{0}$ branch
points is contained in a union of finitely many varieties of dimension at most
$R-(m+1)$. Then, with $R$ sufficiently large and $(R,{\bf C})$ as in Condition
3.8, it follows as in Lemma 3.4 that these varieties are not sufficient to
yield defining equations for every cover in ${\tensf H}_{G,R}({\bf
C})(\overline{k})$ which is defined over $k$. Furthermore, the additional
Condition 3.8(b) is sufficient for the proof of Lemma 3.2 over $k$, if $G$ is
non-cyclic. For cyclic $G={\mathbb{Z}}/n{\mathbb{Z}}$, Lemma 3.2 holds as soon
as the genus-$0$ extension $k(\sqrt[n]{T})/k(T)$ is $k$-regular, i.e., as soon
as $e^{2ipi/n}\in k$. On the other hand, Galois groups of genus-$1$ cyclic
covers are only ${\mathbb{Z}}/n{\mathbb{Z}}$ for $n\in\\{2,3,4,6\\}$, and it
is easy to verify that for $e^{2ipi/n}\notin k$, these genus-$1$ covers cannot
be defined over $k$ either, as a special case of Branch Cycle Lemma (see
[Fri77] and [Völ96, Lemma 2.8]). ∎

###### Proof of Theorem 3.7(a-1).

Let $k$ be an ample field of characteristic zero. Equivalently to the
definition of “ample”, every absolutely irreducible variety over $k$ with a
simple $k$-rational point has a Zariski-dense set of $k$-rational points, see
[Jar11, Lemma 5.3.1]. On the other hand, the $1/2$-Riemann Existence Theorem
yields plenty of class $r$-tuples ${\bf C}$ of $G$ such that ${\tensf
H}_{G,r}({\bf C})(k)$ is non-empty. For example, all ${\bf C}$ corresponding
to an arbitary long repetition of the tuple
$(x_{1},x_{1}^{-1},\ldots,x_{n},x_{n}^{-1})$, where $x_{i}$ runs through all
non-identity elements of $G$, are fine. For $Z(G)=\\{1\\}$, this then implies
the existence of a $k$-rational point on $\mathcal{H}^{in}_{G,r}(\bf C)$, and
therefore in fact of a Zariski-dense set of $k$-rational points. Since these
$k$-rational points are exactly the equivalence classes of covers $f$ defined
over $k$, Condition 3.8 holds, even with $m=0$, and therefore the assertion of
Theorem 3.9 holds over $k$. Finally, by an easy and classical argument, the
case of arbitrary $G$ can be reduced to the above assumption $Z(G)=\\{1\\}$
upon embedding $G$ as a quotient into a group with trivial center. ∎

###### Proof of Theorem 3.7(a-2).

As a first example, let $G$ be an elementary abelian $2$-group. Since $G$ is
abelian, every tuple $(C_{1},\ldots,C_{R})$ of conjugacy classes in $G$ with
non-empty Nielsen class is a rigid tuple. Furthermore, since all non-identity
elements of $G$ are of order $2$, every conjugacy class is trivially rational
(i.e., unchanged if taken to a power relatively prime to the order of its
elements). It then follows from the rigidity method that for every choice
$(t_{1},\ldots,t_{R})$ of $R$ distinct points in $\mathbb{P}^{1}(k)$, there
exists a Galois cover of $\mathbb{P}^{1}$, defined over $k$, with inertia
canonical invariant $(C_{1},\ldots,C_{R})$ and branch points
$(t_{1},\ldots,t_{R})$. In particular, the set of all these covers cannot be
obtained by a set of defining equations of dimension $<R$. Hence $G$ fulfills
Condition 3.8 over $k$. The assertion of Theorem 3.9 therefore holds for $G$
over an arbitrary field of characteristic zero.

As a next step, let $G$ be an arbitrary abelian group of even order. As
before, all class tuples with non-empty Nielsen class are rigid. Let
$(C_{1},\ldots,C_{R})$ be a class tuple of $G$ containing each element of
order $\geq 3$ exactly once, and each element of order $2$ an arbitrary even
number of times. This then yields a product-$1$ tuple generating $G$, and if
the branch points for each set of generators of a cyclic subgroup
${\mathbb{Z}}/e{\mathbb{Z}}$ are chosen appropriately to form a full set of
conjugates (under the action of
${\textrm{Gal}}({\mathbb{Q}}(e^{2ipi/e})/{\mathbb{Q}}$)) in $k(e^{2ipi/e})$,
then the associated ramification type is a rational ramification type,
implying that there is again a $k$-Galois cover with inertia canonical
invariant $(C_{1},\ldots,C_{R})$ and with the prescribed branch point set.
Note that the branch points for elements of order $2$ are still allowed to be
chosen freely in $k$. Since there are less than $|G|$ other branch points, it
follows as above that the set of Galois covers with these ramification data
cannot be obtained by a set of defining equations of dimension $\leq R-|G|$.
Therefore again, Condition 3.8 is fulfilled. ∎

###### Proof of Theorem 3.7(a-3).

Now, let $G=A\times H$, where $A$ is an abelian group of even order and $H$ is
any non-solvable group which occurs as a regular Galois group over $k$. We use
the non-solvability assumption only to obtain that there are then no Galois
covers of genus $\leq 1$ with group $H$, and a fortiori none with group $G$.
Take a tuple $(C_{1},\ldots,C_{R})$ of classes of $A\leq G$ as in the previous
case, and prolong it by a fixed tuple $(C_{R+1},\ldots,C_{S})$ of classes
which occurs as some ramification type for $H$ over $k$. With the appropriate
choice of branch point set for the $H$-cover, we obtain a Galois cover with
group $A\times H$, where once again the branch points with involution inertia
in $A$ can be chosen freely (outside of the fixed branch points of the
$H$-cover). Increasing $R$ as above (whilst fixing $(C_{R+1},\ldots,C_{S})$),
we again obtain that Condition 3.8(a) is fulfilled, and so the assertion of
Theorem 3.9 holds over $k$ for Galois covers of genus $\geq 2$. Since $G$ has
no Galois covers of genus $\leq 1$, Theorem 1.1(a) holds for $G$ over all
fields of characteristic $0$. ∎

###### Proof of Theorem 3.7(a-4).

Let $H_{g,d}$ denote the moduli space of simply branched covers (i.e., all
non-trivial inertia groups are generated by transpositions) of degree $d$ and
genus $g$. It is known ([AC81]) that $H_{g,5}$ is unirational for all $g\geq
6$, and in fact, this holds even over the smallest field of definition
$\mathbb{Q}$ (and therefore over all fields of characteristic zero).

Note that $H_{g,5}$ parameterizes $S_{5}$-covers with inertia canonical
invariant $(C_{1},\ldots,C_{8+2g})$, where each $C_{i}$ is the class of
transpositions. This space is of dimension $8+2g$, and unirationality (over
$k$) implies that its function field is of finite index in some
$k(T_{1},\ldots,T_{8+2g})$ with independent transcendentals $T_{i}$. But of
course, every $k$-rational value of $(T_{1},\ldots,T_{8+2g})$ then leads to a
$k$-rational point on $H_{g,5}$ (and thus, a cover defined over $k$), and the
set of such $k$-rational points on a unirational variety is always Zariski-
dense.

This implies that Condition 3.8(a) is fulfilled, and since $S_{5}$ does not
possess Galois covers of genus $\leq 1$, Theorem 1.1(a) holds for $S_{5}$ over
all fields of characteristic $0$, thus concluding the proof. It would be
interesting to find out if something similar can be obtained for larger
$S_{n}$. ∎

##  4\. Regular Parametricity - Part Two, Proof of Theorem 1.2

Throughout the section fix an algebraically closed field $k$ of characteristic
$0$ and a Galois cover $f:X\rightarrow\mathbb{P}^{1}$ with group $G$ and
ramification type $(r,\bf C)$. Assume $G$ is not a subgroup of
$PGL_{2}(\mathbb{C})$. We treat here the case $r\geq 4$. The case $r=3$ is
more technical and is treated with similar tools in Appendix A. Let ${\bf
t}=\\{t_{1},\ldots,t_{r}\\}$ be the branch point set of $f$. Let
$e_{f}(t_{0})$ denote the ramification index of $t_{0}\in\mathbb{P}^{1}$ under
$f$, and set $e_{i}=e_{f}(t_{i})$ for $i=1,\ldots,r$. Given $T_{0}\in
k(U)\setminus\mathbb{P}^{1}$ and $t_{0}\in k$, let $e(q|t_{0})$ denote the
ramification index under $T_{0}$ of $q\in T_{0}^{-1}(t_{0})$. The proof is
based on the following estimate on the number of branch points of a pullback,
which strengthens the bounds in [Dèb18, Theorem 3.1(b-2)].

######  Lemma 4.1.

Let $T_{0}\in k(U)\setminus k$ be of degree $n$ such that the pullback
$f_{T_{0}}$ is connected. Let $a_{i}$ be the number of preimages $q\in
T_{0}^{-1}(t_{i})$ with $e_{i}\mid e(q|t_{i})$ for $i=1,\ldots,r$, and let
$U_{T_{0},f}$ be the set of points $q\in\mathbb{P}^{1}$ such that
$e(q|t_{0})\neq e_{f}(t_{0})$ for $t_{0}=T_{0}(q)$. Then the number
$r_{T_{0}}$ of branch points of $f_{T_{0}}$ is at least

$r_{T_{0}}\geq(r-4)n+4+\sum_{i=1}^{r}(e_{i}-2)a_{i}+\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)}.$

Moreover, equality holds if and only if $T_{0}$ is unramified away from ${\bf
t}$ and its ramification indices over $t_{i}$ are either $e_{i}$ or not
divisible by $e_{i}$, for $i=1,\ldots,r$.

