On Malle’s conjecture for nilpotent groups

Abstract

We develop an abstract framework for studying the strong form of Malle’s conjecture [J. Number Theory 92 (2002), pp. 315–329; Experiment. Math. 13 (2004), pp. 129–135] for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle’s conjecture for any nilpotent group $G$ such that all elements of order $p$ are central, where $p$ is the smallest prime divisor of $\# G$.

We also give an upper bound for any nilpotent group $G$ tight up to logarithmic factors, and tight up to a constant factor in case all elements of order $p$ pairwise commute. Finally, we give a new heuristical argument supporting Malle’s conjecture in the case of nilpotent groups in their regular representation.

1. Introduction

1.1. Malle’s conjectures

Let $G$ be a non-trivial, finite group and let $K$ be a number field. In this paper we are interested in the counting function

$$\begin{equation*} N(X, G, K) \coloneq \#\{L/K \text{ Galois} : \operatorname {Gal}(L/K) \cong G, \ |N_{K/\mathbb{Q}}(\Delta (L/K))| \leq X\}, \end{equation*}$$

where $\Delta (L/K)$ denotes the relative discriminant of $L/K$. As part of a broader conjecture, Malle 3738 conjectured the following.

Conjecture 1 (Strong form of Malle’s conjecture.).

Let $G$ be a non-trivial, finite group. Then there exist constants $a(G) \in \mathbb{Q}_{>0}$, $b(G, K) \in \mathbb{Z}_{>0}$ and $c(G, K) > 0$ such that

$$\begin{align} N(X, G, K) \sim c(G, K) X^{a(G)} (\log X)^{b(G, K) - 1}. {\tag{1.1}} \end{align}$$

Malle 37 proposed the following recipe for the constant $a(G)$

$$\begin{equation*} a_{\mathrm{M}}(G) = \frac{p}{(p - 1) \# G} \end{equation*}$$

with $p$ being the smallest prime divisor of $\# G$. In follow-up work Malle 38 also included a proposal for $b(G, K)$

$$\begin{equation*} b_{\mathrm{M}}(G, K) = \#\{C \in \operatorname {Conj}(G) : C \text{ non-trivial and } c^p = \mathrm{id} \ \forall c \in C\}/\sim , \end{equation*}$$

where $\operatorname {Conj}(G)$ denotes the set of conjugacy classes of $G$ and where two conjugacy classes $C$ and $C'$ are equivalent if there exists $\sigma \in \operatorname {Gal}(\overline{K}/K)$ such that $\sigma \ast C = C'$. Here the action is given by $\sigma \ast C = C^{\chi (\sigma )}$ with $\chi : \operatorname {Gal}(\overline{K}/K) \rightarrow \hat{\mathbb{Z}}^\ast$ the cyclotomic character.

Malle does not give a recipe to compute the constant $c(G, K)$. In some cases the constant $c(G, K)$ is a product of local densities. This is known for $S_3$, $S_4$ and $S_5$ (and conjectured for $S_n$) and is commonly referred to as the Malle–Bhargava principle.

In case $G$ is allowed to be an arbitrary finite group, counterexamples are known to the strong form of Malle’s conjecture, see the work of Klüners 27. In the counterexample of Klüners, Malle gives the wrong value of $b_{\mathrm{M}}(G, K)$. However, the value of $a_{\mathrm{M}}(G)$ is still correct in Klüners’ counterexample. It is widely believed, but unproven, that Malle’s prediction for $a_{\mathrm{M}}(G)$ is correct for all non-trivial, finite groups $G$ (for this reason we will simply write $a(G)$ from now on). Even such a result seems to be out of reach currently, since it implies a positive answer to the inverse Galois problem. Klüners’ counterexample led Türkelli 51 to propose a corrected $b(G, K)$.

The strong form of Malle’s conjecture (including the more general situation where $G$ is not necessarily considered in its regular representation) has been verified in a limited amount of cases, see 1516 for $S_3$, 53 for $G$ abelian, 10 for $D_4 \subseteq S_4$, 28 for generalized quaternion groups, 5 for $S_4$, 6 for $S_5$, 9 for $S_3 \subseteq S_6$, 22 for nonic Heisenberg extensions, 3952 for direct products $G \times A$ with $G = S_3, S_4, S_5$ and $A$ abelian. There is also the recent work 4, which counts $D_4 \subseteq S_4$ when the extensions are ordered by Artin conductor instead of discriminant.

It is worth mentioning that the weak form of Malle’s conjecture, which asserts that

$$\begin{equation*} X^{a(G)} \ll N(X, G, K) \ll _\epsilon X^{a(G) + \epsilon }, \end{equation*}$$

is much better understood. There are no known counterexamples to the weak form, even when $G$ is allowed to be an arbitrary finite group. The weak form is known for nilpotent groups by the work of Klüners–Malle 30. Alberts 12 and Alberts–O’Dorney 3 made further progress in the solvable case.

1.2. Main results

We prove the strong form of Malle’s conjecture for a large family of nilpotent groups.

Theorem 1.1.

Let $K$ be a number field and let $G$ be a non-trivial, finite, nilpotent group. Let $p$ be the smallest prime divisor of $\# G$ and assume that all elements of order $p$ are central. Then there exists a constant $c > 0$ such that

$$\begin{equation*} N(X, G, K) \sim c X^{\frac{p}{(p - 1) \# G}} (\log X)^{b_{\mathrm{M}}(G, K) - 1}, \end{equation*}$$

where $b_{\mathrm{M}}(G, K)$ is the Malle constant, which equals the number of elements of order $p$ divided by $[K(\zeta _p) : K]$ in this case.

In the process, we give a parametrization of $G$-extensions that may prove fruitful for future investigations. We will say more about this parametrization in the next subsection.

Klüners 29 has previously proven an upper bound of the correct order of magnitude in the situation of Theorem 1.1, namely

$$\begin{equation*} N(X, G, K) \ll X^{\frac{p}{(p - 1) \# G}} (\log X)^{b_{\mathrm{M}}(G, K) - 1}. \end{equation*}$$

This follows from 29, Theorem 1.4 and 29, Lemma 5.13.

Let us construct some $2$-groups $G$ satisfying the hypotheses of Theorem 1.1. Take for example any finite, abelian $2$-group $A$. Then there is an action of $\mathbb{Z}/2\mathbb{Z}$ on $A$ by inversion, which gives an action of $\mathbb{Z}/4\mathbb{Z}$ on $A$ by projecting first to $\mathbb{Z}/2\mathbb{Z}$. If one takes $G = A \rtimes \mathbb{Z}/4\mathbb{Z}$, then $G$ fulfills all the conditions of Theorem 1.1. Note that such $G$ can have arbitrarily large nilpotency class.

We remark that our techniques do not give an explicit handle on the constant $c > 0$ guaranteed by Theorem 1.1. Even in the case where $G$ is cyclic of prime order it is a rather non-trivial task to provide an explicit value of the constant $c$, see 11. The following classical result of Wright 53 is an immediate corollary of Theorem 1.1.

Corollary 1.2.

Let $K$ be a number field and let $A$ be a non-trivial, finite, abelian group. Let $p$ be the smallest prime divisor of $\# A$. Then there exists a constant $c > 0$ such that

$$\begin{equation*} N(X, A, K) \sim c X^{\frac{p}{(p - 1) \# A}} (\log X)^{b(A, K) - 1}, \end{equation*}$$

where $b(A, K)$ is the number of elements of order $p$ divided by $[K(\zeta _p) : K]$.

Our proof is substantially shorter than Wright’s proof 53 and makes limited use of class field theory. Our parametrization of $G$-extensions immediately implies Theorem 1.3.

Theorem 1.3.

Let $K$ be a number field and let $G$ be a non-trivial, finite, nilpotent group. Let $p$ be the smallest prime divisor of $\# G$. Then

$$\begin{equation*} N(X, G, K) \ll X^{\frac{p}{(p - 1) \# G}} (\log X)^{i(G, K) - 1}, \end{equation*}$$

where $i(G, K)$ is the number of elements of order $p$ in $G$ divided by $[K(\zeta _p) : K]$.

Theorem 1.3 improves, in case of nilpotent groups in the regular representation, on recent work of Klüners–Wang 31 and Klüners 29 (see Proposition 5.8 and the text preceding it). The result in Theorem 1.3 is sharp up to logarithmic factors. In general we have the inequality $i(G, K) \geq b_{\mathrm{M}}(G, K)$. In fact, the upper bound in Theorem 1.3 matches Malle’s prediction precisely when we are in the situation of Theorem 1.1.

It should be possible to use our techniques to prove a more general version of Theorem 1.3 valid for arbitrary representations of nilpotent groups, but we shall not pursue this further here. Theorem 1.4 shows that we can achieve a sharp upper bound, up to a constant factor, provided that the elements of order $p$ commute with each other.

Theorem 1.4.

Let $K$ be a number field and let $G$ be a non-trivial, finite, nilpotent group. Let $p$ be the smallest prime divisor of $\# G$. Suppose that all elements of order $p$ commute with each other. Then

$$\begin{equation*} N(X, G, K) \ll X^{\frac{p}{(p - 1) \# G}} (\log X)^{b_{\mathrm{M}}(G, K) - 1}. \end{equation*}$$

The corresponding lower bound for $N(X, G, K)$ follows from 1, Corollary 1.7, where we take $T \coloneq \{g \in G : g^p = \mathrm{id}\}$ (the assumptions imply that $T$ is indeed abelian and normal). Earlier work of Klüners–Malle 30 provides a slightly weaker lower bound correct up to logarithmic factors.

To give examples of groups $G$ where Theorem 1.4 applies (but Theorem 1.1 does not), consider an $\mathbb{F}_2$ vector space $V$ of dimension $2^n$ and pick an ordered basis $\{b_0, \dots , b_{2^n - 1}\}$. Let $\mathbb{Z}/2^n\mathbb{Z}$ act on $V$ by cycling the ordered basis and extending linearly. Then we can take $G_n = V \rtimes \mathbb{Z}/2^{n + 1}\mathbb{Z}$, where $\mathbb{Z}/2^{n + 1}\mathbb{Z}$ acts on $V$ by first projecting to $\mathbb{Z}/2^n\mathbb{Z}$. We remark that $G_1$ is isomorphic to the small group $\langle 16, 3 \rangle$ from the GAP database and that $G_2$ is isomorphic to the small group $\langle 128, 48 \rangle$. Note that $G_n$ can have arbitrarily large nilpotency class.

1.3. Method of proof

The main innovation of this paper is a new parametrization of $G$-extensions, which is given in Section 2. The simplest case of the parametrization is that of multiquadratic fields over $\mathbb{Q}$. The straightforward way to parametrize multiquadratic fields is the following. Let $\mathbf{v} = (a_1, \dots , a_n)$ be a vector such that the $a_i$ are squarefree and linearly independent in $\mathbb{Q}^\ast /\mathbb{Q}^{\ast 2}$. To each such vector $\mathbf{v}$, we associate the field $\operatorname {Par}(\mathbf{v}) = \mathbb{Q}(\sqrt {a_1}, \dots , \sqrt {a_n})$. In this way we have created a map $\mathrm{Par}$ to multiquadratic fields of degree $2^n$. This map has two convenient properties

•

the pre-images $\operatorname {Par}^{-1}(K)$ all have the same size (and are finite);

•

the map $\mathrm{Par}$ is surjective.

However, we will nevertheless argue that this does not provide a convenient way to count multiquadratic extensions. Indeed, the discriminant of $\operatorname {Par}(\mathbf{v})$ is equal to $\operatorname {rad}(a_1 \cdot \ldots \cdot a_n)^{2^{n - 1}}$ up to factors of $2$, where $\mathrm{rad}$ is the radical of an integer. Let us now consider the counting function

$$\begin{equation*} N'(X, \mathbb{F}_2^n, \mathbb{Q}) = \sum _{\substack{a_1, \dots , a_n \text{ squarefree} \\ \operatorname {rad}(a_1 \cdot \ldots \cdot a_n)^{2^{n - 1}} \leq X}} 1, \end{equation*}$$

which is a good prototype for $N(X, \mathbb{F}_2^n, \mathbb{Q})$. To evaluate $N'(X, \mathbb{F}_2^n, \mathbb{Q})$, we introduce the new variables

$$\begin{equation*} b_S = \prod _{\substack{p \\ p \mid a_i \Leftrightarrow i \in S}} p \end{equation*}$$

for every non-empty subset $S$ of $\{1, \dots , n\}$. Then the variables $b_S$ are squarefree and pairwise coprime. From the variables $b_S$ we can reconstruct the $a_i$ using the formula

$$\begin{align} a_i = \prod _{\substack{S \subseteq \{1, \dots , n\} \\ i \in S}} b_S. {\tag{1.2}} \end{align}$$

Under this change of variables, the sum becomes

$$\begin{equation*} N'(X, \mathbb{F}_2^n, \mathbb{Q}) = \sum _{\substack{b_S \text{ squarefree and pairwise coprime} \\ \prod _{\emptyset \subsetneq S \subseteq \{1, \dots , n\}} b_S^{2^{n - 1}} \leq X}} 1 = \sum _{d \leq X^{1/2^{n - 1}}} \mu ^2(d) \cdot \omega (d)^{2^n - 1}, \end{equation*}$$

which can be evaluated using classical techniques in analytic number theory. The crux of this idea is that the discriminant is simply

$$\begin{equation*} \prod _{\emptyset \subsetneq S \subseteq \{1, \dots , n\}} b_S^{2^{n - 1}} \end{equation*}$$

in the variables $b_S$, while the discriminant is a more complicated function in terms of the variables $a_i$. We remark that this change of variables also plays a role in the study of $2$-Selmer ranks and $4$-ranks, see for instance 2124. Inspired by this, we parametrize multiquadratic fields by tuples of squarefree integers $(b_S)_{\emptyset \subsetneq S \subseteq \{1, \dots , n\}}$ that are pairwise coprime. The parametrization map is then given by $\operatorname {Par}(a_1, \dots , a_n)$, where the $a_i$ are defined through equation 1.2.

