$\text{red}_{\mathfrak{l}_{2}}(P)=\gamma([1])$ for some CM point $P$ and
$\text{red}_{\mathfrak{l}_{2}}(P_{m}(a))=x_{m}(a)\tau_{n}(\text{red}_{\mathfrak{l}_{2}}(P))$.
∎

######  Theorem 6.6 (Second reciprocity law).

Under the assumption $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-DT})$, PO and
"Ihara’s lemma", there exists an $n$-admissible form
$\mathcal{D}^{\prime\prime}=(\Delta_{B}\mathfrak{l}_{1}\mathfrak{l}_{2},g^{\prime\prime})$
such that the following equalities hold

$v_{\mathfrak{l}_{1}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{2}))=v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{1}))=\theta_{\infty}(\mathcal{D}^{\prime\prime})\in\mathcal{O}_{f,n}[[\Gamma]].$

###### Proof.

Let
$\mathcal{J}^{[\mathfrak{l}_{1}]}_{n}=\text{Pic}^{0}({\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n})$.
Then $\gamma$ induces a map

$\gamma_{*}:\mathbf{Z}[X_{B^{\prime\prime}}(\mathcal{U}^{\prime\prime}Y)]\smash{\mathop{\longrightarrow}}\mathcal{J}^{[\mathfrak{l}_{1}]}_{n}(k_{\mathfrak{l}^{2}_{2}})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}_{1}]}_{g}\smash{\mathop{\longrightarrow}}H^{1}_{{\rm
fin}}(K_{\mathfrak{l}_{2}},T_{f,n})\simeq\mathcal{O}_{f,n}.$

By [VO13, Proposition 11.9] or [Lon12, Lemma 7.20] based on Ihara’s general
theorem [Iha99, Section 3 (G)] and Proposition 6.4, $\gamma_{*}$ is in fact
surjective and hence gives rise to a modular form $g^{\prime\prime}\in
S^{B^{\prime\prime}}_{2}(\mathcal{U}^{\prime\prime}Y,\mathcal{O}_{f,n})$.
Moreover it is an eigenform for the Hecke algebra
${\mathbb{T}}_{B^{\prime\prime}}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ such
that its Hecke eigenvalues are given by
$\lambda^{\prime\prime}:{\mathbb{T}}_{B^{\prime\prime}}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})\smash{\mathop{\longrightarrow}}\mathcal{O}_{f,n}$
which is determined by
$\lambda^{\prime\prime}(T_{v})=\lambda_{f}(T_{v})\,(\mbox{mod }{\varpi^{n}})$
for $v\nmid{\mathfrak{p}}\mathfrak{n}\mathfrak{l}_{1}\mathfrak{l}_{2}$,
$\lambda^{\prime\prime}(U_{v})=\lambda_{f}(U_{v})\,(\mbox{mod }{\varpi^{n}})$
for $v\mid{\mathfrak{p}}\mathfrak{n}$ and
$\lambda^{\prime\prime}(U_{\mathfrak{l}_{i}})=\epsilon_{\mathfrak{l}_{i}}$ for
$i\in\\{1,2\\}$. Such an eigenform is unique up to a unit in
$\mathcal{O}_{f,n}$ again by the multiplicity one result Theorem 9.9.

It follows from the above discussion that

(6.4) $\displaystyle
v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{1})_{m})$
$\displaystyle=\sum_{[a]_{m}\in G_{m}}\langle
x_{m}(a)\tau_{n},g^{\prime\prime}\rangle\pi([a]_{m})$
$\displaystyle=\sum_{[a]_{m}\in G_{m}}g^{\prime\prime}(x_{m}(a))\pi([a]_{m})$
$\displaystyle=\theta_{m}(\mathcal{D}^{\prime\prime}).$

By switching the role of $\mathfrak{l}_{1}$ and $\mathfrak{l}_{2}$, we get the
conclusion of the theorem. ∎

##  7\. Euler system argument

In this chapter we carry out the Euler system argument to bound the Selmer
group. The method employed here is originally due to [BD05]. The main result
Theorem 7.17 is proved under the assumptions: $(\text{CR}^{+})$,
$(\mathfrak{n}^{+}\text{-min})$ and PO. The assumption
$(\mathfrak{n}^{+}\text{-min})$ is stronger than the desired assumption
$(\mathfrak{n}^{+}\text{-DT})$ but we will later relax it to
$(\mathfrak{n}^{+}\text{-DT})$ in chapter 10.

###  7.1. Special elements in the Galois representation

Throughout this section we will assume the assumptions PO, $(\text{CR}^{+})$
and $(\mathfrak{n}^{+}\text{-min})$. Let $H=\bar{\rho}^{*}_{f}(G_{F})$ be the
image of the residual Galois representation. We examine the existence of
certain special elements in the image. We have the following lemma.

######  Lemma 7.1.

The group $H$ contains

  1. (1)

the scalar matrix $-1$;

  2. (2)

an element $h\in H$ such that $\mathop{\rm Tr}\nolimits(h)=\det(h)+1$ with
$\det(h)\neq\pm 1\in\mathbb{F}_{{\mathfrak{p}}}$.

###### Proof.

For (1), let $Z$ be the center of $H$. From $(\text{CR}^{+})(1)$, we note that
$Z$ is contained in the set of scalar matrices. If $p\mid\\#H$, then it is
well-known that $H$ contains a conjugate of $\mathop{\rm
SL}\nolimits_{2}(\mathbb{F}_{p})$ and the result follows. So we assume that
$H$ has order prime to $p$ but then $H/Z$ has the following possibilities:

$H/Z$ is cyclic. This is impossible in view of $(\text{CR}^{+})(1)$.

$H/Z$ is the three exceptional groups $A_{4},S_{4},A_{5}$. However if we let
$I_{{\mathfrak{p}}}$ be the image of the inertia at ${\mathfrak{p}}$ in $H/Z$,
then we know it is a cyclic group of order $>5$ by the assumption that
$\\#(\mathbb{F}^{\times}_{{\mathfrak{p}}})^{k-1}>5$. Then these possibilities
can not happen.

$H/Z$ is the dihedral group. Since
$\\#(\mathbb{F}^{\times}_{{\mathfrak{p}}})^{k-1}$ is even,
$I_{{\mathfrak{p}}}$ is contained in a cyclic group of even order and hence
there is an element of order $2$ in $H/Z$. We can then conclude that there is
an element of order $2$ in $Z$.

For (2), if $p\nmid{\bf{N}}(\mathfrak{l})^{2}-1$ for some
$\mathfrak{l}\mid\mathfrak{n}^{-}$, then we can simply choose
$\mathrm{Frob}_{\mathfrak{l}}$ as our element $h$. If this is not the case,
then by our assumption $\text{CR}^{+}(3)$, $\bar{\rho}^{*}_{f}(G_{F})$ is
ramified at $\mathfrak{l}$ and hence $H$ contains the image of the inertia
which is a subgroup of order divisible by $p$. Therefore $H/Z$ contains the
group $\mathop{\rm PSL}\nolimits_{2}(F)$ for some finite extension
$F/\mathbb{F}_{p}$ and is contained in $\mathop{\rm PGL}\nolimits_{2}(F)$.
Therefore $[\mathop{\rm SL}\nolimits_{2}(F),\mathop{\rm
SL}\nolimits_{2}(F)]=\mathop{\rm SL}\nolimits_{2}(F)$ is contained in $H$. We
take $C=\mathop{\rm GL}\nolimits_{2}(F)\cap H$. Then $C$ sits in between
$\mathop{\rm SL}\nolimits_{2}(F)$ and $\mathop{\rm GL}\nolimits_{2}(F)$. Let
$\gamma\in H$ be the image of a generator of the inertia group at
${\mathfrak{p}}$. Then we have $\det(\gamma)=u$ generates
$\mathbb{F}^{\times}_{{\mathfrak{p}}}$ and moreover $\mathop{\rm
Tr}\nolimits(\gamma)=u^{k/2}+u^{1-k/2}\in\mathbb{F}_{{\mathfrak{p}}}$ is non-
zero. By the previous discussion, we have
$H\subset\bar{\mathbb{F}}_{{\mathfrak{p}}}\mathop{\rm GL}\nolimits_{2}(F)$. We
conclude from $\mathop{\rm
Tr}\nolimits(\gamma)=u^{k/2}+u^{1-k/2}\in\mathbb{F}_{{\mathfrak{p}}}$ that
$\gamma\in C$. Thus $C\supset\mathbb{F}_{{\mathfrak{p}}}SL_{2}(F)$. The result
follows as then $H$ contains $\mathop{\rm
GL}\nolimits_{2}(\mathbb{F}_{{\mathfrak{p}}})$. ∎

As a consequence of the above two lemmas, we prove the following key theorem
for our Euler system argument.

######  Theorem 7.2.

Let $t\leq n$ be a postive integer. Let $\kappa\in H^{1}(K,A_{f,t})$ be a non-
zero element, then there exist infinitely many $n$-admissible primes
$\mathfrak{l}$ such that $\partial_{\mathfrak{l}}(\kappa)=0$ and
$v_{\mathfrak{l}}:\langle\kappa\rangle\smash{\mathop{\longrightarrow}}H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,t})$ is injective. Here $\langle\kappa\rangle$ is
the $\mathcal{O}_{f,t}$-submodule of $H^{1}(K,A_{f,t})$ generated by
$\langle\kappa\rangle$.

###### Proof.

First of all, since $\mathfrak{l}$ is an $n$-admissible prime, $H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,1})\hookrightarrow H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,t})$ is injective. Thus without loss of generality
we can assume that $t=1$ and $\kappa\in H^{1}(K,A_{f,1})$.

Let $L_{n}=\bar{F}^{\mathop{\rm ker}\nolimits(\rho_{n})}$ be the field cut out
by $\rho_{n}:=\rho^{*}_{f}\,(\mbox{mod }{\varpi^{n}})$. We remark that since
$D_{K/F}$ is prime to $\mathfrak{n}p$ and $\rho_{n}$ is unramified away form
$\mathfrak{n}p$, the field $L_{n}$ is linearly disjoint from $K$.

We put $M=L_{n}K$, then $\mathop{\rm Gal}\nolimits(M/F)=\mathop{\rm
Gal}\nolimits(K/F)\times\mathop{\rm Gal}\nolimits(L_{n}/F)$ which is of the
form $\langle\tau\rangle\times\mathop{\rm Aut}\nolimits(A_{f,n})$. By Lemma
7.1(1), $\mathop{\rm Gal}\nolimits(L_{n}/F)$ contains a subgroup of the form
$\langle\pm 1\rangle$. Then by the Hochschild-Serre spectral sequence

$H^{r}(\mathop{\rm Gal}\nolimits(L_{n}/F)/\langle\pm 1\rangle,H^{s}(\langle\pm
1\rangle,A_{f,n}))\Rightarrow H^{r+s}(L_{n}/F,A_{f,n}),$

we see $H^{i}(L_{n}/F,A_{f,n})=H^{i}(M/K,A_{f,n})=0$ for all $i$. Thus

$H^{1}(K,A_{f,1})\smash{\mathop{\longrightarrow}}H^{1}(M,A_{f,1})$

is injective . We identify ${\kappa}$ with its non-zero image $\kappa\in
H^{1}(M,A_{f,1})=\mathop{\rm Hom}\nolimits(G_{M},A_{f,1})$. Let $M_{\kappa}$
be the field cut out by ${\kappa}$ over $M$ and
$A_{\kappa}:={\kappa}(\mathop{\rm Gal}\nolimits(M_{\kappa}/M))\subset A_{f,1}$
be a $\mathbb{F}_{p}[\mathop{\rm Gal}\nolimits(M/F)]$ submodule of $A_{f,1}$.
Notice $A_{\kappa}$ is of dimension 2 by (LABEL:Irr).

Without loss of generality, we can assume $\tau\kappa=\delta\kappa$ for some
$\delta\in\\{\pm 1\\}$. Then $M_{\kappa}/F$ is Galois and we can identify
$\mathop{\rm Gal}\nolimits(M_{\kappa}/F)$ with $A_{\kappa}\rtimes\mathop{\rm
Gal}\nolimits(M/F)$ where $(\tau^{j},\sigma)\in\mathop{\rm Gal}\nolimits(M/F)$
acts on $v\in A_{\kappa}$ by $(\tau^{j},\sigma)v=\delta^{j}\rho_{1}(\sigma)v$.

By Lemma 7.1, we can find an element $({\bf v},\tau,\sigma)\in\mathop{\rm
Gal}\nolimits(M_{\kappa}/F)\simeq A_{\kappa}\rtimes(\mathop{\rm
Gal}\nolimits(K/F)\times\mathop{\rm Gal}\nolimits(L_{n}/F))$ such that

  1. (1)

$\rho_{n}(\sigma)$ has eigenvalues $\delta$ and $\lambda$ in
$(\mathcal{O}_{f,n})^{\times}$ where $\lambda\neq\pm 1$ and has order prime to
$p$;

  2. (2)

${\bf v}$ is in the $\delta$-eigenspace of $\sigma$.

