# On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of
parallel weights

Haining Wang  Department of Mathematics, McGill University, 805 Sherbrooke St
W, Montreal, QC H3A 0B9, Canada<EMAIL_ADDRESS>
###### Abstract.

In this article, we study the Iwasawa theory for Hilbert modular forms over
the anticyclotomic extension of a CM field. We prove a one sided divisibility
result toward the Iwasawa main conjecture. The proof relies on the first and
second reciprocity law relating theta elements to Heegner points Euler system.
As a by-product we also prove certain Bloch-Kato type result in the rank $0$
case and a parity conjecture.

###### Key words and phrases:

_Iwasawa theory, $p$-adic L-functions, Shimura curves, totally real field._

###### 2000 Mathematics Subject Classification:

Primary 11G40, Secondary 11G18

###### Contents

  1. 1 Introduction
  2. 2 Selmer groups
  3. 3 Automorphic forms on quaternion algebra
  4. 4 Shimura curves
  5. 5 Level raising and Euler system of Heegner points
  6. 6 Explicit reciprocity laws
  7. 7 Euler system argument
  8. 8 Some arithmetic consequences
  9. 9 Multiplicity one for quaternion alegbra
  10. 10 $\mu$-invariant and $\Phi$-functor

##  1\. Introduction

###  1.1. Main results

Let $F$ be a totally real field over $\mathbf{Q}$ of degree $d$ with ring of
integers $\mathcal{O}_{f}$ and $\mathfrak{n}$ be a non-zero integral ideal of
$\mathcal{O}_{F}$. We write $F_{\infty}=\prod_{v\mid\infty}F_{v}$. Let

$\Gamma_{0}(\mathfrak{n})=\\{\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\in\mathop{\rm
GL}\nolimits_{2}(\hat{\mathcal{O}}_{F}):c\equiv 0\,\mbox{mod
}{\mathfrak{n}}\\}.$

We take $f\in S_{k}(\Gamma_{0}(\mathfrak{n}),1)$ to be a Hilbert cusp form of
level $\mathfrak{n}$, parallel even weight $k$ for $k\geq 2$ and trivial
central character. We recall the definition of Hilbert modular forms following
[Hid88]. For a complex function $f:\mathop{\rm
GL}\nolimits_{2}(\mathbf{A}_{F})\rightarrow\mathbf{C}$ and an element in
$u\in\mathop{\rm GL}\nolimits_{2}(\mathbf{A}_{F})$, we define

$f\mid_{k}u(x)=j(u_{\infty},z_{0})^{-k}\det(u_{\infty})^{(k-1)}f(xu^{-1})$

for $x\in\mathop{\rm GL}\nolimits_{2}(\mathbf{A}_{F})$,
$z_{0}=(i,\cdots,i)\in{\mathcal{H}}^{d}$ and

$j:\mathop{\rm
GL}\nolimits_{2}(F_{\infty})\times{\mathcal{H}}^{d}\rightarrow\mathbf{C},(\begin{pmatrix}{a_{v}}&{b_{v}}\\\
{c_{v}}&{d_{v}}\end{pmatrix})_{v\mid\infty}\times(z_{v})_{v\mid\infty}\rightarrow\prod_{v\mid\infty}(c_{v}z_{v}+d_{v})_{v\mid\infty}.$

Then $f$ belongs to the space $S_{k}(\Gamma_{0}(\mathfrak{n}),1)$ of Hilbert
cusp forms of parallel weight $k$ and trivial central character if

  1. (1)

$f\mid_{k}u=f$ for all $u\in\Gamma_{0}(\mathfrak{n})C_{\infty}$ with
$C_{\infty}=F_{\infty}\mathop{\rm SO}\nolimits_{2}(F_{\infty})$.

  2. (2)

For $\gamma\in\mathop{\rm GL}\nolimits_{2}(F)$ and $z\in Z$ the center of
$\mathop{\rm GL}\nolimits_{2}(\mathbf{A}_{F})$, $f(\gamma zg)=f(g)$.

  3. (3)

For $x\in\mathop{\rm GL}\nolimits_{2}(\mathbf{A}^{f}_{F})$, the function
$f_{x}:{\mathcal{H}}^{d}\rightarrow\mathbf{C},uz_{0}\rightarrow
j(u,z_{0})^{k}\det(u)^{1-k}f(xu)$ is holomorphic where $u$ is any element of
$\mathop{\rm GL}\nolimits_{2}(F_{\infty})$.

  4. (4)

For all $x\in\mathop{\rm GL}\nolimits_{2}(\mathbf{A}_{F})$,
$\int_{F\backslash\mathbf{A}_{F}}f(\begin{pmatrix}{1}&{a}\\\
{0}&{1}\end{pmatrix}x)d(a)=0$ with a Harr measure $d(a)$ on
$F\backslash\mathbf{A}_{F}$.

We denote the space of Hilbert modular cusp forms of weight $k$, level
$\Gamma_{0}(\mathfrak{n})$, with trivial central character by
$S_{k}(\Gamma_{0}(\mathfrak{n}),1)$. The space
$S_{k}(\Gamma_{0}(\mathfrak{n}),1)$ is acted on by the algebra of Hecke
operators ${\mathbb{T}}_{0}(\mathfrak{n})$. A typical element in this algebra
acts on a Hilbert modular form by the double coset

$[\Gamma_{0}(\mathfrak{n})\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix}\Gamma_{0}(\mathfrak{n})]$

for a place $v$ in $F$.

We assume that $f$ is a new(primitive) form in the sense of Shimura [Shi78].
Let $K$ be a totally imaginary quadratic extension over $F$ with relative
discriminant $D_{K/F}$. Then $K$ determines a decomposition of $\mathfrak{n}$
into two integral ideals: $\mathfrak{n}=\mathfrak{n}^{+}\mathfrak{n}^{-}$
where $\mathfrak{n}^{+}$ is only divisible by primes split in $K$ and
$\mathfrak{n}^{-}$ is only divisible by primes that are non-split. In this
article we assume that

(ST) $\displaystyle\text{ the number of prime divisors of $\mathfrak{n}^{-}$
has the same parity as }d.$

The assumption (ST) puts us in the situation that

  1. (1)

the Hilbert modular form comes from a modular form on a totally definite
quaternion algebra via the Jacquet-Langlands correspondence;

  2. (2)

the sign of the functional equation of the Rankin Selberg $L$-function
$L(f/K,\chi,s)=L(f\otimes\theta_{\chi},s)$ of $f$ and $\theta_{\chi}$, the
theta series attached to an anticyclotomic character $\chi$, is $+1$ cf.
[CV07, Remark 1.2].

Let $p$ be an odd prime. We fix an embedding
$\iota_{p}:\overline{\mathbf{Q}}\rightarrow\mathbf{C}_{p}$. Let $E_{f}$ be the
$p$-adic Hecke field for the Hilbert modular form $f$ i.e. the field over
$\mathbf{Q}_{p}$ generated by the Hecke eigenvalues of $f$ induced by
$\iota_{p}$. We let $\mathcal{O}_{f}$ be the ring of integers of $E_{f}$ and
$\varpi$ be a uniformizer of $\mathcal{O}_{f}$. Denote by ${\bf a}_{v}(f)$ the
Hecke eigenvalue of $f$ at the prime $v$. We assume the following ordinary
assumption at $p$ for $f$:

(ORD) $\displaystyle{\bf a}_{v}(f)\text{ is a unit in }\mathcal{O}_{f}\text{
for all }v\mid p.$

Let $\rho_{f}:\mathop{\rm Gal}\nolimits(\bar{F}/F)\rightarrow GL_{2}(E_{f})$
be the $p$-adic Galois representation associated to $f$ cf. [Wil88], [Tay89].
We know that $\det\rho_{f}=\epsilon^{k-1}$ where $\epsilon$ is the $p$-adic
cyclotomic character of $G_{F}=\mathop{\rm Gal}\nolimits(\bar{F}/F)$, and that
if $\mathfrak{l}$ is a prime not dividing $p\mathfrak{n}$, then the
characteristic polynomial of $\rho_{f}(\mathrm{Frob}_{\mathfrak{l}})$ is given
by

$x^{2}-{\bf{a}}_{\mathfrak{l}}(f)x+{{\bf{N}}(\mathfrak{l})}^{k-1},$

here $\mathrm{Frob}_{\mathfrak{l}}$ is the Frobenius element at
$\mathfrak{l}$. In this article we will consider the selfdual twist of
$\rho_{f}$, namely $\rho_{f}^{*}:=\rho_{f}\otimes\epsilon^{\frac{2-k}{2}}$.
Let $V_{f}$ be the underlying representation space for $\rho_{f}^{*}$ and
$T_{f}$ be a $G_{F}$ stable $\mathcal{O}_{f}$ lattice in $V_{f}$ and
$A_{f}={V_{f}}/{T_{f}}$. By (ORD) and a result of Wiles [Wil88, Theorem 2],
for $v$ a place dividing $p$ in $F$, the restriction of $\rho_{f}^{*}$ at
$G_{F_{v}}$ is of the form

$\rho^{*}_{f}\mid_{G_{F_{v}}}=\left(\begin{array}[]{cc}\chi^{-1}_{v}\epsilon^{\frac{k}{2}}&*\\\
0&\chi_{v}\epsilon^{\frac{2-k}{2}}\\\ \end{array}\right)$

where $\chi_{v}$ is an unramified character of $G_{F_{v}}$ such that
$\chi_{v}(\mathrm{Frob}_{v})=\alpha_{v}$ with $\alpha_{v}$ the unit root of
the Hecke polynomial $x^{2}-{\bf a}_{v}(f)x+{\bf{N}}(v)^{k-1}$.This induces an
exact sequence of representations of $G_{F_{v}}$

$0\smash{\mathop{\longrightarrow}}{F^{+}_{v}V_{f}}\smash{\mathop{\longrightarrow}}{V_{f}}\smash{\mathop{\longrightarrow}}{F^{-}_{v}V_{f}}\smash{\mathop{\longrightarrow}}0$

where $G_{F_{v}}$ acts on $F^{+}_{v}V_{f}$ (resp. $F^{-}_{v}V_{f}$) via
$\epsilon^{\frac{k}{2}}\chi^{-1}_{v}$ (resp.
$\chi_{v}\epsilon^{\frac{2-k}{2}}$). Define $F^{+}_{v}A_{f}$ to be the image
of $F^{+}_{v}V_{f}$ in $A_{f}$.

Let ${\mathfrak{p}}$ be a fixed prime in $F$ over a rational prime $p$ such
that ${\mathfrak{p}}\nmid\mathfrak{n}D_{K/F}$. Let $H_{m}$ be the ring class
field of conductor ${\mathfrak{p}}^{m}$ and let $G_{m}=\mathop{\rm
Gal}\nolimits(H_{m}/k)$ be the ring class group. The group $G_{m}$ corresponds
to
$K^{\times}\backslash\hat{K}^{\times}/\hat{\mathcal{O}}^{\times}_{{\mathfrak{p}}^{m}}\hat{F}^{\times}$
via class field theory and here $\mathcal{O}_{{\mathfrak{p}}^{m}}$ is the
order in $K$ of conductor ${\mathfrak{p}}^{m}$. Our convention of ring class
field follows that of [AN10, Section 1.1]. Let
$d_{{\mathfrak{p}}}=[F_{{\mathfrak{p}}}:\mathbf{Q}_{p}]$ be the inertia degree
of ${\mathfrak{p}}$ and we put $K_{\infty}$ to be the maximal
$\mathbf{Z}_{p}^{d_{{\mathfrak{p}}}}$-extension of $K$ in the tower of ring
class fields of $K$ with ${\mathfrak{p}}$-power conductor. Let $K_{m}$ be the
$m$-th layer of $K_{\infty}$. Let $\Gamma=\mathop{\rm
Gal}\nolimits(K_{\infty}/K)$ and let $\Lambda=\mathcal{O}_{f}[[\Gamma]]$ be
the $d_{{\mathfrak{p}}}$ variable Iwasawa algebra over $\mathcal{O}_{f}$. We
also put $\Gamma_{m}=\mathop{\rm Gal}\nolimits(K_{m}/K)$. In this article we
study the structure of the minimal Selmer group

$\text{Sel}(K_{\infty},A_{f}):=\mathop{\rm
ker}\nolimits\\{H^{1}(K_{\infty},A_{f})\rightarrow\prod_{v\nmid
p}H^{1}(K_{\infty,{v}},A_{f})\times\prod_{v\mid
p}H^{1}(K_{\infty,v},A_{f}/F^{+}_{{\mathfrak{p}}}A_{f})\\}.$

On the analytic side, based on the work of Chida-Hsieh [CH12], Hung [Hun14]
has constructed an element $\theta_{\infty}(f)\in\Lambda$ which we will refer
to as the theta element such that for each finite character
$\chi:\Gamma\smash{\mathop{\longrightarrow}}\mu_{p^{\infty}}$ of conductor
${\mathfrak{p}}^{s}$, it satisfies

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

Here

  1. (1)

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

  2. (2)

We fix a decomposition
$\mathfrak{n}^{+}=\mathfrak{N}^{+}{\overline{\mathfrak{N}^{+}}}$ where
${\overline{\mathfrak{N}^{+}}}$ is the complex conjugation of
$\mathfrak{N}^{+}$.

