A genus formula for the positive \'{e}tale wild kernel

TL;DR

This paper establishes a genus formula relating the positive étale wild kernel groups of a number field and its Galois extension, extending the understanding of their structure in algebraic number theory.

Contribution

It introduces a genus formula for the positive étale wild kernel, connecting the kernel groups of a number field and its Galois extensions, a novel result in the field.

Findings

Derived a genus formula linking kernel groups of extensions and base fields.

Established the relationship between Galois group actions and kernel group orders.

Enhanced the theoretical framework for studying étale wild kernels in number fields.

Abstract

Let $F$ be a number field and let $i\geq 2$ be an integer. In this paper, we study the positive \'{e}tale wild kernel $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E/F$ is a Galois extension of number fields with Galois group $G$, we prove a genus formula relating the order of the groups $ (\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}E)_{G}$ and $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$.

Equations

$$H^2(G_{F,S},{\mathbf{Z}}_p(i))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{+}^2(G_{F,S},{\mathbf{Z}}_2(i))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{sgn}}_F:H^1(F,{\mathbf{Z}}_2(i))/2\cong D_F^{(i)}/F^{{\bullet }^2}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$W{K}_{2i-2}^{\text{mbox}\acute{e}t,+}F\cong W{K}_{2i-2}^{\text{mbox}\acute{e}t}F.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{+}^1(F,{\mathbf{Z}}_2(i))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\frac{|(W{K}_{2i-2}^{\text{mbox}\acute{e}t,+}E{)}_G|}{|W{K}_{2i-2}^{\text{mbox}\acute{e}t,+}F|}=\frac{|X_{E/F}^{(i)}|.{\prod }_{v\in S_f}|H_1(G_v,H^2(E_w,{\mathbf{Z}}_2(i)))|}{|H_1(G,H^0(E,{\mathbf{Q}}_2/{\mathbf{Z}}_2(1-i){)}^{\vee })|.[H_{+}^1(F,{\mathbf{Z}}_2(i)):{\mathcal{H}}_{+}^{1,\mathcal{N}}(F,{\mathbf{Z}}_2(i))]}$$

$$\kappa :H_2(G,{\oplus }_{w\in S_f}{H}^2(E_w,{\mathbf{Z}}_2(i)))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\frac{|(W{K}_{2i-2}^{\text{mbox}\acute{e}t,+}E{)}^G|}{|W{K}_{2i-2}^{\text{mbox}\acute{e}t,+}F|}=\frac{2^{r(E/F)-1+t}}{[D_F^{+(i)}:D_F^{+(i)}\cap N_G{E}^{\bullet }]}$$

$$M\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{+}^n(G_{F,S},M):=H^{n-1}(G_{F,S},M_{+}).$$

$$\cdots \text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^p(G,H_{+}^q(G_{E,S},M))⟹H_{+}^{p+q}(G_{F,S},M).$$

$$H_p(G,H_{+}^q(G_{E,S},M))⟹H_{+}^{p+q}(G_{F,S},M).$$

$$H_{+}^2(G_{E,S},M{)}_G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{+}^2(G_{E,S},{\mathbf{Z}}_2(i){)}_G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\begin{array}[]{cccccc}H^{n}(F_{v},M)&\times&H^{2-n}(F_{v},M^{\ast})&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces&H^{2}(F_{v},\mu_{2^{\infty}})\simeq{\bf Q}_{2}/{\bf Z}_{2},&\mbox{if $v$ is finite}\\ &&&&\\ \widehat{H}^{n}(F_{v},M)&\times&\widehat{H}^{2-n}(F_{v},M^{\ast})&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces&H^{2}(F_{v},\mu_{2^{\infty}}),&\mbox{if $v$ is infinite}\end{array}$$

$${\oplus }_{v\in S_f}{H}^0(F_v,M)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{III}_{S}^{n}(M):=\ker(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 29.21576pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-29.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{H^{n}(G_{F,S},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 47.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 47.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bigoplus_{v\in S_{f}}H^{n}(F_{v},M)}$}}}}}}}\ignorespaces}}}}\ignorespaces)$$

$$\mathrm{III}_{S}^{n,+}(M):=\ker(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 29.21576pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-29.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{H^{n}_{+}(G_{F,S},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 47.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 47.21576pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bigoplus_{v\in S_{f}}H^{n}(F_{v},M)}$}}}}}}}\ignorespaces}}}}\ignorespaces).$$

