# On Malle’s conjecture for the product of symmetric and nilpotent groups

Hrishabh Mishra  Chennai Mathematical Institute, H1, SIPCOT IT Park,
Kelambakkam, Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>and
Anwesh Ray  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a finite nilpotent group and $n\in\\{3,4,5\\}$. Consider
$S_{n}\times G$ as a subgroup of $S_{n}\times S_{|G|}\subset S_{n|G|}$, where
$G$ embeds into the second factor of $S_{n}\times S_{|G|}$ via the regular
representation. Over any number field $k$, we prove the weak form of Malle’s
conjecture (cf. [Mal02, p.316]) for $S_{n}\times G$ viewed as a subgroup of
$S_{n|G|}$. Our result requires that $G$ satisfies some mild conditions.

##  1\. Introduction

###  1.1. Motivation and historical context

Let $k$ be a number field and let $n\in\mathbb{Z}_{\geq 1}$. Let $\mathcal{G}$
be a transitive subgroup of $S_{n}$. Given a number field extension $K/k$,
denote by $\widetilde{K}$ its Galois closure over $k$. Suppose that
$[K\mathrel{\mathop{\mathchar 58\relax}}k]=n$. We enumerate the embeddings
$\iota_{j}\mathrel{\mathop{\mathchar 58\relax}}K\hookrightarrow\bar{K}$ for
$j=1,\dots,n$, and for $\sigma\in\mathcal{G}$, consider the composite

$\iota_{\sigma(j)}\mathrel{\mathop{\mathchar
58\relax}}K\xrightarrow{\iota_{j}}\bar{K}\xrightarrow{\sigma}\bar{K}.$

This gives rise to a natural permutation representation and we realize
$\operatorname{Gal}(\widetilde{K}/k)$ as a transitive subgroup of $S_{n}$. Set
$\operatorname{disc}(K/k)$ to denote the relative discriminant of $K/k$ and
take

$\operatorname{Disc}(K)\mathrel{\mathop{\mathchar
58\relax}}=|\operatorname{Norm}_{k/\mathbb{Q}}(\operatorname{disc}(K/k))|.$

For $X>0$, let $N_{k}(\mathcal{G};X)$ be the number of extensions $K/k$ with
$[K\mathrel{\mathop{\mathchar 58\relax}}k]=n$ for which
$\operatorname{Gal}(\widetilde{K}/k)$ is isomorphic to $\mathcal{G}$ as a
permutation subgroup of $S_{n}$, and $\operatorname{Disc}(K)\leq X$. The
quantity $N_{k}(\mathcal{G};X)$ depends not only on $\mathcal{G}$ and $k$, but
also on the embedding of $\mathcal{G}$ in $S_{n}$, and this is suppressed in
our notation. For $g\in S_{n}$, set

(1.1) $\operatorname{ind}(g)\mathrel{\mathop{\mathchar
58\relax}}=n-\text{number of orbits for }g.$

We set $1=1_{\mathcal{G}}$ to denote the identity element in $\mathcal{G}$ and
$\mathcal{G}^{\ast}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{G}\backslash\\{1\\}$.

Malle made predictions about the asymptotic growth of $N_{k}(\mathcal{G};X)$
as $X\rightarrow\infty$. The following conjecture (cf. [Mal02, p.316]) is
referred to as the _weak form of Malle’s conjecture_

######  Conjecture 1.1.

For all $\epsilon>0$,

$X^{a(\mathcal{G})}\ll_{k,\mathcal{G}}N_{k}(\mathcal{G};X)\ll_{k,\mathcal{G},\epsilon}X^{a(\mathcal{G})+\epsilon},$

where $a(\mathcal{G})\mathrel{\mathop{\mathchar
58\relax}}=\left(\operatorname{min}\\{\operatorname{ind}(g)\mid
g\in\mathcal{G}^{\ast}\\}\right)^{-1}.$

There is a stronger version of this conjecture, which states that

$N_{k}(\mathcal{G};X)\sim c(k,\mathcal{G})X^{a(\mathcal{G})}(\log
X)^{b(k,\mathcal{G})-1},$

where $b(k,\mathcal{G})\in\mathbb{Z}_{\geq 0}$ is a constant that depends on
$k$ and $\mathcal{G}$ (see [Mal04] for a precise statement). This form of
Malle’s conjecture has been shown to be false, and Klüners provided an
explicit counterexample, cf. [Klü05]. However, the weak version of the
conjecture is still widely believed to be true. The strong form is however
known for various families of groups, some of the well known cases are listed
below.

  * •

The conjecture has been proven for the abelian groups by Maki [Mäk85] and
Wright [Wri89].

  * •

For the groups $S_{n}$ for $n\leq 5$, the conjecture was settled by
Davenport–Heilbronn [DH71] for $n=3$, for $n=4,5$ over $\mathbb{Q}$ by
Bhargava [Bha05, Bha10] and for general number fields $k$ by
Bhargava–Shankar–Wang [BSW15].

  * •

For the dihedral group $D_{4}\subset S_{4}$ the conjecture was resolved by by
Cohen, Diaz Y. Diaz, and Olivier [CDO02].

  * •

For finite nilpotent groups $G$ sitting in $S_{|G|}$, satisfying additional
conditions, the conjecture was recently resolved by Koymans and Pagano [KP23].
The weak version of the conjecture for all finite nilpotent groups was
previously settled by Klüners and Malle [KM04].

  * •

Wang [Wan21] proved the strong form of the conjecture for $S_{n}\times A$,
where $n\in\\{3,4,5\\}$ and $A$ is a finite abelian group satisfying some mild
conditions. Here, $A$ is viewed as a subgroup of $S_{|A|}$ via the regular
representation, and $S_{n}\times A$ embeds naturally in $S_{n|A|}$ via the
inclusions

$S_{n}\times A\hookrightarrow S_{n}\times S_{|A|}\hookrightarrow S_{n|A|}.$

The conditions on $A$ were subsequently relaxed due to work of Masri, Thorne,
Wei-Lun and Wang [MTTW20].

###  1.2. Main result

Let $G$ be a finite nilpotent group and $\ell_{G}$ the smallest prime factor
of $|G|$. We recall that $a(G)=\frac{\ell_{G}}{(\ell_{G}-1)|G|}$ and
$a(S_{n})=1$ for all $n\geq 2$. It is easy to see that $a(S_{n}\times
G)=\frac{1}{|G|}$. We now state our main result.

######  Theorem 1.2.

Let $G$ be a non-trivial finite nilpotent group and $k$ a number field.
Consider the regular representation
$\operatorname{reg}_{G}\mathrel{\mathop{\mathchar 58\relax}}G\hookrightarrow
S_{|G|}$. Here, $S_{n}\times G$ embeds naturally in $S_{n|G|}$ via natural
inclusions

$S_{n}\times
G\xrightarrow{\operatorname{id}\times\operatorname{reg}_{G}}S_{n}\times
S_{|G|}\hookrightarrow S_{n|G|}.$

Then

$X^{\frac{1}{|G|}}\ll N_{k}(S_{n}\times
G;X)\ll_{\epsilon}X^{\frac{1}{|G|}+\epsilon}$

in the following cases:

  1. (1)

$n=3$, if $2\nmid|G|$,

  2. (2)

$n=4$, if $2,3\nmid|G|$,

  3. (3)

$n=5$, if $2,3,5\nmid|G|$.

Thus, Conjecture 1.1 is true in this setting.

This extends the main result of [Wan21]. The result is proven via a synthesis
of the methods of Klüners–Malle [KM04], Wang [Wan21] and Koymans–Pagano
[KP23]. We remark that the extra assumption from [KP23] is not required since
we do not attempt to prove the strong form of Malle’s conjecture. The
conditions on $G$ are consistent with those in [Wan21].

###  1.3. Organization

Including the introduction the paper consists of four sections. In section 2,
a number of preliminary results are proven on the discriminants of the
compositum of two number fields. Furthermore, some analytic tools are
introduced here. Many of the results here are based on those in [Wan21,
section 2]. In section 3, we recall a partial parameterization for finite
nilpotent extensions of $k$ with fixed Galois group $G$. This parameterization
is due to Koymans and Pagano [KP23], and does not require any additional
hypothesis on $G$. We use this to prove a local uniformity estimate for
nilpotent groups (cf. Proposition 3.6). The proof of the main theorem in given
in section 4.

###  1.4. Outlook

It is possible that the methods developed in this article could eventually
yield a proof of the strong form of Malle’s conjecture for $S_{n}\times G$.
The main difficulty is to obtain precise asymptotics for $G$-extensions
satisfying prescribed local conditions. This would potentially lead to a
sharper assertion than that of Proposition 4.4. We were informed that our
method can indeed be further developed to prove the strong form of Malle’s
conjecture, by leveraging the high-powered results in forthcoming work of
Alberts, Lemke-Oliver, Wang and Wood.

### Acknowledgments

The authors would like to thank Peter Koymans and Jiuya Wang for their
insightful comments.

##  2\. Preliminary results

We begin this section by recalling some notation. Throughout, $k$ will be a
fixed number field. Set $\Omega_{k}$ to denote the set of all primes of $k$.
For any non-zero ideal $\mathfrak{I}\subset\mathcal{O}_{k}$, set
$|\mathfrak{I}|\mathrel{\mathop{\mathchar
58\relax}}=|\operatorname{Norm}_{k/\mathbb{Q}}(\mathfrak{I})|$. Given any
finite extensions $K/k$ we denote by $\mathrm{disc}(K/k)$ the relative
discriminant ideal in $k$ and set $\mathrm{Disc}(K)\mathrel{\mathop{\mathchar
58\relax}}=|\mathrm{disc}(K/k)|$. We use $\mathfrak{p}$ to denote a finite
prime in $k$. Take $\mathrm{disc}_{\mathfrak{p}}(K/k)$ to be the ideal
$\mathfrak{p}^{\operatorname{ord}_{\mathfrak{p}}(\mathrm{disc}(K/k))}$ and
$\mathrm{Disc}_{\mathfrak{p}}(K)$ its absolute norm. We denote the inertia
group at a prime $\mathfrak{p}$ by $I_{\mathfrak{p}}(K)$. Let $G$ be a finite
nilpotent group and $n\in\\{3,4,5\\}$ be an integer. Note that $G$ sits inside
$S_{|G|}$ via the regular representation. Therefore, we view the product
$S_{n}\times G$ as a subgroup of $S_{n}\times S_{|G|}\subset S_{n|G|}$.

We prove a number of preliminary results for the product of $S_{n}\times G$.
We note that the case where $G$ is an abelian group is treated in [Wan21].
Throughout, it shall be assumed that $G$ satisfies the conditions of Theorem
1.2.

######  Proposition 2.1.

Let $K/k$ and $L/k$ be number field extensions with
$n\mathrel{\mathop{\mathchar 58\relax}}=[K\mathrel{\mathop{\mathchar
58\relax}}k]$ and $m\mathrel{\mathop{\mathchar
58\relax}}=[L\mathrel{\mathop{\mathchar 58\relax}}k]$. Assume that
$[KL\mathrel{\mathop{\mathchar 58\relax}}k]=[K\mathrel{\mathop{\mathchar
58\relax}}k][L\mathrel{\mathop{\mathchar 58\relax}}k]$. Then we have that

$\mathrm{Disc}(KL)\leq\mathrm{Disc}(K)^{m}\mathrm{Disc}(L)^{n}.$

###### Proof.

The stated result is [Wan21, Theorem 2.1]. ∎

If $K/k$ and $L/k$ are tamely ramified at $\mathfrak{p}$, then the inertia
groups at $\mathfrak{p}$ are cyclic. Let $n\mathrel{\mathop{\mathchar
58\relax}}=[K\mathrel{\mathop{\mathchar 58\relax}}k]$ and
$m\mathrel{\mathop{\mathchar 58\relax}}=[L\mathrel{\mathop{\mathchar
58\relax}}k]$. Let $g_{K}\in S_{n}$ and $g_{L}\in S_{m}$ respectively denote
generators for these groups. Then,
$\mathrm{disc}_{\mathfrak{p}}(K)=p^{\mathrm{ind}(g_{K})}$ and
$\mathrm{disc}_{\mathfrak{p}}(L)=p^{\mathrm{ind}(g_{L})}$, where
$\mathrm{ind}(g)$ denotes the index of $g$ (cf. (1.1)). The next result is
used to determine $\mathrm{disc}_{\mathfrak{p}}(KL)$ in terms of
$\mathrm{disc}_{\mathfrak{p}}(K)$ and $\mathrm{disc}_{\mathfrak{p}}(L)$ at the
tamely ramified places. Denote by $\widetilde{K}$ (resp. $\widetilde{L}$) the
Galois closure of $K$ (resp. $L$).