###### Proof.

Let $b_{i}$ denote the number of preimages $q$ in $T_{0}^{-1}(t_{i})$ such
that $e(q|t_{i})$ is not divisible by $e_{i}$, for $i=1,\ldots,r$. Note that
since $f_{T_{0}}$ is connected, Abhyankar’s lemma implies that

(7) $r_{T_{0}}=\sum_{i=1}^{r}b_{i}.$

By the Riemann–Hurwitz formula for $T_{0}$ one has:

(8) $2n-2=\sum_{t_{0}\in\mathbb{P}^{1}}\sum_{q\in
T_{0}^{-1}(t_{0})}(e(q|t_{0})-1)\geq\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)}+\sum_{i=1}^{r}(e_{i}-1)a_{i},$

with equality if and only if $e(q|t_{i})$ is either $e_{i}$ or non-divisible
by $e_{i}$, for all points $q\in T_{0}^{-1}(t_{i}),i=1,\ldots,r$. The same
Riemann–Hurwitz formula also implies

$2n-2\geq\sum_{i=1}^{r}(n-a_{i}-b_{i})=rn-\sum_{i=1}^{r}(a_{i}+b_{i}),$

with equality if and only if $T_{0}$ is unramified away from $\bf t$. Combined
with (8) this gives:

$\displaystyle 2n-2$ $\displaystyle\geq rn-\sum_{i=1}^{r}(a_{i}+b_{i})$
$\displaystyle\geq rn+\sum_{i=1}^{r}(e_{i}-2)a_{i}+\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)}-(2n-2)-\sum_{i=1}^{r}b_{i}.$

Combined with (7) this gives

(9)
$r_{T_{0}}=\sum_{i=1}^{r}b_{i}\geq(r-4)n+4+\sum_{i=1}^{r}(e_{i}-2)a_{i}+\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)},$

with equality if and only if $T_{0}$ is unramified away from ${\bf t}$ and
every ramification index $e(q|t_{i})$ for $q\in T_{0}^{-1}(t_{i})$,
$i=1,\ldots,r$ which is divisible by $e_{i}$, is equal to $e_{i}$. ∎

Let $E$ denote the multiset $\\{e_{1},\ldots,e_{r}\\}$.

###### Proof of Theorem 1.2 when $r\geq 4$ and
$E\neq\\{2,2,2,3\\},\\{2,2,2,4\\}$.

The case where $f$ is of genus $1$ follows from Remark 3.3, so henceforth we
shall assume that the genus of $X$ is at least $2$. Let $(g_{1},\ldots,g_{r})$
be a tuple in the Nielsen class of ${\bf C}$, corresponding to $f$. As any
permutation of the tuple ${\bf C}=(C_{1},\ldots,C_{r})$ has a non-empty
Nielsen class, without loss of generality we may assume that $g_{i}$ is a
branch cycle over $t_{i}$ and the orders $e_{1},\ldots,e_{r}$ of
$g_{1},\ldots,g_{r}$ are ordered in decreasing order. This tuple can be
modified to a tuple ($P_{y}$) $y,y^{-1}g_{1},\ldots,g_{r}$ for any $y\neq
g_{1}$. Such a tuple generates $G$ and has product $1$, giving a non-empty
Nielsen class corresponding to a ramification type which we denote by
$(r+1,{\bf D}_{y})$. We will show that for a suitable choice of $y\in G$, no
connected pullback $f_{T_{0}}$, along $T_{0}\in k(U)\setminus k$ of degree
$n>1$, has ramification type $(r+1,{\bf D}_{y})$. Since a cover with
ramification type $(r+1,{\bf D}_{y})$ is a pullback of $f$ only if $y$ is a
power of some element in $C_{1},\ldots,C_{r}$, by varying $y$, we may assume
that every conjugacy class in $G$ is a power of one of $C_{1},\ldots,C_{r}$.

As in Lemma 4.1, let $a_{i}$ (resp. $b_{i}$) denote the number of preimages
$q$ in $T_{0}^{-1}(t_{i})$ such that the ramification index $e(q|t_{i})$ is
divisible by $e_{i}$ (resp. is not divisible by $e_{i}$) for $i=1,\ldots,r$.
For $r\geq 6$, as $n\geq 2$, the lower bound on $r_{T_{0}}$ from Lemma 4.1 is
at least $r+2$, as desired.171717We note that for $r\geq 6$, this claim also
follows for any choice of $y$ from [Dèb18, Theorem 3.1.(b-2)], since the
latter implies that every pullback of $\bf C$ has at least $r+2$ branch
points, and hence is not in ${\tensf H}_{G,r}({\bf D}_{y}).$

The case $r=5$: We may assume $r_{T_{0}}=6$. By Lemma 4.1

$n\leq 2-\sum_{i=1}^{r}(e_{i}-2)a_{i}-\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)}.$

As $n>1$, this forces $n=2$, $e(q|t_{0})=1$ for all $q\in U_{T_{0},f}$, and
$\sum_{i=1}^{r}(e_{i}-2)a_{i}=0$. Thus, every ramification index of $T_{0}$
over $t_{i}$ is either $e_{i}$ or $1$, and $T_{0}$ is unramified away from
${\bf t}$. Since $T_{0}$ is of degree $2$, it ramifies over exactly two branch
points with ramification index $2$. Hence without loss of generality we may
assume $e_{4}=e_{5}=2$, so that the inertia canonical invariant of $f_{T_{0}}$
is ${\bf C}_{T_{0}}=(C_{1},C_{1},C_{2},C_{2},C_{3},C_{3})$. If $e_{1}>2$, by
picking $y$ to be an involution, then every cover with ramification type
$(r+1,{\bf D}_{y})$ is not a pullback of $f$, as ${\bf D}_{y}$ has more
conjugacy classes of involutions than ${\bf C}_{T_{0}}$ does. Similarly, if
$e_{1}=2$ and $G$ contains an element $y$ of order $>2$, then every cover
whose ramification type coincides with $(r+1,{\bf D}_{y})$ is not a pullback
of $f$, as its Nielsen class has a conjugacy class of non-involutions. If
$e_{1}=2$ and all elements of $G$ are of order $2$, then $G$ is an elementary
abelian $2$-group generated by three conjugacy classes ($C_{1},C_{2},C_{3}$).
As $G$ is not a subgroup of $\operatorname{PGL}_{2}(\mathbb{C})$, this forces
$G\cong(\mathbb{Z}/2\mathbb{Z})^{3}$. In the latter case, we may choose $y$ to
be an involution whose conjugacy class is different from
$C_{1},C_{2},C_{3},C_{4},C_{5}$, so that every cover with ramification type
$(r+1,{\bf D}_{y})$ is not a pullback of $f$.

The case $r=4$: Assume $r_{T_{0}}=5$. By Lemma 4.1

(10) $\sum_{i=1}^{r}(e_{i}-2)a_{i}+\sum_{q\in
U_{T_{0},f}}\bigl{(}e(q|t_{0})-1\bigr{)}\leq 1.$

Let ${\bf t}_{2}$ (resp. ${\bf t}_{>2}$) denote the set of $t_{i}$ with
$e_{i}=2$ (resp. $e_{i}>2$). Since every point $q\in T_{0}^{-1}(t_{i})$,
$t_{i}\in{\bf t}_{2}$, with ramification index $e(q|t_{0})>2$ contributes at
least $2$ to (10), we deduce that $e(q|t_{i})=1$ or $2$ for all $q\in
T_{0}^{-1}(t_{i})$, $t_{i}\in{\bf t}_{2}$. On the other hand for $t_{i}\in{\bf
t}_{>2}$, (10) gives $a_{i}=0$ with the possible exception of a single
$t_{\iota}$ for which $a_{t_{\iota}}=1$ and $e_{\iota}=3$. Moreover, if such
$\iota$ exists, then the contribution of the sum over $U_{T_{0},f}$ in (10) is
$0$, and hence

(11) $\begin{array}[]{l}\text{$e(q|t_{i})=1$ for all points $q\in
T_{0}^{-1}(t_{i})$, $t_{i}\in{\bf t}_{>2}$, with the possible }\\\
\text{exception of a single $q_{\iota}\in T_{0}^{-1}(t_{\iota})$ where
$e_{\iota}=e(q_{\iota}|t_{\iota})=3$.}\end{array}$

Otherwise $a_{i}=0$ for all $t_{i}\in{\bf t}_{>2}$, in which case (10) shows
that

(12) $\begin{array}[]{l}\text{$e(q|t_{i})=1$ for all points $q\in
T_{0}^{-1}(t_{i})$, $t_{i}\in{\bf t}_{>2}$, with the possible }\\\
\text{exception of a single $q_{\eta}\in T_{0}^{-1}(t_{\eta})$ where
$e(q_{\eta}|t_{\eta})=2\neq e_{\eta}$.}\end{array}$

If $G$ is of odd order, (11) and (12) force $T_{0}$ to have at most one
ramification point, which by the Riemann–Hurwitz formula forces a
contradiction to $n>1$. Henceforth assume $G$ is of even order.

