On the square root of the inverse different

Abstract

Let $F_{\pi }$ be a finite Galois-algebra extension of a number field F, with group G. Suppose that $F_{\pi }/F$ is weakly ramified and that the square root $A_\pi $ of the inverse different $\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, $|G|$ is odd.) Erez has conjectured that the class $(A_\pi )$ of $A_\pi $ in the locally free class group $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of $\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class $W(F_{\pi }/F)$ associated with $F_\pi /F$. This conjecture has been verified in many cases. We establish a precise formula for $(A_\pi )$ in terms of $W(F_{\pi }/F)$ in all cases where $A_\pi $ is defined and $F_\pi /F$ is tame, and are thereby able to deduce that, in general, $(A_\pi )$ is not equal to $W(F_\pi /F)$.

1 Introduction

Let F be a number field with absolute Galois group $\Omega _F$ . Suppose that G is a finite group on which $\Omega _F$ acts trivially, and let $\pi : \Omega _F \to G$ be a surjective homomorphism. Let $F_{\pi }$ be the corresponding G-Galois-algebra extension of F. (We note that since $\pi $ is surjective, $F_{\pi }$ is in fact a number field, and not merely a Galois algebra.) Write $\mathfrak {D}_\pi $ for the different of $F_{\pi }/F$ and $O_{\pi }$ for the ring of integers of $F_{\pi }$ . If $\mathfrak {P}$ is any prime of $O_{\pi }$ , the power $v_{\mathfrak {P}}(\mathfrak {D}_{\pi })$ of $\mathfrak {P}$ occurring in $\mathfrak {D}_\pi $ is given by

$$\begin{align*}v_{\mathfrak{P}}(\mathfrak{D}_\pi) = \sum_{i = 0}^{\infty} \left(|G^{(i)}_{\mathfrak{P}}| - 1\right), \end{align*}$$

where $G^{(i)}_{\mathfrak {P}}$ denotes the ith ramification group at $\mathfrak {P}$ (see [23, Chapter IV, Proposition 4]). This implies that if, for example, $|G|$ is odd, then the inverse different $\mathfrak {D}_{\pi }^{-1}$ has a square root, i.e., there exists a unique fractional ideal $A_\pi $ of $O_{\pi }$ such that

$$\begin{align*}A_{\pi}^{2} = \mathfrak{D}_{\pi}^{-1}. \end{align*}$$

(Let us remark at once that if $|G|$ is even, then $\mathfrak {D}_{\pi }^{-1}$ may well—but of course need not—also have a square root.)

Recall that $F_{\pi }/F$ is said to be weakly ramified if $G^{(2)}_{\mathfrak {P}} = 1$ for all prime ideals $\mathfrak {P}$ of $O_{\pi }$ . Erez has shown that $F_{\pi }/F$ is weakly ramified if and only if $A_\pi $ is a locally free $O_FG$ -module (see [10, Theorem 1]). Hence, if $F_{\pi }/F$ is weakly ramified, it follows that $A_\pi $ is a locally free ${\mathbf Z} G$ -module, and so defines an element $(A_\pi )$ in the locally free class group $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of $\mathbf {Z} G$ . The following result is due to Erez (see [10, Theorem 3]).

Theorem 1.1 Suppose that $F_{\pi }/F$ is tamely ramified and that $|G|$ is odd. Then $A_\pi $ is a free $\mathbf {Z} G$ -module.

Based on this and other results, Vinatier has made the following conjecture (cf. [30, Conjecture] and [4, Section 1.2]).

Conjecture 1.2 Suppose that $F_{\pi }/F$ is weakly ramified and that $|G|$ is odd. Then $A_\pi $ is a free $\mathbf {Z} G$ -module.

The first detailed study of the Galois structure of $A_\pi $ when $|G|$ is even is due to the third author and Vinatier [4]. By studying the Galois structure of certain torsion modules first considered by Chase [6], they proved the following result, and thereby were able to exhibit the first examples for which $(A_\pi ) \neq 0$ in $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ (see [4, Theorem 2]).

Theorem 1.3 Suppose that $F_{\pi }/F$ is tame and locally abelian (i.e., the decomposition group at every ramified prime of $F_{\pi }/F$ is abelian). Assume also that $A_\pi $ exists. Then $(A_\pi ) = (O_{\pi })$ in $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ .

A well-known theorem of M. Taylor [26] asserts that, if $F_\pi /F$ is tame, then

(1.1) $$ \begin{align} (O_\pi) = W(F_\pi/F), \end{align} $$

where $W(F_\pi /F)$ denotes the Cassou–Noguès–Fröhlich root number class, which is defined in terms of Artin root numbers attached to nontrivial irreducible symplectic characters of G. (In particular, if $|G|$ is odd, and so has no nontrivial irreducible symplectic characters, then $W(F_{\pi }/F) = 0$ .)

We therefore see that Theorem 1.3 may be viewed as saying that if $F_{\pi }/F$ is tame and locally abelian, and if $A_\pi $ exists, then we have

$$\begin{align*}(A_{\pi}) = (O_{\pi}) = W(F_{\pi}/F). \end{align*}$$

In light of the results described above, as well as those contained in [7], Erez has made the following (unpublished) conjecture.

Conjecture 1.4 Suppose that $F_{\pi }/F$ is weakly ramified and that $A_\pi $ exists. Then

$$\begin{align*}(A_\pi) = W(F_\pi/F). \end{align*}$$

Conjecture 1.4 includes Vinatier’s Conjecture 1.2 as a special case, and was the motivation for the work described in [4]. It also explains almost all previously obtained results on the $\mathbf {Z} G$ -structure of $A_\pi $ . In a different direction, the conjecture is related to the recent work of Bley, Hahn, and the second author [3] concerning metric structures arising from $A_\pi $ (for more details of which, see the Ph.D. thesis [17] of the fourth author).

In this paper, we show that, in general, Conjecture 1.4 fails for tame extensions. For each tame extension $F_\pi /F$ , we use the signs at infinity of certain symplectic Galois–Jacobi sums to define an element ${\mathcal J}^*_\infty (F_\pi /F) \in \operatorname {\mathrm {Cl}}({\mathbf Z} G)$ . The class ${\mathcal J}^*_\infty (F_\pi /F)$ is of order at most $2$ , and is often, but not always, equal to zero. We prove the following result.

Theorem 1.5 Suppose that $F_{\pi }/F$ is tame and that $A_\pi $ exists. Then

$$\begin{align*}(A_\pi) - (O_\pi) = {\mathcal J}^*_\infty(F_\pi/F), \end{align*}$$

i.e., (see (1.1))

(1.2) $$ \begin{align} (A_\pi) = W(F_\pi/F) + {\mathcal J}^*_\infty(F_\pi/F). \end{align} $$

Our proof of Theorem 1.5 combines methods from [1, 2] involving relative algebraic K-theory with the use of non-abelian Galois–Jacobi sums, the explicit computation by Fröhlich and Queyrut of the local root numbers of dihedral representations and a detailed representation-theoretic analysis of the failure (in the relevant cases) of induction functors to commute with Adams operators. In particular, it is interesting to compare our use of Galois–Jacobi sums with the methods of [4], where abelian Jacobi sums play a critical role.

Remark 1.6 It remains an open question as to whether (1.2) continues to hold if the tameness hypothesis is relaxed.

For any integer $m \geq 1$ , we write $H_{4m}$ for the generalized quaternion group of order $4m$ . The following result, which is obtained by combining Theorem 1.5 with the work of Fröhlich on root numbers (see [11]), gives infinitely many counterexamples to Conjecture 1.4.

Theorem 1.7 Let F be an imaginary quadratic field such that $\operatorname {\mathrm {Cl}}(O_F)$ contains an element of order $4$ . Then, for any sufficiently large prime $\ell $ with $\ell \equiv 3\ \pmod {4}$ , there are infinitely many tame, $H_{4\ell }$ -extensions $F_\pi /F$ such that $A_\pi $ exists and $(A_\pi ) \neq (O_\pi )$ in $\operatorname {\mathrm {Cl}}({\mathbf Z} H_{4\ell })$ .

An outline of the contents of this paper is as follows. In Section 2, we recall certain basic facts about relative algebraic K-theory from [1, 2]. In Section 3, we discuss how ideals in Galois algebras give rise to elements in certain relative K-groups. Section 4 contains a description of the Stickelberger factorization of certain tame resolvends (see [2, Section 7]) in the case of both rings of integers and square roots of inverse differents, while Section 5 develops properties of Stickelberger pairings and explains how these may be used to give explicit descriptions of the tame resolvends considered in the previous section. In Section 6, we recall a number of facts concerning Galois–Gauss sums. We define Galois–Jacobi sums, and we establish some of their basic properties. In Section 7, we compute the signs of local Galois–Jacobi sums at symplectic characters by combining an analysis of the behavior of Adams operators with respect to induction functors together with the theorem of Fröhlich and Queyrut. In Section 9, we prove Theorem 1.5. Finally, in Section 10, we prove Theorem 1.7.

Notation and conventions

For any field L, we write $L^c$ for an algebraic closure of L, and we set $\Omega _L:= \operatorname {\mathrm {Gal}}(L^c/L)$ . If L is a number field or a non-archimedean local field (by which we shall always mean a finite extension of ${\mathbf Q}_p$ for some prime p), then $O_L$ denotes the ring of integers of L. If L is an archimedean local field, then we adopt the usual convention of setting $O_L = L$ .

Throughout this paper, F will denote a number field. For each place v of F, we fix an embedding $F^c \to F_{v}^{c}$ , and we view $\Omega _{F_v}$ as being a subgroup of $\Omega _F$ via this choice of embedding. We write $I_v$ for the inertia subgroup of $\Omega _{F_v}$ when v is finite.

If H is any finite group, we write $\operatorname {\mathrm {Irr}}(H)$ for the set of irreducible $F^c$ -valued characters of H and $R_H$ for the corresponding ring of virtual characters. We write ${\mathbf 1}_H$ (or simply ${\mathbf 1}$ if there is no danger of confusion) for the trivial character in $R_H$ .

Let L be a number field or local field, and suppose that $\Gamma $ is any group on which $\Omega _L$ acts continuously. (We shall usually, but not always, be primarily concerned with the case of trivial $\Omega _L$ -action; see below for further remarks on this.) We identify $\Gamma $ -torsors over L (as well as their associated algebras, which are Hopf–Galois extensions associated with $A_{\Gamma }:= (L^c\Gamma )^{\Omega _{L}}$ ) with elements of the set $Z^1(\Omega _L, \Gamma )$ of $\Gamma $ -valued continuous $1$ -cocycles of $\Omega _L$ (see [24, I.5.2]). If $\pi \in Z^1(\Omega _L, \Gamma )$ , then we write $L_\pi /L$ for the corresponding Hopf–Galois extension of L, and $O_\pi $ for the integral closure of $O_L$ in $L_\pi $ . (Thus, $O_{\pi } = L_{\pi }$ if L is an archimedean local field.) Each such $L_{\pi }$ is a principal homogeneous space of the Hopf algebra $\operatorname {\mathrm {Map}}_{\Omega _L}(\Gamma , L^c)$ of $\Omega _L$ -equivariant maps from $\Gamma $ to $L^c$ . It may be shown that if $\pi _1, \pi _2 \in Z^1(\Omega _L,\Gamma )$ , then $L_{\pi _1} \simeq L_{\pi _2}$ if and only if $\pi _1$ and $\pi _2$ differ by a coboundary. The set of isomorphism classes of $\Gamma $ -torsors over L may be identified with the pointed cohomology set $H^1(L,\Gamma ):=H^1(\Omega _L,\Gamma )$ . We write $[\pi ] \in H^1(L,\Gamma )$ for the class of $L_{\pi }$ in $H^1(L,\Gamma )$ . If L is a number field or a non-archimedean local field, we write $H^1_t(L,\Gamma )$ for the subset of $H^1(L,\Gamma )$ consisting of those $[\pi ] \in H^1(L,\Gamma )$ for which $L_{\pi }/L$ is at most tamely ramified. If L is an archimedean local field, we set $H^1_t(L,\Gamma ) = H^1(L, \Gamma )$ . We denote the subset of $H^1_t(L,\Gamma )$ consisting of those $[\pi ] \in H^1_t(L,\Gamma )$ for which $L_{\pi }/L$ is unramified at all (including infinite) places of L by $H^{1}_{nr}(L,\Gamma )$ . (So, with this convention, if L is an archimedean local field, we have $H^{1}_{nr}(L, \Gamma ) = 0$ .)

We remark that if $\Omega _L$ acts trivially on $\Gamma $ , then we recover classical Galois theory: $\pi $ is a homomorphism, $L_{\pi }/L$ is simply an extension of $\Gamma $ -Galois algebras, and $L_{\pi }$ is a field if $\pi $ is surjective. For the most part, this is the only case that will be needed in this paper. There is, however, one important exception. This occurs in Section 4 when we describe a certain decomposition (a Stickelberger factorization) of resolvends of normal basis generators of tame local extensions. (This is a non-abelian analogue of Stickelberger’s factorization of abelian Gauss sums. See [2, Definition 7.2] for further remarks on this choice of terminology.)

If A is any algebra, we write $Z(A)$ for the center of A. If A is an R-algebra for some ring R, and $R \to R_1$ is an extension of R, we write $A_{R_1}:= A \otimes _{R} R_1$ to denote extension of scalars from R to $R_1$ .

2 Relative algebraic K-theory

The purpose of this section is briefly to recall a number of basic facts concerning relative algebraic K-theory that we shall need. For a more extensive discussion of these topics, the reader is strongly encouraged to consult [2, Section 5] as well as [1, Sections 2 and 3] and [25, Chapter 15].

Let R be a Dedekind domain with field of fractions L of characteristic zero, and suppose that G is a finite group upon which $\Omega _L$ acts trivially. Let ${\mathfrak A}$ be any finitely generated R-algebra satisfying ${\mathfrak A} \otimes _R L \simeq LG$ .

For any extension $\Lambda $ of R, we write $K_0({\mathfrak A}, \Lambda )$ for the relative algebraic K-group that arises via the extension of scalars afforded by the map $R \to \Lambda $ . Each element of $K_0({\mathfrak A}, \Lambda )$ is represented by a triple $[M, N;\xi ]$ , where M and N are finitely generated, projective ${\mathfrak A}$ -modules, and $\xi : M \otimes _{R} \Lambda \xrightarrow {\sim } N \otimes _R \Lambda $ is an isomorphism of ${\mathfrak A} \otimes _R \Lambda $ -modules.

Recall that there is a long exact sequence of relative algebraic K-theory (see [25, Theorem 15.5])

(2.1) $$ \begin{align} K_1({\mathfrak A}) \xrightarrow{\iota} K_1({\mathfrak A} \otimes_R \Lambda) \xrightarrow{\partial^{1}_{{\mathfrak A},\Lambda}} K_0({\mathfrak A}, \Lambda) \xrightarrow{\partial^{0}_{{\mathfrak A},\Lambda}} K_0({\mathfrak A}) \to K_0({\mathfrak A} \otimes_R \Lambda). \end{align} $$

The first and last arrows in this sequence are induced by the extension of scalars map $R \to \Lambda $ , whereas the map $\partial ^{0}_{{\mathfrak A}, \Lambda }$ sends the triple $[M, N;\xi ]$ to the element $[M] - [N] \in K_0({\mathfrak A})$ .

The map $\partial ^{1}_{{\mathfrak A}, \Lambda }$ is defined as follows. The group $K_1({\mathfrak A} \otimes _R \Lambda )$ is generated by elements of the form $(V,\phi )$ , where V is a finitely generated, free ${\mathfrak A} \otimes _R \Lambda $ -module, and $\phi : V \xrightarrow {\sim } V$ is an ${\mathfrak A} \otimes _R \Lambda $ -isomorphism. To define $\partial ^{1}_{{\mathfrak A}, \Lambda }((V,\phi ))$ , we choose any projective ${\mathfrak A}$ -submodule T of V such that $T \otimes _{{\mathfrak A}} \Lambda = V$ , and we set

$$\begin{align*}\partial^{1}_{{\mathfrak A}, \Lambda}((V,\phi)) := [T,T;\phi]. \end{align*}$$

It may be shown that this definition is independent of the choice of T.

Let $\operatorname {\mathrm {Cl}}({\mathfrak A})$ denote the locally free class group of ${\mathfrak A}$ . If $\Lambda $ is a field (as will in fact always be the case in this paper), then (2.1) yields an exact sequence

(2.2) $$ \begin{align} K_1({\mathfrak A}) \xrightarrow{\iota} K_1({\mathfrak A} \otimes_R \Lambda) \xrightarrow{\partial^{1}_{{\mathfrak A},\Lambda}} K_0({\mathfrak A}, \Lambda) \xrightarrow{\partial^{0}_{{\mathfrak A},\Lambda}} \operatorname{\mathrm{Cl}}({\mathfrak A}) \to 0, \end{align} $$

and this is the form of the long exact sequence of relative algebraic K-theory that we shall use in this paper.

We shall make heavy use of the fact that computations in relative K-groups and in locally free class groups may be carried out using functions on the characters of G. Suppose that L is either a number field or a local field, and write $R_G$ for the ring of virtual characters of G. The group $\Omega _L$ acts on $R_G$ via the rule given by

$$\begin{align*}\chi^{\omega}(g) = \omega(\chi(g)), \end{align*}$$

where $\omega \in \Omega _L$ , $\chi \in \operatorname {\mathrm {Irr}}(G)$ , and $g \in G$ . For each element $a \in (L^cG)^{\times }$ , we define $\operatorname {\mathrm {Det}}(a) \in \operatorname {\mathrm {Hom}}(R_G, (L^{c})^{\times })$ as follows. If T is any representation of G with character $\phi $ , then we set $\operatorname {\mathrm {Det}}(a)(\phi ):= \det (T(a))$ . It may be shown that this definition is independent of the choice of representation T, and so depends only on the character  $\phi $ .