By the Chebotarev density theorem, there exist infinitely many unramified
$\mathfrak{l}$ with $\mathfrak{l}\nmid\mathfrak{n}$ such that
$\mathrm{Frob}_{\mathfrak{l}}(M_{\kappa}/F)=({\bf v},\tau,\sigma)$. Then it
follows that $\mathfrak{l}$ is $n$-admissible for $f$ as we have
$\mathrm{Frob}_{\mathfrak{l}}(M/F)=(\tau,\sigma)$. Let $\mathfrak{L}$ be any
place above $\mathfrak{l}$ in $M$. Then we have
$\mathrm{Frob}_{\mathfrak{L}}(M_{\kappa}/M)=({\bf v},\tau,\sigma)^{d}={\bf
v}+\delta\sigma{\bf v}\cdots+\delta\sigma^{d-1}{\bf v}=d{\bf v}$ where $d$ is
the residue degree of $\mathfrak{L}$ over $\mathfrak{l}$ which is necessarily
even. Hence
${\kappa}(\mathrm{Frob}_{{\mathfrak{L}}}(M_{\kappa}/M))=d{\kappa}({\bf v})\neq
0$ and
$v_{\mathfrak{l}}(\kappa)=({\kappa}(\mathrm{Frob}_{{\mathfrak{L}}}(M_{\kappa}/M)))_{{\mathfrak{L}}\mid\mathfrak{l}}$
is not zero. ∎

Let $\Delta$ be a square free product of primes in $F$ such that
$\Delta/\mathfrak{n}^{-}$ is a product of n-admissible primes.

######  Definition 7.3 ($n$-admissible set).

A finite set $S$ of primes in $F$ is said to be $n$-admissible for $f$ if:

  1. (1)

$S$ consists of $n$-admissible primes of $f$.

  2. (2)

The map $\text{Sel}_{\Delta}(K,T_{f,n})\rightarrow\oplus_{\mathfrak{l}\in
S}H^{1}_{{\rm fin}}(K_{\mathfrak{l}},T_{f,n})$ is injective.

######  Remark.

It follows from Theorem 7.2 that any collection of $n$-admissible primes can
be enlarged to an $n$-admissible set.

###  7.2. Freeness of Selmer groups

Let $S$ be an $n$-admissible set for $f$ and $L/K$ be a finite extension
contained in $K_{\infty}/K$. In this section all the results are proved under
the assumptions $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-min})$, and PO.

######  Lemma 7.4.

The map
$\text{Sel}_{\Delta}(L,T_{f,n})\smash{\mathop{\longrightarrow}}\oplus_{\mathfrak{l}\in
S}H^{1}_{{\rm fin}}(L_{\mathfrak{l}},T_{f,n})$ is injective.

###### Proof.

Let $C$ be the kernel of this map. Since $\mathop{\rm Gal}\nolimits(L/K)$ is a
$p$-group, there exists a non-trivial fixed point of its action. It follows
from Proposition 2.10 that this fixed point is in the kernel of
$\text{Sel}_{\Delta}(K,T_{f,n})\smash{\mathop{\longrightarrow}}\oplus_{\mathfrak{l}\in
S}H_{{\rm fin}}^{1}(K_{\mathfrak{l}},T_{f,n})$ which is a contradiction to the
$n$-admissibility of $S$. ∎

We have the following Poitou-Tate exact sequence.

######  Lemma 7.5.

$0\smash{\mathop{\longrightarrow}}\text{Sel}_{\Delta}(L,T_{f,n})\smash{\mathop{\longrightarrow}}\text{Sel}^{S}_{\Delta}(L,T_{f,n})\smash{\mathop{\longrightarrow}}\oplus_{\mathfrak{l}\in
S}H^{1}_{{\rm
sing}}(L_{\mathfrak{l}},T_{f,n})\smash{\mathop{\longrightarrow}}{\text{Sel}_{\Delta}(L,A_{f,n})}^{\\!\vee}\smash{\mathop{\longrightarrow}}0.$

###### Proof.

This is standard, see [Rub00, Theorem 1.7.3]. The right exactness follows from
the $n$-admissibility of $S$. ∎

######  Proposition 7.6.

The Selmer group $\text{Sel}^{S}_{\Delta}(L,T_{f,n})$ is a free
$\mathcal{O}_{f,n}[\mathop{\rm Gal}\nolimits(L/K)]$-module of rank $|S|$.

###### Proof.

Let $L=K_{m}$ for some $m$ large enough, and let $\mathfrak{m}_{n,m}$ be the
maximal ideal of $\mathcal{O}_{f,n}[\mathop{\rm Gal}\nolimits(K_{m}/K)]$.
Notice $\mathcal{O}_{f,n}[\mathop{\rm Gal}\nolimits(K_{m}/K)]$ is of the form

$\mathcal{O}_{f,n}[x_{1},\cdots,x_{d}]/(x_{1}^{p^{m}}-1,\cdots,x_{d}^{p^{m}}-1)$

which implies it is local complete intersection and thus Gorentein.

By Lemma 2.10, we have
$\text{Sel}^{S}_{\Delta}(K,T_{f,1})=\text{Sel}^{S}_{\Delta}(K_{m},T_{f,n})[\mathfrak{m}_{n,m}]$.
Hence

${\text{Sel}^{S}_{\Delta}(K_{m},T_{f,n})}^{\\!\vee}/\mathfrak{m}_{n,m}\simeq{\text{Sel}^{S}_{\Delta}(K,T_{f,1})}^{\\!\vee}$

where the latter module is free of rank $|S|$ over $\mathcal{O}_{f,1}$ by
Lemma 7.5. Then by Nakayama’s lemma,
${\text{Sel}^{S}_{\Delta}(K_{m},T_{f,n})}^{\\!\vee}$ has a basis of $|S|$
elements over $\mathcal{O}_{f,n}$. But by Lemma 7.5 and a cardinality
consideration, ${\text{Sel}^{S}_{\Delta}(K_{m},T_{f,n})}^{\\!\vee}$ is free of
rank $|S|$. By the Gorensteiness of $\mathcal{O}_{f,n}[\mathop{\rm
Gal}\nolimits(K_{m}/K)]$, it follows that
$\text{Sel}^{S}_{\Delta}(K_{m},T_{f,n})$ is also free of rank $|S|$. ∎

We summarize some easy consequence of the above lemma in the following
corollary.

######  Corollary 7.7.

Let $S$ be an $n$-admissible set, then

  1. (1)

$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})$ is free of rank $\\#S$
over $\Lambda/\varpi^{n}$.

  2. (2)

$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})/\mathfrak{m}_{\Lambda}={\text{Sel}}^{S}_{\Delta}(K,T_{f,1})$.

  3. (3)

$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})/\varpi{\Lambda}=\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,1})$.

###  7.3. Divisiblity of the Euler system and Selmer group

Let $\xi:\Lambda\smash{\mathop{\longrightarrow}}\mathcal{O}_{\xi}$ be an
$\mathcal{O}_{f}$-algbera homomorphism, where $\mathcal{O}_{\xi}$ is a
discrete valuation ring of characteristic $0$. Let $\varpi_{\xi}$ be a
uniformizer of $\mathcal{O}_{\xi}$. Let $M$ be an $\mathcal{O}_{\xi}$-module.
For each $x\in M$, we denote by
$\text{ord}_{\varpi_{\xi}}(x)=\sup\\{m\in\mathbf{Z}\cup\\{\infty\\}\mid
x\in\varpi_{\xi}^{m}M\\}$ the divisibility index of $x$. Note $x=0$ if and
only if $\text{ord}_{\varpi_{\xi}}(x)=\infty$.

For a finitely generated $\Lambda$-module, we will write
$\text{char}_{\Lambda}(M)$ the characteristic ideal of $M$. The following
result original due to Bertonili-Darmon [BD05] and in our case due to Longo
[Lon12] gives a criterion of identifying elements in
$\text{char}_{\Lambda}(M)$.

######  Lemma 7.8.

Let $M$ be a finitely generated $\Lambda$-module and $L\in\Lambda$. Suppose
that for any homomorphism
$\xi:\Lambda\smash{\mathop{\longrightarrow}}\mathcal{O}_{\xi}$, we have
$\operatorname{length}_{\mathcal{O}_{\xi}}(M\otimes_{\xi}\mathcal{O}_{\xi})\leq\text{ord}_{\varpi_{\xi}}(\xi(L))$.
Then $L\in\text{char}_{\Lambda}(M)$.

###### Proof.

It follows from the statement of this Lemma that $\xi(L)\in\mathop{\rm
Fitt}\nolimits_{\mathcal{O}_{\xi}}(M_{\xi})$ for all $\xi$, then the lemma
follows from [Lon12, lemma 7.4]. ∎

Let $n$ be a positive integer and $\Delta$ be a square-free product of primes
in $F$ as in Definition 5.1. For any
$\xi:\Lambda\rightarrow\mathcal{O}_{\xi}$, we define two integers for each
$n$-admissible form $\mathcal{D}=(\Delta,f_{n})$:

(7.1) $\displaystyle
s_{\mathcal{D}}=\operatorname{length}_{\mathcal{O}_{\xi}}{\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi};$
$\displaystyle
t_{\mathcal{D}}=\text{ord}_{\varpi_{\xi}}\xi(\theta_{\infty}(\mathcal{D}))\text{
where
$\xi(\theta_{\infty}(\mathcal{D}))\in\mathcal{O}_{f,n}[[\Gamma]]\otimes_{\xi}\mathcal{O}_{\xi}=\mathcal{O}_{\xi}/\varpi_{\xi}^{n}$}.$

We will prove the following analogue of [PW11, Proposition 4.3] using an
inductive argument. The one divisibility result will immediately follow from
it.

######  Theorem 7.9.

Assume $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-min})$, PO and "Ihara’s
lemma" hold. Let $t_{0}\leq n$ be a non-negative ineteger and let
$\mathcal{D}_{n+t_{0}}=(\Delta,f_{n+t_{0}})$ be an $(n+t_{0})$-admissible
form. We set $\mathcal{D}_{n}=\mathcal{D}_{n+t_{0}}\,(\mbox{mod
}{\varpi^{n}})=(\Delta,f_{n+t_{0}}\,(\mbox{mod }{\varpi^{n}}))$. Suppose that
$t_{\mathcal{D}_{n}}\leq t_{0}$, then $s_{\mathcal{D}_{n}}\leq
2t_{\mathcal{D}_{n}}$.

####  7.3.1. Proof of the Theorem 7.9

We will prove this theorem by induction on $t_{\mathcal{D}_{n}}$. If
$t_{\mathcal{D}_{n}}=\infty$ or $s_{\mathcal{D}_{n}}=0$, then the claim is
trivially true. So we consider the case when

(7.2) $\displaystyle
t_{\mathcal{D}_{n}}<\infty\Rightarrow\xi(\theta_{\infty}(\mathcal{D}_{n}))\neq
0;$ $\displaystyle
s_{\mathcal{D}_{n}}>0\Rightarrow\text{Sel}_{\Delta}(K_{\infty},A_{f,n})\otimes_{\xi}\mathcal{O}_{\xi}\neq\\{0\\}.$

From now on we will denote

$t=t_{\mathcal{D}_{n}},$ $s=s_{\mathcal{D}_{n}}.$

Let
$\mathfrak{S}^{\xi}_{n}=\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})\otimes_{\xi}\mathcal{O}_{\xi}$
and for some $n$-admissible prime $\mathfrak{l}$, let
$\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})\in\mathfrak{S}^{\xi}_{n+t}$ be
the image of $\kappa_{\mathcal{D}_{n+t}}(\mathfrak{l})$. We define an integer
$e_{\mathcal{D}_{n+t}}(\mathfrak{l})$ by

$e_{\mathcal{D}_{n+t}}(\mathfrak{l})=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})).$

By the first reciprocity law inTheorem 6.2, we have

(7.3)
$e_{\mathcal{D}_{n+t}}(\mathfrak{l})\leq\text{ord}_{\varpi_{\xi}}(\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})))=\text{ord}_{\varpi_{\xi}}(\xi(\theta_{\infty}(\mathcal{D}_{n+t})))=\text{ord}_{\varpi_{\xi}}(\xi(\theta_{\infty}(\mathcal{D}_{n})))=t.$

Let $\tilde{\kappa}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})$ be such that
$\varpi_{\xi}^{e_{{\mathcal{D}_{n+t}}}(\mathfrak{l})}\tilde{\kappa}_{\mathcal{D}_{n+t},\xi}{(\mathfrak{l})}=\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})$
and let $\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})$ be the image
of $\tilde{\kappa}_{\mathcal{D}_{n+t}}(\mathfrak{l})$ under the map
$\mathfrak{S}^{\xi}_{n+t}\smash{\mathop{\longrightarrow}}\mathfrak{S}^{\xi}_{n}$.

######  Lemma 7.10.

We have

  1. (1)

$\text{ord}_{\varpi_{\xi}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))=0$,

  2. (2)

$\text{ord}_{\varpi_{\xi}}(\partial_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})))=t-e_{\mathcal{D}_{n+t}}(\mathfrak{l})$,

  3. (3)

$\partial_{v}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))=0\text{
for all $v\nmid\Delta\mathfrak{l}$}$,

  4. (4)

${\text{res}}_{v}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))\in\widehat{H}^{1}_{\text{ord}}(K_{\infty,v},T_{f,n})$
for all $v\mid\Delta\mathfrak{l}$.

###### Proof.

(1) follows from the control theorem of Selmer groups Corollary 7.7:
$\mathfrak{S}^{\xi}_{n+t}/\varpi_{\xi}\rightarrow\mathfrak{S}^{\xi}_{n}/\varpi_{\xi}$
is an isomorphism and hence
$\text{ord}_{\varpi_{\xi}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))=\text{ord}_{\varpi_{\xi}}(\tilde{\kappa}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))=0$.

