On the self-duality of rings of integers in tame and abelian extensions

TL;DR

This paper explores the self-duality properties of rings of integers in tame, abelian Galois extensions of number fields, focusing on the relationships among various ideal classes and their duals.

Contribution

It establishes new connections between the classes of the ring of integers, inverse different, and square root of inverse different in abelian extensions, highlighting their self-duality properties.

Findings

A_{L/K}^2 = \, ext{inverse different} = \, ext{dual of} \, ext{ring of integers}

A_{L/K} is self-dual with respect to the trace pairing

Relationships among ideal classes are characterized in abelian Galois extensions

Abstract

Let $L/K$ be a tame and Galois extension of number fields with group $G$. It is well-known that any ambiguous ideal in $L$ is locally free over $\mathcal{O}_KG$ (of rank one), and so it defines a class in the locally free class group of $\mathcal{O}_KG$, where $\mathcal{O}_K$ denotes the ring of integers of $K$. In this paper, we shall study the relationship among the classes arising from the ring of integers $\mathcal{O}_L$ of $L$, the inverse different $\mathfrak{D}_{L/K}^{-1}$ of $L/K$, and the square root of the inverse different $A_{L/K}$ of $L/K$ (if it exists), in the case that $G$ is abelian. They are naturally related because $A_{L/K}^2 = \mathfrak{D}_{L/K}^{-1} = \mathcal{O}_L^*$, and $A_{L/K}$ is special because $A_{L/K} = A_{L/K}^*$, where $*$ denotes dual with respect to the trace of $L/K$.

Equations

$$[{\mathcal{O}}_L{]}_{\mathbb{Z}G}=[A_{L/K}{]}_{\mathbb{Z}G}$$

$$[{\mathcal{O}}_L]=[A_{L/K}]$$

$$R({\mathcal{O}}_{K}G)$$

$$R_{\mathrm{sd}}({\mathcal{O}}_{K}G)$$

$${\mathcal{A}}^t({\mathcal{O}}_{K}G)=\{[A_{L/K}]:\text{tame L/K with }\mathrm{Gal}(L/K)\simeq G\}.$$

$${\Psi }_k\in \text{mbox}Aut(\mathrm{Cl}({\mathcal{O}}_{K}G));{\Psi }_k([X])=[X_k],$$

$$s*x=\phi_{k}^{-1}(s)\cdot x\mbox{ for $s\in G$ and $x\in X$},$$

$${\Psi }_{-1}([{\mathcal{O}}_L])$$

$${\Psi }_{-1}([A_{L/K}])$$

$${\Psi }_{-1}([{\mathcal{O}}_L])=[{\mathfrak{D}}_{L/K}^{-1}{]}^{-1}\text{mbox}and{\Psi }_{-1}([A_{L/K}])=[A_{L/K}{]}^{-1},$$

$$[A_{L/K}]=[{\mathcal{O}}_L]{\Psi }_2([{\mathcal{O}}_L]),$$

$${\Xi }_k:R({\mathcal{O}}_{K}G)⟶R({\mathcal{O}}_{K}G);$$

$${\Xi }_k^{\prime }:R({\mathcal{O}}_{K}G)⟶R({\mathcal{O}}_{K}G);$$

$$R_{\mathrm{sd}}({\mathcal{O}}_{K}G)=\ker ({\Xi }_{-1})\text{mbox}and{\mathcal{A}}^t({\mathcal{O}}_{K}G)=\mathrm{Im}({\Xi }_2^{\prime }),$$

$$R({\mathcal{O}}_{K}G)\supset R_{\mathrm{sd}}({\mathcal{O}}_{K}G)\supset {\mathcal{A}}^t({\mathcal{O}}_{K}G)$$

$$\prod _{j=0}^{n_G(2)-1}{\Psi }_{2^j}(c)=\prod _{j=0}^{n_G(2)-1}{\Psi }_{2^j}(d{)}^{-1}{\Psi }_{2^{j+1}}(d)={\Psi }_{2^0}(d{)}^{-1}{\Psi }_{2^{n_G(2)}}(d)=1.$$

$${\Psi }_{\mathbb{Z}}=\{{\Psi }_k:k\in \mathbb{Z}\text{mbox}coprimeto|G|\},$$

$${\mathrm{Cl}}^0({\mathcal{O}}_{K}G)=\ker (\mathrm{Cl}({\mathcal{O}}_{K}G)⟶\mathrm{Cl}({\mathcal{O}}_K)),$$

$$({\mathrm{Cl}}^0({\mathcal{O}}_K{C}_p{)}^{{\Psi }_{\mathbb{Z}}}{)}^{(p-1)/2}\subset R({\mathcal{O}}_K{C}_p{)}^{{\Psi }_{\mathbb{Z}}}.$$

$$T({\mathcal{O}}_K{C}_p)\subset {\mathrm{Cl}}^0({\mathcal{O}}_K{C}_p{)}^{{\Psi }_{\mathbb{Z}}}.$$

$$T(\mathcal{O}_{K}C_{p})\simeq\begin{cases}C_{(p+1)/2}\mbox{ or }C_{p+1}&\mbox{if $\left(\frac{D}{p}\right)=-1$},\\ C_{(p-1)/2}\mbox{ or }C_{p-1}&\mbox{if $\left(\frac{D}{p}\right)=1$},\end{cases}$$

$$\begin{cases}R(\mathcal{O}_{K}C_{p})\supsetneq R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})&\mbox{if $\left(\frac{D}{p}\right)=-1$ and $p\neq 3$},\\ R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\supsetneq\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})&\mbox{if $\left(\frac{D}{p}\right)=-1$ and $p\equiv-1\pmod{8}$},\end{cases}$$

$$\left(\frac{2}{p}\right)=(-1{)}^{\frac{p^2-1}{8}},\left(\frac{-1}{p}\right)=(-1{)}^{\frac{p-1}{2}},\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1{)}^{\frac{p-1}{2}\frac{q-1}{2}},$$

$$\left(\frac{D}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{|D|}{p}\right)=-1,\text{mbox}whence{R}_{\mathrm{sd}}({\mathcal{O}}_K{C}_p)⊋{\mathcal{A}}^t({\mathcal{O}}_K{C}_p)$$

$$R({\mathcal{O}}_{K}G)/R_{\mathrm{sd}}({\mathcal{O}}_{K}G)\simeq \mathrm{Im}({\Xi }_{-1}).$$

$$d_{n}(K)=\begin{cases}[K(\zeta_{n}):K]/2&\mbox{when $(\zeta_{n}\mapsto\zeta_{n}^{-1})\in\mathrm{Gal}(K(\zeta_{n})/K)$},\\ [K(\zeta_{n}):K]&\mbox{when $(\zeta_{n}\mapsto\zeta_{n}^{-1})\notin\mathrm{Gal}(K(\zeta_{n})/K)$}.\end{cases}$$

$$T_{\mathrm{cyc}}^{\ast }({\mathcal{O}}_{K}G)=\prod _{\begin{array}{c}H\le G\\ H\text{mbox}cyclic\end{array}}{T}_H^{\ast }({\mathcal{O}}_{K}G).$$

$$\delta(G)=\begin{cases}2&\mbox{when $|G|$ is a power of two},\\ 1&\mbox{otherwise},\end{cases}$$