  3. (3)

$\epsilon_{{\mathfrak{p}}}(f)$ is the local root number of $f$ 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 $f$. It is
known as the Gross period and is given by [Hun14, (5.2)].

In this article we prove half of the anticyclotomic Iwasawa main conjecture
for Hilbert modular forms under similar hypothesis at $p$ as in [CH15b]. Let
$\mathfrak{n}_{\bar{\rho}}$ be the Artin conductor of the residual Galois
representation $\bar{\rho}_{f}$.

######  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)

$\frac{\mathfrak{n}}{\mathfrak{n}_{\bar{\rho}}}$ is coprime to
$\mathfrak{n}_{\bar{\rho}}$.

For primes dividing $\mathfrak{n}^{+}$, we assume that

######  Hypothesis  ($\mathfrak{n}^{+}\text{-DT}$).

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

At places above $p$, we impose the condition

######  Hypothesis  (PO).

${\bf a}_{v}^{2}(f)\not\equiv 1(\,\mbox{mod }{p})\text{ for all $v\mid p$ if
$k=2$ }.$

In order to prove the second reciprocity law, we assume that the so-called
"Ihara’s Lemma" is true in our setting. We refer the reader to Assumption 6.3
for the precise statement.

We put $L_{p}(K_{\infty},f)=\theta_{\infty}(f)^{2}$ as an element in
$\Lambda$. It is well known that the Selmer group
$\text{Sel}(K_{\infty},A_{f})$ is a cofinitely generated $\Lambda$-module and
let $\text{char}_{\Lambda}\text{Sel}(K_{\infty},A_{f})^{\lor}$ be the
characteristic ideal of its Pontryagain dual. The Iwasawa main conjecture for
Hilbert modular forms over the anticyclotomic extension $K_{\infty}$ predicts
the following

######  Conjecture 1.

$\text{char}_{\Lambda}\text{Sel}(K_{\infty},A_{f})^{\lor}=(L_{p}(K_{\infty},f)).$

It can be seen as a family version of the Bloch-Kato conjecture which relates
the size of the Selmer groups with the $L$-function. We prove in this article
the following result (one sided divisibility) toward the Iwasawa main
conjecture for Hilbert modular forms over the anticyclotomic extension.

######  Theorem 1.

Assume $(\text{CR}^{+})$, $(\text{PO})$, $(\mathfrak{n}^{+}\text{-DT})$ and
the "Ihara’s lemma" are true. Then

$\text{char}_{\Lambda}\text{Sel}(K_{\infty},A_{f})^{\lor}\mid(L_{p}(K_{\infty},f)).$

Recall for each finite generated module $X$ over $\Lambda$, $X$ is pseudo-
isomorphic to
$\Lambda^{r}\oplus\prod_{i}\Lambda/(\varpi^{\mu_{i}})\oplus\prod_{j}\Lambda/(f^{m_{j}}_{j})$
for some distinguished polynomials $f_{j}$ [Lan90, Theorem 3.1]. The algebraic
Iwasawa $\mu$-invariant $\mu$ of $X$ is defined to be $\mu=\sum_{j}\mu_{j}$.
An immediate corollary of this theorem and the result of Hung [Hun14, Theorem
B] is the following corollary.

######  Corollary 1.

Assume the same hypothesis in Theorem 1. Then the $\Lambda$-module
$\text{Sel}(K_{\infty},A_{f})$ is cotorsion with trivial algebraic
$\mu$-invariant.

The ingredients of the proof of this theorem can also be used to prove the
following result in the direction of the Bloch-Kato conjecture. We refer the
reader to [BK90] for more about the conjecture and for the definition of their
Selmer group.

######  Theorem 2.

Assume $(\text{CR}^{+})$, $(\text{PO})$, and $(\mathfrak{n}^{+}\text{-DT})$
hold. Let $\chi$ be a finite order character of $\Gamma_{m}=\mathop{\rm
Gal}\nolimits(K_{m}/K)$ and assume

$L(f/K,\chi,k/2)\neq 0,$

then the Bloch-Kato Selmer group $\text{Sel}_{f}(K,V_{f,\chi})=0$.

Using this result, we obtain as a corollary the proof of the parity conjecture
below.

######  Theorem 3.

Assume $(\text{CR}^{+})$, $(\text{PO})$, and $(\mathfrak{n}^{+}\text{-DT})$
hold. Let $\chi$ be as in Theorem 2, then

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

###  1.2. Strategy of the proof

To prove this one-sided divisibility result, we follow the scheme of [BD05]
and its modification in [PW11], [CH15b]. We give an account of the strategy to
prove Theorem 1 in this section and summarize the structure of this article.

  * •

In section $2$ we collect some basic facts about Galois cohomology over the
anticyclotomic tower and introduce various Selmer groups associated to the
Galois representation $\rho^{*}_{f}$. Notably we introduce the
$\Delta$-ordinary, $S$-relaxed Selmer group
$\text{Sel}_{\Delta}^{S}(L,T_{f,n})$ for some $L$ contained in $K_{\infty}$.
In section $3$, we study Hilbert modular forms on a definite quaternion
algebra $B$ over $F$. In section $4$, we study the Shimura curve obtained from
$B$ by interchanging the invariant at an infinite place of $F$ and a place
inert in $K$. In particular, we give a modular description of the component
group of its special fiber at a prime of bad reduction. We will construct in
section 5 an Euler system
$\kappa_{\mathcal{D}}(\mathfrak{l})\in\widehat{H}^{1}(K_{\infty},T_{f,n})$
indexed by $(\mathfrak{l},\mathcal{D})$. Here $\mathfrak{l}$ is an
$n$-admissible prime in the sense of Definition 2.1 and
$\mathcal{D}=(\Delta,g)$ consists of $\Delta=\mathfrak{n}^{-}S$ where $S$ is a
square free product of even number of n-admissible primes and $g$ is a weight
two modular form on the definite quaternion algebra of discriminant $\Delta$
whose Hecke eignevalues are congruent to those of $f$ modulo $\varpi^{n}$. The
construction is based on a level raising principle initiated by Ribet [Rib90],
see also [BD05, Theorem 5.15]. Roughly speaking, for each admissible form
$\mathcal{D}=(\Delta,g)$ and an $n$-admissible prime $\mathfrak{l}$, one can
prove an isomorphism $T_{f,n}\simeq
T_{p}J^{[\mathfrak{l}]}_{n}/\mathcal{I}^{[\mathfrak{l}]}_{g}$ where
$T_{p}J^{[\mathfrak{l}]}_{n}$ is the $p$-adic Tate module of the Jacobian
$J^{[\mathfrak{l}]}_{n}$ of a Shimura curve and
$\mathcal{I}^{[\mathfrak{l}]}_{g}$ is an ideal corresponding to $g$ in a
suitable Hecke alegbra. Then the Euler system
$\kappa_{\mathcal{D}}(\mathfrak{l})$ is produced using the Euler system of
Heegner points on $J^{[\mathfrak{l}]}_{n}$.

  * •

Section 6 is devoted to the proof of the so called first and second
reciprocity laws. The first reciprocity law relates the residue of
$\kappa_{\mathcal{D}}(\mathfrak{l})\in\widehat{\text{Sel}}_{\Delta\mathfrak{l}}(K_{\infty},T_{f,n})$
at $\mathfrak{l}$ to an element
$\theta_{\infty}(\mathcal{D})\in\mathcal{O}_{f,n}[[\Gamma]]$. This element
$\theta_{\infty}(\mathcal{D})\in\mathcal{O}_{f,n}[[\Gamma]]$ is closely
related to the theta element $\theta_{\infty}(f)$ via congruences. For two
admissible primes $\mathfrak{l}_{1}$ and $\mathfrak{l}_{2}$, the second
reciprocity law is the assertion that there exists an $n$-admissible form
$\mathcal{D}^{\prime\prime}=(\Delta_{B}\mathfrak{l}_{1}\mathfrak{l}_{2},g^{\prime\prime})$
such that
$v_{\mathfrak{l}_{1}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{2}))=v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{1}))=\theta_{\infty}(\mathcal{D}^{\prime\prime})$.
Here $v_{\mathfrak{l}_{1}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{2}))$ (resp.
$v_{\mathfrak{l}_{2}}(\kappa_{\mathcal{D}}(\mathfrak{l}_{1}))$) is the image
of $\kappa_{\mathcal{D}}(\mathfrak{l}_{2})$ (resp.
$\kappa_{\mathcal{D}}(\mathfrak{l}_{1})$) in the finite part of the local
Galois cohomology group which is of rank $1$ over $\Lambda$. The proof of the
first reciprocity law is quite straightforward while the proof of the second
reciprocity law relies on the "Ihara’s lemma" which we treat as Assumption
6.3.

  * •

The first reciprocity law and second reciprocity law allow one to use an
induction argument to prove the important Theorem 7.9. We sketch the structure
of the proof here. Let $\xi:\Lambda\rightarrow\mathcal{O}_{\xi}$ be a
character of the Iwasawa algebra of $\Gamma$ over $\mathcal{O}_{f}$. For any
$\mathcal{O}_{\xi}$ module $M$ and 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 $x$ . For each positive
integer $n$ and $n$-admissible form $\mathcal{D}=(\Delta,f)$, we define two
integers:

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

The first one measures the length of the specialization of the Selmer group
and the second one measures the divisibility of the specialization of the
theta element. We would like to show $s_{\mathcal{D}}\leq 2t_{\mathcal{D}}$.
We do induction on the number $t_{\mathcal{D}}$. It follows from the first
reciprocity law that if $t_{\mathcal{D}}=0$, then $s_{\mathcal{D}}=0$ and we
have the base case. The underlying principle in the base case is the rank $0$
case of the Bloch-Kato conjecture which we will treat in chapter 8. For the
induction step, we choose a suitable $n$-admissible prime $\mathfrak{l}_{1}$
and define

(1.2)
$e_{\mathcal{D}}(\mathfrak{l}_{1})=\text{ord}_{\varpi_{\xi}}(\kappa_{\mathcal{D},\xi}(\mathfrak{l}_{1}))$

which is the divisibility index of the specialization class
$\kappa_{\mathcal{D},\xi}(\mathfrak{l}_{1})$ in the Selmer group. By the
choice of $\mathfrak{l}_{1}$, we have
$e_{\mathcal{D}}(\mathfrak{l}_{1})<t_{\mathcal{D}}$. Now we choose a second
$n$-admissible prime $\mathfrak{l}_{2}$ and define an $n$-admissible form
$\mathcal{D}^{\prime\prime}=(\Delta\mathfrak{l}_{1}\mathfrak{l}_{2},g)$. By
carefully choosing the admissible prime $\mathfrak{l}_{2}$, we can show
$t_{\mathcal{D}^{\prime\prime}}=e_{\mathcal{D}}(\mathfrak{l}_{1})<t_{\mathcal{D}}$
using the second reciprocity law. Thus by the induction hypothesis, we have
$s_{\mathcal{D}^{\prime\prime}}\leq 2t_{\mathcal{D}^{\prime\prime}}$. To
finish the proof, we show $s_{\mathcal{D}}-s_{\mathcal{D}^{\prime\prime}}\leq
2(t_{\mathcal{D}}-t_{\mathcal{D}^{\prime\prime}})$ by comparing the Selmer
group ordinary at $\Delta$ with the Selmer group ordinary at
$\Delta\mathfrak{l}_{1}\mathfrak{l}_{2}$.

The induction argument is based on the following principle which plays a
prominent role in the recent proof of the Kolyvagin’s conjecture by Wei Zhang
[Zha14]: level raising will make the theta element more primitive and at the
same time will bring down the rank of the Selmer group. The one divisibility
of Iwasawa main conjecture follows directly from Theorem 7.9.

  * •

Beyond the applications on the proof of the one divisibility result, the first
reciprocity law can also be used in proving the Bloch-Kato type result Theorem
2 stated above. We note that the proof of the first reciprocity law does not
rely on the "Ihara’s lemma" cf. Assumption 6.3. With this Bloch-Kato type
result at hand, we prove the parity conjecture in Theorem 3 in our setting
using the general method of Nekovar in [Nek07]. We remark that this result is
not new and is treated by Nekovar in [Nek08]. However our method is different
with Nekovar’s method, while Nekovar uses Hida family of Hilbert modular
forms, we deform the character $\chi$ that we twist our Hilbert modular form
by. A similar argument is used in [CH15a, Theorem 6.4] to prove the Parity
conjecture in their setting. Chapter 8 is devoted to prove these results. In
chapter 9, we prove a multiplicity one result for the space of quaternionic
Hilbert modular forms as a module over the Hecke algebra upon localizing at
certain maximal ideal. This multiplicity one result is crucial in proving the
first and second reciprocity law mentioned earlier.