(2) follows from the first reciprocity law in Theorem 6.2. (3) and (4) follow
from the definition of the relevant Selmer group. ∎

######  Lemma 7.11.

Let $\eta_{\mathfrak{l}}:\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}},T_{f,n})\otimes_{\xi}\mathcal{O}_{\xi}\smash{\mathop{\longrightarrow}}{\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}$
be the map defined by $\eta_{\mathfrak{l}}(c)(x)=\langle
c,v_{\mathfrak{l}}(x)\rangle$ for
$x\in{\text{Sel}_{\Delta}(K_{\infty},A_{f,n})[\mathop{\rm
ker}\nolimits{\xi}]}$ and $c\in\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}},T_{f,n})$. Then
$\eta_{\mathfrak{l}}(\partial_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})))=0.$

###### Proof.

Note that
${\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}={\text{Sel}_{\Delta}(K_{\infty},A_{f,n})[\mathop{\rm
ker}\nolimits\xi]}^{\\!\vee}$. Consider the sum
$\sum_{\mathfrak{q}}\langle{\text{res}}_{\mathfrak{q}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})),{\text{res}}_{\mathfrak{q}}(x)\rangle=0$
by the global class field theory.

Notice that for $\mathfrak{q}\nmid\Delta\mathfrak{l}$,
$\partial_{\mathfrak{q}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))=0$,
and for $\mathfrak{q}\mid\Delta$, both ${\text{res}}_{\mathfrak{q}}(x)$ and
${\text{res}}_{\mathfrak{q}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))$
are ordinary. It follows that the only possible non-trivial term in the above
sum is
$\eta_{\mathfrak{l}}({\text{res}}_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})))=\langle{\text{res}}_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})),{\text{res}}_{\mathfrak{l}}(x)\rangle$.
This shows that this term itself is zero. ∎

######  Lemma 7.12 (Base case).

If $t=0$, then $s=0$.

###### Proof.

If $t=\text{ord}_{\varpi_{\xi}}\xi(\theta_{\infty}(\mathcal{D}_{n}))=0$(in
this case $e_{\mathcal{D}_{n+t}}(\mathfrak{l})=t=0$ by (7.3) and
$\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})=\kappa_{\mathcal{D}_{n},\xi}(\mathfrak{l})$),
then $\xi(\theta_{\infty}(\mathcal{D}_{n}))$ is a unit and hence by the first
reciprocity law, for any admissible prime $\mathfrak{l}$,
$\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}_{n},\xi}(\mathfrak{l}))$ spans
the rank one space $\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}},T_{f,n})\otimes_{\xi}\mathcal{O}_{\xi}$. It
follows from Lemma 7.11 that the map $\eta_{\mathfrak{l}}$ is zero. Now if
$s\neq 0$,
${\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}$
is non-trivial and by Nakayama’s lemma

$({\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi})/\mathfrak{m}_{\xi}={(\text{Sel}_{\Delta}(K_{\infty},A_{f,n})[\mathfrak{m}_{\Lambda}])}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}$

is nontrivial. By Theorem 7.2 we can choose an admissible prime $\mathfrak{l}$
and a nontrivial element
$c\in\text{Sel}_{\Delta}(K_{\infty},A_{f,n})[\mathfrak{m}_{\Lambda}]=\text{Sel}_{\Delta}(K,A_{f,1})$
such that $v_{\mathfrak{l}}(c)$ is non-zero. It then follows that
$\eta_{\mathfrak{l}}(\partial_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l})))(c)$
is non-trivial which is a contradiction to the previous lemma. ∎

Now we assume that $t>0$. Let $\Xi$ be the set of primes $\mathfrak{l}$ such
that

  1. (1)

$\mathfrak{l}$ is an $n+t_{0}$-admissible and $\mathfrak{l}\nmid\Delta$;

  2. (2)

the integer $e_{\mathcal{D}_{n+t}}(\mathfrak{l})$ is minimal for the set of
$\mathfrak{l}$ in (1).

Let $e=e_{\mathcal{D}_{n+t}}(\mathfrak{l})$ for any $\mathfrak{l}$ satisfies
the above conditions.

######  Lemma 7.13.

We have $e<t$.

###### Proof.

As we already know $e\leq t$, we assume $e=t$ and derive a contradiction.
Since we assumed $s>0$, the Selmer group
${\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}$
is non-trivial. We may use the same argument as in Lemma 7.12 to conclude that
there is a non-trivial class
$x\in\text{Sel}_{\Delta}(K_{\infty},A_{f,n})[\mathfrak{m}_{\Lambda}]\subset
H^{1}(K,A_{f,1})$. Then by Theorem 7.2, there are $n+t_{0}$ admissible primes
$\mathfrak{l}$ such that $v_{\mathfrak{l}}(x)\neq 0$. Now by the assumption
$e=t$ and Lemma 7.10, we have
$\partial_{\mathfrak{l}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}))$
is indivisible and hence its image is non-zero in
$H^{1}(K,A_{f,1})\otimes_{\xi}\mathcal{O}_{\xi}$. But this is a contradiction
to Lemma 7.11. ∎

We choose $\mathfrak{l}_{1}\in\Xi$ and enlarge it to an
$(n+t_{0})$-adimissible set. Let $\kappa_{1}$ be the image of
$\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})$ under the map

$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})\otimes_{\xi}\mathcal{O}_{\xi}/\varpi_{\xi}\hookrightarrow
H^{1}(K,T_{f,1})\otimes_{\xi}\mathcal{O}_{\xi}.$

By Lemma 7.10(1), $\kappa_{1}$ is nontrivial viewing it as an element in
$H^{1}(K,T_{f,1})\otimes_{\xi}\mathcal{O}_{\xi}$. So by Theorem 7.2, we can
pick an $(n+t_{0})$-admissible prime $\mathfrak{l}_{2}$ such that
$v_{\mathfrak{l}_{2}}(\kappa_{1})\neq 0\in
H^{1}(K_{\mathfrak{l}_{2}},T_{f,1})\otimes_{\xi}\mathcal{O}_{\xi}$.

It follows from above choice that

$\text{ord}_{\varpi_{\xi}}(v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})))=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1}))\leq\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2}))\leq\text{ord}_{\varpi_{\xi}}(v_{\mathfrak{l}_{1}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})))$

where the middle inequality is due to the fact that $\mathfrak{l}_{1}$ is an
element in $\Xi$.

From the second explicit reciprocity law, we can find an $n+t_{0}$-admissible
form
$\mathcal{D}^{\prime\prime}_{n+t_{0}}=(\Delta\mathfrak{l}_{1}\mathfrak{l}_{2},g_{n+t_{0}})$
such that

$v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}_{n+t}}(\mathfrak{l}_{1}))=v_{\mathfrak{l}_{1}}(\kappa_{\mathcal{D}_{n+t}}(\mathfrak{l}_{2}))=\theta_{\infty}(\mathcal{D}^{\prime\prime}_{n+t})$

where
$\mathcal{D}^{\prime\prime}_{n+t}=\mathcal{D}^{\prime\prime}_{n+t_{0}}\,(\mbox{mod
}{\varpi^{n+t}})$. This implies that the above sequence of inequalities are
all in fact equalities and in particular that
$e=e_{\mathcal{D}_{n+t}}(\mathfrak{l}_{1})=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1}))=e_{\mathcal{D}_{n+t}}(\mathfrak{l}_{2})=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})).$
It then follows that $\mathfrak{l}_{2}\in\Xi$. Let
$\mathcal{D}^{\prime\prime}_{n}=(\Delta\mathfrak{l}_{1}\mathfrak{l}_{2},g_{n+t_{0}}\,(\mbox{mod
}{\varpi^{n}}))$, then we have

$t_{\mathcal{D}^{\prime\prime}_{n}}=\text{ord}_{\varpi_{\xi}}(\xi(\theta_{\infty}(\mathcal{D}^{\prime\prime}_{n})))=\text{ord}_{\varpi_{\xi}}(v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})))=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1}))=e<t\leq
t_{0}.$

Then by the induction hypothesis applied to $\mathcal{D}^{\prime\prime}_{n}$,
we have $s_{\mathcal{D}^{\prime\prime}_{n}}\leq
2t_{\mathcal{D}^{\prime\prime}_{n}}$. To finish the proof, we prove the
following inequality

$s_{\mathcal{D}_{n}}\leq
s_{\mathcal{D}^{\prime\prime}_{n}}+2(t-t_{\mathcal{D}^{\prime\prime}_{n}}).$

We denote by $S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}$ the submodule of
$\text{Sel}_{\Delta}(K_{\infty},A_{f,n})$ consisting of classes that are
locally trivial at primes over $\mathfrak{l}_{1}$ and $\mathfrak{l}_{2}$. By
definition we have the following exact sequence

(7.4) $\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}_{1}},T_{f,n})\oplus\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}_{2}},T_{f,n})\xrightarrow{P}{\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\smash{\mathop{\longrightarrow}}{S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}}^{\\!\vee}\smash{\mathop{\longrightarrow}}0$

where $P$ is induced by
$\langle,\rangle_{\mathfrak{l}_{1}}\oplus\langle,\rangle_{\mathfrak{l}_{2}}$.
Similarly, we have

(7.5) $\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}_{1}},T_{f,n})\oplus\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}_{2}},T_{f,n})\xrightarrow{P}{\text{Sel}_{\Delta\mathfrak{l}_{1}\mathfrak{l}_{2}}(K_{\infty},A_{f,n})}^{\\!\vee}\smash{\mathop{\longrightarrow}}{S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}}^{\\!\vee}\smash{\mathop{\longrightarrow}}0.$

We tensor the first exact sequence by $\mathcal{O}_{\xi}$ and we fix
isomorphisms

$(\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}_{1}},T_{f,n})\oplus\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}_{2}},T_{f,n}))\otimes_{\xi}\mathcal{O}_{\xi}\simeq\mathcal{O}^{2}_{\xi}/\xi(\varpi)^{n}$

and

$(\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}_{1}},T_{f,n})\oplus\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}_{2}},T_{f,n}))\otimes_{\xi}\mathcal{O}_{\xi}\simeq\mathcal{O}^{2}_{\xi}/\xi(\varpi)^{n}.$

By Lemma 7.11, we get from the first exact sequence above,

(7.6)
$\frac{\mathcal{O}_{\xi}}{(\partial_{\mathfrak{l}_{1}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})))}\oplus\frac{\mathcal{O}_{\xi}}{(\partial_{\mathfrak{l}_{2}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})))}\xrightarrow{P}{\text{Sel}_{\Delta}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}\rightarrow{S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}\rightarrow
0.$

######  Lemma 7.14.

We have an isomorphism

${\text{Sel}_{\Delta\mathfrak{l}_{1}\mathfrak{l}_{2}}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}\simeq{S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}.$

###### Proof.

Let
$s\in\text{Sel}_{\Delta\mathfrak{l}_{1}\mathfrak{l}_{2}}(K_{\infty},A_{f,n})[\mathop{\rm
ker}\nolimits(\xi)]$. We consider the sum

$\sum_{v}\langle{\text{res}}_{v}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})),{\text{res}}_{v}(s)\rangle_{v}=0.$

For $v\nmid\mathfrak{l}_{2}$, the summand
$\langle{\text{res}}_{v}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})),{\text{res}}_{v}(s)\rangle_{v}=0$
and it follows that

$\langle
v_{\mathfrak{l}_{2}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})),{\text{res}}_{\mathfrak{l}_{2}}(s)\rangle_{\mathfrak{l}_{2}}=0.$

The same argument can be used to show that $\langle
v_{\mathfrak{l}_{1}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})),{\text{res}}_{\mathfrak{l}_{1}}(s)\rangle_{\mathfrak{l}_{1}}=0$.
Hence (7.5) factors through

$\frac{\mathcal{O}_{\xi}}{(v_{\mathfrak{l}_{1}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})))}\oplus\frac{\mathcal{O}_{\xi}}{(v_{\mathfrak{l}_{2}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})))}\xrightarrow{P}{\text{Sel}_{\Delta\mathfrak{l}_{1}\mathfrak{l}_{2}}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}\rightarrow{S_{[\mathfrak{l}_{1},\mathfrak{l}_{2}]}}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}\rightarrow
0.$

But
$\text{ord}_{\varpi_{\xi}}(v_{\mathfrak{l}_{1}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})))=t_{\mathcal{D}^{\prime\prime}_{n+t}}-e=0$
and the claim follows. ∎

Now Theorem 7.9 follows by pluging in the isomorphism to the exact sequence
(7.6) and noticing that
$\text{ord}_{\varpi_{\xi}}(\partial_{\mathfrak{l}_{1}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{1})))=\text{ord}_{\varpi_{\xi}}(\partial_{\mathfrak{l}_{2}}(\kappa^{\prime}_{\mathcal{D}_{n+t},\xi}(\mathfrak{l}_{2})))=t-e=t-t_{\mathcal{D}^{\prime\prime}_{n}}$.

###  7.4. One divisibility of the Iwasawa main conjecture

Now we deduce the one divisibility of the main conjecture from the above
theorem. Let $\pi$ be the unitary automorphic representation associated to
$f$. Following [CH12], Hung [Hun14] has defined
$\theta_{\infty}(f)=\theta_{\infty}(f,1)\in\Lambda$. We briefly sketch its
construction below and deduce our main result from it.