Let $a=\\#{\bf t}_{>2}$. Since the genus of $X$ is more than $1$ and $r=4$,
the Riemann–Hurwitz formula for $f$ implies that $a\geq 1$. Assume first that
$a>1$. In this case we let $y\in G$ be an involution. The number of elements
of order $>2$ in $(P_{y})$ is $a$ or $a-1$. In case there exists $\iota$ as
above, (11) forces every point $t_{i}\in{\bf t}_{>2}\setminus\\{t_{\iota}\\}$
to have at least three unramified preimages and hence forces $f_{T_{0}}$ to
have at least $3(a-1)$ branch points with ramification index $>2$. As
$3(a-1)>a$ for $a>1$, $f_{T_{0}}$ does not not have ramification type
$(r+1,{\bf D}_{y})$. The same argument applies if (12) holds and $n\geq 3$. If
$n=2$ and (12) holds, then $f_{T_{0}}$ has at least $2(a-1)+1$ branch points
with ramification index $>2$. Similarly $2(a-1)+1>a$ and hence $f_{T_{0}}$
does not have ramification $(r+1,{\bf D}_{y})$.

Henceforth assume $a=1$, that is $E=\\{e,2,2,2\\}$. Note that $e>4$ by
assumption. Condition (11) does not hold since $e>3$, and hence (12) does. The
latter implies that the inertia canonical conjugacy classes of $f_{T_{0}}$
over points in $T_{0}^{-1}(t_{1})$ are either $C_{1}$ or $C_{1}^{2}$, hence of
order $e$ or $e/2$. Thus if we pick $y$ to be of order different from $e$ and
$e/2$, then the only element of $(P_{y})$ that can appear in such conjugacy
class is $yx_{1}$. Thus, (12) implies that

(13) $\begin{array}[]{l}\text{$T_{0}^{-1}(t_{1})$ consists of a single point
$q_{\eta}$ with ramification $e(q_{\eta}|t_{1})=2$ such that}\\\ \text{the
inertia canonical invariant of $f_{T_{0}}$ over $q_{\eta}$ is the conjugacy
class of $yx_{1}$.}\end{array}$

The latter then has to be of order $\tilde{e}=e/2$ or $e$.

At first consider the case where the conjugacy classes $C_{2},C_{3},C_{4}$ do
not coincide. Without loss of generality we may then assume that $C_{2}$ is
different from $C_{3}$ and $C_{4}$. Picking $y=x_{3}$, (13) implies that
$n=2$, and $(r+1,{\bf D}_{y})$ is a ramification type consisting of conjugacy
classes of orders $2,\tilde{e},2,2,2$, with $\tilde{e}=e$ or $e/2$. In
particular, $\tilde{e}>2$ and $C_{2}$ appears at most once in ${\bf D}_{y}$.
Since $n=2$ and $f_{T_{0}}$ is assumed to have $5$ branch points, $T_{0}$ has
to ramify over exactly one of the places $t_{2},t_{3},t_{4}$, say $t_{k}$. If
$k=2$, then $C_{2}$ does not appear in the ramification type of $f_{T_{0}}$,
contradicting its appearance in ${\bf D}_{y}$. If $k=3$ or $4$, then $t_{2}$
is unramified under $T_{0}$ and hence $C_{2}$ appears at least twice in the
ramification type of $f_{T_{0}}$, but only once in ${\bf D}_{y}$,
contradiction.

Now consider the case $C_{2}=C_{3}=C_{4}$. Note that since $G$ is non-cyclic,
$C_{2}$ is not a power of $C_{1}$. Assume next that $e$ is even. We may then
pick $y$ to be an involution which is a power of $x_{1}$. As $y$ is not of
order $e$ or $e/2$, we get that $y\not\in C_{1}\cup C_{1}^{2}\cup C_{2}$,
contradicting that those are the only conjugacy classes that may appear in the
ramification type of $f_{T_{0}}$. Next assume $e$ is odd, and pick a prime $p$
dividing $e$. If $p=e$, then $G$ is solvable by Burnside’s theorem. Letting
$N$ be a minimal normal subgroup of $G$, it follows that $N$ contains exactly
one of $C_{1}$ and $C_{2}$. The product $1$ relation then gives a
contradiction in $G/N$. Thus, we may assume $p$ is a proper divisor of $e$. In
this case, if we pick $y$ to be a power of $x_{1}$ of order $p$, then (13)
implies that the only conjugacy classes appearing in the ramification type of
$f_{T_{0}}$ are $C_{1},C_{1}^{2}$ or $C_{2}$, neither of which contains $y$,
contradiction. ∎

######  Remark 4.2.

In the following we shall use Magma for computations with small order groups.
More specifically, we use the command ExtensionsOfElementaryAbelianGroup to
run over extensions of a given group by an elementary abelian group.

Finally, we show that if $E$ is $\\{3,2,2,2\\}$ or $\\{4,2,2,2\\}$, then
either $G\subseteq\operatorname{PGL}_{2}(\mathbb{C})$ or there is no group $G$
whose maximal conjugacy classes are the classes of a product $1$ tuple
$x_{1},\ldots,x_{4}$ with orders in $E$. Since in both cases $\\#G$ is
divisible by at most two primes, $G$ is solvable by Burnside’s theorem. Let
$N$ be a minimal normal subgroup of $G$, so that
$N\cong(\mathbb{Z}/2\mathbb{Z})^{u}$ or $(\mathbb{Z}/3\mathbb{Z})^{u}$, for
$u\geq 1$.

Case {3,2,2,2}: If $N\cong(\mathbb{Z}/3\mathbb{Z})^{u}$, the images of
$x_{2},x_{3},x_{4}$ in $G/N$ remain of order $2$ and hence
$G/N\cong(\mathbb{Z}/2\mathbb{Z})^{2}$. Since moreover $G/N$ acts transitively
on the $\mathbb{Z}/3\mathbb{Z}$ subgroups of $N$, it follows that $u=1$ or
$2$. A check using Magma (Remark 4.2) shows that all such group extensions $G$
contain an element of order $6$.

In the case $N\cong(\mathbb{Z}/2\mathbb{Z})^{u}$, $N$ contains exactly one of
the conjugacy classes $C_{2},C_{3},C_{4}$ since otherwise we get a
contradiction to the product $1$ relation in $G/N$. Since $G/N$ is generated
by three elements of orders $3,2$ and $2$ with product $1$, it is isomorphic
to $S_{3}$. As $G/N$ acts transitively on the $\mathbb{Z}/2\mathbb{Z}$ copies
in $N$, we have $u=1$ or $2$. Once again a Magma check shows that all such
group extensions $G$ contain an element of order $4$ or $6$, contradicting
that $C_{i}$, $i=1,\ldots,4$ are the only maximal ones.

Case {4,2,2,2}: In this case $G$ is a $2$-group, and
$N\cong(\mathbb{Z}/2\mathbb{Z})^{u}$. The conjugacy classes of involutions in
$G$ are $C_{1}^{2},C_{2},C_{3}$, and $C_{4}$. If $N$ does not contain
$C_{1}^{2}$, then the product one relation in $G/N$ implies that $N$ contains
exactly one of the conjugacy classes $C_{2},C_{3},C_{4}$. In this case $G/N$
is dihedral of order $8$, acting transitively on the $\mathbb{Z}/2\mathbb{Z}$
copies in $N$. Hence $u=1$ or $2$. A Magma check shows that such a $G$ either
contains an element of order $8$, or has more than four conjugacy classes of
involutions, or has more than two conjugacy classes of elements of order $4$
(in which case there is more than one conjugacy class of cyclic subgroups of
order $4$).

If $N$ contains $C_{1}^{2}$, then all elements of $G/N$ are involutions, and
hence $G/N$ is an elementary abelian $2$-group. Since we may assume $N$ is a
proper subgroup of $G$, the product $1$ relation in $G/N$ implies that $N$ one
or two of the conjugacy classes $C_{2},C_{3},C_{4}$. In the former case, $G/N$
is a $2$-group acting transitively on involutions in $N$, forcing $u=1$. In
this case, a Magma check shows that for such group extensions either the
number of conjugacy classes of involutions is more than $4$ or the number of
conjugacy classes of order $4$ elements is more than $2$. If $N$ contains two
of $C_{2},C_{3},C_{4}$, then $G/N\cong\mathbb{Z}/2\mathbb{Z}$. Thus by
minimality of $N$, we get that $u\leq 2$ and hence that $G$ is isomorphic to a
subgroup of $\operatorname{PGL}_{2}(\mathbb{C})$.

##  5\. Regular Parametricity and Genericity

For this section, let $k$ be an arbitrary field of characteristic zero and
$(T,U,V)$ a triple of indeterminates. Given a non-trivial finite group $G$,
let $F/k(T)$ be a $k$-regular Galois extension of group $G$ and branch point
set ${\bf{t}}=\\{t_{1},\dots,t_{r}\\}$. We also denote the genus of $F$ by $g$
and the ramification indices of $t_{1},\dots,t_{r}$ by $e_{1},\dots,e_{r}$,
respectively. The unordered $r$-tuple $(e_{1},\dots,e_{r})$ is finally denoted
by e. The main topic of the present section is the following question: given a
field extension $L/k$, is the extension $F/k(T)$ $L$-parametric?