$$R({\mathcal{O}}_{K}G)=R_{\mathrm{sd}}({\mathcal{O}}_{K}G)\text{mbox}ifandonlyif\mathrm{Cl}({\mathcal{O}}_K)=1\text{mbox}and{T}_{\mathrm{cyc}}^{\ast }({\mathcal{O}}_{K}G)=1,$$

$$T({\mathcal{O}}_K{C}_p{)}^{d_p(K)}\subset R({\mathcal{O}}_K{C}_p{)}^{{\Psi }_{\mathbb{Z}}},\text{mbox}where{d}_p(K)\text{mbox}divides(p-1)/2,$$

$$T({\mathcal{O}}_K{C}_p{)}^{p-1}\subset T({\mathcal{O}}_K{C}_p{)}^{2d_p(K)}\subset \mathrm{Im}({\Xi }_{-1})\subset T({\mathcal{O}}_K{C}_p)$$

$$T(\mathcal{O}_{K}C_{p})\simeq\begin{cases}C_{(p+1)/4}&\mbox{if $K=\mathbb{Q}(\sqrt{-1})$ and $p\equiv 3\pmod{8}$},\\ C_{(p+1)/6}&\mbox{if $K=\mathbb{Q}(\sqrt{-3})$ and $p\equiv 5\pmod{12}$}.\end{cases}$$

Full text

1. 1 Introduction

2. 1.1 Basic set-up and notation

3. 1.2 Statements of the main theorems

4. 2 Comparison between $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$ and $\mathcal{A}^{t}(\mathcal{O}_{K}G)$

5. 2.1 Proof of Theorem 1.9 (b)

6. 2.2 Proof of Theorem 1.10

7. 3 Comparison between $R(\mathcal{O}_{K}G)$ and $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$

8. 3.1 Locally free class group

9. 3.2 McCulloh’s characterization

10. 3.3 Generalized Swan subgroups

11. 3.4 Preliminaries

12. 3.5 Proof of Theorem 1.12

13. 3.6 Proof of Theorem 1.13

14. 4 Acknowledgments

1. Introduction

Let $L/K$ be a Galois extension of number fields with group $G$. There are two ambiguous ideals in $L$, namely ideals in $L$ which are invariant under the action of $G$, whose Galois module structure has been studied extensively in the literature. The first is the ring of integers $\mathcal{O}_{L}$ of $L$, the study of which is a classical problem; see [8]. The second is the square root $A_{L/K}$ (if it exists) of the inverse different ideal $\mathfrak{D}_{L/K}^{-1}$ of $L/K$, the study of which was initiated by B. Erez in [6]. By Hilbert’s formula [20, Chapter IV, Proposition 4], this ideal $A_{L/K}$ exists when $|G|$ is odd, for example. Also, we note that $A_{L/K}$ is special because it is the unique ideal in $L$ (if it exists) which is self-dual with respect to the trace $\mathrm{Tr}_{L/K}$ of $L/K$.

It is natural to ask whether the Galois module structures of $\mathcal{O}_{L}$ and $A_{L/K}$ coincide. More specifically, suppose that $L/K$ is tame. Then, any ambiguous ideal $\mathfrak{A}$ in $L$ is locally free over $\mathcal{O}_{K}G$ of rank one by [26, Theorem 1]. Hence, it determines a class $[\mathfrak{A}]_{\mathbb{Z}G}$ in $\mathrm{Cl}(\mathbb{Z}G)$ as well as a class $[\mathfrak{A}]$ in $\mathrm{Cl}(\mathcal{O}_{K}G)$, where $\mathrm{Cl}(-)$ denotes locally free class group. Provided that $A_{L/K}$ exists, we ask:

Question 1.1**.**

Does $[\mathcal{O}_{L}]_{\mathbb{Z}G}=[A_{L/K}]_{\mathbb{Z}G}$ hold in $\mathrm{Cl}(\mathbb{Z}G)$?

Question 1.2**.**

Does $[\mathcal{O}_{L}]=[A_{L/K}]$ hold in $\mathrm{Cl}(\mathcal{O}_{K}G)$?

Since $A_{L/K}$ is self-dual with respect to $\mathrm{Tr}_{L/K}$ and $\mathfrak{D}_{L/K}^{-1}$ is the dual of $\mathcal{O}_{L}$ with respect to $\mathrm{Tr}_{L/K}$ by definition, we have that

[TABLE]

In other words, for Questions 1.1 and 1.2 to admit an affirmative answer, the ideal $\mathcal{O}_{L}$ is necessarily stably self-dual as a $\mathbb{Z}G$-module and an $\mathcal{O}_{K}G$-module, respectively. It is then natural to also ask:

Question 1.3**.**

Does $[\mathcal{O}_{L}]_{\mathbb{Z}G}=[\mathfrak{D}_{L/K}^{-1}]_{\mathbb{Z}G}$ hold in $\mathrm{Cl}(\mathbb{Z}G)$?

Question 1.4**.**

Does $[\mathcal{O}_{L}]=[\mathfrak{D}_{L/K}^{-1}]$ hold in $\mathrm{Cl}(\mathcal{O}_{K}G)$?

On the one hand, a theorem of M. J. Taylor [22] implies that Question 1.3 admits an affirmative answer; this fact was re-established by S. U. Chase [4]. Using tools from [4], L. Caputo and S. Vinatier showed in [3] that Question 1.1 also admits an affirmative answer as long as $L/K$ is locally abelian.

On the other hand, both Questions 1.2 and 1.4 have never been considered in the literature. The main purpose of this paper is to show that for $K\neq\mathbb{Q}$, they both admit a negative answer in general; see Theorem 1.9 below.

1.1. Basic set-up and notation

Fix a number field $K$ as well as a finite group $G$. Let us define

[TABLE]

where “sd” stands for “self-dual”. For $G$ of odd order, further define

[TABLE]

Let us remark that both classes $[\mathcal{O}_{L}]$ and $[A_{L/K}]$ depend upon the choice of the isomorphism $\mathrm{Gal}(L/K)\simeq G$. For $K\neq\mathbb{Q}$, we shall prove that even the weakened versions of Questions 1.2 and 1.4 below admit a negative answer in general; see Theorem 1.9 below.

Question 1.5**.**

Does $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)=\mathcal{A}^{t}(\mathcal{O}_{K}G)$ hold when $|G|$ is odd?

Question 1.6**.**

Does $R(\mathcal{O}_{K}G)=R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$ hold?

In what follows, for simplicity, suppose that $G$ is abelian. We shall implicitly suppose also that $G$ has odd order whenever we write $\mathcal{A}^{t}(\mathcal{O}_{K}G)$. Then, the three subsets of $\mathrm{Cl}(\mathcal{O}_{K}G)$ in question are related to the so-called Adams operations on $\mathrm{Cl}(\mathcal{O}_{K}G)$ as follows; also see [1] and [2] for other connections between Adams operations and Galois module structures.