######  Proposition 2.2.

With respect to notation above, assume that $\widetilde{K}\cap\widetilde{L}=k$
and that $p$ is tamely ramified in $\widetilde{K}$ and $\widetilde{L}$. Then
the following assertions hold.

  1. (1)

Let $e_{K}$ and $e_{L}$ be the ramification indices of $\widetilde{K}$ and
$\widetilde{L}$ at $p$ with $(e_{K},e_{L})=1$. Then denote a generator of an
inertia group of $KL$ at $p$ by $g_{KL}$, we have

$\mathrm{ind}(g_{KL})=\mathrm{ind}(g_{K})\cdot m+\mathrm{ind}(g_{L})\cdot
n-\mathrm{ind}(g_{K})\cdot\mathrm{ind}(g_{L}).$

  2. (2)

Let the generator of an inertia group of $K$ at $p$ be $g_{K}=\prod_{i}c_{i}$,
and the generator of an inertia group of $L$ at $p$ be $g_{L}=\prod_{j}d_{j}$.
Then the generator $g_{KL}$ of an inertia group of $KL$ at $p$ satisfies

$\mathrm{ind}(g_{KL})=mn-\sum_{i,j}\mathrm{gcd}(|c_{i}|,|d_{j}|).$

Here $n=[K\mathrel{\mathop{\mathchar 58\relax}}k]$ and
$m=[L\mathrel{\mathop{\mathchar 58\relax}}k]$.

###### Proof.

The result above is [Wan21, Theorem 2.2, 2.3]. ∎

######  Lemma 2.3.

Given $K/k$ and $L/k$ with $\widetilde{K}\cap\widetilde{L}=k$. The local étale
algebra of the compositum $(KL)_{\mathfrak{p}}$ at a prime $p$ could be
determined by the local étale algebras $(K)_{\mathfrak{p}}$ and
$(L)_{\mathfrak{p}}$. In particular, the relative discriminant ideal
$\mathrm{disc}_{\mathfrak{p}}(KL)$ as an invariant of $(KL)_{\mathfrak{p}}$
could be determined by $(K)_{\mathfrak{p}}$ and $(L)_{\mathfrak{p}}$.

###### Proof.

The result is [Wan21, Theorem 2.4]. ∎

It is convenient to now present a group-theoretic version of Proposition 2.2.
This version will be later employed in index computation within the product of
appropriate symmetric and nilpotent groups.

######  Lemma 2.4.

Let $\sigma\in S_{n}$ and $\tau\in S_{m}$ with disjoint cycle decomposition
$\sigma=\prod_{i}c_{j}$ and $\tau=\prod_{j}d_{j}$. Suppose that
$(|c_{i}|,|d_{j}|)=1$ for all $i,j$ and consider $(\sigma,\tau)\in S_{n}\times
S_{m}\subset S_{nm}$ then

$\mathrm{ind}(\sigma,\tau)=\mathrm{ind}(\sigma)\cdot m+\mathrm{ind}(\tau)\cdot
n-\mathrm{ind}(\sigma)\cdot\mathrm{ind}(\tau).$

###### Proof.

Since $(|c_{i}|,|d_{j}|)=1$ for all $i,j$ the element $(c_{i},d_{j})\in
S_{nm}$ is a cycle of length $|c_{i}||d_{j}|$ for each $i,j$. These cycles in
$S_{nm}$ are disjoint from each other and
$(\sigma,\tau)=\prod_{(i,j)}(c_{i},d_{j})$. Hence, the number of cycles in the
decomposition of $(\sigma,\tau)$ is the product of the number of cycles in the
decomposition of $\sigma$ and $\tau$. We recall that the number of such cycles
is $n-\mathrm{ind}(\sigma)$ for $\sigma\in S_{n}$ and $m-\mathrm{ind}(\tau)$
for $\tau\in S_{m}$. Therefore,

$nm-\mathrm{ind}(\sigma,\tau)=(n-\mathrm{ind}(\sigma))(m-\mathrm{ind}(\tau))$

and the Lemma follows. ∎

Let $\ell_{1},\dots,\ell_{c}$ be the primes that divide $|G|$ and denote by
$G(\ell_{i})$ the $\ell_{i}$-Sylow subgroup of $G$. Since $G$ is a nilpotent
group, we have the product decomposition $G\mathrel{\mathop{\mathchar
58\relax}}=\prod_{i=1}^{c}G(\ell_{i})$. Denote by $\ell_{G}$ the smallest
prime factor of $|G|$. Given $\sigma\in S_{n}$ and $g\in G$ we compute the
index of $(\sigma,g)\in S_{n|G|}$ using the Lemma 2.4. The method employed by
Wang for abelian groups, as detailed in [Wan21, Lemmas 2.5, 2.6 and 2.7]
generalizes to nilpotent groups. For the sake of completeness, we provide
complete details. In what follows, we set $G^{\ast}$ to denote
$G\backslash\\{1_{G}\\}$ and $\operatorname{ind}(G)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{min}\\{\operatorname{ind}(g)\mid g\in G^{\ast}\\}$.

######  Proposition 2.5.

Let $G$ be a finite nilpotent group.

  1. (1)

When $n=3$, suppose that $2\nmid|G|$. Then we have that

$\mathrm{ind}((12),g)/|G|>2\text{ and }\mathrm{ind}((123),g)/|G|>1$

for all $g\in G$.

  2. (2)

For $n=4$ assume that $2,3\nmid|G|$. Then

$\mathrm{ind}((12),g)/|G|>2\text{ and }\mathrm{ind}((12)(34),g)/|G|>1,$
$\mathrm{ind}((123),g)/|G|>3\text{ and }\mathrm{ind}((1234),g)/|G|>2$

for all $g\in G$.

  3. (3)

When $n=5$ assume that $2,3,5\nmid|G|$. Then we have that

$\mathrm{ind}(\sigma,g)/|G|\geq 1+\mathrm{ind}(\sigma)-1/7$

for all $\sigma\in S_{5}$ and $g\in G$. Further if $\sigma$ is not conjugate
to $(12345)$ in $S_{5}$ then

$\mathrm{ind}(\sigma,g)/|G|\geq 12/7+\mathrm{ind}(\sigma).$

###### Proof.

Since $g$ sits inside $S_{m}$ via the regular representation, we have that

(2.1) $\operatorname{ind}(g)=|G|\left(1-\frac{1}{e(g)}\right)$

where $e(g)$ denotes the order of $g$. Note that since it is assumed that
$\ell_{G}>2$, and hence $e(g)>2$ for all $g\neq 1_{G}$. It follows that
$\operatorname{ind}(g)>\frac{|G|}{2}$ for all $g\in G^{\ast}$ and

$\operatorname{ind}(G)=|G|\left(1-\frac{1}{\ell_{G}}\right).$

We begin with (1). For $(12)\in S_{3}$, using Lemma 2.4 we deduce that for all
$g\in G$ we have that $\mathrm{ind}((12),g)=|G|+2\cdot\mathrm{ind}(g)$ as
$\mathrm{ind}(12)=1$. We deduce that $\mathrm{ind}((12),g)/|G|\geq
1+2\cdot\mathrm{ind}(G)/|G|>2$ since $\ell_{G}>2$.

For $(123)\in S_{3}$, if $3\nmid|G|$, then again using Lemma 2.4 we obtain
that for all $g\in G$, $\mathrm{ind}((123),g)=2|G|+\mathrm{ind}(g)$ as
$\mathrm{ind}(123)=2$. Hence the inequality holds in this case. If $3\mid|G|$,
then $G=G(3)\times\widetilde{G}$. Given any $g\in G$ we can write it as
$(g(3),\widetilde{g})\in S_{|G(3)|}\times S_{\widetilde{G}}\subset S_{|G|}$.
Further we note that
$\mathrm{ind}((123),g)=\mathrm{ind}((123),(g(3),\widetilde{g}))=\mathrm{ind}(((123),g(3)),\widetilde{g})$
in $S_{|G|}$. Since $3\nmid|\widetilde{G}|$ we can use Lemma 2.4 to deduce
that
$\mathrm{ind}(((123),g(3)),\widetilde{g})=\mathrm{ind}((123,g(3))\cdot(|\widetilde{G}|-\mathrm{ind}(\widetilde{g}))+3\cdot|G(3)|\cdot\mathrm{ind}(\widetilde{g})$.
Now, since the smallest possible value of $\mathrm{ind}((123,g(3))$ is
$2|G(3)|$ when $g(3)$ is identity and smallest possible value for
$\mathrm{ind}(\widetilde{g})$ is $\mathrm{ind}(\widetilde{G})$ by definition.
Hence,
$\mathrm{ind}((123,g(3))\cdot(|\widetilde{G}|-\mathrm{ind}(\widetilde{g}))+3\cdot|G(3)|\cdot\mathrm{ind}(\widetilde{g})\geq
2|G|+|G(3)|\cdot\mathrm{ind}(\widetilde{G})>|G|$. This concludes the proof.

(2) We note that $\mathrm{ind}(12),$ $\mathrm{ind}(123),$
$\mathrm{ind}(1234),$ and $\mathrm{ind}(12)(34)$ are $1,2,3,$ and $2$
respectively. Further $2,3\nmid|G|$ so we can use Lemma 2.4 and we deduce that
for all $g\in G$, $\mathrm{ind}((12),g)=|G|+3\cdot\mathrm{ind}(g)>2|G|$,
$\mathrm{ind}((123),g)=2|G|+2\cdot\mathrm{ind}(g)>3|G|$ as
$\mathrm{ind}(g)\geq\mathrm{ind}(G)=\frac{\ell_{G}-1}{\ell_{G}}|G|\geq\frac{4}{5}|G|$,
next $\mathrm{ind}(1234,g)=3|G|+\mathrm{ind}(g)>2|G|$ and
$\mathrm{ind}((12)(34),g)=2|G|+2\cdot\mathrm{ind}(g)>|G|$.

(3) Using Lemma 2.4 we conclude that
$\mathrm{ind}(\sigma,g)=\mathrm{ind}(\sigma)|G|+5\cdot\mathrm{ind}(g)-\mathrm{ind}(\sigma)\cdot\mathrm{ind}(g)\geq\mathrm{ind}(\sigma)\cdot|G|+(5-\mathrm{ind}(\sigma))\cdot\mathrm{ind}(g)\geq\mathrm{ind}(\sigma)\cdot|G|+(5-\mathrm{ind}(\sigma))\cdot\mathrm{ind}(G)\geq\mathrm{ind}(\sigma)\cdot|G|+(5-\mathrm{ind}(\sigma))\cdot(\ell_{G}-1)|G|/\ell_{G}\geq\mathrm{ind}(\sigma)\cdot|G|+(5-\mathrm{ind}(\sigma))\cdot
6|G|/7$ since $\ell_{G}\geq 7$. For $\sigma\in S_{5}$ not conjugate to
$(12345)$ we have that $(5-\mathrm{ind}(\sigma))\geq 2$ and hence
$\mathrm{ind}(\sigma)\cdot|G|+(5-\mathrm{ind}(\sigma))\cdot
6|G|/7\geq\mathrm{ind}(\sigma)\cdot|G|+12|G|/7$. This concludes the proof of
Proposition. ∎

###  2.1. Bound for product of two groups

In this section, we develop upon the results in [Wan21, Section 3] in a
framework that is appropriate to our applications. Let $S_{1}$ and $S_{2}$ be
multisets with entries in positive integers. We associate the following
counting functions with these multi-sets for $i=1,2$

(2.2) $F_{i}(X)\mathrel{\mathop{\mathchar 58\relax}}=\\#\\{s_{i}\in
S_{i}\mathrel{\mathop{\mathchar 58\relax}}s_{i}\leq X\\}.$

We suppose that $F_{i}(X)$ is finite for all $X>0$ and $i=1,2$. Given positive
integers $a$ and $b$, consider the following product counting function,

(2.3) $P_{a,b}(X)\mathrel{\mathop{\mathchar 58\relax}}=\\#\\{(s_{1},s_{2})\in
S_{1}\times S_{2}\mathrel{\mathop{\mathchar 58\relax}}s_{1}^{a}s_{2}^{b}\leq
X\\}.$

The next result shows that given asymptotic bounds for $F_{1}(X),F_{2}(X)$ one
can obtain results about bounds on the product counting function.