$$\begin{array}[]{ccccc}\mathrm{III}_{S}^{n,+}(M)&\times&\mathrm{III}_{S}^{3-n}(M^{\ast})&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces&{\bf Q}_{2}/{\bf Z}_{2}.\end{array}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\oplus }_{v\in S_f}{H}^1(F_v,M)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{III}}_S^{1,+}(M{)}^{\vee }\cong {\mathrm{III}}_S^2(M^{\ast })\text{mbox}and{\mathrm{III}}_S^{2,+}(M{)}^{\vee }\cong {\mathrm{III}}_S^1(M^{\ast }).$$

$$\begin{array}[]{cccc}&F^{\bullet}/F^{\bullet^{2}}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces&\oplus_{v\;real}{\bf Z}/2{\bf Z}\\ &x&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 21.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces&(\mathrm{sgn}_{v}(x))_{v\;real}\end{array}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^1(F,\mathbf{Z}/2(i))$$

Full text

1. Introduction

Let $F$ be a number field and let $p$ be a prime number. For a finite set $S$ of primes of $F$ containing the $p$-adic primes and the infinite primes, let $G_{F,S}$ be the Galois group of the maximal algebraic extension $F_{S}$ of $F$ which is unramified outside $S$. It is well known that for all integer $i\geq 2$, the kernel of the localization map

[TABLE]

is independent of the choice of the set $S$ [Sc 79, §6, Lemma 1] [Ko 03, page 336]. This kernel is called the étale wild kernel [Ng 92, Ko 93], and denoted by $WK_{2i-2}^{\mbox{\'{e}t}}F$. It is the analogs of the $p$-part of the classical wild kernel $WK_{2}F$ which occurs in Moore’s exact sequence of power norm symbols (cf. [M 71]).

Let $E/F$ be a Galois extension with Galois group $G$. For a fixed odd prime $p$, several authors have studied Galois co-descent and proved genus formulas [Ko-Mo 00, Gri 05, As-As, As], for the étale wild kernel, which is analogue to the Chevalley genus formula for the class groups. In this paper we settle the case $p=2$. For this purpose, we use a slight variant of cohomology, the so-colled totally positive Galois cohomology [Ka 93, §5]. More precisely, for all integer $i$, we are interested in the kernel of the map

[TABLE]

Here $S_{f}$ denotes the set of finite primes in $S$ and $H^{j}_{+}(.,.)$ denotes the $j$-th totally positive Galois cohomology groups (Section 2.1). When $i=1$, this kernel is isomorphic to the $2$-primary part of the narrow $S$-class group of $F$. For $i\geq 2$, we show that this kernel is independent of the set $S$; it is referred to as the positive étale wild kernel, and is denoted by $WK_{2i-2}^{\mbox{\'{e}t},+}F$. It is analogue to the narrow $S$-class group of $F$, and fits into an exact sequence (Proposition 2.6)

[TABLE]

where $D_{F}^{(i)}$ (see [Ko 03, Defintion 2.3]) is the étale Tate kernel and $D_{F}^{+(i)}/F^{\bullet^{2}}$ is the kernel of the signature map

[TABLE]

Here $r_{1}$ denotes the number of real places of $F$. In particular,

• •

for $i$ even, we have

[TABLE]

• •

for $i$ odd, we have an exact sequence

[TABLE]

where $\delta_{i}(F)$ is the $2$-rank of the cokernel of the signature map $\mathrm{sgn}_{F}$.

For a place $v$ of $F$, let $G_{v}$ denote the decomposition group of $v$ in $E/F$. Define the plus normic subgroup $H^{1,\mathcal{N}}_{+}(F,{\bf Z}_{2}(i))$ to be the kernel of the map

[TABLE]

where $N_{G_{v}}=\sum_{\sigma\in G_{v}}\sigma$ is the norm map, and if $v$ is a prime of $F$, we denote by $w$ a prime of $E$ above $v$.We prove the following genus formula for the positive étale wild kernel.

Theorem**.**

Let $E/F$ be a Galois extension of number fields with Galois group $G$. Then for every $i\geq 2$, we have

[TABLE]

The group $X^{(i)}_{E/F}$ (Definition 3.2) has order at most $|H_{2}(G,H^{0}(E,{\bf Q}_{2}/{\bf Z}_{2}(1-i))^{\vee})|$, and is trivial if the canonical morphism

[TABLE]

is surjective.In particular, if $E/F$ is a relative quadratic extension of number fields, the order of the group $X^{(i)}_{E/F}$ is at most $2$. In this case we give, in the last section, a genus formula involving the positive Tate kernel $D^{+(i)}_{F}$. Roughly speaking, let $F_{\infty}$ (resp. $F_{v,\infty}$) denote the cyclotomic ${\bf Z}_{2}$-extension of $F$ (resp. $F_{v}$) and let $R_{E/F}$ be the set of both finite primes tamely ramified in $E/F$ and $2$-adic primes such that $E_{w}\cap F_{v,\infty}\neq E_{w}$. Then for any odd integer $i\geq 2$, we have

1. (i)

if $E\subseteq F_{\infty}$, the positive étale wild kernel satisfies Galois codescent; 2. (ii)

if $E\nsubseteq F_{\infty}$,

[TABLE]

where $r(E/F)=|R_{E/F}|$ and $t\in\{0,1\}$. Moreover, $t=0$ if $R_{E/F}\neq\emptyset$.

