# The Lichtenbaum conjecture for abelian extensions of imaginary quadratic
fields

Chaochao Sun  Chaochao Sun, School of Mathematics and Statistics, Linyi
University, Linyi, China 276005<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the cohomological Lichtenbaum conjecture of abelian
extensions of imaginary quadratic fields up to a finite set of bad primes.

###### Key words and phrases:

Lichtenbaum conjecture, abelian extension , imaginary quadratic field

###### 2000 Mathematics Subject Classification:

Primary ; Secondary

##  1\. Introduction

In this paper, we prove the Lichtenbaum conjecture of abelian extensions of
imaginary quadratic fields. For this we compare the elliptic unites with the
etale cohomology by a similiar “Soulé map”. This method has been used by Kings
(see[17]) to prove the Tamagawa conjecture for CM elliptic curves.

The Lichtenbaum conjecture on special values of zeta functions of number
fields at negative integers (see[20]) stems from generalizing the analytic
class number formula. Let $F$ be a number field, $m\geq 2$. Let $R_{m}(F)$ be
the Beilinson regulator of $F$. The cohomological Lichtenbaum conjecture is,
up to sign and up to powers of 2,

$\zeta_{F}(1-m)^{*}=R_{m}(F)\prod_{p}\frac{\\#H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))}{\\#H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))_{\mbox{tors}}}.$

In particular, the cohomological Lichtenbaum conjecture holds for abelian
number fields $F$.

Borel in [3] made a breakthrough on it, proving that the first non-vanishing
coefficient of Talyor series of zeta function at negative integers is equal to
Borel regulator up to a rational factor. Beilinson in [1] formulated far
reaching conjecture describing special values of $L$ function of motives in
terms of Beilinson regulator. In number field case, Borel regulator map is
twice of Beilinson regulator map(see [12]Thm.10.9).

In order to understand the rational factor of special value of $L$ functions
of motives, Bloch and Kato in [2] formulated a conjecture, that is the Bloch-
Kato conjecture of special value of $L$ function, which is expressed in terms
of Haar measures and Tamagawa numbers. The Bloch-Kato conjecture for Artin
motive implies the cohomological Lichtenbaum conjecture(see [14, Theorem1.4.1]
). Fontaine and Perrin-Riou [11] proposed another formulation in terms of
determinants of perfect complexes and Galois cohomology. The two formulations,
are equivalent at least when the field of coefficients is $\mathbb{Q}$, which
is shown by T. Nguyen Quang Do in [21]. The abelian cohomological Lichtenbaum
conjecture was first proved in [18], but with some errors which were
subsequently corrected in the appendix of [5]. J.Coates [6] reproved the
Bloch-Kato conjecture of $\mathbb{Q}$ in a view of Iwasawa theory. Since
rational field $\mathbb{Q}$ and imaginary quadratic field behave similar in
some ways, such as cyclotomic units and elliptic units, this lead us to
consider the Lichtenbaum conjecture of abelian extensions of imaginary
quadratic fields in a similar way.

In the sections 2-4, we recall the known results that we need. Section 5 is
main work of us to calculate an index by Iwasawa theory of imaginary quadratic
fields. Section 6 collect all the foregoing results to obtain the Lichtenbaum
conjecture of abelian extension of imaginary quadratic fields.

Acknowledgements I thank ?? for reading the manuscript and giving many
valuable suggestions.

##  2\. Special value of $L$-funcion of Imaginary quadratic field

Let $K$ be an imaginary quadratic field. Take an abelian extension $F$ of $K$
with conductor $\mathfrak{f}$. let $E$ be an elliptic curve over $F$, with
complex multiplication by $\mathcal{O}_{K}$, satisfying $F(E_{\text{tor}})$
being an abelian extension of $K$. This implies $F\supset K(1)$. The condition
$F(E_{\text{tor}})$ being an abelian extension of $K$ is equivalent to that
there exists a Grössencharacter $\varphi$ of type $(1,0)$ of $K$ such that the
Grössencharacter $\psi$ of $E$ over $F$ can be written as

$\psi=\varphi\circ N_{F/K},$

where $N_{F/K}$ denotes the norm map from $F$ to $K$.

Let $\chi\neq 1$ be an irreducible character of $\text{Gal}(F/K)$, and
$\widetilde{\chi}$ the associated Hecke character mod $\mathfrak{f}$. Then the
Artin $L$-function of character $\chi$ and the Hecke $L$-function of Hecke
character $\widetilde{\chi}$ satisfy the identity

$L(F/K,\chi,s)=\prod_{\mathfrak{p}\in
S}\frac{1}{1-\chi(\varphi_{\mathfrak{P}})\mathfrak{N}(\mathfrak{p})^{-s}}L(\widetilde{\chi},s),$

where $S=\\{\mathfrak{p}|\mathfrak{f}\ |\ \chi(I_{\mathfrak{P}})=1\\}$.

For simplicity, we take $F=K(\mathfrak{f})$, the ray class field of $K$ modulo
$\mathfrak{f}$. Fix an embedding of $K(\mathfrak{f})$ into $\mathbb{C}$ such
that $j(E)=j(\mathcal{O}_{K})$. There is a unique isomorphism

$\theta_{E}:\mathcal{O}_{K}\cong\text{End}_{K(\mathfrak{f})}(E)$

such that $\theta_{E}^{*}(\alpha)=\alpha\omega$ for any invariant differential
$\omega\in\Omega_{E/K(\mathfrak{f})}$ and for all $\alpha\in\mathcal{O}_{K}$.
Viewing $E$ as an elliptic curve over $\mathbb{C}$, we have
$E\cong\mathbb{C}/\Gamma$ where the period lettice
$\Gamma=\Omega\mathcal{O}_{K}$ for some $\Omega\in\mathbb{C}$.

Let $1\neq\mathfrak{m}|\mathfrak{f}$, and $\rho_{\mathfrak{m}}\in I_{K}$ be an
idèle with ideal $\mathfrak{m}$. We can choose a $f_{\mathfrak{m}}\in K^{*}$
with

$v_{\mathfrak{p}}(f_{\mathfrak{m}})\leq 0\ \text{if}\
\mathfrak{p}\nmid\mathfrak{m}\ \text{and}\
v_{\mathfrak{p}}(f_{\mathfrak{m}}^{-1}-(\rho_{\mathfrak{f}})_{\mathfrak{p}}^{-1})\geq
0\ \text{if}\ \mathfrak{p}|\mathfrak{m}.$

Let $G_{\mathfrak{f}}=\text{Gal}(F/K)$. For $g\in G_{\mathfrak{f}}$, let
${}^{g}E$ be the curve obtained by base change

$\begin{CD}{}^{g}E@>{}>{}>E\\\ @V{}V{}V@V{}V{}V\\\ F@>{g}>{}>F\\\
\end{CD}\quad.$

Let $\mathcal{A}=R_{K(\mathfrak{f})/K}E$ be the Weil restriction of the
elliptic curve. $\mathcal{A}$ is an abelian variety over $K$ with CM by a
semisimple $K$-algebra $T$ and Serre-Tate character $\varphi_{\mathcal{A}}$.
Deninger proves that any Hecke character, $\varphi$ of weight $w>0$ is of the
form $\prod_{i=1}^{w}\varphi_{\lambda_{i}}$ where
$\lambda_{i}\in\text{Hom}(T,\mathbb{C})$ and
$\varphi_{\lambda_{i}}=\lambda_{i}\circ\varphi_{\mathcal{A}}$([9]Prop.1.3.1).

Let $[\ ]:\mathbb{C}\to\mathbb{C}/\Gamma_{g}$ be the projection and observe
that
${}^{g}E(\mathbb{C})_{\mathfrak{f}}=\mathfrak{f}^{-1}\Gamma_{g}/\Gamma_{g}$.
Let $\beta=([\Omega f_{\mathfrak{f}}^{-1}])$ be the divisor associated with
the $\mathfrak{f}$-torsion point $[\Omega f_{\mathfrak{f}}^{-1}]$ on $E$ which
is rational over $K(\mathfrak{f})$.

Twisting a complex character $\eta$ by the norm character gives a Hecke
character of $K$ of weight $2$

$\eta N_{K/\mathbb{Q}}=\varphi_{\lambda_{1}}\varphi_{\lambda_{2}}.$

By functoriality, we can increase $\mathfrak{f}$ so that it is a multiple of
the conductors of $\varphi_{\lambda_{1}}$ and $\varphi_{\lambda_{2}}$. Fix a
set of ideals $\\{\mathfrak{b}_{g}\subseteq\mathcal{O}_{K}\\}_{g\in
G_{\mathfrak{f}}}$ with Artin symbol
$(\mathfrak{b}_{g},K(\mathfrak{f})/K)=g\in G_{\mathfrak{f}}$. For integral
ideals $\mathfrak{a}$ of $K$ prime to the conductors of
$\varphi_{\lambda_{1}}$ and $\varphi_{\lambda_{2}}$, define
$\Lambda(\mathfrak{a})\in K(\mathfrak{f})^{\times}$ by

$\varphi_{\mathcal{A}}(\mathfrak{a})^{*}\omega^{\sigma_{\mathfrak{a}}}=\Lambda(\mathfrak{a})\omega$

where $\omega\in H^{0}(E,\Omega^{1})$ has period lattice $\Gamma$,
$\varphi_{\mathcal{A}}(\mathfrak{a})\in T^{\times}$ is viewed as an isogeny
$E\to{}^{\sigma_{\mathfrak{a}}}E$ and $\sigma_{\mathfrak{a}}$ is the Artin
automorphism of $\mathfrak{a}$. For all $g\in G_{\mathfrak{f}}$, $\omega^{g}$
has period lattice $\Gamma_{g}$ and we can identify ${}^{g}E(\mathbb{C})$ with
$\mathbb{C}/\Gamma_{g}$ via Abel-Jacobi map to obtain a divisor

$\beta_{g}=(\Lambda(\mathfrak{b}_{g}))\Omega f_{\mathfrak{m}}^{-1}$

on ${}^{g}E(K(\mathfrak{f}))$ with $G_{\mathfrak{f}}$ action given by
${}^{h}\beta_{g}=\beta_{hg}$.