For arbitrary nilpotent extensions (but still over $\mathbb{Q}$) our parametrization keeps track of the image of the inertia subgroup $I_p$ of $\operatorname {Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ in our $G$-extension for every prime $p$. This involves the choice of an embedding $i_p: \operatorname {Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p) \rightarrow \operatorname {Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. For arbitrary groups $G$ it is unfortunately not true that a homomorphism $\psi : \operatorname {Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow G$ is determined by the restrictions $\psi |_{i_p(I_p)}$.

Indeed, in general one needs to know the restriction of $\psi$ to $j_p(I_p)$ for every embedding $j_p: \operatorname {Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p) \rightarrow \operatorname {Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to make this conclusion. However, for nilpotent groups $G$ it suffices to fix one embedding $i_p: \operatorname {Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p) \rightarrow \operatorname {Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, as a homomorphism $\psi : \operatorname {Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow G$ is completely determined by the restrictions $\psi |_{i_p(I_p)}$ in this case, see Proposition 2.5. Our parametrization uses this property of nilpotent extensions in an essential way.

It is precisely for this reason that the source of our parametrization map is still tuples of squarefree integers that are pairwise coprime and are indexed by the elements in $G - \{\mathrm{id}\}$. We remark that this matches the multiquadratic case, since there is a bijection between non-empty subsets of $\{1, \dots , n\}$ and $\mathbb{F}_2^n - \{\mathrm{id}\}$. This allows us to give a similar formula as above for the discriminant, see Proposition 2.10.

The fact that our parametrization is in terms of tuples of squarefree integers is also most convenient for analytic purposes. For example, it allows us to prove Theorem 1.3 using essentially the trivial bound, namely by ignoring all the embedding problems. This demonstrates the power of the parametrization, since it improves on the existing results in the literature by bounding trivially! In some sense one may view Theorem 1.3 as the correct trivial bound for Malle’s conjecture.

It also enables us to write down the various embedding problems explicitly, which we do in Section 3. We use this description to give a heuristic argument why Malle’s conjecture is correct for nilpotent extensions. We believe that this may also be of value to possible future work on this topic. Indeed, it makes clear exactly what the difficulties are in proving Malle’s conjecture for arbitrary nilpotent extensions. Roughly speaking, at every central extension we pinpoint a certain set of Frobenius elements that need to be equidistributed in a subgroup that we correspondingly identify. This explains how the constant $b_{\mathrm{M}}(G, K)$ arises from the construction of a nilpotent group as successive central extensions, see Section 3.

The parametrization also plays an essential role in the proof of Theorem 1.1 and Theorem 1.4. Using the parametrization and some analysis, we reduce these theorems to counting multicyclic extensions with some extra local conditions. This idea also provides a new and short proof of Wright’s theorem 53.

1.4. Related results

There is also the related problem of counting the number of degree $n$ extensions with bounded discriminant, which was first treated by Schmidt 43. His upper bound was drastically improved by Ellenberg–Venkatesh 18, Couveignes 14 and Lemke Oliver–Thorne 36.

Malle’s conjecture has strong ties with the Cohen–Lenstra conjectures 12. There is the classical work of Davenport–Heilbronn 16 on $3$-torsion of class groups of quadratic fields, which was later extended by 8 and 49 in the form of a secondary main term. Davenport and Heilbronn obtain their results by counting certain $S_3$-extensions.

Fouvry and Klüners 2021 dealt with the $4$-rank of quadratic fields building on earlier work of Gerth 23, and Heath-Brown 24 on $2$-Selmer groups. There is also a rich literature on upper bounds for $2$-torsion elements in class groups of which we mention 3334 for multiquadratic extensions, 7 for $S_n$-extensions and 4546 for monogenic $S_n$-extensions. Furthermore, the average size of the $2$-torsion of $S_n$-extensions has been determined in 25 conditional on a tail estimate. Over function fields Malle’s conjecture and the Cohen–Lenstra conjectures are better understood due to the results in 1719.

Recently, Smith 4748 dealt with the $2$-part of class groups of quadratic fields, which was extended by the authors to the $\ell$-part of class groups of degree $\ell$ cyclic fields 32. His techniques can be adapted to give a lower bound for the number of $D_{2^n}$-extensions of $\mathbb{Q}$ of the correct order of magnitude: this is another instance that highlights the clear ties between Malle’s conjecture and the Cohen–Lenstra conjectures. It seems plausible that this can be extended to an asymptotic once one extends the results on ray class groups of Pagano–Sofos 42.

It is natural to wonder whether it is possible to combine Smith’s techniques with our parametrization to tackle Malle’s conjecture for all finite, nilpotent groups. It is the authors’ belief that this should be possible for a wide class of nilpotent groups, and we hope to return to this topic. However, one convenient property of $D_{2^n}$ is that the embedding problem for odd ramified primes $p$ often comes down to the condition that $p$ has residue field degree $1$. In terms of our parametrization this is very convenient, since it gives good inductive control over the relevant Frobenius elements.

This fails badly for arbitrary nilpotent $G$-extensions, where one may face “loops” in the following sense. We might simultaneously want to prove equidistribution of $\mathrm{Frob}_p$ in a quotient of $G$ depending on $q$, while we also need equidistribution of $\mathrm{Frob}_q$ in a quotient of $G$ depending on $p$. The authors currently do not know how to overcome this difficulty.

1.5. Organization of the paper

The paper is divided as follows. The core of the paper is Section 2, where we provide a new parametrization of $G$-extensions. In Section 3 we give a new heuristic in support of Malle’s conjecture for nilpotent groups in their regular representation. Our main theorems are proven in Section 5 with Section 4 providing some analytic tools.

2. Arbitrary nilpotent groups and number fields

In our first subsection we restrict ourselves to the case that $G$ is a finite $l$-group, but we work with an arbitrary number field $K$. This is then extended to general nilpotent $G$ in the second subsection.

2.1. Finite $l$-groups

We use the abbreviation $[n] \coloneq \{1, \dots , n\}$ throughout the paper. For a set $S$ and a prime number $l$, we write $\mathbb{F}_l^S$ for the free $\mathbb{F}_l$ vector space on the set $S$. Let us fix a separable closure $\mathbb{Q}^{\mathrm{sep}}$ of $\mathbb{Q}$ once and for all. All our number fields are implicitly taken inside this fixed separable closure. Furthermore, we write $G_K \coloneq \operatorname {Gal}(\mathbb{Q}^{\mathrm{sep}}/K)$ for the absolute Galois group of a number field $K$. Similarly, for each prime $p$ we fix a separable closure $\mathbb{Q}_p^{\mathrm{sep}}$ of $\mathbb{Q}_p$ (which allows us to define $G_K \coloneq \operatorname {Gal}(\mathbb{Q}_p^{\mathrm{sep}}/K)$ for any extension $K$ of $\mathbb{Q}_p$ inside $\mathbb{Q}_p^{\mathrm{sep}}$) together with an embedding

$$\begin{equation*} i_p: \mathbb{Q}^{\mathrm{sep}} \to \mathbb{Q}_p^{\mathrm{sep}}. \end{equation*}$$

This yields an inclusion

$$\begin{equation*} i_p^{*}: G_{\mathbb{Q}_p} \to G_{\mathbb{Q}}. \end{equation*}$$

Let $K$ be a number field, picked inside $\mathbb{Q}^{\mathrm{sep}}$. Denote by $\Omega _K$ the set of all places of $K$. For each finite place $\mathfrak{q}$ in $\Omega _K$, lying above a rational prime $q$, the restriction of the map $i_q^{*}$ provides us with an inclusion

$$\begin{equation*} i_{\mathfrak{q}}^{*}: G_{K_{\mathfrak{q}}} \to G_{K}. \end{equation*}$$

Denote by $k_{\mathfrak{q}}$ the residue field of $K$ at $\mathfrak{q}$. Write

$$\begin{equation*} I_{\mathfrak{q}} \coloneq \operatorname {ker}(G_{K_{\mathfrak{q}}} \to G_{k_{\mathfrak{q}}}) \end{equation*}$$

for the inertia subgroup.

Let now $l$ be any prime number. In what follows the group $\mathbb{F}_l$ will always be implicitly interpreted as a Galois module with trivial action, whenever the notation suggests an implicit action of a group on $\mathbb{F}_l$. We denote by $H_{\mathrm{unr}}^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)$ the image, via inflation, of $H^1(G_{k_{\mathfrak{q}}}, \mathbb{F}_l)$ in $H^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)$.

For a subset $S \subseteq \Omega _K$, containing all the archimedean places of $K$, we consider the map

$$\begin{equation*} \Phi _K(l, S): H^1(G_K, \mathbb{F}_l) \to \bigoplus _{\mathfrak{q} \in \Omega _K-S} \frac{H^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)}. \end{equation*}$$

In case $S$ consists exactly of the archimedean places, we will denote the resulting map simply by $\Phi _K(l)$. We start by recalling the following classical fact.

Proposition 2.1.

The abelian groups $\operatorname {ker}(\Phi _K(l))$ and $\operatorname {coker}(\Phi _K(l))$ are finite.

Proof.

By class field theory we have a canonical identification

$$\begin{equation*} \operatorname {ker}(\Phi _K(l)) = \operatorname {Cl}(K, m_{\infty })^{\vee }[l], \end{equation*}$$

where $m_{\infty }$ is the modulus consisting of all archimedean places. Therefore we have that

$$\begin{equation*} \#\operatorname {ker}(\Phi _K(l))=\#\operatorname {Cl}(K,m_{\infty })^{\vee }[l] \leq l^{[K:\mathbb{Q}]} \cdot \#\operatorname {Cl}(K)^{\vee }[l], \end{equation*}$$

where the first factor $l^{[K:\mathbb{Q}]}$ can be dropped when $l$ is odd. Therefore the finiteness of $\operatorname {ker}(\Phi _K(l))$ follows from the finiteness of $\operatorname {Cl}(K)$. The finiteness of $\operatorname {coker}(\Phi _K(l))$ is established in 44, Theorem 5, eq. (14) and (16). In the notation of Shafarevich, we have

$$\begin{equation*} (\mathfrak{U}/\mathfrak{U}_S^l)^\vee = \bigoplus _{\mathfrak{q} \in \Omega _K-S} \frac{H^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)}, \quad (J/J^l K)^\vee = H^1(G_K, \mathbb{F}_l) \end{equation*}$$

and the dual of his homomorphism $\phi _l$ is our $\Phi _K(l, S)$. Combining 44, Theorem 5, eq. (14) and (16) yields

$$\begin{equation*} \dim _{\mathbb{F}_l} \operatorname {coker}(\Phi _K(l)) = \sigma (S), \end{equation*}$$

which is a finite number defined on 44, p. 129.

■

The following important fact falls as an immediate consequence of Proposition 2.1.

Proposition 2.2.

There exists a finite set of places $S$, containing all archimedean places, such that $\Phi _K(l, S)$ is surjective.

Proof.