For each $k\in\mathbb{Z}$ coprime to $|G|$, the $k$th Adams operation is defined by

[TABLE]

where $X$ denotes an arbitrary locally free $\mathcal{O}_{K}G$-module of rank one, and $X_{k}$ denotes the $\mathcal{O}_{K}$-module $X$ on which $G$ acts via

[TABLE]

where $\phi_{k}$ is the automorphism on $G$ given by $\phi_{k}(s)=s^{k}$. For example, when $X=\mathcal{O}_{L}$, where $L/K$ is a tame and Galois extension with $h:\mathrm{Gal}(L/K)\simeq G$, then we have $X_{k}=\mathcal{O}_{L^{\prime}}$, where $L^{\prime}=L$ but with $h^{\prime}:\mathrm{Gal}(L^{\prime}/K)\simeq G$ defined by $h^{\prime}=\phi_{k}\circ h$; similarly when $X=A_{L/K}$. In the case that $k=-1$, we have

[TABLE]

by [7, Appendix IX, Proposition 3]. Let $*$ denote dual with respect to $\mathrm{Tr}_{L/K}$. Since $\mathfrak{D}_{L/K}^{-1}=\mathcal{O}_{L}^{*}$ and $A_{L/K}=A_{L/K}^{*}$, we then deduce that

[TABLE]

where the latter equality was proven in [23, Theorem 1.2 (a)] as well. In the case that $|G|$ is odd and $k=2$, we further have

[TABLE]

which was shown in [24, Theorem 1.2.4] and is also essentially a special case of [1, Theorem 1.4].

Now, it is known by [16] that $R(\mathcal{O}_{K}G)$ is a subgroup of $\mathrm{Cl}(\mathcal{O}_{K}G)$. Writing the operation in $\mathrm{Cl}(\mathcal{O}_{K}G)$ multiplicatively, we then have well-defined maps

[TABLE]

which are in fact homomorphisms because $\mathrm{Cl}(\mathcal{O}_{K}G)$ is an abelian group. In addition, the above discussion implies that

[TABLE]

which are hence subgroups of $R(\mathcal{O}_{K}G)$. In particular, we have a chain

[TABLE]

of subgroups in $\mathrm{Cl}(\mathcal{O}_{K}G)$. From (1.1), we deduce the following criteria which distinguish classes in these three subgroups.

Proposition 1.7**.**

Suppose that $G$ is abelian and let $c\in R(\mathcal{O}_{K}G)$.

1. (a)

Assume that $\Psi_{-1}(c)=c$. Then, we have $c\in R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$ if and only if $|c|$ divides two. 2. (b)

Assume that $|G|$ is odd and that $\Psi_{2}(c)=c$. Then, we have $c\in A^{t}(\mathcal{O}_{K}G)$ only if $c^{n_{G}(2)}=1$, where $n_{G}(2)$ is the multiplicative order of $2$ mod $|G|$.

Proof.

Part (a) follows directly from (1.1). As for part (b), suppose that $|G|$ is odd and that $c\in A^{t}(\mathcal{O}_{K}G)$. By (1.1), we know that $c=d^{-1}\Psi_{2}(d)$ for some $d\in R(\mathcal{O}_{K}G)$. This implies that

[TABLE]

It follows that $c^{n_{G}(2)}=1$ whenever $\Psi_{2}(c)=c$ holds. ∎

For notation, let us also define

[TABLE]

which is plainly a group isomorphic to $(\mathbb{Z}/|G|\mathbb{Z})^{\times}$, and

[TABLE]

where the map is that induced by augmentation. Our idea is to use Proposition 1.7 as well as classes in $\mathrm{Cl}^{0}(\mathcal{O}_{K}G)^{\Psi_{\mathbb{Z}}}$, namely, classes in $\mathrm{Cl}^{0}(\mathcal{O}_{K}G)$ which are invariant under $\Psi_{\mathbb{Z}}$, to answer Questions 1.5 and 1.6.

Finally, for each $n\in\mathbb{N}$, let $C_{n}$ denote a cyclic group of order $n$, and let $\zeta_{n}$ denote a primitive $n$th root of unity. Given any multiplicative group $\Gamma$, write $\Gamma^{n}$ for the set of $n$th powers of elements in $\Gamma$.

1.2. Statements of the main theorems

First, consider $G=C_{p}$, where $p$ is an odd prime. For $K\neq\mathbb{Q}$, in order to answer Questions 1.5 and 1.6 in the negative, by (1.2), we must exhibit non-trivial classes in $R(\mathcal{O}_{K}C_{p})$. This was done in [10] and a key ingredient is the inclusion

[TABLE]

This was shown in the proof [10, Proposition 4] using the characterization of $R(\mathcal{O}_{K}C_{p})$ due to L. R. McCulloh in [15]. Using (1.3), we deduce that:

Proposition 1.8**.**

Let $p$ be an odd prime and let $c\in\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})^{\Psi_{\mathbb{Z}}}$.

1. (a)

If $|c|$ does not divide $p-1$, then $c^{(p-1)/2}\in R(\mathcal{O}_{K}C_{p})\setminus R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})$. 2. (b)

If $|c|=2$ and $p\equiv-1\pmod{8}$, then $c\in R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\setminus\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$.

Proof.

Observe that $c^{(p-1)/2}\in R(\mathcal{O}_{K}C_{p})$ by (1.3). Part (a) is then clear from Proposition 1.7 (a). As for part (b), suppose that $|c|=2$. If $p\equiv-1\pmod{4}$, then $c=c^{(p-1)/2}\in R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})$ by Proposition 1.7 (a). If $p\equiv-1\pmod{8}$ in addition, then $c\notin\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$ by Proposition 1.7 (b), because in this case $2$ is a square mod $p$ but $-1$ is not, whence $n_{C_{p}}(2)$ is necessarily odd. ∎

Using Proposition 1.8 and some further ideas from (1.3), we shall prove:

Theorem 1.9**.**

Suppose that $K\neq\mathbb{Q}$. Then we have:

1. (a)

$R(\mathcal{O}_{K}C_{p})\supsetneq R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})$* for infinitely many odd primes $p$.* 2. (b)

$R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\supsetneq\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$* for infinitely many odd primes $p$.*

Proof.

We shall prove part (b) in Subsection 2.1. For part (a), we may deduce it using results in [10] as follows. Let $p$ be an odd prime. Let $T(\mathcal{O}_{K}C_{p})$ denote the Swan subgroup of $\mathrm{Cl}(\mathcal{O}_{K}C_{p})$; see [25] or [5, Section 53] for the definition. Then, as shown in the proof of [10, Proposition 4], we have

[TABLE]

Using Chebotarev’s density theorem, it was further shown in [10, Theorem 5 and Proposition 9] that $T(\mathcal{O}_{K}C_{p})$ contains a class of order coprime to $p-1$ for infinitely many $p$. The claim now follows from Proposition 1.8 (a). ∎

Since the proof of Theorem 1.9 uses Chebotarev’s density theorem, it does not give explicit primes $p$ satisfying the conclusion. In the special case that $K/\mathbb{Q}$ is abelian with $K$ imaginary, by slightly modifying the proof, we shall give explicit primes $p$ such that $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\supsetneq\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$. See [11] for explicit conditions on $K$, in which $p$ is ramified, such that the $p$-rank of $T(\mathcal{O}_{K}C_{p})$ is at least one, so $R(\mathcal{O}_{K}C_{p})\supsetneq R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})$ by (1.4) and Proposition 1.8 (a).

Theorem 1.10**.**

Suppose that $K/\mathbb{Q}$ is abelian with $K$ imaginary, and let $m$ be the conductor of $K$. Then, we have $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\supsetneq\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$ for all primes $p$ satisfying $p\equiv-1\pmod{8}$ and $p\equiv-1\pmod{2m}$.