For the special values of these Artin $L$-functions at negative integers,
there is a proposition ([9](3.4),[15])

######  Proposition 2.1.

The $L$-series $L(\chi,s)$ has a first order zero for every $s=-j<0$, and the
special value is given by the formula

$L^{\prime}(\chi,-j)=(-1)^{j}\frac{\Phi(\mathfrak{f}_{\chi})(j!)^{2}}{\Phi(\mathfrak{f})}\bigg{(}\frac{\sqrt{d_{K}}\mathcal{N}\mathfrak{f}_{\chi}}{2\pi
i}\bigg{)}^{j}\chi(\rho_{\mathfrak{f}_{\chi}})\sum_{g\in
G_{\mathfrak{f}}}\chi(g)A(\Gamma_{g})^{1+j}\mathcal{M}_{j}(\beta_{g}),$

where $\Phi$ is the totient function and $d_{K}$ is the discriminant of $K$.
For any $\mathbb{Z}$-basis of $\Gamma_{g}$ with
$Im(v/u)>0,A(\Gamma_{g})=(\bar{u}v-\bar{v}u)/2\pi i$, and for divisors on the
$\mathfrak{f}_{\chi}$-torsion of $E$, $\mathcal{M}_{j}$ is defined by
linearity from

$\mathcal{M}_{j}(x)=\sum_{0\neq\gamma\in\Gamma_{g}}\frac{(x,\gamma)_{g}}{|\gamma|^{2(1+j)}},\
x\in E_{\mathfrak{f}_{\chi}}$

with the Pontryagin $(,)_{g}:\mathbb{C}/\Gamma_{g}\times\Gamma_{g}\to U(1)$
given by
$(z,\gamma)_{g}=\exp(A(\Gamma_{g})^{-1}(z\bar{\gamma}-\bar{z}\gamma))$.

##  3\. Motivic elements

Let $X$ be a smooth variety over $\mathbb{Q}$. The motivic cohomology of $X$
is the bigraded family of rational vector spaces

$H_{\mathcal{M}}^{j}(X,j)=(K_{2j-i}(X)\otimes\mathbb{Q})^{(j)},$

where the subscript $(j)$ denotes the simultaneous eigenspace of all Adams
operators $\psi^{k},k\geq 1$ belonging to the eigenvalue $k^{j}$. The groups
$K_{*}(X)$ are the usual Quillen $K$-groups.

The original construction of Eisenstein classes in motivic cohomology was
given by Beilinson in[1]. The classes obtained are called the Eisenstein
symbols. Deninger [8]constructs motivic elements from the divisor $\beta$
using a variation of the Eisention symbol

$\mathcal{E}_{\mathcal{M}}^{k}:\mathbb{Q}[E]^{0}\to
H_{\mathcal{M}}^{k+1}(E^{k},k+1)$

which is defined only for divisors of degree 0. The Eisenstein symbol used in
the work of Huber, Kings and Scholl[13, 17, 27] are related with Deninger’s.
Their relation has been demonstrated by Johnson in [15]. Choose an integer
$N\geq 2$ and define a degree 0 divisor on ${}^{g}E(\mathbb{C})$

$\alpha_{g}=N^{2}(0)-\sum_{p\in{}^{g}E(\mathbb{C})_{N}}(p).$

Deninger shows that

######  Proposition 3.1.

The degree zero divisor

$\beta^{\prime}_{g}=\beta_{g}-(\deg\beta_{g})(0)+\frac{\deg\beta_{g}}{N^{2}}\left(1-\frac{1}{N^{4+2j}}\right)^{-1}\alpha_{g}$

is defined over ${}^{g}E(K(\mathfrak{f}))$ with
${}^{h}\beta^{\prime}_{g}=\beta^{\prime}_{hg}$ and

$\mathcal{M}_{j}(\beta^{\prime}_{g})=\mathcal{M}_{j}(\beta^{\prime}_{g}).$

Johnson [15] extended Deninger’s Eisenstein symbol to any degree, which is
necessary in order to have the desired shape in the $\ell$-adic computation.

######  Proposition 3.2.

For $k>0$, there is a variation of the Eisenstein symbol
$\mathcal{E}is^{k}_{\mathcal{M}}:\mathbb{Q}[E[\mathfrak{f}]\setminus 0]\to
H^{k+1}_{\mathcal{M}}(E^{k},k+1)$ which is defined for divisors of any degree.
Moreover,

$\mathcal{E}is(\beta^{\prime})=\mathcal{E}is(\beta).$

Deninger constructs the Kronecker map, which is a projector given by

${H_{\mathcal{M}}^{2j+1}(E^{2j},2j+1)}$${H_{\mathcal{M}}^{2j+1}(E^{j},2j+1)}$${H^{1}_{\mathcal{M}}(\text{Spec}(K(\mathfrak{f})),j+1),}$$\scriptstyle{(\delta^{j})^{*}}$$\scriptstyle{\mathcal{K}_{\mathcal{M}}}$$\scriptstyle{\text{pr}_{*}}$

where $\delta=(\text{id},\theta_{E}(\sqrt{d_{K}})):E\to
E\times_{K(\mathfrak{f})}E$ and this $j$-times gives $\delta^{j}:E^{j}\to
E^{2j}$, pr is the projection $E^{j}\to\text{Spec}(K(\mathfrak{f}))$. Then the
Kronecker map is

$\mathcal{K}_{\mathcal{M}}=\text{pr}_{*}\circ(\delta^{j})^{*}.$

Recall that the regulator $\rho_{\infty}$ is an isomorphism

$K_{2j+1}(\mathcal{O}_{K(\mathfrak{f})})\otimes_{\mathbb{Z}}\mathbb{R}\xrightarrow{\rho_{\infty}}\left(\bigoplus_{\sigma\in\mathcal{T}}\mathbb{C}/\mathbb{R}\cdot(2\pi
i)^{j+1}\cdot\sigma\right)^{+},$

where $\mathcal{T}=\text{Hom}(K(\mathfrak{f}),\mathbb{C})$.

Thus Johnson defines

$\xi_{\mathfrak{m}}(j):=\mathcal{K}_{\mathcal{M}}\mathcal{E}is^{2j}(\rho_{\mathfrak{m}}\cdot\beta).$

Note that the definition does not construct motivic element for small
$\mathfrak{m}$. The following lemma[15](Lem.3.1.8) atones for this omission.

######  Lemma 3.3.

$w_{\mathfrak{m}}/w_{\mathfrak{pm}}\text{Tr}_{K(\mathfrak{pm})/K(\mathfrak{m})}\xi_{\mathfrak{mp}}(j)=\begin{cases}\xi_{\mathfrak{m}}(j),&\mbox{if
}\ \mathfrak{p}\mid\mathfrak{m}\neq 1\\\
(1-\text{Fr}_{\mathfrak{p}}^{-1})\xi_{\mathfrak{m}}(j),&\mbox{if}\ \
\mathfrak{p}\nmid\mathfrak{m}\neq 1.\end{cases}$

With this lemma one can define motivic elements for small ideals via the Trace
map. In particular, for any prime $\mathfrak{q}$ of $K$ with
$w_{\mathfrak{q}}=1$, one define

${}_{\mathfrak{q}}\xi_{1}(j):=(1-\text{Fr}_{\mathfrak{q}}^{-1})^{-1}w_{K}\text{Tr}_{K(\mathfrak{q})/K(1)}\xi_{\mathfrak{q}}(j)$

for a family of motivic element at level 1.

Johnson proves the following theorem which is a modification of a result of
Deninger[9].