###  5.1. Main results

In Theorem 5.1 below, which generalizes and complements Theorem 1.4 from §1,
we give three explicit field extensions $L_{1}/k$, $L_{2}/k$, and $L_{3}/k$,
independent of either the extension $F/k(T)$ or the group $G$, such that the
answer to the above question is in general negative if $L$ is taken among the
fields $L_{1}$, $L_{2}$, and $L_{3}$.

######  Theorem 5.1.

(a) Assume $g\geq 1$. Then, $F/k(T)$ is not $L_{1}$-parametric, where
$L_{1}=K(U)$ and $K\supset k$ is any overfield which is ample. More precisely,
infinitely many $K$-regular Galois extensions of $L_{1}$ of group $G$ are not
specializations of $FL_{1}/L_{1}(T)$.

(b) Assume one of the following three conditions holds:

(i) $G=S_{n}$ ($n\geq 4$),

(ii) $G=A_{n}$ ($n\geq 4$),

(iii) $G=D_{n}$ ($n\geq 2$ even).

Then, $F/k(T)$ is not $L_{2}$-parametric, where $L_{2}=K((V))(U)$ and
$K\supset k$ is any overfield which is algebraically closed. More precisely,
infinitely many $K((V))$-regular Galois extensions of $L_{2}$ of group $G$ are
not specializations of $FL_{2}/L_{2}(T)$.

(c) Assume either one of the following two conditions holds:

(i) $G$ is cyclic of even order, $r=2$, and
${\bf{t}}\not\subset\mathbb{P}^{1}(k)$,

(ii) $G$ is odd dihedral, $r=3$, and ${\bf{t}}\not\subset\mathbb{P}^{1}(k)$.

Then, $F/k(T)$ is not $L_{3}$-parametric, where $L_{3}=k(U)$. More precisely,
infinitely many ${k}$-regular Galois extensions of $L_{3}$ of group $G$ are
not specializations of $FL_{3}/L_{3}(T)$.

######  Remark 5.2.

Given a field $K$, the field $K((V))$ is ample. Hence, (b) is a special case
of (a) under the extra assumption $g\geq 1$ (e.g., if $G=S_{n}$ with $n\geq
5$, see below). However, as we shall see in §5.2.2, we shall use a different
approach, which works regardless of the value of $g$, to prove (b).

Since the case $g=0$ can occur only in the following five cases:

\- $G$ is cyclic and ${\bf{e}}=(|G|,|G|)$,

\- $G$ is dihedral and ${\bf{e}}=(2,2,|G|/2)$,

\- $G=A_{4}$ and ${\bf{e}}=(2,3,3)$,

\- $G=S_{4}$ and ${\bf{e}}=(2,3,4)$,

\- $G=A_{5}$ and ${\bf{e}}=(2,3,5)$,

the following three cases are the only ones which are not covered by Theorem
5.1:

(1) $G$ is cyclic of even order, $r=2$, and
${\bf{t}}\subset\mathbb{P}^{1}(k)$,

(2) $G$ is cyclic of odd order and $r=2$,

(3) $G$ is odd dihedral, $r=3$, and ${\bf{t}}\subset\mathbb{P}^{1}(k)$.

We handle them in Proposition 5.3 below:

######  Proposition 5.3.

Assume one of the conditions (1), (2), and (3) holds. Then, $F/k(T)$ is
generic. More precisely, for every field extension $L/k$ and every Galois
extension $E/L$ of group contained in $G$, one has $E=(FL)_{t_{0}}$ for
infinitely many points $t_{0}\in\mathbb{P}^{1}(L)$.

Theorem 5.1 and Proposition 5.3 are proved in §5.2 and §5.3, respectively.

A first consequence of Theorem 5.1 and Proposition 5.3 is the following: to
prove that the extension $F/k(T)$ is generic, it suffices to check the
$L$-parametricity property for each of the three explicit extensions
$L_{1}/k$, $L_{2}/k$, and $L_{3}/k$ given in Theorem 5.1. They actually
provide a full description of all $k$-regular Galois extensions of $k(T)$ that
are generic. Specifically, we have this corollary which generalizes Theorem
1.5 from §1:

######  Corollary 5.4.

The following three conditions are equivalent:

(a) the extension $F/k(T)$ is generic,

(b) the extension $F/k(T)$ is $\overline{k}((V))(U)$-parametric and
${k}(U)$-parametric,

(c) one of the following three conditions holds:

(i) $G$ is cyclic of even order $n$ such that $e^{2i\pi/n}\in k$, $r=2$, and
${\bf{t}}\subset\mathbb{P}^{1}(k)$,

(ii) $G$ is cyclic of odd order $n$ such that $e^{2i\pi/n}+e^{-2i\pi/n}\in k$
and $r=2$,

(iii) $G$ is dihedral of order $2n$ with $n\geq 3$ odd and
$e^{2i\pi/n}+e^{-2i\pi/n}\in k$, $r=3$, and
${\bf{t}}\subset\mathbb{P}^{1}(k)$.

In particular, $G$ occurs as the group of a $k$-regular Galois extension of
$k(T)$ that is generic iff one of the following three conditions holds:

$\bullet$ $G$ is cyclic of even order $n$ and $e^{2i\pi/n}\in k$,

$\bullet$ $G$ is cyclic of odd order $n$ and $e^{2i\pi/n}+e^{-2i\pi/n}\in k$,

$\bullet$ $G$ is dihedral of order $2n$ with $n\geq 3$ odd and
$e^{2i\pi/n}+e^{-2i\pi/n}\in k$.

###### Proof.

The corollary is a combination of Theorem 5.1, Proposition 5.3, and the
following two statements on regular realizations of cyclic and odd dihedral
groups, which are consequences of the rigidity method and the Branch Cycle
Lemma:

\- given $n\geq 2$, there is a $k$-regular Galois extension of $k(T)$ of group
$\mathbb{Z}/n\mathbb{Z}$ and with two branch points iff
$e^{2i\pi/n}+e^{-2i\pi/n}\in k$; both branch points can be chosen to be in
$\mathbb{P}^{1}(k)$ iff $e^{2i\pi/n}\in k$,

\- given $n\geq 3$ odd, there is a $k$-regular Galois extension of $k(T)$ of
group $D_{n}$, with three branch points, and all branch points in
$\mathbb{P}^{1}(k)$ iff $e^{2i\pi/n}+e^{-2i\pi/n}\in k$. ∎

######  Remark 5.5.

If $k$ is algebraically closed, then, the assumptions of Theorem 5.1(c) cannot
happen. In particular, condition (b) above can then be replaced by

(b)’ the extension $F/k(T)$ is ${k}((V))(U)$-parametric.

###  5.2. Proof of Theorem 5.1

We break the proof into three parts, corresponding to the three statements
(a), (b), and (c) of Theorem 5.1.

####  5.2.1. Proof of (a)

Assume $g\geq 1$ and let $K\supset k$ be an overfield as in (a). As shown in
the proof of Theorem 3.7(a-1), Condition 3.8 holds for every finite group and
every ample field of characteristic zero. Hence, by Theorem 3.9, there are
infinitely many $K$-regular Galois extensions $E/K(U)$ of group $G$ each of
which satisfies $E\not=(FK(U))_{t_{0}}$ for each $t_{0}\in K(U)\setminus K$.
Pick such an extension $E/K(U)$ and suppose there is
$t_{0}\in\mathbb{P}^{1}(K(U))$ such that $E=(FK(U))_{t_{0}}$. One then has
$t_{0}\in\mathbb{P}^{1}(K)$ by the previous property. Lemma 2.1 then yields
$E=(FK)_{t_{0}}K(U)$. As $E/K(U)$ is $K$-regular, one has $(FK)_{t_{0}}=K$ and
then $E=K(U)$, which cannot happen. Hence, (a) holds.

####  5.2.2. Proof of (b)

Let $K$ be an algebraically closed field containing $k$, set $M=K((V))$, and,
for each $u\in M$, denote by $\mathfrak{P}_{u}$ the prime ideal of $M[U]$
generated by $U-u$. Below, we show that there are infinitely many $M$-regular
Galois extensions of $M(U)$ of group $G$ which are not specializations of
$FM(U)/M(U)(T)$, provided one of the following three conditions holds:

(i) $G=S_{n}$ ($n\geq 4$),

(ii) $G=A_{n}$ ($n\geq 4$),

(iii) $G=D_{n}$ ($n\geq 2$ even).

First, we need the following lemma, which is a function field analog of
[KLN17, Proposition 6.3] (which is stated over number fields):

######  Lemma 5.6.

For all but finitely many $u\in M$, the Galois group of the completion at
$\mathfrak{P}_{u}$ of every specialization of $FM(U)/M(U)(T)$ is cyclic.

###### Proof.

The proof is similar to that in the number field case and relies on [KLN17,
Theorem 4.1], which is the main result of that paper. For the convenience of
the reader, we offer a full proof below, with the necessary adjustments.