######  Proposition 2.6.

Let $a,b$ be positive integers and $\alpha\in(0,1)$ is a positive real number
satisfying the relation

(2.4) $b-a\cdot\alpha>0.$

Furthermore, suppose that

$F_{1}(X)\leq C_{1}X\text{ and }F_{2}(X)\leq C_{2}X^{\alpha}$

for some constants $C_{1},C_{2}>0$. Then, we have the following upper bound

(2.5) $P_{a,b}(X)\leq\frac{(b+(1-\alpha)a)\cdot
C_{1}C_{2}}{b-a\cdot\alpha}X^{1/a}.$

###### Proof.

Given $r\in(0,\infty)$ we set $n_{i}(r)$ to be the number of times $r$ occurs
in the set $S_{i}$. Then from the definition we deduce that,

(2.6) $\begin{split}P(X)&=\sum_{r_{1}^{a}r_{2}^{b}\leq
X}n_{1}(r_{1})n_{2}(r_{2})=\sum_{r^{b}\leq X}n_{2}(r)F_{1}((X/r^{b})^{1/a})\\\
&\leq C_{1}X^{1/a}\sum_{r^{b}\leq X}\frac{n_{2}(r)}{r^{b/a}}.\end{split}$

Using Abel’s summation formula and $b-a\cdot\alpha>0$.

(2.7) $\begin{split}\sum_{r^{b}\leq X}\frac{n_{2}(r)}{r^{b/a}}&\leq
C_{2}X^{\frac{\alpha}{b}-\frac{1}{a}}+C_{2}\int_{1}^{X}\frac{t^{\alpha}}{t^{b/a+1}}dt,\\\
&\leq C_{2}\left(1+\int_{1}^{\infty}t^{\alpha-b/a-1}dt\right).\end{split}$

The integral is bounded above by $\frac{a}{b-a\cdot\alpha}$. Substituting the
the upper bound (2.7) back into (2.6), we obtain the result. ∎

A lower bound for $P(X)$ is given below.

######  Proposition 2.7.

Suppose there exist real constants $c_{1},c_{2}>0$ such that

$c_{1}X\leq F_{1}(X)\text{ and }c_{2}X^{\alpha}\leq F_{2}(X)$

for some $\alpha\in(0,1)$ and $b-a\cdot\alpha>0$ then we have that,

(2.8) $cX^{1/a}\leq P(X),$

where $c\mathrel{\mathop{\mathchar
58\relax}}=\min\\{c_{1}c_{2},c_{1}c_{2}a/(b-a\cdot\alpha)\cdot(1-2^{\alpha-b/a})\\}$.

###### Proof.

From the definition of $P(X)$ we deduce that,

(2.9) $\begin{split}P(X)&=\sum_{r_{1}^{a}r_{2}^{b}\leq
X}n_{1}(r_{1})n_{2}(r_{2})=\sum_{r^{b}\leq X}n_{2}(r)F_{1}((X/r^{b})^{1/a})\\\
&\geq c_{1}X^{1/a}\sum_{r^{b}\leq X}\frac{n_{2}(r)}{r^{b/a}}.\end{split}$

Using Abel’s summation formula,

$\sum_{r\leq
X^{1/b}}\frac{n_{2}(r)}{r^{b/a}}\geq\frac{c_{2}\underline{X}^{\alpha}}{\underline{X}^{b/a}}+c_{2}\int_{1}^{\underline{X}}\frac{t^{\alpha}}{t^{b/a+1}}dt,$

where $\underline{X}$ denotes the largest integer smaller or equal to
$X^{1/b}$ such that $n_{2}(\underline{X})>0$. Noting that the integral
$\int_{1}^{\underline{X}}\frac{t^{\alpha}}{t^{b/a+1}}dt=a/(b-a\cdot\alpha)\cdot(1-\underline{X}^{\alpha-b/a})$.
Hence, $\int_{1}^{\underline{X}}\frac{t^{\alpha}}{t^{2}}dt\geq
a/(b-a\cdot\alpha)\cdot(1-2^{\alpha-b/a})$ for $\underline{X}\geq 2$. This
concludes the proof. ∎

Let $G$ be an appropriate finite nilpotent group satisfying condition in
Theorem 1.2 for some $n\in\\{3,4,5\\}$. Consider a property $\mathcal{P}_{1}$
of degree $n$, $S_{n}$-extensions of $k$, and a property $\mathcal{P}_{2}$ of
$G$-extensions of $k$ and an ordering of such extensions by some invariants
$\operatorname{inv}_{1}$ and $\operatorname{inv}_{2}$ valued in positive
integers respectively. Assume that the following bounds are satisfied

(2.10) $\begin{split}&\\#\\{K\in\mathcal{P}_{1}\mid\text{ and
}\operatorname{inv}_{1}(K)\leq X\\}\ll_{k,S_{n}}X\\\
&\\#\\{L\in\mathcal{P}_{2}\mid\text{ and }\operatorname{inv}_{2}(L)\leq
X\\}\ll_{k,\epsilon}X^{\alpha+\epsilon}.\end{split}$

Then, as an immediate application of previous results by taking the multi-sets
$S_{1},S_{2}$ to be the multi-sets consisting of $\operatorname{inv}_{1}(K)$
and $\operatorname{inv}_{2}(L)$, as $K$ and $L$ range over $\mathcal{P}_{1}$
and $\mathcal{P}_{2}$ respectively. The product counting function,

(2.11) $\begin{split}P_{|G|,n,k}(X)=&\\#\\{(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq
S_{n},\operatorname{Gal}(L/k)\simeq G,\\\
&K\in\mathcal{P}_{1},L\in\mathcal{P}_{2},\text{ and
}\operatorname{inv}_{1}(K)^{|G|}\operatorname{inv}_{2}(L)^{n}\leq
X\\},\end{split}$

satisfies the upper bound,

$P_{|G|,n,k}(X)\ll_{k,S_{n},G,\epsilon}X^{\frac{1}{|G|}+\epsilon},$

whenever $n-|G|\alpha>0$. Similarly, if we have lower bounds of the form,

(2.12) $\begin{split}X\ll_{k,S_{n}}\\#\\{K\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,\operatorname{Gal}(\widetilde{K}/k)\simeq
S_{n},K\in\mathcal{P}_{1},\text{ and }\operatorname{inv}_{1}(K)\leq X\\}\\\
X^{\alpha}\ll_{k,G}\\#\\{L\mathrel{\mathop{\mathchar
58\relax}}[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(L/k)\simeq G,L\in\mathcal{P}_{2},\text{
and }\operatorname{inv}_{2}(L)\leq X\\}.\end{split}$

Then we deduce that,

$X^{\frac{1}{|G|}}\ll_{k,S_{n},G}P_{|G|,n,k}(X),$

whenever $n-|G|\alpha>0$. We take note of these observations below.

######  Corollary 2.8.

With respect to notation above, assume that

  1. (1)

the bounds (2.10) and (2.12) are satisfied for
$\alpha\mathrel{\mathop{\mathchar 58\relax}}=a(G)$,

  2. (2)

$n-|G|a(G)>0$.

Then we have the following bounds,

$X^{\frac{1}{|G|}}\ll_{k,S_{n},G}P_{|G|,n,k}(X)\ll_{k,S_{n},G,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

###### Proof.

The result follows from (2.10) and (2.12) and Propositions 3.5 and 3.5. ∎

##  3\. Local uniformity estimates

In this section, we state local uniformity estimates for degree $n$ extensions
of $k$ with Galois group $S_{n}$ given in [Wan21, Section 4] for $n=3,4,$ and,
$5$. We refer to an extension $K/k$ as a degree $n$, $S_{n}$-extension if
$[K\mathrel{\mathop{\mathchar 58\relax}}k]=n$ and
$\operatorname{Gal}(\widetilde{K}/k)\simeq S_{n}$. Furthermore, we use the
parameterization for nilpotent extensions developed in [KP23] to prove a
similar estimate for $G$-extensions of $K$, where $G$ is any finite nilpotent
group. We refer to an extension $L/k$ as a $G$-extension if $L$ is Galois over
$k$ and $\operatorname{Gal}(L/k)\simeq G$. Let $\mathfrak{p}$ be a finite
prime of $k$, and $K/k$ be an $S_{4}$-extension then, we say that
$\mathfrak{p}$ is overramified in $K/k$ if $\mathfrak{p}$ factors into primes
in $K/k$ as $\mathfrak{B}^{4},\mathfrak{B}^{2},$ or
$\mathfrak{B}_{1}^{2}\mathfrak{B}_{2}^{2}$ for finite $p$. For an archimedean
$\mathfrak{p}$, it must factor into a product of two ramified places.

The local uniformity estimates for $S_{n}$ are stated below.

######  Theorem 3.1.

Let $\mathfrak{q}\subset\mathcal{O}_{k}$ be any (non-zero) squarefree ideal,
and write $\mathfrak{q}=\prod_{i=1}^{f}\mathfrak{p}_{i}$, where
$\mathfrak{p}_{1},\dots\mathfrak{p}_{f}$ are distinct prime ideals. Then, the
following assertions hold.

  1. (1)

Let $M_{3,\mathfrak{q}}(k,X)$ be the number of non-cyclic cubic extensions
$K/k$ such that

     * •

$K/k$ is totally ramified at $\mathfrak{p}_{1},\dots,\mathfrak{p}_{f}$,

     * •

$\mathrm{Disc}(K)\leq X$.

Then, for any $\epsilon>0$, we have that

$M_{3,\mathfrak{q}}(k,X)=O_{\epsilon}(X/|\mathfrak{q}|^{2-\epsilon}).$

  2. (2)

The number of $S_{4}$ quartic extensions $K/k$ which are overramified at all
primes $\mathfrak{p}_{1},\dots,\mathfrak{p}_{f}$ and $\mathrm{Disc}(K)\leq X$
is denoted $M_{4,\mathfrak{q}}(k,X)$. Then, for any $\epsilon>0$, we obtain
the bound

$M_{4,\mathfrak{q}}(k,X)=O_{\epsilon}(X/|\mathfrak{q}|^{2-\epsilon}).$

  3. (3)

The number of $S_{5}$ quintic extensions $K/k$ which are totally ramified at
all primes $\mathfrak{p}_{1},\dots,\mathfrak{p}_{f}$ and $\mathrm{Disc}(K)\leq
X$ is denoted $M_{5,\mathfrak{q}}(k,X)$ and one has the following bound

$M_{5,\mathfrak{q}}(k,X)=O_{\epsilon}(X/|\mathfrak{q}|^{2/5-\epsilon}).$

In each case, the implied constant is independent of the ideal $q$.

###### Proof.

The statement (1) is [DW88][Proposition 6.2] and for the proof of (2),(3) we
refer to [Wan21][Theorem 4.3, Theorem 1.3] respectively. ∎

###  3.1. A parameterization for nilpotent extensions

Let $G$ be a finite nilpotent group. We recall that $G$ is a product of its
Sylow subgroups $G=\prod_{j=1}^{c}G(\ell_{j})$. Fix an algebraic closure
$\bar{k}$ of $k$, and set $\operatorname{G}_{k}$ to denote the absolute Galois
group $\operatorname{Gal}(\bar{k}/k)$. In this section, we briefly recall the
parameterization of $G$-extensions of $k$ from [KP23, section 2]. This
parameterization comes in handy in the proof of the local uniformity estimate
for $G$-extensions.

We begin by considering the case when $G$ is an $\ell$-group, i.e., $|G|$ is a
power of $\ell$. Let $S_{\infty}\subset\Omega_{k}$ be the archimedian primes
of $k$. We view $k$ is a subfield of $\bar{\mathbb{Q}}$ and set
$\operatorname{G}_{k}$ to denote the Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}/k)$. Likewise, for each prime
$\mathfrak{q}\in\Omega_{k}$, denote by $k_{\mathfrak{q}}$ the completion of
$K$ at $q$ and set
$\operatorname{G}_{k_{\mathfrak{q}}}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{\mathbb{Q}}_{\mathfrak{p}}/k_{\mathfrak{q}})$.
For each prime $q\in\Omega_{k}$, choose an embedding
$\iota_{\mathfrak{q}}\mathrel{\mathop{\mathchar
58\relax}}\bar{k}\hookrightarrow\bar{k}_{\mathfrak{q}}$. The choice of
embedding induces an inclusion of
$\iota_{\mathfrak{q}}^{\ast}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{k_{\mathfrak{q}}}\hookrightarrow\operatorname{G}_{k}$.
Denote by $\operatorname{I}_{q}\subset\operatorname{G}_{k_{\mathfrak{q}}}$ the
inertia group at $q$. We set $\mathbb{F}_{\ell}$ to be the field with
$\ell$-elements, and set
$H^{1}_{\operatorname{nr}}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})$
is the image of the inflation map

$H^{1}(\operatorname{G}_{k_{\mathfrak{q}}}/\operatorname{I}_{q},\mathbb{F}_{\ell})\xrightarrow{\operatorname{inf}}H^{1}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell}).$

The classes in
$H^{1}_{\operatorname{nr}}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})$
are the unramified classes at $\ell$. Let
$\mathcal{G}_{k}^{\operatorname{pro}-\ell}$ denote the maximal pro-$\ell$
quotient of $\operatorname{G}_{k}$.