######  Theorem 3.4.

For every ideal $1\neq\mathfrak{m}\mid\mathfrak{f}$, there are motivic
elements

$\xi_{\mathfrak{m}}(j)\in H^{1}_{\mathcal{M}}(K(\mathfrak{f}),j+1)$

with the property that if $\chi$ is a rational character of $G_{\mathfrak{f}}$
of conductor $\mathfrak{m}$, then

$e_{\chi}(\rho_{\infty}(\xi_{\mathfrak{m}}(j)))=\frac{(2\mathcal{N}\mathfrak{m})^{j-1}\Phi(\mathfrak{f})}{(-1)^{1-j}(2j)!\Phi(\mathfrak{m})}L^{\prime}(\chi,-j)\eta_{\mathbb{Q}},$

where $\eta_{\mathbb{Q}}$ is a basis of the $\chi$-component of
$(H_{B}^{0}(\text{Spec}(F)(\mathbb{C}),\mathbb{Q}(j))^{+})^{*},$ and

$\Phi(\mathfrak{f})=|(\mathcal{O}_{K}/\mathfrak{f})^{\times}|=\mathcal{N}\mathfrak{f}\prod_{\mathfrak{p}\mid\mathfrak{f}}(1-\mathcal{N}\mathfrak{p}^{-1}).$

Moreover, these elements forms a norm compatible system, and for
$\mathfrak{m}=1$ we have a family of elements, ${}_{\mathfrak{q}}\xi_{1}(j)$,
indexed by the primes of $K$ which are defined via the norm map and satisfying
the above formula for any choice of $\mathfrak{q}$ with $w_{\mathfrak{q}}=1.$

##  4\. $\ell$-adic regulator of motivic element

This section need to introduce elliptic units, so let us review the
construction of them. Let $L=\mathbb{Z}w_{1}+\mathbb{Z}w_{2}$ be a lattice in
$\mathbb{C}$ such that $\tau:=w_{1}/w_{2}$ has positive imaginary part. The
Dedekind eta-function is defined as

$\eta(\tau)=e^{\frac{\pi i\tau}{12}}\prod_{n=1}^{\infty}(1-q_{\tau}^{n});\quad
q_{\tau}=e^{2\pi i\tau}.$

Put

$\eta(w_{1},w_{2})=w_{2}^{-1}2\pi\eta(w_{1}/w_{2})^{2}.$

This function depends on the choice of basis but

$\Delta(L)=\Delta(\tau)=\eta(w_{1},w_{2})^{12}$

does not. Define a theta-function

$\theta(z,\tau)=ie^{\frac{\pi
iz}{2}\frac{z-\bar{z}}{\tau-\bar{\tau}}}q_{\tau}^{1/12}q_{z}^{-1/2}(1-q_{z})\prod_{n=1}^{\infty}(1-q_{z}q_{\tau}^{n})(1-q_{z}^{-1}q_{\tau}^{n}),$

where $q_{z}=e^{2\pi iz}$ and

$\theta(z;w_{1},w_{2})=\theta(z/w_{2},w_{1}/w_{2}).$

For any pair lattices $L\subseteq L^{\prime}$ of index prime to 6 with
oriented bases $\omega=(w_{1},w_{2})$ and
$\omega^{\prime}=(w^{\prime}_{1},w^{\prime}_{2})$, Robert [26] shows that
there exists a unique choice of 12-th root of unity
$C(\omega,\omega^{\prime})$ such that the functions

$\delta(L,L^{\prime}):=C(\omega,\omega^{\prime})\eta(w_{1},w_{2})^{[L^{\prime}:L]}/\eta(w^{\prime}_{1},w^{\prime}_{2})$

and

$\vartheta(z;L,L^{\prime})=C(\omega,\omega^{\prime})\theta(z;\omega)^{[L^{\prime}:L]}/\theta(z;\omega^{\prime})=\delta(L,L^{\prime})\prod_{u\in
T}(\wp(z;L)-\wp(u;L))^{-1}$

only depend on the lattices $L,L^{\prime}$ and so that $\vartheta$ satisfies
the distribution relation

(4.1)
$\vartheta(z;M,M^{\prime})=\prod_{i=1}^{[M:L]}\vartheta(z+t_{i};L,L^{\prime})$

for any lattice $L\subseteq M$ such that $M\cap L^{\prime}=L$ (and where
$M^{\prime}=M+L^{\prime}$), where $t_{i}\in M$ are a set of representatives of
$M/L$. The set $T$ is any set of representatives of
$(L^{\prime}\setminus\\{0\\})/(\pm 1\ltimes L)$ and $\wp$ is the Weierstrass
$\wp$-function associated to $L$. In particular $\vartheta(z;L,L^{\prime})$ is
an elliptic function.

Kato reproves Robert’s result in a scheme theoretic context, which is
demonstrated as follows.

######  Lemma 4.1.

(Kato[16](15.4.)) Let $E$ be an elliptic curve over a field $F$ with
$\mathcal{O}_{K}\cong End_{F}(E)$ and $\mathfrak{a}$ an ideal in
$\mathcal{O}_{K}$ prime to $6$. Then there is a unique function

$\Theta_{\mathfrak{a}}\in\Gamma(E\setminus\ker([\mathfrak{a}]),\mathcal{O}^{\times})$

with

(i) $div(\Theta_{\mathfrak{a}})=N\mathfrak{a}\cdot(0)-\ker([\mathfrak{a}])$

(ii) For any $m\in\mathbb{Z}$ prime to $\mathfrak{a}$ we have
$N_{m}(\Theta_{\mathfrak{a}})=\Theta_{\mathfrak{a}}$, where $N_{m}$ is the
norm map associated to the finite flat morphism
$E\setminus\ker([m\mathfrak{a}])\to E\setminus\ker([\mathfrak{a}])$ given by
multiplication with $m$.

For $F=\mathbb{C}$ we have

$\Theta_{\mathfrak{a}}(z)=\vartheta(z;L,\mathfrak{a}^{-1}L).$

Given $\mathfrak{f}\neq 1$ and any $\mathfrak{a}$ which is prime to
$6\mathfrak{f}$, the elliptic unit is defined by

${}_{\mathfrak{a}}z_{\mathfrak{f}}=\vartheta(1;\mathfrak{f},\mathfrak{a}^{-1}\mathfrak{f})$

and for $\mathfrak{f}=1$ we define a family of elements indexed by all ideals
$\mathfrak{a}$ of $K$ by

$u(\mathfrak{a})=\frac{\Delta(\mathcal{O}_{K})}{\Delta(\mathfrak{a}^{-1})}.$

######  Lemma 4.2.

The complex numbers ${}_{\mathfrak{a}}z_{\mathfrak{f}}$ and $u(\mathfrak{a})$
satisfy the following properties

(i) (Rationality) ${}_{\mathfrak{a}}z_{\mathfrak{f}}\in K(\mathfrak{f}),\
u(\mathfrak{a})\in K(1)$;

(ii) (Integrality)

$\displaystyle{}_{\mathfrak{a}}z_{\mathfrak{f}}\in\begin{cases}\mathcal{O}_{K(\mathfrak{f})}^{\times},&\mathfrak{f}\
\mbox{divisible by primes }\ \mathfrak{p}\neq\mathfrak{q}\\\
\mathcal{O}_{K(\mathfrak{f}),\\{v|\mathfrak{f}\\}}^{\times},&\mathfrak{f=p}^{n}\
\mbox{for some prime}\ \mathfrak{p}.\end{cases}$ $\displaystyle
u(\mathfrak{a})\cdot\mathcal{O}_{K(1)}=\mathfrak{a}^{-12}\mathcal{O}_{K(1)}$

(iii) (Galois action) For $(\mathfrak{c,fa})=1$ with Artin symbol
$\sigma_{\mathfrak{c}}\in Gal(K(\mathfrak{f})/K)$ we have

${}_{\mathfrak{a}}z_{\mathfrak{f}}^{\sigma_{\mathfrak{c}}}=\vartheta(1;\mathfrak{c}^{-1}\mathfrak{f},\mathfrak{c}^{-1}\mathfrak{a}^{-1}\mathfrak{f});\quad
u(\mathfrak{a})^{\sigma_{\mathfrak{c}}}=u(\mathfrak{ac})/u(\mathfrak{c}).$

This implies

${}_{\mathfrak{a}}z_{\mathfrak{f}}^{N\mathfrak{c}-\sigma_{\mathfrak{c}}}={}_{\mathfrak{c}}z_{\mathfrak{f}}^{N\mathfrak{a}-\sigma_{\mathfrak{a}}};\quad
u(\mathfrak{a})^{1-\sigma_{\mathfrak{c}}}=u(\mathfrak{c})^{1-\sigma_{\mathfrak{a}}}.$

(iv) (Norm compatibility) For a prime ideal $\mathfrak{p}$ we have

$N_{K(\mathfrak{pf})/K(\mathfrak{f})}({}_{\mathfrak{a}}z_{\mathfrak{pf}})^{w_{\mathfrak{f}}/w_{\mathfrak{pf}}}=\begin{cases}{}_{\mathfrak{a}}z_{\mathfrak{f}},&\mbox{if
}\ \mathfrak{p}\mid\mathfrak{f}\neq 1\\\
{}_{\mathfrak{a}}z_{\mathfrak{f}}^{1-\sigma_{\mathfrak{p}}^{-1}},&\mbox{if }\
\mathfrak{p}\nmid\mathfrak{f}\neq 1\\\
u(\mathfrak{p})^{(\sigma_{\mathfrak{a}}-N\mathfrak{a})/12},&\mbox{if}\ \
\mathfrak{f}=1.\end{cases}$

In order to compute the image of $\xi_{\mathfrak{m}}(j)$ under the étale Chern
class map $\rho_{et}$, one need the following commutative diagram