Let $t_{0}$ be in $\mathbb{P}^{1}(M(U))$ and $u$ in $M$. If
$({F}M(U))_{t_{0}}/M(U)$ is unramified at $\mathfrak{P}_{u}$, then, the Galois
group of its completion at $\mathfrak{P}_{u}$ clearly is cyclic. We may then
assume the extension $({F}M(U))_{t_{0}}/M(U)$ is ramified at
$\mathfrak{P}_{u}$. In particular, $t_{0}$ is not a branch point of
${F}/k(T)$. Indeed, if it was, then, it would be in
$\mathbb{P}^{1}(\overline{k})$. By Lemma 2.1, one would have

$({F}M(U))_{t_{0}}=({F}\overline{k})_{t_{0}}M(U)=M(U),$

which cannot happen as $({F}M(U))_{t_{0}}/M(U)$ ramifies at
$\mathfrak{P}_{u}$. Up to dropping finitely many values of $u$ (depending only
on ${F}M(U)/M(U)(T)$), one may use the Specialization Inertia Theorem of
[Leg16, §2.2] to get that $t_{0}$ meets some branch point of ${F}/k(T)$, say
$t$, modulo $\mathfrak{P}_{u}$181818We refer to [Leg16, Definition 2.2] for
more details about the terminology.. As above, one has $({F}M(U))_{t}=M(U)$.
Denote the inertia group of ${F}k(t)/k(t)(T)$ at the prime ideal $\langle
T-t\rangle$ by $I_{t}$. Up to dropping finitely many values of $u$ (depending
only on ${F}M(U)/M(U)(T)$), [KLN17, Theorem 4.1] then shows that the Galois
group of the completion at $\mathfrak{P}_{u}$ of $({F}M(U))_{t_{0}}/M(U)$
embeds into $I_{t}$. As $I_{t}$ is cyclic, we are done. ∎

Then, to prove (b), it suffices, by Lemma 5.6, to show that the following
holds:

($*$) for each $u\in M$, there is a $M$-regular Galois extension of $M(U)$ of
group $G$ and whose completion at $\mathfrak{P}_{u}$ has non-cyclic Galois
group.

First, assume condition (i) holds. Given $u\in M$, consider the Galois
extension

$E_{1}/M(U)=M(U)(\sqrt{U-u},\sqrt{V})/M(U).$

Pick distinct elements $\alpha_{1},\dots,\alpha_{n-4}$ of $M[U]$ and set

$P_{1}(Y)=(Y^{2}-(U-u))(Y^{2}-V)(Y-\alpha_{1})\cdots(Y-\alpha_{n-4}).$

Then, denote the trivial extension $M(U)/M(U)$ by $E_{2}/M(U)$. Pick distinct
elements $\beta_{1},\dots,\beta_{n}$ of $M[U]$ and set

$P_{2}(Y)=(Y-\beta_{1})\cdots(Y-\beta_{n}).$

Finally, let $E_{3}/M(U)$ be a Galois extension of group $S_{n}$. Pick
$\gamma_{0},\dots,\gamma_{n-1}\in M[U]$ such that

$P_{3}(Y)=Y^{n}+\gamma_{n-1}Y^{n-1}+\cdots+\gamma_{1}Y+\gamma_{0}$

is separable and $E_{3}$ is the splitting field of $P_{3}(Y)$ over $M(U)$. By
polynomial interpolation, there are polynomials $a_{0}(T),\dots,a_{n-1}(T)\in
M[U][T]$ such that

$Y^{n}+a_{n-1}(i)Y^{n-1}+\cdots+a_{1}(i)Y+a_{0}(i)=P_{i}(Y)$

for each $i\in\\{1,2,3\\}$. Let $\mathcal{F}$ be the splitting field over
$M(U)(T)$ of

$Y^{n}+a_{n-1}(T)Y^{n-1}+\cdots+a_{1}(T)Y+a_{0}(T).$

By the above and since $P_{1}(Y)$, $P_{2}(Y)$, and $P_{3}(Y)$ are separable,
the points 1, 2, and 3 are not branch points of $\mathcal{F}/M(U)(T)$ and the
corresponding specialized extensions are $E_{1}/M(U)$, $E_{2}/M(U)$, and
$E_{3}/M(U)$, respectively. In particular, $\mathcal{F}/M(U)(T)$ is
$M(U)$-regular (since $E_{2}/M(U)$ is trivial) and it has Galois group $S_{n}$
(since this is true for $E_{3}/M(U)$). By the compatibility between the
Hilbert specialization property and the weak approximation property of
$\mathbb{P}^{1}$, there are infinitely many points $t_{0}$ in $M(U)$ such that
$\mathcal{F}_{t_{0}}/M(U)$ has Galois group $S_{n}$ and its completion at
$\mathfrak{P}_{u}$ is the completion of $E_{1}/M(U)$ at $\mathfrak{P}_{u}$,
that is, the extension

$M((U-u))(\sqrt{U-u},\sqrt{V})/M((U-u)),$

which has Galois group $(\mathbb{Z}/2\mathbb{Z})^{2}$. Moreover, given
$u^{\prime}\in M\setminus\\{u\\}$, for infinitely many such points $t_{0}$,
one may require that the completion of $\mathcal{F}_{t_{0}}/M(U)$ at
$\mathfrak{P}_{u^{\prime}}$ is the completion at $\mathfrak{P}_{u^{\prime}}$
of $E_{2}/M(U)$, that is, the extension

$M((U-u^{\prime}))/M((U-u^{\prime})).$

Consequently, $\mathcal{F}_{t_{0}}/M(U)$ is $M$-regular for such a $t_{0}$.
Hence, condition ($*$) holds.

Now, assume condition (ii) holds. Given $u\in M$, set $u^{\prime}=u-V^{2}$ and
consider the Galois extension

$E/M(U)=M(U)(\sqrt{(U-u^{\prime})V},\sqrt{(U-u)(U-u^{\prime})})/M(U),$

which is of group $(\mathbb{Z}/2\mathbb{Z})^{2}$. Clearly, the completion of
$E/M(U)$ at $\mathfrak{P}_{u}$ has Galois group $(\mathbb{Z}/2\mathbb{Z})^{2}$
and that at $\mathfrak{P}_{u^{\prime}}$ is totally ramified of degree 2. By a
classical result of Mestre (see [Mes90] and [KM01, Theorem 3]), the extension
$E/M(U)$ occurs as a specialization of some $M(U)$-regular Galois extension
$\mathcal{F}/M(U)(T)$ of group $A_{n}$. As under condition (i), there are
infinitely many points $t_{0}$ in $M(U)$ such that $\mathcal{F}_{t_{0}}/M(U)$
has Galois group $A_{n}$ and its completion at $\mathfrak{P}_{u}$ (resp., at
$\mathfrak{P}_{u^{\prime}}$) is the completion of $E/M(U)$ at
$\mathfrak{P}_{u}$ (resp., at $\mathfrak{P}_{u^{\prime}}$), which is not
cyclic (resp., which is totally ramified). In particular, by the last
condition, $\mathcal{F}_{t_{0}}/M(U)$ is $M$-regular, thus proving condition
($*$).

Finally, assume condition (iii) holds. Set

$P(U,Y)=Y^{2n}-UY^{n}+V^{n}\in M[U,Y]$

and let $y\in\overline{M(U)}$ be a root of $P(U,Y)$. Clearly, $y^{n}$ is equal
to $(U\pm\sqrt{U^{2}-4V^{n}})/2,$ thus showing that the fields $M(U)(y^{n})$
and $M(U)(\sqrt{U^{2}-4V^{n}})$ coincide. Moreover, $y^{n}$ is of valuation
w.r.t. $\infty$ equal to $\pm 1$. Then, by the Capelli lemma (see, e.g.,
[Lan02, Chapter VI, §9, Theorem 9.1]), the polynomial $Y^{n}-y^{n}$ is
irreducible over $\overline{M}(U)(\sqrt{U^{2}-4V^{n}})$. As a consequence, the
polynomial $P(U,Y)$ is irreducible over $\overline{M}(U)$; denote the field
$M(U)(y)$ by $E$. Clearly, $e^{2i\pi/n}y$ is a root of $P(U,Y)$ and it is
easily checked that the same holds for $V/y$. Consequently, the distinct
elements $e^{2i\pi m/n}y$, $m\in\\{0,\dots,n-1\\}$, and $V/(e^{2i\pi m/n}y)$,
$m\in\\{0,\dots,n-1\\}$, of $\overline{M(U)}$ are the distinct roots of
$P(U,Y)$, which are in $E$. Hence, the $M$-regular extension $E/M(U)$ is
Galois. Moreover, there are $\sigma$ and $\tau$ in ${\rm{Gal}}(E/M(U))$ such
that $\sigma(y)=e^{2i\pi/n}y$ and $\tau(y)=V/y$. Since
$\langle\sigma,\tau\rangle=D_{n}$, the group ${\rm{Gal}}(E/M(U))$ is then
equal to $D_{n}$. Now, set $u=2V^{n/2}\in M$ (as $n$ is even). Then, the
completion of $E/M(U)$ at $\mathfrak{P}_{u}$ contains

$M((U-u))(\sqrt{U^{2}-4V^{n}})=M((U-u))(\sqrt{(U-u)(U+u)})$

(which is the completion at $\mathfrak{P}_{u}$ of the quadratic subfield of
$E/M(U)$) and

$M((U-u))(\sqrt{V})$

(as $\sqrt{V}$ is a root of the specialized polynomial $P(u,Y)$). As the
former (resp., the latter) is (totally) ramified (resp., unramified) of degree
2 over $M((U-u))$, the group $(\mathbb{Z}/2\mathbb{Z})^{2}$ is a quotient of
the Galois group of the completion of $E/M(U)$ at $\mathfrak{P}_{u}$, which
cannot be cyclic. Hence, up to applying suitable changes of variable,
condition ($*$) holds.

