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 [Reference Agboola and McCulloh2, Section 5] as well as [Reference Agboola and Burns1, Sections 2 and 3] and [Reference Swan25, 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 [Reference Swan25, 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 [Reference Agboola and McCulloh2, 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., [Reference Curtis and Reiner9, 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

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

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., [Reference Fröhlich15, Chapter IV] or [Reference Taylor27, 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., [Reference Fröhlich15, Chapter II] or [Reference Taylor27, 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 $K_0(O_LG, L^c)$ .

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., [Reference Agboola and McCulloh2, 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 [Reference Ullom29]). 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 [Reference Caputo and Vinatier4, p. 109, footnote 1], the proof of [Reference Erez10, 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 [Reference Agboola and McCulloh2, 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. [Reference Tsang28, 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 [Reference Agboola and McCulloh2, 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 $L_{\pi }/L$ .

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. [Reference Agboola and McCulloh2, 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 [Reference Tsang28, 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 $\operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }^{*}_{s})) \cdot \operatorname {\mathrm {Det}}({\mathbf r}_G({\varphi }_{s}))^{-1} \in \operatorname {\mathrm {Det}}(LG)^{\times }$ .

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 [Reference Agboola and McCulloh2, Definition 9.1] and [Reference Tsang28, Definition 2.5.1].

We shall make use of the following alternative descriptions of the above Stickelberger pairing using the standard inner product on $R_G$ (see [Reference Agboola and McCulloh2, 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 $R_G$ .

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 [Reference Erez10, 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 $\chi \in R_{\Gamma }$ .

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

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

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.

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

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

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 [Reference Fröhlich14, 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.)

The following result is entirely analogous to [Reference Fröhlich14, 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.

Proof (a) Recall from [Reference Agboola and McCulloh2, 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 [Reference Fröhlich14, 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 [Reference Fröhlich14, 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 [Reference Fröhlich13]).

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 [Reference Taylor27, 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.

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. [Reference Martinet and Fröhlich18, 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.

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 [Reference Serre22, Section 8.5, Exercise 8.10] and the argument of [Reference Serre22, 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 [Reference Navarro, Sanus and Tiep20, 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 [Reference Fröhlich and Queyrut16].

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 [Reference Martinet and Fröhlich18, 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 [Reference Fröhlich and Queyrut16, 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. [Reference Fröhlich and Queyrut16, 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. [Reference Fröhlich14, 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. [Reference Curtis and Reiner9, 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. [Reference Martinet and Fröhlich18, 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 [Reference Fröhlich14, 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 ${\mathbf Q}^c$ . [Reference Fröhlich14, 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.

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 [Reference Martinet and Fröhlich18, 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. [Reference Fröhlich14, 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 [Reference Fröhlich14, 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

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

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., [Reference McCulloh19, 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

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

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. [Reference Fröhlich14, Chapter II, Lemma 2.1]). Then, reasoning analogously to the proof of [Reference Fröhlich14, 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 [Reference Caputo and Vinatier4].

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 [Reference Fröhlich11]).

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 [Reference Fröhlich14, 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 [Reference Fröhlich11, p. 405].)

For each prime $\ell $ , the symbol $\eta _\ell $ will denote a primitive $\ell $ th root of unity. Then, following [Reference Fröhlich11, Theorem 4], we consider the following conditions on primes.

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 [Reference Fröhlich11, equation (8.5)]).