Given a subset $S\subset\Omega_{k}$ that contains $S_{\infty}$, consider the
natural restriction map

$\Phi(\ell,S)\mathrel{\mathop{\mathchar
58\relax}}H^{1}(\operatorname{G}_{k},\mathbb{F}_{\ell})\rightarrow\bigoplus_{q\in\Omega_{k}\setminus
S}\frac{H^{1}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})}{H^{1}_{\operatorname{nr}}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})}.$

There is a finite set of primes $S_{\operatorname{clean}}(\ell)$ containing
$S_{\infty}$ such that the map $\Phi(\ell,S_{\operatorname{clean}}(\ell))$ is
surjective, and such that the kernel
$\Phi(\ell,S_{\operatorname{clean}}(\ell))$ is finite (cf. [KP23, Proposition
2.1 and 2.2]. Let $\widetilde{\Omega}_{k}(\ell)$ be the subset of
$\Omega_{k}\setminus S_{\operatorname{clean}}(\ell)$ consisting of primes $q$
with

$\frac{H^{1}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})}{H^{1}_{\operatorname{nr}}(\operatorname{G}_{k_{\mathfrak{q}}},\mathbb{F}_{\ell})}\neq
0.$

Every prime
$q\in\Omega_{k}\setminus\left(S_{\operatorname{clean}}(\ell)\cup\widetilde{\Omega}_{k}(\ell)\right)$
is unramified in all finite $\ell$-extensions of $K$ (cf. [KP23][Proposition
2.3]). For $\mathfrak{q}\in\widetilde{\Omega}_{k}(\ell)$ there exists a
character $\chi_{\mathfrak{q}}\in
H^{1}(\operatorname{G}_{K},\mathbb{F}_{\ell})$ such that
$\Phi(\ell,S_{\operatorname{clean}}(\ell))$ has nontrivial coordinate
precisely at $\mathfrak{q}$ and at no other prime in
$\widetilde{\Omega}_{k}(\ell)$. The set of characters $\chi_{\mathfrak{q}}$
(as $\mathfrak{q}$ ranges over $\widetilde{\Omega}_{k}(\ell)$) is a linearly
independent set. There is a positive integer $t$ and a basis

$J\mathrel{\mathop{\mathchar 58\relax}}=\\{\chi_{i}\mid i=1,\dots,t\\}$

of $\operatorname{ker}\left(\Phi(\ell,S_{\operatorname{clean}}(\ell))\right)$
such that

$\\{\chi_{\mathfrak{q}}\mid\mathfrak{q}\in\widetilde{\Omega}_{k}(\ell)\\}\cup
J$

is a basis of $H^{1}(\operatorname{G}_{K},\mathbb{F}_{\ell})$.

Let $[0,1]^{t}$ denote the set of all vectors $\vec{v}=(v_{1},\dots,v_{t})$
whose entries are either $0$ or $1$. Given $T\in[0,1]^{t}$, set $\pi_{j}(T)$
to denote its $j$-th coordinate. Set
$\mathcal{S}_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=[0,1]^{[t]}\times\mathcal{S}_{\ell}^{\prime}$, where
$\mathcal{S}_{\ell}^{\prime}$ is the set consisting of squarefree integral
ideals of $\mathcal{O}_{k}$ supported entirely on
$\widetilde{\Omega_{k}}(\ell)$. Denote by
$\operatorname{Prim}\left(\mathcal{S}_{\ell}^{G^{\ast}}\right)$ the set of
vectors supported on $G^{\ast}$ having pairwise coprime coordinates. Here, two
pairs $(T,\mathfrak{b})$ and $(T^{\prime},\mathfrak{b}^{\prime})$ are coprime
if $\mathfrak{b}$ and $\mathfrak{b}^{\prime}$ are coprime and there does not
exist a $j\in[t]$, such that $\pi_{j}(T)=\pi_{j}(T^{\prime})=1$. Elements of
$\operatorname{Prim}\left(\mathcal{S}_{\ell}^{G^{\ast}}\right)$ are vectors
$(v_{g})_{g\in G^{\ast}}$ where $v_{g}=(v_{g}{(1)},v_{g}{(2)})$. Let
$\operatorname{Epi}_{\operatorname{top-
gr}}\left(\mathcal{G}_{k}^{\operatorname{pro}-\ell};G\right)$ denote the set
of surjective homomorphisms
$\mathcal{G}_{k}^{\operatorname{pro}-\ell}\twoheadrightarrow G$.

######  Proposition 3.2.

There is a surjective map

$P_{G}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Prim}\left(\mathcal{S}_{\ell}^{G^{\ast}}\right)\twoheadrightarrow\operatorname{Epi}_{\operatorname{top-
gr}}\left(\mathcal{G}_{k}^{\operatorname{pro}-\ell};G\right)\cup\\{\cdot\\}.$

###### Proof.

The above result is [KP23, Proposition 2.8]. ∎

The pre-image of $\operatorname{Epi}_{\operatorname{top-
gr}}\left(\mathcal{G}_{k}^{\operatorname{pro}-\ell};G\right)$ is denoted by
$\operatorname{Prim}\left(\mathcal{S}_{\ell}^{G^{\ast}}\right)(\operatorname{solv.})$.
For an epimorphism $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{G}_{k}^{\operatorname{pro}-\ell}\twoheadrightarrow G$, set
$\operatorname{Disc}(\psi)$ to denote the relative discriminant of the
corresponding extension of $k$ that is fixed by the kernel of $\psi$. Given an
ideal $\mathfrak{b}$, denote by
$\operatorname{free}_{S_{\operatorname{clean}}(\ell)}(\mathfrak{b})$ to be the
largest ideal that divides $\mathfrak{b}$ and is supported outside
$S_{\operatorname{clean}}(\ell)$.

######  Proposition 3.3.

Let $v=(v_{g}(1),v_{g}(2))_{g\in G^{\ast}}$ be an element in
$\operatorname{Prim}\left(\mathcal{S}_{\ell}^{G^{\ast}}\right)(\operatorname{solv.})$,
then,

$\operatorname{free}_{S_{\operatorname{clean}}(\ell)}\left(\operatorname{Disc}(P_{G}(v))\right)=\prod_{g\in
G^{\ast}}v_{g}(2)^{|G|\left(1-\frac{1}{\\#\langle g\rangle}\right)}.$

###### Proof.

The result above is [KP23, Proposition 2.10]. ∎

Next, we consider the more general setting, where $G$ is a nilpotent group. By
taking the product of all the maps $P_{G(\ell_{j})}$, we get a map

$P_{G}\mathrel{\mathop{\mathchar
58\relax}}\prod_{j\in[c]}\operatorname{Prim}\left(\mathcal{S}_{\ell_{j}}^{G(\ell_{j})-\operatorname{id}}\right)\twoheadrightarrow\operatorname{Epi}_{\operatorname{top}-\operatorname{gr}}\left(\operatorname{G}_{k},G\right)\cup\\{\cdot\\}.$

We set $S$ to be union $\bigcup_{j\in[c]}S_{\operatorname{clean}}(\ell_{j})$,
and take $t_{j}$ to be the value of $t$ for the prime $\ell_{j}$. Define

$\mathcal{S}\mathrel{\mathop{\mathchar
58\relax}}=\\{0,1\\}^{[t_{1}+\dots+t_{c}]}\times\mathcal{S}^{\prime},$

where $\mathcal{S}^{\prime}$ is the set of squarefree ideals supported outside
$S$. Given a subset $\Delta\subset G^{\ast}$, let
$\operatorname{Prim}\left(\mathcal{S}^{\Delta}\right)$ be the set of all
tuples $(v_{g}(1),v_{g}(2))_{g\in\Delta}$ satisfying the following properties

  * •

writing $\Pi_{i}\mathrel{\mathop{\mathchar 58\relax}}[\sum
t_{i}]\rightarrow[t_{i}]$ for the natural projection map, then,
$\Pi_{i}(v_{g}(1))$ are pairwise coprime,

  * •

the $v_{g}(2)$ are pairwise coprime,

  * •

if $\mathfrak{p}$ divides $v_{g}(2)$ and $\ell$ is a prime that divides the
order of $g$, then,

$\\#\left(\mathcal{O}_{k}/\mathfrak{p}\right)\equiv 1\mod{\ell}.$

There is a natural bijection

$\operatorname{Prim}\left(\mathcal{S}^{G^{\ast}}\right)\xrightarrow{\sim}\prod_{j\in[c]}\operatorname{Prim}\left(\mathcal{S}_{\ell_{j}}^{G(\ell_{j})-\operatorname{id}}\right),$

cf. [KP23, p.326] for further details. Thus, in the general case we get a map

$P_{G}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Prim}\left(\mathcal{S}^{G^{\ast}}\right)\twoheadrightarrow\operatorname{Epi}_{\operatorname{top}-\operatorname{gr}}\left(\operatorname{G}_{k},G\right)\cup\\{\cdot\\}$

and set

$\operatorname{Prim}\left(\mathcal{S}^{G^{\ast}}\right)(\operatorname{solv.})\mathrel{\mathop{\mathchar
58\relax}}=P_{G}^{-1}\left(\operatorname{Epi}_{\operatorname{top}-\operatorname{gr}}\left(\operatorname{G}_{k},G\right)\right).$

The next result allows us to read off the $S$-free discriminant of $P_{G}(v)$
for

(3.1) $v=(v_{g,j}(1),v_{g,j}(2))_{j\in[c],g\in
G(\ell_{j})-\operatorname{id}}\in\operatorname{Prim}\left(\mathcal{S}^{G^{\ast}}\right)(\operatorname{solv.}).$

######  Proposition 3.4.

For $v$ as above, we have that

$\operatorname{free}_{S}\left(\operatorname{Disc}(P_{G}(v))\right)=\operatorname{free}_{S}\left(\prod_{G^{\ast}}v_{g}(2)^{|G|\left(1-\frac{1}{\\#\langle
g\rangle}\right)}\right).$

###### Proof.

The result is [KP23, Proposition 2.13] and is a direct consequence of
Proposition 3.3. ∎

Next, we recall some useful estimates from [KP23, section 4]. Let $F$ be a
number field and $L$ be a finite abelian extension of $F$. Let $\mathfrak{S}$
be a subset of $\operatorname{Gal}(L/F)$. Denote by $\mathcal{I}_{F}$ the
multiplicative group of non-zero fractional ideals of $F$. For
$I\in\mathcal{I}_{F}$, set $\omega_{\mathfrak{S}}(I)$ to denote the number of
prime divisors $\mathfrak{p}$ of $I$ such that $\mathfrak{p}$ is unramified in
$L$, and such that the Frobenius $\operatorname{Frob}_{\mathfrak{p}}$ belongs
to $\mathfrak{S}$. Let $z$ be a complex number and $\mathcal{P}$ be a finite
set of prime ideals of $\mathcal{O}_{F}$. Consider the sum

$A_{z}(x)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{\begin{subarray}{c}|I|\leq x,\\\
\mathfrak{p}|I\Rightarrow\operatorname{Frob}_{\mathfrak{p}}\in\mathfrak{S}\text{
and
}\mathfrak{p}\notin\mathcal{P}\end{subarray}}\mu^{2}(I)z^{\omega_{\mathfrak{S}}(I)},$

######  Proposition 3.5.

Let $F,L,\mathcal{P}$ ans $\mathfrak{S}$ be as above. Then for all positive
real numbers $R$, and all $z\in\mathbb{C}$ such that $|z|\leq R$

(3.2) $A_{z}(x)=Cx(\log
x)^{\frac{z\\#\mathfrak{S}}{\\#\operatorname{Gal}(L/F)}-1}+O_{R,F,L,\mathfrak{S},\mathcal{P}}\left(x(\log
x)^{\frac{\operatorname{Re}(z)\\#\mathfrak{S}}{\\#\operatorname{Gal}(L/F)}-2}\right),$

where $C>0$ depends only on $z,F,L,\mathfrak{S},\mathcal{P}$.