######  Remark 5.7.

(a) Under condition (i) or condition (iii), shorter proofs of condition ($*$)
could have been given, by using that the involved groups have a generic
polynomial over $K$, and a classical result of Saltman (see [Sal82, Theorem
5.9]) asserting that, under the existence of such a polynomial, Grunwald
problems (that is, approximation of finitely many local extensions by a global
one of given group) can be solved. However, as we intend to apply our approach
to generic polynomials (see Appendix B), we have deliberately avoided such
tools above.

(b) A property shared by all groups in Theorem 5.1(b) that we have used in our
local approach is that they have a non-cyclic abelian subgroup. Under the sole
condition that $G$ has a non-cyclic abelian subgroup, the same tools show that
there are infinitely many $M$-regular Galois extensions of $M(U)$ of group
contained in $G$ which are not specializations of $FM(U)/M(U)(T)$ 191919In
particular, the extension $F/k(T)$ is not strongly $M(U)$-parametric in the
sense of the upcoming Definition 5.9..

Indeed, $G$ then contains $(\mathbb{Z}/p\mathbb{Z})^{2}$ for some prime number
$p$. By Lemma 5.6, it suffices to show that, for each $u\in M$, there is a
$M$-regular Galois extension of $M(U)$ of group $(\mathbb{Z}/p\mathbb{Z})^{2}$
and whose completion at $\mathfrak{P}_{u}$ has Galois group
$(\mathbb{Z}/p\mathbb{Z})^{2}$. Fix $u\in M$ and set

$E_{u}=M(U)(\sqrt[p]{U-u},\sqrt[p]{V+U-u}).$

Then, the group of the Galois extension $E_{u}/M(U)$ is
$(\mathbb{Z}/p\mathbb{Z})^{2}$ since the subextension
$M(U)(\sqrt[p]{U-u})/M(U)$ (resp., $M(U)(\sqrt[p]{V+U-u})/M(U)$) is totally
ramified at $\mathfrak{P}_{u}$ (resp., unramified at $\mathfrak{P}_{u}$) of
degree $p$. The same argument shows that the completion of $E_{u}/M(U)$ at
$\mathfrak{P}_{u}$ is also of Galois group $(\mathbb{Z}/p\mathbb{Z})^{2}$.
Finally, we claim that $E_{u}/M(U)$ is $M$-regular. Indeed, if it was not,
then, one would have

$E_{u}=M(U)(\sqrt[p]{V},\sqrt[p]{U-u}).$

But this last equality cannot happen as the branch point sets of the
extensions $E_{u}/M(U)$ and $M(U)(\sqrt[p]{V},\sqrt[p]{U-u})/M(U)$ are
$\\{u,u-V,\infty\\}$ and $\\{u,\infty\\}$, respectively.

####  5.2.3. Proof of (c)

First, assume $G=\mathbb{Z}/n\mathbb{Z}$ for some even $n\geq 2$ and $r=2$. As
$n$ is even, there is a $k(U)$-regular Galois extension of ${k}(U)(T)$ of
group $G$, with a branch point in $\mathbb{P}^{1}(k(U))$, and with another
branch point of ramification index $n$. Then, by [Leg16, Corollary 3.4], there
is a prime $\mathfrak{P}$ of $k[U]$ such that, for all but finitely many $u$
in $k$, there is a Galois extension $E_{u}/k(U)$ of group $G$, which ramifies
at $\mathfrak{P}_{u}=\langle U-u\rangle$, and whose ramification index at
$\mathfrak{P}$ is $n$. In particular, $E_{u}/k(U)$ is $k$-regular (by the last
condition). Suppose $E_{u}/k(U)$ is a specialization of $Fk(U)/k(U)(T)$ for
infinitely many $u\in k$. Without loss, we may assume $\infty\not\in{\bf{t}}$.
For $i\in\\{1,2\\}$, denote the minimal polynomial of $t_{i}$ over $k$ by
$m_{i}(T)$. Then, by [Leg16, Corollary 2.12 and Remark 3.11], the reduction
modulo $\mathfrak{P}_{u}$ of $m_{1}(T)m_{2}(T)$ (viewed as a polynomial in
$k[U][T]$) has a root in the residue field $k[U]/\mathfrak{P}_{u}$ for some
$u\in k$. As this residue field is $k$, the polynomial $m_{1}(T)m_{2}(T)$ has
a root in $k$. Hence, by the Branch Cycle Lemma, $t_{1}$ and $t_{2}$ are in
$\mathbb{P}^{1}(k)$.

Now, assume $G=D_{n}$ for some odd $n\geq 3$ and $r=3$. As $n$ is odd, the
ramification indices $e_{1}$, $e_{2}$, and $e_{3}$ are 2, 2, and $n$,
respectively (up to reordering). In particular, by the Branch Cycle Lemma (and
as $n\not=2$), $t_{3}$ is in $\mathbb{P}^{1}(k)$. By [FJ08, §16.2 and
Proposition 16.4.4], every $k$-regular quadratic extension of $k(U)$ embeds
into a $k$-regular Galois extension of $k(U)$ of group $G$. Hence, if all but
finitely many $k$-regular Galois extensions of $k(U)$ of group $G$ are
specializations of $Fk(U)/k(U)(T)$, then, as $G$ has a unique subgroup of
index 2, all but finitely many $k$-regular quadratic extensions of $k(U)$ have
to occur as specializations of the quadratic subextension of $Fk(U)/k(U)(T)$.
As this quadratic subextension has only two branch points (namely, $t_{1}$ and
$t_{2}$), one may use a similar argument as in the cyclic case to get that
these branch points have to be in $\mathbb{P}^{1}(k)$.

###  5.3. Proof of Proposition 5.3

Assume one of the following three conditions holds:

(1) $G$ is cyclic of even order, $r=2$, and
${\bf{t}}\subset\mathbb{P}^{1}(k)$,

(2) $G$ is cyclic of odd order and $r=2$,

(3) $G$ is odd dihedral, $r=3$, and ${\bf{t}}\subset\mathbb{P}^{1}(k)$.

Let $L/k$ be a field extension and $E/L$ a Galois extension of group contained
in $G$. Below, we show that the extension $E/L$ occurs as the specialized
extension of the extension $FL/L(T)$ at $t_{0}$ for infinitely many points
$t_{0}\in\mathbb{P}^{1}(L)$.

By the twisting lemma [Dèb99], there is a $k$-regular extension
$(FL)_{E}/L(T)$ such that $(FL)_{E}\overline{L}=F\overline{L}$ and, given
$t_{0}\in\mathbb{P}^{1}(L)\setminus{\bf{t}}$, if there is a prime ideal lying
over $\langle T-t_{0}\rangle$ in $(FL)_{E}/L(T)$ with residue degree 1, then,
$E/L$ is the specialized extension of $FL/L(T)$ at $t_{0}$. In each case, the
genus of $F$ is $0$ (if (3) holds, this follows from ${\bf{e}}$ being
$(2,2,|G|/2)$). Hence, $(FL)_{E}$ has genus 0 as well. It then suffices to
find $t\in\mathbb{P}^{1}(L)$ for which there is a prime ideal lying over
$\langle T-t\rangle$ in $(FL)_{E}/L(T)$ with residue degree 1.

If (1) holds, then, the unique prime ideal lying over $\langle T-t_{1}\rangle$
in $(FL)_{E}/L(T)$ has residue degree 1. If (2) holds, the desired conclusion
follows from $G$ being of odd order and the genus being equal to 0; see, e.g.,
[Ser92, §1.1] for more details. Finally, assume (3) holds. As already seen,
the ramification indices $e_{1}$, $e_{2}$, and $e_{3}$ are 2, 2, and $n$,
where $|G|=2n$, respectively (up to reordering). As
$\\{t_{1},t_{2}\\}\subset\mathbb{P}^{1}(k)$, we may assume the quadratic
subfield of $F$ is $k(\sqrt{T})$ (up to applying a suitable change of
variable). Hence, there is $d\in L\setminus\\{0\\}$ such that $(FL)_{E}$
contains the quadratic field $L(\sqrt{dT})$. Set $Y=\sqrt{dT}$. The extension
$(FL)_{E}/L(Y)$ is of degree $n$ and it has only two branch points; it is then
Galois of group $\mathbb{Z}/n\mathbb{Z}$ and of genus 0. As $n$ is odd, there
exists $y_{0}\in L$ such that the specialized extension of $(FL)_{E}/L(Y)$ at
$y_{0}$ is $L/L$. Consequently, there is a prime ideal lying over $\langle
T-(y_{0})^{2}/d\rangle$ in $(FL)_{E}/L(T)$ with residue degree 1, thus
concluding the proof of Proposition 5.3.

###  5.4. On Schinzel’s problem and its variants

In this subsection, we explain how to obtain a conjectural counter-example to
the problem of Schinzel recalled as Question 1.6, Question 1.7, and the
Working Hypothesis of [Dèb18].