  * •

The proof of the one divisibility result Theorem 1 relies on the freeness
result of the Selmer group theorem cf. Proposition 7.6. This freeness result
in turn is based on the control theorem for Selmer groups cf. Proposition
2.10. However the control theorem only holds when we replace the hypothesis
$(\mathfrak{n}^{+}\text{-DT})$ by the stronger one:

######  Hypothesis  ($\mathfrak{n}^{+}\text{-min}$).

if $\mathfrak{l}\mid\mathfrak{n}^{+}$, then $\bar{\rho}_{f}$ is ramified at
$\mathfrak{l}$.

Thus we will first prove Theorem 1 under the hypothesis
$(\mathfrak{n}^{+}\text{-min})$. However we can relax the assumption
$(\mathfrak{n}^{+}\text{-min})$ to $(\mathfrak{n}^{+}\text{-DT})$ using the
one divisibility result proved under $(\mathfrak{n}^{+}\text{-min})$. This is
carried out in chapter 10 based on the method invented by Pollack-Weston cf.
[PWK15]. Using their method we bypass the control theorem to establish the
freeness result Lemma 10.12. Note their result fixes a gap in [CH15b] who
mistakenly proved the control theorem under the hypothesis
$(\mathfrak{n}^{+}\text{-DT})$ cf. [CH15b, Proposition 1.9(2)]. This gap is
also implicit in [PW11].

###  1.3. Historical remark

The method to construct Euler system and to prove the one divisibility result
employed in this paper originates from the work of Bertolini and Darmon
[BD05], [BD96] where the one divisibility result for elliptic curve over
$\mathbf{Q}$ is proved. We also note that Bertolini-Darmon assume the
corresponding modular forms are $p$-isolated, which is assumption that
excludes the possibility of congruences between the modular form in question
with the others. The study of the $\mu$-invariant of the corresponding
$p$-adic $L$-function is carried out by Vastal [Vat03], [Vat02] where the
celebrated Mazur’s conjecture for the definite case (Gross points) is handled.
The method in [BD05] is subsequently strengthened to handled the more general
weight $2$ modular form case by Pollack-Weston in [PW11] and the $p$-isolated
condition is relaxed. However the proof there contains a little gap concerning
the freeness of Selmer groups, fortunately the authors themselves are able to
resolve this issue [PWK15] see also the last chapter in the present article.
For modular forms of small weight (comparing to $p$), Chida-Hsieh has
constructed the higher weight theta element in [CH12] using congruence between
modular forms of different weights when evaluated at CM points. The one
divisibility of the main conjecture is proved in [CH15b] adapting a similar
method as in [PW11] and the study of the $\mu$-invariant of the $p$-adic
$L$-function is carried out in [CH12]. In another direction, Longo first
proved the one divisbility result in [Lon12] for Hilbert modular forms of
parallel weight $2$ under similar assumption as in [BD05], some of the
assumptions are removed by Van Order in [VO13]. As for the $\mu$-invariant,
Cornut-Vastal successfully extended the results in[Vat02] to the totally real
field [CV07]. In [Hun14], the higher weight theta element for Hilbert modular
forms is studied, in particular a similar non-vanishing result to [CV07] is
proved. The present article treat the one divisibility of the Iwasawa main
conjecture for small parallel weight Hilbert modular forms based on the work
of [CH15b] and [Hun14].

###  1.4. Notations and conventions

Let $L$ be a number field. We denote by $\mathcal{O}_{L}$ its ring of
integers. If $v$ is a place of $L$, we identify it with a non-zero prime ideal
in $\mathcal{O}_{L}$ and we denote its absolute norm by ${\bf{N}}(v)$. We
write the completion of $L$ at $v$ as $L_{v}$ and its valuation ring as
$\mathcal{O}_{L,v}$. We will write $\varpi_{v}$ as the uniformizer of
$\mathcal{O}_{L,v}$. If $M/L$ is an algebraic extension and $v$ is a place in
$L$, we will set $M_{v}=M\otimes\mathcal{O}_{L,v}$ and similarly
$\mathcal{O}_{M,v}=\mathcal{O}_{M}\otimes\mathcal{O}_{L,v}$. We use the
standard notations for Galois groups $G_{L}=\mathop{\rm
Gal}\nolimits(\bar{L}/L)$. We denote by $\mathrm{Frob}_{v}$ a Frobinius
element at $v$ in $\mathop{\rm Gal}\nolimits(\bar{L}/L)$. For a place $v$ of
$L$, we use the notation $I_{L_{v}}$ for the inertia group of $L$ at $v$. When
without the risk of confusion, we abbreviate $I_{L_{v}}$ by $I_{v}$. We denote
by $L^{\text{ur}}_{v}$ the maximal unramified extension of $L_{v}$.

Recall we defined in the introduction for $f$, the $p$-adic Hecke field
$E_{f}$, its ring of integers $\mathcal{O}_{f}$, a uniformizer $\varpi$ and
$\mathcal{O}_{f,n}=\mathcal{O}_{f}/\varpi^{n}$.

###  1.5. Acknowledgments

This article is the author’s thesis at Penn state university. There seems to
be an increasing interest towards this thesis and the author finally decided
to revise it and make it available on arXiv. I would like to thank my advisor
Professor Winnie Li for constant help and encouragement during my graduate
studies. As is clear to the reader, this thesis is an extension of a result of
Masataka Chida and Ming-Lun Hsieh. I would like to express my gratitude to my
thesis advisor Professor Ming-Lun Hsieh for generously sharing his knowledge
with me and answering numerous questions not restricted to the topic of this
thesis.

##  2\. Selmer groups

###  2.1. Galois cohomology

In this section we define several versions of Selmer groups that are related
to the Galois representations attached $f$ and prove some standard results
about the behavior of the local cohomology over the tower of anticyclotomic
extensions. Let $L$ be a finite algebraic extension of $F$ and $M$ be a
discrete $G_{F}$ module. For a place $\mathfrak{l}$ in $F$, we define the
local cohomology groups

$H^{1}(L_{\mathfrak{l}},M)=\oplus_{\lambda\mid\mathfrak{l}}H^{1}(L_{\lambda},M)$

where the sum is taken over the places in $L$ above $\mathfrak{l}$. Similarly
we define

$H^{1}(I_{\mathfrak{l}},M)=\oplus_{\lambda\mid\mathfrak{l}}H^{1}(I_{{\lambda}},M)$

for the inertia group $I_{\lambda}=I_{L_{\lambda}}$. Let
$\text{res}_{\mathfrak{l}}:H^{1}(L,M)\rightarrow H^{1}(L_{\mathfrak{l}},M)$ be
the restriction map. We define the finite part of the local cohomology group
as

$H^{1}_{{\rm fin}}(L_{\mathfrak{l}},M)=\mathop{\rm
ker}\nolimits\\{H^{1}(L_{\mathfrak{l}},M)\rightarrow
H^{1}(I_{\mathfrak{l}},M)\\}$

and the singular quotient as

$H^{1}_{{\rm sing}}(L_{\mathfrak{l}},M)=H^{1}(L_{\mathfrak{l}},M)/H^{1}_{{\rm
fin}}(L_{\mathfrak{l}},M).$

We denote by $\partial_{\mathfrak{l}}$ the residual map obtained from the
composition of the restriction map with the natural quotient map

(2.1) $\partial_{\mathfrak{l}}:H^{1}(L,M)\rightarrow
H^{1}(L_{\mathfrak{l}},M)\rightarrow H_{{\rm sing}}^{1}(L_{\mathfrak{l}},M).$

###  2.2. Selmer group

In this subsection, we define certain Slemer groups for the Hilbert modular
form $f$. First we need to recall some properties of the Galois representation
associated to $f$, since we are working in the ordinary setup, this
representation is constructed in [Wil88]. Let
$\rho_{f}:G_{F}\rightarrow\mathop{\rm Aut}\nolimits_{E_{f}}(V_{f})$ be the
Galois representation attached to $f$ and we fix a Galois stable lattice
$T_{f}$ in $V_{f}$ and put $A_{f}=V_{f}/T_{f}$. Denote by
$\rho^{*}_{f}=\rho_{f}\otimes\epsilon^{\frac{2-k}{2}}$ the self dual twist of
$\rho_{f}$. We collect the following properties of $\rho^{*}_{f}$ cf. [Nek06,
Chapter 12]:

  1. (1)

$\rho^{*}_{f}$ is unramified outside of $\mathfrak{n}p$.

  2. (2)

$\rho^{*}_{f}\mid_{G_{F_{v}}}=\left(\begin{array}[]{cc}\chi^{-1}_{v}\epsilon^{\frac{k}{2}}&*\\\
\O&\chi_{v}\epsilon^{\frac{2-k}{2}}\\\ \end{array}\right)$ for any $v$ above
$p$. Here $\chi_{v}$ is an unramfied character such that
$\chi_{v}(\mathrm{Frob}_{v})=\alpha_{v}(f)$ with $\alpha_{v}(f)$ the unit root
of the Hecke polynomial of $f$ at $v$.

  3. (3)

$\rho^{*}_{f}\mid_{G_{F_{\mathfrak{l}}}}=\left(\begin{array}[]{cc}\pm\epsilon&*\\\
\O&\pm 1\\\ \end{array}\right)$ for any $\mathfrak{l}$ dividing $\mathfrak{n}$
exactly once.

Let $\varpi$ be a uniformizer of $\mathcal{O}_{f}$ we set:

  1. (1)

$\mathcal{O}_{f,n}=\mathcal{O}_{f}/\varpi^{n}$;

  2. (2)

$T_{f,n}=T_{f}/\varpi^{n}$;

  3. (3)

$A_{f,n}=\mathop{\rm
ker}\nolimits\\{A_{f}\smash{\mathop{\longrightarrow}\limits^{\varpi^{n}}}A_{f}\\}$.

######  Definition 2.1.

A prime $\mathfrak{l}$ in $F$ is said to be $n$-admissible for $f$ if

  1. (1)

$\mathfrak{l}\nmid p\mathfrak{n}$;

  2. (2)

$\mathfrak{l}$ is inert in $K$;

  3. (3)

${{\bf{N}}(\mathfrak{l})}^{2}-1$ is not divisible by $p$;

  4. (4)

$\varpi^{n}$ divides
${\bf{N}}(\mathfrak{l})^{\frac{k}{2}}+{\bf{N}}(\mathfrak{l})^{\frac{k-2}{2}}-\epsilon_{\mathfrak{l}}\mathbf{a}_{\mathfrak{l}}(f)$
where $\epsilon_{\mathfrak{l}}\in\\{\pm 1\\}$.

Let $L/K$ be a finite extension, we now define $F^{+}_{\mathfrak{l}}A_{f,n}$
and $H^{1}_{\text{ord}}(L_{\mathfrak{l}},A_{f,n})$ for a prime $\mathfrak{l}$
of $F$ which divides $p\mathfrak{n}^{-}$ or is $n$-admissible.

####  2.2.1. $\mathfrak{l}=v\mid p$

We have
$0\smash{\mathop{\longrightarrow}}{F^{+}_{v}V_{f}}\smash{\mathop{\longrightarrow}}{V_{f}}\smash{\mathop{\longrightarrow}}{F^{-}_{v}V_{f}}\smash{\mathop{\longrightarrow}}0$
where $F^{+}_{v}V_{f}$ is the subspace that $G_{F_{v}}$ acts by
$\chi^{-1}_{v}\epsilon^{\frac{k}{2}}$ as in the introduction. We take
$F^{+}_{v}A_{f}$ to be the image of $F^{+}_{v}V_{f}$ in $A_{f}$ and
$F^{+}_{v}A_{f,n}$ be $A_{f,n}\cap F^{+}_{v}A_{f}$.

###  2.3. $\mathfrak{l}\mid\mathfrak{n}^{-}$

In this case we let $F^{+}_{\mathfrak{l}}V_{f}$ to be the subspace of $V_{f}$
such that $G_{F_{\mathfrak{l}}}$ acts by $\pm\epsilon$. Then
$F^{+}_{\mathfrak{l}}A_{f}$ and $F^{+}_{\mathfrak{l}}A_{f,n}$ are defined
similarly as above

####  2.3.1. $\mathfrak{l}$ is $n$-admissible

Since $\mathfrak{l}$ is $n$-admissible, we know that the characteristic
polynomial for the action of $\mathrm{Frob}_{\mathfrak{l}}$ is given by
$x^{2}-\epsilon_{\mathfrak{l}}(1+{\bf{N}}(\mathfrak{l}))x+{\bf{N}}(\mathfrak{l})=(x-\epsilon_{\mathfrak{l}}{\bf{N}}(\mathfrak{l}))(x-\epsilon_{\mathfrak{l}})$
modulo $\varpi^{n}$. We thus let $F^{+}_{\mathfrak{l}}A_{f,n}$ to be the line
that $\mathrm{Frob}_{\mathfrak{l}}$ acts by
$\epsilon_{\mathfrak{l}}{\bf{N}}(\mathfrak{l})$.