2. Positive étale wild kernel

2.1. Totally positive Galois cohomology

Let $F$ be a number field and let $S$ be a finite set of primes of $F$ containing the set $S_{2}$ of dyadic primes and the set $S_{\infty}$ of archimedean primes. For a place $v$ of $F$, we denote by $F_{v}$ the completion of $F$ at $v$, and by $G_{F_{v}}$ the absolute Galois group of $F_{v}$.For a discrete or a compact ${\bf Z}_{2}[[\mathrm{G_{F,S}}]]$-module $M$, we write $M_{+}$ for the cokernel of the map

[TABLE]

where $\mathrm{Ind}^{G_{F}}_{G_{F_{v}}}M$ denotes the induced module. Hence we have the exact sequence

[TABLE]

Following [Ka 93, §5], we define the $n$-$\mathrm{th}$ totally positive Galois cohomology group $H^{n}_{+}(G_{F,S},M)$ of $M$ by

[TABLE]

Recall from [Ka 93, §5] some facts about the totally positive Galois cohomology.

Proposition 2.1**.**

We have the following properties:

1. (i)

There is a long exact sequence

[TABLE] 2. (ii)

$H^{n}_{+}(G_{F,S},M)=0$* for all $n\neq 1,2$.* 3. (iii)

If $E/F$ is an extension unramified outside $S$ with Galois group $G$ then there is a cohomological spectral sequence

[TABLE]

$\square$**

We also have a Tate spectral sequence

[TABLE]

Hence, for a finite $2$-primary Galois module $M$, we have an isomorphism

[TABLE]

([We 06, Lemma 6.4]). In particular, by passing to the inverse limit, the corestriction map induces an isomorphism

[TABLE]

Recall the local duality Theorem (e.g. [Mi 86, Corollary I.2.3]): For $n=0,1,2$ and for every place $v$ of $F$, the cup product

[TABLE]

is a perfect pairing, where $\widehat{H}^{n}(F_{v},.)$ is the Tate cohomology group, $\mu_{2^{\infty}}$ is the group of all roots of unity of $2$-power order, and $(.)^{\ast}$ means the Kummer dual: $M^{\ast}=\mathrm{Hom}(M,\mu_{2^{\infty}})$.We have an analogue of the Poitou-Tate long exact sequence

Proposition 2.2**.**

Let $S_{f}$ denote the set of finite places in $S$. Then there is a long exact sequence

[TABLE]

where the subscript $(.)^{\vee}$ refers to the Pontryagin dual: $M^{\vee}=\mathrm{Hom}(M,{\bf Q}_{2}/{\bf Z}_{2})$.

Proof.

See [Ma 18, Proposition 2.6]. ∎

For a ${\bf Z}_{2}[[G_{F,S}]]$-module $M$ and $n=1,2$ we define the groups $\mathrm{III}_{S}^{n}(M)$ and $\mathrm{III}_{S}^{n,+}(M)$ to be the kernels of the localization maps

[TABLE]

and

[TABLE]

We state a Poitou-Tate duality in the case $p=2$ as a consequence of Proposition 2.2 and local duality (2).

Corollary 2.3**.**

Let $n=1,2$. Then there is a perfect pairing

[TABLE]

Proof.

By Proposition 2.2 we have the exact sequences

[TABLE]

and

[TABLE]

Dualizing these exact sequences and using the local duality $(\ref{local duality})$, we get

[TABLE]

∎

2.2. Signature

In this subsection we recall some properties of the signature map (see e.g. [Ko 03], [As-Mo 18, §1.2]). For any real place $v$ of the number field $F$, let $i_{v}:F\longrightarrow{\bf R}$ denote the corresponding real embedding. The natural signature maps $\mathrm{sgn}_{v}:F^{\bullet}\longrightarrow{\bf Z}/2{\bf Z}$ (where $\mathrm{sgn}_{v}(x)=0$ or $1$ according to $i_{v}(x)>0$ or not) give rise to the following surjective map

[TABLE]

The exact sequence of $G_{F}$-modules

[TABLE]

gives rise to an exact sequence

[TABLE]

where for an abelian group $A$, $A/2$ denotes the cokernel of the multiplication by $2$ on $A$.Since we have

[TABLE]

there exists a subgroup $D_{F}^{(i)}$ (the étale Tate kernel) of $F^{\bullet}$ containing $F^{\bullet^{2}}$ such that

[TABLE]

We will consider the restriction of the above signature map to the quotient $D_{F}^{(i)}/F^{\bullet^{2}}$:

[TABLE]

where $r_{1}$ is the number of real places of $F$.