Write $A$ for the finite subset of archimedean places of $\Omega _K$. Thanks to Proposition 2.1 we can find a finite set of places $A \subseteq S \subseteq \Omega _K$ such that the natural map

$$\begin{equation*} \bigoplus _{\mathfrak{q} \in S - A} \frac{H^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)} \to \operatorname {coker}(\Phi _K(l)) \end{equation*}$$

is surjective. This implies that for any vector

$$\begin{equation*} v \in \bigoplus _{\mathfrak{q} \in \Omega _K-S} \frac{H^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)}, \end{equation*}$$

we can find $w \in \bigoplus _{\mathfrak{q} \in S - A} \frac{H^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_\mathfrak{q}}, \mathbb{F}_l)}$ and a global character $\chi \in H^1(G_K, \mathbb{F}_l)$ such that

$$\begin{equation*} \Phi _K(l)(\chi ) = v + w. \end{equation*}$$

Therefore, since $w$ is entirely supported in $S$, we conclude that

$$\begin{equation*} \Phi _K(l, S)(\chi ) = v. \end{equation*}$$

Hence we have shown that the map $\Phi _K(l, S)$ is surjective with this choice of $S$, which is precisely the desired conclusion.

■

Of course if a set $S$ as in Proposition 2.2 works, then any larger set works as well. We fix once and for all a finite set $S_{\mathrm{clean}}(l)$ as in Proposition 2.2, making sure that it also contains all places above $l$. We denote by $\widetilde{\Omega }_K(l)$ the subset of $\Omega _K - S_{\mathrm{clean}}(l)$ with

$$\begin{align} \frac{H^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)}{H_{\mathrm{unr}}^1(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)} \neq 0. {\tag{2.1}} \end{align}$$

For $\mathfrak{q} \in \Omega _K - S_{\mathrm{clean}}(l)$, it follows from local class field theory that equation 2.1 is equivalent to the condition

$$\begin{equation*} \# \left(\mathcal{O}_K/\mathfrak{q} \right) \equiv 1 \bmod l. \end{equation*}$$

Write $K^{\mathrm{pro}-l}$ for the compositum of all finite Galois extensions $L$ of $K$ with $[L : K]$ a power of $l$.

Proposition 2.3.

Let $\mathfrak{q} \in \Omega _K$ be a finite place coprime to $l$ such that $\# \left(\mathcal{O}_K/\mathfrak{q} \right) \not \equiv 1 \bmod l$. Then $\mathfrak{q}$ is unramified in every finite extension $L/K$ inside $K^{\mathrm{pro}-l}$.

Proof.

It suffices to prove the proposition locally at $\mathfrak{q}$. Take a positive integer $f$ and let $K_{\mathfrak{q}^f}$ be the unique unramified extension of $K_{\mathfrak{q}}$ of degree equal to $f$ with residue field denoted by $k_{\mathfrak{q}^f}$. Then if we have a cyclic totally ramified degree $l$ extension of $K_{\mathfrak{q}^f}$ it follows from local class field theory and the fact that $\mathfrak{q}$ is coprime to $l$

$$\begin{equation*} \#k_{\mathfrak{q}^f} = \#k_{\mathfrak{q}}^f \equiv 1 \bmod l. \end{equation*}$$

Now, if $f$ is a power of $l$ itself, we conclude that $\#k_{\mathfrak{q}}$ is already congruent to $1$ modulo $l$ contrary to our assumption that $\# \left(\mathcal{O}_K/\mathfrak{q} \right) \not \equiv 1 \bmod l$.

■

We remark that Proposition 2.3 certainly applies to any place $\mathfrak{q} \in \Omega _K - S_{\mathrm{clean}}(l) - \widetilde{\Omega }_K(l)$. Let $\mathfrak{q} \in \widetilde{\Omega }_K(l)$. Thanks to Proposition 2.2, there exists a character

$$\begin{equation*} \chi _{\mathfrak{q}} \in H^1(G_K, \mathbb{F}_l) \end{equation*}$$

such that $\Phi _K(l, S_{\mathrm{clean}}(l))(\chi _{\mathfrak{q}})$ has non-trivial coordinate precisely at $\mathfrak{q}$ and at no other places in $\Omega _K-S_{\mathrm{clean}}(l)$. Fix once and for all such a choice of $\chi _{\mathfrak{q}}$ for each $\mathfrak{q} \in \widetilde{\Omega }_K(l)$. By construction

$$\begin{equation*} \{\chi _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)} \end{equation*}$$

is a linearly independent set. Furthermore by Proposition 2.1 we obtain that the subspace

$$\begin{equation*} \langle \{\chi _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)} \rangle \subseteq H^1(G_K, \mathbb{F}_l) \end{equation*}$$

has finite index. Additionally, there exist a positive integer $t$ and a basis

$$\begin{equation*} J \coloneq \{\chi _i\}_{i=1}^t \subseteq \operatorname {ker}(\Phi _K(l, S_{\mathrm{clean}}(l))) \end{equation*}$$

such that $J \cup \{\chi _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)}$ is a basis of $H^1(G_K, \mathbb{F}_l)$. Fix once and for all such a choice of $J$. We denote by

$$\begin{equation*} \mathcal{B}(K, l) \coloneq J \cup \{\chi _{\mathfrak{q}} \}_{\mathfrak{q}} \end{equation*}$$

this fixed choice of a basis. Put

$$\begin{equation*} \mathcal{G}_K^{\mathrm{pro}-l}\coloneq \operatorname {Gal}(K^{\mathrm{pro}-l}/K). \end{equation*}$$

For each finite place $\mathfrak{q} \in \Omega _K$ we denote by

$$\begin{equation*} I_{\mathfrak{q}}(l) \coloneq \operatorname {proj}(G_K \to \mathcal{G}_K^{\mathrm{pro}-l}) \circ i_q^{*}(I_{\mathfrak{q}}). \end{equation*}$$

We have the following basic fact.

Proposition 2.4.

The group $I_{\mathfrak{q}}(l)$ is pro-cyclic for each finite place $\mathfrak{q} \in \Omega _K$ coprime to $l$.

Proof.

Let $L$ be a non-archimedean local field of characteristic $0$ and write $p$ for its residue characteristic. Let $d$ be a positive integer coprime to $p$. Then every finite totally ramified extension of $L$ of degree equal to $d$ can be obtained as $L(\sqrt [d]{\pi })$ for $\pi$ a uniformizer of $L$. For an elementary proof of this well-known fact see 32, Proposition A.5. Applying this repeatedly to all finite unramified extensions of $L$, we conclude that

$$\begin{equation*} \frac{I_L}{I_L^{\mathrm{wild}}} \end{equation*}$$

is a pro-cyclic group, where $I_L$ is the inertia subgroup and $I_L^{\mathrm{wild}}$ is the wild inertia subgroup. Since $\mathfrak{q}$ is coprime to $l$, we conclude that $I_{\mathfrak{q}}(l)$ is a quotient of the pro-cyclic group $\frac{I_{\mathfrak{q}}}{I_{\mathfrak{q}}^{\mathrm{wild}}}$, which gives in particular the desired conclusion.

■

Remark 1.

Since the group $I_{\mathfrak{q}}(l)$ is a pro-cyclic pro-$l$ group, it is either isomorphic to a finite group of order a power of $l$ or isomorphic to $\mathbb{Z}_l$. In case a primitive $l$-th root of unity $\zeta _l$ is in $K$, then we claim that $I_{\mathfrak{q}}(l)$ is infinite. Indeed, we have the infinitely ramified subextension

$$\begin{equation*} K(\sqrt [\infty ]{1}, \sqrt [\infty ]{\alpha }) \end{equation*}$$

of $K^{\mathrm{pro}-l}/K$ (which is contained in $K^{\mathrm{pro}-l}/K$ exactly because $\zeta _l$ is in $K$) given by any $\alpha$ in $K$ with $v_{\mathfrak{q}}(\alpha )=1$. Therefore we conclude that if $K$ possesses a non-trivial $l$-th root of unity, then

$$\begin{equation*} I_{\mathfrak{q}}(l) \simeq _{\mathrm{top.gr.}} \mathbb{Z}_l \end{equation*}$$

for every finite place $\mathfrak{q} \in \Omega _K$ coprime to $l$. Instead if $\zeta _l$ is not in $K$, we observe that Proposition 2.3 shows that the group $I_{\mathfrak{q}}(l)$ is trivial in case $\mathfrak{q}$ is a finite place of $\Omega _K$ that is coprime to $l$ and satisfies $\left(\mathcal{O}_K/\mathfrak{q} \right) \not \equiv 1 \bmod l$.

We fix once and for all a topological generator $\sigma _{\mathfrak{q}}$ of $I_{\mathfrak{q}}(l)$ for all $\mathfrak{q} \in \widetilde{\Omega }_K(l)$ in the following manner. Observe that $\chi _{\mathfrak{q}}(\sigma _{\mathfrak{q}}) \neq 0$ for any topological generator of $I_{\mathfrak{q}}(l)$, since the character $\chi _{\mathfrak{q}}$ ramifies at $\mathfrak{q}$. Hence we can always pick a generator $\sigma _{\mathfrak{q}}$ with the normalization $\chi _{\mathfrak{q}}(\sigma _{\mathfrak{q}})=1$. We make such a choice of $\sigma _{\mathfrak{q}}$ once and for all. In case $\mathfrak{q} \in \Omega _K - S_{\mathrm{clean}}(l) - \widetilde{\Omega }_K(l)$, then the group $I_{\mathfrak{q}}(l)$ is trivial by Proposition 2.3, and we declare $\sigma _{\mathfrak{q}}\coloneq \mathrm{id}$.

Now it follows by construction that

$$\begin{equation*} \chi (\sigma _{\mathfrak{q}}) = \delta _{\chi _{\mathfrak{q}}}(\chi ) \end{equation*}$$

for each $\chi \in \mathcal{B}(K, l)$ and $\mathfrak{q} \in \widetilde{\Omega }_K(l)$, where $\delta$ denotes the Kronecker delta function. Therefore we can complete the set

$$\begin{equation*} \{\sigma _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)} \end{equation*}$$

to a minimal set of generators

$$\begin{equation*} \{\sigma _i\} \cup \{\sigma _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)}, \end{equation*}$$

which is dual to the basis $\mathcal{B}(K, l)$, i.e.

$$\begin{align*} &\chi _i(\sigma _{\mathfrak{q}}) = 0 = \chi _{\mathfrak{q}}(\sigma _i) \quad && \text{ for each } i \in [t] \text{ and } \mathfrak{q} \in \widetilde{\Omega }_K(l), \\ &\chi _i(\sigma _j) = \delta _i(j) \quad && \text{ for each } i, j \in [t], \\ &\chi _{\mathfrak{q}}(\sigma _{\mathfrak{q}'}) = \delta _{\mathfrak{q}}(\mathfrak{q}') \quad && \text{ for every } \mathfrak{q}, \mathfrak{q}' \in \widetilde{\Omega }_K(l). \end{align*}$$

We denote by

$$\begin{equation*} \mathcal{B}^{\vee }(K, l) \coloneq \{\sigma _i\}_{i=1}^{t} \cup \{\sigma _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)}. \end{equation*}$$

Then we have the following.

Proposition 2.5.

The set $\mathcal{B}^{\vee }(K, l)$ is a set of topological generators of $\mathcal{G}_K^{\mathrm{pro}-l}$.

Proof.

By construction $\mathcal{B}^{\vee }(K, l)$ topologically generates $\operatorname {Gal}(L/K)$, where $L$ is by definition the compositum of all degree $l$ cyclic extensions of $K$. But since $\mathcal{G}_K^{\mathrm{pro}-l}$ is a pro-$l$-group, we see that $\operatorname {Gal}(L/K)$ is also the quotient of $\mathcal{G}_K^{\mathrm{pro}-l}$ by its Frattini subgroup. This implies the proposition, since a subset $S$ topologically generates if and only if it topologically generates modulo the Frattini subgroup.

■

Let $L/K$ be a finite Galois extension inside $K^{\mathrm{pro}-l}$ with Galois group $G \coloneq \operatorname {Gal}(L/K)$. Take a $2$-cocycle $\theta$

$$\begin{equation*} \theta : G^2 \to \mathbb{F}_l \end{equation*}$$

with the requirement that $\theta (\mathrm{id}, \mathrm{id}) = 0$. By the same argument as before, every class in $H^2(G, \mathbb{F}_l)$ can be represented by such a $2$-cocycle $\theta$. Consider the group

$$\begin{equation*} (\mathbb{F}_l \times G,*_{\theta }), \end{equation*}$$

where the group law is

$$\begin{equation*} (a_1, g_1)*_{\theta }(a_2, g_2) = (a_1 + a_2 + \theta (g_1, g_2), g_1g_2). \end{equation*}$$

Our assumption on $\theta$ ensures that $(0, \mathrm{id})$ is the trivial element of $(\mathbb{F}_l \times G,*_{\theta })$.

Proposition 2.6.

Let $l$ be a prime number. Let $K$ be a number field and let $L$ be an extension with $G = \operatorname {Gal}(L/K)$ a finite $l$-group. Suppose that $\theta$ is non-trivial in $H^2(G, \mathbb{F}_l)$.