######  Definition 2.2.

In all three cases we will define the ordinary cohomology group by

$H^{1}_{\rm ord}(L_{\mathfrak{l}},A_{f,n})=\mathop{\rm
ker}\nolimits\\{H^{1}(L_{\mathfrak{l}},A_{f,n})\rightarrow
H^{1}(L_{\mathfrak{l}},A_{f,n}/F^{+}_{\mathfrak{l}}A_{f,n})\\}.$

Let $\Delta$ be a square free product of primes in $F$ such that
$\Delta/\mathfrak{n}^{-}$ is a product of even number of $n$-admissible primes
and let $S$ be an integral ideal of $F$ such that $S$ is prime to
$p\Delta\mathfrak{n}$.

######  Definition 2.3.

For $M=A_{f,n}\text{ or }T_{f,n}$, we define the $\Delta$-ordinary,
$S$-relaxed Selmer group $\text{Sel}_{\Delta}^{S}(L,M)$ with respect to the
data $(f,n,\Delta,S)$ to be the group of elements $s\in H^{1}(L,M)$ such that

  1. (1)

$\partial_{\mathfrak{l}}(s)=0\text{ for all }\mathfrak{l}\nmid p\Delta S$;

  2. (2)

${\rm res}_{\mathfrak{l}}(s)\in H^{1}_{\rm ord}(L_{\mathfrak{l}},M)\text{ for
all }\mathfrak{l}\mid p\Delta$;

  3. (3)

${\rm res}_{\mathfrak{l}}(s)$ is arbitrary at $\mathfrak{l}\mid S$.

When $S$ is trivial we will abbreviate $\text{Sel}_{\Delta}^{1}(L,M)$ as
$\text{Sel}_{\Delta}(L,M)$.

###  2.4. Anticyclotomic tower and Selmer groups

Let $m$ be a non-negative integer. Denote by $H_{m}$ the ring class field of
conductor ${\mathfrak{p}}^{m}$ over $K$. Let $G_{m}=\mathop{\rm
Gal}\nolimits(H_{m}/K)$ and $H_{\infty}=\cup_{m=1}^{\infty}H_{m}$. Let
$K_{\infty}$ be the unique subfield of $H_{\infty}$ such that $\mathop{\rm
Gal}\nolimits(K_{\infty}/K)\simeq\mathbf{Z}_{p}^{d_{{\mathfrak{p}}}}$ where
$d_{{\mathfrak{p}}}$ is the inertia degree of ${\mathfrak{p}}$ and we call
$K_{\infty}$ the ${\mathfrak{p}}$-anticyclotomic
$\mathbf{Z}_{p}^{d_{{\mathfrak{p}}}}$-extension over $K$. We put
$\Gamma=\mathop{\rm Gal}\nolimits(K_{\infty}/K)$. Let $K_{m}=K_{\infty}\cap
H_{m}$ be the $m$-th layer of this tower of extensions and
$\Gamma_{m}=\mathop{\rm Gal}\nolimits(K_{m}/K)$ so that
$\Gamma=\varprojlim_{m}\Gamma_{m}$. Let $\Lambda=\mathcal{O}_{f}[[\Gamma]]$ be
the Iwasawa algebra over $\mathcal{O}_{f}$ and $\mathfrak{m}_{\Lambda}$ be its
maximal ideal.

Let

$H^{1}(K_{\infty},A_{f,n})=\varinjlim_{m}H^{1}(K_{m},A_{f,n})$

where the limit is with respect to the restriction maps and

$\widehat{H}^{1}(K_{\infty},T_{f,n})=\varprojlim_{m}H^{1}(K_{m},T_{f,n})$

where the limit is with respect to the corestriction(norm) maps.

We can define various local cohomology groups
$H^{1}_{*}(K_{\infty,\mathfrak{l}},A_{f,n})$ and
$\widehat{H}^{1}_{*}(K_{\infty,\mathfrak{l}},T_{f,n})$ in the similar fashion.
Here $*$ can be any of the finite, singular or ordinary structure defined
above. Now we can define

$\text{Sel}^{S}_{\Delta}(K_{\infty},A_{f,n})=\varinjlim_{m}\text{Sel}^{S}_{\Delta}(K_{m},A_{f,n}),$
$\widehat{\text{Sel}}^{S}_{\Delta}(K_{\infty},T_{f,n})=\varprojlim_{m}\text{Sel}_{\Delta}^{S}(K_{m},T_{f,n}).$

Recall we have defined the minimal Selmer group $\text{Sel}(K_{\infty},M)$ in
the introduction.

######  Proposition 2.4.

Assume $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ hold. Then
$\text{Sel}(K_{\infty},A_{f})$ is of finite index in

$\varinjlim_{n}\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f,n}).$

If furthermore $\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f})$ is
$\Lambda$-cotorsion, then they are equal to each other.

###### Proof.

This follows from the proof of [PW11, Proposition 3.6]. The exact same
computation shows that $\text{Sel}_{\mathfrak{n}^{-}}(K_{\infty},A_{f,n})$ is
of finite index in $\text{Sel}(K_{\infty},A_{f})[\varpi^{n}]$ with index
independent of $n$, then the result follows from the fact that
$\text{Sel}(K_{\infty},A_{f})$ has no finite index submodule if it is
$\Lambda$-cotorsion. ∎

###  2.5. Properties of local Iwasawa cohomologies

In this subsection we prove some standard results on the local cohomology
groups defined before. Since we know that $A_{f,n}$ and $T_{f,n}$ are Cartier
dual to each other, we have the following local Tate pairing for each prime
$\mathfrak{l}$ in $K_{m}$

$\langle,\rangle_{\mathfrak{l}}:H^{1}(K_{m,\mathfrak{l}},T_{f,n})\times
H^{1}(K_{m,\mathfrak{l}},A_{f,n})\rightarrow E_{f}/\mathcal{O}_{f}.$

Taking limit with respect to $m$, we have

$\langle,\rangle_{\mathfrak{l}}:\widehat{H}^{1}(K_{\infty,\mathfrak{l}},T_{f,n})\times
H^{1}(K_{\infty,\mathfrak{l}},A_{f,n})\rightarrow E_{f}/\mathcal{O}_{f}.$

The following lemma is well-known.

######  Lemma 2.5.

Suppose $\mathfrak{l}\nmid p$, $H^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}},A_{f,n})$ and $\widehat{H}^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}},T_{f,n})$ are orthogonal complement to each
other under the pairing $\langle,\rangle_{\mathfrak{l}}$.

###### Proof.

This is a standard result. See [Rub00, Proposition 1.4.2, 1.4.3]. ∎

It follows from the above lemma that $H^{1}_{{\rm
fin}}(K_{\infty,\mathfrak{l}},A_{f,n})$ and $\widehat{H}^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}},T_{f,n})$ are dual to each other under the
above pairing.

One of the differences between the cyclotomic Iwasawa theory and the
anticyclotomic Iwasawa theory is the existence of primes that split completely
in the $\mathbf{Z}_{p}$-extension and as a consequence we have the following
lemma.

######  Lemma 2.6.

Suppose $\mathfrak{l}\nmid p$ is a prime in $F$. Then

  1. (1)

if $\mathfrak{l}$ splits in $K$, then $H^{1}_{\rm
fin}(K_{\infty,\mathfrak{l}},A_{f,n})=\\{0\\}$ and hence $\widehat{H}^{1}_{\rm
sing}(K_{\infty,\mathfrak{l}},T_{f,n})=\\{0\\}$ .

  2. (2)

If $\mathfrak{l}$ is inert in $K$, then $\widehat{H}^{1}_{\rm
sing}(K_{\infty,\mathfrak{l}},T_{f,n})\simeq
H^{1}(K_{\mathfrak{l}},T_{f,n})\otimes\Lambda$ and it follows that
${H}^{1}_{\rm fin}(K_{\infty,\mathfrak{l}},A_{f,n})\simeq\mathop{\rm
Hom}\nolimits(\widehat{H}^{1}_{\rm
sing}(K_{\mathfrak{l}},T_{f,n})\otimes\Lambda,{E_{f}}/{\mathcal{O}_{f}})$.

###### Proof.

Suppose that $\mathfrak{l}=\mathfrak{l}_{1}\mathfrak{l}_{2}$ in $K$, then the
Frobenius element in $\mathop{\rm Gal}\nolimits(K_{\infty}/K)$ above
$\mathfrak{l}_{i}$ is non-trivial. Hence it generates a subgroup of finite
index in $G_{\infty}$ and $K_{\infty,\mathfrak{l}_{i}}$ is a finite product of
unramified $\mathbf{Z}_{p}$ extension of $K_{\mathfrak{l}_{i}}$. Then $(1)$
follows because the pro-$p$ part of $\mathop{\rm
Gal}\nolimits(K^{ur}_{\mathfrak{l}_{i}}/K_{\infty,\mathfrak{l}_{i}})$ is
trivial by definition. If $\mathfrak{l}$ is inert in $K$, then by class field
theory, $\mathfrak{l}$ splits completely in the tower $K_{\infty}/K$. The
conclusion of $(2)$ follows easily from Shapiro’s lemma. ∎

For $n$-admissible primes, we have a simple decomposition of the local
cohomologies.

######  Lemma 2.7.

If $\mathfrak{l}$ is $n$-admissible, then

  1. (1)

both $\widehat{H}^{1}_{\rm sing}(K_{\infty,\mathfrak{l}},T_{f,n})$ and
$\widehat{H}^{1}_{\rm fin}(K_{\infty,\mathfrak{l}},T_{f,n}$) are free of rank
one over $\Lambda/\varpi^{n}\Lambda$.

  2. (2)

We have the following decompositions:

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

and

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

###### Proof.

By the definition of $n$-admissible primes, we have a decomposition
$A_{f,n}=F^{+}_{\mathfrak{l}}A_{f,n}\oplus F^{-}_{\mathfrak{l}}A_{f,n}$. Hence
it induces a decomposition
$H^{1}(K_{\mathfrak{l}},A_{f,n})=H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})\oplus
H^{1}(K_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f,n})$. By definition, we have
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})=H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})$.
On the other hand, we have

$H^{1}_{\rm
fin}(K_{\mathfrak{l}},A_{f,n})=F^{-}_{\mathfrak{l}}A_{f,n}/(\mathrm{Frob}_{\mathfrak{l}}-1)F^{-}_{\mathfrak{l}}A_{f,n}=F^{-}_{\mathfrak{l}}A_{f,n}$

since $\mathfrak{l}$ is $n$-admissible and $\mathrm{Frob}_{\mathfrak{l}}$ acts
trivially on $F^{-}_{\mathfrak{l}}A_{f,n}$. The claim then follows from the
fact that
$H^{1}(K_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f,n})=F^{-}_{\mathfrak{l}}A_{f,n}$
by local Tate duality and Kummer theory. The rest of the lemma then follows
immediately. ∎

######  Proposition 2.8.

Assume $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-DT})$ and $(\text{PO}$)
hold. Then for $\mathfrak{l}$ an $n$-admissible prime or a prime dividing
$p\mathfrak{n}^{-}$, $H^{1}_{\text{ord}}(K_{\infty,\mathfrak{l}},A_{f,n})$ and
$\widehat{H}^{1}_{\text{ord}}(K_{\infty,\mathfrak{l}},T_{f,n})$ are orthogonal
complement to each other.

###### Proof.

We will treat three different cases.

($\mathfrak{l}$ is admissible) In this case the claim follows directly from
the previous lemmas.

($\mathfrak{l}\mid\mathfrak{n}^{-}$ ) By definition we have an exact sequence

(2.2) $\displaystyle 0$
$\displaystyle\smash{\mathop{\longrightarrow}}H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(K_{\mathfrak{l}},A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(K_{\mathfrak{l}},A_{f,n}/F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}$
$\displaystyle
H^{2}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}H^{2}(K_{\mathfrak{l}},A_{f,n})\smash{\mathop{\longrightarrow}}H^{2}(K_{\mathfrak{l}},A_{f,n}/F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}0.$

It follows from this exact sequence and the local Euler characteristic
formula,
$|H^{1}(K_{\mathfrak{l}},A_{f,n})|=|A_{f,n}^{G_{K_{\mathfrak{l}}}}|^{2}$ and
$|H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})|=|A_{f,n}^{G_{K_{\mathfrak{l}}}}|.$
This means we only need to prove the ordinary cohomology groups are orthogonal
to each other.

By definition, we have an exact sequence

$0\smash{\mathop{\longrightarrow}}H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}_{{\rm
sing}}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}0.$

When ${\bf{N}}(\mathfrak{l})^{2}\not\equiv 1\,(\mbox{mod }{\varpi^{n}})$,
$H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})$ annihilates
$H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}T_{f,n})$ since they pair to
$H^{2}(K_{\mathfrak{l}},\mathcal{O}_{f,n}(\epsilon^{2}))=0$. As
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})$ (resp.
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},T_{f,n})$) is contained in the image of
$H^{1}(K_{\mathfrak{l}},F^{+}_{\mathfrak{l}}A_{f,n})$ (resp.
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},T_{f,n})$), the claim in this case
follows.