Example 1.11**.**

Consider the special case when $K=\mathbb{Q}(\sqrt{D})$, where $D$ is a negative square-free integer not divisible by $p$. For simplicity, let us assume that $D\not\in\{-1,-3\}$. Then, by [12, Lemma 3.2 and Theorem 3.4], we have

[TABLE]

where $\left(\frac{\cdot}{\cdot}\right)$ denotes the Legendre symbol. From Proposition 1.8 and (1.4), we then deduce that

[TABLE]

where the second statement may be viewed as a refinement of Theorem 1.10. To see why, note that by quadratic reciprocity, we have

[TABLE]

for any odd prime $q$. Suppose that $p\equiv-1\pmod{8}$ and $p\equiv-1\pmod{2m}$, where $m$ is the conductor of $K$. Since $|D|$ divides $m$, we see that any of its prime divisor is a square mod $p$. It follows that

[TABLE]

by the above, as predicted by Theorem 1.10. Let us note that not much may be deduced from Proposition 1.8 if $\left(\frac{D}{p}\right)=1$, and that the case $D\in\{-1,-3\}$ may be dealt with analogously.

Next, we return to an arbitrary abelian group $G$. Recall that the proof of Theorem 1.9 (a) uses the Swan subgroup $T(\mathcal{O}_{K}G)$ of $\mathrm{Cl}(\mathcal{O}_{K}G)$. The connection between Question 1.6 and the Swan subgroup was already observed in [4] and [21]; they both used the fact that $T(\mathbb{Z}C)=1$ for all finite cyclic groups $C$ to answer Question 1.3 in the positive. We shall investigate this connection further as follows.

Observe that the first equality in (1.1) implies that

[TABLE]

Thus, it suffices to understand $\mathrm{Im}(\Xi_{-1})$. In Subsection 3.3, for each subgroup $H$ of $G$, we shall define a generalized Swan subgroup $T_{H}^{*}(\mathcal{O}_{K}G)$ of $\mathrm{Cl}(\mathcal{O}_{K}G)^{\Psi_{\mathbb{Z}}}$, such that $T_{G}^{*}(\mathcal{O}_{K}G)$ is the usual Swan subgroup $T(\mathcal{O}_{K}G)$. We shall give lower and upper bounds for $\mathrm{Im}(\Xi_{-1})$ in terms of these $T_{H}^{*}(\mathcal{O}_{K}G)$.

Theorem 1.12**.**

Suppose that $G$ is abelian. Let $H$ be a cyclic subgroup of $G$ and let $n$ denote its order.

1. (a)

We have $T_{H}^{*}(\mathcal{O}_{K}G)^{d_{n}(K)}\subset R(\mathcal{O}_{K}G)^{\Psi_{\mathbb{Z}}}$, where

[TABLE]

In particular, we have $T_{H}^{*}(\mathcal{O}_{K}G)^{2d_{n}(K)}\subset\mathrm{Im}(\Xi_{-1})$. 2. (b)

We have $T_{H}^{*}(\mathcal{O}_{K}G)\subset\mathrm{Im}(\Xi_{-1})$ if $n$ is odd and $\zeta_{n}\in K^{\times}$.

Theorem 1.13**.**

Suppose that $G$ is abelian.

1. (a)

We have $\mathrm{Im}(\Xi_{-1})\subset T_{\mathrm{\tiny cyc}}^{*}(\mathcal{O}_{K}G)$ if $\mathrm{Cl}(\mathcal{O}_{K})=1$, where

[TABLE] 2. (b)

We have $\mathrm{Im}(\Xi_{-1})\neq 1$ if $\mathrm{Cl}(\mathcal{O}_{K})^{\delta(G)}\neq 1$ and $\zeta_{\exp(G)}\in K^{\times}$, where

[TABLE]

and $\exp(G)$ denotes the exponent of $G$, provided that $G\neq 1$.

From Theorems 1.12 and 1.13, as well as (1.6), we deduce that

[TABLE]

under the assumption that $G$ is an abelian group of odd order such that all $|G|$th roots of unity are contained in $K$.

Example 1.14**.**

Suppose that $G=C_{p}$, where $p$ is an odd prime. Applying Theorem 1.12 (a) to the full group $G$, we obtain

[TABLE]

and so we may regard Theorem 1.12 (a) as a refinement of (1.3) and (1.4). By Theorem 1.13 (a), when $\mathrm{Cl}(\mathcal{O}_{K})=1$, we then have a chain

[TABLE]

of inclusions. Let us consider a few special examples of $K$ with $\mathrm{Cl}(\mathcal{O}_{K})=1$.

By [12, Lemma 3.2 and Theorem 3.4], we know that

[TABLE]

By [18], we also know that

[TABLE]

In all of the above cases, we deduce that $\mathrm{Im}(\Xi_{-1})=T(\mathcal{O}_{K}C_{p})$, and in particular, from (1.6) we see that the difference between $R(\mathcal{O}_{K}C_{p})$ and $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})$ becomes bigger as $p$ increases.

2. Comparison between $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$ and $\mathcal{A}^{t}(\mathcal{O}_{K}G)$

In this section, we shall prove Theorems 1.9 (b) and 1.10, by using Proposition 1.8 (b) to exhibit the existence of a class in $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}C_{p})\setminus\mathcal{A}^{t}(\mathcal{O}_{K}C_{p})$ for infinitely many odd primes $p$.

In what follows, let $p$ be any odd prime. Define

[TABLE]

is the natural quotient map. Then, we have a surjective homomorphism

[TABLE]

as shown in [10, Theorem 5]. This, together with (1.4), implies that:

Lemma 2.1**.**

If $p\equiv-1\pmod{4}$ and $V_{p}(\mathcal{O}_{K})$ has an element of order four, then $\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})^{\Psi_{\mathbb{Z}}}$ has an element of order two.

In the case that $K$ is not totally real, we shall prove Theorem 1.9 (b) using Lemma 2.1. In the case that $K$ is totally real, however, our method fails in general; see Remark 2.6. Hence, we must look for elements in $\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})^{\Psi_{\mathbb{Z}}}$ of order two lying outside of $T(\mathcal{O}_{K}C_{p})$.

To that end, let $\mathcal{M}(KC_{p})$ denote the maximal order in $KC_{p}$, and for convenience, assume that $p$ is large enough so that $[K(\zeta_{p}):K]=p-1$. Then, we have a natural isomorphism

[TABLE]

where $\chi$ is a fixed non-trivial character on $C_{p}$. This induces an isomorphism

[TABLE]

In particular, we have a surjective homormorphism

[TABLE]

such that the $\Psi_{\mathbb{Z}}$-action on $\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})$ corresponds precisely to the $\Gamma_{p}$-action on $\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})$, where $\Gamma_{p}=\mathrm{Gal}(K(\zeta_{p})/K)$. This implies that:

Lemma 2.2**.**

If $\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})^{\Gamma_{p}}$ has an element of order two, then $\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})^{\Psi_{\mathbb{Z}}}$ also has an element of order two.

To show that $\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})^{\Gamma_{p}}$ contains an element of order two, we shall need the following so-called Chevalley’s ambiguous class formula.