By the Jaquet-Langlands correspondence we can find an automorphic
representation $\pi^{\prime}$ for the group $G=\mathop{\rm
Res}\nolimits_{F/\mathbf{Q}}(B^{\times})$ and an normalized eigenform
$f_{B}\in S_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathbf{C})$ with the
property that $T_{v}f_{B}={\bf a}_{v}(f)f_{B}$ for $v\nmid\mathfrak{n}$ and
$U_{v}f_{B}=\alpha_{v}(f)f_{B}$ for $v\mid\mathfrak{n}$. Here we let $B$ be a
quaternion algebra over $F$ with discriminant $\mathfrak{n}^{-}$ and
$\widehat{R}_{\mathfrak{n}^{+}}$ be an Eichler order of level
$\mathfrak{n}^{+}$. We normalize $f_{B}$ such that its $p$-adic avatar
$\widehat{f_{B}}\neq 0(\,\mbox{mod }{\varpi})$.

For ${\bf{v}}\in L_{k}(\mathbf{C})$ and $h_{B}\in
S_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathbf{C})$, we can define
$\Psi({\bf{v}}\otimes h_{B})\in{\mathcal{A}}_{k}(\mathfrak{n}^{+},\mathbf{C})$
in the space of scalar valued automorphic forms on $\widehat{B}^{\times}$ of
level $\mathfrak{n}^{+}$. Here $\Psi({\bf{v}}\otimes
h_{B})(g)=\langle\rho_{k,\infty}(g_{\infty}){\bf{v}},h_{B}\rangle_{k}$.

Back to our setting, we set $\varphi_{B}=\Psi({\bf v}_{0}\otimes
f_{B})\in{\mathcal{A}}_{k}(\mathfrak{n}^{+},\mathbf{C})[\pi^{\prime}]$ where
${\bf v}_{0}=X^{\frac{k-2}{2}}Y^{\frac{k-2}{2}}$. And define the
${\mathfrak{p}}$-stablization $\varphi_{B}^{\dagger}$ of $\varphi_{B}$ with
respect to $\alpha_{{\mathfrak{p}}}$ as

$\varphi^{\dagger}_{B}=\varphi_{B}-\frac{1}{\alpha_{{\mathfrak{p}}}}\pi^{\prime}(\begin{pmatrix}{1}&{0}\\\
{0}&{\varpi_{{\mathfrak{p}}}}\end{pmatrix})\varphi_{B}.$

The theta element is defined as

$\Theta_{m}(f)={\alpha_{{\mathfrak{p}}}^{-m}}\sum_{a\in\mathop{\rm
Gal}\nolimits(H_{m}/K)}\varphi^{\dagger}_{B}(x_{m}(a))[a]\in\mathbf{C}[\mathop{\rm
Gal}\nolimits(H_{m}/K)].$

By [Hun14, Lemma 5.4 (1)], one can in fact identify $\theta_{m}(f)$ as an
element in $\mathcal{O}_{f}[\mathop{\rm Gal}\nolimits(H_{m}/K)]$. By the same
computation as in Lemma 5.8, we verify

$\pi_{m+1,m}(\Theta_{m+1}(f))=\Theta_{m}(f)$

for the transition map $\pi_{m+1,m}:\mathop{\rm
Gal}\nolimits(H_{m+1}/K)\rightarrow\mathop{\rm Gal}\nolimits(H_{m}/K)$. We
define $\theta_{m}(f)$ as the image of $\Theta_{m}(f)$ under the natural
projection $\mathop{\rm Gal}\nolimits(H_{m}/K)\rightarrow\mathop{\rm
Gal}\nolimits(K_{m}/K).$ In particular we obtain an element
$\theta_{\infty}(f)\in\mathcal{O}_{f}[[\mathop{\rm
Gal}\nolimits(H_{\infty}/K)]]$. We can project this element to
$\mathcal{O}_{f}[[\mathop{\rm Gal}\nolimits(K_{\infty}/K)]]$. We define the
$p$-adic $L$-function $L_{p}(K_{\infty},f)$ by
$L_{p}(K_{\infty},f)=\theta_{\infty}(f)^{2}$ and we have the following
interpolation formula proved by Hung [Hun14].

######  Theorem 7.15.

For each finite character
$\chi:\Gamma\smash{\mathop{\longrightarrow}}\mathbf{C}_{p}$ of conductor
${\mathfrak{p}}^{m}$,

$\chi({\theta^{2}_{\infty}(f)})=\Gamma(\frac{k}{2})^{2}u^{2}_{K}\frac{\sqrt{D_{K}}D^{k-1}_{K}}{D_{F}^{\frac{3}{2}}}{\bf{N}}({\mathfrak{p}})^{m(k-1)}\chi(\mathfrak{N}^{+})\epsilon_{{\mathfrak{p}}}(f){\bf
a}_{{\mathfrak{p}}}(f)^{-m}e_{{\mathfrak{p}}}(f,\chi)^{2}\frac{L(f/K,\chi,k/2)}{\Omega_{f,\mathfrak{n}^{-}}}.$

Here we denote by

  1. (1)

$D_{F}$ resp. $D_{K}$ is the absolute discriminant of $F$ resp. $K$.

  2. (2)

We fix a decomposition
$\mathfrak{n}^{+}=\mathfrak{N}^{+}{\overline{\mathfrak{N}^{+}}}$.

  3. (3)

$\epsilon_{{\mathfrak{p}}}(f)$ is the local root number at ${\mathfrak{p}}$.

  4. (4)

$u_{K}=[\mathcal{O}^{\times}_{K}:\mathcal{O}^{\times}_{F}]$.

  5. (5)

$e_{{\mathfrak{p}}}(f,\chi)$ is the $p$-adic multiplier defined as follows

$e_{\mathfrak{p}}(f,\chi)=\begin{cases}1&\hbox{if $\chi$ is ramified;}\\\
(1-\alpha_{\mathfrak{p}}^{-1}\chi(\mathfrak{P}))(1-\alpha_{\mathfrak{p}}^{-1}\chi(\overline{\mathfrak{P}}))&\hbox{if
$s=0$ and $\mathfrak{p}=\mathfrak{P\overline{P}}$ is split;}\\\
1-\alpha_{\mathfrak{p}}^{-2}&\hbox{if $s=0$ and $\mathfrak{p}$ is inert;}\\\
1-\alpha_{\mathfrak{p}}^{-1}\chi(\mathfrak{P})&\hbox{if $s=0$\hbox{ and
}$\mathfrak{p}=\mathfrak{P}^{2}$ is ramified.}\end{cases}$

  6. (6)

$\Omega_{f,\mathfrak{n}^{-}}$ is a complex period associated to $f$ known as
the Gross period [Hun14, (5.2)].

######  Proposition 7.16.

If $\Delta=\mathfrak{n}^{-}$, there is an $n$-admissible form
$\mathcal{D}^{f}_{n}=(\mathfrak{n}^{-},f^{\dagger,[k-2]}_{n})$, such that

$\theta_{m}(\mathcal{D}^{f}_{n})\equiv\theta_{m}(f)\,(\mbox{mod
}{\varpi^{n}})$

and in particular that

$\theta_{\infty}(\mathcal{D}^{f}_{n})\equiv\theta_{\infty}(f)\,(\mbox{mod
}{\varpi^{n}}).$

###### Proof.

Let $B$ and $f_{B}$ be chosen as in the previous discussion. Let

$f^{\dagger,[k-2]}_{n}(b)=\sqrt{\beta}^{\frac{2-k}{2}}\langle
X^{k-2},{\widehat{f^{\dagger}}}_{B}(b)\rangle_{k}\,\mbox{mod }{\varpi}^{n}.$

Then we have $f^{\dagger,[k-2]}_{n}\in
S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f,n})$. Moreover it is shown in [Hun14,
Lemma 6.11] that $f^{\dagger,[k-2]}_{n}$ is not Eisenstein. Hence we conclude
by [Hun14, Corolarry 6.8] that $f^{\dagger,[k-2]}_{n}(\,\mbox{mod }{\varpi})$
is non-zero. Finally we conclude with [Hun14, Lemma 5.4] for the desired
congruence relation. Thus we know that $f^{\dagger,[k-2]}_{n}$ is
$n$-admissible. ∎

######  Theorem 7.17.

Assume the hyothesis $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-min})$, PO
and "Ihara’s lemma" hold. We have
$\text{char}_{\Lambda}{\text{Sel}(K_{\infty},A_{f})}^{\\!\vee}\supset(L_{p}(K_{\infty},f))$.

###### Proof.

let $\xi:\Lambda\smash{\mathop{\longrightarrow}}\mathcal{O}_{\xi}$ be an
$\mathcal{O}_{f}$-algebra homomorphism. If $\xi(L_{p}(K_{\infty},f))$ is zero,
then then it is in $\mathop{\rm
Fitt}\nolimits_{\mathcal{O}_{\xi}}({\text{Sel}(K_{\infty},A_{f}))}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi}$.
If $\xi(L_{p}(K_{\infty},f))$ is non-zero, then we can choose
$t_{0}>\text{ord}_{\varpi_{\xi}}(\xi(L_{p}(K_{\infty},f)))$. For each positive
$n$, we have the $n+t_{0}$\- admissible form
$\mathcal{D}^{f}_{n+t_{0}}=(\mathfrak{n}^{-},{\bf{f}}^{\dagger,[k-2]}_{n+t_{0}})$
as in Proposition 7.16 and the $n$-admissible form
$\mathcal{D}^{f}_{n}=\mathcal{D}^{f}_{n+t_{0}}\,(\mbox{mod }{\varpi^{n}})$. By
applying Theorem 7.9, we have
$\xi(L_{p}(K_{\infty},f))=\xi(\theta_{\infty}(\mathcal{D}^{f}_{n})^{2})\in\mathop{\rm
Fitt}\nolimits_{\mathcal{O}_{\xi}}({\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f,n})}^{\\!\vee}\otimes_{\xi}\mathcal{O}_{\xi})$
for all $\xi$ and $n$. Then the criterion Lemma 7.8 shows that
$L_{p}(K_{\infty},f)\in\mathop{\rm
Fitt}\nolimits_{\Lambda}{\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})}^{\\!\vee}$.
By [Hun14, Theorem B], $L_{p}(K_{\infty},f)$ is non-zero and hence
$\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})$ is $\Lambda$-cotorsion. The
theorem then follows from Proposition 2.4 which identifies
$\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})$ with
$\text{Sel}(K_{\infty},A_{f})$. ∎

##  8\. Some arithmetic consequences

Let $L$ be a number field and $V$ be a representation of the Galois group
$G_{L}$ over some $p$-adic field $E$. Recall that the Bloch-Kato Selmer group
$\text{Sel}_{f}(L,V)$ is defined by

$\text{Sel}_{f}(L,V)=\mathop{\rm
ker}\nolimits\\{H^{1}(L,V)\rightarrow\prod_{v}H^{1}(L_{v},V)/H^{1}_{f}(L_{v},V)\\}$

where

(8.1) $\displaystyle H^{1}_{f}(L_{v},V)=H^{1}_{{\rm fin}}(L_{v},V)\text{ when
$v\nmid p$},$ $\displaystyle H^{1}_{f}(L_{v},V)=\mathop{\rm
ker}\nolimits\\{H^{1}(L_{v},V)\rightarrow
H^{1}(L_{v},V\otimes_{\mathbf{Q}_{p}}B_{\rm cris})\\}\text{ for $v\mid p$}.$

The Bloch-Kato conjecture relates the rank of the Selmer group and the order
of vanishing of the $L$-function attached to the Galois representation. In
this section we prove certain cases of the refined type Bloch-Kato conjecture
in the rank zero case.

###  8.1. Bloch-Kato type conjecture and parity conjecture

In this chapter, we denote by $H_{{\mathfrak{c}}}$ the ring class field of K
with conductor ${\mathfrak{c}}$. Let $\chi$ be a finite order character of
$\mathop{\rm Gal}\nolimits(H_{{\mathfrak{c}}}/K)$, we consider the twisted
Bloch-Kato Selmer group $\text{Sel}_{f}(K,V_{f}\otimes\chi)$. We have a
canonical isomorphism
$\text{Sel}_{f}(K,V_{f}\otimes\chi)\simeq\text{Sel}_{f}(H_{{\mathfrak{c}}},V_{f})^{\chi^{-1}}$
where the righthand side is the $\chi^{-1}$-isotypic component of
$\text{Sel}_{f}(K,V_{f})$ for the action of $\mathop{\rm
Gal}\nolimits(H_{{\mathfrak{c}}}/K)$. The complex conjugation of $\mathop{\rm
Gal}\nolimits(H_{{\mathfrak{c}}}/K)$ induces an isomorphism between

$\text{Sel}_{f}(H_{{\mathfrak{c}}},V_{f})^{\chi^{-1}}\text{ and
}\text{Sel}_{f}(H_{{\mathfrak{c}}},V_{f})^{\chi}.$

The Bloch-Kato conjecture relates the order of the $L$-function
$L(f/K,\chi,k/2)$ to the rank of these Selmer groups, more precisely we
expect:

$\text{ord}_{s=k/2}L(f/K,\chi,s)=\text{rank }\text{Sel}_{f}(K,V_{f,\chi}).$

In this section we provide some results in this direction. In particular we
prove the so called parity conjecture in this setup.