${H_{\mathcal{M}}^{2j+1}(E^{2j},2j+1)}$${H^{2m+1}(E^{2j},\mathbb{Q}_{\ell}(2j+1))}$${H^{1}_{\mathcal{M}}(\text{Spec}(K(\mathfrak{f})),j+1)}$${H^{1}(K(\mathfrak{f}),\mathbb{Q}_{\ell}(j+1))}$$\scriptstyle{\rho_{et}}$$\scriptstyle{\mathcal{K}_{\mathcal{M}}}$$\scriptstyle{\mathcal{K}_{\ell}}$$\scriptstyle{\rho_{et}}$

By [13, Theorem2.2.4], the étale realization of the Eisenstein symbol can be
computed in term of the pullback of the elliptic polylogarithm along torsion
section. Then,

$\rho_{et}(\xi_{\mathfrak{m}}(j))=\mathcal{K}_{\ell}(\rho_{et}(\mathcal{E}is^{2j}(\rho_{\mathfrak{m}}\cdot\beta)))=\psi(\rho_{\mathfrak{m}})^{-1}\cdot\mathcal{N}\mathfrak{m}^{2j-1}\mathcal{K}_{\ell}(\beta^{*}\mathcal{P}ol_{\mathbb{Q}_{\ell}})^{2j}.$

Kings [17, Thm.4.2.9] gets the explicit $\ell$-adic Eisenstein class for
$\ell\nmid\mathfrak{f}$, i.e.

$(\beta^{*}\mathcal{P}ol_{\mathbb{Q}_{\ell}})^{2j}=\pm\frac{1}{\mathcal{N}\mathfrak{a}([\mathfrak{a}]^{2j}\mathcal{N}\mathfrak{a}-1)(2j)!}\left(\delta\sum_{[\ell^{n}]t_{n}=\beta}\Theta_{\mathfrak{a}}(-t_{n})\tilde{t}_{n}^{\otimes
2j}\right)_{n}$

where $\delta$ is the connecting homomorphism in Kummer sequence, the
$\Theta_{\mathfrak{a}}(-t_{n})$ are elliptic units, $\tilde{t}_{n}$ is the
projection of $t_{n}$ to $E[\ell^{n}]$, and
$\mathfrak{a}\subset\mathcal{O}_{K}$ is chosen prime to $\ell\mathfrak{f}$.

Consider the elliptic curve $E$ over $K(\mathfrak{f})$ with a uniformization
$\mathbb{C}/\Gamma$. We can define a multiplication by $\rho_{\mathfrak{m}}$
on the elliptic curve componentwise. By the choice of torsion point $\beta$,
we see $\rho_{\mathfrak{m}}t_{n}\in E[\ell^{n}]$. Taking this as our
$\tilde{t}_{n}$ we have to multiply King’s result by a factor of
$\rho_{\mathfrak{m}}^{-2j}$.

For $t\in E[\ell^{n}]$, define $\gamma(t)^{k}:=\langle
t,\sqrt{d_{K}}t\rangle^{\otimes k}$, where $\langle,\ \rangle$ is the Weil
pairing. Then the Kronecker map acts on the Tate module by

$\mathcal{K}_{\ell}(\tilde{t}_{n}^{\otimes
2j})=\gamma(\tilde{t}_{n})^{j}=\zeta_{\ell^{n}}^{\otimes j},$

on the integral isogenies $[\mathfrak{a}]$ by

$\mathcal{K}_{\ell}([\mathfrak{a}]^{2j})=\mathcal{N}\mathfrak{a}^{j}$

and on the idele $\rho_{\mathfrak{m}}$ by
$\mathcal{K}_{\ell}(\rho_{\mathfrak{m}}^{-2j})=\mathcal{N}(\rho_{\mathfrak{m}})^{-j}=\mathcal{N}\mathfrak{m}^{-j}.$
The Artin automorphism $\sigma_{\mathfrak{a}}$ acts on the space
$H^{1}(K(\mathfrak{f}),\mathbb{Q}_{\ell})$ by $\mathcal{N}\mathfrak{a}$ and
thus on the space $H^{1}(K(\mathfrak{f}),\mathbb{Q}_{\ell}(j+1))$ by
$\mathcal{N}\mathfrak{a}^{j+2}$. At last, Johnson concludes that (see [15,
p.34])

$\rho_{et}(\xi_{\mathfrak{m}}(j))=\frac{\psi(\rho_{\mathfrak{m}})^{-1}}{\mathcal{N}\mathfrak{a}-\sigma_{\mathfrak{a}}}\cdot\mathcal{N}\mathfrak{m}^{j-1}\left(\delta\sum_{[\ell^{n}]t_{n}=\beta}\Theta_{\mathfrak{a}}(-t_{n})\zeta_{\ell^{n}}^{\otimes
j}\right)_{n}.$

Since there is(see [15, Lemma3.2.4])

$\prod_{\mathfrak{l}|\ell}(1-\mbox{Fr}_{\mathfrak{l}}^{-1})^{-1}\left(\sum_{[\ell^{n}]t_{n}=\Omega
f_{\mathfrak{m}}^{-1}}\Theta_{\mathfrak{a}}(-t_{n})\zeta_{\ell^{n}}^{\otimes
j}\right)_{n}=w_{\mathfrak{m}}(\mbox{Tr}_{K(\ell^{n}\mathfrak{m})/K(\mathfrak{m})}\Theta_{\mathfrak{a}}(-s_{n})\zeta_{\ell^{n}}^{\otimes
j})_{n},$

where $s_{n}$ is a primitive $\ell^{n}$th root of $\beta$, we obtain the
following theorem (see [15, Theorem3.2.1])

######  Theorem 4.3.

For all $1\neq\mathfrak{m}|\mathfrak{f}$, we have that

$\rho_{et}(\xi_{\mathfrak{m}}(j))=\frac{\mathcal{N}\mathfrak{m}^{j-1}w_{\mathfrak{m}}}{(\mathcal{N}\mathfrak{a}-\sigma_{\mathfrak{a}})\prod_{\mathfrak{l}|\ell}(1-\mbox{Fr}_{\mathfrak{l}}^{-1})(2j)!}(\mbox{Tr}_{K(\ell^{n}\mathfrak{m})/K(\mathfrak{m})}\Theta_{\mathfrak{a}}(-s_{n})\zeta_{\ell^{n}}^{\otimes
j})_{n}$

##  5\. “Main conjecture” of Iwasawa theory for quadratic imaginary fields

Fix an elliptic curve $E$ defined over imaginary quadratic field $K$, with
$\text{End}_{K}(E)=\mathcal{O}_{K}$. Fix a prime $\mathfrak{p}$ of $K$ where
$E$ has good reduction. Write
$K_{n}=K(E[\mathfrak{p}^{n}]),n=0,1,2,\cdots,\infty$. We have

$\text{Gal}(K_{\infty}/K)\cong\mathcal{O}_{\mathfrak{p}}^{\times}\cong\text{Gal}(K_{\infty}/K_{1})\times\text{Gal}(K_{1}/K)$

where

$\text{Gal}(K_{1}/K)\cong(\mathcal{O}/\mathfrak{p})^{\times}$

is the prime-to-$p$ part of $\text{Gal}(K_{\infty}/K)$ and

$\text{Gal}(K_{\infty}/K_{1})\cong
1+\mathfrak{p}\mathcal{O}_{\mathfrak{p}}\cong\mathbb{Z}_{p}^{[K_{\mathfrak{p}}:\mathbb{Q}_{p}]}$

is the $p$-part.

Let $F=K(\mathfrak{f}),(\mathfrak{f,p})=1,$ and
$F_{n}=K(\mathfrak{fp}^{n})\supset F(\mu_{p^{n}}),n=0,\cdots,\infty$. Write
$\mathscr{G}_{F}=\text{Gal}(F_{\infty}/F)\subset\mathscr{G}_{\infty}=\text{Gal}(F_{\infty}/K),\Delta=\text{Gal}(F_{1}/K),$
$\Gamma=\text{Gal}(F_{\infty}/F_{1})$. Let $\mathscr{A}_{n}=p$-part of the
class group of $F_{n}$, $\mathscr{A}^{\prime}_{n}=p$-part of the $S$-class
group of $F_{n}$,
$\mathscr{A}_{\infty}=\varprojlim\mathscr{A}_{n},\mathscr{A}^{\prime}_{\infty}=\varprojlim\mathscr{A}^{\prime}_{n}$.
Put $\mathscr{X}_{n}=G_{S}^{\text{ab}}(K_{n})\otimes\mathbb{Z}_{p}$,
$\mathscr{X}_{\infty}=\varprojlim\mathscr{X}_{n}.$ Write
$U^{\prime}(F_{n})=$group of $S$-units of $F_{n}$,
$\overline{U}^{\prime}_{n}=U^{\prime}_{n}\otimes\mathbb{Z}_{p},$
$\overline{U}^{\prime}_{\infty}=\varprojlim\overline{U}^{\prime}_{n},\tilde{U}^{\prime}_{n}=\text{fr}_{\mathbb{Z}_{p}}\overline{U}^{\prime}_{n},\tilde{U}^{\prime}_{\infty}=\text{fr}_{\Lambda}\overline{U}^{\prime}_{\infty}$.