Proposition 2.3**.**

Let $F/K$ be a cyclic extension. Let $\Gamma=\mathrm{Gal}(F/K)$ denote its Galois group and let $N_{F/K}:F\longrightarrow K$ denote its norm. Then, we have

[TABLE]

where $r$ is the number of real places in $K$ which complexify in $F/K$. Here $\mathfrak{p}$ ranges over the prime ideals in $K$ and $e_{\mathfrak{p}}$ is its ramification index in $F/K$.

Proof.

See [9, Chapter II, Remark 6.2.3]. ∎

Lemma 2.4**.**

If $K\neq\mathbb{Q}$ is not totally imaginary, with $[K(\zeta_{p}):K]=p-1$, and $p$ is totally split in $K/\mathbb{Q}$, then $\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})^{\Gamma_{p}}$ has an element of order two.

Proof.

Assume the hypothesis. Let us write $[K:\mathbb{Q}]=r_{1}+2r_{2}$, where $r_{1}$ and $2r_{2}$, respectively, denote the number of real and complex embeddings of $K$. Applying Proposition 2.3 to the field $F=K(\zeta_{p})$, we then obtain

[TABLE]

Indeed, we have $r=r_{1}$ since $K(\zeta_{p})$ is totally imaginary. Further, the prime ideals $\mathfrak{p}$ in $K$ which ramify in $K(\zeta_{p})/K$ are precisely those above $p$. Since $p$ is totally split in $K/\mathbb{Q}$, there are $[K:\mathbb{Q}]$ such $\mathfrak{p}$, and $e_{\mathfrak{p}}=[K(\zeta_{p}):K]=p-1$.

Now, by the Dirichlet’s unit theorem, we know that

[TABLE]

where $\epsilon_{0}$ is a root of unity and $\epsilon_{1},\dots,\epsilon_{r_{1}+r_{2}-1}$ are fundamental units. Hence, we have a natural surjective homomorphism

[TABLE]

and so the order of the quotient group on the right divides

[TABLE]

Notice that $n_{0}$ divides $p-1$ and that $n_{0}=2$ when $K$ is totally real. We then deduce that $|\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})^{\Gamma_{p}}|$ is divisible by

[TABLE]

By hypothesis, we have $r_{1}\geq 1$, and $r_{1}\geq 2$ when $r_{2}=0$. Hence, the number above is always even, and so $\mathrm{Cl}(\mathcal{O}_{K(\zeta_{p})})^{\Gamma_{p}}$ has an element of order two. ∎

2.1. Proof of Theorem 1.9 (b)

Fix an algebraic closure $K^{c}$ of $K$. Let $\widetilde{K}$ denote the Galois closure of $K$ over $\mathbb{Q}$ lying in $K^{c}$ and let $K_{4}$ denote the field obtained by adjoining to $\widetilde{K}$ all fourth roots of elements in $\mathcal{O}_{K}^{\times}$. Notice that $K_{4}/\mathbb{Q}$ is a Galois extension.

The next lemma is motivated by [10, Proposition 9] and it allows us to use Chebotarev’s density theorem to prove Theorem 1.9 (b).

Lemma 2.5**.**

Let $\tau\in\emph{Gal}(K^{c}/\mathbb{Q})$ and let $f\in\mathbb{N}$ denote the smallest natural number such that $\tau^{f}|_{K}=\emph{Id}_{K}$.

1. (a)

Suppose that

[TABLE]

Let $\mathfrak{P}$ be any prime ideal in $K_{4}(\sqrt{2})$, unramified over $\mathbb{Q}$, such that

[TABLE]

and let $p\mathbb{Z}$ be the prime lying below $\mathfrak{P}$. Then, we have $p\equiv-1\pmod{8}$, and the group $V_{p}(\mathcal{O}_{K})$ has an element of order four. 2. (b)

Suppose that

[TABLE]

Let $\mathfrak{P}$ be any prime ideal in $\widetilde{K}(\sqrt{-1},\sqrt{2})$, unramified over $\mathbb{Q}$, such that

[TABLE]

and let $p\mathbb{Z}$ be the prime lying below $\mathfrak{P}$. Then, we have $p\equiv-1\pmod{8}$, and the prime $p$ is totally split in $K/\mathbb{Q}$.

Proof.

In both parts (a) and (b), we clearly have $p\equiv-1$ (mod $8)$ because

[TABLE]

by (1.5). In part (b), the prime $p$ is totally split in $\widetilde{K}/\mathbb{Q}$ and hence in $K/\mathbb{Q}$.

In part (a), let $\mathfrak{p}_{4}$ and $\mathfrak{p}$ denote the prime ideals in $K_{4}$ and $K$, respectively, lying below $\mathfrak{P}$. Note that $f$ is the inertia degree of $\mathfrak{p}$ over $\mathbb{Q}$, and we have

[TABLE]

This means that $\mathfrak{p}$ is totally split in $K_{4}/K$, and so elements in $\mathcal{O}_{K}^{\times}$ reduce to fourth powers in $\mathcal{O}_{K}/\mathfrak{p}$. Hence, we have surjective homomorphisms

[TABLE]

where $\uppi_{\mathfrak{p}}:\mathcal{O}_{K}\longrightarrow\mathcal{O}_{K}/\mathfrak{p}$ is the natural quotient map. But $(\mathcal{O}_{K}/\mathfrak{p})^{\times}\simeq C_{p^{f}-1}$,

and $4$ divides $p^{f}-1$ because $f\geq 2$ is even. It follows that the last quotient group above and in particular $V_{p}(\mathcal{O}_{K})$ has an element of order four. ∎

Proof of Theorem 1.9 (b).

Let $\sigma_{c},\sigma_{r}:K^{c}\longrightarrow\mathbb{C}$ be embeddings such that

[TABLE]

if they exist. Further, define

[TABLE]

where $\rho:\mathbb{C}\longrightarrow\mathbb{C}$ denotes complex conjugation. Observe that:

1. (i)

If $K$ is not totally real, then $\sigma_{c}$ exists, and $\tau_{c}$ satisfies (2.1). 2. (ii)

If $K$ is totally real, then $\sigma_{r}$ exists, and $\tau_{r}$ satisfies (2.2).

In both cases, let $p\equiv-1$ (mod $8$) be a prime given as in Lemma 2.5. Then, we deduce from Lemmas 2.1, 2.2, and 2.4 that $\mathrm{Cl}^{0}(\mathcal{O}_{K}C_{p})^{\Psi_{\mathbb{Z}}}$ has an element of order two. The claim now follows from Proposition 1.8 (b) and Chebotarev’s density theorem. ∎

Remark 2.6*.*

Suppose that $K$ is a real quadratic field such that its fundamental unit $\epsilon$ has norm $-1$ over $\mathbb{Q}$. For any odd prime $p$ which is inert in $K/\mathbb{Q}$, we then have $\epsilon^{p+1}\equiv-1$ (mod $p\mathcal{O}_{K}$), as shown in [13, (1.0.1)]. Letting $n_{p}(\epsilon)$ denote the multiplicative order of $\epsilon$ mod $p\mathcal{O}_{K}$, this implies that

[TABLE]

The first quotient is an odd integer by [13, Theorem 1.3], so then $V_{p}(\mathcal{O}_{K})$ has odd order when $p\equiv-1$ (mod $4$). This means that we cannot use Lemma 2.5 (a) to find primes $p\equiv-1$ (mod $8$) such that $V_{p}(\mathcal{O}_{K})$ has an element of order four.