First we provide the following proposition which relates the divisibility
index of the theta element to the size of the Selmer group studied in this
paper. We remark that the proof of this result only depends on the first
reciprocity law in Theorem 6.2 and in particular we don’t have to assume
"Ihara’s lemma". Let $\chi$ be a finite order character of $\mathop{\rm
Gal}\nolimits(K_{m}/K)$ and let $\varpi_{\chi}$ be the uniformizer for the
ring generated over $\mathcal{O}_{f}$ by the values of $\chi$.

######  Proposition 8.1.

Let $c=\text{ord}_{\varpi_{\chi}}(\chi(\theta_{m}(f)))$. Then
$\varpi_{\chi}^{c}\text{Sel}_{\mathfrak{n}^{-}}(K_{m},A_{f,n})^{\chi}=0$ for
every large $n$.

###### Proof.

For every integer $n>c$, we can find an admissible form
$\mathcal{D}^{f}_{n}=(\mathfrak{n}^{-},f^{\dagger,[k-2]}_{n})$, such that

$\theta_{m}(\mathcal{D}^{f}_{n})\equiv\theta_{m}(f)\,(\mbox{mod
}{\varpi^{n}}).$

By the first reciprocity law and Theorem 7.2, we therefore can find an
admissible prime $\mathfrak{l}$ such that
$\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}^{f}_{n}}(\mathfrak{l}))\equiv\theta_{m}(f)\,(\mbox{mod
}{\varpi_{{\mathfrak{p}}}^{n}})$. Let
$s\in\text{Sel}_{\mathfrak{n}^{-}}(K_{m},A_{f,n})^{\chi}$, we can also choose
the admissible prime $\mathfrak{l}$ such that ${\text{res}}_{\mathfrak{l}}(s)$
is non-zero using Theorem 7.2. Then by considering the sum

$\sum_{v}\langle{\text{res}}_{v}(\kappa_{\mathcal{D}^{f}_{n}}(\mathfrak{l})),{\text{res}}_{v}(s)\rangle=0,$

we conclude that

$\langle\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}^{f}_{n}}(\mathfrak{l})),{\text{res}}_{\mathfrak{l}}(s)\rangle=0$

using the same type of argument as in Lemma 7.11. Hence $\varpi_{\chi}^{c}s=0$
by the non-degeneracy of the local Tate pairing. Therefore we see that
$\varpi_{\chi}^{c}\text{Sel}_{\mathfrak{n}^{-}}(K_{m},A_{f,n})^{\chi}=0$. ∎

Now an immediate corollary from the above theorem is the proof of the
following rank zero case of the refined Bloch-Kato conjecture.

######  Theorem 8.2.

Suppose that $L(f,\chi,k/2)\neq 0$. Then
$\text{Sel}_{f}(K_{m},V_{f})^{\chi}=0$.

###### Proof.

By the interpolation formula Theorem 7.15 and Proposition 8.1, we see that
$\text{Sel}_{\mathfrak{n}^{-}}(K_{m},A_{f,n})^{\chi}$ is bounded independent
of $n$. Therefore it follows that
$\text{Sel}_{\mathfrak{n}^{-}}(K_{m},A_{f})^{\chi}$ is finite. Since
$\text{Sel}_{f}(K_{m},V_{f})^{\chi}\subset\text{Sel}_{\mathfrak{n}^{-}}(K_{m},V_{f})^{\chi}$,
we obtain $\text{Sel}_{f}(K_{m},V_{f})^{\chi}=0$. ∎

###  8.2. The parity conjecture

From the Bloch-Kato type theorem proved in the last section, one can deduce a
version of the parity conjecture for twists of $V_{f}$. The result stated here
is already proved by Nekovar [Nek08] and we give another proof based on the
general treatment of the parity conjecture in [Nek07]. A similar treatment of
the parity conjecture in a slightly different situation is proved in [CH15a,
Theorem 6.4], we follow their method closely. We let $\chi$ be a finite order
character of $\Gamma$ of conductor ${\mathfrak{p}}^{m}$ as above. Let
$V_{f,\chi}=V_{f}\otimes\chi$. We have an isomorphism
$\text{Sel}_{f}(K,V_{f,\chi})\cong\text{Sel}_{f}(K_{m},V_{f})^{\chi^{-1}}$.
The parity conjecture in our context asserts the following:

######  Theorem 8.3.

$\text{ord}_{s=k/2}L(f/K,\chi,s)\equiv\text{rank
}\text{Sel}_{f}(K,V_{f,\chi})\equiv 0\,(\mbox{mod }{2}).$

###### Proof.

Recall that $\Lambda$ is the Iwasawa algebra for $\Gamma$ over
$\mathcal{O}_{f}$. Let $\chi^{uni}:G_{K}\rightarrow\Lambda^{\times}$ be the
universal deformation of the character $\chi$. Denote by $\mathcal{T}$ the
$G_{F}$-module $T\otimes\text{Ind}_{K}^{F}\chi^{univ}$ and let
${\mathcal{V}}=\mathcal{T}\otimes E_{f}$. We also put
$\mathcal{T}^{+}_{p}=F^{+}T\otimes\Lambda$. Then the pair
$(\mathcal{T},\mathcal{T}^{+}_{p})$ and its specializations satisfies the
condition listed in [Nek07, 5.12.(1)-(9)]. Let $\phi$ be a finite order
character of $\Gamma$ sufficiently wildly ramified such that
$\chi\phi(\theta_{\infty}(f))^{2}\neq 0$. Then by Theorem 8.1, we can conclude
that $\text{rank }\text{Sel}_{f}(K,V_{f,\chi\phi})=0$. We put
${\mathcal{V}}_{\phi}$ as the specialization of ${\mathcal{V}}$ at $\phi$. By
[Nek07, 5.3.1], we have $\text{rank
}\text{Sel}_{f}(F,{\mathcal{V}}_{\phi})\equiv\text{rank
}\text{Sel}_{f}(F,{\mathcal{V}}_{triv})\,(\mbox{mod }{2})$ where
${\mathcal{V}}_{triv}$ is the specialization of ${\mathcal{V}}$ at the trivial
character. We also know the sign of the associated Weil-Deligne representation
$\epsilon({\mathcal{V}}_{\phi^{\prime}})$ is independent of the specialization
$\phi^{\prime}$ and is equal to $1$. Notice that we have
$\text{Sel}_{f}(F,{\mathcal{V}}_{\phi^{\prime}})\cong\text{Sel}_{f}(K,V_{f,\chi\phi^{\prime}})$
for any specialization $\phi^{\prime}$ and we conclude by letting
$\phi^{\prime}$ be $\phi$ and be the trivial character respectively. This
shows

$\displaystyle\text{rank }\text{Sel}_{f}(K,V_{f,\chi})\equiv\text{rank
}\text{Sel}_{f}(K,V_{f,\chi\phi})\equiv 0\,(\mbox{mod }{2})$

which finishes the proof. ∎

##  9\. Multiplicity one for quaternion alegbra

The purpose of this section is to prove a multiplicity one result for the
space of automorphic forms on a quaternion algebra localized at certain
maximal ideal of the Hecke algebra. This result plays a crucial role in
establishing the reciprocity laws.

###  9.1. Taylor-Wiles construction for Hilbert modular forms

Let $\pi$ be the automorphic representation attached to $f$. Let
$\pi^{\prime}$ be the preimage of $\pi$ under the Jaquet-Langlands
correspondence. So $\pi^{\prime}$ gives rise to a morphism
$\lambda_{\pi^{\prime}}:{\mathbb{T}}_{B}(\mathfrak{n}^{+}Y)\rightarrow\mathcal{O}_{f}$
with $Y=\widehat{F}^{\times}$. Let
$\mathfrak{n}_{0}=\mathfrak{n}_{\bar{\rho}_{f}}$ be the Artin conductor of the
residual Galois representation $\bar{\rho}_{f}$ of $\rho_{f}$. Let
$\mathfrak{n}^{-}_{1}$ be the product of primes of $\mathfrak{n}^{-}$ not
dividing $\mathfrak{n}_{0}$ and let
$\mathfrak{n}_{\emptyset}=\mathfrak{n}^{-}_{1}\mathfrak{n}_{0}$. By the level
lowering results [Raj01], [Jar99], there exists a morphism
$\lambda_{\emptyset}:{\mathbb{T}}_{B}(\mathfrak{n}_{\emptyset}Y)\rightarrow\mathcal{O}_{f}$
such that
$\lambda_{\emptyset}(T_{v})=\lambda_{\pi^{\prime}}(T_{v})\,(\mbox{mod
}{\varpi})$ for all $v\nmid\mathfrak{n}$. We write
$\prod_{v}v^{m_{v}}=\mathfrak{n}/\mathfrak{n}_{\emptyset}$. We recall our
assumptions $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ namely:

######  Hypothesis  ($\text{CR}^{+}$).

  1. (1)

$p>k+1$ and $\\#(\mathbb{F}^{\times}_{{\mathfrak{p}}})^{k-1}>5$.

  2. (2)

The restriction of $\bar{\rho}_{f}$ to $G_{F(\sqrt{p^{*}})}$ is irreducible
where $p^{*}=(-1)^{\frac{p-1}{2}}p$.

  3. (3)

$\bar{\rho}_{f}$ is ramified at $\mathfrak{l}$ if
$\mathfrak{l}\mid\mathfrak{n}^{-}$ and ${{\bf{N}}(\mathfrak{l})^{2}}\equiv
1\,(\mbox{mod }{p})$.

  4. (4)

$\mathfrak{n}_{0}$ is prime to $\mathfrak{n}/\mathfrak{n}_{0}$.

$(\mathfrak{n}^{+}\text{-DT}):\text{If $\mathfrak{l}\mid\mid\mathfrak{n}^{+}$
and ${\bf{N}}(\mathfrak{l})\equiv 1\,(\mbox{mod }{p})$, then $\bar{\rho}_{f}$
is ramified.}$

By [Jar99, Theorem 1.5] and $(\text{CR}^{+})$, we know that $m_{v}\leq 2$ and
$m_{v}=0$ unless $v\mid\mathfrak{n}^{+}$. Let $\Sigma$ be a subset of the set
of prime factors of $\mathfrak{n}/\mathfrak{n}_{\emptyset}$ and we put
$\mathfrak{n}_{\Sigma}=\mathfrak{n}_{\emptyset}\prod_{v\in\Sigma}v^{m_{v}}$.
Let
${\mathbb{T}}_{\Sigma}={\mathbb{T}}_{B}(\mathfrak{n}_{\Sigma}Y)_{\mathfrak{m}_{\Sigma}}$
be the localization of the Hecke algebra at the maximal ideal
$\mathfrak{m}_{\Sigma}$ generated by:

$\varpi,T_{v}-\lambda_{\emptyset}(T_{v})\text{ for
$v\nmid\mathfrak{n}_{\Sigma}$},U_{v}-\lambda_{\pi^{\prime}}(U_{v})\text{ for
$v\mid\mathfrak{n}_{\Sigma}$}.$

Let
$S_{\Sigma}=S_{k}(\widehat{R}_{\mathfrak{n}_{\Sigma}}^{\times}Y,\mathcal{O}_{f})_{\mathfrak{m}_{\Sigma}}$
where ${R}_{\mathfrak{n}_{\Sigma}}$ is the Eicher order of level
$\mathfrak{n}_{\Sigma}$. This chapter is devoted to proving the following
multiplicity one result.

######  Theorem 9.1.

Assume $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ hold, the module
$S_{\Sigma}$ is free over ${\mathbb{T}}_{\Sigma}$ of rank 1.

The proof of this is based on the refinement of Diamond on the Taylor-Wiles
method [Dia97b]. Notice this result is already shown in [Tay06, Theroem 3.2]
in the case when $m_{v}=2$ for every $v\in\Sigma$. We review his result and
extend it to our situation, this section follows closely to [CH12, Section 6].

There is a Galois representation $\rho_{\Sigma}:G_{F}\rightarrow
GL_{2}({\mathbb{T}}_{\Sigma})$ such that

  * •

$\rho_{\Sigma}$ is unramified outside $p\mathfrak{n}_{\Sigma}$.

  * •

$\mathop{\rm Tr}\nolimits\rho_{\Sigma}(\mathrm{Frob}_{v})=T_{v}$ for all $v$
away from $p\mathfrak{n}_{\Sigma}$.

  * •

There is a character
$\chi_{v}:G_{F_{v}}\rightarrow{\mathbb{T}}^{\times}_{\Sigma}$ such that
$\rho_{\Sigma}\mid_{G_{F_{v}}}=\begin{pmatrix}{\chi^{-1}_{v}\epsilon}&{*}\\\
{0}&{\chi_{v}}\end{pmatrix}$ and $\chi_{v}\mid_{I_{F_{v}}}=\epsilon^{(2-k)/2}$
for $v\mid p$.

  * •

If $v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}^{-}_{1}$, then
$\rho_{\Sigma}\mid_{G_{F_{v}}}=\begin{pmatrix}{\chi^{-1}_{v}\epsilon}&{*}\\\
{0}&{\chi_{v}}\end{pmatrix}$ for some character
$\chi_{v}:G_{F_{v}}\rightarrow{\mathbb{T}}^{\times}_{\Sigma}$ with
$\chi_{v}(\mathrm{Frob}_{v})=U_{v}$.