By [25, Corollary5.20], we have

(5.1)
$\Gamma=\text{Gal}(F_{\infty}/F_{1})=\varprojlim\text{Gal}(F_{n}/F_{1})\cong\varprojlim\text{Gal}(K_{n}/K_{1}).$

For prime $\mathfrak{p}$ of $K$ over $p$, suppose $p\nmid[F_{1}:K]$ and $p$
does not divide the number of roots of unity in the Hilbert class field of
$K$. From (5.1), we have
$\Gamma=\text{Gal}(F_{\infty}/F_{1})\cong\mathbb{Z}_{p}$ or
$\mathbb{Z}_{p}^{2}$. Then
$\mathscr{G}_{\infty}=\text{Gal}(F_{\infty}/K)\cong\Delta\times\Gamma$. Write
$\mathscr{A}(F_{n})$ for the $p$-part of the ideal class group of $F_{n}$,
$\mathscr{E}(F_{n})$ for the group of global units of $F_{n}$, and
$\mathscr{C}(F_{n})$ for the group of Robert elliptic units. Let
$\mathscr{U}(F_{n})$ be the group of local units of
$F_{n}\otimes_{K}K_{\mathfrak{p}}$ which are congruent to 1 modulo the primes
above $\mathfrak{p}$. Let $\overline{\mathscr{E}}(F_{n})$ and
$\overline{\mathscr{C}}(F_{n})$ denote the closure of
$\mathscr{E}(F_{n})\cap\mathscr{U}(F_{n})$ and
$\mathscr{C}(F_{n})\cap\mathscr{U}(F_{n})$ in $\mathscr{U}(F_{n})$. For
$F_{\infty}$, we define

$\displaystyle\mathscr{A}(F_{\infty})=\varprojlim\mathscr{A}(F_{n}),\ \
\overline{\mathscr{E}}(F_{\infty})=\varprojlim\overline{\mathscr{E}}(F_{n}),$
$\displaystyle\overline{\mathscr{C}}(F_{\infty})=\varprojlim\overline{\mathscr{C}}(F_{n}),\
\ \mathscr{U}(F_{\infty})=\varprojlim\mathscr{U}(F_{n}),$

with respect to norm maps. Let $M(F_{n})$ for the maximal abelian
$p$-extension of $F_{n}$ which is unramified outside the primes above
$\mathfrak{p}$, and write $\mathscr{X}(F_{n})=\text{Gal}(M(F_{n})/F_{n})$. We
will also write $\mathscr{A}_{\infty},\
\mathscr{U}_{\infty},\overline{\mathscr{E}}_{\infty},\
\overline{\mathscr{C}}_{\infty}$ and $\mathscr{X}_{\infty}$ for
$\mathscr{A}(F_{\infty}),\
\mathscr{U}(F_{\infty}),\overline{\mathscr{E}}(F_{\infty}),\
\overline{\mathscr{C}}(F_{\infty})$ and $\mathscr{X}(F_{\infty})$,
respectively. Then global class field theory gives us the following exact
sequence(The details can be found in[10]Chapter3 §1.7)

(5.2)
$0\to\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}\to\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}\to\mathscr{X}_{\infty}\to\mathscr{A}_{\infty}\to
0.$

If $Y$ is a $\mathbb{Z}_{p}[\Delta]$-module and $\chi$ is an irreducible
$\mathbb{Z}_{p}$-representation of $\Delta$, define $Y^{\chi}$ to be the
$\chi$-isotypic component of $Y$ and

$e_{\chi}=\frac{1}{\\#\Delta}\sum_{\tau\in\Delta}\text{Tr}(\chi(\tau))\tau^{-1}\in\mathbb{Z}_{p}[\Delta].$

Then $Y^{\chi}=e_{\chi}Y$.

Define the Iwasawa algebra

$\Lambda=\mathbb{Z}_{p}[[\mathscr{G}_{\infty}]]=\varprojlim\mathbb{Z}_{p}[\text{Gal}(F_{n}/K)].$

Then for every irreducible $\mathbb{Z}_{p}$-representation $\chi$ of $\Delta$,

$\Lambda^{\chi}=\mathbb{Z}_{p}[[\mathscr{G}_{\infty}]]^{\chi}=R_{\chi}[[\Gamma]],$

where $R_{\chi}$ is the ring of integers of the unramified extension of
$\mathbb{Q}_{p}$ of degree dim($\chi$). Further, $R_{\chi}[[\Gamma]]$ is
(noncanonically) isomorphic to a power series rings in 1 or 2 variables over
$R_{\chi}$ according to $\Gamma\cong\mathbb{Z}_{p}$ or $\mathbb{Z}_{p}^{2}.$

We have $\Lambda=\bigoplus_{\chi}\Lambda^{\chi}.$ We will call a
$\Lambda$-module $Y$ a torsion if every element of $Y$ is annihilated by an
element of $\Lambda$ which is not a zero-divisor, or equivalently if and only
if $Y^{\chi}$ is a torsion $\Lambda^{\chi}$-module for every $\chi$.

A finitely generated $\Lambda$-module is called pseudo-null if it is
annihilated by an ideal of height 2; if $\Gamma\cong\mathbb{Z}_{p}$ the
pseudo-null modules are exactly the finite modules. A pseudo-isomorphism of
$\Lambda$-modules is a map with pseudo-null kernel and cokernel. It follows
from the well known classification theorem for $\Lambda$-modules that for
every finitely generated torsion $\Lambda$-module $Y$ can find elements
$f_{i}$ of $\Lambda$ and pseudo-isomorphisms

$Y\to\oplus\Lambda/f_{i}\Lambda\quad\text{and}\quad\oplus\Lambda/f_{i}\Lambda\to
Y.$

The characteristic ideal $(\Pi f_{i})/\Lambda$ is a well-defined invariant of
$Y$ which we will denote by char$(Y)$. For every $\chi$,
char$(Y)^{\chi}\subset\Lambda^{\chi}$ is the usual character ideal of the
$\Lambda^{\chi}$-module $Y^{\chi}$, and a generator of char$(Y)^{\chi}$ is
usually called a characteristic power series of $Y^{\chi}$.

Rubin proves that $\mathscr{A}_{\infty},\
\mathscr{U}_{\infty},\overline{\mathscr{E}}_{\infty},\
\overline{\mathscr{C}}_{\infty}$ and $\mathscr{X}_{\infty}$ are all finitely
generated $\Lambda$-modules, and $\mathscr{A}_{\infty}$ and
$\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}$ are torsion
$\Lambda$-modules in [24]. Further, he got the following main conjecture of
Iwasawa theory for imaginary quadratic fields, that is,

######  Theorem 5.1.

(i) Suppose $p$ splits into two distinct primes in $K$. Then

$char(\mathscr{A}_{\infty})=char(\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty})\quad\text{and}\quad
char(\mathscr{X}_{\infty})=char(\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty})$

(ii) Suppose $p$ remains prime or ramifies in $K$. Then

$char(\mathscr{A}_{\infty})\quad\text{divides}\quad
char(\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}).$

If $\chi$ is an irreducible $\mathbb{Z}_{p}$-representation of $\Delta$ which
is nontrivial on the decomposition group of $\mathfrak{p}$ in $\Delta$, then

$char(\mathscr{A}_{\infty})^{\chi}=char(\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty})^{\chi}.$

##  6\. étale cohomology and Iwasawa module

For an odd prime number $p$, let $S=S(F)$ denote the set of primes of $F$
above $p$. Denote the maximal algebraic $S$-ramified extension of $F$ by
$\Omega_{S}$ and set $G_{S}(F)=\text{Gal}(\Omega_{S}/F)$. It is well known
that the $p$-adic cohomology group
$H^{i}(G_{S}(F),\mathbb{Z}_{p}(m))$$=\varprojlim
H^{i}(G_{S}(F),\mathbb{Z}/p^{n}\mathbb{Z}(m)),\ m\in\mathbb{Z}$, coincide with
the étale cohomology groups
$H^{i}_{et}(\text{Spec}\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$. Put
$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}(m)):=H^{1}_{et}(\text{Spec}\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}(m))$.

There is a unified description of the groups
$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}(m))$ in terms of Iwasawa modules
which goes back to Deligne[7] and Soulé[28, 29]. Let $E_{n}=F(\mu_{p^{n}}),\
E_{\infty}=F(\mu_{p^{\infty}}),\ G_{\infty}=\text{Gal}(E_{\infty}/F),\
U^{\prime}_{n}=U^{\prime}_{E_{n}},$ and
$\overline{U}^{\prime}_{\infty}=\varprojlim
U^{\prime}_{n}\otimes\mathbb{Z}_{p}.$ In [28, 29], Soulé proved that
$(\overline{U}^{\prime}_{\infty}(m-1))_{G_{\infty}}\otimes\mathbb{Q}_{p}\cong
K_{2m-1}(F)\otimes\mathbb{Q}_{p}\cong
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}(m))$ for $m\geq 2.$ Kolster et.al.
(see [18]) extended Soulé’s results for all $m\in\mathbb{Z},m\neq 1$, modulo
the standard conjecture.

