On the square root of the inverse different

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