###### Proof.

For the proof of the above estimate we refer to [KP23][Theorem 4.1]. ∎

###  3.2. Uniformity estimate for nilpotent groups

Let $G$ be a non-trivial finite nilpotent group. Given an integral ideal
$\mathfrak{q}$ of $\mathcal{O}_{k}$, set $N_{q}(G;X)$ to be the number of
Galois extensions $L/k$ with $\operatorname{Gal}(L/k)=G$,
$\mathrm{Disc}(L)\leq X$, and such that $\mathfrak{q}$ divides
$\mathrm{disc}(L/k)$. We recall that $\ell_{G}$ denotes the smallest prime
divisor of $|G|$ and $I(G)$ is the subset of elements of $G$ with order
$\ell_{G}$. Set $H(G)\mathrel{\mathop{\mathchar
58\relax}}=I(G)\cup\\{\operatorname{id}\\}$. We establish an extension a
uniformity estimate for nilpotent groups. This generalizes [Wan21][Theorem
4.12]. Recall that

$i(G,k)=\frac{\\#I(G)}{[k(\mu_{\ell_{G}})\mathrel{\mathop{\mathchar
58\relax}}k]}.$

######  Proposition 3.6.

Let $G$ be a non-trivial finite nilpotent group and $k$ be a number field. Let
$\mathfrak{q}$ be a non-zero ideal of $\mathcal{O}_{k}$, and recall that
$|\mathfrak{q}|$ denotes the absolute norm to $\mathbb{Q}$. Then, for all
$\epsilon>0$, we have the following asymptotic estimate

$N_{q}(G;X)\ll_{G,k,\epsilon}\left(\frac{X}{|\mathfrak{q}|^{1-\epsilon}}\right)^{a(G)}(\log
X)^{i(G,k)-1}.$

###### Proof.

Without loss of generality, we may assume that $\mathfrak{q}$ is not divisible
by any prime in $S$. For $g\in G$, set $e_{g}\mathrel{\mathop{\mathchar
58\relax}}=|G|\left(1-\frac{1}{\\#\langle g\rangle}\right)$. Observe that for
$g\in I(G)$, we have that $e_{g}=a(G)^{-1}$. It follows from the
parameterization in the previous subsection and Proposition 3.4 that it
suffices to bound the following

(3.3)
$\\#\left\\{v=(v_{g}(1),v_{g}(2))\in\operatorname{Prim}\left(\mathcal{S}^{G^{\ast}}\right)\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{q}\mid\Delta_{v}\text{ and }|\Delta_{v}|\leq X\right\\},$

where

$\Delta_{v}\mathrel{\mathop{\mathchar 58\relax}}=\prod_{g\in
G-\\{\operatorname{id}\\}}v_{g}(2)^{e_{g}}.$

Suppose $(\mathfrak{q}_{1},\mathfrak{q}_{2})$ is a pair of integral ideals
such that $\mathfrak{q}=\mathfrak{q}_{1}\mathfrak{q}_{2}$. We can write

$\begin{split}&v^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{G-H(G)}(v)\mathrel{\mathop{\mathchar 58\relax}}=\prod_{g\in
G-H(G)}(v_{g}(1),v_{g}(2));\\\ &v^{\prime\prime}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{I(G)}(v)\mathrel{\mathop{\mathchar 58\relax}}=\prod_{h\in
I(G)}(v_{h}(1),v_{h}(2)).\end{split}$

The elements $v^{\prime}$ and $v^{\prime\prime}$ belong to
$\operatorname{Prim}\left(\mathcal{S}^{G-H(G)}\right)$ and
$\operatorname{Prim}\left(\mathcal{S}^{I(G)}\right)$ respectively. We set
$N_{\mathfrak{q}_{2}}(v^{\prime},X)$ to denote the number of
$v^{\prime\prime}=\prod_{h\in I(G)}(v_{1}(h),v_{2}(h))_{h\in
I(G)}\in\operatorname{Prim}\left(\mathcal{S}^{I(G)}\right)$ such that

  * •

$\mathfrak{q}_{2}\mid\prod_{h\in I(G)}v_{h}(2)^{a(G)^{-1}}$,

  * •

$|\prod_{h\in I(G)}v_{h}(2)^{a(G)^{-1}}|\leq\left(\frac{X}{|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|}\right)$.

Let $\mathcal{S}\left(\widetilde{\Omega}_{K}(\ell_{G})\right)$ be the set of
all squarefree ideals $\mathcal{I}$ supported in
$\widetilde{\Omega}_{K}(\ell_{G})$. We note that for $h\in I(G)$, the only
prime that divides the order of $h$ is $\ell_{G}$. All primes $\mathfrak{p}$
that divide $v_{h}(2)$ must satisfy the congruence

$\\#\left(\mathcal{O}_{K}/\mathfrak{p}\right)\equiv 1\mod{\ell_{G}}.$

Therefore for $h\in I(G)$, the ideal $v_{h}(2)$ belongs to
$\mathcal{S}\left(\widetilde{\Omega}_{K}(\ell_{G})\right)$.

Since we prove an upper bound, we may as well neglect the coprimality
conditions between $(v_{g}(1),v_{g}(2))$ for $g\in G-H(G)$. It suffices to
bound the following sum

(3.4) $\sum_{\begin{subarray}{c}v^{\prime},|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|\leq X\\\ \mathfrak{q}_{1}\mid\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}\end{subarray}}N_{\mathfrak{q}_{2}}(v^{\prime},X).$

Set $\widetilde{\mathfrak{q}}_{2}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\mid\mathfrak{q}_{2}}\mathfrak{p}$ to be the
squarefree part of $\mathfrak{q}_{2}$ and write
$\widetilde{\mathfrak{q}}_{2}I_{2}=\prod_{h\in I(G)}v_{h}(2)$. For $h\in
I(G)$, recall that the ideals $v_{h}(2)$ are all squarefree and mutually
coprime. Furthermore, they are supported on
$\widetilde{\Omega}_{K}(\ell_{G})$. Therefore,
$\widetilde{\mathfrak{q}}_{2}I_{2}$ is squarefree and supported in
$\widetilde{\Omega}_{K}(\ell_{G})$. It is easy to see that

(3.5) $N_{\mathfrak{q}_{2}}(v^{\prime},X)\leq\sum_{\begin{subarray}{c}I_{2}\\\
|\mathfrak{q}_{2}\left(I_{2}\right)^{a(G)^{-1}}|\leq\frac{X}{|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|},\\\ \widetilde{\mathfrak{q}}_{2}I_{2}\text{
supported in
}\widetilde{\Omega}_{K}(\ell_{G})\end{subarray}}\mu^{2}(I_{2})\\#I(G)^{\omega(\widetilde{\mathfrak{q}}_{2})\omega(I_{2})},$

cf. the proof of [KP23][Theorem 5.1] for further details.

Set $L$ to be the cyclotomic extension $k(\mu_{\ell_{G}})$ and
$\mathfrak{S}=\\{1\\}\subset\operatorname{Gal}(L/k)$. Note that an ideal $I$
is supported on $\widetilde{\Omega}_{K}(\ell_{G})$ if and only if all prime
ideals $\mathfrak{p}|I$ split in $L$. Setting $z\mathrel{\mathop{\mathchar
58\relax}}=\\#I(G)$, rewrite (3.5) as

$N_{\mathfrak{q}_{2}}(v^{\prime},X)\leq
z^{\omega(\mathfrak{q}_{2})}A_{z}\left(Y\right),$

where

$Y\mathrel{\mathop{\mathchar
58\relax}}=\left(\frac{X^{a(G)}}{|\mathfrak{q}_{2}|^{a(G)}|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}a(G)}|}\right).$

Using the estimate (3.2) we obtain the following bound

$N_{\mathfrak{q}_{2}}(v^{\prime},X)\ll_{G,k,\epsilon}Y(\log
Y)^{\frac{z\\#\mathfrak{S}}{[k(\mu_{\ell_{G}})\mathrel{\mathop{\mathchar
58\relax}}k]}-1}.$

Noting that

$i(G,k)=\frac{\\#I(G)}{[k(\mu_{\ell_{G}})\mathrel{\mathop{\mathchar
58\relax}}k]}=\frac{z\\#\mathfrak{S}}{[k(\mu_{\ell_{G}})\mathrel{\mathop{\mathchar
58\relax}}k]},$

and that

$\log(Y)\sim a(G)\log(X),$

we obtain the following upper bound

(3.6)
$N_{\mathfrak{q}_{2}}(v^{\prime},X)\ll_{G,k,\epsilon}\left(\frac{1}{|\mathfrak{q}_{2}|}\right)^{a(G)}\frac{X^{a(G)}}{|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}a(G)}|}(\log X)^{i(G,K)-1}.$

For any $\epsilon>0$, the integral ideal $\mathfrak{q}$ can be written as the
product of two proper ideals in at most
$O_{\epsilon}(|\mathfrak{q}|^{\epsilon})$ ways. From (3.6), we obtain

(3.7) $\begin{split}&\sum_{\begin{subarray}{c}v^{\prime},|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|\leq X\\\ \mathfrak{q}_{1}\mid\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}\end{subarray}}N_{\mathfrak{q}_{2}}(v^{\prime},X)\\\
\ll_{G,k,\epsilon}&|\mathfrak{q}|^{\epsilon/2}\operatorname{max}_{(\mathfrak{q}_{1},\mathfrak{q}_{2})}\left(\left(\frac{X}{\mathfrak{q}_{2}}\right)^{a(G)}(\log
X)^{i(G,k)-1}\sum_{\begin{subarray}{c}v^{\prime}\\\ |\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|\leq X,\\\ \mathfrak{q}_{1}\mid\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}\end{subarray}}\frac{1}{|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}a(G)}|}\right).\end{split}$

Observe that $e_{g}a(G)>1$ for $g\in G-H(G)$. Setting
$t\mathrel{\mathop{\mathchar 58\relax}}=|G|-\\#H(G)$, enumerate the elements
of $G-H(G)$ as $\\{g_{1},\dots,g_{t}\\}$. Setting,
$e_{i}\mathrel{\mathop{\mathchar 58\relax}}=e(g_{i})$, we find that

$\sum_{\begin{subarray}{c}v^{\prime}\\\ |\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}|\leq X,\\\ \mathfrak{q}_{1}\mid\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}\end{subarray}}\frac{1}{|\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}a(G)}|}\leq\sum_{\begin{subarray}{c}I_{1},\dots,I_{t}\\\
|\prod_{i}I_{i}^{e_{i}}|\leq X\\\
\mathfrak{q}_{1}|\prod_{i}I_{i}^{e_{i}}\end{subarray}}\frac{1}{|\prod_{i}I_{i}^{e_{i}a(G)}|},$

where $I_{1},\dots,I_{t}$ are coprime integral ideals. Since $g_{i}\notin
I(G)$, we find that $e_{i}a(G)>1$. We write $I_{i}=\mathfrak{q}_{1,i}J_{i}$,
where $J_{i}$ is coprime to $\mathfrak{q}_{1}$, and $\mathfrak{q}_{1}$ divides
$\prod_{i}\mathfrak{q}_{1,i}^{e_{i}}$. Then, the product
$\prod_{i}I_{i}^{e_{i}a(G)}$ can be written as

$\prod_{i}I_{i}^{e_{i}a(G)}=\prod_{i}\mathfrak{q}_{1,i}^{e_{i}a(G)}\prod_{i}J_{i}^{e_{i}a(G)}.$

Note that the sum

$\sum_{\begin{subarray}{c}I_{1},\dots,I_{t}\\\
\mathfrak{q}_{1}|\prod_{i}I_{i}^{e_{i}}\end{subarray}}\frac{1}{|\prod_{i}I_{i}^{e_{i}a(G)}|}$

converges and in fact, is bounded above by

$\prod_{i=1}^{t}\sum_{I}\frac{1}{|I|^{e_{i}a(G)}}=\prod_{i=1}^{t}\zeta_{K}(e_{i}a(G)).$

Let $\mathfrak{p}_{1},\dots,\mathfrak{p}_{k}$ be the prime divisors of
$\mathfrak{q}_{1}$, which factorizes as
$\mathfrak{q}_{1}=\mathfrak{p}_{1}^{a_{1}}\dots\mathfrak{p}_{k}^{a_{k}}$. Let
$\mathfrak{T}$ be the set of all tuples

$\mathfrak{T}=\\{(\mathfrak{a}_{1},\dots,\mathfrak{a}_{t})\mid\mathfrak{a}_{i}\text{
are coprime,
}\prod_{i}\mathfrak{a}_{i}=\mathfrak{p}_{1}\dots\mathfrak{p}_{k}\\}.$

Writing $\mathfrak{a}_{i}=\mathfrak{p}_{i_{1}}\dots\mathfrak{p}_{i_{r}}$, set

$\widetilde{\mathfrak{a}}_{i}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{j=1}^{r}\mathfrak{p}_{i_{j}}^{\lceil
a_{i_{j}}/e_{i}\rceil}.$

We write

$\begin{split}&\sum_{\begin{subarray}{c}I_{1},\dots,I_{t}\\\
\mathfrak{q}_{1}|\prod_{i}I_{i}^{e_{i}}\end{subarray}}\frac{1}{|\prod_{i}I_{i}^{e_{i}a(G)}|}\\\
=&\sum_{(\mathfrak{a}_{1},\dots,\mathfrak{a}_{k})\in\mathfrak{T}}\prod_{i=1}^{t}\left(\sum_{\begin{subarray}{c}I_{i}\\\
\widetilde{\mathfrak{a}}_{i}|I_{i}\end{subarray}}\frac{1}{|I_{i}|^{e_{i}a(G)}}\right)\\\
=&\sum_{(\mathfrak{a}_{1},\dots,\mathfrak{a}_{k})\in\mathfrak{T}}\prod_{i=1}^{t}\left(\sum_{J_{i}}\frac{1}{|\widetilde{\mathfrak{a}}_{i}|^{e_{i}a(G)}|J_{i}|^{e_{i}a(G)}}\right).\end{split}$