(a) The natural projection map $\pi : G_K \twoheadrightarrow G$ can be lifted to a surjective homomorphism

$$\begin{equation*} \psi : G_K \to (\mathbb{F}_l \times G,*_{\theta }) \end{equation*}$$

if and only if $\theta$ is trivial in $H^2(G_{K_v}, \mathbb{F}_l)$ for each place $v$ that ramifies in $L/K$. Moreover, if $\psi$ is a lift, then the $\mathbb{F}_l$-coordinate of $\psi$ is a continuous $1$-cochain $\phi (\psi ): G_K \rightarrow \mathbb{F}_l$ with

$$\begin{equation*} \operatorname {d}(-\phi (\psi )) = \theta . \end{equation*}$$

Conversely, given any such continuous $1$-cochain $\phi : G_K \rightarrow \mathbb{F}_l$ with $\operatorname {d}(-\phi ) = \theta$, the assignment

$$\begin{equation*} \psi (\phi )(g) = (\phi (g), \pi (g)) \end{equation*}$$

is an epimorphism lifting the canonical projection $\pi : G_K \to G$ to an epimorphism $G_K \to (\mathbb{F}_l \times G,*_{\theta })$. The two assignments are mutual inverses.

(b) In case one has a lift $\psi$ as in part (a), then there is a unique one satisfying

$$\begin{equation*} \phi (\psi )(\sigma ) = 0 \text{ for all } \sigma \in \mathcal{B}^{\vee }(K, l). \end{equation*}$$

Proof.

We start with part (a). We claim that a map $\phi (\psi ): G_K \to \mathbb{F}_l$ is the first coordinate of a homomorphism

$$\begin{equation*} \psi : G_K \twoheadrightarrow (\mathbb{F}_l \times G,*_{\theta }), \quad g \mapsto (\phi (\psi )(g), \pi (g)) \end{equation*}$$

if and only if

$$\begin{equation*} \operatorname {d}(-\phi (\psi )) = \theta . \end{equation*}$$

Indeed, since $\psi$ is a homomorphism, we obtain

$$\begin{multline*} (\phi (\psi )(g_1g_2), \pi (g_1g_2)) = \psi (g_1g_2) = \psi (g_1) \psi (g_2) \\ = (\phi (\psi )(g_1) + \phi (\psi )(g_2) + \theta (g_1, g_2), \pi (g_1g_2)), \end{multline*}$$

which is equivalent to

$$\begin{equation*} \operatorname {d}(-\phi (\psi ))(g_1, g_2) = \phi (\psi )(g_1g_2) - \phi (\psi )(g_1) - \phi (\psi )(g_2) = \theta (g_1, g_2) \end{equation*}$$

as claimed.

Now suppose that there exists $\phi (\psi )$ with $\operatorname {d}(-\phi (\psi )) = \theta$. We claim that $(\phi (\psi ), \pi )$ is surjective. Let us first show that all characters $(\mathbb{F}_l \times G,*_{\theta }) \rightarrow \mathbb{F}_l$ must come from $G$. If not, then the kernel of such a hypothetical character provides a splitting of $\theta$, which implies that $\theta$ is trivial contrary to our assumptions. Hence, since $\pi$ is surjective, the image of $(\phi (\psi ), \pi )$ generates modulo the Frattini of $(\mathbb{F}_l \times G,*_{\theta })$, and therefore equals $(\mathbb{F}_l \times G,*_{\theta })$.

Furthermore, we see that the lifting $\psi$ exists if and only if the inflation of $\theta$ to $H^2(G_K, \mathbb{F}_l)$ is trivial if and only if $\theta$ is trivial in $H^2(G_{K_v}, \mathbb{F}_l)$ for every place $v$ of $K$. Thanks to 32, Proposition 4.4, the vanishing at the finite places unramified in $L/K$ is already guaranteed: notice that in 32, Section 4 the number $l$ is assumed to be an odd prime but Proposition 4.4 of 32 also holds for $l = 2$ with an identical proof. Let us now show $\theta$ is locally trivial at any unramified, infinite place of $L/K$. If $v$ is an archimedean complex place, then this is clear. If $v$ is instead an archimedean real place and the extension $L/K$ is unramified at $v$, then this means that $L_w = K_v$ for each place $w$ of $L$ above $v$ and thus the embedding problem is locally trivial at $v$. This ends the proof of part (a).

We now prove part (b). The uniqueness follows at once from part (a) combined with the fact that $\mathcal{B}^{\vee }(K, l)$ is a system of topological generators for $\mathcal{G}_K^{\mathrm{pro}-l}$. Indeed, an epimorphism $\psi$ as in part (a) is entirely determined by its values on a set of topological generators. We next show the existence: here we will take advantage of the fact that $\mathcal{B}^{\vee }(K, l)$ is a minimal set of topological generators. Take a map $\phi : G_K \rightarrow \mathbb{F}_l$ satisfying

$$\begin{equation*} \operatorname {d}(-\phi ) = \theta . \end{equation*}$$

The resulting epimorphism $\psi (\phi ):\mathcal{G}_K^{\mathrm{pro}-l} \twoheadrightarrow (\mathbb{F}_l \times G, *_{\theta })$ corresponds to a finite extension. As such we conclude that $\phi (\sigma _{\mathfrak{q}}) = 0$ for all but finitely many $\mathfrak{q} \in \widetilde{\Omega }_K(l)$. Therefore the sum

$$\begin{equation*} \sum _{i = 1}^t \phi (\sigma _i) \cdot \chi _i + \sum _{\mathfrak{q} \in \widetilde{\Omega }_K(l)} \phi (\sigma _{\mathfrak{q}}) \cdot \chi _{\mathfrak{q}} \end{equation*}$$

is a well-defined element of $H^1(G_K, \mathbb{F}_l)$. Hence, since $\mathcal{B}(K, l)$ and $\mathcal{B}^{\vee }(K, l)$ are dual to each other, we obtain that

$$\begin{equation*} \phi - \sum _{i = 1}^t \phi (\sigma _i) \cdot \chi _i - \sum _{\mathfrak{q} \in \widetilde{\Omega }_K(l)} \phi (\sigma _{\mathfrak{q}}) \cdot \chi _{\mathfrak{q}} \end{equation*}$$

vanishes at $\sigma _i$ for all $i \in [t]$ and at $\sigma _\mathfrak{q}$ for all $\mathfrak{q} \in \widetilde{\Omega }_K(l)$. This ends the proof of part (b).

■

We denote the unique $1$-cochain as in part (b) of Proposition 2.6 by $\phi (G, \theta )$. In case $\theta$ is trivial as a $2$-cocycle, then we choose $\phi (G, \theta ) \coloneq 0$. With this choice, we see that $\phi (G, \theta )$ satisfies part (b) of Proposition 2.6, since the trivial character is the unique cyclic degree $l$ character vanishing at all $\sigma \in \mathcal{B}^{\vee }(K, l)$.

Denote by $\mathcal{S}_l$ the set $\{0,1\}^{[t]} \times \mathcal{S}_l'$, where $\mathcal{S}_l'$ is the set of squarefree integral ideals in $\mathcal{O}_K$ entirely supported in $\widetilde{\Omega }_K(l)$. To an element $(T, \mathfrak{b}) \in \mathcal{S}_l$ we attach the character

$$\begin{equation*} \chi _{(T, \mathfrak{b})} \coloneq \sum _{\substack{i \in [t] \\ \pi _i(T) = 1}} \chi _i + \sum _{\substack{\mathfrak{q} \mid \mathfrak{b} \\ \mathfrak{q} \in \widetilde{\Omega }_K(l)}} \chi _{\mathfrak{q}}. \end{equation*}$$

Two pairs $(T, \mathfrak{b}), (T', \mathfrak{b}') \in \mathcal{S}_l$ are said to be coprime in case $\mathfrak{b}, \mathfrak{b'}$ are coprime ideals and there does not exist a $j \in [t]$ such that $\pi _j(T) = \pi _j(T') = 1$. Formulated differently, the two pairs are coprime exactly when

$$\begin{equation*} \{\sigma \in \mathcal{B}^{\vee }(K, l): \chi _{(T, \mathfrak{b})}(\sigma ) \neq 0 \} \cap \{\sigma ' \in \mathcal{B}^{\vee }(K, l): \chi _{(T', \mathfrak{b}')}(\sigma ') \neq 0 \}=\emptyset . \end{equation*}$$

Let $V$ be any finite set. We denote by $\operatorname {Prim}(\mathcal{S}_l^{\mathbb{F}_l^V - \{(0, \ldots , 0)\}})$ the subset of $\mathcal{S}_l^{\mathbb{F}_l^V - \{(0, \ldots , 0)\}}$ consisting of vectors possessing pairwise coprime coordinates. We conclude this subsection by giving a bijection

$$\begin{equation*} \operatorname {Pow}_l(V): \operatorname {Prim}(\mathcal{S}_l^{\mathbb{F}_l^V - \{(0, \ldots , 0)\}}) \to H^1(G_K, \mathbb{F}_l)^{V}, \end{equation*}$$

which sends a vector $(v_g)_{g \in \mathbb{F}_l^V - \{(0, \dots , 0)\}}$ to

$$\begin{equation*} \operatorname {Pow}_l(V)((v_g)_{g \in \mathbb{F}_l^V - \{(0, \dots , 0)\}}) \coloneq \left(\sum _{g \in \mathbb{F}_l^V - \{(0, \ldots , 0)\}} \pi _j(g) \cdot \chi _{v_g} \right)_{j \in V}, \end{equation*}$$

where $\pi _j$ is the projection map on the $j$-th coordinate. Let us prove that this map is indeed a bijection.

Proposition 2.7.

Let $V$ be a finite set. Then the map $\operatorname {Pow}_l(V)$ is a bijection.

Proof.

Assume without loss of generality that $V = [r]$. To a vector $(\chi _1, \ldots , \chi _r)$ in $H^1(G_K, \mathbb{F}_l)^r$ we attach a point

$$\begin{equation*} \Pi _l(\chi _1, \ldots , \chi _r) \coloneq (v_g(1), v_g(2))_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}} \end{equation*}$$

in $\operatorname {Prim}(\mathcal{S}_l^{\mathbb{F}_l^r - \{(0, \ldots , 0)\}})$ as follows. For each $\mathfrak{q} \in \widetilde{\Omega }_K(l)$ we let $\mathfrak{q}$ divide the entry $v_g(2)$ if and only if

$$\begin{equation*} (\chi _1(\sigma _{\mathfrak{q}}), \ldots , \chi _r(\sigma _{\mathfrak{q}})) = g. \end{equation*}$$

Likewise for each $j \in [t]$ we put $\pi _j(v_g(1)) = 1$ if and only if

$$\begin{equation*} (\chi _1(\sigma _j), \ldots , \chi _r(\sigma _j)) = g. \end{equation*}$$

By construction $(v_g(1), v_g(2))_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}}$ is in $\operatorname {Prim}(\mathcal{S}_l^{\mathbb{F}_l^r - \{(0, \ldots , 0)\}})$. Using that $\mathcal{B}^{\vee }(K, l)$ is a system of topological generators, we deduce that

$$\begin{equation*} \operatorname {Pow}_l([r])((v_g(1), v_g(2))_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}}) = (\chi _1, \ldots , \chi _r), \end{equation*}$$

since the equality holds by construction when evaluated in an element of $\mathcal{B}^{\vee }(K, l)$. Conversely let $\sigma \in \mathcal{B}^{\vee }(K, l)$ and let $(v_g)_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}} \in \operatorname {Prim}(\mathcal{S}_l^{\mathbb{F}_l^r - \{(0, \ldots , 0)\}})$. There exists at most one $g_0 \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}$ such that $\chi _{v_{g_0}}(\sigma ) \neq 0$. Suppose that such a $g_0$ exists. Then

$$\begin{equation*} \operatorname {Pow}_l([r])((v_g)_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}})(\sigma ) = g_0, \end{equation*}$$

which implies that

$$\begin{equation*} \Pi _l \circ \operatorname {Pow}_l([r])((v_g)_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}}) = (v_g)_{g \in \mathbb{F}_l^r - \{(0, \ldots , 0)\}}. \end{equation*}$$

Hence $\Pi _l$ and $\operatorname {Pow}_l([r])$ are mutual inverses, which finishes the proof of the proposition.

■

We now have the necessary tools to parametrize nilpotent extensions. We carry this out in the next subsection.

2.2. The parametrization in general

Let $r \in \mathbb{Z}_{\geq 1}$. A sequence of pairs

$$\begin{equation*} \{(G_i, \theta _i)\}_{i \in [r]} \end{equation*}$$

is called an admissible sequence if it satisfies the following inductive rules:

•

$G_0$ is the trivial group by convention. Furthermore, $G_i$ is an $l$-group and $\theta _i: G_{i - 1}^2 \to \mathbb{F}_l$ is a $2$-cocycle with $\theta _i(\mathrm{id}, \mathrm{id}) = 0$ for each $i \in [r]$;

•

we have$$\begin{equation*} G_i = (\mathbb{F}_l \times G_{i - 1}, *_{\theta _i}) \end{equation*}$$

for all $i \in [r]$;

•

$\theta _i$ is the zero map if and only if the class of $\theta$ in $H^2(G_{i - 1}, \mathbb{F}_l)$ is trivial.