  * •

If
$v\mid\mathfrak{n}^{-}_{1},\rho_{\Sigma}\mid_{G_{F_{v}}}=\begin{pmatrix}{\pm\epsilon}&{*}\\\
{0}&{\pm 1}\end{pmatrix}$.

Let $v\mid\mathfrak{n}/\mathfrak{n}^{-}_{1}$ be a prime such that
$v\not\in\Sigma$. We define a level raising map $L_{v}:S_{\Sigma}\rightarrow
S_{\Sigma\cup\\{v\\}}$ and an element $u_{\Sigma,v}\in{\mathbb{T}}_{\Sigma}$
as follows:

  * •

if $m_{v}=2$, we put $u_{\Sigma,{v}}=0$,

$L_{v}(f)={\bf{N}}(v)f-\begin{pmatrix}{1}&{0}\\\
{0}&{\varpi_{v}}\end{pmatrix}T_{v}f+\begin{pmatrix}{1}&{0}\\\
{0}&{\varpi^{2}_{v}}\end{pmatrix}f.$

  * •

If $m_{v}=1$, we have a congruence
$x^{2}-T_{v}x+{\bf{N}}(v)\equiv(x-\epsilon_{v})(x-{\bf{N}}(v)\epsilon_{v})\,(\mbox{mod
}{\varpi})$. By Hensel’s lemma $\epsilon_{v}$ lifts to a unique root
$u_{\Sigma,v}$ of $x^{2}-T_{v}x+{\bf{N}}(v)$ since ${\bf{N}}(v)\not\equiv
1\,(\mbox{mod }{p})$. And we put

$L_{v}(f)=u_{\Sigma,{v}}f-\begin{pmatrix}{1}&{0}\\\
{0}&{\varpi_{v}}\end{pmatrix}f.$

  * •

One can verify immediately that $L_{v}\circ u_{\Sigma,v}=U_{v}\circ L_{v}$.

######  Lemma 9.2.

The map $L_{v}$ is injective.

###### Proof.

This is [Tay06, Lemma 3.1]. Both of the above cases rely on the the strong
approximation theorem and the fact that $\mathfrak{m}_{\Sigma}$ is non-
Eisenstein. ∎

It follows then that the map
${\mathbb{T}}_{\Sigma\cup\\{v\\}}\rightarrow{\mathbb{T}}_{\Sigma}$ induced by
sending $U_{v}$ to $u_{\Sigma,v}$ is surjective. In particular we have a
surjective map
${\mathbb{T}}_{\Sigma}\twoheadrightarrow{\mathbb{T}}_{\emptyset}$. Let
$\lambda_{\Sigma}:{\mathbb{T}}_{\Sigma}\rightarrow{\mathbb{T}}_{\emptyset}\rightarrow\mathcal{O}_{f}$
be the composite morphism and let $I_{\lambda_{\Sigma}}$ be its kernel. Set

$S_{\Sigma}[\lambda_{\Sigma}]=\\{c\in S_{\Sigma}:I_{\lambda_{\Sigma}}c=0\\}.$

######  Lemma 9.3.

The localized Hecke algebra ${\mathbb{T}}_{\Sigma}$ is reduced.

###### Proof.

We need to show that $U_{v}$ is semisimple in ${\mathbb{T}}_{\Sigma}$. When
$v\in\Sigma$ and $m_{v}=2$, $U_{v}=0$ see [Tay06, Corollary 1.8]. And the case
$m_{v}=1$ follows from our assumption $(\mathfrak{n}^{+}\text{-DT})$ since
$U_{v}=\begin{pmatrix}{T_{v}}&{-1}\\\ {{\bf{N}}(v)}&{0}\end{pmatrix}$ under a
suitable basis. ∎

Hence by strong multiplicity one and the above lemma, we conclude that
$S_{\Sigma}[\lambda_{\Sigma}]$ is rank one over $\mathcal{O}_{f}$. Let
$S_{\Sigma}[\lambda_{\Sigma}]$ be the dual module defined by

$S_{\Sigma}[\lambda_{\Sigma}]^{\bot}=\\{c\in
S_{\Sigma}[\lambda_{\Sigma}]\otimes E_{f}:\langle
c,x\rangle_{\mathfrak{n}_{\Sigma}}\in\mathcal{O}_{f},\forall x\in
S_{\Sigma}[\lambda_{\Sigma}]\\}.$

Then the congruence module for $\lambda_{\Sigma}$ is defined by
$C(\mathfrak{n}_{\Sigma})=S_{\Sigma}[\lambda_{\Sigma}]^{\bot}/S_{\Sigma}[\lambda_{\Sigma}]$
and
$\mu_{\Sigma}=\lambda_{\Sigma}(\text{Ann}_{{\mathbb{T}}_{\Sigma}}(I_{\lambda_{\Sigma}}))$
is called the congruence ideal. Then we have the following criterion for the
freeness of $S_{\Sigma}$ over ${\mathbb{T}}_{\Sigma}$.

######  Lemma 9.4.

$\\#C(\mathfrak{n}_{\Sigma})\leq\\#(\mathcal{O}_{f}/\mu_{\Sigma})$ and
equality holds if $S_{\Sigma}$ is free over ${\mathbb{T}}_{\Sigma}$.

###### Proof.

This is the numerical criterion in [Dia97b, Theorem 2.4]. ∎

######  Lemma 9.5.

If $v\|\mathfrak{n}/\mathfrak{n}_{\emptyset}$ and $v\not\in\Sigma$, then
$\\#C(\mathfrak{n}_{\Sigma\cup\\{v\\}})=\\#C(\mathfrak{n}_{\Sigma})\\#(\mathcal{O}_{f}/((\lambda_{\emptyset}(u_{\emptyset,v})^{2}-1)\mathcal{O}_{f})$.

###### Proof.

Let $L^{*}_{v}:S_{\Sigma}\rightarrow S_{\Sigma\cup\\{v\\}}$ be the adjoint map
of $L_{v}$ with respect to $\langle.\rangle_{\mathfrak{n}_{\Sigma}}$ and
$\langle.\rangle_{\mathfrak{n}_{\Sigma\cup\\{v\\}}}$. Then by Lemma 9.2 and
the above discussion, we have
$L_{v}(S_{\Sigma}[\lambda_{\Sigma}])=S_{\Sigma\cup\\{v\\}}[\lambda_{\Sigma\cup\\{v\\}}]$
and dually
$L^{*}_{v}(S_{\Sigma\cup\\{v\\}}[\lambda_{\Sigma\cup\\{v\\}}]^{\bot})=S_{\Sigma}[\lambda_{\Sigma}]^{\bot}$.

A direct computation shows that
$L^{*}_{v}=[\mathcal{U}\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix}\mathcal{U}^{\prime}]-[\mathcal{U}\mathcal{U}^{\prime}]$
for $\mathcal{U}=\widehat{R}^{\times}_{\mathfrak{n}_{\Sigma}}$ and
$\mathcal{U}^{\prime}=\widehat{R}^{\times}_{\mathfrak{n}_{\Sigma\cup\\{v\\}}}$.
Thus we have

(9.1) $\displaystyle L^{*}_{v}\circ
L_{v}=u_{\Sigma,v}[\mathcal{U}\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix}\mathcal{U}]u_{\Sigma,v}-(1+{\bf{N}}(v))u_{\Sigma,v}[\mathcal{U}\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix}\mathcal{U}]$
$\displaystyle-(1+{\bf{N}}(v))u_{\Sigma\cup\\{v\\}}[\mathcal{U}\mathcal{U}]+[\mathcal{U}\begin{pmatrix}{1}&{0}\\\
{0}&{\varpi_{v}}\end{pmatrix}\mathcal{U}]$
$\displaystyle=(1+u_{\Sigma,v}^{2})T_{v}-2u_{\Sigma,v}(1+{\bf{N}}(v)).$

Since $u^{2}_{\Sigma,v}-T_{v}u_{\Sigma,v}+{\bf{N}}(v)=0$, we obtain

$L^{*}_{v}\circ
L_{v}=u_{\Sigma,v}^{-1}(u^{2}_{\Sigma,v}-1)(u^{2}_{\Sigma,v}-{\bf{N}}(v)).$

Since by $(\mathfrak{n}^{+}\text{-DT})$, we have
$u_{\Sigma,v}^{2}-{\bf{N}}(v)\equiv 1-{\bf{N}}(v)\not\equiv 0\,(\mbox{mod
}{m_{\Sigma}})$. Hence

(9.2) $\displaystyle\\#C(\mathfrak{n}_{\Sigma\cup\\{v\\}})$
$\displaystyle=\\#(S_{\Sigma\cup\\{v\\}}[\lambda_{\Sigma\cup\\{v\\}}]^{\bot}/S_{\Sigma\cup\\{v\\}}[\lambda_{\Sigma\cup\\{v\\}}])$
$\displaystyle=\\#(S_{\Sigma}[\lambda_{\Sigma}]^{\bot}/L^{*}_{v}\circ
L_{v}(S_{\Sigma}[\lambda_{\Sigma}]))$
$\displaystyle=\\#S_{\Sigma}[\lambda_{\Sigma}]^{\bot}/(S_{\Sigma}[\lambda_{\Sigma}]).\\#S_{\Sigma}[\lambda_{\Sigma}]/L^{*}_{v}\circ
L_{v}(S_{\Sigma}[\lambda_{\Sigma}])$
$\displaystyle=\\#C(\mathfrak{n}_{\Sigma}).\\#\mathcal{O}_{f}/(\lambda_{\emptyset}(u_{\emptyset,v})^{2}-1)\mathcal{O}_{f}.$

∎

###  9.2. Deformation ring and its tangent space

Let $\mathcal{D}_{\Sigma}:\text{CNL}_{\mathcal{O}_{f}}\rightarrow\text{SET}$
be the functor that sends a complete Noetherian local
$\mathcal{O}_{f}$-algebra $A$ to the set $\mathcal{D}_{\Sigma}(A)$ of
representations $\rho_{A}:G_{F}\rightarrow\mathop{\rm GL}\nolimits_{2}(A)$
such that

  * •

$\rho_{A}\equiv\bar{\rho_{f}}\,(\mbox{mod }{\varpi_{A}})$.

  * •

$\rho_{A}$ is minimally ramified outside $\mathfrak{n}^{-}_{1}\Sigma$ in the
sense of [Dia97a].

  * •

For $v\mid p$, there is a character $\chi_{v}:G_{F}\rightarrow A^{\times}$
such that

$\rho_{A}\mid_{G_{F_{v}}}=\begin{pmatrix}{\chi_{v}^{-1}\epsilon}&{*}\\\
{0}&{\chi_{v}}\end{pmatrix}$

with $\chi_{v}\mid_{I_{F_{v}}}=\epsilon^{(2-k)/2}$.

  * •

For $v\mid\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\emptyset}$, there exists a
unramified character $\chi_{v}:G_{F_{v}}\rightarrow A^{\times}$ such that

$\rho_{A}\mid_{G_{F_{v}}}=\begin{pmatrix}{\chi_{v}^{-1}\epsilon}&{*}\\\
{0}&{\chi_{v}}\end{pmatrix}.$

  * •

For $v\mid\mathfrak{n}^{-}$, then

$\rho_{A}\mid_{G_{F_{v}}}=\begin{pmatrix}{\pm\epsilon}&{*}\\\ {0}&{\pm
1}\end{pmatrix}$

with $\varpi_{A}\mid*$.

It is well-known that this deformation functor is representable by
$R_{\Sigma}$ and we have a universal deformation
$\rho_{\Sigma}:G_{F}\rightarrow\mathop{\rm GL}\nolimits_{2}(R_{\Sigma})$ for
the residual Galois representation. In particular the universal property of
$R_{\Sigma}$ induces two maps $R_{\Sigma}\rightarrow R_{\emptyset}$ and
$R_{\Sigma}\rightarrow{\mathbb{T}}_{\Sigma}$.

######  Lemma 9.6.

The map $R_{\Sigma}\rightarrow{\mathbb{T}}_{\Sigma}$ is surjective.

###### Proof.

This is clear because:

  * •

If $v\nmid\mathfrak{n}_{\Sigma}$, then $\mathop{\rm
Tr}\nolimits\rho_{R_{\Sigma}}(\mathrm{Frob}_{v})\rightarrow
T_{v}\in{\mathbb{T}}_{\Sigma}$.

  * •

If $v^{2}\mid\mathfrak{n}$, then $U_{v}=0\in{\mathbb{T}}_{\Sigma}$.

  * •

If $v\mid\mathfrak{n}^{-}_{1}$, then $U_{v}=\pm 1\in{\mathbb{T}}_{\Sigma}$.

  * •

Finally if $v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}^{-}_{1}$,

$\rho_{\Sigma}\mid_{G_{F_{v}}}=\begin{pmatrix}{\chi^{-1}_{v}\epsilon}&{*}\\\
{0}&{\chi_{v}}\end{pmatrix}$

such that $\chi_{v}(\mathrm{Frob}_{v})\rightarrow
U_{v}\in{\mathbb{T}}_{\Sigma}$.