Let $D_{F}^{+(i)}/F^{\bullet^{2}}$ be the kernel and $({\bf Z}/2)^{\delta_{i}(F)}$ be the cokernel of $\mathrm{sgn}_{F}$, respectively. So we have an exact sequence

[TABLE]

If $i$ is an even integer, the signature map

[TABLE]

is trivial [As-Mo 18, Proposition 1.2], and then $D_{F}^{+(i)}=D_{F}^{(i)}$.

2.3. Positive étale wild kernel

Following [Ng 92, Ko 93], the étale wild kernel $WK_{2i-2}^{\mbox{\'{e}t}}F$ is the group

[TABLE]

For $i\geq 2$, it is well known that the étale wild kernel $WK_{2i-2}^{\mbox{\'{e}t}}F$ is independent of the set $S$ containing the $p$-adic primes and the infinite primes ([Sc 79, §6, Lemma 1], [Ko 03, page 336]).There have been much work on the Galois co-descent for the étale wild kernel at odd primes [Ko-Mo 00, Gri 05, As-As, As]. The case $p=2$ has been studied essentially in the classical case $i=2$ [Ko-Mo 00, Ko-Mo 03, Gri 05]. The situation for $p=2$ is more complicated, since the cohomology groups $H^{k}({\bf R},{\bf Z}_{2}(i))$ and $H^{k}({\bf R},{\bf Q}_{2}/{\bf Z}_{2}(i))$ do not necessarily vanish, and the group $H^{2}(G_{E,S},{\bf Z}_{2}(i))$ does not satisfy Galois co-descent. This motivates the following definition of the positive étale wild kernel.Let $S_{f}$ denote the set of finite primes in $S$ and let $\mathcal{O}_{F,S}$ be the ring of $S$-integers of $F$. For all $i\in{\bf Z}$, recall the last three terms of the Poitou-Tate exact sequence (Proposition 2.2):

[TABLE]

Definition 2.4**.**

Let $i\in{\bf Z}$. We define the positive étale wild kernel $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$ to be the kernel of the localization map

[TABLE]

Remark 2.5**.**

For $i=1$, the group $WK_{0}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$ is isomorphic to the $2$-part of the narrow $S$-class group $A_{F,S}^{+}$ of $F$. In particular it depends on the set $S$. Indeed, on the one hand Corollary 2.3 shows that

[TABLE]

On the other hand

[TABLE]

It follows that $WK_{0}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}\cong A_{F,S}^{+}$.

Hence the positive étale wild kernel plays a similar role as the $2$-primary part of the narrow $S$-class group. We restrict our study to the case $i\geq 2$, and we will show that $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$ is independent of the set $S$ containing the $2$-adic primes and the infinite primes. However, all the results remain true for $i\neq 1$ if we assume the finiteness of the Galois cohomology group $H^{2}(G_{F,S},{\bf Z}_{2}(i))$. Note that for $i=0$, this finiteness is equivalent to the Leopoldt conjecture, and for $i\geq 2$ this is true as a consequence of the finiteness of the $K$-theory groups $K_{2i-2}\mathcal{O}_{F,S}$ and the connection between $K$-theory and étale cohomology via Chern characters [So 79, Dw-Fr 85].The following proposition gives the link between the kernels $WK^{\mbox{\'{e}t},+}_{2i-2}\mathcal{O}_{F,S}$ and $WK^{\mbox{\'{e}t}}_{2i-2}F$.

Proposition 2.6**.**

For all integer $i\geq 2$, there exists an exact sequence

[TABLE]

In particular,

• •

if $i$ is even, there is an isomorphism

[TABLE]

• •

if $i$ is odd, we have an exact sequence:

[TABLE]

where $\delta_{i}(F)$ is the $2$-rank of the cokernel of the signature map $\mathrm{sgn}_{F}$.

Proof.