For the remainder of this section we fix an admissible sequence $\{(G_i, \theta _i)\}_{i \in [r]}$. Set

$$\begin{equation*} G \coloneq G_r. \end{equation*}$$

The aim of this section is to construct a surjective map

$$\begin{equation*} P_G: \operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}}) \twoheadrightarrow \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G) \cup \{\bullet \}, \end{equation*}$$

which restricts to a bijection between

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})(\mathrm{solv.}) \coloneq P_G^{-1}(\operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G)) \end{equation*}$$

and $\operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G)$. Furthermore, we explain how to read the ramification data on the right hand side from the left hand side of this parametrization.

We start by defining a map

$$\begin{equation*} \tilde{P}_G:H^1(G_K, \mathbb{F}_l)^r \to \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G) \cup \{\bullet \} \end{equation*}$$

as follows. Let $v \coloneq (\chi _1, \ldots , \chi _r)$ be an element of $H^1(G_K, \mathbb{F}_l)^{r}$. If $\chi _{1}$ is the trivial character, we declare $\tilde{P}_G(v) = \bullet$. So we assume from now on that $\chi _{1}$ is non-trivial. Equivalently,

$$\begin{equation*} \chi _{1} \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_1). \end{equation*}$$

Hence $\chi _{1}^*(\theta _2) = \theta _2(\chi _1(-), \chi _1(-))$ is now a $2$-cocycle on $G_{K}$. If $\chi _{1}^*(\theta _2)$ is not trivial in $H^2(G_{K}, \mathbb{F}_l)$, then we declare $\tilde{P}_G(v) = \bullet$. Now assume that $\chi _{1}^*(\theta _2)$ is zero in $H^2(G_{K}, \mathbb{F}_l)$; we distinguish two cases. If $\theta _2$ is already a trivial $2$-cocycle on $G_1$, we have that $\phi (G_1, \chi _{1}^{*}(\theta _2)) = 0$ and

$$\begin{equation*} (\chi _2, \chi _1) \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_2) \end{equation*}$$

if and only if $\chi _{1}$ and $\chi _{2}$ are linearly independent. In case $\chi _{1}$ and $\chi _{2}$ are linearly dependent, we set $\tilde{P}_G(v) = \bullet$ and otherwise we proceed. If instead $\theta _2$ is a non-trivial $2$-cocycle on $G_1$, we always have that

$$\begin{equation*} (\phi (G_1, \chi _{1}^{*}(\theta _2)) + \chi _{2}, \chi _{1}) \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_2) \end{equation*}$$

by Proposition 2.6.

Now we continue in this fashion inductively. At step $i < r$ we have either already assigned $v$ to • or we have obtained an epimorphism $\psi _i \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_i)$. Then we get a $2$-cocycle $\psi _i^*(\theta _{i + 1})$, which gives a class in $H^2(G_{K}, \mathbb{F}_l)$. If this class is non-trivial in $H^2(G_{K}, \mathbb{F}_l)$, we send $v$ to •.

In case $\psi _i^*(\theta _{i + 1})$ is trivial in $H^2(G_K, \mathbb{F}_l)$, we distinguish two cases. If $\psi _i^*(\theta _{i + 1})$ is already trivial in $H^2(G_i, \mathbb{F}_l)$, we have that $\phi (G_i, \psi _i^*(\theta _{i + 1})) = 0$. Then

$$\begin{equation*} (\chi _{i + 1}, \psi _i) \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_{i + 1}) \end{equation*}$$

if and only if $\chi _{i + 1}$ is linearly independent from the characters $\chi _{j}$ satisfying

$$\begin{equation*} \phi (G_{j - 1}, \psi _{j - 1}^*(\theta _j)) = 0. \end{equation*}$$

Indeed, observe that in this case we have $G_{i + 1} = \mathbb{F}_l \times G_i$, therefore, since we have surjectivity if and only if we have surjectivity modulo the Frattini, we need to have that $\chi _{i + 1}$ is linearly independent from the characters coming from $G_i$. Thus our claim comes down to the claim that such characters are precisely spanned by the set

$$\begin{equation*} \{\chi _j\}_{j \leq i: \theta _j = 0}. \end{equation*}$$

This is justified by the following observation. Let $H$ be a finite $l$-group and let $\theta :H^2 \to \mathbb{F}_l$ be a $2$-cocycle. Then $\theta$ is trivial in $H^2(H, \mathbb{F}_l)$ if and only if the dimension of $(\mathbb{F}_l \times H,*_{\theta })$ modulo its Frattini subgroup is one larger than that of $H$ modulo its Frattini subgroup. To see the non-trivial direction observe that if one takes a character $\chi : (\mathbb{F}_l \times H,*_{\theta }) \to \mathbb{F}_l$ that does not come from $H$, then its kernel gives a splitting of the sequence.

If $\chi _{i + 1}$ is linearly dependent on these characters $\chi _j$, we send $v$ to •. Otherwise we go to step $i + 1$.

Now suppose that $\theta _{i + 1}$ is a non-trivial class of $H^2(G_i, \mathbb{F}_l)$. Then we always obtain by means of Proposition 2.6 a new epimorphism

$$\begin{equation*} (\phi (G_i, \psi _i^*(\theta _{i + 1})) + \chi _{i + 1}, \psi _i) \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G_{i + 1}) \end{equation*}$$

and we go to step $i + 1$. Continuing in this fashion we obtain either • or an element of

$$\begin{equation*} \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G), \end{equation*}$$

which is by definition $\tilde{P}_G(v)$. We put

$$\begin{equation*} P_G \coloneq \tilde{P}_G \circ \operatorname {Pow}_l([r]). \end{equation*}$$

We remark that $\phi (G_i, \theta )$ is only defined in case $G_i$ is a Galois group. Fortunately, this small abuse of notation does not present any issues. Indeed, the epimorphism $\psi _i$ realizes the implicit identification between $G_i$ and the corresponding Galois group.

We additionally remark that in the construction of the map $P_G$ we have implicitly used that $G$ is set-theoretically defined to be $\mathbb{F}_l^r$ with the identity element being $(0, \ldots , 0)$, which is a consequence of our convention that $2$-cocycles vanish on $(\mathrm{id}, \mathrm{id})$. Hence it makes sense to invoke the map $\operatorname {Pow}_l([r])$.

Proposition 2.8.

The map

$$\begin{equation*} P_G: \operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}}) \twoheadrightarrow \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G) \cup \{\bullet \} \end{equation*}$$

is a surjection, which restricts to a bijection between $\operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})(\mathrm{solv.})$ and surjective homomorphisms $\operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_{K}^{\mathrm{pro}-l}, G)$.

Proof.

This follows upon combining Proposition 2.6 and Proposition 2.7.

■

The parametrization $P_G$ allows us to read off very neatly the image of the elements $\{\sigma _{\mathfrak{q}} \}_{\mathfrak{q} \in \widetilde{\Omega }_K(l)}$ from the tuples of squarefree ideals.

Proposition 2.9.

Let $(v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}}$ be an element of $\operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})(\mathrm{solv.})$. Let $\mathfrak{q} \in \Omega _K - S_{\mathrm{clean}}(l)$. If $\mathfrak{q} \mid v_{g_0}(2)$ for a (necessarily) unique $g_0 \in G - \{\mathrm{id}\}$ then

$$\begin{equation*} P_G((v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}})(\sigma _{\mathfrak{q}}) = g_0. \end{equation*}$$

If $\mathfrak{q}$ does not divide any of the elements of the vector $(v_g(2))_{g \in G - \{\mathrm{id}\}}$, then

$$\begin{equation*} P_G((v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}})(\sigma _{\mathfrak{q}}) = \mathrm{id}. \end{equation*}$$

Proof.

For the case $\mathfrak{q} \in \widetilde{\Omega }_K(l)$ the conclusion follows immediately from the fact that the map $\Pi _l$ in the proof of Proposition 2.7 is inverse to the map $\operatorname {Pow}_l([r])$. Otherwise, we have that $\mathfrak{q} \in \Omega _K - S_{\mathrm{clean}}(l) - \widetilde{\Omega }_K(l)$ so that $\sigma _{\mathfrak{q}} = \mathrm{id}$ by definition. By construction of $\mathcal{S}_l$ it follows that such $\mathfrak{q}$ do not divide any $v_g(2)$. Hence we are always in the second case of the proposition. Therefore the statement also holds for such $\mathfrak{q}$.

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We next read off the value of the discriminant under the bijection $P_G$. For any continuous homomorphism $\psi$ of $G_{K}$ with values in some finite group, we denote by $\operatorname {Disc}(\psi )$ the relative discriminant (which is an ideal of $\mathcal{O}_K$) of the corresponding extension. For a non-zero integral ideal $\mathfrak{b}$ in $\mathcal{O}_K$ we write $\text{free}_S(\mathfrak{b})$ for the largest ideal dividing $\mathfrak{b}$ and entirely supported outside of $S$.

Proposition 2.10.

Let $(v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}}$ be an element of

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})(\mathrm{solv.}). \end{equation*}$$

Then

$$\begin{equation*} \operatorname {free}_{S_{\mathrm{clean}}(l)}(\operatorname {Disc}(P_G((v_g)_{g \in G - \{\mathrm{id}\}}))) = \prod _{g \in G - \{\mathrm{id}\}} v_g(2)^{\#G \cdot (1-\frac{1}{\#\langle g \rangle })}. \end{equation*}$$

Proof.

We show that the $\mathfrak{q}$-adic valuation matches for any prime $\mathfrak{q}$ of $\mathcal{O}_K$. This is certainly true for the places $\mathfrak{q}$ in $S_{\mathrm{clean}(l)}$, but also for the places $\mathfrak{q}$ outside $\widetilde{\Omega }_K(l)$ by Proposition 2.3. Now take a place $\mathfrak{q}$ in $\widetilde{\Omega }_K(l)$. Since $\mathfrak{q}$ is coprime to $l$ we know that

$$\begin{align*} &v_{\mathfrak{q}}(\operatorname {Disc}(P_G((v_g)_{g \in G - \{\mathrm{id}\}}))) \\ & \qquad = \frac{\#G}{\# \langle P_G((v_g)_{g \in G - \{\mathrm{id}\}})(\sigma _{\mathfrak{q}}) \rangle } \cdot \left(\# \langle P_G((v_g)_{g \in G - \{\mathrm{id}\}})(\sigma _{\mathfrak{q}}) \rangle - 1\right). \end{align*}$$

Thanks to Proposition 2.9 we deduce that $P_G((v_g)_{g \in G - \{\mathrm{id}\}})(\sigma _{\mathfrak{q}})$ is trivial in case $\mathfrak{q}$ does not divide any $v_g$ and equals $g_0$ in case $\mathfrak{q}$ divides $v_{g_0}$. This is precisely the desired conclusion.

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Our final goal for this subsection is to generalize Propositions 2.8, 2.9 and 2.10 to arbitrary finite, nilpotent groups. Recall that a finite group $G$ is nilpotent if and only if it decomposes as a direct product of its Sylow subgroups. Let $c$ be a positive integer. Let $l_1, \ldots , l_c$ be distinct prime numbers. For each $j \in [c]$ fix an admissible sequence

$$\begin{equation*} \{(G_i(l_j), \theta _i(l_j))\}_{i \in [r_j]} \end{equation*}$$

and write $G(l_j) \coloneq G_{r_j}(l_j)$ for the resulting $l_j$-group. We put

$$\begin{equation*} G \coloneq \prod _{j = 1}^c G(l_j). \end{equation*}$$

To parametrize $G$-extensions, we reduce to $l$-groups by means of Proposition 2.11.

Proposition 2.11.

We have an identification

$$\begin{equation*} \operatorname {Epi}_{\mathrm{top.gr.}}(G_K, G) = \prod _{j \in [c]} \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_K^{\mathrm{pro}-l_j}, G(l_j)) \end{equation*}$$

through the natural map.

Proof.

Let $H$ be any group and $\psi = (\psi _j)_{j \in [c]}:H \to G$ be any group homomorphism. We have to show that if $\psi _j$ is surjective for each $j \in [c]$, then $\psi$ is surjective. Observe that if each $\psi _j$ is surjective, then $\#\operatorname {Im}(\psi )$ is divisible by $\#G(l_j)$ for each $j \in [c]$. Since these values are coprime, we find out that $\#\operatorname {Im}(\psi )$ is divisible by $\#G$, which means precisely that $\psi$ is surjective.