The map $\operatorname {\mathrm {Det}}$ is essentially the same as the reduced norm map

(2.3) $$ \begin{align} \operatorname{\mathrm{nrd}}: (L^{c}G)^{\times} \to Z(L^cG)^{\times} \end{align} $$

(see [2, Remark 4.2]): (2.3) induces an isomorphism

(2.4) $$ \begin{align} \operatorname{\mathrm{nrd}}: K_1(L^cG) \xrightarrow{\sim} Z(L^cG)^{\times} \simeq \operatorname{\mathrm{Hom}}(R_G, (L^{c})^{\times}), \end{align} $$

and we have $\operatorname {\mathrm {Det}}(a)(\phi ) = \operatorname {\mathrm {nrd}}(a)(\phi )$ .

Suppose now that we are working over a number field F (i.e., $L = F$ above). We define the group of finite ideles $J_f(K_1(FG))$ to be the restricted direct product over all finite places v of F of the groups $\operatorname {\mathrm {Det}}(F_vG)^{\times } \simeq K_1(F_vG)$ with respect to the subgroups $\operatorname {\mathrm {Det}}(O_{F_v}G)^{\times }$ . (We shall require no use of the infinite places of F in the idelic descriptions given below. See, e.g., [9, pp. 226–228] for details concerning this point.)

For each finite place v of F, we write

$$\begin{align*}\operatorname{\mathrm{loc}}_v: \operatorname{\mathrm{Det}}(FG)^{\times} \to \operatorname{\mathrm{Det}}(F_vG)^{\times} \subseteq \operatorname{\mathrm{Hom}}_{\Omega_{F_v}}(R_G, (F_{v}^c)^{\times}) \end{align*}$$

for the obvious localisation map.

Let E be any extension of F. Then the homomorphism

$$\begin{align*}\operatorname{\mathrm{Det}}(FG)^{\times} \to J_{f}(K_1(FG)) \times \operatorname{\mathrm{Det}}(EG)^{\times};\quad x \mapsto ((\operatorname{\mathrm{loc}}_v(x))_v, x^{-1}) \end{align*}$$

induces a homomorphism

$$\begin{align*}\Delta_{{\mathfrak A},E}: \operatorname{\mathrm{Det}}(FG)^{\times} \to \frac{J_{f}(K_1(FG))}{\prod_{v \nmid \infty} \operatorname{\mathrm{Det}}({\mathfrak A}_v)^{\times}} \times \operatorname{\mathrm{Det}}(EG)^{\times}. \end{align*}$$

Theorem 2.1 (a) There is a natural isomorphism

$$\begin{align*}\operatorname{\mathrm{Cl}}({\mathfrak A}) \xrightarrow{\sim} \frac{J_{f}(K_1(FG))}{\operatorname{\mathrm{Det}}(FG)^{\times} \prod_{v \nmid \infty} \operatorname{\mathrm{Det}}({\mathfrak A}_v)^{\times}}. \end{align*}$$

(b) There is a natural isomorphism

$$\begin{align*}h_{{\mathfrak A},E}: K_0({\mathfrak A}, E) \xrightarrow{\sim} \operatorname{\mathrm{Coker}}(\Delta_{{\mathfrak A},E}). \end{align*}$$

(c) Let v be a finite place of F, and suppose that $L_v$ is any extension of $F_v$ . Then there are isomorphisms

$$\begin{align*}K_0({\mathfrak A}_v, L_v) \simeq K_1(L_vG)/\iota(K_1({\mathfrak A}_v)) \simeq \operatorname{\mathrm{Det}}(L_vG)^{\times}/ \operatorname{\mathrm{Det}}({\mathfrak A}_v)^{\times}. \end{align*}$$

Proof Part (a) is due to Fröhlich (see, e.g., [15, Chapter I] or [12]). Part (b) is proved in [1, Theorem 3.5], and a proof of part (c) is given in [2, Lemma 5.7].

Remark 2.2 Suppose that $x \in K_0({\mathfrak A}, E)$ is represented by the idele $[(x_{v})_{v}, x_{\infty }] \in J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(EG)^{\times }$ . Then $\partial ^0(x) \in \operatorname {\mathrm {Cl}}({\mathfrak A})$ is represented by the idele $(x_{v})_{v} \in J_{f}(K_1(FG))$ .

Remark 2.3 Suppose that $[M,N;\xi ] \in K_0(O_FG, E)$ and that M and N are locally free ${\mathfrak A}$ -modules of rank one. An explicit representative in $J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(EG)^{\times }$ of $h_{{\mathfrak A},E}([M,N;\xi ])$ may be constructed as follows.

For each finite place v of F, fix ${\mathfrak A}_v$ -bases $m_v$ of $M_v$ and $n_v$ of $N_v$ . Fix also an $FG$ -basis $n_{\infty }$ of $N_F$ , and choose an isomorphism $\theta : M_F \xrightarrow {\sim } N_F$ of $FG$ -modules.

The element $\theta ^{-1}(n_{\infty })$ is an $FG$ -basis of $M_F$ . Hence, for each place v, we may write

$$ \begin{align*} m_v &= \mu_v \cdot \theta^{-1}(n_{\infty}), \\ n_v &= \nu_v \cdot n_{\infty}, \end{align*} $$

where $\mu _v, \nu _v \in (F_vG)^{\times }$ .

If we write $\theta _E: M_E \xrightarrow {\sim } N_E$ for the isomorphism afforded by $\theta $ via extension of scalars, then we see that the isomorphism $\xi \circ \theta ^{-1}_{E}: N_E \xrightarrow {\sim } N_E$ is given by $n_{\infty } \mapsto \nu _{\infty } \cdot n_{\infty }$ for some $\nu _{\infty } \in (EG)^{\times }$ .

A representative of $h_{{\mathfrak A},E}([M,N;\xi ])$ is given by the image of $[(\mu _v \cdot \nu ^{-1}_{v})_{v}, \nu _{\infty }]$ in $J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(EG)^{\times }$ .

Remark 2.4 We see from Theorem 2.1(b) and (c) that there are isomorphisms

$$\begin{align*}K_0({\mathfrak A}, F) \simeq \frac{J_{f}(K_1(FG))}{\prod_{v \nmid \infty} \operatorname{\mathrm{Det}}({\mathfrak A}_{v})^{\times}} \simeq \frac{\operatorname{\mathrm{Hom}}_{\Omega_F}(R_G, J_{f}(F^c))}{\prod_{v \nmid \infty} \operatorname{\mathrm{Det}}({\mathfrak A}_{v})^{\times}} \simeq \oplus_{v \nmid \infty} K_{0}({\mathfrak A}_{v}, F_{v}). \end{align*}$$

There is a natural injection

$$\begin{align*} K_0({\mathfrak A}, F) &\to K_0({\mathfrak A}, F^c) \\ [M,N;\xi] &\mapsto [M,N;\xi_{F^{c}}], \end{align*}$$

where $\xi _{F^c}: M_{F^c} \xrightarrow {\sim } N_{F^c}$ is the isomorphism obtained from $\xi :M_F \xrightarrow {\sim } N_{F}$ via extension of scalars from F to $F^c$ . It is not hard to check that this map is induced by the inclusion map

$$\begin{align*} J_{f}(K_1(FG)) &\to J_{f}(K_1(FG)) \times \operatorname{\mathrm{Det}}(F^{c}G)^{\times} \\ (x_{v})_{v} &\mapsto [(x_{v})_{v}, 1]. \end{align*}$$

We now recall the description of the restriction of scalars map on relative K-groups and locally free class groups in terms of the isomorphism given by Theorem 2.1(b).

Suppose that ${\mathcal F}/F$ is a finite extension and that E is an extension of ${\mathcal F}$ . Then restriction of scalars from $O_{{\mathcal F}}$ to $O_F$ yields homomorphisms

$$\begin{align*}K_0({\mathfrak A}_{O_{{\mathcal F}}}, E) \to K_0({\mathfrak A}, E) \end{align*}$$

and

$$\begin{align*}\operatorname{\mathrm{Cl}}({\mathfrak A}_{O_{{\mathcal F}}}) \to \operatorname{\mathrm{Cl}}({\mathfrak A}), \end{align*}$$

which may be described as follows (see, e.g., [15, Chapter IV] or [27, Chapter 1]).

Let $\{\omega \}$ be any transversal of $\Omega _{{\mathcal F}} \backslash \Omega _F$ . Then the map

$$ \begin{align*} J_{f}(K_1({\mathcal F} G)) \times \operatorname{\mathrm{Det}}(EG)^{\times} &\to J_{f}(K_1(FG)) \times \operatorname{\mathrm{Det}}(EG)^{\times}\\ [(y_v)_v, y_{\infty}] &\mapsto \prod_{\omega} [(y_v)_v, y_{\infty}]^{\omega} \end{align*} $$

induces homomorphisms

(2.5) $$ \begin{align} {\mathcal N}_{{\mathcal F}/F}: K_0({\mathfrak A}_{O_{{\mathcal F}}}, E) \to K_0({\mathfrak A}, E) \end{align} $$

and

(2.6) $$ \begin{align} {\mathcal N}_{{\mathcal F}/F}: \operatorname{\mathrm{Cl}}({\mathfrak A}_{O_{{\mathcal F}}}) \to \operatorname{\mathrm{Cl}}({\mathfrak A}). \end{align} $$

These homomorphisms are independent of the choice of $\{\omega \}$ and are equal to the natural maps on relative K-groups (resp. locally free class groups) afforded by restriction of scalars from $O_{{\mathcal F}}$ to $O_F$ .

We conclude this section by recalling the definitions of certain induction maps on relative algebraic K-groups and on locally free class groups of group rings (see, e.g., [15, Chapter II] or [27, Chapter I]).

Suppose that G is a finite group and that H is a subgroup of G. Let E be an algebraic extension of F. Then extension of scalars from $O_FH$ to $O_FG$ yields natural homomorphisms

(2.7) $$ \begin{align} \operatorname{\mathrm{Ind}}^G_H: K_0(O_FH, E) \to K_0(O_FG, E) \end{align} $$

and

(2.8) $$ \begin{align} \operatorname{\mathrm{Ind}}^G_H:\operatorname{\mathrm{Cl}}(O_FH) \to \operatorname{\mathrm{Cl}}(O_FG). \end{align} $$

It may be shown that these homomorphisms are induced (via the isomorphisms described in Theorem 2.1) by the maps

$$ \begin{align*} &\operatorname{\mathrm{Ind}}^G_H: \operatorname{\mathrm{Hom}}(R_H, J(F^c)) \to \operatorname{\mathrm{Hom}}(R_G, J(F^c)), \\ &\operatorname{\mathrm{Ind}}^G_H:\operatorname{\mathrm{Hom}}(R_H, (F^{c})^{\times}) \to \operatorname{\mathrm{Hom}}(R_G, (F^{c})^{\times}) \end{align*} $$

given by

(2.9) $$ \begin{align} (\operatorname{\mathrm{Ind}}^G_Hf)(\chi) = f(\chi \mid_H), \quad \chi \in R_G. \end{align} $$

It is not hard to check from the definitions that the following diagram commutes:

(2.10)

3 Galois algebras and ideals

Let L be either a number field or a local field, and suppose that $\pi \in Z^1(\Omega _L, G)$ is a continuous G-valued $\Omega _L$ $1$ -cocycle. We may define an associated G-Galois L-algebra $L_{\pi }$ by

$$\begin{align*}L_{\pi} := \operatorname{\mathrm{Map}} _{\Omega_L}(^{\pi}G, L^c), \end{align*}$$

where $^{\pi }G$ denotes the set G endowed with an action of $\Omega _L$ via the cocycle $\pi $ (i.e., $g^{\omega } = \pi (\omega ) \cdot g$ for $g \in {^{\pi }G}$ and $\omega \in \Omega _L$ ), and $L_{\pi }$ is the algebra of $L^c$ -valued functions on $^{\pi }G$ that are fixed under the action of $\Omega _L$ . The group G acts on $L_{\pi }$ via the rule

$$\begin{align*}a^g(h) = a(h \cdot g) \end{align*}$$

for all $g \in G$ and $h \in {^{\pi }G}$ .

The Wedderburn decomposition of the algebra $L_{\pi }$ may be described as follows. Set

$$\begin{align*}L^{\pi} := (L^{c})^{\operatorname{\mathrm{Ker}}(\pi)}, \end{align*}$$

so $\operatorname {\mathrm {Gal}}(L^{\pi }/L) \simeq \pi (\Omega _L)$ . Then

(3.1) $$ \begin{align} L_{\pi} \simeq \prod_{\pi(\Omega_L) \backslash G} L^{\pi}, \end{align} $$

and this isomorphism depends only on the choice of a transversal of $\pi (\Omega _L)$ in G. It may be shown that every G-Galois L-algebra is of the form $L_{\pi }$ for some $\pi $ and that $L_{\pi }$ is determined up to isomorphism by the class $[\pi ]$ of $\pi $ in the pointed cohomology set $H^1(L, G)$ . In particular, every Galois algebra may be viewed as being a subalgebra of the $L^c$ -algebra $\operatorname {\mathrm {Map}}(G, L^c)$ .

Definition 3.1 The resolvend map ${\mathbf r}_G$ on $\operatorname {\mathrm {Map}}(G, L^c)$ is defined as

$$ \begin{align*} {\mathbf r}_G: \operatorname{\mathrm{Map}}(G, L^c) &\rightarrow L^cG \\ a &\mapsto \sum_{g \in G} a(g) \cdot g^{-1}. \end{align*} $$

(This is an isomorphism of $L^cG$ -modules, but it is not an isomorphism of $L^c$ -algebras because it does not preserve multiplication.)

Suppose now that $L_{\pi }/L$ is a G-extension and that ${\mathcal L} \subseteq L_{\pi }$ is a nonzero projective $O_LG$ -module. Then there are isomorphisms

$$\begin{align*}\operatorname{\mathrm{Map}}(G, L^c) \simeq {\mathcal L} \otimes_{O_L} L^c, \quad L^cG \simeq O_LG \otimes_{O_L} L^c, \end{align*}$$

and so the triple $[{\mathcal L}, O_LG; {\mathbf r}_G]$ yields an element of $K_0(O_LG, L^c)$ .

Proposition 3.2 Let $F_{\pi }/F$ be a G-extension of a number field F, and suppose that ${\mathcal L}_i \subseteq F_{\pi }$ ( $i =1,2$ ) are nonzero projective $O_FG$ -modules. For each place v of F, choose a basis $l_{i,v}$ of ${\mathcal L}_{i,v}$ over $O_{F_v}G$ , as well as a basis $l_{\infty }$ of $F_{\pi }$ over $FG$ .

1. (a) The element $[{\mathcal L}_i, O_FG; {\mathbf r}_G] \in K_0(O_FG, F^c)$ is represented by the image of the idele $[({\mathbf r}_G(l_{i,v}) \cdot {\mathbf r}_G(l_{\infty })^{-1})_v, {\mathbf r}_G(l_{\infty })^{-1}] \in J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(F^cG)^{\times }$ .

2. (b) The element

   $$\begin{align*}[{\mathcal L}_1, O_FG; {\mathbf r}_G] - [{\mathcal L}_2, O_FG; {\mathbf r}_G] \in K_0(O_FG, F^c) \end{align*}$$
   is represented by the image of the idele
   $$\begin{align*}[({\mathbf r}_G(l_{1,v}) \cdot {\mathbf r}_G(l_{2,v}^{-1}))_v, 1] \in J_{f}(K_1(FG)) \times \operatorname{\mathrm{Det}}(F^cG)^{\times}. \end{align*}$$

3. (c) We have that

   $$\begin{align*}[{\mathcal L}_1, O_FG; {\mathbf r}_G] - [{\mathcal L}_2, O_FG; {\mathbf r}_G] \in K_0(O_FG, F) \subseteq K_0(O_FG, F^c). \end{align*}$$

Proof For each finite place v of F, write

$$\begin{align*}l_{i,v} = x_{i,v} \cdot l_{\infty}, \end{align*}$$

with $x_{i,v} \in (F_vG)^{\times }$ . Then it follows from Remark 2.3 that $[{\mathcal L}_i, O_FG; {\mathbf r}_G] \in K_0(O_FG, F^c)$ is represented by the image of the idele $[(x_{i,v})_{v}, {\mathbf r}_G(l_{\infty })^{-1}] \in J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(F^cG)^{\times }$ . However,

$$\begin{align*}x_{i,v} = {\mathbf r}_G(l_{i,v}) \cdot {\mathbf r}_G(l_{\infty})^{-1} \end{align*}$$

(because the resolvend map is an isomorphism of $F^cG$ and $F^{c}_{v}G$ -modules), and this implies (a). Part (b) now follows directly from (a).

To show part (c), we first recall that

$$\begin{align*}K_0(O_FG, F) \simeq \oplus_{v \nmid \infty} K_0(O_{F_v}G, F_v) \simeq \oplus_{v \nmid \infty} \operatorname{\mathrm{Det}}(F_vG)^{\times}/\operatorname{\mathrm{Det}}(O_{F_v}G)^{\times} \end{align*}$$

and that an element $c \in K_0(O_FG, F^c)$ lies in $K_0(O_FG, F)$ if it has an idelic representative lying in $J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(FG)^{\times } \subseteq J_{f}(K_1(FG)) \times \operatorname {\mathrm {Det}}(F^cG)^{\times }$ (see Remark 2.4).

Now, a standard property of resolvends implies that

$$\begin{align*}{\mathbf r}_G(l_{i,v})^{\omega} = {\mathbf r}_G(l_{i,v}) \cdot \pi(\omega) \end{align*}$$

for every $\omega \in \Omega _{F_v}$ (see, e.g., [2, 2.2]), and so we see that $({\mathbf r}_G(l_{1,v}) \cdot {\mathbf r}_G(l_{2,v}^{-1}))_v \in (F_vG)^{\times }$ for each v. (In fact, as we may take $l_{1,v} = l_{2,v}$ for almost all v, we may suppose that $({\mathbf r}_G(l_{1,v}) \cdot {\mathbf r}_G(l_{2,v}^{-1}))_v =1$ for almost all v.) Hence, it now follows from (b) that $[{\mathcal L}_1, O_FG; F^c] - [{\mathcal L}_2, O_FG; F^c] \in K_0(O_FG, F) $ , as claimed.