On the one hand, the exact sequence

[TABLE]

gives rise to an exact commutative diagram

[TABLE]

where the vertical maps are the localization maps. Since

[TABLE]

for all infinite place $v$ of $F$, we get

[TABLE]

Observe that the composite

[TABLE]

is the signature map

[TABLE]

and then

[TABLE]

On the other hand, by the definition of totally positive Galois cohomology, we have the exact sequence

[TABLE]

where for $n=1\,\mbox{or}\,2$, $H^{n}_{v}$ denotes the cohomology group $H^{n}(F_{v},{\bf Z}_{2}(i))$. Since

[TABLE]

for all infinite place $v$ of $F$, we get

[TABLE]

Therefore, we have the following exact commutative diagram

[TABLE]

By the snake lemma and (7), we obtain the exact sequence

[TABLE]

Since $\bigoplus_{v\mid\infty}H^{1}(F_{v},{\bf Z}_{2}(i))$ is isomorphic to $({\bf Z}/2)^{r_{1}}$ if $i$ is odd, and is trivial if $i$ is even, we obtain

• •

for $i$ even, $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}\cong WK^{\mbox{\'{e}t}}_{2i-2}F$, and

• •

for $i$ odd, we have the exact sequence

[TABLE]

where $\delta_{i}(F)$ is the $2$-rank of the cokernel of the signature map $\mathrm{sgn}_{F}$.

∎

The following corollary shows that $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$ is in fact independent of the set $S$ containing $S_{2\infty}=S_{2}\cup S_{\infty}$ for $i\geq 2$.

Corollary 2.7**.**

For $i\geq 2$, the positive étale wild kernel $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$ is independent of the set $S$ containing $S_{2\infty}$.

Proof.

Since $\delta_{i}(F)$ and $WK^{\mbox{\'{e}t}}_{2i-2}F$ are independent of the set $S$ [Ko 03, page 336 ], the exact sequence (8) shows that the order of $WK^{\mbox{\'{e}t},+}_{2i-2}\mathcal{O}_{F,S}$ is also independent of $S$. Therefore it suffices to prove that there exists an injective map from $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S_{2\infty}}$ to $WK_{2i-2}^{\mbox{\'{e}t},+}\mathcal{O}_{F,S}$. But this follows from the exact commutative diagram

[TABLE]

where the middle vertical map is the inflation map. ∎

From now on, we make the notation

[TABLE]

We finish this subsection by giving a description of $WK_{2i-2}^{\mbox{\'{e}t},+}F$ as an Iwasawa module. Let $X^{\prime}_{\infty}$ be the Galois group of the maximal unramified $p$-extension of the cyclotomic ${\bf Z}_{p}$-extension of $F$, which is completely decomposed at all primes above $p$. It is well known that ([Sc 79, Lemma 1,§6]) for any odd prime $p$

[TABLE]

In the next proposition, we prove an analogue result in the case $p=2$. Let $F_{\infty}$ be the cyclotomic ${\bf Z}_{2}$-extension of $F$ with Galois group $\Gamma=\mathrm{Gal}(F_{\infty}/F)$, and let $X^{\prime,+}_{\infty}$ be the Galois group of the maximal $2$-extension of $F_{\infty}$, which is unramified at finite places and completely decomposed at all primes above $2$.

Proposition 2.8**.**

Let $i\geq 2$ be an integer. If either $i$ is odd, or $i$ is even and $\sqrt{-1}\in F$, then

[TABLE]

In particular, in both cases we recover that the group $WK_{2i-2}^{\mbox{\'{e}t},+}F$ is independent of the set $S$ containing $S_{2\infty}$.

Proof.

First observe that, if $i$ is odd or $i$ is even and $\sqrt{-1}\in F$, then

[TABLE]

Indeed, in both cases, ${\bf Q}_{2}/{\bf Z}_{2}(1-i)$ is a $\Gamma$-module and using [Se 68, §XIII.1, Proposition 1], we see that

[TABLE]

where $\gamma$ is a topological generator of $\Gamma$. Hence

[TABLE]

Now we consider the following exact commutative diagram

[TABLE]

where $j=1-i$, $H^{1}(F_{v,\infty},{\bf Q}_{2}/{\bf Z}_{2}(j))=\bigoplus_{w\mid v}H^{1}(F_{w,\infty},{\bf Q}_{2}/{\bf Z}_{2}(j))$, and $\Gamma_{v}$ denotes the decomposition group of $v$ in $F_{\infty}/F$. By (10), we have

[TABLE]

and then

[TABLE]

Hence, using the duality

[TABLE]

(Corollary 2.3), we obtain the isomorphism

[TABLE]