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Thanks to Proposition 2.11 we can now bundle together the various maps $P_{G(l_j)}$ into one map

$$\begin{equation*} P_G: \prod _{j \in [c]} \operatorname {Prim}(\mathcal{S}_{l_j}^{G(l_j) - \{\mathrm{id}\}}) \twoheadrightarrow \operatorname {Epi}_{\mathrm{top.gr.}}(G_K, G) \cup \{\bullet \} \end{equation*}$$

by simply taking the product map. Put

$$\begin{equation*} S \coloneq \bigcup _{j \in [c]} S_{\mathrm{clean}}(l_j). \end{equation*}$$

For every $j \in [c]$, let $\{\chi _{j, i}\}_{i \in [t_j]}$ be a basis for the space of characters $G_K \rightarrow \mathbb{F}_{l_j}$ only ramified at $S$. Define $\mathcal{S}$ to be $\{0, 1\}^{[t_1 + \dots + t_c]} \times \mathcal{S}'$, where $\mathcal{S}'$ is the set of squarefree ideals supported outside $S$. Let

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}}) \end{equation*}$$

be the set of tuples $(v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}}$ satisfying the following properties

•

writing $\pi _i$ for the natural projection map $[t_1 + \dots + t_c] \rightarrow [t_i]$, we have that the $\pi _i(v_g(1))$ are pairwise coprime;

•

the $v_g(2)$ are pairwise coprime;

•

if $\mathfrak{p}$ divides $v_g(2)$ and $l$ is a prime dividing the order of $g$, then$$\begin{equation*} \# (\mathcal{O}_K/\mathfrak{p}) \equiv 1 \bmod l. \end{equation*}$$

For an element

$$\begin{equation*} (v_{g, j}(1), v_{g, j}(2))_{j \in [c], g \in G(l_j) - \{\mathrm{id}\}} \in \prod _{j \in [c]} \operatorname {Prim}(\mathcal{S}_{l_j}^{G(l_j) - \{\mathrm{id}\}}) \end{equation*}$$

and for $g \coloneq (g_1, \ldots , g_c) \in G - \{\mathrm{id}\}$ we define

$$\begin{equation*} v_g(1) \coloneq (v_{g_1, 1}(1), \dots , v_{g_c, c}(1)), \quad v_g(2) \coloneq \prod _{\substack{\mathfrak{p} \\ \forall j \in [c] \forall h \in G(l_j) - \{\mathrm{id}\} : \mathfrak{p} \mid v_{h, j}(2) \Leftrightarrow g_j = h}} \mathfrak{p}. \end{equation*}$$

In this way we have created a very convenient bijection between

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}}) \cong \prod _{j \in [c]} \operatorname {Prim}(\mathcal{S}_{l_j}^{G(l_j) - \{\mathrm{id}\}}). \end{equation*}$$

We need one additional piece of notation, namely we define

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}})(\mathrm{solv.}) \coloneq \prod _{j \in [c]} \operatorname {Prim}(\mathcal{S}_{l_j}^{G(l_j) - \{\mathrm{id}\}})(\mathrm{solv.}), \end{equation*}$$

which we shall often implicitly view as a subset of $\operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}})$. Proposition 2.12 generalizes Proposition 2.9.

Proposition 2.12.

Let $v \coloneq (v_{g, j}(1), v_{g, j}(2))_{j \in [c], g \in G(l_j) - \{\mathrm{id}\}}$ be an element of

$$\begin{equation*} \operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}})(\mathrm{solv.}). \end{equation*}$$

Take some $\mathfrak{q} \in \Omega _K - S$. Let $T$ be the subset of $[c]$ such that $j \in T$ if and only if there exists a (necessarily) unique $g_0^j \in G(l_j) - \{\mathrm{id}\}$ with $\mathfrak{q} \mid v_{g_0^j, j}(2)$. Then we have

$$\begin{equation*} P_G(v)(\sigma _{\mathfrak{q}}) = (g_0^j)_{j \in T} \times (\mathrm{id})_{k \in [c] - T}. \end{equation*}$$

In particular if $\mathfrak{q}$ does not divide any of the elements $v_{g, j}(2)$, i.e. $T=\emptyset$, then

$$\begin{equation*} P_G(v)(\sigma _{\mathfrak{q}}) = \mathrm{id}. \end{equation*}$$

Proof.

This follows at once from Proposition 2.8, applied to each $G(l_j)$-factor.

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Proposition 2.13 generalizes Proposition 2.10. In the new coordinates we have a rather simple formula for the discriminant.

Proposition 2.13.

Notations as above. Let $(v_{g, j}(1), v_{g, j}(2))_{j \in [c], g \in G(l_j) - \{\mathrm{id}\}}$ be an element of $\operatorname {Prim}(\mathcal{S}^{G - \{\mathrm{id}\}})(\mathrm{solv.})$. Then

$$\begin{multline*} \operatorname {free}_S(\operatorname {Disc}(P_G((v_{g, j}(1), v_{g, j}(2))_{j \in [c], g \in G(l_j) - \{\mathrm{id}\}}))) \\ = \operatorname {free} _S \left( \prod _{g \in G - \{\mathrm{id}\}} v_g(2)^{\#G \cdot (1-\frac{1}{\#\langle g \rangle })} \right). \end{multline*}$$

Proof.

This follows from Proposition 2.12 with exactly the same argument as used to establish Proposition 2.10 as a consequence of Proposition 2.9.

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3. Local conditions and conjugacy classes

3.1. Some group theory

Let $l$ be a prime number and let $G$ be a finite $l$-group given by an admissible sequence $\{(G_i, \theta _i)\}_{i \in [r]}$ with $G \coloneq G_r$. Our first goal is to study the formation of a conjugacy class in $G$, through the various groups $G_i$, with $i \in [r]$. For $g \in G$ we denote by $\operatorname {Conj}_G(g)$ its conjugacy class. For each $0 \leq i \leq r$ we write

$$\begin{equation*} \pi _i: G \to G_i \end{equation*}$$

for the natural projection map.

For now we take any finite $l$-group $H$, a $2$-cocycle $\theta$ representing a class in $H^2(H, \mathbb{F}_l)$, with $\theta (\mathrm{id}, \mathrm{id}) = 0$, and an element $h \in H$. Denote the centralizer of $h$ by $\operatorname {Cent}_H(h)$. Then lifting elements of $\operatorname {Cent}_H(h)$ to $(\mathbb{F}_l \times H, *_{\theta })$ and taking the commutator with any lift $h_0$ of $h$ induces a homomorphism

$$\begin{equation*} [-, h_0]_\theta : \frac{\operatorname {Cent}_H(h)}{\langle h \rangle } \to \mathbb{F}_l, \end{equation*}$$

which does not depend on the choice of lifts. Put

$$\begin{equation*} \widetilde{\operatorname {Cent}}_H(h, \theta ) \coloneq \operatorname {ker}([-, h_0]_{\theta }), \end{equation*}$$

which is by definition a subgroup of $\frac{\operatorname {Cent}_H(h)}{\langle h \rangle }$. Let

$$\begin{equation*} \pi _{\theta }: (\mathbb{F}_l \times H, *_{\theta }) \to H \end{equation*}$$

be the natural projection map. Proposition 3.1 describes the relationship between $\operatorname {Cent}_H(h)$ and $\widetilde{\operatorname {Cent}}_H(h, \theta )$.

Proposition 3.1.

Let $H, h, \theta$ be as above this proposition. Then

$$\begin{equation*} \left[\frac{\operatorname {Cent}_H(h)}{\langle h \rangle }:\widetilde{\operatorname {Cent}}_H(h, \theta )\right] \in \{1, l\}. \end{equation*}$$

The index equals $1$ if and only if the elements in $\pi _{\theta }^{-1}(h)$ are pairwise non-conjugate in $(\mathbb{F}_l \times H, *_{\theta })$. The index equals $l$ if and only if the elements of $\pi _{\theta }^{-1}(h)$ sit inside a unique conjugacy class in $(\mathbb{F}_l \times H, *_{\theta })$.

Proof.

Indeed, take a lift $h_0 \coloneq (a, h)$ in $\pi _{\theta }^{-1}(h)$. Observe that if we have

$$\begin{equation*} [(b, h'), h_0]*_\theta h_0 = (b, h')*_\theta h_0*_\theta (b, h')^{-1} = (a', h) \end{equation*}$$

for some $a' \in \mathbb{F}_l$, then it follows that $h' \in \operatorname {Cent}_H(h)$. From the left hand side we see that if the index is $l$, then $a'$ can take any possible value. If the index is instead equal to $1$, then $a'$ must be equal to $a$.

■

We also have a similar proposition for the exponent of an element.

Proposition 3.2.

Let $H, h, \theta$ be as above. Then either $\pi _{\theta }^{-1}(h)$ consists entirely of elements with order equal to $l \cdot \# \langle h \rangle$ or it consists entirely of elements with order equal to $\#\langle h \rangle$.

Proof.

The class $\theta$ restricted to $\langle h \rangle$ gives an element of $H^2(\langle h \rangle , \mathbb{F}_l) = \operatorname {Ext}(\langle h \rangle , \mathbb{F}_l)$. If the class is $0$, then the sequence is split and we have that all the elements of $\pi _{\theta }^{-1}(h)$ have the same order as $h$. If the class is non-zero, then the sequence has the shape

$$\begin{equation*} 0 \to \mathbb{F}_l \to \mathbb{Z}/l \cdot \#\langle h \rangle \mathbb{Z} \to \langle h \rangle \to 0, \end{equation*}$$

and hence all elements of $\pi _{\theta }^{-1}(h)$ have order $l$ times bigger than that of $h$.

■

In case an $h$ as in Proposition 3.2 satisfies the second conclusion we say that $h$ is $\theta$-stable. We now return to our previous setup. To $g \in G$ we attach the following quantity

$$\begin{equation*} j_G(g, \{(G_i, \theta _i)\}_{i \in [r]}) \coloneq \#\left\{i \in [r] : \frac{\operatorname {Cent}_{G_{i - 1}}(\pi _{i - 1}(g))}{\langle \pi _{i - 1}(g) \rangle } \neq \widetilde{\operatorname {Cent}}_{G_{i - 1}}(\pi _{i - 1}(g), \theta _i)\right\}. \end{equation*}$$

This quantity turns out to be the exponent of $l$ in the size of the conjugacy class $\operatorname {Conj}_G(g)$. We call the $i$’s counted by $j_G(g, \{(G_i, \theta _i)\}_{i \in [r]})$ the breaks for $g$ with respect to $\{(G_i, \theta _i)\}_{i \in [r]}$.

Proposition 3.3.

We have

$$\begin{equation*} \#\operatorname {Conj}_G(g) = l^{j_G(g, \{(G_i, \theta _i)\}_{i \in [r]})}. \end{equation*}$$

Proof.

We have a filtration of subgroups

$$\begin{equation*} G = \operatorname {Cent}_G^0(g) \supseteq \operatorname {Cent}_G^1(g) \supseteq \dots \supseteq \operatorname {Cent}_G^r(g) = \operatorname {Cent}_G(g), \end{equation*}$$

where we define

$$\begin{equation*} \operatorname {Cent}_G^i(g) \coloneq \pi _i^{-1}(\operatorname {Cent}_{G_i}(\pi _i(g))) \end{equation*}$$

for every integer $0 \leq i \leq r$. We have that

$$\begin{equation*} \#\operatorname {Conj}_G(g) = \frac{\#G}{\#\operatorname {Cent}_G(g)} = \prod _{0 \leq i \leq r - 1} \frac{\#\operatorname {Cent}_G^i(g)}{\#\operatorname {Cent}_G^{i + 1}(g)}. \end{equation*}$$

Observe that

$$\begin{equation*} [\operatorname {Cent}_G^{i}(g):\operatorname {Cent}_G^{i + 1}(g)] = \left[\frac{\operatorname {Cent}_{G_i}(\pi _i(g))}{\langle \pi _i(g) \rangle }:\widetilde{\operatorname {Cent}}_{G_i}(\pi _i(g), \theta _{i + 1})\right]. \end{equation*}$$

Therefore the desired conclusion follows at once from Proposition 3.1.