It is a classical result, due to E. Noether, that a G-extension $F_{\pi }/F$ is tamely ramified if and only if $O_{\pi }$ is a locally free (and therefore projective) $O_FG$ -module. Ullom has shown that if $F_{\pi }/F$ is tame, then in fact all G-stable ideals in $O_{\pi }$ are locally free. He also showed that if any G-stable ideal B, say, in a G-extension $F_{\pi }/F$ is locally free, then all second ramification groups at primes dividing B are equal to zero (see [29]). If $F_{\pi }/F$ is any G-extension for which $|G|$ is odd (and so the square root $A_\pi $ of the inverse different automatically exists), then Erez has shown that $A_\pi $ is a locally free $O_FG$ -module if and only if all second ramification groups associated with $F_{\pi }/F$ vanish, i.e., if and only if $F_{\pi }/F$ is weakly ramified. In fact, as pointed out by the third author and Vinatier [4, p. 109, footnote 1], the proof of [10, Theorem 1] shows that if $F_{\pi }/F$ is any weakly ramified extension such that $A_{\pi }$ exists, then $A_{\pi }$ is locally free.

Definition 3.3 Suppose that $[\pi ] \in H^{1}_{t}(F,G)$ and that $A_\pi $ exists. We define

$$\begin{align*}{\mathfrak c} = {\mathfrak c}(\pi) := [A_\pi, O_FG;{\mathbf r}_G] - [O_{\pi}, O_FG;{\mathbf r}_G] \in K_0(O_FG, F) \subseteq K_0(O_FG, F^c). \end{align*}$$

4 Local decomposition of tame resolvends

Our goal in this section is to recall certain facts from [2, Section 7] concerning Stickelberger factorizations of resolvends of normal integral basis generators of tame local extensions, and to describe similar results concerning resolvends of basis generators of the square root of the inverse different (when this exists). Roughly speaking, the underlying idea is that any tame Galois extension of local fields arises as the compositum of an unramified field extension with a totally ramified Hopf–Galois extension (which, in particular, need not be normal).

Let L be a local field, and fix a uniformizer ${\varpi } = {\varpi }_L$ of L. Set $q:= |O_L/{\varpi }_L O_L|$ .

Fix also a compatible set of roots of unity $\{ \zeta _m \}$ , and a compatible set $\{ {\varpi }^{1/m} \}$ of roots of ${\varpi }$ . (Hence, if m and n are any two positive integers, then we have $(\zeta _{mn})^m = \zeta _n$ , and $({\varpi }^{1/mn})^{m} = {\varpi }^{1/n}$ .)

Let $L^{nr}$ (resp. $L^{t}$ ) denote the maximal unramified (resp. tamely ramified) extension of L. Then

$$\begin{align*}L^{nr}= \bigcup_{\stackrel{m \geq 1}{(m,q)=1}} L(\zeta_m),\quad L^t = \bigcup_{\stackrel{m \geq 1}{(m,q)=1}} L(\zeta_m, {\varpi}^{1/m}). \end{align*}$$

The group $\Omega ^{nr}:= \operatorname {\mathrm {Gal}}(L^{nr}/L)$ is topologically generated by a Frobenius element $\phi \in \operatorname {\mathrm {Gal}}(L^t/L)$ which may be chosen to satisfy

$$\begin{align*}\phi(\zeta_m) = \zeta_{m}^{q}, \qquad \phi({\varpi}^{1/m}) = {\varpi}^{1/m} \end{align*}$$

for each integer m coprime to q. Our choice of compatible roots of unity also uniquely specifies a topological generator $\sigma $ of $\Omega ^r := \operatorname {\mathrm {Gal}}(L^t/L^{nr})$ by the conditions

$$\begin{align*}\sigma({\varpi}^{1/m}) = \zeta_m \cdot {\varpi}^{1/m}, \qquad \sigma(\zeta_m) = \zeta_m \end{align*}$$

for all integers m coprime to q. The group $\Omega ^{t}:=\operatorname {\mathrm {Gal}}(L^{t}/L)$ is topologically generated by $\phi $ and $\sigma $ , subject to the relation

(4.1) $$ \begin{align} \phi \cdot \sigma \cdot \phi^{-1} = \sigma^{q}. \end{align} $$

The reader may find it helpful to keep in mind the following explicit example, due to Tsang (cf. [28, Proposition 4.2.2]), while reading the next two sections.

Example 4.1 (Tsang)

Suppose that L contains the eth roots of unity with $(e,q)=1$ , and set $M := L({\varpi }_{L}^{1/e})$ . Write ${\varpi }_M := {\varpi }_{L}^{1/e}$ , then ${\varpi }_M$ is a uniformizer of M. Set $H := \operatorname {\mathrm {Gal}}(M/L) = \langle s \rangle $ , say.

Let n be an integer with $0 \leq |n| \leq e-1$ , and let us consider the ideal

$$\begin{align*}{\varpi}^{n}_{M} O_M = {\varpi}_{L}^{n/e}O_M \end{align*}$$

as an $O_LH$ -module. Set

$$\begin{align*}\alpha = \frac{1}{e}\sum_{i=0}^{e-1} {\varpi}_{M}^{n+i} = \frac{1}{e}\sum_{i=0}^{e-1} {\varpi}_{L}^{(n+i)/e}. \end{align*}$$

We wish to explain why

$$\begin{align*}O_LH \cdot \alpha = {\varpi}_{M}^{n} \cdot O_{M}, \end{align*}$$

and to give some motivation for the definition of the Stickelberger pairings in Definition 5.1.

Suppose that $s({\varpi }_M) = \zeta \cdot {\varpi }_M$ , where $\zeta $ is a primitive eth root of unity. Then, for each $0 \leq j \leq e-1$ , we have

$$\begin{align*}s^j(\alpha) = \frac{1}{e}\sum_{i=0}^{e-1}\zeta^{(i+n)j} {\varpi}_M^{i+n}. \end{align*}$$

Multiplying both sides of this last equality by $\zeta ^{-(l+n)j}$ , where $0 \leq l \leq e-1$ , gives

$$\begin{align*}s^j(\alpha) \zeta^{-(l+n)j} = \frac{1}{e} \sum^{e-1}_{i=0} \zeta^{(i-l)j} {\varpi}_{M}^{i+n}. \end{align*}$$

Now, sum over j to obtain

$$\begin{align*}\sum_{j=0}^{e-1} s^j(\alpha) \zeta^{-(l+n)j} = \frac{1}{e} \sum_{i=0}^{n} {\varpi}_{M}^{i+n} \sum_{j=0}^{e-1} \zeta^{(i-l)j} = {\varpi}_{M}^{l+n}. \end{align*}$$

So, if for any $\chi \in \operatorname {\mathrm {Irr}}(H)$ , we choose the unique integer $(\chi ,s)_{H,n}$ in the set

$$\begin{align*}\{l+n \mid 0 \leq l \leq e-1 \} \end{align*}$$

such that $\chi (s) = \zeta ^{(\chi ,s)_{H,n}}$ , then we see that

(4.2) $$ \begin{align} \operatorname{\mathrm{Det}}({\mathbf r}_H(\alpha))(\chi) = \sum_{j=0}^{e-1} s^j(\alpha) \zeta^{-(l+n)j} = {\varpi}_{M}^{(\chi,s)_{H,n}}. \end{align} $$

The cases $n =0$ and $n = (1-e)/2$ (for e odd) correspond to the ring of integers and the square root of the inverse different, respectively, and we see the appearance of the relevant Stickelberger pairing (see Definition 5.1) in each case.

It follows from (4.2) that

$$\begin{align*}B_n := \{ {\varpi}_{M}^{l+n} : 0 \leq l \leq e-1 \} \subseteq O_{L}H \cdot \alpha. \end{align*}$$

As $B_n$ is an $O_L$ -basis of the ideal ${\varpi }_{M}^{n} \cdot O_{M}$ , and as $\zeta _{e} \in O_L$ , we see that

$$\begin{align*}O_LH \cdot \alpha = {\varpi}^{n}_{M} \cdot O_{M}, \end{align*}$$

i.e., $\alpha $ is a free generator of ${\varpi }^{n}_{M} \cdot O_M$ as an $O_{L}H$ -module.

Definition 4.2 If $g \in G$ , we set

$$\begin{align*}\beta_g := \frac{1}{|g|} \sum_{i=0}^{|g|-1} {\varpi}^{i/|g|}; \end{align*}$$

note that $\beta _g$ depends only on $|g|$ , and so in particular we have

$$\begin{align*}\beta_{g} = \beta_{\gamma^{-1}g\gamma} \end{align*}$$

for every $\gamma \in G$ . We define ${\varphi }_{g} \in \operatorname {\mathrm {Map}}(G, L^{c})$ by setting

$$\begin{align*}{\varphi}_{g}(\gamma) = \begin{cases} \sigma^{i}(\beta_g), &\text{if }\gamma = g^i, \\ 0, &\text{if }\gamma \notin \langle g \rangle. \end{cases} \end{align*}$$

Then

(4.3) $$ \begin{align} {\mathbf r}_G({\varphi}_{g}) = \sum_{i=0}^{|g|-1} {\varphi}_{g}(g^i) g^{-i} = \sum_{i=0}^{|g|-1} \sigma^{i}(\beta_g) g^{-i}. \end{align} $$

Suppose now that $\pi \in Z^1(\Omega _L, G)$ , with $[\pi ] \in H^1_t(L,G)$ . Write $s:= \pi (\sigma )$ and $t:= \pi (\phi )$ . We define, $\pi _{r}, \pi _{nr} \in \operatorname {\mathrm {Map}}(\Omega ^t, G)$ by setting

(4.4) $$ \begin{align} &\pi_r(\sigma^m \phi^n) = \pi(\sigma^m) = s^m , \end{align} $$

(4.5) $$ \begin{align} &\pi_{nr}(\sigma^m \phi^n) = \pi(\phi^n) = t^n , \end{align} $$

so that

$$\begin{align*}\pi = \pi_{r} \cdot \pi_{nr}. \end{align*}$$

It may be shown that in fact $\pi _{nr} \in \operatorname {\mathrm {Hom}}(\Omega ^{nr}, G)$ , and so corresponds to a unramified G-extension $L_{\pi _{nr}}$ of L. It may also be shown that $\pi _{r} |_{\Omega _{r}}\in \operatorname {\mathrm {Hom}}(\Omega ^r, G)$ , corresponding to a totally (tamely) ramified extension $L^{nr}_{\pi _{r}}/L^{nr}$ . If we write $[{\widetilde {\pi }}]$ for the image of $[\pi ]$ under the natural restriction map $H^1(L, G) \to H^1(L^{nr},G)$ , then $[{\widetilde {\pi }}] = [\pi _{r}]$ . The element ${\varphi }_s$ is a normal integral basis generator of the extension $L^{nr}_{\pi _{r}}/L^{nr}$ . (See [2, Section 7] for proofs of these assertions.) If in addition $|s|$ is odd, then the inverse different of $L_{\pi }/L$ has a square root $A_\pi $ , and

$$\begin{align*}A_\pi = {\varpi}^{(1-|s|)/2|s|} \cdot O_{\pi}. \end{align*}$$

We can now state the Stickelberger factorization theorem for resolvends of normal integral bases.

Theorem 4.3 If $a_{nr} \in L_{\pi _{nr}}$ is any normal integral basis generator of $L_{\pi _{nr}}/L$ , then the element $a \in L_{\pi }$ defined by

(4.6) $$ \begin{align} {\mathbf r}_G(a_{nr}) \cdot {\mathbf r}_G({\varphi}_s) = {\mathbf r}_G(a) \end{align} $$

is a normal integral basis generator of $L_{\pi }/L$ .

Proof See [2, Theorem 7.9].

We shall now describe a similar result (due to Tsang when G is abelian) concerning $O_LG$ -generators of the square root of the inverse different of a tame extension of L.

Definition 4.4 Suppose that $g \in G$ and that $|g|$ is odd. Set

$$\begin{align*}\beta^{*}_{g} = \frac{1}{|g|} \sum_{i=0}^{|g|-1} {\varpi}^{\frac{1}{|g|}(i + \frac{1-|g|}{2})}. \end{align*}$$

Define ${\varphi }^{*}_{g} \in \operatorname {\mathrm {Map}}(G, L^{c})$ by

$$\begin{align*}{\varphi}^{*}_{g}(\gamma) = \begin{cases} \sigma^{i}(\beta^{*}_{g}), &\text{if }\gamma = g^i, \\ 0, &\text{if }\gamma \notin \langle g \rangle. \end{cases} \end{align*}$$

Then

(4.7) $$ \begin{align} {\mathbf r}_G({\varphi}^{*}_{g}) = \sum_{i=0}^{|g|-1} {\varphi}_{g}(g^i) g^{-i} = \sum_{i=0}^{|g|-1} \sigma^{i}(\beta^{*}_{g}) g^{-i}. \end{align} $$

Theorem 4.5 (Cf. [2, Theorem 7.9])

If $a_{nr}$ is any choice of n.i.b. generator of $L_{\pi _{nr}}/L$ , then the element b of $L_{\pi }$ defined by

(4.8) $$ \begin{align} {\mathbf r}_G(b) = {\mathbf r}_G(a_{nr}) \cdot {\mathbf r}_G({\varphi}^{*}_{s}) \end{align} $$

satisfies $A_\pi = O_{L}G \cdot b$ .

Proof To ease notation, set $N:= L^{nr}$ and $H := \langle s \rangle $ .

Write $[{\widetilde {\pi }}] \in H^1(N, G)$ for the image of $[\pi ] \in H^1(L, G)$ under the restriction map $H^1(L, G) \to H^1(N, G)$ . Then $A_{{\widetilde {\pi }}} = O_{N}\cdot A_\pi $ because $N/L$ is unramified. Hence, to establish the desired result, it suffices to show that

(4.9) $$ \begin{align} A_{{\widetilde{\pi}}} = O_{N}G \cdot b. \end{align} $$

As ${\mathbf r}_G(a_{nr}) \in (O_{N}G)^{\times }$ , (4.9) is equivalent to the equality

(4.10) $$ \begin{align} A_{{\widetilde{\pi}}} = O_{N}G \cdot {\varphi}^{*}_{s}. \end{align} $$

Now,

(4.11) $$ \begin{align} N_{{\widetilde{\pi}}} \simeq \prod_{H \backslash G} N^{{\widetilde{\pi}}}, \end{align} $$

where $N^{{\widetilde {\pi }}} = N({\varpi }^{1/|s|})$ (cf. (3.1)), and this isomorphism induces a decomposition

(4.12) $$ \begin{align} A_{{\widetilde{\pi}}} = \prod_{H \backslash G} A^{{\widetilde{\pi}}}, \end{align} $$

where

$$\begin{align*}A^{{\widetilde{\pi}}} = A(N^{{\widetilde{\pi}}}) = {\varpi}^{(1-|s|)/2|s|} \cdot O_{N} \end{align*}$$

is the square root of the inverse different of the extension $N^{{\widetilde {\pi }}}/N$ .

It therefore follows from the definition of ${\varphi }^{*}_{s}$ that (4.10) holds if and only if

(4.13) $$ \begin{align} A^{{\widetilde{\pi}}} = O_{N}H \cdot \beta^{*}_{s}. \end{align} $$

This last equality follows exactly as in [28, Proposition 4.2.2], and a proof is given by taking $n = (1-e)/2$ (for e odd) in Example 4.1.

Proposition 4.6 Suppose that $[\pi ] \in H^{1}_{t}(L,G)$ and that $s := \pi (\sigma )$ is of odd order. Then the class

$$\begin{align*}{\mathfrak c}(\pi) := [A_\pi, O_LG;{\mathbf r}_G] - [O_{\pi}, O_LG;{\mathbf r}_G] \in K_0(O_LG, L) \simeq \operatorname{\mathrm{Det}}(LG)^{\times}/\operatorname{\mathrm{Det}}(O_LG)^{\times} \end{align*}$$

is represented by $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }^{*}_{s})) \cdot \operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }_{s}))^{-1} \in \operatorname {\mathrm {Det}}(LG)^{\times }$ .

Proof This is a direct consequence of Theorems 4.3 and 4.5, together with the proof of Proposition 3.2(c).

5 Stickelberger pairings and resolvends

Our goal in this section is to describe explicitly the elements $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }_s))$ and $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }^{*}_{s}))$ constructed in the previous section. We begin by recalling the definition of two Stickelberger pairings. The first of these is due to McCulloh, whereas the second is due to Tsang in the case of abelian G. See [2, Definition 9.1] and [28, Definition 2.5.1].

Definition 5.1 Let $\zeta = \zeta _{|G|}$ be a fixed, primitive, $|G|$ th root of unity. Suppose first that G is cyclic. For $g \in G$ and $\chi \in \operatorname {\mathrm {Irr}}(G)$ , write $\chi (g) = \zeta ^r$ for some integer r.

1. (1) We define

   $$\begin{align*}\langle \chi, g \rangle_G = \{r/|G|\}, \end{align*}$$
   where $ 0 \leq \{r/|G|\} <1$ denotes the fractional part of $r/|G|$ .

   Alternatively (cf. Example 4.1, but note that there we worked with the primitive eth root of unity $\zeta _{e}$ , where e is the exponent of G), if we choose r to be the unique integer in the set $\{l : 0 \leq l \leq |G|-1\}$ such that $\chi (g) = \zeta ^r$ , then

   $$\begin{align*}\langle \chi, g \rangle_G = r/|G|. \end{align*}$$

2. (2) Suppose that $|G|$ is odd, and choose $r \in [(1-|G|)/2, (|G|-1)/2]$ to be the unique integer such that $\chi (g) = \zeta ^r$ . Define

   $$\begin{align*}\langle \chi, g \rangle^*_G = r/|G|. \end{align*}$$

   We extend each of these to pairings

   $$\begin{align*}\mathbf{Q} R_G \times \mathbf{Q} G \to \mathbf{Q} \end{align*}$$
   via linearity. Finally, we extend the definitions to arbitrary finite groups G by setting
   $$\begin{align*}\langle \chi, s \rangle _G := \langle \chi \mid_{\langle s \rangle}, s \rangle_{\langle s \rangle} \end{align*}$$
   and
   $$\begin{align*}\langle \chi, s \rangle ^*_G := \langle \chi \mid_{\langle s \rangle}, s \rangle^{*}_{\langle s \rangle}, \end{align*}$$
   where the second definition of course only makes sense when the order $|s|$ of s is odd.