∎

Let $F_{\infty}=\cup_{n}F_{n}$ be the cyclotomic ${\bf Z}_{2}$-extension of $F$ and for $n\geq 0$, $G_{n}=\mathrm{Gal}(F_{n}/F)$. The above description of the positive étale wild kernel, leads immediately to the following corollary:

Corollary 2.9**.**

If either $i$ is odd, or $i$ is even and $\sqrt{-1}\in F$, then the positive étale wild kernel satisfies Galois co-descent in the cyclotomic ${\bf Z}_{2}$-extension:

[TABLE]

Compare to [Ko-Mo 00, Theorem 2.18], which deals with the case $i=2$ and $\sqrt{-1}\in F$. If $p$ is odd, the Galois co-descent holds in the cyclotomic tower as a consequence of Schneider’s description of the étale wild kernel.

3. Genus formula

Let $E/F$ be a Galois extension of number fields with Galois group $G$. Let $S$ denote the set of infinite places, $2$-adic places and those which ramify in $E/F$. We denote also by $S$ the set of places of $E$ above places in $S$. In the sequel we assume that $i\geq 2$. By the definition of $WK_{2i-2}^{\mbox{\'{e}t},+}F$ and Proposition 2.2, we have the exact sequence

[TABLE]

where $\widetilde{\oplus}_{v\in S_{f}}H^{2}(F_{v},{\bf Z}_{2}(i))$ denotes the kernel of the surjective map

[TABLE]

Then the corestriction map induces the exact commutative diagram

[TABLE]

where the middle vertical map is an isomorphism by $(\ref{cor iso})$. Using the snake lemma, we get

• •

$\mathrm{coker}N_{i}\cong\ker N_{i}^{\prime}$.

• •

$\ker N_{i}\cong\mathrm{coker}(\widetilde{\alpha})$, where $\widetilde{\alpha}$ is the homology map

[TABLE]

We first determine $\mathrm{coker}N_{i}$, and then we give a criterion of the surjectivity of the morphism $N_{i}$. For this, the exact commutative diagram

[TABLE]

shows that

[TABLE]

Now we give a description of $H_{1}(G_{v},H^{2}(E_{w},{\bf Z}_{2}(i)))$ and $H_{1}(G,H^{0}(E,{\bf Q}_{2}/{\bf Z}_{2}(1-i))^{\vee})$. We need some notation

[TABLE]

Proposition 3.1**.**

Let $i\geq 2$ be an integer and let $v$ be a finite place of $F$. If either $i$ is odd, or $i$ is even and $\sqrt{-1}\in F$, then we have

[TABLE]

In particular, the map $N_{i}$ is surjective if and only if $L_{\infty}^{+}\cap E_{\infty}=F_{\infty}$.

Proof.

Using the assumption and (10), we have

[TABLE]

Then $(1)$ can be proved with the same argument of Proposition $2.1$ of [As-As]. The second assertion is a direct consequence of the first one. ∎

To prove a genus formula for the positive étale wild kernel, we give a description of

[TABLE]

Consider the following exact commutative diagram

[TABLE]

Then we have an exact sequence

[TABLE]

Definition 3.2**.**

We define the module $X^{(i)}_{E/F}$ as

[TABLE]

where $\kappa$ is the homology map

[TABLE]

So

[TABLE]

We have the following comparison between $|(WK_{2i-2}^{\mbox{\'{e}t},+}E)_{G}|$ and $|WK_{2i-2}^{\mbox{\'{e}t},+}F|$.

Proposition 3.3**.**

For any integer $i\geq 2$, we have

[TABLE]

Proof.

On the one hand, by $(\ref{exact sequence of kappa alpha})$ and the definition of $X^{(i)}_{E/F}$, we have an exact sequence

[TABLE]

On the other hand the exact commutative diagram $(\ref{exact commutaive diagram widetilde})$ shows that

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

∎

Now we are going to compute $|\mathrm{coker\alpha}|$. For every $q\in{\bf Z}$, we have an isomorphism

[TABLE]

given by cup-product ([CKPS, Proposition 3.1]), where $\widehat{H}^{\ast}(.,.)$ denotes the Tate cohomology. Then the commutative diagram

[TABLE]

shows that

[TABLE]

Definition 3.4**.**

Let $H^{1,\mathcal{N}}_{+}(F,{\bf Z}_{2}(i))$ denote the kernel of the map

[TABLE]

where $N_{G_{v}}=\sum_{\sigma\in G_{v}}\sigma$ is the norm map.