We note that $\prod_{i=1}^{t}\widetilde{\mathfrak{a}_{i}}^{e_{i}}$ is
divisible by $\mathfrak{q}_{1}$, and therefore,

$\sum_{\begin{subarray}{c}I_{1},\dots,I_{t}\\\
\mathfrak{q}_{1}|\prod_{i}I_{i}^{e_{i}}\end{subarray}}\frac{1}{|\prod_{i}I_{i}^{e_{i}a(G)}|}\leq\left(\frac{\\#\mathfrak{T}}{|\mathfrak{q}_{1}|^{a(G)}}\right)\prod_{i=1}^{t}\left(\sum_{J_{i}}\frac{1}{|J_{i}|^{e_{i}a(G)}}\right)=\left(\frac{\\#\mathfrak{T}}{|\mathfrak{q}_{1}|^{a(G)}}\right)\prod_{i=1}^{t}\zeta_{K}(e_{i}a(G)).$

We note that
$\\#\mathfrak{T}=t^{\omega(\mathfrak{q}_{1})}=o(|\mathfrak{q}|^{\epsilon/2})$.
Combining the above estimates with (3.7), we thus obtain

$\sum_{\begin{subarray}{c}v^{\prime},|\prod_{g\in G-H(G)}v_{g}(2)^{e_{g}}|\leq
X\\\ \mathfrak{q}_{1}\mid\prod_{g\in
G-H(G)}v_{g}(2)^{e_{g}}\end{subarray}}N_{\mathfrak{q}_{2}}(v^{\prime},X)\\\
\ll_{G,K,\epsilon}|\mathfrak{q}|^{\epsilon}\left(\frac{X}{|\mathfrak{q}|}\right)^{a(G)}(\log
X)^{i(G,K)-1}.$

This concludes the proof of the result. ∎

##  4\. Proof of the main theorem 1.2

In this section, $G$ denotes a non-trivial finite nilpotent group. We also
suppose that $G$ satisfies the appropriate conditions of Theorem 1.2 for
$S_{3},S_{4},$ or $S_{5}$. We set

$\begin{split}\mathcal{F}_{k}(S_{n}\times G;X)\mathrel{\mathop{\mathchar
58\relax}}=\\#\\{&(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(K/k)\simeq S_{n},\\\
&\operatorname{Gal}(L/k)\simeq G\text{ and }\mathrm{Disc}(KL)\leq
X\\}\end{split}$

and prove asymptotic upper bounds for $\\#\mathcal{F}_{k}(S_{n}\times G;X)$ as
$X\rightarrow\infty$. Then, it is shown that the same upper bounds hold for
$N_{k}(S_{n}\times G;X)$. We first establish the lower bound in Theorem 1.2.

######  Theorem 4.1.

Let $G$ and $k$ be as in Theorem 1.2, then,

$N_{k}(S_{n}\times G;X)\gg_{G,k}X^{\frac{1}{|G|}}.$

###### Proof.

Using the upper bound on $\mathrm{Disc}(KL)$ from Proposition 2.1, we note
that

$\displaystyle N_{k}(S_{n}\times G;X)\geq$
$\displaystyle\\#\\{(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq S_{n}\text{ and }$
$\displaystyle\operatorname{Gal}(L/k)\simeq
G,\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}\leq X\\}.$

Malle’s conjecture is known for $S_{n}$ ($n\in\\{3,4,5\\})$ [DH71, Bha05,
Bha10]. For nilpotent groups, the weak form of the conjecture is known in
generality, cf. [KM04]. Thus, we find that

$\begin{split}&N_{k}(S_{n};X)\gg_{n,k}X,\text{ and }\\\
&N_{k}(G;X)\gg_{G,k}X^{a(G)}.\end{split}$

The result then follows from the above bounds and Corollary 2.8 setting
$\operatorname{inv}_{1}(K)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Disc}(K)$ and
$\operatorname{inv}_{1}(L)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Disc}(L)$. ∎

Next, we establish the asymptotic upper bound from Theorem 1.2. We examine a
set of invariants denoted by $\\{\mathrm{Disc}_{Y}\\}_{Y>0}$ for $S_{n}\times
G$-extensions, defined below.

######  Definition 4.2.

Let $Y>0$, and suppose $(K,L)\in\mathcal{F}_{k}(S_{n}\times G;X)$ we define

$\mathrm{Disc}_{Y}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\in\Omega_{k}}\mathrm{Disc}_{Y,\mathfrak{p}}(KL),$

where

$\mathrm{Disc}_{Y,\mathfrak{p}}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\mathrm{Disc}_{\mathfrak{p}}(KL)\text{
}\mathrm{if}\text{ }|\mathfrak{p}|\leq Y,\\\
\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}\mathrm{Disc}_{\mathfrak{p}}(L)^{n}\text{
}\mathrm{if}\text{ }|\mathfrak{p}|>Y.\end{cases}$

We set

(4.1) $\begin{split}N_{k,Y}(S_{n}\times G;X)\mathrel{\mathop{\mathchar
58\relax}}=&\\#\\{(K,L)\mid[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar 58\relax}}k]=|G|,\\\
&\operatorname{Gal}(\widetilde{K}/k)\simeq S_{n},\operatorname{Gal}(L/k)\simeq
G\text{ and }\mathrm{Disc}_{Y}(KL)<X\\}.\end{split}$

We remark that $\mathrm{Disc}(KL)\leq\mathrm{Disc}_{Y}(KL)$ for all $Y>0$
using Proposition 2.1. Hence

$N_{k,Y}(S_{n}\times G;X)\leq N_{k}(S_{n}\times G;X)$

for all $Y>0$. The invariants $\mathrm{Disc}_{Y}$ approximate $\mathrm{Disc}$
as $Y\to\infty$. The counting function associated with $\mathrm{Disc}_{Y}$ is
denoted as $N_{k,Y}(S_{n}\times G;X)$. We establish asymptotic upper bounds
for $N_{k,Y}(S_{n}\times G;X)$ for $Y>0$ and leverage these bounds to derive
the intended upper bound for $N_{k}(S_{n}\times G;X)$.

Let $S_{Y}$ be the finite set of all primes of $k$ with absolute norm
$|\mathfrak{p}|\leq Y$. Let $\Sigma_{Y}$ denote the set of tuples
$((\mathcal{K}_{\mathfrak{p}}),\mathcal{L}_{\mathfrak{p}}))_{\mathfrak{p}\in
S_{Y}}$ of pairs of local étale extensions over $k_{\mathfrak{p}}$ of degree
$n$ and $|G|$ respectively. We will think of these tuples as specifying local
conditions at all $\mathfrak{p}\in S_{Y}$ for $S_{n}$ and $G$-extensions.
Accordingly, given a pair $(K,L)\in\mathcal{F}_{k}(S_{n}\times G;X)$ and a
tuple
$((\mathcal{K}_{\mathfrak{p}}),\mathcal{L}_{\mathfrak{p}}))_{\mathfrak{p}\in
S_{Y}}\in\Sigma_{Y}$ we write
$(K,L)\in((\mathcal{K}_{\mathfrak{p}}),\mathcal{L}_{\mathfrak{p}}))_{\mathfrak{p}\in
S_{Y}}$, if for each $\mathfrak{p}\in S_{Y}$,

  1. (1)

the local étale algebra $(K)_{\mathfrak{p}}=\mathcal{K}_{\mathfrak{p}}$, and

  2. (2)

the local étale algebra $(L)_{\mathfrak{p}}=\mathcal{L}_{\mathfrak{p}}$.

Next, we have the following Lemma relating $\mathrm{Disc}_{Y}$ and
$\mathrm{Disc}$ under given local specifications,

######  Lemma 4.3.

Let $Y>0$ and $\Pi\mathrel{\mathop{\mathchar
58\relax}}=((\mathcal{K}_{\mathfrak{p}}),\mathcal{L}_{\mathfrak{p}}))_{\mathfrak{p}\in
S_{Y}}\in\Sigma_{Y}$. If $(K,L)\in\Pi$ then,

$\mathrm{Disc}_{Y}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}}{d_{\Pi}}$

where $d_{\Pi}$ depends only on $\Pi$.

###### Proof.

From the definition of $\mathrm{Disc}_{Y}$, it follows that,

$\mathrm{Disc}_{Y}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\in
S_{Y}}\mathrm{Disc}_{\mathfrak{p}}(KL)\prod_{\mathfrak{p}\notin
S_{Y}}\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}\mathrm{Disc}_{\mathfrak{p}}(L)^{n}.$

Hence, it follows that

$\mathrm{Disc}_{Y}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}}{d_{\Pi}}$

with $d_{\Pi}=\prod_{\mathfrak{p}\in
S_{Y}}\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}\mathrm{Disc}_{\mathfrak{p}}(L)^{n}\mathrm{Disc}_{\mathfrak{p}}(KL)^{-1}$.
From Lemma 2.3 we deduce that
$\mathrm{Disc}_{\mathfrak{p}}(K)$,$\mathrm{Disc}_{\mathfrak{p}}(L)$ and
$\mathrm{Disc}_{\mathfrak{p}}(KL)$ are determined by $(K)_{\mathfrak{p}}$ and
$(L)_{\mathfrak{p}}$ and hence $d_{\Pi}$ depends only on $\Pi$. ∎

######  Proposition 4.4.

Let $Y>0$ then we have that,

$N_{k,Y}(S_{n}\times
G,X)\ll_{k,S_{n},G,Y,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

###### Proof.

We count extensions by specifying local conditions at each prime
$\mathfrak{p}\in S_{Y}$. Let $\Pi\in\Sigma_{Y}$, we consider the counting
function

$\displaystyle N_{k,Y,\Pi}(S_{n}\times G;X)=$
$\displaystyle\\#\\{(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq
S_{n},\operatorname{Gal}(L/k)\simeq G,$ $\displaystyle\text{ and
}(K,L)\in\Pi,\mathrm{Disc}_{Y}(KL)\leq X\\}.$

Using Lemma 4.3 we obtain the following equality,

$\displaystyle N_{k,Y,\Pi}(S_{n}\times G;X)=$
$\displaystyle\\{(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq
S_{n},\operatorname{Gal}(L/k)\simeq G,$ $\displaystyle\text{ and
}(K,L)\in\Pi,\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}\leq d_{\Pi}X\\}.$

We use Corollary 2.8 along with the bounds

$\begin{split}&N_{k}(S_{n};X)\ll_{n,k,\epsilon}X^{1+\epsilon},\text{ and }\\\
&N_{k}(G;X)\ll_{G,k,\epsilon}X^{a(G)+\epsilon}.\end{split}$

to get that

$N_{k,Y,\Pi}(S_{n}\times
G;X)\ll_{k,S_{n},G,\Pi,\epsilon}X^{\frac{1}{|G|}+\epsilon},$

as the set $S_{Y}$ is finite and since there are only finitely many local
étale extensions over $k_{\mathfrak{p}}$ of degree $n,|G|$ respectively, the
set $\Sigma_{Y}$ is also finite. Hence we conclude that

$\displaystyle N_{k,Y}(S_{n}\times G;X)=$
$\displaystyle\sum_{\Pi\in\Sigma_{Y}}N_{k,Y,\Pi}(S_{n}\times G;X)$
$\displaystyle\ll_{k,S_{n},G,Y,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

∎

We have determined upper bounds for $N_{k,Y}(S_{n}\times G;X)$ across all
$Y>0$ mirroring the bound we aim to establish for $N_{k,Y}(S_{n}\times G;X)$.
Next, we analyze the difference

$N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times G;X)$

and obtain an upper bound for this expression, thereby concluding the proof of
the upper bound for $N_{k}(S_{n}\times G;X)$. Before proceeding, we state the
following important observation which follows from index computation in
Proposition 2.4 and local uniformity estimates in Theorem 3.1. Given a square-
free ideal $\mathfrak{q}$ in $\mathcal{O}_{k}$ and $\sigma\in S_{n}$ we denote
by $N_{k,\mathfrak{q},\sigma}(S_{n};X)$ the counting function for number of
$S_{n}$ degree $n$ extensions $K$ of $k$ such that $\widetilde{K}$ is tamely
ramified at all primes $\mathfrak{p}\mid\mathfrak{q}$ with inertia groups
equal to $\langle\sigma\rangle$ up to conjugacy. We consider the set,

$S_{\sigma}\mathrel{\mathop{\mathchar 58\relax}}=\\{s\geq
0\mathrel{\mathop{\mathchar
58\relax}}N_{k,\mathfrak{q},\sigma}(S_{n};X)\ll_{\epsilon}X/|\mathfrak{q}|^{s-\epsilon}\text{
for all squarefree }\mathfrak{q}\text{ in }k\\},$

and we define $s_{\sigma}=\sup S_{\sigma}$. The Theorem 3.1 gives explicit
lower bounds on $s_{\sigma}$ for certain $\sigma\in C(S_{n})$ when $n=3,4,$ or
$5$.