We shall make use of the following alternative descriptions of the above Stickelberger pairing using the standard inner product on $R_G$ (see [2, Proposition 9.2]). For each element $s \in G$ , write $\zeta _{|s|} = \zeta _{|G|}^{|G|/|s|}$ , and define a character $\xi _s$ of $\langle s \rangle $ by $\xi _{s}(s^i) = \zeta ^{i}_{|s|}$ . Set

$$\begin{align*}\Xi_s := \frac{1}{|s|} \sum_{j=1}^{|s|-1} j \xi^{j}_{s}. \end{align*}$$

For $|s|$ odd, we also define

$$\begin{align*}\Xi^{*}_{s} := \frac{1}{|s|} \sum_{j=1}^{(|s|-1)/2} j(\xi^{j}_{s} - \xi^{-j}_{s}). \end{align*}$$

Let $(-,-)_G$ denote the standard inner product on $R_G$ .

Proposition 5.2

1. (a) If $s \in G$ and $\chi \in R_G$ , we have

   $$\begin{align*}\langle \chi, s \rangle_G = (\operatorname{\mathrm{Ind}}^{G}_{\langle s \rangle} (\Xi_{s}), \chi)_G. \end{align*}$$

2. (b) If furthermore $|s|$ is odd, then we have

   $$\begin{align*}\langle \chi, s \rangle^*_G = (\operatorname{\mathrm{Ind}}^{G}_{\langle s \rangle} (\Xi^{*}_{s}), \chi)_G. \end{align*}$$

3. (c) If $|s|$ is odd, then

   $$\begin{align*}\Xi^{*}_{s} - \Xi_s = -\sum_{j = 1}^{(|s|-1)/2} \xi^{-j}_{s}. \end{align*}$$

4. (d) For s odd, write

   $$\begin{align*}d(s) := -\sum_{j = 1}^{(|s|-1)/2} \xi^{-j}_{s}. \end{align*}$$
   Then, for each $\chi \in R_G$ , we have
   $$\begin{align*}\langle \chi, s \rangle^{*}_{G} - \langle \chi, s \rangle_G = (\operatorname{\mathrm{Ind}}^{G}_{\langle s \rangle} (d(s)), \chi)_G. \end{align*}$$

Proof Part (a) is proved in [2, Proposition 9.2]. The proof of (b) is the same mutatis mutandis. Part (c) follows directly from the definitions of $\Xi _s$ and $\Xi ^{*}_{s}$ , and then (d) follows from (a) and (b).

We may use Proposition 5.2 to describe the relationship between the two Stickelberger pairings in Definition 5.1 when $|s|$ is odd.

In the sequel, for any finite group $\Gamma $ (which will be clear from context), and any natural number k, we write $\psi _k$ for the kth Adams operator on $R_{\Gamma }$ . Thus, if $\chi \in R_{\Gamma }$ and $\gamma \in \Gamma $ , then one has $\psi _k(\chi )(\gamma ) = \chi (\gamma ^k)$ . In particular, we recall that, for all k, $\psi _{k}$ commutes with the restriction and inflation functors, as well as with the action of $\Omega _{{\mathbf Q}}$ on $R_\Gamma $ (see [10, Proposition–Definition 3.5]). If L is a number field or a local field, we also write $\psi _k$ for the homomorphism

$$\begin{align*}\operatorname{\mathrm{Hom}}(R_{\Gamma}, (L^{c})^{\times}) \to \operatorname{\mathrm{Hom}}(R_{\Gamma}, (L^c)^{\times}) \end{align*}$$

defined by setting

$$\begin{align*}\psi_k(f)(\chi) = f(\psi_k(\chi)) \end{align*}$$

for $f \in \operatorname {\mathrm {Hom}}(R_{\Gamma }, (L^{c})^{\times })$ and $\chi \in R_{\Gamma }$ .

Proposition 5.3 Suppose that $s \in G$ is of odd order, and set $H:= \langle s \rangle $ .

1. (a) If $1 \leq j \leq |s|-1$ , then

   $$ \begin{align*} (\Xi^{*}_{s}, \xi^j )_H &= (\Xi_{s}, \xi^{2j} - \xi^j )_H \\ &= (\Xi_{s}, \psi_2(\xi^{j}) - \xi^j)_H. \end{align*} $$

2. (b) (Tsang) For each $\chi \in R_G$ , we have

   $$\begin{align*}\langle \chi, s\rangle^*_G = \langle \psi_2(\chi) - \chi, s \rangle_G. \end{align*}$$

Proof (a) If $1 \leq j \leq |s|/2$ , then we have

$$\begin{align*}(\Xi_{s}, \xi_{s}^{2j} - \xi_{s}^{j})_H = \frac{2j-j}{|s|} = \frac{j}{|s|}, \end{align*}$$

whereas if $|s|/2 \leq j \leq s-1$ , then

$$\begin{align*}(\Xi_{s}, \xi_{s}^{2j} - \xi_{s}^{j} )_H = \frac{(2j - |s|) -j}{|s|} = \frac{j - |s|}{|s|}. \end{align*}$$

Thus, in each case, we have

$$\begin{align*}(\Xi^{*}_{s}, \xi_{s}^{j} )_H = (\Xi_{s}, \xi_{s}^{2j} - \xi_{s}^{j} )_H, \end{align*}$$

and this establishes the claim.

(b) Proposition 5.2(b), together with Frobenius reciprocity, gives

$$ \begin{align*} \langle \chi, s \rangle^*_G &= (\operatorname{\mathrm{Ind}}^{G}_{\langle s \rangle} (\Xi^{*}_{s}), \chi)_G \\ &= (\Xi^{*}_{s}, \chi \mid_H)_H. \end{align*} $$

The desired result now follows from part (a), together with the fact that, for any $\chi \in R_G$ , we have the equality

$$\begin{align*}\psi_2(\chi) \mid_{H} = \psi_2(\chi \mid_H).\\[-35pt] \end{align*}$$

Remark 5.4 Proposition 5.3(b) (due to Tsang) shows very clearly why the second Adams operator $\psi _2$ appears when one studies the Galois structure of the square root of the inverse different as opposed to the ring of integers. This appearance of the second Adams operator was first observed by Erez (see [10, Proposition–Definition 3.5 and Theorem 3.6]) in the initial work on this topic.

The following result describes the elements $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }_{s}))$ and $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }^{*}_{s}))$ in terms of Stickelberger pairings. In what follows, we retain the notation and conventions of Section 4.

Proposition 5.5 Suppose that $\chi \in R_G$ and $s \in G$ .

1. (a) We have

   $$\begin{align*}\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_{s}))(\chi) = {\varpi}^{\langle \chi, s \rangle_G}. \end{align*}$$

2. (b) If $|s|$ is odd, then we have

   $$\begin{align*}\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}^{*}_{s}))(\chi) = {\varpi}^{\langle \chi, s \rangle^{*}_{G}}. \end{align*}$$

3. (c) For $|s|$ odd, we have

   $$ \begin{align*} [\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}^{*}_{s})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_s))^{-1}] (\chi) &= {\varpi}^{\langle \chi, s \rangle^{*}_{G} -\langle \chi, s \rangle_{G}}\\ &={\varpi}^{\langle \psi_{2}(\chi) - 2\chi, s \rangle_{G}}\\ &= \frac{\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_{s}))(\psi_2(\chi))}{\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_{s}))(2\chi)}. \end{align*} $$
   That is to say,
   $$\begin{align*}\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}^{*}_{s})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_s))^{-1} = \psi_2(\operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_{s}))) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_G({\varphi}_{s}))^{-2}. \end{align*}$$

Proof Part (a) is proved in [2, Proposition 10.5(a)]. The proof of (b) is very similar, using [28, Proposition 4.2.2], which in fact shows the result for G abelian. Part (c) follows from parts (a) and (b), and Proposition 5.3.

Corollary 5.6 Suppose that $[\pi ] \in H^1_t(L,G)$ and that $s:= \pi (\sigma )$ is of odd order. Then a representing homomorphism for the class

$$\begin{align*}{\mathfrak c}(\pi) = [A_\pi, O_LG;{\mathbf r}_G] - [O_{\pi}, O_LG; {\mathbf r}_G] \end{align*}$$

in

$$\begin{align*}K_0(O_LG, L) \simeq \frac{\operatorname{\mathrm{Det}}(LG)^{\times}}{\operatorname{\mathrm{Det}}(O_LG)^{\times}} \simeq \frac{\operatorname{\mathrm{Hom}}_{\Omega_{L}}(R_G, (L^{c})^{\times})}{\operatorname{\mathrm{Det}}(O_LG)^{\times}} \end{align*}$$

is the map $f_{\pi } \in \operatorname {\mathrm {Hom}}_{\Omega _{L}}(R_G, (L^{c})^{\times })$ given by

$$\begin{align*}f_{\pi}(\chi) = {\varpi}^{\langle \psi_{2}(\chi) - 2\chi, s \rangle_{G}}. \end{align*}$$

Proof This follows from Propositions 4.6 and 5.5(c).

6 Galois–Gauss and Galois–Jacobi sums

Let L be a local field of residual characteristic p. Suppose that $[\pi ] \in H^1_t(L, G)$ , and recall that we have (see (3.1))

$$\begin{align*}L_{\pi} \simeq \prod_{\pi(\Omega_L)\backslash G} L^{\pi}. \end{align*}$$

Set $H := \pi (\Omega _L) = \operatorname {\mathrm {Gal}}(L^{\pi }/L)$ , and write $\tau ^{*}(L^{\pi }/L,\, -) \in \operatorname {\mathrm {Hom}}(R_{H}, ({\mathbf Q}^c)^{\times })$ for the adjusted Galois–Gauss sum homomorphism associated with $L^{\pi }/L$ (see [14, Chapter IV, equation (1.7)]). Recall that this is defined by

$$\begin{align*}\tau^{*}(L^{\pi}/L,\, -) := \tau(L^{\pi}/L,\, -) \cdot y(L^{\pi}/L, -)^{-1} \cdot z(L^{\pi}/L, -), \end{align*}$$

where $\tau (L^{\pi }/L,\, -)$ denotes the Galois–Gauss sum homomorphism and $y(L^{\pi }/L, -)$ and $z(L^{\pi }/L, -)$ are homomorphisms taking values in roots of unity in ${\mathbf Q}^c$ . We define $\tau ^{*}(L_{\pi }/L,\, -) \in \operatorname {\mathrm {Hom}}(R_{G}, ({\mathbf Q}^c)^{\times })$ by composing $\tau ^{*}(L^{\pi }/L,\, -)$ with the natural map $R_G \to R_H$ .

For a finite group $\Gamma $ , we write $\operatorname {\mathrm {Irr}}_p(\Gamma )$ for the set of ${\mathbf Q}_{p}^{c}$ -valued irreducible characters of $\Gamma $ and $R_{\Gamma , p}$ for the free abelian group on $\operatorname {\mathrm {Irr}}_p(\Gamma )$ . We fix a local embedding $\operatorname {\mathrm {Loc}}_p: {\mathbf Q}^c \to {\mathbf Q}^c_p$ , and we shall identify $\operatorname {\mathrm {Irr}}(\Gamma )$ with $\operatorname {\mathrm {Irr}}_p(\Gamma )$ via this choice of embedding.

For each rational prime $l \neq p$ , we fix a semilocal embedding $\operatorname {\mathrm {Loc}}_l: {\mathbf Q}^c \to ({\mathbf Q}^c)_l := {\mathbf Q}^c \otimes _{{\mathbf Q}} {\mathbf Q}_{l}$ . (Caveat: note that this is not the same thing as a local embedding ${\mathbf Q}^c \to {\mathbf Q}^c_l$ !) For each rational prime l, write ${\mathbf Q}^t_l$ for the maximal, tamely ramified extension of ${\mathbf Q}_l$ .

We shall require the following results. (We remind the reader that the definition of the Adams operators $\psi _k$ was recalled just prior to the statement of Proposition 5.3.)

Proposition 6.1 Fix a rational prime l.

1. (a) Let K be an unramified extension of ${\mathbf Q}_{l}$ . Then, for any integer k, we have that

   $$\begin{align*}\psi_k(\operatorname{\mathrm{Det}}(O_{K}G)^{\times}) \subseteq \operatorname{\mathrm{Det}}(O_{K}G)^{\times}. \end{align*}$$

2. (b) Let $\Gamma $ be a finite group with abelian p-Sylow subgroups. Then, for any integer k,

   $$\begin{align*}\psi_k(\operatorname{\mathrm{Det}}(O_{{\mathbf Q}_{p}^{t}} \Gamma)^{\times}) \subseteq \operatorname{\mathrm{Det}}(O_{{\mathbf Q}_{p}^{t}} \Gamma)^{\times}. \end{align*}$$

3. (c) Suppose that $l \neq p$ . Then

   $$\begin{align*}\operatorname{\mathrm{Loc}}_l(\tau^{*}(L_{\pi}/L,\, -)) \in \operatorname{\mathrm{Det}}(O_{{\mathbf Q}(\mu_p),l} G)^{\times}. \end{align*}$$

Proof Parts (a) and (b) are results of Cassou–Noguès and Taylor. For part (a), see, e.g., [27, Chapter 9, Theorem 1.2], and note that for this particular result, we do not need to assume that $(k, |G|) = 1$ . For part (b), see [5, p. 83, Remark].

Part (c) follows from [14, Chapter IV, Theorem 30], where analogous results are proved for $\tau ^{*}(L^{\pi }/L,\,-)$ ; the corresponding results for $\tau ^{*}(L_{\pi }/L,\,-)$ are then a direct consequence of the definition of $\tau ^{*}(L_{\pi }/L,\,-)$ .

The following result is entirely analogous to [14, Chapter IV, Lemma 2.1]. Recall that if $f \in \operatorname {\mathrm {Hom}}(R_{\Gamma }, ({\mathbf Q}^{c}_{p})^{\times })$ , then $\omega \in \Omega _{{\mathbf Q}_{p}}$ acts on f by the rule

$$\begin{align*}f^{\omega}(\chi) = f(\chi^{\omega^{-1}})^{\omega}. \end{align*}$$

Lemma 6.2 Let $L/{\mathbf Q}_p$ be a finite extension, and let $\{\nu \}$ be any right transversal of $\Omega _L$ in $\Omega _{{\mathbf Q}_{p}}$ . Suppose that $f \in \operatorname {\mathrm {Hom}}_{\Omega _{L^{\operatorname {\mathrm {nr}}}}}(R_{\Gamma }, ({\mathbf Q}_{p}^{c})^{\times })$ . Then (cf. (2.5) and (2.6)):

$$\begin{align*}{\mathcal N}_{L/{\mathbf Q}_{p}} f := \prod_{\nu} f^{\nu} \in \operatorname{\mathrm{Hom}}_{\Omega_{{\mathbf Q}_{p}^{\operatorname{\mathrm{nr}}}}}(R_{\Gamma}, ({\mathbf Q}_{p}^{c})^{\times}). \end{align*}$$

Proof It suffices to show that this result holds with respect to a particular choice of transversal of $\Omega _L$ in $\Omega _{{\mathbf Q}_{p}}$ .

We first observe that, as $\Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ is normal in $\Omega _{{\mathbf Q}_{p}}$ , $\Omega _L \cdot \Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ is a subgroup of $\Omega _{{\mathbf Q}_{p}}$ . We choose a right transversal $\{\omega \}$ of $\Omega _L \cdot \Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ in $\Omega _{{\mathbf Q}_{p}}$ .

Next, we choose a right transversal $\{\sigma \}$ of $\Omega _L \cap \Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ in $\Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ . It follows that $\{\sigma \}$ is also a right transversal of $\Omega _L$ in $\Omega _L \cdot \Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ . We now deduce that $\{\sigma \omega \}$ is a right transversal of $\Omega _L$ in $\Omega _{{\mathbf Q}_{p}}$ . We also note that

$$\begin{align*}\Omega_{L} \cap \Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}} = \Omega_{L^{\operatorname{\mathrm{nr}}}} \cap \Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}} \end{align*}$$

and that (since $\Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ is normal in $\Omega _{{\mathbf Q}_{p}}$ )

$$\begin{align*}\omega^{-1}_{i} (\Omega_{L^{\operatorname{\mathrm{nr}}}} \cap \Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}}) \omega_i = \omega^{-1}_{i} \Omega_{L^{\operatorname{\mathrm{nr}}}} \omega_i \cap \Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}} \end{align*}$$

for any $\omega _i \in \{\omega \}$ .

Now, suppose that $f \in \operatorname {\mathrm {Hom}}_{\Omega _{L^{\operatorname {\mathrm {nr}}}}}(R_{\Gamma }, ({\mathbf Q}_{p}^{c})^{\times })$ and that $\omega _i \in \{\omega \}$ . Then

$$\begin{align*}f^{\omega_{i}} \in \operatorname{\mathrm{Hom}}_{\omega^{-1}_{i}\Omega_{L^{\operatorname{\mathrm{nr}}}} \omega_i}(R_{\Gamma}, ({\mathbf Q}_{p}^{c})^{\times}), \end{align*}$$

and so

$$\begin{align*}f^{\omega_{i}} \in \operatorname{\mathrm{Hom}}_{(\omega^{-1}_{i}\Omega_{L^{\operatorname{\mathrm{nr}}}} \omega_{i}) \cap \Omega_{{\mathbf Q}^{nr}_{p}}}(R_{\Gamma}, ({\mathbf Q}_{p}^{c})^{\times}). \end{align*}$$

Now, observe that for fixed $\omega _i \in \{\omega \}$ , $\{\omega ^{-1}_{i} \sigma \omega _i\}_{\sigma }$ is a right transversal of $\omega ^{-1}_{i}\Omega _{L^{\operatorname {\mathrm {nr}}}} \omega _i \cap \Omega _{{\mathbf Q}^{nr}_{p}}$ in $\Omega _{{\mathbf Q}^{\operatorname {\mathrm {nr}}}_{p}}$ , and so

$$\begin{align*}\prod_{\sigma} (f^{\omega_{i}})^{\omega^{-1}_{i} \sigma \omega_{i}} \in \operatorname{\mathrm{Hom}}_{\Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}}}(R_{\Gamma}, ({\mathbf Q}^{c}_{p})^{\times}). \end{align*}$$

Hence, finally, we obtain

$$\begin{align*}\prod_{\omega, \sigma} (f^{\omega})^{\omega^{-1}\sigma \omega} = \prod_{\omega, \sigma} f^{\sigma \omega} \in \operatorname{\mathrm{Hom}}_{\Omega_{{\mathbf Q}^{\operatorname{\mathrm{nr}}}_{p}}}(R_{\Gamma}, ({\mathbf Q}^{c}_{p})^{\times}), \end{align*}$$

as required.