■

Proposition 3.4 provides the crucial link between group theoretic data and the local conditions imposed, through Proposition 2.6, on tuples in $\operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})$. We invoke the notation of Section 2. Let $L/K$ be a finite Galois extension inside $K^{\mathrm{pro}-l}/K$. Let now $G\coloneq \operatorname {Gal}(L/K)$ and let $\theta$ be a $2$-cocycle representing a class in $H^2(\operatorname {Gal}(L/K), \mathbb{F}_l)$, with $\theta (\mathrm{id}, \mathrm{id}) = 0$. Let $\mathfrak{q} \in \Omega _K$ be a finite place coprime to $l$. Recall that

$$\begin{equation*} \frac{G_{K_{\mathfrak{q}}}}{I_{\mathfrak{q}}} \end{equation*}$$

is a pro-cyclic group, equipped with a canonical generator $\operatorname {Frob}_{\mathfrak{q}}$. We will fix once and for all the $\operatorname {proj}(G_K \to \mathcal{G}_K^{\mathrm{pro}-l}) \circ i_{\mathfrak{q}}^{*}$-image of a lift to $G_{K_{\mathfrak{q}}}$ of such an element. In this way we obtain an element in $\mathcal{G}_K^{\mathrm{pro}-l}$ that we will denote also by $\operatorname {Frob}_{\mathfrak{q}}$: this slight abuse of notation will cause no confusion. As such we have naturally an element

$$\begin{equation*} \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\operatorname {Frob}_{\mathfrak{q}}) \in \frac{N_G(\langle \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}}) \rangle )}{ \langle \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})\rangle }. \end{equation*}$$

Here $N_G(-)$ denotes the normalizer of a subgroup in $G$. For any non-trivial finite group $G$, we denote by $l_G$ the smallest prime divisor of $\#G$ and by $I(G)$ the subset of $g \in G - \{\mathrm{id}\}$ such that $g^{l_G} = \mathrm{id}$.

Proposition 3.4.

Let $G = \operatorname {Gal}(L/K)$ be a finite $l$-group and let $\mathfrak{q} \in \Omega _K$ be a finite place coprime to $l$. Assume that $\operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})$ is an element of $I(G)$, which we shall also call $\sigma _{\mathfrak{q}}$. Then

$$\begin{equation*} \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\operatorname {Frob}_{\mathfrak{q}}) \in \frac{\operatorname {Cent}_G(\sigma _{\mathfrak{q}})}{\langle \sigma _{\mathfrak{q}} \rangle }. \end{equation*}$$

Moreover, if $\operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})$ is $\theta$-stable, then

$$\begin{equation*} \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\operatorname {Frob}_{\mathfrak{q}}) \in \frac{\widetilde{\operatorname {Cent}}_G(\sigma _{\mathfrak{q}}, \theta )}{\langle \sigma _{\mathfrak{q}} \rangle } \end{equation*}$$

if and only if $\theta$ is trivial in $H^2(G_{K_{\mathfrak{q}}}, \mathbb{F}_l)$.

Proof.

Let $G$ be any finite non-trivial group. Then we claim that $g \in I(G)$ implies $N_G(\langle g \rangle ) = \operatorname {Cent}_G(g)$. Indeed, conjugation induces a homomorphism

$$\begin{equation*} N_G(\langle g \rangle ) \to \operatorname {Aut}_{\mathrm{gr.}}(\langle g \rangle ) \simeq _{\mathrm{gr.}} \mathbb{F}_{l_G}^{*}. \end{equation*}$$

Since the latter group has size $l_G - 1$, and $l_G$ is the smallest prime divisor of $\#G$, we have that $\#G$ is coprime to $l_G-1$. Therefore the above homomorphism is actually trivial. This means exactly that $N_G(\langle g \rangle ) = \operatorname {Cent}_G(g)$ as claimed. Hence we have already shown the first part of this proposition, namely that

$$\begin{equation*} \operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\operatorname {Frob}_{\mathfrak{q}}) \in \frac{\operatorname {Cent}_G(\sigma _{\mathfrak{q}})}{\langle \sigma _{\mathfrak{q}} \rangle }. \end{equation*}$$

Assume now that $\operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})$ is also $\theta$-stable. Observe that since $\sigma _{\mathfrak{q}}$ lands in $I(G)$, we in particular conclude that $\mathfrak{q}$ ramifies in $L/K$. It follows from Proposition 2.3 that

$$\begin{equation*} q \coloneq \#\left(\mathcal{O}_K/\mathfrak{q} \right) \ \equiv 1 \ \mathrm{mod} \ l. \end{equation*}$$

Also observe that the maximal pro-$l$ quotient of $G_{K_{\mathfrak{q}}}$ is isomorphic to

$$\begin{equation*} \mathbb{Z}_l \rtimes q^{\mathbb{Z}_l}, \end{equation*}$$

where $q$ acts by multiplication by $q$ on $\mathbb{Z}_l$. Here $I_{K_{\mathfrak{q}}}$ is sent to $\mathbb{Z}_l \rtimes \{1\}$, while a lift of $\operatorname {Frob}_{\mathfrak{q}}$ is sent to $\{0\} \rtimes \{q\}$. Since $\operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})$ is $\theta$-stable and lands in $I(G)$, we have that the lifting problem imposed by $\theta$ factors through

$$\begin{equation*} l \cdot \mathbb{Z}_l \rtimes \{1\}. \end{equation*}$$

Since $q$ is $1$ modulo $l$, the resulting quotient is simply

$$\begin{equation*} \mathbb{F}_l \times q^{\mathbb{Z}_l}. \end{equation*}$$

Recalling once more that $\operatorname {proj}(\mathcal{G}_K^{\mathrm{pro}-l} \to G)(\sigma _{\mathfrak{q}})$ is $\theta$-stable, we see that the lifting problem is solvable if and only if the restriction of $\theta$ to

$$\begin{equation*} \operatorname {proj}(G_K \to G) \circ i_{\mathfrak{q}^*}(G_{K_{\mathfrak{q}}}) \end{equation*}$$

is in

$$\begin{equation*} \operatorname {Ext}(\operatorname {proj}(G_K \to G) \circ i_{\mathfrak{q}^*}(G_{K_{\mathfrak{q}}}), \mathbb{F}_l). \end{equation*}$$

This is precisely equivalent to asking

$$\begin{equation*} \operatorname {Frob}_{\mathfrak{q}} \in \frac{\widetilde{\operatorname {Cent}}_G(\sigma _{\mathfrak{q}}, \theta )}{\langle \sigma _{\mathfrak{q}} \rangle } \end{equation*}$$

as was to be shown.

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The final lemma of this subsection provides a simple way to compute the Malle constant $b_{\mathrm{M}}(G, K)$ for nilpotent $G$.

Lemma 3.5.

Let $G$ be a non-trivial, finite group and let $K$ be a number field. Then we have

$$\begin{equation*} b_{\mathrm{M}}(G, K) = \frac{\#\{C \in \operatorname {Conj}(G) : C \subseteq I(G)\}}{[K(\zeta _{l_G}) : K]}. \end{equation*}$$

Proof.

Recall that there is a natural action of $\operatorname {Gal}(\overline{K}/K)$ on

$$\begin{equation*} X \coloneq \{C \in \operatorname {Conj}(G) : C \subseteq I(G)\}, \end{equation*}$$

which sends a conjugacy class $C$ to $C^{\chi (\sigma )}$ with $\chi : \operatorname {Gal}(\overline{K}/K) \rightarrow \hat{\mathbb{Z}}^\ast$ the cyclotomic character. By definition $b_{\mathrm{M}}(G, K)$ equals the number of orbits of this group action. But our action clearly factors through $\operatorname {Gal}(K(\zeta _{l_G})/K)$. We claim that the induced action of $\operatorname {Gal}(K(\zeta _{l_G})/K)$ on $X$ is free, which implies the lemma.

So suppose that there exists $\sigma \in \operatorname {Gal}(K(\zeta _{l_G})/K)$ such that

$$\begin{equation*} C^{\chi (\sigma )} = C. \end{equation*}$$

This implies that there exists a non-trivial $g \in C$ such that $g$ and $g^{\chi (\sigma )}$ are conjugate, say

$$\begin{equation*} g = h^{-1} g^{\chi (\sigma )} h, \end{equation*}$$

and hence $h \in N_G(\langle g \rangle ) = \operatorname {Cent}_G(g)$ by the argument given at the start of Proposition 3.4. We conclude that $g = g^{\chi (\sigma )}$, which forces $\sigma$ to be the identity as desired.

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3.2. Interpretation of Malle’s constant

The goal of this subsection is to give a heuristic supporting Malle’s conjecture in the nilpotent case. Our heuristic is based on a combination of the parametrization given in Proposition 2.8 and the examination of the local conditions carried out in Subsection 3.1. To simplify the notation, we shall limit ourselves to the case where $G$ is an $l$-group. We leave it to the reader to generalize the material below to arbitrary nilpotent groups $G$.

Ignoring the finitely many bad places in $S_{\mathrm{clean}}(l)$, it follows from Proposition 2.8 and Proposition 2.10 that

$$\begin{equation*} \#\{\psi \in \operatorname {Epi}_{\mathrm{top.gr.}}(\mathcal{G}_K^{\mathrm{pro}-l}, G): |N_{K/\mathbb{Q}} \operatorname {Disc}(\psi )| \leq X\} \end{equation*}$$

should have order of magnitude

$$\begin{align*} &\#\{(v_g(1), v_g(2))_{g \in G - \{\mathrm{id}\}} \\ & \qquad \in \operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})(\mathrm{solv.}): \prod _{g \in G - \{\mathrm{id}\}}(|N_{K/\mathbb{Q}}v_g(2)|)^{\#G \cdot (1-\frac{1}{\#\langle g \rangle })} \leq X\}. \end{align*}$$

We now focus on the variables $(v_g(1), v_g(2))$ with $g \in I(G)$. Upon combining Proposition 3.4 and Proposition 2.9, we see that the primes $\mathfrak{q}$ dividing $v_g(2)$ impose a local condition only at the breaks for $g$ in the admissible sequence $\{(G_i, \theta _i)\}_{i \in [r]}$. The local conditions at the points that are not breaks are automatically satisfied in virtue of Proposition 3.4. Now pretend that the values

$$\begin{equation*} \operatorname {Frob}_{\mathfrak{q}} \in \frac{\operatorname {Cent}_{G_{i - 1}}(\sigma _{\mathfrak{q}})}{\langle \sigma _{\mathfrak{q}} \rangle } \end{equation*}$$

are jointly equidistributed at every break point $i$. Then, in virtue of Proposition 3.4, we get the following sum

$$\begin{equation*} \approx \sum _{\prod _{g \in G - \{\mathrm{id}\}}(|N_{K/\mathbb{Q}}v_g(2)|)^{\#G \cdot (1-\frac{1}{\#\langle g \rangle })} \leq X} \left(\prod _{g \in I(G)} \frac{1}{l^{\omega (v_g(2)) j_G(g, \{(G_i, \theta _i)\}_{i \in [r]})}} \right), \end{equation*}$$

where the sum runs over all points $(v_g(1), v_g(2))$ in $\operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})$. Thanks to Proposition 3.3 the latter expression equals

$$\begin{align} \sum _{\prod _{g \in G - \{\mathrm{id}\}}(|N_{K/\mathbb{Q}}v_g(2)|)^{\#G \cdot (1-\frac{1}{\#\langle g \rangle })} \leq X} \left(\prod _{g \in I(G)} \frac{1}{\#\operatorname {Conj}_G(g)^{\omega (v_g(2))}} \right), {\tag{3.1}} \end{align}$$

where the sum still ranges over all points $(v_g(1), v_g(2))$ in $\operatorname {Prim}(\mathcal{S}_l^{G - \{\mathrm{id}\}})$. Standard analytic techniques, see Theorem 4.1, show that the sum in equation 3.1 is asymptotic to

$$\begin{equation*} c(G, K) \cdot X^{a(G)} \cdot \log (X)^{\beta (G, K) - 1}, \quad \beta (G, K) \coloneq \sum _{g \in I(G)} \frac{1}{\# \operatorname {Conj}_G(g) \cdot [K(\zeta _l) : K]}, \end{equation*}$$

where $\beta (G, K)$ is the Malle constant by Lemma 3.5. We remark that to turn this simple heuristic into an argument one also has to pay careful attention to the local conditions at the primes dividing variables outside of $I(G)$ and to the local conditions at the primes in $S_{\mathrm{clean}}(l)$. These will affect the constant $c(G, K)$ in the asymptotic. We finish this section by explaining the interplay between this heuristic and the proofs of our main theorems.

During the proof of Theorem 5.1 we simply ignore the local conditions, at the cost of losing track of the conjugation in $G$: for this reason we get $i(G, K)-1$ instead of $b_{\mathrm{M}}(G, K) - 1$.

Correspondingly for those $G$ for which $i(G, K) - 1$ and $b_{\mathrm{M}}(G, K) - 1$ coincide we have that all the elements of $I(G)$ are central, and consistently with Proposition 3.4 we have no local conditions coming from the variables in $I(G)$. In this case we are able to prove an asymptotic in Theorem 5.7.