∎

Consider the composite morphism $R_{\Sigma}\rightarrow
R_{\emptyset}\rightarrow{\mathbb{T}}_{\emptyset}\smash{\mathop{\longrightarrow}\limits^{\lambda_{\emptyset}}}\mathcal{O}_{f}$.
Denote by $\wp_{\Sigma}$ the kernel of the composite morphism. Set
$W_{\rho}={\rm ad}^{0}\rho_{\lambda_{\emptyset}}\otimes
E_{f}/\mathcal{O}_{f}$. Define the subspace
$W^{+}_{\rho}=\\{\begin{pmatrix}{a}&{b}\\\ {0}&{-a}\end{pmatrix}\\}\subset
W_{\rho}=\\{\begin{pmatrix}{a}&{b}\\\ {c}&{-a}\end{pmatrix}\\}$. We will
define the Selmer group $\text{Sel}_{\Sigma}(W_{\rho})$ by choosing local
conditions $\\{{\mathcal{L}}_{v}\\}$ for various $v$.

  * •

For $v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\emptyset}$, we let
${\mathcal{L}}_{v}=H^{1}(F_{v},W^{+}_{\rho})$.

  * •

For $v\mid\mathfrak{n}^{-}_{1}$, we choose a lift of Frobenius denoted by
$\mathrm{Frob}_{v}$, and let ${\mathcal{L}}_{v}=\mathop{\rm
ker}\nolimits\\{H^{1}(G_{F_{v}},W_{\rho})\rightarrow
H^{1}(\langle\mathrm{Frob}_{v}\rangle,W_{\rho})\\}$.

  * •

For $v$ at other places, ${\mathcal{L}}_{v}=H^{1}_{f}(G_{F_{v}},W_{\rho})$ for
the usual local Bloch-Kato group cf. (8.1).

Then the adjoint Selmer group is defined by

$\text{Sel}_{\Sigma}(W_{\rho})=\mathop{\rm
ker}\nolimits\\{H^{1}(F,W_{\rho})\rightarrow\prod_{v}H^{1}(F_{v},W_{\rho})/{\mathcal{L}}_{v}\\}.$

The Selmer group controls the tangent space of the deformation ring
$R_{\Sigma}$ in the following manner:

(9.3) $\mathop{\rm
Hom}\nolimits(\wp_{\Sigma}/\wp_{\Sigma}^{2},E_{f}/\mathcal{O}_{f})\simeq\text{Sel}_{\Sigma}(W_{\rho}).$

We denote by $\Sigma^{(2)}$ the subset of prime factors $v$ of
$\mathfrak{n}/\mathfrak{n}_{\emptyset}$ with $m_{v}=2$. Assume
$\Sigma^{(2)}\subset\Sigma$ we have the following divisibility:

######  Lemma 9.7.

$\\#\text{Sel}_{\Sigma}(W_{\rho})/\\#\text{Sel}_{\Sigma^{(2)}}(W_{\rho})\mid\prod_{v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\emptyset}}\\#H^{1}(I_{F_{v}},W^{+}_{\rho})^{\langle\mathrm{Frob}_{v}\rangle}.$

###### Proof.

Put $W^{-}_{\rho}=W_{\rho}/W^{+}_{\rho}$ and let
$v\mid\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\emptyset}$. Observe that we have
the following commutative diagram:

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>H^{1}(G_{F_{v}}/I_{F_{v}},W_{\rho})@>{}>{}>H^{1}(G_{F_{v}},W_{\rho})@>{\pi}>{}>H^{1}(I_{F_{v}},W_{\rho})^{\langle\mathrm{Frob}_{v}\rangle}@>{}>{}>0\\\
@V{}V{}V@V{\pi_{1}}V{}V@V{\pi_{2}}V{}V\\\
0@>{}>{}>H^{1}(G_{F_{v}}/I_{F_{v}},W^{-}_{\rho})@>{}>{}>H^{1}(G_{F_{v}},W^{-}_{\rho}<EMAIL_ADDRESS>\end{CD}$

By $(\mathfrak{n}^{+}\text{-DT})$, we know ${\bf{N}}(v)\not\equiv
1\,(\mbox{mod }{p})$ and hence $H^{0}(G_{F_{v}}/I_{F_{v}},W^{-}_{\rho})$,
$H^{0}(G_{F_{v}},W^{-}_{\rho})$ and $H^{1}(G_{F_{v}}/I_{F_{v}},W^{-}_{\rho})$
are all zero. It follows then from the snake lemma that:

  * •

$\mathop{\rm
ker}\nolimits(\pi_{1})={\mathcal{L}}_{v}=H^{1}(G_{F_{v}},W^{+}_{\rho}).$

  * •

$H^{1}(G_{F_{v}}/I_{F_{v}},W^{+}_{\rho})=H^{1}(G_{F_{v}}/I_{F_{v}},W_{\rho}).$

  * •

$H^{1}(I_{F_{v}},W^{+}_{\rho})^{\langle\mathrm{Frob}_{v}\rangle}=H^{1}(G_{F_{v}},W^{+}_{\rho})/H^{1}(G_{F_{v}}/I_{F_{v}},W^{+}_{\rho})={\mathcal{L}}_{v}/H^{1}(G_{F_{v}}/I_{F_{v}},W_{\rho}^{+}).$

Then our claim follows from the tautological exact sequence:

$0\rightarrow\text{Sel}_{\Sigma^{(2)}}(W_{\rho})\rightarrow\text{Sel}_{\Sigma}(W_{\rho})\rightarrow\prod_{v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\emptyset}}{\mathcal{L}}_{v}/H^{1}(G_{F_{v}}/I_{F_{v}},W_{\rho}^{+})$.

∎

As a corollary we obtain

######  Corollary 9.8.

$\\#(\wp_{\Sigma}/\wp^{2}_{\Sigma})\mid\\#(\wp_{\Sigma^{(2)}}/\wp^{2}_{\Sigma^{(2)}})\prod_{v\mid\mathfrak{n}/\mathfrak{n}_{\emptyset}}\\#(\mathcal{O}_{f}/(\lambda^{2}_{\emptyset}(u_{\emptyset,v})-1)\mathcal{O}_{f})$.

###### Proof.

By a standard calculation using the Euler characteristic formula:

$\begin{aligned}
\\#H^{1}(I_{F_{v}},W^{+}_{\rho})^{\langle\mathrm{Frob}_{v}\rangle}&=\\#H^{1}(G_{F_{v}},W^{+}_{\rho})/\\#H^{1}(G_{F_{v}}/I_{F_{v}},(W^{+}_{\rho})^{I_{F_{v}}})\\\
&=\\#H^{1}(G_{F_{v}},W^{+}_{\rho})/\\#H^{0}(G_{F_{v}},W^{+}_{\rho})\\\
&=\\#H^{0}(G_{F_{v}},W^{+}_{\rho}(1)).\\\ \end{aligned}$

A direct calculation shows that

$\\#H^{0}(G_{F_{v}},W^{+}_{\rho}(1))=\\#\mathcal{O}_{f}/({\bf{N}}(v)-1)(\lambda_{\emptyset}^{2}(u_{\emptyset,v})-1)\mathcal{O}_{f}.$

This calculation combined with Lemma 9.7 and (9.3) conclude this corollary. ∎

Now we can prove the main result of this chapter.

######  Theorem 9.9.

Assume $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ hold, the module
$S_{\Sigma}$ is free over ${\mathbb{T}}_{\Sigma}$ of rank 1.

###### Proof.

By [Tay06, Theorem 3.2], we have
$\\#\wp_{\Sigma^{(2)}}/\wp^{2}_{\Sigma^{(2)}}\mid\\#C(\mathfrak{n}_{\Sigma^{(2)}}).$
Since we know
$\\#C(\mathfrak{n}_{\Sigma})=\\#C(\mathfrak{n}_{\Sigma^{(2)}})\prod_{v\|\mathfrak{n}_{\Sigma}/\mathfrak{n}_{\Sigma^{(2)}}}\\#\mathcal{O}_{f}/(\lambda^{2}(u_{\emptyset,v})-1)\mathcal{O}_{f}$
by Lemma 9.5 and
$\\#(\wp_{\Sigma}/\wp^{2}_{\Sigma})\mid\\#(\wp_{\Sigma^{(2)}}/\wp^{2}_{\Sigma^{(2)}})\prod_{v\mid\mathfrak{n}/\mathfrak{n}_{\emptyset}}\\#(\mathcal{O}_{f}/(\lambda^{2}_{\emptyset}(u_{\emptyset,v})-1)\mathcal{O}_{f})$
by Corollary 9.8, we have
$\\#\wp_{\Sigma}/\wp^{2}_{\Sigma}\mid\\#C(\mathfrak{n}_{\Sigma})$. Then the
desired multiplity one result follows from [Dia97b, Theorem 2.4]. ∎

##  10\. $\mu$-invariant and $\Phi$-functor

Recall we have proved the one divisibility result towards the anticyclotomic
Iwasawa main conjecture in Theorem 7.17 under a slightly stronger condition
$(\mathfrak{n}^{+}\text{-min})$ than $(\mathfrak{n}^{+}\text{-DT})$. In this
section, we relax $(\mathfrak{n}^{+}\text{-min})$ and
$(\mathfrak{n}^{+}\text{-DT})$. The reason for assuming a stronger condition
in Theorem 7.17 is that the Euler system method we employ relies on the
freeness result Proposition 7.6. In order to relax the condition
$(\mathfrak{n}^{+}\text{-min})$, we need to recover the freeness of the Selmer
group. A closer look at the Euler system argument in Section $7.3$ shows that
we in fact only need the freeness of a larger Selmer group, namely
$\widehat{\text{Sel}}_{\Delta}^{S\mathfrak{n}^{+}}(K_{\infty},T_{f,n}).$

###  10.1. Greenberg-Vastal method

To relax the assumption $(\mathfrak{n}^{+}\text{-min})$ to
$(\mathfrak{n}^{+}\text{-DT})$, we study the Iwasawa $\mu$-invariant. And the
following method is based on the unpublished note by Pollack-Weston
communicated to us by Chan-Ho Kim and their preprint [PWK15]. The key
observation is the following theorem.

######  Theorem 10.1.

Under the condition $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$,
$\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})$ is $\Lambda$ cotorsion and
the algebraic $\mu$-invariant

$\mu({\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})}^{\\!\vee})$

vanishes.

###### Proof.

Let $g$ be a Hilbert modular form congruent to $f$ modulo $\varpi$ which
satisfies $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$. Notice the
existence of such $g$ is guaranteed by level-lowering [Jar99]. Then the
theorem for $g$ follows immediately from Theorem 7.17 and the main result of
[Hun14, Theorem B] concerning the analytic $\mu$-invariant. Since all primes
dividing $\mathfrak{n}^{+}$ splits in $K$, we can apply the same method as
[GV00, Cor 2.3] and conclude that
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{g})$ is
$\Lambda$-cotorsion with vanishing $\mu$-invariant. Since $g$ is obtained from
$f$ by level lowering, an adaption of the method of [GV00, Prop 2.8] shows
that
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{f})[\varpi]=\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{g})[\varpi]$.
The point is that when we relaxed the local condition at $\mathfrak{n}^{+}$
for our Selmer
groups$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{f})[\varpi]$
(resp.
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{g})[\varpi]$)
only depends on the residual representation $A_{f,1}=A_{g,1}$. Since
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{g})$ is known
to be cotorsion with vanishing $\mu$-invariant,
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{g})[\varpi]$
is finite. This implies in turn that
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{f})[\varpi]$
is finite. Then we can conclude that
$\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{f})$ is
cotorsion with vanishing $\mu$-invariant. Now the same reasoning as in [GV00,
Cor 2.3] allows us to conclude that
$\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})$ has the same properties. ∎

For any place $v$ of $F$, we put
${\mathcal{H}}_{v}=\varinjlim_{m}\prod_{\omega\mid
v}H^{1}(K_{m,\omega},A_{f})$.

######  Lemma 10.2.

Let $S$ be finite set of places of $F$ that are away from $\mathfrak{n}$, then
we have an exact sequence:

$0\rightarrow\text{Sel}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}}(K_{\infty},A_{f})\rightarrow\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})\rightarrow\prod_{v\in
S}{\mathcal{H}}_{v}\rightarrow 0.$

###### Proof.

This follows from the definition except for the surjectivity of the last map.
The same method as in [PW11, Proposition A.2] applies showing the
surjectivity. ∎

######  Lemma 10.3.

Let $\mathfrak{l}$ be an $n$-admissible prime for $f$, then we have
${\mathcal{H}}_{\mathfrak{l}}\cong{(\Lambda/\varpi^{t})}^{\\!\vee}$ for some
$t\geq n$.