Proposition 6.3 Let $a_{\pi }$ be any n.i.b. generator of $L_{\pi }/L$ . Suppose also that the square root $A_\pi $ of the inverse different of $L_{\pi }/L$ exists (i.e., that $s := \pi (\sigma )$ is of odd order) and that $A_\pi = O_L G \cdot b_{\pi }$ . Then:

1. (a) $ {\mathcal N}_{L/{\mathbf Q}_{p}}[ \operatorname {\mathrm {Det}}({\mathbf r}_G(b_{\pi }))^{-1} \cdot \psi _2(\operatorname {\mathrm {Det}}({\mathbf r}_G(a_{\pi }))) \cdot \operatorname {\mathrm {Det}}({\mathbf r}_G(a_{\pi }))^{-1}] \in \operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}}G)^{\times }. $

2. (b)

   1. (i) $\operatorname {\mathrm {Loc}}_p[(\tau ^{*}(L_{\pi }/L,\, -))]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_{p}}[\operatorname {\mathrm {Det}}({\mathbf r}_{G}(a_{\pi })) ] \in \operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}} G)^{\times }$ .

   2. (ii) $\operatorname {\mathrm {Loc}}_p[\psi _2(\tau ^{*}(L_{\pi }/L,\,-))]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_{p}}[\psi _2(\operatorname {\mathrm {Det}}({\mathbf r}_{G}(a_{\pi })))]\in \operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}} G)^{\times }$ .

3. (c) $\operatorname {\mathrm {Loc}}_p[\psi _2(\tau ^{*}(L_{\pi }/L,\, -)) \cdot (\tau ^{*}(L_{\pi }/L,\, -))^{-1}]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname {\mathrm {Det}}({\mathbf r}_{G}(b_{\pi }))] \in \operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}} G)^{\times }.$

4. (d) $ \operatorname {\mathrm {Loc}}_p[\psi _2(\tau ^{*}(L_{\pi }/L,\,-)) \cdot (\tau ^{*}(L_{\pi }/L,\, -))^{-2}]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname {\mathrm {Det}}({\mathbf r}_{G}(b_{\pi })) \cdot \operatorname {\mathrm {Det}}({\mathbf r}_{G}(a_{\pi }))^{-1}] $ belongs to $\operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}} G)^{\times }$ .

5. (e) With the notation of Proposition 4.6, the element

   $$\begin{align*}\operatorname{\mathrm{Loc}}_p[\psi_2(\tau^{*}(L_{\pi}/L,\,-)) \cdot (\tau^{*}(L_{\pi}/L,\, -))^{-2}]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname{\mathrm{Det}}({\mathbf r}_{G}({\varphi}^{*}_{s})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_{G}({\varphi}_{s}))^{-1}] \end{align*}$$

   belongs to $\operatorname {\mathrm {Det}}(O_{{\mathbf Q}^{t}_{p}} G)^{\times }$ .

Proof (a) Recall from [2, Definition 7.12] that for any n.i.b. generator $a_{\pi }$ of $L_{\pi }/L$ , one has

$$\begin{align*}{\mathbf r}_{G}(a_{\pi}) = u\cdot {\mathbf r}_{G}(a_{nr}) \cdot {\mathbf r}_{G}({\varphi}_{s}), \end{align*}$$

where $u \in (O_LG)^{\times }$ and ${\mathbf r}_{G}(a_{nr})\in (O_{L^{nr}}G)^{\times }$ . Furthermore, $u\cdot a_{nr}$ is also an n.i.b generator of $L_{\pi _{nr}}/L$ .

Hence,

$$\begin{align*}\operatorname{\mathrm{Det}}({\mathbf r}_G(a_{\pi}) \cdot {\mathbf r}_G({\varphi}_{s})^{-1}) = \operatorname{\mathrm{Det}}(u \cdot a_{\operatorname{\mathrm{nr}}}) \in \operatorname{\mathrm{Det}}(O_{L^{\operatorname{\mathrm{nr}}}}G)^{\times}, \end{align*}$$

and Lemma 6.2 implies that also

$$\begin{align*}{\mathcal N}_{L/{\mathbf Q}_{p}}[\operatorname{\mathrm{Det}}({\mathbf r}_G(a_{\pi}) \cdot {\mathbf r}_G({\varphi}_{s})^{-1})] \in \operatorname{\mathrm{Det}}(O_{{\mathbf Q}_{p}^{nr}}G)^\times. \end{align*}$$

It now follows from Proposition 6.1 that the product

(6.1) $$ \begin{align} {\mathcal N}_{L/{\mathbf Q}_{p}}[ (\operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi})) \cdot \operatorname{\mathrm{Det}}( {\mathbf r}_{G}({\varphi}_{s}))^{-1})^{-1} \cdot \psi_{2} \bigl(\operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi})) \cdot \operatorname{\mathrm{Det}}( {\mathbf r}_{G}({\varphi}_{s}))^{-1}\bigr) ] \end{align} $$

belongs to $\operatorname {\mathrm {Det}}(O_{{\mathbf Q}_{p}^{nr}}G)^\times $ .

Part (a) now follows from (6.1), together with Proposition 5.5(c) and the Stickelberger factorization of ${\mathbf r}_G(b_{\pi })$ (see Theorem 4.5).

(b) Let $O^\pi $ denote the integral closure of $O_L$ in $L^\pi $ and fix an element $\alpha \in L^{\pi }$ such that $O^{\pi } = O_LH \cdot \alpha $ . It follows from [14, Chapter IV, Theorem 31] that there exists an element $w\in (O_{{\mathbf Q}_{p}^{t}}H)^\times $ such that

(6.2) $$ \begin{align} \mathrm{Loc}_{p}(\tau^{\ast} (L^{\pi}/L, - ))^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_{p}} \operatorname{\mathrm{Det}}({\mathbf r}_{H}(\alpha)) = \operatorname{\mathrm{Det}}(w). \end{align} $$

Under our hypotheses, the inertia subgroup of H is cyclic of order $|s|$ coprime to p. Hence, Proposition 6.1(b) implies that

(6.3) $$ \begin{align} \mathrm{Loc}_p[\psi_2(\tau^{*}(L^{\pi}/L,\,-))]^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_{p}}[\psi_2( \operatorname{\mathrm{Det}}({\mathbf r}_{H}(\alpha)) )] \end{align} $$

belongs to $\psi _{2}(\operatorname {\mathrm {Det}}(O_{{\mathbf Q}_{p}^{t}}H)^\times ) \subseteq \operatorname {\mathrm {Det}}(O_{{\mathbf Q}_{p}^{t}}H)^\times \subseteq \operatorname {\mathrm {Det}}(O_{{\mathbf Q}_{p}^{t}}G)^\times $ .

Next, we construct a map $a_{\pi } \in \mathrm {Map}(G, L^c)$ associated with $\alpha $ by setting

$$\begin{align*}a_{\pi}(\gamma) := \left\{ \begin{array}{ll} \tilde{\gamma}(\alpha), & \quad \mbox{if } \quad \gamma=\pi(\tilde{\gamma}) \text{ for } \tilde{\gamma} \in \Omega_{L}, \\ 0, & \quad \mbox{otherwise}.\end{array} \right. \end{align*}$$

It is easy to see from (3.1) that $a_{\pi } \in L_\pi $ and satisfies that $O_{\pi } = O_LG \cdot a$ . In particular, for each $\chi \in R_{G}$ , we have

$$\begin{align*}\operatorname{\mathrm{Det}}_\chi( \mathbf{r}_G(a_{\pi})) = \operatorname{\mathrm{Det}}_{\chi} \bigl( \sum_{\gamma \in G} a_{\pi}(\gamma)\gamma^{-1} \bigr) = \operatorname{\mathrm{Det}}_{\chi} \bigl( \sum_{\gamma\in H} \tilde{\gamma}(\alpha)\gamma^{-1} \bigr) = \operatorname{\mathrm{Det}}_{\mathrm{res} \chi}({\mathbf r}_{H}(\alpha)), \end{align*}$$

with $\mathrm {res}:= \mathrm {res}^{G}_H: R_+{G} \to R_{H}$ . This implies that

(6.4) $$ \begin{align} \begin{array}{rl} {\mathcal N}_{L/{\mathbf Q}_{p}}[\operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi}))] =\, & {\mathcal N}_{L/{\mathbf Q}_{p}}[\operatorname{\mathrm{Det}}({\mathbf r}_{H}(\alpha))] ,\\ {\mathcal N}_{L/{\mathbf Q}_{p}}[\psi_2(\operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi})))] =\, & {\mathcal N}_{L/{\mathbf Q}_{p}}[\psi_2(\operatorname{\mathrm{Det}}({\mathbf r}_{H}(\alpha)))] .\\ \end{array} \end{align} $$

We now see from the definition of $\tau ^{\ast }(L_{\pi }/L, - )$ that (i) follows from (6.2) and (6.4), whereas part (ii) is a consequence of (6.3) and (6.4).

(c) Follows from (a) and (b) above.

(d) Follows from (b)(i) together with (c).

(e) Follows from (d) above.

Proposition 6.3(d) and (e) motivates the following definition.

Definition 6.4 We retain the notation established above. Define the adjusted Galois–Jacobi sum homomorphism associated with  $L_{\pi }/L$ , $J^*(L_{\pi }/L,\,-) \in \operatorname {\mathrm {Hom}}(R_{G}, ({\mathbf Q}^c)^{\times })$ , by

$$\begin{align*}J^*(L_{\pi}/L,\,-) := \psi_2(\tau^*(L_{\pi}/L,\,-)) \cdot (\tau^*(L_{\pi}/L,\,-))^{-2}. \end{align*}$$

It follows from the Galois action formulae for Galois–Gauss sums (see [14, pp. 119 and 152]) that in fact $J^*(L_{\pi }/L,\,-) \in \operatorname {\mathrm {Hom}}_{\Omega _{{\mathbf Q}}}(R_{\Gamma }, ({\mathbf Q}^c)^{\times })$ .

Remark 6.5 Let $\tau (L^{\pi }/L,\,-) \in \operatorname {\mathrm {Hom}}(R_{H}, ({\mathbf Q}^{c})^{\times })$ denote the (unadjusted) Galois–Gauss sum associated with $L^{\pi }/L$ , and write $\tau (L_{\pi }/L,\,-) \in \operatorname {\mathrm {Hom}}(R_{G}, ({\mathbf Q}^{c})^{\times })$ for the composition of $\tau (L^{\pi }/L,\,-)$ with the natural map $R_G \to R_H$ . We remark that the Galois–Jacobi sum $J(L_{\pi }/L,\, -) \in \operatorname {\mathrm {Hom}}(R_{G}, ({\mathbf Q}^c)^{\times })$ defined by

$$\begin{align*}J(L_{\pi}/L,\, -) := \psi_2(\tau(L_{\pi}/L,\, -)) \cdot (\tau(L_{\pi}/L,\, -))^{-2} \end{align*}$$

is a special case of the non-abelian Jacobi sums first introduced by Fröhlich (see [13]).

Proposition 6.6

1. (a) Suppose that $l \neq p$ . Then

   $$\begin{align*}\operatorname{\mathrm{Loc}}_l(J^*(L_{\pi}/L, -)) \in \operatorname{\mathrm{Det}}({\mathbf Z}_l G^{\times}). \end{align*}$$

2. (b) Using the notation of Proposition 6.3, we have

   $$\begin{align*}\operatorname{\mathrm{Loc}}_p(J^*(L_{\pi}/L,\, -))^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname{\mathrm{Det}}({\mathbf r}_{G}(b_{\pi})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi}))^{-1}] \in \operatorname{\mathrm{Det}}({\mathbf Z}_p G^{\times}). \end{align*}$$
   Hence,
   $$\begin{align*}\operatorname{\mathrm{Loc}}_p(J^*(L_{\pi}/L,\, -))^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname{\mathrm{Det}}({\mathbf r}_{G}({\varphi}^{*}_{s})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_{G}({\varphi}_s))^{-1}] \in \operatorname{\mathrm{Det}}({\mathbf Z}_p G^{\times}). \end{align*}$$

Proof (a) Recall that $J^*(L_{\pi }/L,\, -) \in \operatorname {\mathrm {Hom}}_{\Omega _{{\mathbf Q}}}(R_{G}, ({\mathbf Q}^c)^{\times })$ and that ${\mathbf Q}(\mu _p)/{\mathbf Q}$ is unramified at l. It therefore follows from Proposition 6.1(a) and (c), together with Taylor’s fixed point theorem for determinants (see [27, Chapter 8, Theorem 1.2]), that

$$\begin{align*}\operatorname{\mathrm{Loc}}_l(J^*(L_{\pi}/L,\, -)) \in [\operatorname{\mathrm{Det}}(O_{{\mathbf Q}_{l}(\mu_p)} G^{\times})]^{\Omega_{{\mathbf Q}_{l}}} = \operatorname{\mathrm{Det}} ({\mathbf Z}_l G^{\times}), \end{align*}$$

as claimed.

(b) As both of the functions $\operatorname {\mathrm {Loc}}_p(J^*(L_{\pi }/L,\,-))$ and ${\mathcal N}_{L/{\mathbf Q}_p}[\operatorname {\mathrm {Det}}({\mathbf r}_{G}(b_{\pi })) \cdot \operatorname {\mathrm {Det}}({\mathbf r}_{G}(a_{\pi }))^{-1}]$ lie in $\operatorname {\mathrm {Hom}}_{\Omega _{{\mathbf Q}_{p}}}(R_{G}, ({\mathbf Q}^{c}_{p})^{\times })$ , we see from Proposition 6.3(d) that

$$\begin{align*}&\operatorname{\mathrm{Loc}}_p(J^*(L_{\pi}/L,\, -))^{-1} \cdot {\mathcal N}_{L/{\mathbf Q}_p}[\operatorname{\mathrm{Det}}({\mathbf r}_{G}(b_{\pi})) \cdot \operatorname{\mathrm{Det}}({\mathbf r}_{G}(a_{\pi}))^{-1}] \\&\quad\in [\operatorname{\mathrm{Det}}(O_{{\mathbf Q}^{t}_{p}} G^{\times})]^{\Omega_{{\mathbf Q}_{p}}} = \operatorname{\mathrm{Det}}({\mathbf Z}_p G^{\times}). \end{align*}$$

The final assertion now follows at once from the Stickelberger factorizations of ${\mathbf r}_G(a_\pi )$ and ${\mathbf r}_G(b_\pi )$ (see Theorems 4.3 and 4.5).

7 Symplectic Galois–Jacobi sums I

In this section, we fix data $L, G$ , and $\pi $ as in Section 6. We write $\mathrm {Symp}(G)$ for the set of irreducible symplectic characters of G. For each $\chi \in \operatorname {\mathrm {Irr}}(G)$ , we write $\tau (L_\pi /L, \chi )$ for the associated (unadjusted) Galois–Gauss sum, and

$$\begin{align*}J(L_{\pi}/L,\, -) := \psi_2(\tau(L_{\pi}/L,\, -)) \cdot (\tau(L_{\pi}/L,\, -))^{-2} \end{align*}$$

for the (unadjusted) Galois–Jacobi sum (see Remark 6.5).

We shall prove the following result concerning symplectic Galois–Jacobi sums.

Theorem 7.1 Suppose that $\chi \in \operatorname {\mathrm {Symp}}(G)$ . Then $J(L_\pi /L, \chi )$ is a strictly positive real number.

We see from the decomposition (3.1) that it is enough to prove this result after replacing the Galois algebra $L_\pi $ by the field $L^{\pi }$ and the group G by the Galois group $\pi (\Omega _L) = \operatorname {\mathrm {Gal}}(L^{\pi }/L)$ . In the sequel, we shall therefore restrict to the case where $L_\pi /L$ is a finite Galois extension of p-adic fields and G is its Galois group.

To prove Theorem 7.1, it is therefore enough to show that for each $\chi $ in $\mathrm {Symp}(G)$ , the quotient $\tau (L, \psi _{2}(\chi ))/\tau (L, 2\chi )$ is a strictly positive real number.

To verify this, we recall that since each such $\chi $ is real-valued, the definition of the local root number $W(L,\chi )$ implies that

$$\begin{align*}\tau(L, \chi) = W(L, \chi) \cdot \mathbf{N}_{L}\mathfrak{f}(L_\pi/L, \chi)^{1/2}\end{align*}$$

(cf. [18, Chapter II, Section 4, Definition]). Hence, since $\mathbf {N}_{L}\mathfrak {f}(L_\pi /L, \chi )^{1/2}>0$ , it is enough to prove the following result.

Theorem 7.2 Let $E/F$ be a tamely ramified Galois extension of non-archimedean local fields that has odd ramification degree and set $G := \operatorname {\mathrm {Gal}}(E/F)$ . Then, for each $\chi $ in $\mathrm {Symp}(G)$ , one has $W(F,\psi _2(\chi )) = W(F,2\chi ) = 1$ .

This sort of result is, in principle, hard to prove both because root numbers of symplectic characters are difficult to compute and because Adams operators do not in general commute with induction functors. We therefore prove two preliminary results that help address these problems.

The first of these results is entirely representation-theoretic in nature.