Finally in the proof of Theorem 5.3, thanks to the fact that the elements of $I(G)$ are pairwise commuting, we have to control the behavior of $\operatorname {Frob}_{\mathfrak{q}}$ in the quotient $\frac{G}{I(G) \cup \{\mathrm{id}\}}$ for each $\mathfrak{q}$ dividing a variable in $I(G)$. This is very convenient, since the corresponding field is constructed out of the variables outside of $I(G)$, and those have a very large weight in the formula for the discriminant given in Proposition 2.10. As such, they can almost be treated as fixed, and the required joint equidistribution of Frobenius elements is provable by appealing to the Chebotarev density theorem. Hence in this case we can partially turn the above heuristic into a rigorous argument: since we control only the local conditions at the places dividing variables in $I(G)$, we naturally end up with an upper bound of the correct order of magnitude.

4. Analytic considerations

In this section we provide the analytic tools used to prove our main theorems. The material in this section is a generalization of the material in Montgomery–Vaughan 41, Section 7.4 and is an application of the Selberg–Delange method. Let $K$ be a number field, let $L$ be an abelian extension of $K$ and let $S \subseteq \operatorname {Gal}(L/K)$. Write $\mathcal{I}_K$ for the group of non-zero fractional ideals of $K$. For a squarefree ideal $I$ of $\mathcal{O}_K$ we define $\omega (I)$ for the number of prime divisors $\mathfrak{p}$ of $I$ and $\omega _S(I)$ for the number of prime divisors $\mathfrak{p}$ of $I$ that are unramified in $L$ and satisfy $\operatorname {Frob}_{\mathfrak{p}} \in S$. Given a complex number $z$ and a collection of prime ideals $\mathcal{P}$, we are interested in the sum

$$\begin{equation*} A_z(x) \coloneq \sum _{\substack{N_{K/\mathbb{Q}}(I) \leq x \\ \mathfrak{p} \mid I \Rightarrow \operatorname {Frob}_\mathfrak{p} \in S \text{ and } \mathfrak{p} \not \in \mathcal{P}}} \mu ^2(I) z^{\omega (I)} = \sum _{\substack{N_{K/\mathbb{Q}}(I) \leq x \\ \mathfrak{p} \mid I \Rightarrow \operatorname {Frob}_\mathfrak{p} \in S \text{ and } \mathfrak{p} \not \in \mathcal{P}}} \mu ^2(I) z^{\omega _S(I)}, \end{equation*}$$

where $\mu$ is the Möbius function of $\mathcal{O}_K$. Write

$$\begin{align} F(s, z) = \sum _{\substack{I \in \mathcal{I}_K \\ \mathfrak{p} \mid I \Rightarrow \operatorname {Frob}_\mathfrak{p} \in S \text{ and } \mathfrak{p} \not \in \mathcal{P}}} \frac{\mu ^2(I) z^{\omega (I)}}{N_{K/\mathbb{Q}}(I)^s} = \sum _{n = 1}^\infty \frac{a_z(n)}{n^s} {\tag{4.1}} \end{align}$$

for its Dirichlet series with coefficients $a_z(n)$. Then we have for $s = \sigma + it$

$$\begin{align*} F(s, z) &= \prod _{\substack{\mathfrak{p} \not \in \mathcal{P} \\ \operatorname {Frob}_{\mathfrak{p}} \in S}} \left(1 + \frac{z}{N_{K/\mathbb{Q}}(\mathfrak{p})^{s}}\right) \\ &= \prod _{\mathfrak{p} \not \in \mathcal{P}} \left(1 + \frac{z \sum _{\sigma \in S} \frac{1}{\# \operatorname {Gal}(L/K)}\sum _{\chi \in \operatorname {Gal}(L/K)^\vee } \chi (\operatorname {Frob}_\mathfrak{p}) \overline{\chi (\sigma )}}{N_{K/\mathbb{Q}}(\mathfrak{p})^{s}}\right) \text{ for } \sigma > 1, \end{align*}$$

where $\operatorname {Gal}(L/K)^\vee$ is by definition $\operatorname {Hom}(\operatorname {Gal}(L/K), \mathbb{C}^\ast )$. We assume that the Euler product

$$\begin{align} \prod _{\mathfrak{p} \in \mathcal{P}} \left(1 + \frac{1}{N_{K/\mathbb{Q}}(\mathfrak{p})^s}\right) {\tag{4.2}} \end{align}$$

converges absolutely in the region $\sigma > 1 - \delta$ for some constant $\delta > 0$. Then we approximate the Dirichlet series $F(s, z)$ with

$$\begin{equation*} G(s, z) \coloneq \prod _{\sigma \in S} \prod _{\chi \in \operatorname {Gal}(L/K)^\vee } L(s, \chi )^{\frac{z \overline{\chi (\sigma )}}{\# \operatorname {Gal}(L/K)}}, \end{equation*}$$

where

$$\begin{equation*} L(s, \chi ) = \prod _{\mathfrak{p}} \left(1 - \frac{\chi (\operatorname {Frob}_{\mathfrak{p}})}{N_{K/\mathbb{Q}}(\mathfrak{p})^s}\right)^{-1} \text{ for } \sigma > 1. \end{equation*}$$

We recall that $L(s, \chi )^z$ is by definition $e^{z \log L(s, \chi )}$. Note that $\log L(s, \chi )$ exists since the region $\sigma > 1$ is simply connected and $L(s, \chi )$ does not vanish in this region. We choose our determination of the logarithm in such a way that it agrees with the real logarithm for real $s$. It follows from equation 4.2 that we have the fundamental relation

$$\begin{equation*} F(s, z) = G(s, z) H(s, z), \end{equation*}$$

where $H(s, z)$ is defined by an absolutely convergent Euler product in the region $\sigma > 1 - \delta$ for some $\delta > 0$. In particular, if $|z| \leq R$, then there exists some constant $\delta (R) > 0$ such that $H(s, z)$ is a bounded non-zero holomorphic function on $\sigma > 1 - \delta (R)$.

Theorem 4.1.

Let $K$, $L$, $S$ and $\mathcal{P}$ as above. Then we have for all positive real numbers $R$ and all $|z| \leq R$

$$\begin{equation*} A_z(x) = C x(\log x)^{\frac{z \# S}{\# \operatorname {Gal}(L/K)} - 1} + O_{R, K, L, S, \mathcal{P}}\left(x (\log x)^{\frac{\operatorname {Re}(z) \# S}{\# \operatorname {Gal}(L/K)} - 2}\right), \end{equation*}$$

where $C > 0$ is a real constant depending only on $z$, $K$, $L$, $S$ and $\mathcal{P}$.

Proof.

Since the proof is similar to Montgomery–Vaughan 41, Theorem 7.17, we shall only sketch the necessary modifications. Set $a = 1 + 1/\log x$. An effective version of Perron’s formula, see 41, Corollary 5.3, shows that

$$\begin{align*} & A_z(x) - \frac{1}{2\pi i} \int _{a - iT}^{a + iT} F(s, z) \frac{x^s}{s} ds \\ & \qquad \ll \sum _{\frac{1}{2}x < n < 2x} |a_z(n)| \min \left(1, \frac{x}{T|x - n|}\right) + \frac{x^a}{T} \sum _{n = 1}^\infty |a_z(n)| n^{-a}, \end{align*}$$

where we recall that $a_z(n)$ is defined by equation 4.1 and $T$ is a parameter at our disposal. We choose $T = \exp (\sqrt {\log x})$ and estimate the error terms as in 41. To do so, we need to have a good estimate for the sum

$$\begin{equation*} \sum _{|n - x| \leq \frac{x}{(\log x)^{(2R)^{[K : \mathbb{Q}]} + R + 1}}} |a_z(n)| \leq \sum _{|n - x| \leq \frac{x}{(\log x)^{(2R)^{[K : \mathbb{Q}]} + R + 1}}} (2R)^{[K : \mathbb{Q}] \omega (n)}, \end{equation*}$$

where we assume without loss of generality that $R$ is an integer greater than $1$. The latter sum is estimated in 41, Theorem 7.17 with Dirichlet’s hyperbola method.

We next move the path of integration. Note that $F(s, z)$ has a branch point at $s = 1$ if $z$ is not an integer. For this reason, we move the path of integration in such a way to avoid this branch point. Put $b = 1 - c/\log T$, where $c$ is a small positive constant. Let $\mathcal{C}_1$ be the polygonal path with vertices $a - iT, b - iT, b - i/\log x$, let $\mathcal{C}_2$ be the line segment from $b - i/\log x$ to $1 - i/\log x$, followed by a semicircle $\{1 + e^{i \theta }/\log x: -\pi /2 \leq \theta \leq \pi /2\}$, and a line segment from $1 + i/\log x$ to $b + i/\log x$, and finally let $\mathcal{C}_3$ be the polygonal path with vertices $b + i/\log x, b + iT, a + iT$.

Let $\mathcal{D}$ be the region enclosed by $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ and the line segment from $a - iT$ to $a + iT$. If $c$ is sufficiently small, then $L(s, \chi )$ has no zeroes in the region $\mathcal{D}$ by 26, Theorem 5.10. Clearly, $H(s, z)$ also has no zeroes in $\mathcal{D}$ provided that $c$ is sufficiently small. Since the union of $\mathcal{D}$ with the region $\text{Re}(s) > 1$ is still simply connected, $\log L(s, \chi )$ and $\log H(s, z)$ are also well-defined in this region.

The main term comes from the integral over $\mathcal{C}_2$, and is extracted in exactly the same way as in the proof of 41, Theorem 7.17. Finally, Montgomery–Vaughan estimate the integrals on the paths $\mathcal{C}_1$ and $\mathcal{C}_3$ by appealing to bounds for $\zeta (s)$, see their 41, Theorem 6.7. Hence we need to supply similar bounds for $\zeta _K(s)$ and $L(s, \chi )$. These can be derived by following the proof of 41, Theorem 6.7, where we use 26, Proposition 5.7, (2) as a replacement for 41, Lemma 6.4.

■

Write $\ast$ for the Dirichlet convolution on $\mathcal{I}_K$. In Section 5 we will combine Theorem 4.1 with the following general lemma on convolutions.

Lemma 4.2.

Let $f, g: \mathcal{I}_K \rightarrow \mathbb{R}$ be functions such that

$$\begin{gather*} \sum _{N_{K/\mathbb{Q}}(I) \leq x} f(I) = C_1x (\log x)^A + O(x (\log x)^{A - \delta }), \\ \sum _{N_{K/\mathbb{Q}}(I) \leq x} g(I) = C_2x (\log x)^B + O(x (\log x)^{B - \delta }) \end{gather*}$$

for some real numbers $A, B > -1$, $C_1, C_2 > 0$ and $0 < \delta < 1$. Then there is $C_3 > 0$ such that

$$\begin{equation*} \sum _{N_{K/\mathbb{Q}}(I) \leq x} (f \ast g)(I) = C_3 x (\log x)^{A + B + 1} + O(x (\log x)^{A + B + 1 - \delta }). \end{equation*}$$

Proof.

It follows from Dirichlet’s hyperbola method that

$$\begin{equation*} \sum _{N_{K/\mathbb{Q}}(I) \leq x} (f \ast g)(I) = \sum _{N_{K/\mathbb{Q}}(IJ) \leq x} f(I) g(J) \end{equation*}$$

equals

$$\begin{multline} \sum _{N_{K/\mathbb{Q}}(I) \leq \sqrt {x}} \sum _{N_{K/\mathbb{Q}}(J) \leq \frac{x}{N_{K/\mathbb{Q}}(I)}} f(I) g(J) + \sum _{N_{K/\mathbb{Q}}(J) \leq \sqrt {x}} \sum _{N_{K/\mathbb{Q}}(I) \leq \frac{x}{N_{K/\mathbb{Q}}(J)}} f(I) g(J) \\ - \sum _{N_{K/\mathbb{Q}}(I) \leq \sqrt {x}} \sum _{N_{K/\mathbb{Q}}(J) \leq \sqrt {x}} f(I) g(J). {\tag{4.3}} \end{multline}$$

The latter sum is at most $O(x (\log x)^{A + B})$. Since the first two sums in equation 4.3 play a symmetric role, we shall only treat the first sum. The first sum equals

$$\begin{align} \sum _{N_{K/\mathbb{Q}}(I) \leq \sqrt {x}} \frac{C_2xf(I)}{N_{K/\mathbb{Q}}(I)} \left(\log \frac{x}{N_{K/\mathbb{Q}}(I)}\right)^B = C_2x \sum _{N_{K/\mathbb{Q}}(I) \leq \sqrt {x}} \frac{f(I)}{N_{K/\mathbb{Q}}(I)} \left(\log x - \log N_{K/\mathbb{Q}}(I)\right)^B {\tag{4.4}} \end{align}$$

up to an error of size bounded by