The descent modules $\overline{U}^{\prime}_{\infty}(m-1)_{G_{\infty}}$ is very
useful. It has been shown by Kolster et.al. for $m\neq 0,1$ and by Kuz’min[19]
for $m=1$ that $\overline{U}^{\prime}_{\infty}(m-1)_{G_{\infty}}$ injects into
$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$. Assume that $F$ is abelian
field. Consider the projective limit of Sinnott’s circular $S$-units
$C^{\prime}_{n}\subset U^{\prime}_{n}$. Let
$\overline{C}^{\prime}_{\infty}=\varprojlim
C^{\prime}_{n}\otimes\mathbb{Z}_{p}.$ Again, Kolster et.al. proved
$\overline{C}^{\prime}_{\infty}(m-1)_{G_{\infty}}$ injects into
$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$ with finite index. If $F$ is
an abelian extension of imaginary quadratic field, we use the elliptic units
to get a similar result as them.

One essential tool used is Poitou-Tate’s long exact sequence, which gives a
description of the kernels and cokernels of localization maps in Galois or
étale cohomology. Let us recall the main facts to be used. Denote by
SH${}_{S}^{i}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))$ and
SH${}_{S}^{i}(F,\mathbb{Z}_{p}(m))$ respectively the kernels of the
localization maps

$H^{i}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))\to\bigoplus_{v\in
S}H^{i}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))$

and

$H^{i}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))\to\bigoplus_{v\in
S}H^{i}(F_{v},\mathbb{Z}_{p}(m)).$

By Poitou-Tate, we have canonical isomorphisms(for any $m\in\mathbb{Z}$)

$H^{i}(F_{v},\mathbb{Z}_{p}(m))\cong
H^{j}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\quad(0\leq i,j\leq 2,\
i+j=2)$

and

$\text{{\fontWCA SH}}_{S}^{i}(F,\mathbb{Z}_{p}(m))\cong\text{{\fontWCA
SH}}_{S}^{i}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\quad(0\leq i,j\leq 2,\
i+j=3),$

where $(\cdot)^{*}$ denotes the Pontryagin dual.

We will use Poitou-Tate’s exact sequence in the following form

$0\to\text{{\fontWCA SH}}_{S}^{2}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))\to\bigoplus_{v\in
S}H^{1}(F_{v},\mathbb{Z}_{p}(m))\\\ \cong\bigoplus_{v\in
S}H^{1}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\to\text{{\fontWCA
SH}}_{S}^{2}(F,\mathbb{Z}_{p}(m))\to 0.$

Note that for $m=1$, this is essentially the class field theory exact
sequence. It is known that $H^{2}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0$
for all $m\neq 1$, so

$\text{{\fontWCA
SH}}_{S}^{2}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))$

for all $m\neq 1$. If $m\geq 2$, then
$H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0$, which is
essentially equivalent to the finiteness of $K_{2m-2}(\mathcal{O}_{F}).$

We will use Iwasawa theory to describe the groups
$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$ as modules of descent. In the
process of descent, we will use the following generalized Tate’s lemma.

######  Lemma 6.1.

( Tate’s lemma.) For any $m\neq 0$,
$H^{1}(\mathscr{G}_{F},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0.$ ???

###### Proof.

If $\mathscr{G}_{F}\cong\mathbb{Z}_{p}\times\Delta$, the lemma is the case
[18, Lemma2.1].

Suppose $\mathscr{G}_{F}\cong\mathbb{Z}_{p}^{2}\times\Delta$. Let
$L=F(\mu_{p^{\infty}})$. Then
$\text{Gal}(L/F)\cong\mathbb{Z}_{p}\times\Delta^{\prime}$ and $F_{\infty}/L$
is $\mathbb{Z}_{p}$-extension. Write $H=\text{Gal}(F_{\infty}/L)$. Hence there
are

$H^{1}(\text{Gal}(L/F),(\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))^{H})=H^{1}(\text{Gal}(L/F),\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0$

and $H^{1}(H,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0.$ The cohomology exact
sequence

$0\to
H^{1}(\text{Gal}(L/F),(\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))^{H})\xrightarrow{Inf}H^{1}(\mathscr{G}_{F},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))\xrightarrow{Res}H^{1}(H,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))$

shows that $H^{1}(\mathscr{G}_{F},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m))=0.$ ∎

######  Theorem 6.2.

For $m\neq 1$, we have canonical isomorphisms ???

$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\cong\mathscr{X}_{\infty}(m-1)_{\mathscr{G}_{F}},$
$H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}\cong\mathscr{X}_{\infty}(m-1)^{\mathscr{G}_{F}}$

###### Proof.

The proof is inspired by Coates’s in [6]. Let $F_{S}$ for the maximal
extension of $F$ which is unramified outside of $S=\\{p,\infty\\}$. Let
$G_{S}(F_{\infty})$ be the subgroup of $G_{S}$ fixing $F_{\infty}$. Both
$\mathscr{G}_{F}$ and $G_{S}(F_{\infty})$ act on
$\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)$. Also $\mathscr{G}_{F}$ has
cd${}_{p}\mathscr{G}_{F}=1$. Hence the Hochschild-Serre spectral sequence

$E^{i,j}_{2}=H^{i}(\mathscr{G}_{F},H^{j}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)))\Rightarrow
H^{i+j}(G_{S},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))$

degenerates into the two short exact sequences

$\displaystyle 0$ $\displaystyle\to
H^{1}(\mathscr{G}_{F},H^{i-1}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)))\to
H^{i}(G_{S},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))$ $\displaystyle\to
H^{i}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{\mathscr{G}_{F}}\to
0$

with $i=1,2.$ Since $G_{S}(F_{\infty})$ acts trivially on
$\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)$ , we have

$H^{1}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))=\text{Hom}(\mathscr{X}_{\infty},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))=(\mathscr{X}_{\infty}(m-1))^{*}.$

Moreover, for $m\neq 1$, we have
$H^{1}(\mathscr{G}_{F},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))=0$ by Lemma6.1.
Finally, there is

$H^{2}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))=H^{2}(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p})(1-m)=0\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\textbf{???}}$

The assertions of the theorem now follow from these remarks and the above two
exact sequences. This completes the proof. ∎

######  Theorem 6.3.

For $m\neq 0,1$, we have a canonical exact sequence

$0\to(\overline{U}^{\prime}_{\infty}(m-1))_{\mathscr{G}_{F}}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))\to\mathscr{A}^{\prime}_{\infty}(m-1)^{\mathscr{G}_{F}}\to
0.$

###### Proof.

The exact sequence

$0\to\mathbb{Z}/p^{n}\mathbb{Z}(1)\to
F_{n}^{\times}\xrightarrow{p^{n}}F_{n}^{\times}\to 0$

leads to injective map

$\varprojlim_{n}\overline{U}^{\prime}_{n}\to\varprojlim_{n}H^{1}(\mathcal{O}_{F_{n}}[1/p],\mathbb{Z}_{p}(1)).$

Twisting the above map by Tate module $\mathbb{Z}_{p}(m-1)$ and taking
coinvariants with respect to $\mathscr{G}_{F}$, we have the injective
homomorphism

$0\to(\overline{U}^{\prime}_{\infty}(m-1))_{\mathscr{G}_{F}}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m)).$

∎

We define a map in the sprit of Soulé

$e_{p}:\overline{\mathscr{C}}_{\infty}(m-1)\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m)).$

Write

$H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))=\varprojlim_{r}H^{1}(\mathcal{O}_{F}[1/p],\mu_{p^{r}}^{\otimes
m}).$

For a norm compatible system of elliptic units $(\theta_{n})_{n}$ and an
element $(t_{n})_{n}\in\mu_{p^{n}}^{\otimes m-1}$, put

$e_{p}((\theta_{r}\otimes t_{n})_{n}):=(N_{F_{n}/F}(\theta_{n}\otimes
t_{n}))_{n}$

where $\theta_{n}\otimes t_{n}$ is an element in

$\mathcal{O}_{F_{n}}[1/p]^{*}/(\mathcal{O}_{F_{n}}[1/p]^{*})^{p^{n}}\otimes\mu_{p^{n}}^{\otimes
m-1}\subset H^{1}(\mathcal{O}_{F_{n}}[1/p],\mu_{p^{n}}^{\otimes m}),$

where the inclusion comes from Kummer theory and $N_{F_{n}/F}$ is the norm map
on the cohomology. The map $e_{p}$ factors through the covariants under
$\mathscr{G}_{F}$, so we have the following map

######  Theorem 6.4.

Let $F$ be an abelian extension over $K$ and $m>1$. Then the map

$e_{p}:\overline{\mathscr{C}}_{\infty}(m-1)_{\mathscr{G}_{F}}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$

is injective.