In the sequel, for any finite group $\Gamma $ and character $\phi $ in $R_\Gamma $ , we write $\mathrm {Tr}(\phi )$ for the real-valued character $\phi + \overline {\phi }$ .

Lemma 7.3 Let $\Delta $ be a subgroup of a finite group $\Gamma $ , fix a character $\phi $ of $\Delta $ , and consider the virtual character

$$\begin{align*}\mathrm{I}_{\Gamma}^2(\phi) := \psi_2(\mathrm{Ind}_\Delta^{\Gamma}(\phi)) - \mathrm{Ind}_\Delta^{\Gamma}(\psi_2(\phi)).\end{align*}$$

For elements $\gamma $ and $\delta $ of $\Gamma $ , we set $\gamma ^\delta := \delta \gamma \delta ^{-1}$ .

1. (a) Let $\mathcal {T}$ be a set of coset representatives of $\Delta $ in $\Gamma $ . Then, for every $\gamma \in \Gamma $ , one has

   $$ \begin{align*} (\mathrm{I}_{\Gamma}^2(\phi))(\gamma) = \sum_{\tau}\phi((\gamma^\tau)^2), \end{align*} $$
   where the sum runs over all $\tau \in \mathcal {T}$ for which $(\gamma ^\tau )^2 \in \Delta $ and $\gamma ^\tau \notin \Delta $ .

2. (b) If $\Delta $ is a subnormal subgroup of $\Gamma $ of odd index, then $\mathrm {I}_{\Gamma }^2(\phi ) = 0$ .

3. (c) Assume that $\Gamma $ is a semidirect product of a supersolvable group by an abelian normal subgroup $\Upsilon $ .

   1. (i) Then, for every irreducible character $\mu $ of $\Gamma $ , there exists a subgroup $\Upsilon '$ of $\Gamma $ that contains $\Upsilon $ and a linear character $\lambda $ of $\Upsilon '$ such that $\mu = \mathrm { Ind}_{\Upsilon '}^\Gamma (\lambda )$ .

   In addition, if $\Upsilon \subseteq \Delta $ , the index of $\Delta $ in $\Gamma $ is a power of $2$ and $\Gamma $ has cyclic Sylow $2$ -subgroups, then the following claims are also valid.

   1. (ii) If $\phi $ is real-valued, then $\mathrm {I}_{\Gamma }^2(\phi )$ is an integral linear combination of characters of the form $\mathrm {Ind}_{\Delta '}^\Gamma \lambda $ and $\mathrm {Tr}(\phi ')$ , where $\Delta '$ runs over subgroups of $\Gamma $ that contain $\Delta $ , $\lambda $ over homomorphisms $\Delta ' \to \{\pm 1\}$ and $\phi '$ over elements of $R_\Gamma $ .

   2. (iii) If $\phi $ is induced from a proper normal subgroup of $\Delta $ of $2$ -power index that contains $\Upsilon $ , then $\mathrm {I}_{\Gamma }^2(\phi )=0$ .

4. (d) Assume that $\Gamma $ is generalized quaternion, $\Delta $ is the cyclic subgroup of $\Gamma $ of index $2$ , and $\phi $ is irreducible (and hence linear). Then $\phi ^2$ is trivial on the center Z of $\Gamma $ and

   $$\begin{align*}\psi_2\bigl(\mathrm{Ind}_\Delta^\Gamma\phi\bigr) = \mathrm{Inf}_{\Gamma/Z}^\Gamma\bigl(\mathrm{Ind}_{\Delta/Z}^{\Gamma/Z}(\phi^2)\bigr) + \mathrm{Inf}_{\Gamma/\Delta}^\Gamma(\chi_{\Gamma/\Delta}) - \mathbf{1}_\Gamma,\end{align*}$$
   where we regard $\phi ^2$ as a character of $\Delta /Z$ and write $\chi _{\Gamma /\Delta }$ for the unique nontrivial homomorphism $\Gamma /\Delta \to ({\mathbf Q}^{c})^{\times }$ .

Proof Part (a) follows directly from the explicit formula for induced characters and the fact that for each $\gamma \in \Gamma $ , and $\tau \in \mathcal {T}$ , one has $(\gamma ^\tau )^2\in \Delta $ whenever $\gamma ^\tau \in \Delta $ .

To prove part (b), we fix a chain of subgroups

(7.1) $$ \begin{align} \Delta = \Gamma(1) \subset \cdots \subset \Gamma(t-1) \subset \Gamma(t) = \Gamma\end{align} $$

such that each $\Gamma (i)$ is normal in $\Gamma (i+1)$ . Then the equality

(7.2) $$ \begin{align} \mathrm{I}^2_\Gamma(\phi) = \sum_{i=1}^{i = t-1} \mathrm{Ind}_{\Gamma(i+1)}^\Gamma \bigl( \mathrm{I}_{\Gamma(i+1), \Gamma(i)}^2(\mathrm{Ind}^{\Gamma(i)}_\Delta\phi) \bigr), \end{align} $$

where

$$\begin{align*}\mathrm{I}_{\Gamma(i+1), \Gamma(i)}^2 (\chi) = \psi_{2}(\mathrm{Ind}^{\Gamma(i+1)}_{\Gamma(i)} \chi ) - \mathrm{Ind}^{\Gamma(i+1)}_{\Gamma(i)} (\psi_{2}(\chi)), \end{align*}$$

reduces us to the case $\Delta $ is normal in $\Gamma $ . In this case, the claim follows immediately from the formula in part (a) and the fact that under the stated conditions, for every $\gamma \in \Gamma $ and $\tau \in \mathcal {T}$ , one has $(\gamma ^\tau )^2 \in \Delta \Longleftrightarrow \gamma ^\tau \in \Delta $ .

Turning to part (c), we note first that under the stated hypothesis on $\Gamma $ , claim (c)(i) follows from [22, Section 8.5, Exercise 8.10] and the argument of [22, Section 8.2, Proposition 25].

To verify (c)(ii) and (c)(iii), we assume the additional hypotheses on $\Gamma $ and note, in particular, that since $\Gamma $ has cyclic Sylow $2$ -subgroups, Cayley’s normal $2$ -complement theorem implies that $\Gamma $ , and therefore also its quotient $\Gamma /\Upsilon $ , has a normal $2$ -complement. Writing $\Upsilon _1/\Upsilon $ for the normal $2$ -complement of $\Gamma /\Upsilon $ , the given assumptions imply $\Upsilon _1\subseteq \Delta $ and so, since $\Gamma /\Upsilon _1$ is cyclic of $2$ -power order, there exists a chain of subgroups (7.1) in which $\Gamma (i)$ has index $2$ in $\Gamma (i+1)$ for each i. The corresponding equality (7.2) then reduces claims (c)(ii) and (c)(iii) to the case that $\Delta $ has index 2 in $\Gamma $ . In this case, $|\mathcal {T}| = 2$ and, for every $\gamma \in \Gamma $ and $\tau \in \mathcal {T}$ , one has $(\gamma ^\tau )^2\in \Delta $ and, in addition, $\gamma ^\tau \notin \Delta \Longleftrightarrow \gamma \notin \Delta $ and so the formula in part (a) implies

(7.3) $$ \begin{align} (I_\Gamma^2(\phi))(\gamma) = \begin{cases} 0, &\text{ if }\gamma\in \Delta,\\ \sum_{\tau\in \mathcal{T}}\phi((\gamma^\tau)^2), &\text{ if }\gamma\notin \Delta.\end{cases}\end{align} $$

Now, by (c)(i), every irreducible character of $\Gamma $ has the form $\mu = \mathrm {Ind}_{\Upsilon '}^\Gamma (\lambda )$ , where $\Upsilon '$ is a suitable subgroup of $\Gamma $ that contains $\Upsilon $ and $\lambda $ a linear character of $\Upsilon '$ . Furthermore, if $\Upsilon '\not \subset \Delta $ , then the index of $\Upsilon '$ in $\Gamma $ is odd, so $\mu $ has odd degree and so, by [20, Theorem A], is real-valued if and only if it is a homomorphism of the form $\Upsilon ' \to \{\pm 1\}$ . Claim (c)(ii) follows directly from this fact and the observation that $I_\Gamma ^2(\phi )$ is real-valued if $\phi $ is real-valued.

To prove claim (c)(iii), we assume that $\phi = \mathrm {Ind}_{\Delta '}^\Delta \phi '$ , where $\Delta '$ is a normal subgroup of $\Delta $ that contains $\Upsilon $ and is of $2$ -power index. In this case, the formula (7.3) implies that if $I_\Gamma ^2(\phi )$ is nonzero, then there exists an element of $\Gamma \setminus \Delta $ whose square belongs to $\Delta '$ . However, since $\Upsilon _1 \subseteq \Delta '$ , the image in the (cyclic) group $\Gamma /\Delta '$ of any element in $\Gamma \setminus \Delta $ has order divisible by $4$ and so its square cannot belong to $\Delta '$ . This proves (c)(iii).

Next, under the hypotheses of (d), for every $\gamma \in \Gamma $ , one has $\gamma ^2 \in \Delta $ and hence

$$\begin{align*}\bigl(\psi_2(\mathrm{Ind}_\Delta^\Gamma\phi)\bigr)(\gamma) = (\mathrm{Ind}_\Delta^\Gamma\phi)(\gamma^2) = \phi^2(\gamma) + \phi^2(\gamma^{-1}).\end{align*}$$

In particular, since $\phi ^2(z) = 1$ for every $z\in Z$ , this formula implies that $\psi _2(\mathrm {Ind}_\Delta ^\Gamma \phi )$ is the inflation of a character function on the dihedral group $\Gamma /Z$ , and then the displayed formula in part (d) is verified by an easy explicit computation.

In the sequel, for each finite Galois extension $E/F$ of p-adic fields, and each complex character $\chi $ of $\operatorname {\mathrm {Gal}}(E/F)$ , we abbreviate the root number $W(F,\chi )$ to $W(\chi )$ .

Part (c) of the following result relies on the central result of Fröhlich and Queyrut in [16].

Proposition 7.4 Let $E/F$ be a finite Galois extension of p-adic fields. Set $G := \operatorname {\mathrm {Gal}}(E/F)$ and assume that the inertia subgroup of G has odd order.

1. (a) For all $\phi $ in $R_G$ , one has $W(\mathrm {Tr}(\phi )) = 1$ .

2. (b) If H is a normal subgroup of G and $G/H$ is cyclic, then for each $\phi $ in $R_H$ , one has

   $$\begin{align*}W(\mathrm{Ind}_H^G\phi) = \begin{cases} W(\phi), &\text{if }G/H\text{ has odd order,}\\ W(\phi)W(\chi_{G/H})^{\phi(1)}, &\text{if }G/H\text{ has even order,}\end{cases}\end{align*}$$
   where, in the second case, $\chi _{E'/F}$ is the nontrivial character of $\operatorname {\mathrm {Gal}}(E'/F)$ , with $E'$ the quadratic extension of F in E.

3. (c) Assume that G is dihedral of order congruent to $2$ modulo $4$ , write L for the unique quadratic extension of F in E, and set $H := \operatorname {\mathrm {Gal}}(E/L)$ . Then, for each homomorphism $\phi : H\to ({\mathbf Q}^{c})^{\times }$ , one has $W(\mathrm {Ind}_H^G\phi ) = W(\chi _{G/H})$ , where $\chi _{G/H}$ is the nontrivial character of $G/H$ .

Proof It is enough to prove claim (a) in the case where $\phi $ is a character of G, represented by a homomorphism $T_\phi : G \to \mathrm {GL}_d({\mathbf Q}^{c})$ . In this case, the general result of [18, Chapter II, Section 4, Corollary] implies that

$$ \begin{align*} W(\mathrm{Tr}(\phi)) = W(\phi) W(\bar{\phi}) = \mathrm{det}_{\phi}(\rho_F(-1)), \end{align*} $$

where $\mathrm {det}_\phi $ is the homomorphism $G^{\mathrm {ab}} \to ({\mathbf Q}^{c})^{\times }$ induced by sending each g in G to $\mathrm {det}(T_\phi (g))$ and $\rho _F$ is the reciprocity map $F^\times \to G^{\mathrm {ab}}$ . In addition, $-1$ belongs to $O^\times _{F}$ and so is sent by $\rho _F$ to an element of the inertia subgroup of $G^{\mathrm {ab}}$ of order dividing 2. In particular, since this inertia group has odd order, one has $\rho _F(-1) = 1$ and so $\mathrm {det}_{\phi }(\rho _F(-1))=1$ . This proves claim (a).

To prove part (b), we use the inductivity of local root numbers in degree zero to compute

$$ \begin{align*}W(\mathrm{Ind}_H^G\phi) =&\, W(\mathrm{Ind}_{H}^{G}(\phi - \phi(1)\mathbf{1}_H))W(\mathrm{Ind}_H^G1_H)^{\phi(1)}\\ =&\, W(\phi - \mathbf{1}_H)W(\mathrm{Ind}_{H}^G\mathbf{1}_H)^{\phi(1)}\\ =&\, W(\phi)W(\mathbf{1}_H)^{-1}\prod_{\theta\in (G/H)^*}W(\theta)^{\phi(1)},\end{align*} $$

where $(G/H)^*$ denotes the group of homomorphisms $G/H \to ({\mathbf Q}^{c})^{\times }$ , and the last equality is true because $\mathrm {Ind}_H^G\mathbf {1}_H$ is equal to the sum of $\theta $ over $(G/H)^*$ . Now, if $G/H$ is odd (resp. even), then the only real-valued functions in $(G/H)^*$ are $\mathbf {1}_G$ (resp. $\mathbf {1}_G$ and $\chi _{G/H}$ ) and all other homomorphisms occur in complex conjugate pairs. The result of part (b) therefore follows from the above displayed formula after isolating the conjugate pairs in the product that occurs in the final term, applying the result of part (a) to each of these pairs, and noting that $W(\mathbf {1}_H) = W(\mathbf {1}_G) = 1$ .

To prove part (c), we recall that by a result of Fröhlich and Queyrut [16, Section 4, Theorem 3], one has $W(\phi ) = \phi (\rho _L(x))$ , where $\rho _L$ is the reciprocity map $L^\times \to H$ and x is any element of $L\setminus F$ with $x^2 \in F^\times $ . In addition, since $\phi $ is of dihedral type, it is trivial on restriction to $F^\times $ (cf. [16, Section 3, Lemma 1]) and so $\phi (\rho _L(x))^2 = \phi (\rho _L(x^2)) = \phi (1) = 1$ . On the other hand, the order of $\phi $ is odd (since it divides $|H| = |G|/2$ which, under the given hypothesis on $|G|$ , is odd) and so $\phi (\rho _L(x))^2 = 1$ implies $\phi (\rho _L(x)) =1$ and hence also $W(\phi )=1$ .

This last equality then combines with a straightforward application of the general result of part (b) to prove the formula in part (c).

We are now ready to prove Theorem 7.2. At the outset, we note that G is the semidirect product of its inertia subgroup I by the cyclic quotient group $G/I$ . We further note that, by assumption, the group I is cyclic of odd order, and hence, in particular, that G is supersolvable.

Fix $\chi $ in $\mathrm {Symp}(G)$ . Then, since $\chi $ is tamely ramified, one has $W(\chi )\in \{\pm 1\}$ (cf. [14, Chapter III, Theorem 21(iii)]) and so $W(2\chi ) = W(\chi )^2 = 1$ . It is therefore enough for us to prove that $W(\psi _2(\chi )) = 1$ .

Next, we note that, by Lemma 7.3(c)(i), there exists a subgroup J of G that contains I and a linear character $\phi $ of J such that one has $\chi = \mathrm {Ind}_{J}^G\phi $ . In particular, since J contains I and $G/I$ is cyclic, there exists a normal subgroup H of G with $J \trianglelefteq H \trianglelefteq G$ and such that $H/J$ is cyclic of $2$ -power order and $G/H$ is cyclic of odd order.

Then one has $\chi = \mathrm {Ind}_H^G\chi '$ with $\chi ' := \mathrm {Ind}_J^H\phi $ and we claim that $\chi '$ belongs to $\mathrm {Symp}(H)$ . To see this, we note that $\chi '$ is an irreducible character of H (since $\chi $ is irreducible) and so, by the Frobenius–Schur theorem (cf. [9, Theorem (73.13)]), the sum $c_H(\chi ') := |H|^{-1}\sum _{h \in H}\chi (h^2)$ belongs to $\{-1, 0, 1\}$ and is equal to $-1$ if and only if $\chi '$ is symplectic. In addition, since H is normal in G and of odd index, one has $g^2 \in H \Longleftrightarrow g \in H$ for each $g \in G$ and so

$$ \begin{align*}c_G(\chi) = c_G(\mathrm{Ind}_H^G\chi') =&\, |G|^{-1}\sum_{g \in G}(\mathrm{Ind}_H^G\chi')(g^2)\\ =&\, |G|^{-1}\sum_{\tau \in \mathcal{T}}\sum_{h \in H}(\chi')^\tau(h^2)\\ =&\, |\mathcal{T}|^{-1}\sum_{\tau\in \mathcal{T}}c_H((\chi')^\tau),\end{align*} $$

where $\mathcal {T}$ is a set of coset representatives of H in G and $(\chi ')^\tau $ is the irreducible character of H that sends each element h to $\chi '(h^\tau )$ . In particular, since both $c_G(\chi ) = -1$ (as $\chi \in \mathrm {Symp}(G)$ ) and each $c_H((\chi ')^\tau )$ belongs to $\{-1,0,1\}$ , the displayed equality implies that $c_H((\chi ')^\tau ) = -1$ for all $\tau $ . Thus, one has $c_H(\chi ') = -1$ and so $\chi ' \in \mathrm {Symp}(H)$ , as claimed.

Now, since $G/H$ is cyclic of odd order, one has $W(\psi _2(\chi )) = W(\mathrm {Ind}_H^G(\psi _2(\chi ')) = W(\psi _2(\chi '))$ , where the first equality follows from Lemma 7.3(b) and the second from Proposition 7.4(b). Thus, if necessary after replacing G by H (and $\chi $ by $\chi '$ ), we can assume in the sequel that $\chi $ has $2$ -power degree.