To do this, consider the polynomial
$P(T)=T^{3}+(5/4)T^{2}-2T-7\in\mathbb{Q}[T]$ and the number field
$K=\mathbb{Q}(\sqrt{-3},\sqrt[3]{11})$. Let
$f_{1}:X_{1}\rightarrow\mathbb{P}^{1}_{\mathbb{Q}}$ and
$f_{2}:X_{2}\rightarrow\mathbb{P}^{1}_{\mathbb{Q}}$ be the $\mathbb{Q}$-Galois
covers given by the affine equations $Y^{2}-P(T)$ and $Y^{2}-T$, respectively.

First, consider the elliptic curve $C:Y^{2}=P(T).$ Subject to the Birch and
Swinnerton-Dyer conjecture, for every non-square $u_{0}\in K$, the twisted
elliptic curve $C_{u_{0}}:Y^{2}=u_{0}P(T)$ has positive rank, and then
infinitely many $K$-rational points; see [DD09] for more details. Moreover, as
$C$ has rank 2 over $\mathbb{Q}(\sqrt{-3})$ (checked with Magma), $C$ has
infinitely many $K$-rational points. Hence, the following equivalent two
conditions hold:

(i) the containment
${\tensf{SP}}(f_{2}\otimes_{\mathbb{Q}}K)\subseteq{\tensf{SP}}(f_{1}\otimes_{\mathbb{Q}}K)$
holds,

(ii) for each $u_{0}\in K^{*}$, the polynomial $Y^{2}-u_{0}P(T)$ has a zero
$(y,t)$ in $K^{2}$ such that $y\not=0$.

Now, the cover $f_{1}$ has four branch points while $f_{2}$ has only two. In
particular, by Theorem 2.4, the following condition holds:

(iii) given a field $L$ of characteristic zero, $f_{2}\otimes_{\mathbb{Q}}L$
is not a rational pullback of $f_{1}\otimes_{\mathbb{Q}}L$.

By using Lemma 2.1, one sees that (iii) is equivalent to

(iv) given a field $L$ of characteristic zero, $L(\sqrt{U})/L(U)$ does not
occur as a specialization of $L(U)(T)(\sqrt{P(T)})/L(U)(T)$.

Finally, this shows that the following condition holds:

(v) the polynomial $Y^{2}-UP(T)$ has no zero in $K(U)^{2}$.

Indeed, suppose $Y^{2}-UP(T)$ has a zero $(Y(U),T(U))$ in $K(U)^{2}$. As
$P(T)$ is irreducible over $K$ (checked with Magma), $T(U)$ is not a root of
$P(T)$. Hence, $K(\sqrt{U})/K(U)$ is the specialization of
$K(U)(T)(\sqrt{P(T)})/K(U)(T)$ at $T(U)$, which cannot happen by (iv).

By using (i)-(v), one then obtains the following theorem:

######  Theorem 5.8.

Subject to the Birch and Swinnerton-Dyer conjecture,

(a) the answer to Question 1.6 is negative for the number field $K$ and the
polynomial $Y^{2}-UP(T)$,

(b) the answer to Question 1.7 is negative for the number field $K$ and the
covers $f_{1}\otimes_{\mathbb{Q}}K$ and $f_{2}\otimes_{\mathbb{Q}}K$,

(c) the following Working Hypothesis fails for the number field $K$ and the
sole $K(U)$-Galois cover $X\rightarrow\mathbb{P}^{1}_{K(U)}$ given by the
affine equation $Y^{2}-UP(T)$:

(WH) Let $L$ be a number field and
$f_{i}:X_{i}\rightarrow\mathbb{P}^{1}_{L(U)}$, $i=1,\ldots,N$, be
$L(U)$-covers. Assume none of the $L(U)$-curves $X_{1},\ldots,X_{N}$ has an
unramified202020by “unramified on $X_{i}$” we mean w.r.t. the cover
$f_{i}:X_{i}\rightarrow\mathbb{P}^{1}_{L(U)}$; similarly below “unramified on
$X_{i}|_{u_{0}}$” means w.r.t. the cover
$f_{i}|_{u_{0}}:X_{i}|_{u_{0}}\rightarrow\mathbb{P}^{1}_{L}$, $i=1,\ldots,N$.
$\mathbb{C}(U)$-rational point. Then, for infinitely many $u_{0}\in L$, the
covers $f_{1},\ldots,f_{N}$ have good reduction at $U=u_{0}$ and none of the
reduced curves $X_{1}|_{u_{0}},\ldots,X_{N}|_{u_{0}}$ has an unramified
$L$-rational point.

###  5.5. $L$-parametricity versus $L(U)$-parametricity

Let us now introduce the following strong variant of Definition 2.2(e):

######  Definition 5.9.

Given an overfield $L\supset k$, say that $F/k(T)$ is strongly $L$-parametric
if every Galois extension $E/L$ of group contained in $G$ is a specialization
of $FL/L(T)$.

######  Remark 5.10.

Similarly, one could say that the extension $F/k(T)$ is strongly generic if
$F/k(T)$ is strongly $L$-parametric for every field extension $L/k$. Clearly,
one would have

$F/k(T)$ strongly generic $\Rightarrow$ $F/k(T)$ generic.

However, Theorem 5.1 and Proposition 5.3 show that the converse holds.

To conclude this paper, we discuss connections between the
$L(U)$-parametricity (resp., the strongly $L(U)$-parametricity) and the
$L$-parametricity (resp., the strongly $L$-parame-tricity) properties for
overfields $L\supset k$.

Recall that, by [Dèb18, Remark 2.3], any $k$-regular Galois extension of
$k(T)$ is strongly $k$-parametric if it is strongly $k(U)$-parametric212121The
argument given there shows that this implication also holds if “strongly” is
removed twice.. However, there is no general converse:

$\bullet$ if $k$ is PAC222222Recall that a field $\kappa$ is Pseudo
Algebraically Closed (PAC) if every non-empty geometrically irreducible
$\kappa$-variety has a Zariski-dense set of $\kappa$-rational points. See,
e.g., [FJ08] for more on PAC fields., then, by [Dèb99], every $k$-regular
Galois extension of $k(T)$ is strongly $k$-parametric but, as noted in [Leg15,
Remark 7.2], some of them are not $k(U)$-parametric,

$\bullet$ if $k=\mathbb{C}$, then, by [Dèb18, Corollary 2.5] and Theorem
1.1(a) (see also Remark 2.3), the following three conditions are equivalent:

(a) $G$ is the group of a Galois extension of $k(T)$ that is strongly
$k(U)$-parametric,

(b) $G$ is the group of a Galois extension of $k(T)$ that is
$k(U)$-parametric,

(c) $G\subset{\rm{PGL}}_{2}(\mathbb{C}$).

Below, we provide two new situations for which there is no general converse.
The first one relies on Theorem 5.8:

Subject to the Birch and Swinnerton-Dyer conjecture, the $\mathbb{Q}$-regular
quadratic extension $\mathbb{Q}(T)(\sqrt{P(T)})/\mathbb{Q}(T)$ is strongly
$K$-parametric but not $K(U)$-parametric, where $P(T)=T^{3}+(5/4)T^{2}-2T-7$
and $K=\mathbb{Q}(\sqrt{-3},\sqrt[3]{11})$.

Our second example is devoted to Laurent series fields:

######  Proposition 5.11.

Assume $k$ is algebraically closed. Then, there is a Galois extension of
${k}(T)$ of group $G$ that is strongly $L$-parametric, where $L=K((V))$ and
$K\supset k$ is any overfield which is algebraically closed.

In contrast, by Theorem 5.1(a) and Theorem 5.1(b), no Galois extension of
${k}(T)$ of group $G$ is $L(U)$-parametric, provided $G$ is neither cyclic nor
odd dihedral232323The assumption that $k$ is algebraically closed is not
needed here..

Proposition 5.11 is a straightforward combination of the RET and the following
lemma, which does not use the assumption that $k$ is algebraically closed:

######  Lemma 5.12.

Let $K\supset k$ be an algebraically closed overfield and set $L=K((V))$.
Then, a given $k$-regular Galois extension of $k(T)$ of group $G$ and inertia
canonical invariant $(C_{1},\dots,C_{r})$ is strongly $L$-parametric iff the
following condition is satisfied:

($**$) for each element order $n$ in $G$, there is $i\in\\{1,\dots,r\\}$ such
that elements of $C_{i}$ have order divisible by $n$.

###### Proof.

Galois extensions of $L$ of group contained in $G$ are exactly extensions
$L(\sqrt[n]{V})/L$, where $n$ runs over the set of all element orders in $G$.
Since such an extension $L(\sqrt[n]{V})/L$ is (totally) ramified of index $n$
at the unique maximal ideal $\mathfrak{P}$ of $K[[V]]$, a given $k$-regular
Galois extension $\mathcal{F}/k(T)$ of group $G$ is strongly $L$-parametric
iff, for each element order $n$ in $G$, the extension $\mathcal{F}L/L(T)$ has
a specialization of ramification index $n$ at $\mathfrak{P}$.