2.2. Proof of Theorem 1.10

First, we need the following group-theoretic lemmas.

Lemma 2.7**.**

Let $\Gamma$ be a finite abelian $p$-group, where $p$ is a prime. Let $\Delta$ be any cyclic subgroup of $\Gamma$ whose order is maximal among all cyclic subgroups of $\Gamma$. Then, there exists a subgroup $\Delta^{\prime}$ of $\Gamma$ such that $\Gamma=\Delta\times\Delta^{\prime}$.

Proof.

See the proof of [14, Chapter I, Theorem 8.2], for example. ∎

Lemma 2.8**.**

Let $\Gamma$ be a group isomorphic to $k$ copies of $C_{n}$, where $k,n\in\mathbb{N}$, and let $\Delta$ be any cyclic subgroup of order $n$. Then, there exists a subgroup $\Delta^{\prime}$ of $\Gamma$ such that $\Gamma=\Delta\times\Delta^{\prime}$. Moreover, for any $x\in\Gamma$, there exists a surjective homomorphism from $\Gamma/\langle x\rangle$ to $k-1$ copies of $C_{n}$.

Proof.

The first claim is a direct consequence of Lemma 2.7 and plainly $\Delta^{\prime}$ is necessarily isomorphic to $k-1$ copies of $C_{n}$. The second claim follows as well because any $x\in\Gamma$ is contained in some cyclic subgroup $\Delta$ of order $n$. ∎

Proof of Theorem 1.10.

By Proposition 1.8 (b) and Lemma 2.1, it is enough to show that $V_{p}(\mathcal{O}_{K})$ has an element of order four.

Set $d=[K:\mathbb{Q}]$ and note that $K\subset\mathbb{Q}(\zeta_{m})$ by hypothesis. First, since $K$ is imaginary, by the Dirichlet’s unit theorem, we know that

[TABLE]

where $\epsilon_{0}$ is a root of unity and $\epsilon_{1},\dots,\epsilon_{d/2-1}$ are fundamental units. Now, the hypothesis $p\equiv-1$ (mod $m$) implies that $p$ is unramified in $\mathbb{Q}(\zeta_{m})/\mathbb{Q}$ and

[TABLE]

Since $K$ is imaginary, the inertia degree of $p$ in $K/\mathbb{Q}$ is equal to two, and so

[TABLE]

From Lemma 2.8, we then deduce that there is a surjective homomorphism

[TABLE]

Let $\delta=2$ if $m$ is odd, and $\delta=1$ if $m$ is even. Then, the order of $\langle\epsilon_{0}\rangle$ divides $\delta m$, and we see that there are surjective homomorphisms

[TABLE]

The last cyclic group has order dividing four, because

[TABLE]

by hypothesis. Thus, indeed $V_{p}(\mathcal{O}_{K})$ has an element of order four. ∎

3. Comparison between $R(\mathcal{O}_{K}G)$ and $R_{\mathrm{\tiny sd}}(\mathcal{O}_{K}G)$

In this section, we shall prove Theorems 1.12 and 1.13. A key ingredient is

the characterization of $R(\mathcal{O}_{K}G)$ due to L. R. McCulloh [16], which works for all abelian groups $G$; see Subsection 3.2 below. We remark that the proof of (1.3) given in [10] uses his older characterization of $R(\mathcal{O}_{K}G)$ from [15], which works only for elementary abelian groups $G$.

In the subsequent subsections, except in Subsection 3.3, we shall assume that $G$ is abelian. We shall further use the following notation.

Notation 3.1**.**

Let $M_{K}$ denote the set of finite primes in $K$. The symbol $F$ shall denote either $K$ or the completion $K_{v}$ of $K$ at some $v\in M_{K}$, and

[TABLE]

For each $v\in M_{K}$, we shall regard $K^{c}$ as lying in $K_{v}^{c}$ via a fixed embedding $K^{c}\longrightarrow K_{v}^{c}$ extending the natural embedding $K\longrightarrow K_{v}$.

3.1. Locally free class group

The class group $\mathrm{Cl}(\mathcal{O}_{K}G)$ admits an idelic description as follows; see [5, Chapter 6], for example.

Let $J(KG)$ denote the restricted direct product of $(K_{v}G)^{\times}$ with respect to the subgroups $(\mathcal{O}_{K_{v}}G)^{\times}$ for $v\in M_{K}$. We have a surjective homomorphism

[TABLE]

where we define

[TABLE]

This in turn induces an isomorphism

[TABLE]

Each component $(K_{v}G)^{\times}$ as well as $(KG)^{\times}$ also admit a Hom-description as follows. Write $\widehat{G}$ for the group of irreducible $K^{c}$-valued characters on $G$. We then have canonical identifications

[TABLE]

induced by the association $\alpha\mapsto(\chi\mapsto\alpha(\chi))$, where we define

[TABLE]

Finally, we note that via (3.1) and (3.2), for each $k\in\mathbb{Z}$ coprime to $|G|$, the $k$th Adams operation $\Psi_{k}$ on $\mathrm{Cl}(\mathcal{O}_{K}G)$ is induced by $\chi\mapsto\chi^{k}$ on $\widehat{G}$.

3.2. McCulloh’s characterization

The characterization of $R(\mathcal{O}_{K}G)$ due to L. R. McCulloh [16] is given in terms of the so-called Stickelberger transpose. We shall recall its definition below.

Definition 3.2**.**

Let $G(-1)$ denote the group $G$ on which $\Omega_{F}$ acts by

[TABLE]

where $\kappa(\omega^{-1})\in\mathbb{Z}$, which is unique modulo $\exp(G)$, is such that

[TABLE]

Note that if $\zeta_{n}\in F$, then $\Omega_{F}$ fixes all elements in $G(-1)$ of order dividing $n$.

Definition 3.3**.**

Given $\chi\in\widehat{G}$ and $s\in G$, define

[TABLE]

Extend this to a pairing $\langle\hskip 2.84526pt,\hskip 2.84526pt\rangle:\mathbb{Q}\widehat{G}\times\mathbb{Q}G\longrightarrow\mathbb{Q}$ via $\mathbb{Q}$-linearity, and define

[TABLE]

called the Stickelberger map.

As shown in [16, Proposition 4.5], the Stickelberger map preserves the $\Omega_{F}$-action. Set $A_{\widehat{G}}=\Theta^{-1}(\mathbb{Z}G)$. Then, applying the functor $\mbox{Hom}(-,(F^{c})^{\times})$ and taking $\Omega_{F}$-invariants yield a homomorphism

[TABLE]

This is the Stickelberger transpose map defined in [16].

For brevity, define

[TABLE]

Observe that we have a diagram

[TABLE]

where $rag$ is restriction to $A_{\widehat{G}}$ via the identification (3.2).

Now, let $J(\Lambda(KG))$ denote the restricted direct product of $\Lambda(K_{v}G)^{\times}$ with respect to the subgroups $\mbox{Map}_{\Omega_{F}}(G(-1),\mathcal{O}_{F^{c}}^{\times})$ for $v\in M_{K}$. We then have the following partial characterization of $R(\mathcal{O}_{K}G)$; see [16] for the full characterization.

Lemma 3.4**.**

Given $c=(c_{v})\in J(KG)$, if there exists $g=(g_{v})\in J(\Lambda(KG))$ such that $rag(c_{v})=\Theta^{t}(g_{v})$ for all $v\in M_{K}$, then $j(c)\in R(\mathcal{O}_{K}G)$.