######  Lemma 4.5.

Let $n\in\\{3,4,5\\}$, let $G$ be a finite nilpotent group. Then,

$\mathrm{ind}(\sigma,g)/m-\mathrm{ind}(\sigma)+s_{\sigma}>1,$

for all $(\sigma,g)\in S_{n}\times G$.

###### Proof.

Using computations in Proposition 2.4 and bounds on $s_{\sigma}$ from Theorem
3.1 we conclude that the above inequality holds. ∎

######  Remark 4.6.

We use the fact that the constants $s_{\sigma}$ are not smaller than the
corresponding constants that appear in Theorem 3.1 for suitable $\sigma$, we
denote the corresponding constants by $r_{\sigma}$. For other conjugacy
classes $\sigma$ not covered in Theorem 3.1 we use the fact that the constant
$s_{\sigma}$ is non-negative, in this case, we define $r_{\sigma}$ to be zero.

####  4.0.1. The difference $N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times G;X)$

We fix $Y>0$. From definition 4.2 it is obvious that this difference is given
by,

(4.2) $\begin{split}&N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times G;X)\\\
=&\\#\\{(K,L)\mid[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq S_{n},\\\
&\operatorname{Gal}(L/k)\simeq G,\text{ and
}\mathrm{Disc}(KL)<X<\mathrm{Disc}_{Y}(KL)\\}.\end{split}$

Our method for bounding the above counting function will closely resemble the
proof outlined in Proposition 4.4, wherein we express this counting function
as the aggregate of various counting functions parameterized by specific local
conditions at various primes in $k$. We proceed by describing these local
conditions of our interest, we call such specification of local conditions a
standard tuple.

######  Definition 4.7.

A standard tuple is a tuple $(S,\mathcal{L})$ satisfying following properties:

  1. (1)

$S$ is a finite set of primes in $k$,

  2. (2)

$\mathcal{L}$ is a tuple $(h_{\mathfrak{p}},g_{\mathfrak{p}})_{\mathfrak{p}\in
S}$ such that:

    1. (a)

for each $\mathfrak{p}\mid n!|G|$ the pair
$(h_{\mathfrak{p}},g_{\mathfrak{p}})_{\mathfrak{p}\in S}$ is a pair of
ramified local étale algebras over $k_{\mathfrak{p}}$ of degree $n$ and $|G|$
respectively,

    2. (b)

for each $\mathfrak{p}\nmid n!|G|$ the pair
$(h_{\mathfrak{p}},g_{\mathfrak{p}})_{\mathfrak{p}\in S}$ is a pair of
conjugacy classes of inertia generators in $S_{n}$ and $G$ respectively.

Let $\Sigma_{k}$ denote the set of all standard tuples for the number field
$k$. Given $(K,L)$ and a specification of local conditions
$\Pi\mathrel{\mathop{\mathchar 58\relax}}=(S,\mathcal{L})\in\Sigma_{k}$ as
above, we write $(K,L)\in\Pi$ if following conditions are satisfied.

  1. (1)

If $\mathfrak{p}\in\Omega_{k}$ is such that both $K$, $L$ are ramified at
$\mathfrak{p}$ then $\mathfrak{p}\in S$.

  2. (2)

For each $\mathfrak{p}\in S$ such that for each $\mathfrak{p}\nmid n!|G|$, we
have that $(I_{\mathfrak{p}}(K),I_{\mathfrak{p}}(L))=(\langle
h_{\mathfrak{p}}\rangle,\langle g_{\mathfrak{p}}\rangle)$.

  3. (3)

For each $\mathfrak{p}\in S$ such that for each $\mathfrak{p}\mid n!|G|$, we
have that
$((K)_{\mathfrak{p}},(L)_{\mathfrak{p}})=(h_{\mathfrak{p}},g_{\mathfrak{p}})$.

Write $K\in\Pi$ if for each $\mathfrak{p}\in S$ we have that,
$(K)_{\mathfrak{p}}=h_{\mathfrak{p}}\text{ or }I_{\mathfrak{p}}(K)=\langle
h_{\mathfrak{p}}\rangle$ and we write $L\in\Pi$ if for each $\mathfrak{p}\in
S$ we have that, $(L)_{\mathfrak{p}}=g_{\mathfrak{p}}\text{ or
}I_{\mathfrak{p}}(L)=\langle g_{\mathfrak{p}}\rangle$. Following the notation
in [Wan21], we write $\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})$ to denote the
exponent of $\mathrm{Disc}_{\mathfrak{p}}(KL)$ where
$(K)_{\mathfrak{p}}=h_{\mathfrak{p}}$ or $I_{\mathfrak{p}}(K)=\langle
h_{\mathfrak{p}}\rangle$ and $(L)_{\mathfrak{p}}=g_{\mathfrak{p}}$ or
$I_{\mathfrak{p}}(L)=\langle g_{\mathfrak{p}}\rangle$. The following result
shows that $\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})$ is completely determined
by $(h_{\mathfrak{p}},g_{\mathfrak{p}})$.

######  Lemma 4.8.

In the above setting, $\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})$ depends only
on the tuple $(h_{\mathfrak{p}},g_{\mathfrak{p}})$.

###### Proof.

If $\mathfrak{p}\nmid n!|G|$ then using the [Wan21, Theorem 2.3], the indices
$\mathrm{ind}(h_{\mathfrak{p}})$ and $\mathrm{ind}(g_{\mathfrak{p}})$
completely determine $\mathrm{ind}(h_{\mathfrak{p}},g_{\mathfrak{p}})$ and
hence $\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})$. When $\mathfrak{p}\mid
n!|G|$, the Lemma 2.3 implies that $\mathrm{Disc}_{\mathfrak{p}}(KL)$ is
completely determined by the pair $((K)_{\mathfrak{p}},(L)_{\mathfrak{p}})$
which is the pair $(h_{\mathfrak{p}},g_{\mathfrak{p}})$. ∎

As a consequence of the above and our definition of a standard tuple, we can
prove the following important result.

######  Lemma 4.9.

Let $\Pi\mathrel{\mathop{\mathchar 58\relax}}=(S,\mathcal{L})\in\Sigma_{k}$ be
a standard tuple and suppose that $(K,L)\in\Pi$ then,

$\mathrm{Disc}(KL)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}}{d_{\Pi}}$

where $d_{\Pi}$ depends only on $\Pi$.

###### Proof.

Let $\mathfrak{p}\notin S$ then either $K/k$ or $L/k$ is not ramified at
$\mathfrak{p}$. Without loss of generality, we assume that $K/k$ is not
ramified at $\mathfrak{p}$. We deduce from the upper bound in Proposition 2.1
that the
$\mathrm{disc}_{\mathfrak{p}}(KL/k)\mid\mathrm{disc}_{\mathfrak{p}}(L/k)^{n}$.
We recall that in the tower $KL/L/k$ we have the following formula,

$\mathrm{disc}(KL/k)=\mathrm{Disc}(KL/L)\mathrm{disc}(L/k)^{n}.$

Therefore,
$\mathrm{disc}_{\mathfrak{p}}(KL/k)=\mathrm{disc}_{\mathfrak{p}}(L/k)^{n}$.
Hence, $\mathrm{Disc}_{\mathfrak{p}}(KL)=\mathrm{Disc}_{\mathfrak{p}}(L)^{n}$
or $\mathrm{Disc}_{\mathfrak{p}}(KL)=\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}$
for all $\mathfrak{p}\in S$ depending upon whether $\mathfrak{p}$ is
unramified in $K$ or $L$. We obtain that,

$\mathrm{Disc}(KL)=\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})-n\cdot\exp(g_{\mathfrak{p}})-|G|\cdot\exp(h_{\mathfrak{p}})}.$

Lemma 4.8 implies that the quantity

$d_{\Pi}=\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{n\cdot\exp(g_{\mathfrak{p}})+|G|\cdot\exp(h_{\mathfrak{p}})-\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})},$

depends only on $\Pi$. ∎

For $\Pi\in\Sigma_{k}$, we set

(4.3) $\displaystyle N_{k,\Pi}(S_{n}\times G;X)\mathrel{\mathop{\mathchar
58\relax}}=\\#\\{(K,L)\in\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}(K,L)\in\Pi\text{ and }\mathrm{Disc}(KL)<X\\},$

where

(4.4) $\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}=\\{(K,L)\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(\widetilde{K}/k)\simeq S_{n}\text{ and
}\operatorname{Gal}(L/k)\simeq G\\}.$

From Lemma 4.9 we deduce that

(4.5) $\begin{split}N_{k,\Pi}(S_{n}\times
G;X)=&\\#\\{(K,L)\in\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}(K,L)\in\Pi\text{ and }\\\
&\mathrm{Disc}(K)^{|G|}\mathrm{Disc}(L)^{n}<Xd_{\Pi}\\}\\\
=&\\#\\{(K,L)\in\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}(K,L)\in\Pi\text{ and }\\\ &\prod_{\mathfrak{p}\notin
S}\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}\mathrm{Disc}_{\mathfrak{p}}(L)^{n}<{X}/{\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})}}\\}\end{split}$

We also consider the same counting function with invariants
$\mathrm{Disc}_{Y}$ for $Y>0$. We denote these by $N_{k,Y,\Pi}(S_{n}\times
G;X)$. Then we have the following expression for any $Y>0$,

(4.6) $\begin{split}N_{k}(S_{n}\times G;X)-N_{k,Y}&(S_{n}\times G;X)\\\
&=\sum_{\Pi\in\Sigma_{k}}N_{k,\Pi}(S_{n}\times G;X)-N_{k,Y,\Pi}(S_{n}\times
G;X).\end{split}$

Suppose that $(K,L)\in\Pi$. We recall from the conditons on $S$ that any prime
$\mathfrak{p}\notin S$ must be unramified in either $K$ or $L$. Therefore, we
find that for $\mathfrak{p}\notin S$,

$\operatorname{Disc}_{\mathfrak{p}}(KL)=\operatorname{Disc}_{\mathfrak{p}}(K)^{|G|}\operatorname{Disc}_{\mathfrak{p}}(L)^{n}.$

Thus, if $S\subset S_{Y}$, then we have that

(4.7) $\mathrm{Disc}(KL)=\mathrm{Disc}_{Y}(KL)$

for all $(K,L)\in\Pi$.

######  Lemma 4.10.

Let $\Pi\in\Sigma_{k}$ be such that $\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|\leq Y$. Then,

$N_{k,\Pi}(S_{n}\times G;X)=N_{k,Y,\Pi}(S_{n}\times G;X).$

###### Proof.

The condition that $\prod_{\mathfrak{p}\in S}|\mathfrak{p}|\leq Y$ implies
that $|\mathfrak{p}|\leq Y$ for all $\mathfrak{p}\in S$. Consequently, we get
that $S\subseteq S_{Y}$, and the conclusion follows from (4.7). ∎

Therefore, only $\Pi\in\Sigma_{k}$ with $\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|>Y$ have a positive contribution in the above sum (4.5). This
motivates the following definition.

######  Definition 4.11.

Let $(K,L)\in\mathcal{E}_{k}$ (cf. (4.4)) and
$\Pi=(S,\mathcal{L})\in\Sigma_{k}$. Assume that $(K,L)\in\Pi$. Then, the
_restricted discriminant_ of $K/k$ is defined as

$\mathrm{Disc}_{res}^{\Pi}(K)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\notin S}\mathrm{Disc}_{\mathfrak{p}}(K).$

Define the restricted discriminant $\mathrm{Disc}_{res}^{\Pi}(L)$ for $L/k$ in
the same way.

We now prove asymptotic upper bounds for the number of $S_{n}$ degree $n$
extensions and $G$-extensions when ordered by restricted discriminant. We now
define the relevent counting functions. Given $\Pi\in\Sigma_{k}$, we set

$N_{k,\Pi,res}(S_{n};X)\mathrel{\mathop{\mathchar
58\relax}}=\\#\\{K\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,\operatorname{Gal}(K/k)\simeq S_{n},K\in\Pi,\text{ and
}\mathrm{Disc}_{res}^{\Pi}(K)\leq X\\},$

and

$N_{k,\Pi,res}(G;X)\mathrel{\mathop{\mathchar
58\relax}}=\\#\\{L\mathrel{\mathop{\mathchar
58\relax}}[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(L/k)\simeq G,L\in\Pi,\text{ and
}\mathrm{Disc}_{res}^{\Pi}(L)\leq X\\}.$

Set $C(S_{n})$ to be the set of conjugacy classes in $S_{n}$. Given $\sigma\in
C(S_{n})$, we take $\mathfrak{q}_{\sigma}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\in S,h_{\mathfrak{p}}=\sigma}\mathfrak{p}$.
Recall that $r_{\sigma}$ is defined in Lemma 4.5.