First, assume condition ($**$) holds. Let $n$ be an element order in $G$. Pick
$i\in\\{1,\dots,r\\}$ such that the order $e$ of every element of $C_{i}$ is a
multiple of $n$, and set $e=nm$. By [Leg16, Theorem 3.1], there are infinitely
many points $t_{0}\in L$ such that the inertia group at $\mathfrak{P}$ of
$(\mathcal{F}L)_{t_{0}}/L$ is generated by an element of $C_{i}^{m}$. In
particular, the ramification index at $\mathfrak{P}$ of such a specialization
is $n$. Hence, $\mathcal{F}/{k}(T)$ is strongly $L$-parametric. Conversely,
assume $\mathcal{F}/k(T)$ is strongly $L$-parametric. Let $n$ be an element
order in $G$. By the above characterization, $\mathcal{F}L/L(T)$ has a
specialization of ramification index $n$ at $\mathfrak{P}$. Then, by the
Specialization Inertia Theorem, the inertia canonical invariant of
$\mathcal{F}/k(T)$ contains the conjugacy class of an element of $G$ of order
divisible by $n$. Hence, condition ($**$) holds. ∎

##  Appendix A Proof of Theorem 1.2 for $r=3$

Let $k$ be a fixed algebraically closed field of characteristic $0$. We start
with the case where the multiset of ramification indices
$E:=\\{e_{1},e_{2},e_{3}\\}$ of our cover is different from

(14)
$\\{2,4,5\\},\\{2,4,6\\},\\{2,3,7\\},\\{2,3,8\\},\\{3,3,4\\},\\{3,3,5\\}.$

In this case we shall use the following estimate:

######  Lemma A.1.

Let $f:X\rightarrow\mathbb{P}^{1}$ be a cover corresponding to a product $1$
tuple $x_{1}x_{2}x_{3}=1$ with $x_{i}$ of order $e_{i}$, $i=1,2,3$. If
$f_{T_{0}}$ is a pullback of $f$ along $T_{0}\in k(U)\setminus k$ with
corresponding tuple $x_{1}x_{1}^{-1}x_{2}x_{2}^{-1}=1$ and degree $n:=\deg
T_{0}$, then

$\frac{1}{2e_{3}}\geq\biggl{(}\frac{1}{2}-\frac{1}{n}\biggr{)}\biggl{(}1-\frac{1}{e_{1}}-\frac{1}{e_{2}}\biggr{)}.$

###### Proof.

Denote by $e_{i,j}$, $j=1,\ldots,b_{i},$ the ramification indices of $T_{0}$
over $t_{i}$ which are not multiples of $e_{i}$, and by $e(q|t_{i})$ the
ramification index of $q\in T_{0}^{-1}(t_{i})$ under $T_{0}$, for $i=1,2,3$.
The Riemann–Hurwitz formula for $T_{0}$ then gives

$\displaystyle 2n-2\geq\sum_{i=1}^{3}\sum_{q\in
T_{0}^{-1}(t_{i})}(e(q|t_{i})-1)$
$\displaystyle\geq\sum_{i=1}^{3}\biggl{(}\frac{n-\sum_{j=1}^{b_{i}}e_{i,j}}{e_{i}}(e_{i}-1)+\sum_{j=1}^{b_{i}}(e_{i,j}-1)\biggr{)}$
$\displaystyle=\sum_{i=1}^{3}\biggl{(}n\frac{e_{i}-1}{e_{i}}-b_{i}+\frac{1}{e_{i}}\sum_{j=1}^{b_{i}}e_{i,j}\biggr{)}.$

Since $f_{T_{0}}$ has four branch points, by Abhyankar’s lemma
$b_{1}+b_{2}+b_{3}=4$. Thus the previous estimate gives:

(15) $2\geq
n\biggl{(}1-\sum_{i=1}^{3}\frac{1}{e_{i}}\biggr{)}+\sum_{i=1}^{3}\sum_{j=1}^{b_{i}}\frac{e_{i,j}}{e_{i}}.$

Moreover, since $f_{T_{0}}$ has four branch points with ramification indices
$e_{1},e_{1},e_{2},e_{2}$, Abhyankar’s lemma implies that
$e_{i}/\gcd(e_{i,j},e_{i})=e_{1}$ or $e_{2}$, where each of the values is
obtained for exactly two of the $e_{i,j}$’s. In particular, $e_{i,j}/e_{i}$ is
at least $1/e_{1}$ for two of the $e_{i,j}$’s and at least $1/e_{2}$ for the
other two. Thus (15) gives

$2\geq
n\biggl{(}1-\sum_{i=1}^{r}\frac{1}{e_{i}}\biggr{)}+\frac{2}{e_{1}}+\frac{2}{e_{2}},$

which is equivalent to the desired assertion. ∎

######  Remark A.2.

We shall also repeatedly use Burnside’s theorem: If $G$ is a group of order
$p^{a}q^{b}$ for some primes $p,q$ and integers $a,b$, then $G$ is solvable.
If moreover, $p=q$ and $G$ has only two maximal conjugacy classes of cyclic
groups, and hence $G/[G,G]$ has at most $2$ maximal conjugacy classes, then
$G/[G,G]$ is cyclic. By Burnside’s basis theorem this implies that $G$ is
cyclic.

###### Proof of Theorem 1.2 when $r=3$ and $E$ is not in (14).

Suppose $f:X\rightarrow\mathbb{P}^{1}$ is a $G$-cover and $x_{1}x_{2}x_{3}=1$
is a product $1$ tuple corresponding to $f$. As oppose to the case $r\geq 4$,
we assume the orders $e_{1},e_{2},e_{3}$ of $x_{1},x_{2},x_{3}$ are ordered
increasingly. Since the genus of $X$ is non-zero as
$G\not\leq\operatorname{PGL}_{2}(\mathbb{C})$, and since the case where $X$ is
of genus $1$ follows from Remark 3.3, we shall assume that the genus of $X$ is
at least $2$. This implies by the Riemann–Hurwitz formula for $f$ that

(RHf) $\sum_{i=1}^{3}1/e_{i}<1.$

Let $C_{i}$ be the conjugacy class of $x_{i}$, for $i=1,2,3$. If every
ramification type with four branch points occurs as the ramification type of a
pullback of $f$, then every conjugacy class in $G$ is a power of one of
$C_{1},C_{2},C_{3}$. Hence, we divide the proof into cases according to
maximality of conjugacy classes.

Case 1: Assume next that $G$ has at least three maximal conjugacy classes of
cyclic subgroups. We show that no pullback $f_{T_{0}}$ has a product $1$ tuple
$(R_{1})$ $x_{1},x_{1}^{-1},x_{2},x_{2}^{-1}$. Denote by $n$ the degree of
$T_{0}\in k(U)\setminus k$. Since each conjugacy class is a power of a
conjugacy class in $\bf C$, there are exactly three maximal conjugacy classes
of cyclic groups. In particular, the cyclic subgroup generated by $C_{1}$
(resp. by $C_{2}$) is not a power of $C_{3}$. Hence, $b_{3}=0$, that is, all
ramification indices of $T_{0}$ over $t_{3}$ are divisible by $e_{3}$. In
particular, $e_{3}$ divides $n$. The only tuples $(e_{1},e_{2},e_{3})$
satisfying the latter, (RHf), and the assertion of Lemma A.1, are242424We
carry out such numeric calculations using Magma as follows. First note that
the inequality in Lemma A.1 for $e_{3}\geq e_{2}\geq e_{1}\geq 4$ and
$e_{3}\mid n$, implies $e_{1}=e_{2}=e_{3}=n=4$. The inequality for $e_{1}=3$,
$e_{3}\geq e_{2}\geq 4$, and $e_{3}\mid n$ implies $e_{2}=e_{3}=n=4$. The
inequality for $e_{3}\geq 5$, $e_{2}=e_{1}=3$, and $e_{3}\mid n$ implies
$e_{3}=n=5$. For $e_{1}=2$, $e_{2}\geq 4$, $e_{3}\geq 6$ and $e_{3}\mid n$, it
implies $e_{2}=4$, $e_{3}=n=6$. For $e_{1}=2$, $e_{2}=3$, $e_{3}\geq 8$ and
$e_{3}\mid n$, it implies $e_{3}=n=8$. If $e_{1}=e_{2}=2$, then $G$ is
generated by two involutions and hence dihedral. The remaining list of
possibilities is finite, and is computed using Magma. In similar sequel
inequalities, similar computations are carried out.  either in (14) or
$(3,4,4)$ with $n=4$, or $(4,4,4)$ with $n=4$, or $(2,5,5)$ with $n=5$. A
direct inspection shows that in the latter three cases, no pullback along a
function $T_{0}$ of that degree gives the tuple $(R_{1})$.

Since a finite group $G$ is never the union of conjugates of a proper subgroup
$H\leq G$, the group $G$ has a unique maximal conjugacy class of cyclic groups
if and only if $G$ is cyclic. We can therefore assume from now that $G$ has
two maximal conjugacy classes of cyclic groups and one of the classes in ${\bf
C}$ is not maximal. As $e_{3}\geq e_{i}$ for $i=1,2$, the (unique) non-maximal
conjugacy class of cyclic groups among $C_{1},C_{2},C_{3}$ is either $C_{1}$
or $C_{2}$.

Case 2: Assume $C_{2}$ is the only non-maximal conjugacy class. In this case,
since $C_{2}$ has to be a power of $C_{3}$, we have $e_{2}\mid e_{3}$.
Moreover as all ramification indices of $T_{0}$ over $t_{3}$ are multiples of
$e_{3}/e_{2}$, we have $(e_{3}/e_{2})\mid n$. The only tuples
$(e_{1},e_{2},e_{3})$ and values of $n$ which satisfy $e_{2}\mid e_{3}$,