Next, we note that, since G is supersolvable, an induction theorem of Martinet (cf. [18, Chapter III, Theorem 5.2]) implies that either $\chi = \mathrm {Tr}(\mathrm {Ind}_{H'}^G\phi ')$ , where $\phi '$ is a linear character of some subgroup $H'$ of G, or that $\chi $ is the induction to G of a quaternion character of a subgroup. In view of Proposition 7.4(a), we can therefore also assume in the sequel that there exists a subgroup $J_1$ of G that has $2$ -power index, and hence contains I, and a quaternion character $\phi _1$ of $J_1$ such that $\chi = \mathrm {Ind}_{J_1}^G\phi _1$ .

This implies that $J_1$ has a quotient Q isomorphic to a generalized quaternion group and that

(7.4) $$ \begin{align} \phi_1 = \mathrm{Inf}^{J_{1}}_{Q} (\mathrm{Ind}^{Q}_{P} \theta), \end{align} $$

where P is the cyclic subgroup of Q of index $2$ and $\theta $ a homomorphism $P\to ({\mathbf Q}^{c})^{\times }$ . Let $J_1'$ denote the inverse image of P under the quotient map $J_1 \to Q$ , and set $\phi _1':= \mathrm { Inf}^{J_1'}_{P} \theta $ (so $\phi _1'$ is a linear character of $J_1'$ ). Then the subgroup $J_1'$ is of index $2$ in $J_1$ , and (7.4) implies that

(7.5) $$ \begin{align} \phi_1 = \mathrm{Ind}^{J_{1}}_{J_1'} \phi_1'. \end{align} $$

Now, as $J_1'$ has $2$ -power index in G, it contains I. Thus, since $G/I$ is cyclic, one has $J_1' \trianglelefteq G$ and $G/J_1'$ is cyclic of $2$ -power order. In particular, since the degree $(\psi _2(\phi _1))(1) = \phi _1(1)$ is even, one therefore has

$$\begin{align*}W(\psi_2(\chi)) = W(\psi_2(\mathrm{Ind}_{J_1}^G\phi_1)) = W(\mathrm{Ind}_{J_1}^G(\psi_2(\phi_1))) = W(\psi_2(\phi_1)),\end{align*}$$

where the second equality follows from Lemma 7.3(c)(iii) (after taking account of (7.5)) and the third from Proposition 7.4(b).

In addition, since Q is the Galois group of a tamely ramified extension of p-adic fields that has odd ramification degree, it is the semidirect product of a cyclic (inertia) subgroup of odd order by a cyclic group. In particular, since such a group can have no quotient isomorphic to $H_8$ , the group Q must be isomorphic to $H_{4m}$ , with m odd. In view of (7.4), we can therefore apply Lemma 7.3(d) (with $\Gamma , \Delta $ , and $\phi $ taken to be $Q, P$ , and $\theta $ ) to deduce that

$$\begin{align*}W(\psi_2(\phi_1)) = W(\psi_2(\mathrm{Ind}_P^Q\theta)) = W(\mathrm{ Ind}_{P/N}^{Q/N}(\lambda))W(\chi_{Q/P}), \end{align*}$$

where N denotes the center of Q (so N is the unique subgroup of P of order 2) and $\lambda $ denotes $\theta ^2$ , regarded as a homomorphism $P/N \to ({\mathbf Q}^{c})^{\times }$ .

Finally, since the group $Q/N$ is generalized dihedral with $|Q/N| = 2m \equiv 2$ modulo $4$ , and the inertia subgroup of $Q/N$ has odd order, the theorem of Fröhlich and Queyrut implies (via Proposition 7.4(c)) that $W(\mathrm {Ind}_{P/N}^{Q/N}(\lambda )) = W(\chi _{Q/P})$ . Upon substituting this fact into the last two displayed formulas, we deduce that $W(\psi _2(\chi )) = W(\chi _{Q/P})^2 = 1$ .

This completes the proof of Theorem 7.1.

8 Symplectic Galois–Jacobi sums II

We retain the notation of the previous two sections. For any real number x, we write $\operatorname {\mathrm {sgn}}(x) \in \{ \pm 1\}$ for the sign of x. In this section, we shall examine $\operatorname {\mathrm {sgn}}(J^*(L_\pi /L, \chi ))$ for $\chi \in \operatorname {\mathrm {Symp}}(G)$ . This will in turn lead to the definition of ${\mathcal J}^{\ast }_\infty (F_\pi /F) \in \operatorname {\mathrm {Cl}}({\mathbf Z} G)$ for F a number field and $[\pi ] \in H^1_t(F,G)$ .

Recall that for each $\chi \in R_G$ , the adjusted Galois–Gauss sum is defined (in [14, Chapter IV, Section 1]) by setting

$$\begin{align*}\tau^*(L, \chi):=\tau(L, \chi)y(L,\chi)^{-1}z(L,\chi), \end{align*}$$

for suitable roots of unity $y(L, \chi )$ and $z(L,\chi )$ in ${\mathbf Q}^c$ . [14, Chapter IV, Theorem 29(i)] implies that $y(L,\chi ) =1$ for all $\chi $ in $\mathrm {Symp}(G)$ . One can also check (directly from the definitions) that $z(L,\psi _2(\chi )) = z(L,\chi )^2$ and hence that $z(L,\chi ) = z(L,\psi _2(\chi )) = 1$ for each $\chi $ in $\mathrm { Symp}(G)$ .

Recall that Theorem 7.1 asserts that $J(L_\pi /L, \chi )>0$ whenever $\chi \in \operatorname {\mathrm {Symp}}(G)$ . The following result is now a direct consequence of the definition of the adjusted Galois–Jacobi sum $J^*(L_\pi /L, \chi )$ .

Theorem 8.1 Suppose that $\chi \in \operatorname {\mathrm {Symp}}(G)$ . Then

$$\begin{align*}\operatorname{\mathrm{sgn}}(J^*(L_\pi/L, \chi) = \operatorname{\mathrm{sgn}}(y(L_\pi/L, \psi_2(\chi))). \end{align*}$$

The following Propostion shows that $\operatorname {\mathrm {sgn}}(y(L_\pi /L, \psi _2(\chi ))) = -1$ is possible.

Proposition 8.2 Let $M/L$ be a tamely ramified Galois extension with $\Gamma := \operatorname {\mathrm {Gal}}(M/L) \simeq H_{4m}$ , with m odd. Suppose that the inertia subgroup $\Gamma _{0}$ of $\Gamma $ is odd. Then, for each $\chi \in \operatorname {\mathrm {Symp}}(G)$ , we have $y(M/L, \psi _2(\chi )) = -1$ .

Proof For ease of notation, we write, e.g., $y(\chi )$ rather than $y(M/L,\chi )$ .

To prove the desired result, we shall use Lemma 7.3. Let $\Delta $ be the cyclic subgroup of $\Gamma $ of index $2$ . Then all irreducible symplectic characters of $\Gamma $ can be written in the form $\chi =\mathrm {Ind}_\Delta ^{\Gamma }\phi $ , where $\phi $ is a linear character of $\Delta $ . It is easy to see that the order of $\phi $ does not divide $2$ (for otherwise $\mathrm {Ind}_\Delta ^{\Gamma }\phi $ would be an orthogonal character of $\Gamma $ ; see [18, Chapter III, Theorem 3.1]), and that $\phi $ (and hence also $\phi ^2$ ) is nontrivial on $\Gamma _{0}$ (since $\Gamma _{0}$ has odd order).

Let Z denote the center of $\Gamma $ , and let $\chi _{\Gamma /\Delta }$ denote the unique nontrivial homomorphism $\Gamma /\Delta \rightarrow ({\mathbf Q}^{c})^{\times }$ . Using the formula in Lemma 7.3(d), one can compute that

$$ \begin{align*} y(\psi_{2}(\chi)) =&\, y(\psi_2(\mathrm{Ind}_\Delta^{\Gamma} \phi))\\ =&\, y( \mathrm{Inf}_{\Gamma/Z}^{\Gamma}(\mathrm{Ind}_{\Delta/Z}^{\Gamma/Z}(\phi^2))) \cdot y( \mathrm{Inf}_{\Gamma/\Delta}^{\Gamma}(\chi_{\Gamma/\Delta})) \cdot y( \mathbf{1}_{\Gamma})^{-1} \\ =&\, (-1)^{\deg(n_0)}\mathrm{det}_{n_0}(\sigma) \cdot (-1)\chi_{\Gamma/\Delta}(\sigma) \cdot (-1) \mathbf{1}_{\Gamma}(\sigma)^{-1}\\ =&\, 1 \cdot 1 \cdot (-1) = -1, \end{align*} $$

where $\phi ^2$ is regarded as a character of $\Delta /Z$ , $\sigma $ is the Frobenius element in $\Gamma /\Gamma _{0}$ lifted to $\Gamma $ , and $n_0:= n(\mathrm {Inf}_{\Gamma /Z}^{\Gamma }(\mathrm {Ind}_{\Delta /Z}^{\Gamma /Z}(\phi ^2)))$ denotes the unramified part (cf. [14, Chapter I, equation (5.6)]) of $\mathrm {Inf}_{\Gamma /Z}^{\Gamma }(\mathrm {Ind}_{\Delta /Z}^{\Gamma /Z}(\phi ^2))$ . The third equality above holds since clearly $\mathrm {Inf}_{\Gamma /\Delta }^{\Gamma }(\chi _{\Gamma /\Delta })$ and $\mathbf {1}_{\Gamma }$ are both linear and unramified. The fourth equality follows from the fact that $n_0=0$ (since $\phi ^2$ is irreducible and ramified, by [14, Chapter III, Proposition 1.3(ii)] the unramified part $n(\mathrm {Ind}_{\Delta /Z}^{\Gamma /Z}(\phi ^2))=0$ and therefore $n_0=0$ ).

The above discussion motivates the following definition.

Definition 8.3 We define $J^{*}_{\infty }(L_\pi /L,-) \in \operatorname {\mathrm {Hom}}_{\Omega _{{\mathbf Q}}}(R_G, J({\mathbf Q}^c))$ by its values on $\chi \in \operatorname {\mathrm {Irr}}(G)$ as follows:

$$\begin{align*}J^{*}_{\infty}(L_\pi/L, \chi)_v = \begin{cases} \operatorname{\mathrm{sgn}}(J^{*}(L_\pi/L, \chi)), \quad &\text{if }\chi \in \operatorname{\mathrm{Symp}}(G) \text{ and }v|\infty,\\ 1, \quad &\text{otherwise.} \end{cases} \end{align*}$$

We write $J^{*}_{\infty }(L_\pi /L)$ for the element of $K_{0}({\mathbf Z} G, {\mathbf Q})$ represented by the homomorphism $J^{*}_{\infty }(L_\pi /L,-)$ . Similarly, we also write $J^{*}(L_\pi /L)$ for the element of $K_{0}({\mathbf Z} G, {\mathbf Q})$ represented by $J^{*}(L_\pi /L,-)$ .

Theorem 8.4 We have

$$\begin{align*}J^{*}(L_{\pi}/L,-) \cdot J^{*}_{\infty}(L_{\pi}/L,-)^{-1} \in \operatorname{\mathrm{Det}}({\mathbf Q}^c G), \end{align*}$$

and so

$$\begin{align*}\partial^0(J^{*}(L_\pi/L)) = \partial^0(J^{*}_{\infty}(L_\pi/L)). \end{align*}$$

Proof To ease notation, set $f = J^{*}(L_{\pi }/L,-) \cdot J^{*}_{\infty }(L_{\pi }/L,-)^{-1} $ .

Then, since $f \in \operatorname {\mathrm {Hom}}_{\Omega _{{\mathbf Q}}}(R_G, ({\mathbf Q}^{c})^{\times })$ , the Hasse–Schilling–Maass Norm Theorem (cf. [8, Theorem (7.48)]) implies that the first equality is equivalent to asserting that $f(\chi )$ is a strictly positive real number for every $\chi $ in $\operatorname {\mathrm {Symp}}(G)$ . This in turn follows at once from the definition of $J^{*}_{\infty }(L_{\pi }/L,-)$ .

The second equality is now an immediate consequence of the fact that $\partial ^0(\operatorname {\mathrm {Det}}({\mathbf Q}^c G)) = 0$ .

Suppose now that F is a number field and that $[\pi ] \in H^1_t(F, G)$ . We also recall that $F_{\pi , v}:= F_{\pi } \otimes _{F} F_{v} \simeq F_{v, \pi _v}$ (see, e.g., [19, equation (2.4)]).

Definition 8.5 We set

$$\begin{align*}J^{*}(F_\pi/F):= \sum_{v \nmid \infty} J^{*}(F_{v,\pi_{v}}/F_{v}) \in K_0({\mathbf Z} G, {\mathbf Q}) \end{align*}$$

and

$$\begin{align*}J^{*}_{\infty}(F_\pi/F):= \sum_{v \nmid \infty} J^{*}_{\infty}(F_{v,\pi_{v}}/F_{v}) \in K_0({\mathbf Z} G, {\mathbf Q}). \end{align*}$$

(Note that the infinite sums make sense as $J^{*}_{\infty }(F_{v,\pi _{v}}/F_{v}) = J^{*}(F_{v,\pi _{v}}/F_{v}) =0$ for all places v that are unramified in $F_{\pi }/F$ .)

We define ${\mathcal J}^*(F_{\pi }/F) \in \operatorname {\mathrm {Cl}}({\mathbf Z} G)$ by

$$\begin{align*}{\mathcal J}^*(F_{\pi}/F) := \partial^0(J^{*}(F_\pi/F)), \quad {\mathcal J}^*_\infty(F_{\pi}/F) := \partial^0(J^{*}_{\infty}(F_\pi/F)) \end{align*}$$

(see (2.2)).

Proposition 8.6 Suppose that F is a number field and that $[\pi ] \in H^1_t(F,G)$ . Then

$$\begin{align*}{\mathcal J}^*(F_\pi/F) = {\mathcal J}^*_{\infty}(F_\pi/F). \end{align*}$$

Proof This is a direct consequence of Theorem 8.4 and Definition 8.5.

9 Proof of Theorem 1.5

Let $[\pi ] \in H^1_t(F,G)$ , and write

$$\begin{align*}{\mathfrak c}(\pi) = [A_{\pi}, O_FG;{\mathbf r}_G] - [O_{\pi}, O_FG; {\mathbf r}_G] \in K_0(O_FG,F) \subseteq K_0(O_FG, F^c). \end{align*}$$

For each finite place v of F, we write $[\pi _v]$ for the image of $[\pi ]$ in $H^{1}_{t}(F_v, G)$ .

Recall that

$$\begin{align*}K_0(O_FG, F) \simeq \frac{\operatorname{\mathrm{Hom}}_{\Omega_F}(R_G, J_{f}(F^c))}{\prod_{v \nmid \infty} \operatorname{\mathrm{Det}}(O_{F_v}G)^{\times}}. \end{align*}$$

A representing homomorphism in $\operatorname {\mathrm {Hom}}_{\Omega _F}(R_G, J_{f}(F^c))$ of ${\mathfrak c}(\pi )$ is $f = (f_{v})_{v}$ defined by

$$\begin{align*}f_{v}(\chi) = {\varpi}_{v}^{\langle \psi_{2}(\chi) - 2\chi, s_{v} \rangle_{G}}, \end{align*}$$

using the notation of Corollary 5.6. Let $\operatorname {\mathrm {Ram}}(\pi )$ denote the set of finite places of F at which $F_{\pi }/F$ is ramified. If $v \notin \operatorname {\mathrm {Ram}}(\pi )$ , then $s_v =1$ and so $f_v =1$ .

Definition 9.1 Suppose that $v \in \operatorname {\mathrm {Ram}}(\pi )$ . Then we define ${\mathfrak c}(\pi ;v) \in K_0(O_FG, F)$ to be the element represented by $f^{(v)} = (f^{(v)}_{w})_{w} \in \operatorname {\mathrm {Hom}}_{\Omega _F}(R_G, J_{f}(F^c))$ given by

$$\begin{align*}f^{(v)}_{w}(\chi) = \begin{cases} f_v(\chi) = {\varpi}_{v}^{\langle \psi_{2}(\chi) - 2\chi, s_{v} \rangle_{G}}, &\text{if }w=v, \\ 1, &\text{otherwise.} \end{cases} \end{align*}$$

Lemma 9.2 We have

(9.1) $$ \begin{align} {\mathfrak c}(\pi) = \sum_{v \in \operatorname{\mathrm{Ram}}(\pi)} {\mathfrak c}(\pi; v). \end{align} $$

Proof It follows from the definitions that

$$\begin{align*}f = \prod_{v \in \operatorname{\mathrm{Ram}}(\pi)} f^{(v)}, \end{align*}$$

and this implies the result.

We can now prove Theorem 1.5.

Theorem 9.3 Suppose that $[\pi ] \in H^{1}_{t}(F, G)$ and that $A_{\pi }$ is defined. Then

$$\begin{align*}\partial^{0}({\mathcal N}_{F/{\mathbf Q}}({\mathfrak c}(\pi)) \cdot {\mathcal J}^{*}_{\infty}(F_\pi/F)^{-1} = 0, \end{align*}$$

and so there is an equality

$$\begin{align*}(A_{\pi}) - (O_{\pi}) = {\mathcal J}^{*}_{\infty}(F_\pi/F), \end{align*}$$

i.e., (see (1.1))

$$\begin{align*}(A_{\pi}) - W(F_\pi/F) = {\mathcal J}^{*}_{\infty}(F_\pi/F), \end{align*}$$

in $\operatorname {\mathrm {Cl}}({\mathbf Z} G)$ .