######  Lemma 4.12.

With respect to above notation, we have that

$N_{k,\Pi,res}(S_{n};X)\ll_{k,S_{n},\epsilon}\left(\prod_{\sigma\in
C(S_{n})}|\mathfrak{q}_{\sigma}|^{-r_{\sigma}+\operatorname{ind}(\sigma)+\epsilon}\right)X.$

###### Proof.

Using definition of $N_{k,\Pi,res}(S_{n};X)$ we note that,

(4.8)
$\begin{split}N_{k,\Pi,res}(S_{n};X)&\leq\\#\\{K\mathrel{\mathop{\mathchar
58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,\operatorname{Gal}(K/k)\simeq S_{n},K\in\Pi,\text{ and }\\\
&\mathrm{Disc}(K)\leq X\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(h_{\mathfrak{p}})}\\}\\\
&\leq\\#\\{K\mathrel{\mathop{\mathchar 58\relax}}[K\mathrel{\mathop{\mathchar
58\relax}}k]=n,\operatorname{Gal}(K/k)\simeq S_{n},K\in\Pi,\text{ and }\\\
&\mathrm{Disc}(K)\leq CX\prod_{\sigma\in
C(S_{n})}|\mathfrak{q}_{\sigma}|^{\mathrm{ind}(\sigma)}\\},\end{split}$

where $C>0$ is an absolute constant. This inequality holds because there are
only finitely many primes that can be wildly ramified in $K/k$, and there are
finitely many local étale algebras over $k_{\mathfrak{p}}$ with a fixed degree
for each prime $\mathfrak{p}$ in $k$ and
$\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})$ can be uniformly bounded for all
such primes. By recalling the definition of $r_{\sigma}$ and applying Theorem
3.1 to (4.8), with

$\mathfrak{q}=\prod_{\sigma\in C(S_{n}),r_{\sigma}\neq
0}\mathfrak{q}_{\sigma}$

and counting extensions with $\mathrm{Disc}$ upto $CX\prod_{\sigma\in
C(S_{n})}|\mathfrak{q}_{\sigma}|^{\mathrm{ind}(\sigma)}$, we deduce that

$N_{k,\Pi,res}(S_{n};X)\ll_{k,S_{n},\epsilon}\left(\prod_{\sigma\in
C(S_{n})}|\mathfrak{q}_{\sigma}|^{-r_{\sigma}+\mathrm{ind}(\sigma)+\epsilon}\right)X.$

This concludes the proof of the Lemma. ∎

Next, we use the local uniformity estimates for $G$-extensions to obtain the
asymptotic upper bound for $N_{k,\Pi,res}(G;X)$.

######  Lemma 4.13.

The following asymptotic bound holds

$N_{k,\Pi,res}(G;X)\ll_{k,G,\epsilon}\left(\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{A\epsilon}\right)X^{a(G)+\epsilon}.$

for some constant $A>0$ depending only on $G$ and $k$.

###### Proof.

The proof is a simple application of Proposition 3.6. Note that the following
inequality holds,

$\displaystyle N_{k,\Pi,res}(G;X)\leq\\#\\{L\mathrel{\mathop{\mathchar
58\relax}}[L\mathrel{\mathop{\mathchar
58\relax}}k]=|G|,\operatorname{Gal}(L/k)\simeq G,\text{ and
}\mathrm{Disc}(L)\leq X\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(g_{\mathfrak{p}})}\\}.$

Applying Proposition 3.6 with $\mathfrak{q}=\prod_{\mathfrak{p}\in
S}\mathfrak{p}^{\exp(g_{\mathfrak{p}})}$ and counting extensions with
$\mathrm{Disc}$ upto $X\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(g_{\mathfrak{p}})}$ we deduce the following bound for
the quantity on the right-hand side above and same bound holds for
$N_{k,\Pi,res}(G;X)$,

$N_{k,\Pi,res}(G;X)\ll_{k,G,\epsilon}\left(\frac{X|\mathfrak{q}|}{|\mathfrak{q}|^{1-\epsilon}}\right)^{a(G)}(\log(X|\mathfrak{q}|)^{i(G,k)-1}$

and the Lemma follows by using the fact that
$\log(X)\ll_{\epsilon}X^{\epsilon}$ for some $A>0$. For instance, we can take
$A$ to be the sum $a(G)+i(G,k)-1$. ∎

In the result that follows, set

$\delta\mathrel{\mathop{\mathchar 58\relax}}=\max_{\sigma\in C(S_{n}),g\in
G}(-r_{\sigma}+\mathrm{ind}(\sigma)-\mathrm{ind}(\sigma,g)/|G|).$

Lemma 4.5 implies that $\delta<-1$.

######  Lemma 4.14.

Given $\Pi\in\Sigma_{k}$ and $\mathfrak{q}_{\sigma}$ as in Lemma 4.12. We have
the following asymptotic upper bound

$N_{k,\Pi}(S_{n}\times G,X)\ll_{k,S_{n},G,\epsilon}\left(\prod_{\sigma\in
C(S_{n})}|\mathfrak{q}_{\sigma}|^{\delta+\epsilon}\right)X^{\frac{1}{|G|}+\epsilon}.$

###### Proof.

Considering the definitions of $N_{k,\Pi}(S_{n}\times G;X)$ and the restricted
discriminant, we have:

$\displaystyle N_{k,\Pi}(S_{n}\times G;X)=$
$\displaystyle\\#\\{(K,L)\in\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}(K,L)\in\Pi\text{ and }$ $\displaystyle\prod_{\mathfrak{p}\notin
S}\mathrm{Disc}_{\mathfrak{p}}(K)^{|G|}\mathrm{Disc}_{\mathfrak{p}}(L)^{n}<{X}/{\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})}}\\}$
$\displaystyle=$
$\displaystyle\\#\\{(K,L)\in\mathcal{E}_{k}\mathrel{\mathop{\mathchar
58\relax}}(K,L)\in\Pi\text{ and }$
$\displaystyle\mathrm{Disc}_{res}^{\Pi}(K)^{|G|}\mathrm{Disc}_{res}(L)^{n}<{X}/{\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|^{\exp(h_{\mathfrak{p}},g_{\mathfrak{p}})}}\\}.$

We have previously established upper bounds on the number of suitable $S_{n}$
and $G$-extensions when ordered by the invariant $\mathrm{Disc}_{res}$.
Therefore, a straightforward application of Corollary 2.8 with
$\mathcal{P}_{1},\mathcal{P}_{2}$ properties that $K\in\Pi$, $L\in\Pi$
respectively and $\operatorname{inv}_{1},\operatorname{inv}_{2}$ are
restricted discriminant together with Lemmas 4.12 and 4.13 yields the desired
bound on $N_{k,\Pi}(S_{n}\times G;X)$. ∎

We now give the proof of our main theorem.

###### Proof of Theorem 1.2.

First, we recall that Theorem 4.1 asserts that

$N_{k}(S_{n}\times G;X)\gg_{G,k}X^{\frac{1}{|G|}}.$

Therefore what remains is for us to prove the asymptotic upper bound

$N_{k}(S_{n}\times G;X)\ll_{\epsilon}X^{\frac{1}{|G|}+\epsilon}$

for all $\epsilon>0$. Recall that for $Y>0$, the definition of
$N_{k,Y}(S_{n}\times G;X)$ is given in (4.1). Proposition 4.4 asserts that

$N_{k,Y}(S_{n}\times
G,X)\ll_{k,S_{n},G,Y,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

Thus to complete the proof, it suffices to show that

$N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times
G;X)\ll_{k,S_{n},G,Y,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

From (4.6) we deduce that

(4.9) $\begin{split}N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times
G;X)\leq\sum_{\begin{subarray}{c}\Pi\in\Sigma_{k},\\\ \prod_{\mathfrak{p}\in
S}|\mathfrak{p}|>Y\end{subarray}}N_{k,\Pi}(S_{n}\times G,X),\end{split}$

(see (4.3) for the definition of $N_{k,\Pi}(S_{n}\times G,X)$). Let
$\Pi\in\Sigma_{k}$ be a standard tuple. Set $\mathcal{T}(\Pi)$ to be the
associated tuple $(\mathfrak{q}_{\sigma})_{\sigma\in C(S_{n})}$, where
$C(S_{n})$ denotes the set of conjugacy classes of $S_{n}$ and

$\mathfrak{q}_{\sigma}\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{p}\in S,h_{\mathfrak{p}}=\sigma}\mathfrak{p}.$

We claim that for any tuple of ideals $(\mathfrak{q}_{\sigma})_{\sigma\in
C(S_{n})}$ we show that there are at most $O_{\epsilon}(\prod_{\sigma\in
C(S_{n})}\mathfrak{q}_{\sigma})^{\epsilon}$ standard tuples $\Pi\in\Sigma_{k}$
such that

(4.10) $\mathcal{T}(\Pi)=(\mathfrak{q}_{\sigma})_{\sigma\in C(S_{n})}.$

In greater detail, $\Pi$ is determined by the set of primes $S$ and the tuple
$\mathcal{L}$ of pairs $(h_{\mathfrak{p}},g_{\mathfrak{p}})$ for each
$\mathfrak{p}\in S$. The equation (4.10) implies that there are at most
$2^{\omega(\prod_{\sigma}\mathfrak{q}_{\sigma})}$ choices for the set $S$ and
at most $C(n!|G|)^{\omega(\prod_{\sigma}\mathfrak{q}_{\sigma})}$ choices for
the tuple $\mathcal{L}$ for some absolute constant $C>0$. Hence, we deduce
that at most $O_{\epsilon}(\prod_{\sigma\in
C(S_{n})}\mathfrak{q}_{\sigma})^{\epsilon}$ choices for $\Pi$.

Combining (4.9) with Lemma 4.14, we find that

$\begin{split}N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times
G;X)&\leq\sum_{\begin{subarray}{c}\Pi\in\Sigma_{k},\\\ \prod_{\mathfrak{p}\in
S}|\mathfrak{p}|>Y\end{subarray}}N_{k,\Pi}(S_{n}\times G,X)\\\
&\ll_{k,S_{n},G,\epsilon}\sum_{\begin{subarray}{c}\Pi\in\Sigma_{k},\\\
\prod_{\mathfrak{p}\in
S}|\mathfrak{p}|>Y\end{subarray}}\left(\prod_{\sigma}|\mathfrak{q}_{\sigma}|^{\delta+\epsilon}\right)X^{\frac{1}{|G|}+\epsilon}\\\
&\ll_{k,S_{n},G,\epsilon}\left(\sum_{(\mathfrak{q}_{\sigma}),\prod_{\sigma}|\mathfrak{q}|_{\sigma}>Y}\prod_{\sigma}|\mathfrak{q}_{\sigma}|^{\delta+\epsilon}\right)X^{\frac{1}{|G|}+\epsilon}\\\
&\ll_{k,S_{n},G,\epsilon}\left(\sum_{|\mathfrak{q}|>Y}|\mathfrak{q}|^{\delta+\epsilon}\right)X^{\frac{1}{|G|}+\epsilon}.\end{split}$

As $\delta<-1$, we can select $\epsilon>0$ to ensure convergence of the series
$\sum_{|\mathfrak{q}|>Y}|\mathfrak{q}|^{\delta+\epsilon}$ for all $Y>0$.
Consequently, we infer that

$N_{k}(S_{n}\times G;X)-N_{k,Y}(S_{n}\times
G;X)\ll_{k,S_{n},G,\epsilon}X^{\frac{1}{|G|}+\epsilon}.$

for all $Y>0$. Hence, $N_{k}(S_{n}\times
G;X)\ll_{k,S_{n},G,\epsilon}X^{\frac{1}{|G|}+\epsilon}$. This completes the
proof of the asymptotic upper bound and Theorem 1.2.

In fact, since the sum
$\sum_{|\mathfrak{q}|>Y}|\mathfrak{q}|^{\delta+\epsilon}$ approaches zero as
$Y\to\infty$ for sufficiently small $\epsilon>0$, we deduce that

$\lim_{Y\to\infty}\limsup_{X\to\infty}\frac{N_{k}(S_{n}\times
G;X)-N_{k,Y}(S_{n}\times G;X)}{X^{\frac{1}{|G|}+\epsilon}}=0.$

∎

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