When ${\bf{N}}(\mathfrak{l})^{2}\equiv 1\,(\mbox{mod }{\varpi^{n}})$, a direct
calculation shows that
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})=H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,n})$ (resp.
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},T_{f,n})=H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},T_{f,n})$). In fact, we can show that both
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})$ and $H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,n})$ are of rank $1$ over $\mathcal{O}_{f,n}$ and
there is an injection
$H^{1}_{\text{ord}}(K_{\mathfrak{l}},A_{f,n})\hookrightarrow H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},A_{f,n})$. The orthogonality then follows from Lemma
2.5.

($\mathfrak{l}=v\mid p$) Let $L/K$ be a finite extension in $K_{\infty}$. We
need the following lemma.

######  Lemma 2.9.

For $v$ a place of $L$ above $p$, we have $H^{0}(L_{v},F^{-}_{v}A_{f,n})=0$.

###### Proof.

Notice the cyclotomic character $\epsilon$ is surjective on $I_{L_{v}}$ and
hence
$F^{-}_{v}A_{f,n}=A_{f,n}/F^{+}_{v}A_{f,n}=\mathcal{O}_{n}(\chi_{v}\epsilon^{1-\frac{k}{2}})$
has no fixed part by $I_{L_{v}}$ unless $k=2$. In the case $k=2$,
$(F^{-}_{v}A_{f,n})^{G_{L_{v}}}=(\mathcal{O}_{n}(\chi_{v}))^{\langle\mathrm{Frob}_{v}\rangle}$
and $\chi_{v}(\mathrm{Frob}_{v})=\alpha_{v}^{p^{s}r_{v}}$. Here $p^{s}$ is the
inertia degree of $L$ over $K$ at $v$ and $r_{v}\in\\{1,2\\}$ depending on
whether $v$ splits in $K$. It follows from our assumption (PO) that
$\alpha_{v}^{r_{v}p^{s}}$ is not $1$ on $F^{-}_{v}A_{f,n}$. Then we can
conclude that $H^{0}(L_{v},F^{-}_{v}A_{f,n})=0$. ∎

By the definition of the ordinary cohomology and Lemma 2.9, we have

$H^{1}_{\rm ord}(L_{v},A_{f,n})=H^{1}(L_{v},F^{+}_{v}A_{f,n}).$

The same statement holds if we replace $A_{f,n}$ above by $T_{f,n}$. The claim
in the case $\mathfrak{l}=v\mid p$ then follows from the fact that
$H^{1}(L_{v},F^{+}_{v}A_{f,n})$ is the orthogonal complement of
$H^{1}(L_{v},F^{+}_{v}T_{f,n})$. ∎

###  2.6. Control theorems of Selmer groups

Recall $S$ and $\Delta$ as in Definition 2.3 and let $L/K$ be a finite
extension contained in $K_{\infty}$.

######  Proposition 2.10.

Assume $(\text{CR}^{+})$, $(\mathfrak{n}^{+}\text{-min})$ and $(\text{PO})$
hold, then

  1. (1)

the restriction maps

$H^{1}(K,A_{f,n})\rightarrow H^{1}(L,A_{f,n})^{\mathop{\rm
Gal}\nolimits(L/K)}$

and

$\text{Sel}_{\Delta}^{S}(K,A_{f,n})\rightarrow\text{Sel}_{\Delta}^{S}(L,A_{f,n})^{\mathop{\rm
Gal}\nolimits(L/K)}$

are isomorphisms;

  2. (2)

$\text{Sel}^{S}_{\Delta}(L,A_{f,n})=\text{Sel}^{S}_{\Delta}(L,A_{f})[\varpi^{n}]$.

###### Proof.

The first claim in part (1) follows from $(\text{CR}^{+})(1)$ and the
inflation-restriction exact sequence since by $(\text{CR}^{+})(1)$,
$H^{0}(L,A_{f,n})=0$.

By the definition of the Selmer group and an application of the snake lemma,
it suffices to show the following claims in order to prove the rest of the
assertions in (1):

  1. (1)

$H^{1}(K^{\text{ur}}_{\mathfrak{l}},A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(L^{\text{ur}}_{\mathfrak{l}},A_{f,n})$
is injective for $\mathfrak{l}\nmid p\Delta$;

  2. (2)

$H^{1}(K_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(L_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f,n})$
is injective for $\mathfrak{l}\mid p\Delta$.

When $\mathfrak{l}\nmid p\Delta$ or $\mathfrak{l}\mid\Delta$, the first claim
is clear as $K_{\infty}$ is unramified outside ${\mathfrak{p}}$ and hence
$K^{ur}_{\mathfrak{l}}=L^{ur}_{\mathfrak{l}}$. If $\mathfrak{l}=v\mid p$, then
by the inflation-restriction exact sequence, we only need to show
$F^{-}_{v}A_{f,n}^{G_{L_{\mathfrak{P}}}}=0$ for $\mathfrak{P}$ a place above
$v$. This is Lemma 2.9 and this concludes the proof of (1).

As for (2), we consider the following exact sequence

$0\smash{\mathop{\longrightarrow}}A_{f,n}\smash{\mathop{\longrightarrow}}A_{f}\smash{\mathop{\longrightarrow}}A_{f}\smash{\mathop{\longrightarrow}}0.$

By $(\text{CR}^{+})(1)$ and the fact that $K_{\infty}/K$ is Galois, we have
$A^{G_{L}}_{f,m}=0$ for every $m$. Then $A_{f}^{G_{L}}=0$. Hence by the
inflation-restriction exact sequence again, the map

$\text{Sel}^{S}_{\Delta}(L,A_{f,n})\smash{\mathop{\longrightarrow}}\text{Sel}^{S}_{\Delta}(L,A_{f})[\varpi^{n}]$

is injective. To show surjectivity, by a similar consideration as in (1), we
are led to prove the following claims:

  * •

$H^{1}(L^{\text{ur}}_{\mathfrak{l}},A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(L^{\text{ur}}_{\mathfrak{l}},A_{f})$
is injective for $\mathfrak{l}\nmid p\Delta$.

  * •

$H^{1}(L_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f,n})\smash{\mathop{\longrightarrow}}H^{1}(L_{\mathfrak{l}},F^{-}_{\mathfrak{l}}A_{f})$
is injective for $\mathfrak{l}\mid p\Delta$.

For the first claim, if $\mathfrak{l}\nmid\mathfrak{n}^{+}$,then the Galois
action is trivial and the claim follows immediately. In the case
$\mathfrak{l}\mid\mathfrak{n}^{+}$, the claim follows from our assumption in
$(2)$ since then $H^{0}(L^{\text{ur}}_{\mathfrak{l}},A_{f})$ is divisible.

We prove the second. For $\mathfrak{l}\mid\Delta$, the action of
$G_{L_{\mathfrak{l}}}$ is trivial and hence the claim follows immediately. For
$\mathfrak{l}=v\mid p$, we have proved in Lemma 2.9 that
$H^{0}(L_{v},F^{-}_{v}A_{f,m})=0$ for any $m$ and hence
$H^{0}(L_{v},F^{-}_{v}A_{f})=0$. Therefore we have the injectivity in the case
$\mathfrak{l}=v\mid p$.

∎

##  3\. Automorphic forms on quaternion algebra

In this section we define the notion of automorphic forms on quaternion
algebra over a totally real field. In order to use the congruences between
Hecke eigenvalues of the Hilbert modular forms, by the Jaquet-Langlands
correspondence it suffices to study the Hecke eigenvalues of these
quaternionic automorphic forms.

###  3.1. Hecke algebra on quaternion algebras

Let $B$ be a quaternion algebra over the totally real field $F$ and
$\widehat{B}=B\otimes\hat{\mathbf{Z}}$. Set $\Delta_{B}$ to be the
discriminant of $B$ over $F$. Let $\Sigma$ be a finite set of places of $F$.
We let

$\widehat{B}^{(\Sigma),\times}=\\{x\in\widehat{B}^{\times}:x_{v}=1,\forall
v\in\Sigma\\}$

and

$\widehat{B}^{\times}_{(\Sigma)}=\\{x\in\widehat{B}^{\times}:x_{v}=1,\forall
v\not\in\Sigma\\}.$

A similar notation is used for any open compact subset $\mathcal{U}$ of
$\widehat{B}^{\times}$.

Let $\mathcal{U}$ be an open compact subgroup of $\widehat{B}^{\times}$ and
$\mathcal{U}_{v}$ be its component at $v$. Let
${\mathcal{H}}(B^{\times},\mathcal{U})$ be the Hecke algebra over $\mathbf{Z}$
on $\widehat{B}^{\times}$ which is the space of $\mathbf{Z}$-valued
bi-$\mathcal{U}$ invariant functions on $\widehat{B}^{\times}$ with compact
support. Then ${\mathcal{H}}(B^{\times},\mathcal{U})$ is equipped with the
natural involution product:
$(f*g)(z)=\int_{\widehat{B}^{\times}}f(x)g({x}^{-1}z)dx$ where $dx$ is the
normalized Haar measure such that $\mathop{\rm
vol}\nolimits(\mathcal{U},dx)=1$.

For $x\in\widehat{B}^{\times}$, we denote by $[\mathcal{U}x\mathcal{U}]$ the
characteristic function of the double coset $\mathcal{U}x\mathcal{U}$. For
each finite place $v$ of $F$, we set

$\pi_{v}=\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{1}\end{pmatrix},z_{v}=\begin{pmatrix}{\varpi_{v}}&{0}\\\
{0}&{\varpi_{v}}\end{pmatrix}$

as elements in $\mathop{\rm GL}\nolimits_{2}(F_{v})$. We fix an identification
$i:\widehat{B}^{(\Delta_{B}),\times}\simeq
M_{2}(\widehat{F}^{(\Delta_{B})})^{\times}$.

Let

$\mathfrak{m}^{+}\subset\mathcal{O}_{F}$

be an integral ideal which is prime to $\Delta_{B}$ and let
$R_{\mathfrak{m}^{+}}$ be an Eichler order(intersection of two maximal orders)
of level $\mathfrak{m}^{+}$. Note locally at $v\mid\mathfrak{m}^{+}$ with
$v^{t}\mid\mid\mathfrak{m}^{+}$,
$(R_{\mathfrak{m}^{+}})_{v}=\\{\begin{pmatrix}{a}&{b}\\\
{c}&{d}\end{pmatrix}\in M_{2}(\mathcal{O}_{F,v}):c\equiv 0\,(\mbox{mod
}{\varpi^{t}_{v}})\\}$. Suppose $(p,\Delta_{B})=1$. For a non-negative integer
$n$ and our fixed ${\mathfrak{p}}$, we define a compact open subgroup
$\mathcal{U}_{\mathfrak{m}^{+},{\mathfrak{p}}^{n}}$ by:

(3.1)
$\mathcal{U}_{\mathfrak{m}^{+},{\mathfrak{p}}^{n}}=\\{x\in\widehat{R}^{\times}_{\mathfrak{m}^{+}}:x_{{\mathfrak{p}}}\equiv\begin{pmatrix}{a}&{b}\\\
{0}&{a}\end{pmatrix}\,(\mbox{mod
}{{\mathfrak{p}}^{n}}),a,b\in\mathcal{O}_{F,{\mathfrak{p}}}\\}.$

We set $\mathcal{U}=\mathcal{U}_{\mathfrak{m}^{+},{\mathfrak{p}}^{n}}$ and put
$\Sigma={\mathfrak{p}}\mathfrak{m}^{+}\Delta_{B}$. Notice by definition $i$
induces an identification
$i:(\widehat{B}^{(\Sigma),\times},\mathcal{U}^{(\Sigma)}_{\mathfrak{m}^{+},{\mathfrak{p}}^{n}})\simeq(\mathop{\rm
GL}\nolimits_{2}(\widehat{F}^{(\Sigma)}),\mathop{\rm
GL}\nolimits_{2}(\widehat{\mathcal{O}}_{F}^{(\Sigma)}))$.