The isomorphism (17) shows that

[TABLE]

Hence

[TABLE]

This yields our main result:

Theorem 3.5**.**

Let $E/F$ be a Galois extension of number fields with Galois group $G$. Then for every $i\geq 2$, we have

[TABLE]

$\square$**

Remark 3.6**.**

The above genus formula involves the order of the group $X^{(i)}_{E/F}$ which seems to be difficult to compute. If we make the following hypothesis $(\mathcal{H})$:”the morphism

[TABLE]

*then $X^{(i)}_{E/F}$ is trivial.

This hypothesis is satisfied if there is a prime $v_{0}$ of $F$ such that:*

• •

$H^{0}(E,{\bf Q}_{2}/{\bf Z}_{2}(1-i))\cong H^{0}(E_{w_{0}},{\bf Q}_{2}/{\bf Z}_{2}(1-i))$* and*

• •

$v_{0}$* is an undecomposed $2$-adic prime, or $v_{0}$ is a totally and tamely ramified prime in $E/F$.*

Remark 3.7**.**

The groups $H_{1}(G_{v},H^{2}(E_{w},{\bf Z}_{2}(i)))$ can be easily computed (at least if $i$ is odd, or $i$ is even and $\sqrt{-1}\in F$ by Proposition 3.1). The difficult part here is the norm index $[H^{1}_{+}(F,{\bf Z}_{2}(i)):\mathcal{H}^{1,\mathcal{N}}_{+}(F,{\bf Z}_{2}(i))]$. When $E/F$ is a relative quadratic extension, we obtain a genus formula involving the norm index $[D_{F}^{+(i)}:D_{F}^{+(i)}\cap N_{G}E^{\bullet}]$. Moreover, if $F$ has one $2$-adic prime, we use [As-Mo 18, §4] to give an explicit description of this norm index in terms of the ramification in $E/F$.

4. Relative quadratic extension case

In this section we focus on relative quadratic extensions of number fields $E/F$ with Galois group $G$. For such extensions, we give a genus formula for the positive étale wild kernel involving the norm index $[D_{F}^{+(i)}:D_{F}^{+(i)}\cap N_{G}E^{\bullet}]$, where $D^{+(i)}_{F}$ is the positive Tate kernel. First recall that for every even integer $i\geq 2$, Proposition 2.6 says that the étale wild kernel and the positive étale wild kernel coincide. A genus formula has been obtained by Kolster-Movahhedi [Ko-Mo 00, Theorem 2.18] for $i=2$, and by Griffiths [Gri 05, §4.3], as a generalization, for any even integer $i\geq 2$. This genus formula can be used to determine families of abelian $2$-extensions with trivial $2$-primary Hilbert kernel [Ko-Mo 03, Le 04, Gr-Le 09]. Throughout this section we keep the notations of the previous sections and we assume that the integer $i\geq 2$ is odd.We need to calculate the order of $\mathrm{coker}\alpha$ (see Proposition 3.3). Since $G$ has order $2$, we have the following exact commutative diagram

[TABLE]

Likewise as in the global case, there exists a subgroup $D_{v}^{(i)}$ of $F_{v}^{\bullet}$ containing $F_{v}^{\bullet^{2}}$ such that

[TABLE]

for each $v\in S_{f}$. Then, we have a natural isomorphism

[TABLE]

where $w$ is a prime of $E$ above $v$. Hence,

[TABLE]

On the one hand, there exists a surjective map

[TABLE]

Indeed, the exact sequence

[TABLE]

induces the exact sequence

[TABLE]

Then, by the definition of $D_{F}^{+(i)}$, we have a surjective map

[TABLE]

Hence, the surjectivity of the map (20) follows from the exact commutative diagram

[TABLE]

On the other hand, since $i$ is odd, the canonical sujection map

[TABLE]

is an isomorphism, as a consequence of [Gri 05, Lemma 4.2.1]. Therefore,

[TABLE]

it follows that

[TABLE]

where, in the last equality, $v$ runs through all places of $F$. Then, using the Hasse norm theorem ($G$ is cyclic), we get

[TABLE]

Therefore, we obtain

[TABLE]

Moreover, the order of the group $X^{(i)}_{E/F}$ (see Definition 3.2) is at most $2$:

[TABLE]

Indeed, by the definition of $X^{(i)}_{E/F}$, we have

[TABLE]

since $G$ is cyclic of order $2$. Hence, by Proposition 3.3, we get

[TABLE]

where $t\in\{0,1\}$. First, assume that $E\subseteq F_{\infty}$. Then for every finite prime $v$, we have

[TABLE]

by Proposition 3.1. Since $G$ is cyclic, we also have

[TABLE]

and then, using the commutative diagram (14), we see that

[TABLE]