###### Proof.

First notice that as $\mathfrak{l}$ splits in $K_{\infty}$, for each $m$ we
have

$H^{1}(K_{m,\mathfrak{l}},A_{f})=H^{1}(K_{\mathfrak{l}},A_{f})\otimes\mathcal{O}_{f}[\mathop{\rm
Gal}\nolimits(K_{m}/K)].$

We consider the inflation-restriction exact sequence

$0\rightarrow H^{1}(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}},A_{f})\rightarrow
H^{1}(K_{\mathfrak{l}},A_{f})\rightarrow
H^{1}(I_{\mathfrak{l}},A_{f})^{\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}}/K)}\rightarrow 0.$

First of all,
$H^{1}(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}},A_{f})=A_{f}/(\gamma-1)A_{f}$
for a topological generator of $\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}})$. This shows that
$H^{1}(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}},A_{f})=0$. On the other hand
$H^{1}(I_{\mathfrak{l}},A_{f})\cong\mathop{\rm
Hom}\nolimits(I_{\mathfrak{l}},A_{f})=\mathop{\rm
Hom}\nolimits(\mathbf{Z}_{p}(1),A_{f})$ and thus
$H^{1}(I_{\mathfrak{l}},A_{f})^{\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}})}=\mathop{\rm
Hom}\nolimits(I_{\mathfrak{l}},A_{f})^{\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}})}$. By the
$n$-admissibility of $\mathfrak{l}$, the two eigenvalues $\alpha^{2}$ and
$\beta^{2}$ of $\mathrm{Frob}_{\mathfrak{l}}$ on $A_{f}$ has the property that
$\alpha\equiv\epsilon_{\mathfrak{l}}\,(\mbox{mod }{\varpi^{n}})$ and
$\beta\equiv\epsilon_{\mathfrak{l}}{\bf{N}}(\mathfrak{l})\,(\mbox{mod
}{\varpi^{n}})$. Note also that $\mathrm{Frob}_{\mathfrak{l}}$ acts by
${\bf{N}}(\mathfrak{l})^{2}$ on $\mathbf{Z}_{p}(1)$, it follows therefore
$\mathop{\rm Hom}\nolimits(I_{\mathfrak{l}},A_{f})^{\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}}/K_{\mathfrak{l}})}=\mathcal{O}_{f}/\varpi^{t}$
for the largest power $t$ such that
$\beta\equiv{\bf{N}}(\mathfrak{l})\,(\mbox{mod }{\varpi^{t}})$. Then it is
clear that by taking the direct limit with respect to $m$, we obtain the
desired result. ∎

######  Corollary 10.4.

${\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})}^{\\!\vee}$
is pseudo-isomorphic to a $\Lambda$-module of the form:

$(\bigoplus_{i=1}^{\\#S}\Lambda/\varpi^{t_{i}})\times Y$

with each $t_{i}\geq n$ and $Y$ a torsion $\Lambda$-module with $\mu(Y)=0$.

###### Proof.

This is proved by combining the Lemma 10.2 and Lemma 10.3. ∎

######  Remark.

Note that the above corollary implies in particular that
$\mu({\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})}^{\\!\vee})\geq
n\\#S.$

###  10.2. $\Phi$-functor

For each $n\geq 1$, Pollack-Weston defined the following functor:

(10.1) $\Phi:\\{\text{cofinitely generated
$\Lambda$-module}\\}\rightarrow\\{\text{finitely generated
$\Lambda/\varpi^{n}$-module}\\}$

where $\Phi_{n}(S)=\varprojlim_{m}S[\varpi^{n}]^{\Gamma^{m}}$ for the group
$\Gamma^{m}=\mathop{\rm Gal}\nolimits(K_{\infty}/K_{m})$. And the transition
map is given by the trace map.

The following lemma collects some properties of this functor.

######  Lemma 10.5.

  1. (1)

The functor $\Phi_{n}$ is covariant and left exact.

  2. (2)

$\Phi_{n}(S)=\Phi_{n}(S[\varpi^{n}])$.

  3. (3)

If $S$ is finite, then $\Phi_{n}(S)=0$.

###### Proof.

Only the third point needs some explanation as the others follow from the
definitions. For the third point, notice that $\Gamma^{m}$ fixes $S$ for $m$
sufficiently large. Then the transition map becomes simply multiplication by
$p$. The third point follows. ∎

######  Lemma 10.6.

Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be an exact
sequence of cofinitely generated $\Lambda$-modules.

  1. (1)

If $C$ is finite, then $\Phi_{n}(A)=\Phi_{n}(B)$.

  2. (2)

If $A$ is finite, then $\Phi_{n}(B)=\Phi_{n}(C)$.

###### Proof.

The first part follows immediately from the previous lemma and the fact that
$\Phi$ is left exact.

For the second point, we first take the $\varpi^{n}$ torsion of the exact
sequence to obtain:

$0\rightarrow A[\varpi^{n}]\rightarrow B[\varpi^{n}]\rightarrow
C[\varpi^{n}]\rightarrow A/\varpi^{n}.$

Then for some finite $H$, we have

$0\rightarrow A[\varpi^{n}]\rightarrow B[\varpi^{n}]\rightarrow
C[\varpi^{n}]\rightarrow H\rightarrow 0.$

We split this exact sequence into two short exact sequences:

$0\rightarrow A[\varpi^{n}]\rightarrow B[\varpi^{n}]\rightarrow
C^{\prime}\rightarrow 0$

and

$0\rightarrow C^{\prime}\rightarrow C[\varpi^{n}]\rightarrow H\rightarrow 0.$

By applying the previous lemma, we have
$\Phi_{n}(C^{\prime})\cong\Phi_{n}(C[\varpi^{n}])\cong\Phi_{n}(C)$. For the
first exact sequence, we take $\Gamma^{m}$ invariant to get

$0\rightarrow A[\varpi^{n}]^{\Gamma^{m}}\rightarrow
B[\varpi^{n}]^{\Gamma^{m}}\rightarrow(C^{\prime})^{\Gamma^{m}}\rightarrow
H^{\prime}\rightarrow 0$

for some finite $H^{\prime}$. Again by breaking the this four term exact
sequence into two short exact sequence, we have

$0\rightarrow A[\varpi^{n}]^{\Gamma^{m}}\rightarrow
B[\varpi^{n}]^{\Gamma^{m}}\rightarrow C^{\prime}_{m}\rightarrow 0$

and

$0\rightarrow C^{\prime}_{m}\rightarrow(C^{\prime})^{\Gamma^{m}}\rightarrow
H^{\prime}\rightarrow 0$

where $C^{\prime}_{m}$ is again finite. Taking inverse limit with respect to
$n$ yields:

$0\rightarrow\Phi_{n}(A)\rightarrow\Phi_{n}(B)\rightarrow\varprojlim_{m}C^{\prime}_{m}\rightarrow
0$

where the left exactness is guaranteed by the finiteness of $A$ and the
Mittag-Leffler condition. Then we get
$\Phi_{n}(B)=\varprojlim_{m}C^{\prime}_{m}$. The same reasoning applied to the
second exact sequence yields $\varprojlim
C_{m}^{\prime}\cong\Phi_{n}(C^{\prime})$. Therefore
$\Phi_{n}(B)=\Phi_{n}(C^{\prime})=\Phi_{n}(C)$. ∎

######  Corollary 10.7.

If $A$ and $B$ are two cofinitely generated $\Lambda$-modules which are
pseudo-isomorphic, then $\Phi_{n}(A)\cong\Phi_{n}(B)$

###### Proof.

In light of the definition of pseudo-isomorphism, this follows immediately
from the two previous lemmas. ∎

######  Proposition 10.8.

For a fintely generated cotorsion $\Lambda$-module $Y$ with vanishing
$\mu$-invariant, $\Phi_{n}({Y}^{\\!\vee})=0$ for all $n\geq 1$.

###### Proof.

By the structure theorem of finitely generated cotorsion $\Lambda$-module $Y$,
we can assume that $Y=\Lambda/f\Lambda$ for some $f$ coprime to $p$. Then we
have
$\Phi_{n}({Y}^{\\!\vee})\cong\Phi_{n}({Y}^{\\!\vee}[\varpi^{n}])\cong\Phi_{n}({(Y/\varpi^{n})}^{\\!\vee})$.
As $Y/\varpi^{n}=\Lambda/(f,\varpi^{n})$ is finite, it follows that
$\Phi_{n}({Y}^{\\!\vee})=0$ ∎

######  Proposition 10.9.

If $Y=\Lambda/\varpi^{t}$ with $t\geq n$, then
$\Phi_{n}({Y}^{\\!\vee})\cong\Lambda/\varpi^{n}$.

###### Proof.

By definition, we have

$\begin{aligned}
\Phi_{n}({(\Lambda/\varpi^{t}\Lambda)}^{\\!\vee})&=\varprojlim_{n}{(\Lambda/\varpi^{t})}^{\\!\vee}[\varpi^{n}]^{\Gamma^{m}}\cong\varprojlim_{n}{(\Lambda/\varpi^{n})_{\Gamma^{m}}}^{\\!\vee}\\\
&\cong\varprojlim_{n}{\mathcal{O}_{f,n}[\Gamma_{m}]}^{\\!\vee}.\\\
\end{aligned}$

By the Gorensteiness of ${\mathcal{O}_{f,n}[\Gamma_{m}]}^{\\!\vee}$,
${\mathcal{O}_{f,n}[\Gamma_{m}]}^{\\!\vee}\cong\mathcal{O}_{f,n}[\Gamma_{m}]$
and the transition maps with respect to $n$ are compatible with this
isomorphism. It follows the that

$\begin{aligned}
\Phi_{n}({(\Lambda/\varpi^{t}\Lambda)}^{\\!\vee})&\cong\varprojlim_{n}{\mathcal{O}_{f,n}[\Gamma_{m}]}^{\\!\vee}\cong\varprojlim_{n}\mathcal{O}_{f,n}[\Gamma_{m}]\\\
&\cong\Lambda/\varpi^{n}.\\\ \end{aligned}$ ∎

###  10.3. Finish of the proof

######  Proposition 10.10.

We have
$\Phi_{n}(\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f}))=\widehat{\text{Sel}}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},T_{f,n})$.

###### Proof.

We evaluate the functor $\Phi_{n}$ at
$\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})$:

$\displaystyle\Phi_{n}(\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f}))$
$\displaystyle\cong\varprojlim_{n}\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})[\varpi^{n}]^{\Gamma^{m}}\cong\widehat{\text{Sel}}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},T_{f,n})$

where the last isomorphism is given by the control theorem treated in
Proposition 2.10.

∎

######  Theorem 10.11.

The compact Selmer group
$\widehat{\text{Sel}}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},T_{f,n})$
is free of rank $\\#S$ over $\Lambda/\varpi^{n}$.

######  Remark.

This theorem improves Proposition 7.6.

###### Proof.

By Corollary 10.4, we know
${\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f})}^{\\!\vee}$
is pseudo-isomorphic to

$(\bigoplus_{i=1}^{\\#S}\Lambda/\varpi^{t_{i}})\times Y$

for some $Y$ with vanishing $\mu$-invariant. Combining Proposition 10.8 and
Proposition 10.9, we have

$\Phi_{n}(\text{Sel}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},A_{f}))\cong(\Lambda/\varpi^{n})^{\\#S}.$

Hence
$\widehat{\text{Sel}}_{\mathfrak{n}^{-}}^{S\mathfrak{n}^{+}}(K_{\infty},T_{f,n})\cong(\Lambda/\varpi^{n})^{\\#S}$
by Proposition 10.10. ∎

Now Theorem 7.17 can be improved by assuming the desired assumption
$(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$. Note it suffices to use
a larger Selmer group
$\widehat{\text{Sel}}^{S\mathfrak{n}^{+}}_{\Delta}(K_{\infty},T_{f,n})$ in the
proof of Theorem 7.9. The proof of the following freeness theorem for this
Selmer group will provide all the missing ingredients to prove our main result
Theorem 1 assuming $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ cf.
Corollary 7.7 .

######  Lemma 10.12.

Let $\Delta$ be a square free product of $n$-admissible primes in $F$ with
$\mathfrak{n}^{-}$ and $S$ be a square free product of primes in $F$ away from
$\mathfrak{n}p$. Assume only $(\text{CR}^{+})$, then

  1. (1)

$\widehat{\text{Sel}}_{\Delta}^{\mathfrak{n}^{+}S}(K_{\infty},T_{f,n})/\mathfrak{m}_{\Lambda}=\text{Sel}_{\Delta}^{\mathfrak{n}^{+}S}(K,T_{f,1})$.

  2. (2)

$\widehat{\text{Sel}}_{\Delta}^{\mathfrak{n}^{+}S}(K_{\infty},T_{f,n})$ is
free over $\Lambda/\varpi^{n}$.

###### Proof.

The proof of (1) i.e. the control theorem follows from Corollary 7.7 since we
are relaxing conditions on $\mathfrak{n}^{+}$.

As for $(2)$, we have proved in Theorem 10.11 that
$\widehat{\text{Sel}}_{\mathfrak{n}^{-}}^{\mathfrak{n}^{+}S}(K_{\infty},T_{f,n})$
is free. To prove
$\widehat{\text{Sel}}_{\Delta}^{\mathfrak{n}^{+}S}(K_{\infty},T_{f,n})$ is
free, it suffices to consider the case when
$\Delta=\mathfrak{l}\mathfrak{n}^{-}$ for a single $n$-admissible prime
$\mathfrak{l}$. For this we consider the following exact sequence

$0\rightarrow\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})\rightarrow\widehat{\text{Sel}}^{S\mathfrak{l}}_{\mathfrak{n}^{-}}(K_{\infty},T_{f,n})\rightarrow\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}},T_{f,n})\rightarrow 0.$

This exact sequence results from the decomposition

$\widehat{H}^{1}(K_{\infty,\mathfrak{l}},T_{f,n})=\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}},T_{f,n})\oplus\widehat{H}^{1}_{\text{ord}}(K_{\infty,\mathfrak{l}},T_{f,n})$

in Lemma 2.7. The freeness of
$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})$ follows from the
freeness of
$\widehat{\text{Sel}}^{S\mathfrak{l}}_{\mathfrak{n}^{-}}(K_{\infty},T_{f,n})$
and that of $\widehat{H}^{1}_{{\rm fin}}(K_{\infty},T_{f,n})$. ∎

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