###### Proof.

We first show that the finiteness of
$H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))$ implies the finiteness of
$(\mathscr{A}_{\infty}(m-1))_{\mathscr{G}_{F}}$. Using [24, Lemma6.2] we know
that this implies that $(\mathscr{A}_{\infty}(m-1))^{\mathscr{G}_{F}}$ is
finite. It suffices to show that the kernel of $e_{p}$ on
$\overline{\mathscr{E}}_{\infty}(m-1)$ is finite by [24, Corollary7.8] because
both $\overline{\mathscr{E}}$ and $\overline{\mathscr{C}}$ are
$\Lambda$-modules of rank 1 with
$\overline{\mathscr{E}}/\overline{\mathscr{C}}$ a torsion module. Suppose that
the image of $(\mathscr{E}_{\infty}(m-1))_{\mathscr{G}_{F}}$ under $e_{p}$ has
not rank 1, i.e. is finite. Then the image of
$(\overline{\mathscr{E}}_{\infty}(m-1))_{\mathscr{G}_{F}}$ in
$(\mathscr{U}_{\infty}(m-1))_{\mathscr{G}_{F}}$ must be finite as well because
of

$(\mathscr{U}_{\infty}(m-1))_{\mathscr{G}_{F}}\cong
H^{1}(F\otimes\mathbb{Q}_{p},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}:=\left(\bigoplus_{v|p}H^{1}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))\right)^{*},$

which is the local case of Theorem6.2???. The kernel of the map

$(\overline{\mathscr{E}}_{\infty}(m-1))_{\mathscr{G}_{F}}\to(\mathscr{U}_{\infty}(m-1))_{\mathscr{G}_{F}}$

is
$H_{1}(\mathscr{G}_{F},\mathscr{U}_{\infty}/\overline{\mathscr{E}}_{\infty}(m-1))$.
On the other hand, up to finite groups, there exists
$H_{1}(\mathscr{G}_{F},\mathscr{X}_{\infty}(m-1))\cong
H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}$???. So, we
get a commutative diagram (up to finite groups)

${H_{1}(\mathscr{G}_{F},\mathscr{U}_{\infty}/\overline{\mathscr{E}}_{\infty}(m-1))}$${H_{1}(\mathscr{G}_{F},\mathscr{X}_{\infty}(m-1))}$${(\overline{\mathscr{E}}_{\infty}(m-1))_{\mathscr{G}_{F}}}$${H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))}$${(\mathscr{U}_{\infty}(m-1))_{\mathscr{G}_{F}}}$${H^{1}(F\otimes\mathbb{Q}_{p},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)))^{*}.}$$\scriptstyle{\iota}$$\scriptstyle{e_{p}}$$\scriptstyle{\cong}$

The kernel of the map $\iota$ is a quotient of
$(\mathscr{A}_{\infty}(m-1))^{\mathscr{G}_{F}}$ which is finite. Thus, we get
a contradiction and $e_{p}$ can not be zero on the free part of
$(\overline{\mathscr{E}}_{\infty}(m-1))_{\mathscr{G}_{F}}$. Hence, $e_{p}$ is
non zero on $(\overline{\mathscr{C}}_{\infty}(m-1))_{\mathscr{G}_{F}}$. ∎

Consider the exact sequence

$0\to\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}\to\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}\to\mathscr{X}_{\infty}\to\mathscr{A}_{\infty}\to
0.$

Let $\chi$ be a character of $G=\text{Gal}(F/K)$ (viewed also as a character
of $\Delta=\text{Gal}(F_{1}/K)$). From the above exact sequence, we obtain an
exact sequence of $\chi$-eigenspaces

$0\to(\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi}\to(\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi}\to\mathscr{X}_{\infty}(m-1)^{\chi}\to\mathscr{A}_{\infty}(m-1)^{\chi}\to
0.$

The following algebraic lemma is similar to the results in [6, Lemma3.5.5] or
[18, Lemma5.3].

######  Lemma 6.5.

Let $\chi$ be is an irreducible $\mathbb{Z}_{p}$-representation of $\Delta$
which is nontrivial on the decomposition group of $\mathfrak{p}$ in $\Delta$,
$\alpha:((\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi})^{\Gamma}\to(\mathscr{X}_{\infty}(m-1))^{\chi})^{\Gamma}$
and
$\beta:((\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi})_{\Gamma}\to(\mathscr{X}_{\infty}(m-1))^{\chi})_{\Gamma}$
be induced by cohomology. Then kernels and cokernels of $\alpha$ and $\beta$
are finite, moreover

$\frac{\\#ker\alpha}{\\#coker\alpha}=\frac{\\#ker\beta}{\\#coker\beta}.$

###### Proof.

We follow [18, Lemma5.3] to get this case. By Theorem5.1(ii), the assumption
implies
$A:=(\overline{\mathscr{E}}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi}$
and $B:=\mathscr{A}_{\infty}(m-1)^{\chi}$ have the same characteristic ideal,
hence $A^{\Gamma},A_{\Gamma},B^{\Gamma},B_{\Gamma}$ is finite. Moreover, the
Herbrand quotients are equal:

$\frac{\\#A^{\Gamma}}{\\#A_{\Gamma}}=\frac{\\#B^{\Gamma}}{\\#B_{\Gamma}}.$

Denote $X=(\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\chi}$
and $Y=\mathscr{X}_{\infty}(m-1))^{\chi}$. Consider the following diagram

${0}$${(X/A)^{\Gamma}}$${Y^{\Gamma}}$${B^{\Gamma}}$${(X/A)_{\Gamma}}$${Y_{\Gamma}}$${B_{\Gamma}}$${0}$${X^{\Gamma}}$${X^{\Gamma}}$${X_{\Gamma}}$${X_{\Gamma}}$.$\scriptstyle{=}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}$$\scriptstyle{=}$$\scriptstyle{\delta}$$\scriptstyle{\beta}$

Take the quotients of kernels and cokernels along the diagram

$\frac{\\#ker\gamma}{\\#coker\gamma}\cdot\frac{1}{\\#B^{\Gamma}}\cdot\frac{\\#ker\beta}{\\#coker\beta}=\frac{\\#ker\alpha}{\\#coker\alpha}\cdot\frac{\\#ker\delta}{\\#coker\delta}\cdot\frac{1}{\\#B_{\Gamma}}.$

Since $ker\gamma=A^{\Gamma}$ and

$0\to coker\gamma\to A_{\Gamma}\to ker\delta\to 0$

is exact and $coker\delta=0$, the result follows. ∎

The following lemma is [18, Lemma5.3].

######  Lemma 6.6.

Let $F_{\infty}/F$ be any $\mathbb{Z}_{p}$-extension with Galois group
$\Gamma$, and let

$0\to A\to X\to Y\to B\to 0$

be an exact sequence of $\mathbb{Z}_{p}[[\Gamma]]$-module, such that $X$ and
$Y$ have the same characteristic polynomial. Let $\alpha:X^{\Gamma}\to
Y^{\Gamma}$ and $\alpha:X_{\Gamma}\to Y_{\Gamma}$ be induced by cohomology. If
any of the four groups $A^{\Gamma},A_{\Gamma},B^{\Gamma},B_{\Gamma}$ is
finite, the same is true for kernels and cokernels of $\alpha$ and $\beta$ and
we have

$\frac{\\#ker\alpha}{\\#coker\alpha}=\frac{\\#ker\beta}{\\#coker\beta}.$

The following theorem is to compare the elliptic units with the étale
cohomology of $F$.

######  Theorem 6.7.

Let $F$ be an abelian extension field of imaginary quadratic field $K$, and
$p$ an odd prime with $p\nmid[F:K]$. Suppose $p$ is split in $K$, or $p$ is
inert or ramified over which the elliptic curve $E$ has good reduction. For
$m>1$ and for each character $\chi\neq 1$ of $\text{Gal}(F/K)$, we have

$[H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}:(\overline{\mathscr{C}}_{\infty}(m-1)_{\mathscr{G}_{F}})^{\chi}]=\frac{\\#H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}}{\\#H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))^{\chi}}.$

###### Proof.

Case 1. $p$ is split in $K$. Then we have $(p)=\mathfrak{p}\mathfrak{p}^{*}$.

Consider the Poitou-Tate duality sequence

$\displaystyle 0$ $\displaystyle\to\mathscr{X}(m-1)^{\mathscr{G}_{F}}\to
H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))\to\bigoplus_{v\in
S}\mathscr{X}_{v,\infty}(m-1)_{\mathscr{G}_{F}}$
$\displaystyle\to\mathscr{X}_{\infty}(m-1)_{\mathscr{G}_{F}}\to\mathscr{A}^{\prime}_{\infty}(m-1)_{\mathscr{G}_{F}}\to
0,$

comparing with the sequence

$\displaystyle 0$
$\displaystyle\to(\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))^{\mathscr{G}_{F}}\to\overline{\mathscr{C}}_{\infty}(m-1)_{\mathscr{G}_{F}}$
$\displaystyle\to\mathscr{U}_{\infty}(m-1)_{\mathscr{G}_{F}}\to(\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty}(m-1))_{\mathscr{G}_{F}}\to
0,$

where $\mathscr{X}_{v,\infty}$ is the Galois group over
$F_{v,\infty}:=F_{v}(\mathfrak{p}^{\infty})$ ???of the maximal abelian
$p$-extension of $F_{v,\infty}$.