Proof.

This follows directly from [16, Theorem 6.17]. ∎

For each $v\in M_{K}$, fix a uniformizer $\pi_{v}$ of $K_{v}$. We shall also need:

Lemma 3.5**.**

Let $L/K$ be a tame and Galois extension with $\mathrm{Gal}(L/K)\simeq G$. Then, for each $v\in M_{K}$, there exists $s_{v}\in G$ whose order is the ramification index of $L/K$ at $v$, such that $\Xi_{-1}([\mathcal{O}_{L}])=j(c_{L})$, where $c_{L}=(c_{L,v})\in J(KG)$ is defined by $c_{L,v}(\chi)=\pi_{v}^{\langle\chi,s_{v}\rangle+\langle\chi,s_{v}^{-1}\rangle}$ for $\chi\in\widehat{G}$ via the identification (3.2).

Proof.

We have $L=KG\cdot b$ for some $b\in L$ by the Normal Basis Theorem, and since $\mathcal{O}_{L}$ is locally free over $\mathcal{O}_{K}G$ of rank one, for $v\in M_{K}$, we have

[TABLE]

Following the notation in [16, Section 1], put

[TABLE]

Then, by [16, Proposition 5.4], we may choose $a_{v}$ to be such that

[TABLE]

By [16, Proposition 3.2] and the discussion following it, there exists

[TABLE]

Write $[-1]$ for the involution on $(K_{v}^{c}G)^{\times}$ induced by the involution $s\mapsto s^{-1}$ on $G$. Since $\mathbf{r}_{G}(b)\mathbf{r}_{G}(b)^{[-1]}\in(KG)^{\times}$, we then deduce that

[TABLE]

The claim now follows immediately from (3.4). ∎

3.3. Generalized Swan subgroups

Let $H$ be a subgroup of $G$. Following the definition of the Swan subgroup $T(\mathcal{O}_{K}G)$ given in [25], we shall define a generalized Swan subset/subgroup associated to $H$ as follows.

For each $r\in\mathcal{O}_{K}$ coprime to $|H|$, define

[TABLE]

The next proposition, which generalizes [25, Proposition 2.4 (i)], shows that $(r,\Sigma_{H})$ is locally free over $\mathcal{O}_{K}G$ of rank one and so it defines a class $[(r,\Sigma_{H})]$ in $\mathrm{Cl}(\mathcal{O}_{K}G)$. Define

[TABLE]

to be the collection of all such classes. It follows directly from the definition that $T_{G}^{*}(\mathcal{O}_{K}G)$ is equal to $T(\mathcal{O}_{K}G)$.

Proposition 3.6**.**

Let $r\in\mathcal{O}_{K}$ be coprime to $|H|$. For each $v\in M_{K}$, define

[TABLE]

and set $c_{H,r}=(c_{H,r,v})$. Then we have $\mathcal{O}_{K}G\cdot c_{H,r}=(r,\Sigma_{H})$.

Proof.

For each $v\in M_{K}$, we need to show that

[TABLE]

For $v\nmid r$, we have $r\in\mathcal{O}_{K_{v}}^{\times}$, and this is clear. For $v\mid r$, we have $|H|\in\mathcal{O}_{K_{v}}^{\times}$ because $r$ is coprime to $|H|$, and so

[TABLE]

The reverse inclusion also holds because

[TABLE]

We then see that the claim holds. ∎

In what follows, for simplicity, let us assume that

[TABLE]

Put $Q=G/H$, and let $H_{1},\dots,H_{q}$ denote all the distinct cosets of $H$ in $G$. Notice that we have an augmentation homomorphism

[TABLE]

Then, we have a fiber product diagram of rings, given by

[TABLE]

Here the vertical maps are the canonical quotient maps, and $\overline{\upepsilon}$ is the homomorphism induced by $\upepsilon$. We then have the identification

[TABLE]

In particular, writing

[TABLE]

the corresponding element in $\mathcal{O}_{K}G$ is given by

[TABLE]

Since $Q$ is abelian, from the Mayer-Vietoris sequence (see [5, Section 49B] or [19, (1.12), (4.19), (4.21)]) associated to (3.6), we obtain a homomorphism

[TABLE]

where $D(\mathcal{O}_{K}G)$ denotes the kernel group in $\mathrm{Cl}(\mathcal{O}_{K}G)$ defined as in [19], and

[TABLE]

is equipped with the obvious $\mathcal{O}_{K}G$-module structure via (3.7).

The next proposition, which generalizes [25, Proposition 2.7], shows that

[TABLE]

where $\Lambda_{|H|}$ is regarded as a subring of $\Lambda_{|H|}Q$ in the obvious way (cf. the set $T_{H}(\mathcal{O}_{K}G)$ defined in [17]). This means that under the assumption (3.5), the set $T_{H}^{*}(\mathcal{O}_{K}G)$ is in fact a subgroup of $\mathrm{Cl}(\mathcal{O}_{K}G)$.

Proposition 3.7**.**

Let $r\in\mathcal{O}_{K}$ be coprime to $|H|$. Then we have

[TABLE]

Proof.

For brevity, put $\eta=(r+|H|\mathcal{O}_{K})H$. Note that by definition, we have

[TABLE]

Via the identification (3.7), we may define an $\mathcal{O}_{K}G$-homomorphism

[TABLE]

Below, we shall show that $\mathrm{Im}(\varphi)=(r,\Sigma_{H})$ and $\ker(\varphi)=0$. This would imply that $(\mathcal{O}_{K}G)(\eta)$ and $(r,\Sigma_{H})$ are isomorphic as $\mathcal{O}_{K}G$-modules, from which the claim follows. Given $(x,y)\in(\mathcal{O}_{K}G)(\eta)$, in the notation of (3.8), we have

[TABLE]

First, from (3.9), we immediately see that $\mathrm{Im}(\varphi)\subset(r,\Sigma_{H})$, as well as

[TABLE]

whence $\mathrm{Im}(\varphi)\supset(r,\Sigma_{H})$ holds also. Next, suppose that $(x,y)\in\ker(\varphi)$. It is clear from the definition of $\varphi$ that $x=0$. Then, we deduce from (3.9) that

[TABLE]

This shows that $y=0$, and so $\ker(\varphi)=0$, as desired. ∎

3.4. Preliminaries

Let $H$ be a subgroup of $G$ and let $r\in\mathcal{O}_{K}$ be coprime to $|H|$. Then, via the isomorphism (3.1), we have

[TABLE]

is defined as in Proposition 3.6. Also, note that for $v\mid r$, we have

[TABLE]

for $\chi\in\widehat{G}$ via the identification (3.2). This immediately implies that:

Proposition 3.8**.**

We have $T_{H}^{*}(\mathcal{O}_{K}G)\subset\mathrm{Cl}(\mathcal{O}_{K}G)^{\Psi_{\mathbb{Z}}}$.

Proof.

This follows from (3.11) and the fact that

[TABLE]

for any $k\in\mathbb{Z}$ coprime to $|H|$. ∎

To make connections between $T_{H}^{*}(\mathcal{O}_{K}G)$ and $R(\mathcal{O}_{K}G)$, we shall use Lemmas 3.4 and 3.5. We shall also need the following definitions.