Proof Lemma 9.2 implies that in order to show that

$$\begin{align*}\partial^{0}({\mathcal N}_{F/{\mathbf Q}}({\mathfrak c}(\pi)) \cdot {\mathcal J}^{*}_{\infty}(F_\pi/F)^{-1} = 0, \end{align*}$$

it suffices to show that

$$\begin{align*}\partial^{0}({\mathcal N}_{F/{\mathbf Q}}({\mathfrak c}(\pi;v)) \cdot {\mathcal J}^{*}_{\infty}(F_{v,\pi_{v}}/F_v)^{-1}= 0 \end{align*}$$

for each $v \in \operatorname {\mathrm {Ram}}(\pi )$ . Theorem 8.4 implies that this is equivalent to showing that

$$\begin{align*}\partial^{0}({\mathcal N}_{F/{\mathbf Q}}({\mathfrak c}(\pi;v)) \cdot {\mathcal J}^*(F_{v,\pi_{v}}/F_v)^{-1}= 0 \end{align*}$$

for each $v \in \operatorname {\mathrm {Ram}}(\pi )$ .

We see from the description of $\mathrm {Cl}({\mathbf Z} G)$ given in Theorem 2.1(a) that this last equality will in turn follow if, for each $v \in \operatorname {\mathrm {Ram}}(\pi )$ , we show that

$$\begin{align*}J^{*}(F_{v,\pi_{v}}/F_{v},-)^{-1} \cdot ({\mathcal N}_{F/{\mathbf Q}}(f^{(v)})) \in \prod_{l} \operatorname{\mathrm{Det}}({\mathbf Z}_{l}G)^{\times}. \end{align*}$$

To show this last inclusion, we first observe that Proposition 6.6(a) implies that the inclusion holds at all rational primes l not lying below v.

For each rational prime l that lies below v, we fix an embedding $\operatorname {\mathrm {Loc}}_l: {\mathbf Q}^c \to {\mathbf Q}^c_{l}$ and use it to identify $\operatorname {\mathrm {Irr}}(\Gamma )$ with $\operatorname {\mathrm {Irr}}_l(\Gamma )$ . We recall in particular that such an isomorphism $R_G \to R_{G, l}$ in turn induces an isomorphism $\operatorname {\mathrm {Hom}}_{\Omega _{F}}(R_G, ({\mathbf Q}^c)^{\times }_{l}) \to \operatorname {\mathrm {Hom}}_{\Omega _{F_{v}}}(R_{G, l}, ({\mathbf Q}^c_l)^{\times })$ (cf. [14, Chapter II, Lemma 2.1]). Then, reasoning analogously to the proof of [14, Theorem 19, pp. 114–116], one can deduce from Proposition 6.6(b) that

$$\begin{align*}{\mathcal N}_{F_{v}/{\mathbf Q}_{l}}(f_{v}) \cdot \operatorname{\mathrm{Loc}}_l \bigl({\mathcal N}_{F/{\mathbf Q}}(f^{(v)}) \bigr)^{-1} \in \operatorname{\mathrm{Det}}({\mathbf Z}_{l}G). \end{align*}$$

This establishes the desired inclusion at rational primes lying below v and completes the proof of the desired result.

Remark 9.4 Let us make some remarks concerning Theorem 9.3 when $F_{\pi }/F$ is locally abelian.

Suppose that $v \in \operatorname {\mathrm {Ram}}(\pi )$ . Set $s_v:= \pi (\sigma _v)$ , and write $H_v := \langle s_v \rangle $ . Proposition 5.2(d) with $G = H_v$ and Proposition 5.3(b) imply that for each $\chi \in R_{H_v}$ , we have

$$ \begin{align*} \langle \chi, s_v \rangle^{*}_{H_v} - \langle \chi, s_v \rangle_{H_v} &= (d(s_v), \chi)_{H_v} \\ &= \langle \psi_2(\chi) - \chi, s_v \rangle_{H_v}. \end{align*} $$

Now, suppose also that $F_v$ contains a primitive $|s_v|$ th root of unity. This implies in particular that the extension $F^{\pi _{v}}_{v}/F_v$ is abelian. Let ${\mathfrak b}(\pi ;v) \in K_0(FH_{v}, F)$ be the element represented by $\rho ^{(v)} = (\rho ^{(v)}_{w})_{w} \in \operatorname {\mathrm {Hom}}_{\Omega _{F}}(R_{H_v}, J_{f}(F^c))$ defined by

$$\begin{align*}\rho^{(v)}_{w}(\chi) = \begin{cases} {\varpi}_{v}^{(d(s_{v}), \chi)_{H_{v}}} = {\varpi}_{v}^{\langle \psi_{2}(\chi) - \chi, s_{v} \rangle_{H_v}}, &\text{if }w=v, \\ 1, &\text{otherwise.} \end{cases} \end{align*}$$

Observe that without the hypothesis concerning the number of roots of unity in $F_v$ , we would only have that $\rho ^{(v)} \in \operatorname {\mathrm {Hom}}(R_{H_v}, J_{f}(F^c))$ rather than $\rho ^{(v)} \in \operatorname {\mathrm {Hom}}_{\Omega _{F}}(R_{H_v}, J_{f}(F^c))$ . We also see from the definitions of ${\mathfrak c}(\pi ;v)$ and ${\mathfrak b}(\pi ;v)$ (see also (2.7) and (2.9)) that ${\mathfrak c}(\pi ;v) = \operatorname {\mathrm {Ind}}^{G}_{H_v} {\mathfrak b}(\pi ;v)$ .

Hence, if for every $v \in \operatorname {\mathrm {Ram}}(\pi )$ , $F_v$ contains a primitive $|s_v|$ th root of unity—which is precisely what happens if $F_{\pi }/F$ is locally abelian—then we have

(9.2) $$ \begin{align} {\mathfrak c}(\pi) = \sum_{v \in \operatorname{\mathrm{Ram}}(\pi)} \operatorname{\mathrm{Ind}}^{G}_{H_v} {\mathfrak b}(\pi;v), \end{align} $$

and so (using (2.10))

$$ \begin{align*} \partial^0({\mathfrak c}(\pi)) &= \sum_{v \in \operatorname{\mathrm{Ram}}(\pi)} \partial^0(\operatorname{\mathrm{Ind}}^{G}_{H_v} {\mathfrak b}(\pi;v)) \\ &= \sum_{v \in \operatorname{\mathrm{Ram}}(\pi)} \operatorname{\mathrm{Ind}}^{G}_{H_v} \partial^0({\mathfrak b}(\pi;v)) \\ &=0. \end{align*} $$

We now deduce from Theorem 9.3 that ${\mathcal J}^*_{\infty }(F_\pi /F) = 0$ .

A comparison of (9.2) and (9.1) highlights the crucial difference between the locally abelian case and the general case. In both cases, the class ${\mathfrak c}(\pi )$ may be decomposed into a sum over the places $v \in \operatorname {\mathrm {Ram}}(\pi )$ of classes ${\mathfrak c}(\pi ;v) \in K_0(O_FG, F^c)$ . However, in the locally abelian case, these classes ${\mathfrak c}(\pi ;v)$ are induced from cyclic subgroups of G, whereas in the general case, they are not. This is why Theorem 9.3 may be proved in the locally abelian case using abelian Jacobi sums, thereby showing that in this situation ${\mathcal J}^*_\infty (F_\pi /F) = 0$ ), which is what is done in [4].

10 Proof of Theorem 1.7

Let F be any imaginary quadratic field such that $\operatorname {\mathrm {Cl}}(O_F)$ contains an element of order $4$ . In this section, we shall construct infinitely many counterexamples to Conjecture 1.4 by showing that if $\ell $ is any sufficiently large prime with $\ell \equiv 3\ \pmod {4}$ and G is the generalized quaternion group $H_{4\ell }$ , then there are infinitely many tame G-extensions $F_{\pi }/F$ of fields such that $A_\pi $ exists and ${\mathcal J}^*_\infty (F_\pi /F) \neq 0$ . Hence, for these extensions, $(O_\pi ) \neq (A_\pi )$ in $\operatorname {\mathrm {Cl}}({\mathbf Z} G)$ . This will prove Theorem 1.7.

In what follows, we fix an imaginary quadratic field F such that $\operatorname {\mathrm {Cl}}(O_F)$ contains an element of order $4$ . To prove Theorem 1.7, it will suffice to prove the following result, which we shall derive as a consequence of works of Fröhlich (see [11]).

Lemma 10.1 Suppose that $\ell $ is a sufficiently large prime and that $G\simeq H_{4\ell }$ . Then there exists a G-extension $F_\pi /F$ of fields such that:

1. (a) $F_\pi /F$ is ramified at only a single prime ${\mathfrak p}$ of F with ${\mathfrak p}\nmid \ell $ .

2. (b) The prime ${\mathfrak p}$ does not split in $F_{\pi }/F$ .

3. (c) The ramification index of ${\mathfrak p}$ is equal to $\ell $ .

Before we prove this result, we shall first show that Lemma 10.1 implies Theorem 1.7.

Proof of Theorem 1.7

First, we note that the decomposition subgroup of G at ${\mathfrak p}$ is equal to $H_{4\ell }$ . We also recall that for an odd prime $\ell $ , the generalized quaternion group $H_{4\ell }$ has a single, irreducible, nontrivial symplectic character $\chi $ , say.

If ${\mathfrak q}$ is unramified in $F_\pi /F$ , then one has $\operatorname {\mathrm {sgn}}(y(F_{\pi , {\mathfrak q}}/F_{{\mathfrak q}},\psi _{2}(\chi )))=1$ . On the other hand, Theorem 8.1 and Proposition 8.2 imply that

$$\begin{align*}\operatorname{\mathrm{sgn}}(J^{\ast}(F_{\pi, {\mathfrak p}}/F_{{\mathfrak p}}, \chi)) = \operatorname{\mathrm{sgn}}(y(F_{\pi, {\mathfrak p}}/F_{{\mathfrak p}},\psi_{2}(\chi))) = -1. \end{align*}$$

In particular, if we now assume in addition that $\ell \equiv 3\ \pmod {4}$ , then it follows from [14, Chapter II, Proposition 4.4] that the element ${\mathcal J}^*_\infty (F_\pi /F) \in \operatorname {\mathrm {Cl}}({\mathbf Z} G)$ (see Definitions 8.3 and 8.5 and Proposition 8.6) is nontrivial. (We remark in passing that if instead $\ell \equiv 1\ \pmod {4}$ , then the same argument shows that ${\mathcal J}^*_\infty (F_\pi /F) = 0$ .)

The remainder of this section will be devoted to the construction of the extensions described in Lemma 10.1.

Let L be an unramified, cyclic extension of F of degree $4$ . We write $E/F$ for the quadratic subextension of $L/F$ and write $\varphi _{E/F}$ for the quadratic character of $E/F$ on ideals of F. We also view this as an idele class character of F. If $\omega $ denotes the idele class character of E that cuts out the extension $L/E$ , then $\omega $ is of quaternion type (i.e., the restriction of $\omega $ to $J(F)$ is equal to $\varphi _{E/F}$ —see [11, p. 405].)

For each prime $\ell $ , the symbol $\eta _\ell $ will denote a primitive $\ell $ th root of unity. Then, following [11, Theorem 4], we consider the following conditions on primes.

Property 10.2 Let $\ell $ be an odd prime such that:

1. (a) $[F(\eta _\ell ) : F]$ is even.

2. (b) $E\not \subseteq F(\eta _\ell +\eta _\ell ^{-1})$ .

3. (c) The class number of E is not divisible by $\ell $ .

We remark that these properties are satisfied for all sufficiently large $\ell $ . (We observe, in particular, that in our case, Property 10.2(b) is automatically satisfied for sufficiently large $\ell $ since $E/F$ is unramified.)

Henceforth, we therefore fix a prime $\ell $ satisfying Property 10.2 and abbreviate $\eta _\ell $ to $\eta $ . We then write $\Sigma _{-}$ for the set of primes ${\mathfrak p}$ of F satisfying the following properties (see [11, equation (8.5)]).

Property 10.3 Let ${\mathfrak p}$ be a finite prime of F such that:

1. (a) The prime ${\mathfrak p}$ is inert in $E/F$ (ie. $\varphi _{E/F}({\mathfrak p}) = -1$ ).

2. (b) $N_{F/{\mathbf Q}} \equiv -1\ {\pmod \ell }$ .

In what follows, if ${\mathfrak p} \in \Sigma _{-}$ , we write ${\mathfrak p}_{E}$ for the unique prime of E lying above ${\mathfrak p}$ .

Our argument relies on the following result of Fröhlich (see [11, pp. 432–434]). We state the result and then describe an outline of the proof. We refer the reader to [11] for complete details.

Theorem 10.4 There are infinitely many primes in $\Sigma _{-}$ (in fact, a subset of positive Chebotarev density) for which the following statement is true: there exists a nontrivial idele class character $\theta $ of E of order $\ell $ , and of dihedral type (i.e., the restriction of $\theta $ to $J(F)$ is trivial) which is ramified at ${\mathfrak p}_{E}$ and which is unramified at all other finite places of E.

Proof We remark that necessary conditions for such a $\theta $ to exist are given in [11, Section 8, Lemma 5]. The existence of $\theta $ is demonstrated on pages 433 and 434 of loc. cit. via the following argument.

Recall that $\eta $ is a primitive $\ell $ th root of unity, and set

$$\begin{align*}M:= E(\eta). \end{align*}$$

(Note that this field is denoted by L in [11, p. 433, l. 9], which is an unfortunate clash of notation with the field L defined earlier in loc. cit. (see [11, p. 407]).

Write ${\widetilde {M}}$ for the extension of M obtained by adjoining the elements

$$\begin{align*}\{ y^{1/\ell} \mid y \in O^{\times}_{E} \}. \end{align*}$$

It is shown in loc. cit. that for each prime ${\mathfrak p}$ of F satisfying the following Frobenius conditions, there exists an idele class character $\theta $ of E satisfying the properties we seek.

Property 10.5 For every prime $\mathfrak {P}$ of $\widetilde {M}$ lying above ${\mathfrak p}$ , the Frobenius element $\delta =(\mathfrak {P}, \widetilde {M}/F)$ satisfies:

1. (F1) $\delta ^2=1$ .

2. (F2) $\delta |_{E}$ is nontrivial (so ${\mathfrak p}$ does not split in $E/F$ ).

3. (F3) $\delta \mid _{F(\eta )}$ is nontrivial (so ${\mathfrak p}$ satisfies Property 10.3(b)).

The set of primes ${\mathfrak p}$ of F satisfying Property 10.5 has positive Chebotarev density, and all such primes lie in $\Sigma _{-}$ .

Let $\theta $ be an idele class character of E as constructed in Theorem 10.4, and let $N/E$ denote the extension cut out by $\theta $ . Then $N/E$ is cyclic of order $\ell $ , ramified (necessarily totally) at ${\mathfrak p}_E$ , and at no other primes of E. As $\theta $ is of dihedral type, the extension $N/F$ is dihedral of order $2\ell $ .

Set $\psi := \omega \theta $ . Then $\psi $ is an idele class character of E of quaternion type, and we deduce that $F_{\pi (\psi )}:= NL$ is an $H_{4\ell }$ extension of F. (Note that the field that we call $F_{\pi (\psi )}$ is denoted by the symbol $F_\psi $ in [11].) The extension $F_{\pi (\psi )}/F$ is ramified only at ${\mathfrak p}$ , with ramification index $\ell $ . We have the following diagram of fields and corresponding idele class characters (where we write ${\varphi }$ for ${\varphi }_{E/F}$ ):

To complete the proof of Lemma 10.1, it suffices to show that in Theorem 10.4, there are infinitely many choices of ${\mathfrak p}$ (and so of $\theta $ ) such that the decomposition group of ${\mathfrak p}$ in $F_{\pi (\psi )}/F$ is not abelian. This is equivalent to imposing an additional Frobenius condition on ${\mathfrak p}$ . In order to do this, we require the following lemma.

Lemma 10.6 The extensions $\widetilde {M}/E$ and $L/E$ are linearly disjoint. Hence, $[{\widetilde {M}}L:{\widetilde {M}}] = 2$ .

Proof The extension $\widetilde {M}/E$ has a unique quadratic subextension, viz. the unique quadratic subextension of $M/E$ (recall that $M = E(\eta )$ ). This extension is ramified at places above ${\mathfrak p}$ , and so cannot be equal to the unramified quadratic extension $L/E$ .

We now fix an element $\delta _1 \in \operatorname {\mathrm {Gal}}({\widetilde {M}}L/F)$ which maps under the obvious quotient map onto the element $\delta \in \operatorname {\mathrm {Gal}}({\widetilde {M}}/F)$ constructed in the proof of Theorem 10.4 (see (10.5)), and we consider the set of primes ${\mathfrak p}$ of F satisfying the following Frobenius condition.

Property 10.7 For every prime $\mathfrak {Q}$ of $\widetilde {M}L$ lying above ${\mathfrak p}$ ,

1. (F4) the Frobenius element $(\mathfrak {Q}, \widetilde {M}L/F)$ lies in the conjugacy class of $\delta _1$ .

The set of primes ${\mathfrak p}$ satisfying (10.7) has positive Chebotarev density, and plainly if ${\mathfrak p}$ satisfies (10.7), then it also satisfies (10.5).

Suppose that ${\mathfrak p}$ satisfies (10.7). Then the corresponding extension $F_{\pi (\psi )}/F$ constructed above is an $H_{4\ell }$ -extension unramified outside ${\mathfrak p}$ , in which ${\mathfrak p}$ is nonsplit and ramified, with ramification index $\ell $ . Hence, $F_{\pi (\psi )}/F$ an extension satisfying the conditions of Lemma 10.1.

This completes the proof of Lemma 10.1.

Remark 10.8 It is shown in [11, Theorem 4] that for the extensions $F_{\pi (\psi )}/F$ constructed above satisfying the conditions of Lemma 10.1, we have

$$\begin{align*}W(F_{\pi(\psi)}/F) = \varphi_{E/F}({\mathfrak p}) = -1. \end{align*}$$

This implies that $(O_{\pi (\psi )})\neq 0$ (see (1.1)), and so, since ${\mathcal J}^*_\infty (F_{\pi (\psi )}/F) \neq 0$ , it follows from Theorem 1.5 that $(A_{\pi (\psi )}) = 0$ .

Remark 10.9 Dominik Bullach has explained to us how explicit counterexamples to Conjecture 1.4 can also be derived from Theorem 1.5 by using general results of Neukirch on the embedding problem (see [21]) rather than the explicit computations of Fröhlich in [11].