Let $\mathscr{X}_{p,\infty}=\bigoplus_{v\in S}\mathscr{X}_{v,\infty}.$ Then

$(\mathscr{X}_{p,\infty}(m-1))^{\mathscr{G}_{F}}\cong\bigoplus_{v\in
S}H^{2}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}=0,$

( where $cd_{p}(F_{v})=1$ since char$(F_{v})=p$, see [23, Theorem7.1.8] ) and
hence $\mathscr{U}_{\infty}(m-1)^{\mathscr{G}_{F}}=0$ because of
$\mathscr{U}_{\infty}(m-1)\subset\mathscr{X}_{p,\infty}(m-1)$. Moreover, local
class fields theory yields an exact sequence

$0\to\mathscr{U}_{\infty}\to\mathscr{X}_{p,\infty}\to\bigoplus_{v\in
S}\mathbb{Z}_{p}\to
0,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\textbf{???}}$

which leads to a short exact sequence

$0\to\mathscr{U}_{\infty}(m-1)_{\mathscr{G}_{F}}\to\mathscr{X}_{p,\infty}(m-1)_{\mathscr{G}_{F}}\to\bigoplus_{v\in
S}H^{0}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))\to 0.$

Let us take the $\chi$-eigenspaces of the above sequence. We obtain the
following commutative diagram:

${0}$${(\mathscr{X}(m-1)^{\chi})^{\Gamma}}$${H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}}$${0}$${((\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty})(m-1))^{\chi})^{\Gamma}}$${(\overline{\mathscr{C}}_{\infty}(m-1)^{\chi})_{\Gamma}}$${0}$$\scriptstyle{u}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$
${(\mathscr{X}_{p,\infty}(m-1)^{\chi})_{\Gamma}}$${(\mathscr{X}_{\infty}(m-1)^{\chi})_{\Gamma}}$${(\mathscr{A}^{\prime}_{\infty}(m-1)^{\chi})_{\Gamma}}$${0}$${(\mathscr{U}_{\infty}(m-1)^{\chi})_{\Gamma}}$${((\mathscr{U}_{\infty}/\overline{\mathscr{C}}_{\infty})(m-1))^{\chi})_{\Gamma}}$${0}$${0}$$\scriptstyle{\delta}$$\scriptstyle{\beta}$

For $m>1$, the finiteness of group $H^{2}(G_{S}(F),\mathbb{Z}_{p}(m-1))$
implies the finiteness of $\mathscr{A}^{\prime}_{\infty}(m-1)_{\Gamma}$. By
Lemma6.6, we know that $\alpha$ and $\beta$ have finite kernels and cokernels,
hence the same is true for coker$\delta$. We obtain

$\frac{\\#\mbox{ker}\alpha}{\\#\mbox{coker}\alpha}\cdot\frac{1}{\\#\mbox{coker}\delta}\cdot\frac{1}{\\#(\mathscr{A}^{\prime}_{\infty}(m-1)^{\chi})_{\Gamma}}=\frac{1}{\\#\mbox{coker}\gamma}\cdot\frac{\\#\mbox{ker}\beta}{\\#\mbox{coker}\beta}.$

By Lemma6.6 and Theorem5.1(i), we get

$[H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}:(\overline{\mathscr{C}}_{\infty}(m-1)^{\chi})_{\Gamma}]=\\#\mbox{coker}\delta\cdot\\#(\mathscr{A}^{\prime}_{\infty}(m-1)^{\chi})_{\Gamma}.$

There is

$(\mathscr{A}^{\prime}_{\infty}(m-1)^{\chi})_{\Gamma}\cong\text{{\fontWCA
SH}}_{S}^{2}(F,\mathbb{Z}_{p}(m))^{\chi},$

which fits into an exact sequence

$\displaystyle 0$ $\displaystyle\to\text{{\fontWCA
SH}}_{S}^{2}(F,\mathbb{Z}_{p}(m))^{\chi}\to
H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}$
$\displaystyle\to\bigoplus_{v\in
S}H^{2}(F_{v},\mathbb{Z}_{p}(m))^{\chi}\to(H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*})^{\chi}\to
0.$

By local duality, $H^{2}(F_{v},\mathbb{Z}_{p}(m))\cong
H^{0}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}$, hence

$H^{2}(F_{v},\mathbb{Z}_{p}(m))^{\chi}\cong(H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{\chi^{-1}})^{*}.$

Now
$\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m)^{\chi^{-1}}=\mathbb{Q}_{p}/\mathbb{Z}_{p}^{\chi^{-1}\omega^{m-1}}(1-m)$
is trivial except if $\chi=\omega^{m-1}$, in which case the order of
$H^{0}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*}$ is the same as that of
$H^{0}(F_{v},\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))$???. The same argument
applies to the groups
$(H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1-m))^{*})^{\chi}$ and
$H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1)))^{\chi}$, hence the claim holds.

Case 2. $p$ is inert or ramifies in $K$. If $\chi$ is an irreducible
$\mathbb{Z}_{p}$-representation of $\Delta$ which is nontrivial on the
decomposition group of $\mathfrak{p}$ in $\Delta$, then by Theorem5.1(ii) and
Lemma6.5, through a similar discussion as case 1, we have

$[H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}:(\overline{\mathscr{C}}_{\infty}(m-1)_{\mathscr{G}_{F}})^{\chi}]=\frac{\\#H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))^{\chi}}{\\#H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))^{\chi}}.$

By [17, Lemma2.2.10], we know $\chi$ is non trivial on $\Delta$, where $p$ is
a prime over which $E$ has good reduction???.

∎

##  7\. Lichtenbaum conjecture

Take $\mathfrak{f}$ be the conductor of the abelian extension $F/K$, which is
consider as a subfield of $K(\mathfrak{f})$.

######  Definition 7.1.

Let $B_{j}(K(\mathfrak{f}))$ be the Galois module of
$H^{1}_{\mathcal{M}}(K(\mathfrak{f}),j+1)$ generated by motivic element
$\xi_{\chi}(j)$, where $\chi$ runs over the character of
$G_{\mathfrak{f}}=\text{Gal}(K(\mathfrak{f})/K)$. Let $B_{j}(F)\subset
H^{1}_{\mathcal{M}}(F,j+1)$ be the image of $B_{j}(K(\mathfrak{f}))$ under
corestriction.

######  Theorem 7.2.

Let $F=K(\mathfrak{f})$, $E$ be an elliptic curve with CM by $K$. Let $S$ be a
set of primes of $2$ and bad primes $p$ over which $E$ has bad reduction. Then
the cohomology Lichtenbaum conjecture holds up to the primes in $S$, i.e.,

$\zeta_{F}(1-m)^{*}=R_{m}(F)\prod_{p}\frac{\\#H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))}{\\#H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))_{\mbox{tors}}}.$

###### Proof.

Let $G=\mbox{Gal}(F/K),\widehat{G}$ be the characters group of $G$. Then we
have

(7.1) $\zeta_{F}(s)=\prod_{\chi\in\widehat{G}}L(\chi,s).$

Theorem3.4 immediately gives the covolume of
$\rho_{\infty}(B_{m}(K(\mathfrak{f})))$ in $\mathbb{R}(m-1)^{+}$, that is,

$\text{covol}(\rho_{\infty}(B_{m}(K(\mathfrak{f}))))=\prod_{\chi}\frac{(2\mathcal{N}\mathfrak{f})^{m-1}\Phi(\mathfrak{f}_{\chi})}{(-1)^{1-m}(2m)!\Phi(\mathfrak{f})}L^{\prime}(\chi,1-m).$

But the Beilinson regulator is

$R_{m}(K(\mathfrak{f}))=\frac{\text{covol}(\rho_{\infty}(B_{m}(K(\mathfrak{f}))))}{[K_{2m-1}(K(\mathfrak{f}))/\text{tors}:B_{m}(K(\mathfrak{f}))]}.$

By (7.1), we have

$\zeta_{F}(1-m)^{*}=R_{m}(F)\cdot[K_{2m-1}(K(\mathfrak{f}))/\text{tors}:B_{m}(K(\mathfrak{f}))]\cdot\prod_{\chi}\frac{(-1)^{1-m}(2m)!\Phi(\mathfrak{f})}{(2\mathcal{N}\mathfrak{f})^{m-1}\Phi(\mathfrak{f}_{\chi})}$

Theorem4.3 conclude that

$[K_{2m-1}(K(\mathfrak{f}))/\text{tors}:B_{m}(K(\mathfrak{f}))]\sim_{p}[H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m)):(\overline{\mathscr{C}}_{\infty}(m-1)_{\mathscr{G}_{F}})],$

which are equal up to a $p$-adic unit.

From Theorem6.7, we get that

$\zeta_{F}(1-m)^{*}=R_{m}(F)\cdot\prod_{p}\frac{\\#H^{2}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))}{\\#H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))}\cdot\prod_{\chi}\frac{(-1)^{1-m}(2m)!\Phi(\mathfrak{f})}{(2\mathcal{N}\mathfrak{f})^{m-1}\Phi(\mathfrak{f}_{\chi})}.$

Lemma2.2 in [18] shows that
$H^{0}(F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(m-1))=H^{1}(\mathcal{O}_{F}[1/p],\mathbb{Z}_{p}(m))_{\mbox{tors}}$.
∎

$\mathscr{SUN}$ $\mathscr{CHAOS}$

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