####  3.1.1. Spherical Hecke algebra

We denote by ${\mathbb{T}}^{(\Sigma)}_{B}(\mathfrak{m}^{+})$ the subalgebra of
${\mathcal{H}}(B^{\times},\mathcal{U})$ generated by
$[\mathcal{U}x\mathcal{U}]$ for $x\in\widehat{B}^{(\Sigma),\times}$. Then we
have

${\mathbb{T}}^{(\Sigma)}_{B}(\mathfrak{m}^{+})=\mathbf{Z}[\\{T_{v},S_{v},S_{v}^{-1}:v\not\in\Sigma\\}],$

where

(3.2)
$T_{v}=[\mathcal{U}i^{-1}(\pi_{v})\mathcal{U}],S_{v}=[\mathcal{U}i^{-1}(z_{v})\mathcal{U}].$

####  3.1.2. Complete Hecke algebra

For $v\mid\Delta_{B}$, we choose an element $\pi^{\prime}_{v}\in
B^{\times}_{v}$ whose norm is a uniformizer of $F_{v}$ and put

$U_{v}=[\mathcal{U}\pi^{\prime}_{v}\mathcal{U}].$

For $v\mid\mathfrak{m}^{+}$, we define

$U_{v}=[\mathcal{U}i^{-1}(\pi_{v})\mathcal{U}].$

Finally for the prime ${\mathfrak{p}}$, we define

$U_{{\mathfrak{p}}}=[\mathcal{U}i^{-1}(\pi_{{\mathfrak{p}}})\mathcal{U}],$
$\langle a\rangle=[\mathcal{U}i^{-1}({\bf d}(a))\mathcal{U}]$

where ${\bf d}(a)=\begin{pmatrix}{a}&{0}\\\ {0}&{1}\end{pmatrix}$ for
$a\in\mathcal{O}^{\times}_{F,{\mathfrak{p}}}$.

If $n=0$, we have $\mathcal{U}=\widehat{R}^{\times}_{\mathfrak{m}^{+}}$, then
we let

${\mathbb{T}}_{B}(\mathfrak{m}^{+})={\mathbb{T}}^{(\Sigma)}_{B}(\mathfrak{m}^{+})[\\{U_{v}:v\mid\Delta_{B}\mathfrak{m}^{+}\\}].$

If $n>0$, we have
$\mathcal{U}=\mathcal{U}_{\mathfrak{m}^{+},{\mathfrak{p}}^{n}}$, then we let

${\mathbb{T}}_{B}(\mathfrak{m}^{+},{\mathfrak{p}}^{n})={\mathbb{T}}^{(\Sigma)}_{B}(\mathfrak{m}^{+})[\\{U_{v}:v\mid{\mathfrak{p}}^{n}\Delta_{B}\mathfrak{m}^{+}\\},\\{\langle
a\rangle:a\in\mathcal{O}^{\times}_{F,{\mathfrak{p}}}\\}].$

We refer to these as the complete Hecke algebras. And for $\mathfrak{l}$ a
prime in $\mathcal{O}_{F}$, we let
${\mathbb{T}}^{(\mathfrak{l})}_{B}(\mathfrak{m}^{+},{\mathfrak{p}}^{n})$ be
the subalgebra of the complete Hecke algebra generated by the Hecke operators
away from $\mathfrak{l}$.

###  3.2. Optimal embeddings

Let $K$ be a totally imaginary quadratic extension over $F$ and let
$c:z\rightarrow\bar{z}$ be the nontrivial automorphism of $K$ over $F$. We fix
a CM type $\Sigma$ of $K$ i.e. a subset of $\mathop{\rm
Hom}\nolimits(K,\mathbf{C})$ such that
$\Sigma\cup\overline{\Sigma}=\mathop{\rm Hom}\nolimits(K,\mathbf{C})$ and
$\Sigma\cap\overline{\Sigma}=\emptyset$. We choose $\boldsymbol{\theta}\in K$
that satisfies

  1. (1)

$\mathop{\rm
Im}\nolimits(\sigma(\boldsymbol{\theta}))>0,\forall\sigma\in\Sigma;$

  2. (2)

$\\{1,\boldsymbol{\theta}_{v}\\}$ is a basis of $\mathcal{O}_{K,v}$ over
$\mathcal{O}_{F,v}$ for all $v\mid D_{K/F}{\mathfrak{p}}\mathfrak{n}$;

  3. (3)

$\boldsymbol{\theta}$ is local uniformizer at primes $v$ ramified in $K$.

Then $\boldsymbol{\theta}$ is a generator of $K$ over $F$. We let
$\delta=\boldsymbol{\theta}-\bar{\boldsymbol{\theta}}$.

Let $B$ be a definite quaternion algebra over $F$ of discriminant
$\Delta_{B}$. Assume that $\mathfrak{n}^{-}\mid\Delta_{B}$ and
$\Delta_{B}/\mathfrak{n}^{-}$ is a product of inert primes in $K$, then we can
regard $K$ as a subalgebra of $B$. Let $T_{B}$ and $N_{B}$ be the reduced
trace and reduced norm of $B$. We choose a basis of $B=K\oplus K.J$ over $K$
such that

  1. (1)

$J^{2}=\beta\in F^{\times}$ with $\sigma(\beta)<0$ for all $\sigma\in\Sigma$
and $Jt=\bar{t}J$ for all $t\in K$;

  2. (2)

$\beta\in(\mathcal{O}_{F,v}^{\times})^{2}$ for all
$v\mid{\mathfrak{p}}\mathfrak{n}^{+}$ and $\beta\in\mathcal{O}_{F,v}^{\times}$
for all $v\mid D_{K/F}$.

We further require the fixed isomorphism $i$ satisfies
$\prod_{v\nmid\Delta_{B}}i_{v}:\widehat{B}^{(\Delta_{B})}\simeq
M_{2}(\widehat{\mathbf{Q}}^{(\Delta_{B})})$ such that for each
$v\mid{\mathfrak{p}}\mathfrak{n}^{+}$, $i_{v}:B_{v}\simeq M_{2}(F_{v})$ is
given by

$i_{v}(\boldsymbol{\theta})=\begin{pmatrix}{T(\boldsymbol{\theta})}&{-N(\boldsymbol{\theta})}\\\
{1}&{0}\end{pmatrix}$

where $T(\boldsymbol{\theta})=\boldsymbol{\theta}+\bar{\boldsymbol{\theta}}$,
$N(\boldsymbol{\theta})=\boldsymbol{\theta}\bar{\boldsymbol{\theta}}$;

$i_{v}(J)=\sqrt{\beta}\begin{pmatrix}{-1}&{T(\boldsymbol{\theta})}\\\
{0}&{1}\end{pmatrix}$

and for each place $v\nmid{\mathfrak{p}}\mathfrak{n}^{+}\Delta_{B}$,
$i_{v}(\mathcal{O}_{K,v})\subset M_{2}(\mathcal{O}_{F,v})$.

###  3.3. $P$-adic Hilbert modular forms on quaternion algebra

Let $\Phi$ be a finite extension of $\mathbf{Q}_{p}$ that contains the image
of all embeddings $F\hookrightarrow\bar{\mathbf{Q}}_{p}$. Let $A$ be an
$\mathcal{O}_{\Phi}$ algebra. Let $k\geq 2$ be an even integer and for each
$\sigma\in\Sigma$ we let $L_{k,\sigma}(A)=A[X_{\sigma},Y_{\sigma}]_{k-2}$ be
the space of homogeneous polynomials of degree $k-2$ with two variables over
$A$. Denote $\rho_{k,\sigma}:\mathop{\rm GL}\nolimits_{2}(A)\rightarrow
L_{k,\sigma}(A)$ the representation defined by

$\rho_{k,\sigma}(g)P(X_{\sigma},Y_{\sigma})=\det(g)^{\frac{2-k}{2}}P((X_{\sigma},Y_{\sigma})g).$

Our fixed embedding $F\hookrightarrow\bar{\mathbf{Q}}_{p}$ gives a
decomposition $\Sigma=\cup_{v\mid p}\Sigma_{v}$ where $\Sigma_{v}$ is the set
of those embeddings inducing $v$. We define

$L_{k}(A)=\otimes_{v\mid p}\otimes_{\sigma\in\Sigma_{v}}L_{k,\sigma}(A).$

We can regard $L_{k}(A)$ as an $\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,p})$ module by the action
$\rho_{k}(u_{p})=\otimes_{v\mid
p}\otimes_{\sigma\in\Sigma_{v}}\rho_{k,\sigma}(\sigma(u_{v})).$

Let $\mathcal{U}$ be an open compact set in $\widehat{B}^{\times}$. The space
$S_{k}^{B}(\mathcal{U},A)$ of $p$-adic Hilbert modular forms of level
$\mathcal{U}$ and weight $k$ is defined by

$S_{k}^{B}(\mathcal{U},A)=\\{f:B^{\times}\backslash\widehat{B}^{\times}\rightarrow
L_{k}(A)\mid f(bu)=\rho_{k}(u_{p})^{-1}f(b),u\in\mathcal{U}\\}.$

It is equipped with an action of the Hecke algebra
${\mathbb{T}}_{B}(\mathfrak{n}^{+},{\mathfrak{p}}^{n})$ given by

(3.3)
$\displaystyle[\mathcal{U}x\mathcal{U}]f(b)=\sum_{u\in\mathcal{U}/\mathcal{U}\cap
x\mathcal{U}x^{-1}}f(bux)\text{ for
}x\in\widehat{B}^{({\mathfrak{p}}),\times},$ $\displaystyle\langle a\rangle
f(b)=\rho_{k}(b{\bf d}(a))f(b{\bf d}(a)),$ $\displaystyle
U_{{\mathfrak{p}}}f(b)=\sum_{u\in\mathcal{U}_{{\mathfrak{p}}}/\mathcal{U}_{{\mathfrak{p}}}\cap\pi_{{\mathfrak{p}}}\mathcal{U}_{{\mathfrak{p}}}\pi^{-1}_{{\mathfrak{p}}}}\rho_{k}(u)\hat{\rho}_{k}(\pi_{{\mathfrak{p}}})f(bu\pi_{{\mathfrak{p}}})$

where

$\hat{\rho}_{k}(\pi_{{\mathfrak{p}}})(\otimes_{v\mid
p}\otimes_{\sigma\in\Sigma_{v}}P_{\sigma}(X_{\sigma},Y_{\sigma}))=\otimes_{\sigma\in\Sigma_{{\mathfrak{p}}}}P_{\sigma}(\sigma(\varpi_{{\mathfrak{p}}})X_{\sigma},Y_{\sigma})\otimes(\otimes_{\sigma\not\in\Sigma_{{\mathfrak{p}}}}P_{\sigma}(X_{\sigma},Y_{\sigma})).$

We let
$X_{B}(\mathcal{U})=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}$ be
the Shimura set of level $\mathcal{U}$. The space of weight 2 Hilbert modular
forms $S_{2}^{B}(\mathcal{U},A)=\\{f:X_{B}(\mathcal{U})\rightarrow A\\}$ can
be naturally identified with $A[X_{B}(\mathcal{U})]$ compatible with the Hecke
action if we define Hecke action on the divisor group of the Shimura set via
Picard functoriality.

Let $\tau_{n}\in\widehat{B}^{\times}_{({\mathfrak{p}}\mathfrak{n}^{+})}$ be
given by

(3.4) $\tau_{n,v}=\begin{pmatrix}{0}&{1}\\\
{\varpi_{v}^{\text{ord}_{v}({\mathfrak{p}}^{n}\mathfrak{n}^{+})}}&{0}\end{pmatrix}$

for $v\mid{\mathfrak{p}}^{n}\mathfrak{n}^{+}$. Then $\tau_{n}$ normalizes
$\mathcal{U}$ and it induces an involution on $X_{B}(\mathcal{U})$. We refer
to this involution as the Atkin-Lehner involution. We define a perfect pairing
$\langle,\rangle_{\mathcal{U}}:S_{2}^{B}(\mathcal{U},A)\times
S_{2}^{B}(\mathcal{U},A)\rightarrow A$ by

(3.5) $\langle f,g\rangle_{\mathcal{U}}=\sum_{[b]_{\mathcal{U}}\in
X_{B}(\mathcal{U})}f(b)g(b\tau_{n})\\#(B^{\times}\cap
b\mathcal{U}b^{-1}/F^{\times})^{-1}.$

Notice that the action of the Hecke algebra
${\mathbb{T}}_{B}(\mathfrak{n}^{+},{\mathfrak{p}}^{n})$ is self-adjoint with
respect to this paring.

Set $Y=\widehat{F}^{\times}$ . Then we have an action of $Y$ on the space
$S_{k}^{B}(\mathcal{U},A)$ and we define $S_{k}^{B}(\mathcal{U}Y,A)$ to be the
subspace fixed by the action of $Y$ i.e.

$S_{k}^{B}(\mathcal{U}Y,A)=S_{k}^{B}(\mathcal{U},A)^{Y}.$

We also define ${\mathbb{T}}_{B}^{(\Sigma)}(\mathfrak{m}^{+}Y)$ to be the
quotient of ${\mathbb{T}}_{B}^{(\Sigma)}(\mathfrak{m}^{+})$ by the ideal
generated by $[\mathcal{U}x\mathcal{U}]-1$ for $x\in Y$. We can define
${\mathbb{T}}_{B}(\mathfrak{m}^{+}Y,{\mathfrak{p}}^{n})$ in a similar manner.