Fix a prime $v\in M_{K}$. Recall from (3.2) and (3.3) that

[TABLE]

Given $t\in G$ with $t\neq 1$ and $x\in K_{v}^{\times}$, define

[TABLE]

for $\chi\in\widehat{G}$, where both exponents are integers by Definition 3.3. In the case that $|t|=2$ and $|t|>2$, respectively, define

[TABLE]

for $s\in G(-1)$. In the case that $|t|$ is odd, further define

[TABLE]

for $s\in G(-1)$. We have the following lemmas.

Lemma 3.9**.**

We have $c_{t,v,x,1}\in(K_{v}G)^{\times}$.

Proof.

The map $c_{t,v,x,1}$ preserves the $\Omega_{K_{v}}$-action because

[TABLE]

for all $\chi\in\widehat{G}$ and $s\in G$ by Definition 3.3. ∎

Lemma 3.10**.**

Suppose that $\zeta_{|t|}\in K_{v}^{\times}$. Then we have

[TABLE]

Moreover, for both $i=1,2$, we have

[TABLE]

Proof.

Since $\zeta_{|t|}\in K_{v}^{\times}$, we easily see that $c_{t,v,x,2}$, $g_{t,v,x,1}$, and $g_{t,v,x,2}$ indeed all preserve the $\Omega_{K_{v}}$-action. Since

[TABLE]

by definition, the second also holds by a simple verification. ∎

3.5. Proof of Theorem 1.12

Let $H$ be a cyclic subgroup of $G$ of order $n$ and let $r\in\mathcal{O}_{K}$ be coprime to $n$. Recall (3.10) and that $j(c_{H,r})\in\mathrm{Cl}(\mathcal{O}_{K}G)^{\Psi_{\mathbb{Z}}}$ by Proposition 3.8. We need to show that $j(c_{H,r})^{d_{n}(K)}\in R(\mathcal{O}_{K}G)$ in part (a), and that $j(c_{H,r})\in\mathrm{Im}(\Xi_{-1})$ in part (b). We shall do so using Lemma 3.4.

In what follows, let $t$ be a fixed generator of $H$.

Proof of Theorem 1.12 (a).

Let $D$ be the subgroup of $(\mathbb{Z}/n\mathbb{Z})^{\times}$ such that

[TABLE]

Define $g=(g_{v})\in J(\Lambda(KG))$ by setting $g_{v}=1$ for $v\nmid r$, and

[TABLE]

for $v\mid r$. It is easy to see that $g_{v}$ preserves the $\Omega_{K_{v}}$-action.

Observe that for all $v\in M_{K}$, we have

[TABLE]

Indeed, for $v\nmid r$, this is clear. As for $v\mid r$, we have from (3.13) that

[TABLE]

and from (3.12) that

[TABLE]

for any $\chi\in\widehat{G}$. The equality (3.14) then follows from (3.11). Hence, we have $j(c_{H,r})^{d_{n}(K)}\in R(\mathcal{O}_{K}G)$ by Lemma 3.4, as desired. ∎

Proof of Theorem 1.12 (b).

Suppose that $n$ is odd and that $\zeta_{n}\in K^{\times}$. Then, by Lemma 3.10, we may define $c=(c_{v})\in J(KG)$ by setting $c_{v}=1$ for $v\nmid r$, and $c_{v}=c_{t,v,r,2}$ for $v\mid r$. Also, we have $j(c)\in R(\mathcal{O}_{K}G)$ by Lemma 3.4.

Below, we shall show that $j(c_{H,r})=\Xi_{-1}(j(c))$, whence $j(c_{H,r})\in\mathrm{Im}(\Xi_{-1})$. To that end, let $v\in M_{K}$ and $\chi\in\widehat{G}$. It suffices to show that

[TABLE]

For $v\nmid r$, this is clear. For $v\mid r$, observe that

[TABLE]

because $|t|$ is odd. It then follows from (3.12) that

[TABLE]

From (3.11), we then see that (3.15) indeed holds. ∎

3.6. Proof of Theorem 1.13

In what follows, for each $v\in M_{K}$, let $\pi_{v}$ be a fixed uniformizer of $K_{v}$.

Proof of Theorem 1.13 (a).

Suppose that $\mathrm{Cl}(\mathcal{O}_{K})=1$. For each $v_{0}\in M_{K}$, we may then choose $\pi_{v_{0}}$ to be an element of $\mathcal{O}_{K}$. Then, for any cyclic subgroup $H$ of $G$ of order coprime to $v_{0}$, it makes sense to write

[TABLE]

is as in Proposition 3.6. Plainly $c_{H,\pi_{v_{0}},v}=1$ for all $v\neq v_{0}$.

Now, let $L/K$ be any tame and Galois extension with $\mathrm{Gal}(L/K)\simeq G$, and we shall use the notation as in Lemma 3.5. Let $V$ denote the subset of $M_{K}$ consisting of the primes which ramify in $L/K$. Then, we have

[TABLE]

where we regard $c_{L,v_{0}}$ as an element of $J(KG)$ whose components outside of $v_{0}$ are all $1$. For each $v_{0}\in V$, take $H_{v_{0}}=\langle s_{v_{0}}\rangle$, whose order is coprime to $v_{0}$ because $L/K$ is tame. By (3.11) and (3.12), we have

[TABLE]

By definition, we also have $c_{L,v_{0},v}=c_{H_{v_{0}},\pi_{v_{0}},v}=1$ for $v\neq v_{0}$. It follows that $j(c_{L,v_{0}})=j(c_{H_{v_{0}},\pi_{v_{0}}})$, which is an element of $T_{H_{v_{0}}}^{*}(\mathcal{O}_{K}G)$. This implies that

[TABLE]

as claimed. ∎

Proof of Theorem 1.13 (b).

Suppose that $G\neq 1$. Then, fix an element $t\in G$ with $t\neq 1$, whose order shall be assumed to be odd when $\delta(G)=1$, and fix a character $\chi\in\widehat{G}$ such that $\chi(t)\neq 1$. Now, suppose that $\zeta_{\exp(G)}\in K^{\times}$. Then, via (3.1) and (3.2), evaluation at $\chi$ induces a surjective homomorphism

[TABLE]

Below, we shall show that

[TABLE]

from which the claim would follow.

Now, every class in $\mathrm{Cl}(\mathcal{O}_{K})$ may be represented by a prime ideal $\mathfrak{p}_{0}$ in $\mathcal{O}_{K}$, corresponding to $v_{0}\in M_{K}$, say. Since $\zeta_{\exp(G)}\in K^{\times}$, by Lemmas 3.9 and 3.10, we may define $c=(c_{v})\in J(KG)$ by setting

[TABLE]

and $c_{v}=1$ for $v\neq v_{0}$. Note that $j(c)\in R(\mathcal{O}_{K}G)$ by Lemmas 3.4 and 3.10, whence $\Xi_{-1}(j(c))\in\mathrm{Im}(\Xi_{-1})$. Also, we have

[TABLE]

by (3.12). We then deduce that

[TABLE]

This proves the desired inclusion (3.16). ∎

4. Acknowledgments

The research was supported by the China Postdoctoral Science Foundation Special Financial Grant (grant no.: 2017T100060). The author would like to thank the anoynmous referee for pointing out some unclear arguments in the original manuscript and for suggesting the reference [14] cited in Lemma 2.7.