Then the map

[TABLE]

is an isomorphism, as a consequence of Proposition 3.1 and the fact that $\ker N_{i}\cong\mathrm{coker}\widetilde{\alpha}$. Now, if $E\nsubseteq F_{\infty}$, then

[TABLE]

and if $v\nmid 2$ is a finite ramified prime in $E/F$ or $v$ is a $2$-adic prime such that $E_{w}\cap F_{v,\infty}\neq E_{w}$, then

[TABLE]

Since $G$ is cyclic, $|(WK_{2i-2}^{\mbox{\'{e}t},+}E)_{G}|=|(WK_{2i-2}^{\mbox{\'{e}t},+}E)^{G}|$. We can now formulate the genus formula for a relative quadratic extension:

Proposition 4.1**.**

Let $E/F$ be a relative quadratic extension of number fields with Galois group $G$ and let $R_{E/F}$ be the set of finite primes tamely ramified in $E/F$ or $2$-adic primes such that $E_{w}\cap F_{v,\infty}\neq E_{w}$. Then for any odd positive integer $i\geq 2$

1. (i)

if $E\subseteq F_{\infty}$ then the positive étale wild kernel satisfies Galois codescent, and 2. (ii)

if $E\nsubseteq F_{\infty}$,

[TABLE]

where $r(E/F)=|R_{E/F}|$ and $t\in\{0,1\}$.

$\square$**

Recall that under the hypothesis $(\mathcal{H})$ the group $X_{E/F}^{(i)}$ is trivial. Moreover, if $E/F$ is a relative quadratic extension of number fields, the hypothesis $(\mathcal{H})$ is satisfied precisely when the set $R_{E/F}$ is nonempty. We obtain

Corollary 4.2**.**

Let $E/F$ be a relative quadratic extension of number fields such that $E\nsubseteq F_{\infty}$ with Galois group $G$. If $R_{E/F}\neq\emptyset$ then we have

[TABLE]

Example 4.3**.**

Assume that $F$ has one $2$-adic prime and that the set $R_{E/F}\neq\emptyset$. Let $S=S_{2}\cup R_{E/F}$. Since $E/F$ is quadratic (hence cyclic), we have

[TABLE]

*by Hasse’s local-global norm principle.

Recall that a set $T$ of primes containing $S_{2}$ is $D_{F}^{+(i)}$-primitive for $(F,2)$ if the canonical map*

[TABLE]

is surjective [As-Mo 18, Definition 4.4]. Let $T$ be a maximal $D_{F}^{+(i)}$-primitive set for $(F,2)$ contained in $S$ and $t_{i}^{+}=|T\backslash S_{2}|$. Then using [As-Mo 18, Proposition 4.11], we have

[TABLE]

*In fact, t^{+}_{i}=\dim_{\mathbf{F}_{2}}\mathrm{Im}(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 19.18874pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-19.18874pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{D_{F}^{+(i)}/F^{\bullet^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 37.18874pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 37.18874pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\oplus_{v\in S\backslash S_{2}}D^{(i)}{v}/F{v}^{\bullet^{2}}}}}}}}}}\ignorespaces}}}}\ignorespaces).

Now, let $E$ be a quadratic field such that $E$ is not contained in the cyclotomic ${\bf Z}_{2}$-extension of ${\bf Q}$. Since the narrow class group of ${\bf Q}$ is trivial, the set $R_{E/{\bf Q}}\neq\emptyset$. Moreover, $WK_{2i-2}^{\mbox{\'{e}t},+}{\bf Q}=0$, by Proposition 2.8. By Corollary 4.2 it follows that*

[TABLE]

for all odd integer $i\geq 2$.

Let $rk_{2}(WK_{2i-2}^{\mbox{\'{e}t},+}E)$ denote the $2$-rank of the étale positive wild kernel $WK_{2i-2}^{\mbox{\'{e}t},+}E$. The extension $E/{\bf Q}$ being quadratic, we have

[TABLE]

(see e.g. [Gr-Le 09, Proposition 1.3]). Hence

[TABLE]

Since $\dim_{{\bf F}_{2}}D_{{\bf Q}}^{+(i)}/{\bf Q}^{\bullet^{2}}=1$ [As-Mo 18, §6], $t_{i}^{+}\in\{0,1\}$. Therefore,

[TABLE]

*In particular, $WK_{2i-2}^{\mbox{\'{e}t},+}E$ vanishes if and only if $E$ is unramified outside a set $\{2,\infty,\ell\}$ with $\ell\equiv\pm 3\mod 8$. *