##  4\. Shimura curves

In this section we collect some results on Shimura curve over totally real
field following the exposition of [Nek12]. Let
$\mathfrak{l}\nmid\mathfrak{n}^{+}\Delta_{B}$ be a prime in $F$ inert in $K$.
Let $B^{\prime}$ be an indefinite quaternion algebra with discriminant
$\Delta_{B}\mathfrak{l}$ and split at some $\sigma_{1}\in\Sigma$. We fix an
isomorphism

$\varphi_{B,B^{\prime}}:\widehat{B}^{(\mathfrak{l})}\longrightarrow\widehat{B}^{\prime(\mathfrak{l})}.$

We also let $t^{\prime}:K\rightarrow B^{\prime}$ be an embedding such that it
induces the composite map

(4.1) $\widehat{K}^{(\mathfrak{l})}\xrightarrow{\hskip 5.69046ptt\hskip
5.69046pt}\widehat{B}^{(\mathfrak{l})}\xrightarrow{\hskip
5.69046pt{\varphi_{B,B^{\prime}}}\hskip
5.69046pt}\widehat{B}^{\prime(\mathfrak{l})}.$

###  4.1. Shimura curves and complex uniformization

Let $\mathcal{O}_{B^{\prime}_{\mathfrak{l}}}$ be a maximal order in
$B^{\prime}_{\mathfrak{l}}$. Set
$\mathcal{U}^{\prime}=\varphi_{B,B^{\prime}}(\mathcal{U}^{(\mathfrak{l})})\mathcal{O}^{\times}_{B^{\prime}_{\mathfrak{l}}}$.
Given $(B^{\prime},\mathcal{U}^{\prime})$, there is a Shimura curve
$M_{\mathcal{U}^{\prime}}$ defined over $F$ which is smooth and projective.
The complex points of this curve are given by:

$M_{\mathcal{U}^{\prime}}(\mathbf{C})=B^{\prime\times}\backslash\
{\mathcal{H}}^{\pm}\times\widehat{B}^{\prime\times}/\mathcal{U}^{\prime}$

where
${\mathcal{H}}^{\pm}={\mathbb{P}}^{1}(\mathbf{C})-{\mathbb{P}}^{1}(\mathbf{R}).$

We write $[z,b^{\prime}]_{\mathcal{U}^{\prime}}$ for a point in
$M_{\mathcal{U}^{\prime}}$ corresponding to $z\in{\mathcal{H}}^{\pm}$ and
$b^{\prime}\in B^{\prime}$.

####  4.1.1. Quotient Shimura curves

Recall $Y=\widehat{F}^{\times}$, there is an action of $Y$ on the Shimura
curve $M_{\mathcal{U}^{\prime}}$ and we denote by $M_{\mathcal{U}^{\prime}Y}$
the quotient of the Shimura curve $M_{\mathcal{U}^{\prime}}$ by the action of
$Y$. The complex analytic uniformization of this quotient curve is just

$M_{\mathcal{U}^{\prime}Y}(\mathbf{C})=B^{\prime\times}\backslash{\mathcal{H}}^{\pm}\times\widehat{B}^{\prime\times}/\mathcal{U}^{\prime}Y.$

######  Remark.

This is the curve denoted by $N^{*}_{H}$ in [AN10] and $M_{K^{\prime}}$ in
[Zha01].

###  4.2. $\mathfrak{l}$-adic uniformization of Shimura curves

Let $\mathcal{O}^{ur}_{F,\mathfrak{l}}$ be the ring of integers of the maximal
unramified extension $F^{ur}_{\mathfrak{l}}$ of $F_{\mathfrak{l}}$. There is
an integral model $\mathcal{M}_{\mathcal{U}^{\prime}}$ over
$\mathcal{O}_{F,\mathfrak{l}}$ for $M_{\mathcal{U}^{\prime}}\otimes
F_{\mathfrak{l}}$ [Čer76]. Let ${\mathcal{H}}_{\mathfrak{l}}$ be the
Drinfeld’s $\mathfrak{l}$-adic upper-half plane. Its
$\mathbf{C}_{\mathfrak{l}}$-points are given by
$\mathbb{P}^{1}(\mathbf{C}_{\mathfrak{l}})-\mathbb{P}^{1}(F_{\mathfrak{l}})$.
The formal completion of $\mathcal{M}_{\mathcal{U}^{\prime}}$ along its
special fiber is then identified with

$B^{\times}\backslash\widehat{{\mathcal{H}}_{\mathfrak{l}}}\widehat{\otimes}_{\mathcal{O}_{F_{\mathfrak{l}}}}\widehat{\mathcal{O}}^{ur}_{F,\mathfrak{l}}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}$

where $\widehat{{\mathcal{H}}_{\mathfrak{l}}}$ is a formal model of
${\mathcal{H}}_{\mathfrak{l}}$. Note that the action of $B^{\times}$ on
$\widehat{{\mathcal{H}}_{\mathfrak{l}}}$ factors through $B^{\times}\subset
B^{\times}_{\mathfrak{l}}\simeq\mathop{\rm
GL}\nolimits_{2}(F_{\mathfrak{l}})$, and via the isomorphism
$\varphi_{B,B^{\prime}}$ we can write the above formal completion as

$\widehat{\mathcal{M}_{\mathcal{U}^{\prime}}}=B^{\times}\backslash\widehat{{\mathcal{H}}_{\mathfrak{l}}}\widehat{\otimes}_{\mathcal{O}_{F_{\mathfrak{l}}}}\widehat{\mathcal{O}}^{ur}_{F,{\mathfrak{l}}}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}.$

Thus the associated rigid analytic space has the following uniformization:

$\mathcal{M}_{\mathcal{U}^{\prime}}^{\rm{rig}}(\mathbf{C}_{\mathfrak{l}})=B^{\times}\backslash\widehat{{\mathcal{H}}_{\mathfrak{l}}}(\mathbf{C}_{\mathfrak{l}})\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}.$

####  4.2.1. $\mathfrak{l}$-adic uniformization of quotient Shimura curves

We write an element $g\in Y$ as $g=g_{\mathfrak{l}}g^{(\mathfrak{l})}$, the
action of $g$ on $\widehat{\mathcal{M}_{\mathcal{U}^{\prime}}}$ is via its
action on
$\widehat{{\mathcal{H}}_{\mathfrak{l}}}\widehat{\otimes}_{\mathcal{O}_{F_{\mathfrak{l}}}}\widehat{\mathcal{O}}^{ur}_{F_{\mathfrak{l}}}$
given by
$1\times\mathrm{Frob}_{\mathfrak{l}}^{\text{ord}_{\mathfrak{l}}(N_{B}(g_{\mathfrak{l}}))}$.
We also denote by $\mathcal{M}_{\mathcal{U}^{\prime}Y}$ the quotient of
$\mathcal{M}_{\mathcal{U}^{\prime}}$ by the action of $Y$.

From the above discussion, we get immediately

$\widehat{\mathcal{M}_{\mathcal{U}^{\prime}Y}}=B^{\times}\backslash\widehat{{\mathcal{H}}_{\mathfrak{l}}}\widehat{\otimes}_{\mathcal{O}_{F_{\mathfrak{l}}}}\widehat{\mathcal{O}}^{ur}_{F,{\mathfrak{l}}}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}Y=B^{\times}\backslash\widehat{{\mathcal{H}}_{\mathfrak{l}}}{\otimes_{\mathcal{O}_{F_{\mathfrak{l}}}}}{\mathcal{O}}_{F,\mathfrak{l}^{2}}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}Y^{(\mathfrak{l})}$

where ${\mathcal{O}}_{F,\mathfrak{l}^{2}}$ is the ring of integers of the
unramified quadratic extension of $F_{\mathfrak{l}}$.

###  4.3. Bad reduction of quotient Shimura curves

We recall the structure of the Bruhat-Tits Tree $\mathcal{T}_{\mathfrak{l}}$
for $B^{\times}_{\mathfrak{l}}\simeq\mathop{\rm
GL}\nolimits_{2}(F_{\mathfrak{l}})$. Its set of vertices is identified with
${\mathcal{V}}(\mathcal{T}_{\mathfrak{l}})\simeq
B^{\times}_{\mathfrak{l}}/\mathcal{U}_{\mathfrak{l}}F^{\times}_{\mathfrak{l}}$
while its set of oriented edges is identified with
$\vec{{\mathcal{E}}}(\mathcal{T}_{\mathfrak{l}})\simeq
B^{\times}_{\mathfrak{l}}/\mathcal{U}(\mathfrak{l})_{\mathfrak{l}}F^{\times}_{\mathfrak{l}}$.

The curve
${\mathcal{M}_{\mathcal{U}^{\prime}Y}}\otimes_{\mathcal{O}_{F,\mathfrak{l}}}\mathcal{O}_{F,\mathfrak{l}^{2}}$
is an admissible curve over $\mathcal{O}_{F,\mathfrak{l}^{2}}$ in the sense of
[JL85, Chapter 3]. The special fiber of this curve is organized by the dual
reduction graph ${\mathcal{G}}$ which admits the following description: the
set of vertices which corresponds to the set of irreducible components of the
special fiber is identified with

(4.2) $\displaystyle{\mathcal{V}}({\mathcal{G}})$
$\displaystyle=B^{\times}\backslash{\mathcal{V}}(\mathcal{T}_{\mathfrak{l}})\times\mathbf{Z}/2\mathbf{Z}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}Y^{(\mathfrak{l})}$
$\displaystyle=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}Y\times\mathbf{Z}/2\mathbf{Z}$
$\displaystyle=X_{B}(\mathcal{U}Y)\times\mathbf{Z}/2\mathbf{Z}$

via the map given by

$B^{\times}(b_{\mathfrak{l}}\mathcal{U}_{\mathfrak{l}}Y_{\mathfrak{l}},j,b^{(\mathfrak{l})}U^{(\mathfrak{l})}Y^{(\mathfrak{l})})\rightarrow(B^{\times}b_{\mathfrak{l}}b^{(\mathfrak{l})}\mathcal{U}Y,j,\text{ord}_{\mathfrak{l}}(N_{B}(b_{\mathfrak{l}}))).$

On the other hand the set of oriented edges which corresponds to the set of
singular points on the special fiber is identified with

(4.3) $\displaystyle\vec{{\mathcal{E}}}({\mathcal{G}})$
$\displaystyle=B^{\times}\backslash\vec{{\mathcal{E}}}(\mathcal{T}_{\mathfrak{l}})\times\mathbf{Z}/2\mathbf{Z}\times\widehat{B}^{(\mathfrak{l})\times}/\mathcal{U}^{(\mathfrak{l})}Y^{(\mathfrak{l})}$
$\displaystyle=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}(\mathfrak{l})Y\times\mathbf{Z}/2\mathbf{Z}$
$\displaystyle=X_{B}(\mathcal{U}(\mathfrak{l})Y)\times\mathbf{Z}/2\mathbf{Z}$

where we set

$\mathcal{U}(\mathfrak{l})=\mathcal{U}_{\mathfrak{l}\mathfrak{n}^{+},{\mathfrak{p}}^{n}}$

by adding a level at $\mathfrak{l}$ to our previous $\mathcal{U}$. The last
equality in the above equation is given by a similar formula as in (4.2).

###  4.4. Bad reduction of the Jacobians

Let $J(M_{\mathcal{U}^{\prime}Y})$ be the Jacobian of the quotient Shimura
curve which represents the functor
$\text{Pic}^{0}_{M_{\mathcal{U}^{\prime}Y}/F}\simeq\mathop{\rm
Res}\nolimits_{F^{\prime}/F}\text{Pic}^{0}_{M_{\mathcal{U}^{\prime}Y}/F^{\prime}}$
where $F^{\prime}$ is the field of moduli of $M_{\mathcal{U}^{\prime}Y}$.

Let $L/F$ be a field extension and let $\mathop{\rm
Div}\nolimits^{0}M_{\mathcal{U}^{\prime}Y}(L)$ be the group of divisors on
$M_{\mathcal{U}^{\prime}Y}(L)$ that have degree zero on each geometric
components. For $D\in\mathop{\rm
Div}\nolimits^{0}M_{\mathcal{U}^{\prime}Y}(L)$ we denote by ${\rm cl}(D)$ the
point in $J(M_{\mathcal{U}^{\prime}Y})(L)$ representing $D$.

There is a natural action of
${\mathbb{T}}_{B^{\prime}}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ on the
Jacobian under the Picard functoriality. On the other hand we have an
isomorphism
${\mathbb{T}}_{B}^{(\mathfrak{l})}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})\simeq{\mathbb{T}}_{B^{\prime}}^{(\mathfrak{l})}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$
given by $\varphi_{B,B^{\prime}}$ and thus we get an action of
${\mathbb{T}}_{B}^{(\mathfrak{l})}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$
on the Jacobian. We can extend this action to the complete Hecke algebra by
defining a ring homomorphism

(4.4)
$\varphi_{*}:{\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})\rightarrow{\mathbb{T}}_{B^{\prime}}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$

which sends $U_{\mathfrak{l}}$ to
$[\mathcal{U}^{\prime}\pi^{\prime}_{\mathfrak{l}}\mathcal{U}^{\prime}]$ where
$\pi^{\prime}_{\mathfrak{l}}$ is chosen such that
$N_{B}(\pi^{\prime}_{\mathfrak{l}})=\varpi_{\mathfrak{l}}$.

Let $\mathcal{J}/\mathcal{O}_{F,\mathfrak{l}^{2}}$ be the Néron model of
$J(M_{\mathcal{U}^{\prime}Y})/F_{\mathfrak{l}^{2}}$, $\mathcal{J}_{s}$ be the
special fiber, $\mathcal{J}^{0}_{s}$ be the connected component of
$\mathcal{J}^{0}_{s}$ and $\Phi_{M_{\mathcal{U}^{\prime}Y}}$ be the group of
connected components of $\mathcal{J}_{s}/\mathcal{J}^{0}_{s}$. The Hecke
algebra ${\mathbb{T}}_{B^{\prime}}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ acts
naturally on $\mathcal{J}$ and hence also acts on
$\Phi_{M_{\mathcal{U}^{\prime}Y}}$. Via the morphism $\varphi_{*}$, we can
equip $\Phi_{M_{\mathcal{U}^{\prime}Y}}$ with a structure of
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$-module.

####  4.4.1. First description of $\Phi_{M_{\mathcal{U}^{\prime}Y}}$

The group $\Phi_{M_{\mathcal{U}^{\prime}Y}}$ can be described in terms of the
graph ${\mathcal{G}}$. Fix an orientation of ${\mathcal{G}}$, by which we mean
a section
${\mathcal{E}}({\mathcal{G}})\rightarrow\vec{{\mathcal{E}}}({\mathcal{G}})$ of
the natural projection. We choose an orientation such that the source and
target map
$s,t:{\mathcal{E}}({\mathcal{G}})\rightarrow\vec{{\mathcal{E}}}({\mathcal{G}})\rightarrow{\mathcal{V}}({\mathcal{G}})$
is given by:

(4.5) $\displaystyle
s:{\mathcal{E}}({\mathcal{G}})=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}(\mathfrak{l})Y\rightarrow{\mathcal{V}}({\mathcal{G}})=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}Y\times\mathbf{Z}/2\mathbf{Z},$
$\displaystyle
s(B^{\times}b\mathcal{U}(\mathfrak{l})Y)=(B^{\times}b\mathcal{U}Y,0);$

(4.6) $\displaystyle
t:{\mathcal{E}}({\mathcal{G}})=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}(\mathfrak{l})Y\rightarrow{\mathcal{V}}({\mathcal{G}})=B^{\times}\backslash\widehat{B}^{\times}/\mathcal{U}Y\times\mathbf{Z}/2\mathbf{Z},$
$\displaystyle
t(B^{\times}b\mathcal{U}(\mathfrak{l})Y)=(B^{\times}b\pi^{-1}_{\mathfrak{l}}\mathcal{U}Y,1).$

The chain and cochain complex of ${\mathcal{G}}$ is defined by

(4.7)
$\displaystyle\mathbf{Z}[{\mathcal{E}}({\mathcal{G}})]\xrightarrow{\hskip
5.69046ptd_{*}\hskip 5.69046pt}{}\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]$
$\displaystyle d_{*}=-s_{*}+t_{*};$

(4.8)
$\displaystyle\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]\xrightarrow{\hskip
5.69046ptd^{*}\hskip 5.69046pt}{}\mathbf{Z}[{\mathcal{E}}({\mathcal{G}})]$
$\displaystyle d^{*}=-s^{*}+t^{*}.$

Let $\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]_{0}={\rm im}(d_{*})$, then by
[BLR90, Section 9.6 Theorem 1], there is a natural identification

(4.9)
$\Phi_{M_{\mathcal{U}^{\prime}Y}}\simeq\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]_{0}/d_{*}d^{*}.$

Let
$r_{\mathfrak{l}}:J(M_{\mathcal{U}^{\prime}Y})(F_{\mathfrak{l}^{2}})\rightarrow\Phi_{M_{\mathcal{U}^{\prime}Y}}$
be the reduction map. We can in fact describe this map using the above
identification. More precisely we have the following commutative diagram:

(4.10) $\begin{CD}\mathop{\rm
Div}\nolimits^{0}M_{\mathcal{U}^{\prime}Y}(F_{\mathfrak{l}^{2}})@>{cl}>{}>J(M_{\mathcal{U}^{\prime}Y})(F_{\mathfrak{l}^{2}})\\\
@V{}V{r_{{\mathcal{V}}}}V@V{}V{r_{\mathfrak{l}}}V\\\
\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]_{0}/d_{*}d^{*}@>{\simeq}>{}>\Phi_{M_{\mathcal{U}^{\prime}}Y}\end{CD}$

where $r_{{\mathcal{V}}}$ is the specialization map : for each
$D\in\mathop{\rm
Div}\nolimits^{0}M_{\mathcal{U}^{\prime}Y}(F_{\mathfrak{l}^{2}})$, we extend
$D$ to a Cartier divisor $\tilde{D}$ on
$\mathcal{M}_{\mathcal{U}^{\prime}Y}\otimes\mathcal{O}_{F,\mathfrak{l}^{2}}$,
then
$r_{{\mathcal{V}}}(D)=\sum_{C\in{\mathcal{V}}({\mathcal{G}})}(C,\tilde{D})C$.
Here $(C,\tilde{D})$ is the intersection pairing on
$\mathcal{M}_{\mathcal{U}^{\prime}Y}\otimes\mathcal{O}_{F,\mathfrak{l}^{2}}$.

####  4.4.2. Second description of $\Phi_{M_{\mathcal{U}^{\prime}Y}}$

Now we give a modular description of $\Phi_{M_{\mathcal{U}^{\prime}Y}}$.
Recall we can identify
$\mathbf{Z}[{\mathcal{E}}({\mathcal{G}})]\simeq\mathbf{Z}[X_{B}(\mathcal{U}(\mathfrak{l})Y)]$
with $S^{B}_{2}(\mathcal{U}(\mathfrak{l})Y,\mathbf{Z})$ and identify
$\mathbf{Z}[{\mathcal{V}}({\mathcal{G}})]\simeq\mathbf{Z}[X_{B}(\mathcal{U}Y)]^{\oplus
2}$ with $S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})^{\oplus 2}$. We denote the two
natural maps $s,t$ as $\alpha,\beta$ under this identification, then
$d_{*},d^{*}$ are identified with

(4.11) $\displaystyle
S^{B}_{2}(\mathcal{U}(\mathfrak{l})Y,\mathbf{Z})\xrightarrow{\hskip
5.69046pt\delta_{*}\hskip
5.69046pt}{}S_{2}^{B}(\mathcal{U}Y,\mathbf{Z})^{\oplus 2}$
$\displaystyle\delta_{*}=(-\alpha_{*},\beta_{*});$

and

(4.12) $\displaystyle S_{2}^{B}(\mathcal{U}Y,\mathbf{Z})^{\oplus
2}\xrightarrow{\hskip 5.69046pt\delta^{*}\hskip
5.69046pt}{}S_{2}^{B}(\mathcal{U}(\mathfrak{l})Y,\mathbf{Z})$
$\displaystyle\delta^{*}=-\alpha^{*}+\beta^{*}.$

We define a ring homorphism
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})\rightarrow\mathop{\rm
End}\nolimits(S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})^{\oplus 2})$ by

(4.13) $\displaystyle t\rightarrow\tilde{t}:(x,y)\rightarrow(t(x),t(y))\text{
for
}t\in{\mathbb{T}}_{B}^{(\mathfrak{l})}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n});$
$\displaystyle
U_{\mathfrak{l}}\rightarrow\tilde{U_{\mathfrak{l}}}:(x,y)\rightarrow(-{\bf{N}}(\mathfrak{l})y,x+T_{\mathfrak{l}}y).$

One can check immediately that $\delta_{*}$ is indeed a
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$-module
map. We put
$S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})^{2}_{0}=\delta_{*}S_{2}^{B}(\mathcal{U}(\mathfrak{l})Y,\mathbf{Z})$.

######  Proposition 4.1.

We have the following
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$-module
isomorphism

$\Phi_{M_{\mathcal{U}^{\prime}Y}}\simeq
S_{2}^{B}(\mathcal{U}Y,\mathbf{Z})^{\oplus
2}_{0}/(\tilde{U}_{\mathfrak{l}}^{2}-1).$

###### Proof.

First we notice that a direct calculation shows that

$\delta_{*}\delta^{*}=\begin{pmatrix}{{\bf{N}}(\mathfrak{l})+1}&{-T_{\mathfrak{l}}}\\\
{-T_{\mathfrak{l}}}&{{\bf{N}}(\mathfrak{l})+1}\end{pmatrix}.$

Since $\tau=\begin{pmatrix}{-1}&{-T_{\mathfrak{l}}}\\\ {0}&{-1}\end{pmatrix}$
is an automorphism of $S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})^{\oplus 2}$ and
$\tilde{U}_{\mathfrak{l}}^{2}-1=\delta_{*}\delta^{*}\tau$, we get the desired
isomorphism from the first description of $\Phi_{M_{\mathcal{U}^{\prime}Y}}$
in (4.9). ∎

###  4.5. $\mathfrak{l}$-adic uniformization of Jacobian with purely toric
reduction

In this section, we review the theory of $l$-adic uniformization of the
Jacobian of a Shimura curve at a prime of purely toric reduction following
[Nek12, section 1.7]. Let $B^{\prime}$ be the indefinite quaternion algebra
with discriminant $\mathfrak{l}\Delta$ and let
$M_{n}^{[\mathfrak{l}]}=M_{\mathcal{U}^{\prime}Y}$ be the (quotient) Shimura
curve defined in section 4.1.1. Let $J^{[\mathfrak{l}]}_{n}$ be the Jacobian
of $M_{n}^{[\mathfrak{l}]}$ and let $\Phi^{[\mathfrak{l}]}$ be the component
group of the Néron model of $J^{[\mathfrak{l}]}_{n}$ over
$\mathcal{O}_{F,\mathfrak{l}^{2}}$. Let ${\mathcal{K}}$ be a finite unramified
extension of $F_{\mathfrak{l}^{2}}$. Since $J^{[\mathfrak{l}]}_{n}$ has purely
toric reduction over $\mathcal{O}_{F_{\mathfrak{l}^{2}}}$, we have the
following uniformization exact sequence of $J^{[\mathfrak{l}]}_{n}$:

(4.14) $0\rightarrow
X^{[\mathfrak{l}]}\rightarrow{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes\bar{F}^{\times}_{\mathfrak{l}}\rightarrow
J^{[\mathfrak{l}]}_{n}(\bar{F}_{\mathfrak{l}})\rightarrow 0.$

Here we denote by $X^{[\mathfrak{l}]}$ the character group of the connected
component of the special fiber of $J^{[\mathfrak{l}]}_{n}$. Let $m$ be an
integer, the above exact sequence yields immediately the following exact
sequence of $J^{[\mathfrak{l}]}_{n}$:

(4.15) $0\rightarrow{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes\mu_{m}\rightarrow
J^{[\mathfrak{l}]}_{n}[m]\rightarrow X^{[\mathfrak{l}]}/m\rightarrow 0$

where $\mu_{m}$ is the group of $m$-th root of unity.

Combining Grothendieck’s description of the component group with the canonical
map $J^{[\mathfrak{l}]}_{n}\rightarrow\Phi^{[\mathfrak{l}]}$, we have the
following commutative diagram :

(4.16)
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>X^{[\mathfrak{l}]}@>{}>{}>{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes{\mathcal{K}}^{\times}@>{}>{}>J^{[\mathfrak{l}]}_{n}({\mathcal{K}}^{\times})@>{}>{}>0\\\
\Big{\|}@V{}V{\text{ord}_{\mathfrak{l}}}V@V{}V{}V\\\
0@>{}>{}>X^{[\mathfrak{l}]}@>{}>{}>{X^{[\mathfrak{l}]}}^{\\!\vee}<EMAIL_ADDRESS>
Let $\text{Kum}:J^{[\mathfrak{l}]}_{n}({\mathcal{K}})/m\rightarrow
H^{1}({\mathcal{K}},J^{[\mathfrak{l}]}_{n}[m])$ be the Kummer map, we can fit
the Kummer map in the diagram below:

(4.17)
$\begin{CD}X^{[\mathfrak{l}]}\otimes\mathbf{Z}/m\mathbf{Z}@>{}>{}>{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes{\mathcal{K}}^{\times}\otimes\mathbf{Z}/m\mathbf{Z}@>{}>{}>J^{[\mathfrak{l}]}_{n}({\mathcal{K}})\otimes\mathbf{Z}/m\mathbf{Z}@>{}>{}>0\\\
\Big{\|}\Big{\|}@V{}V{\text{Kum}}V\\\
X^{[\mathfrak{l}]}\otimes\mathbf{Z}/m\mathbf{Z}@>{}>{}>H^{1}({\mathcal{K}},{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes\mu_{m})@>{}>{}>H^{1}({\mathcal{K}},J^{[\mathfrak{l}]}_{n}[m]).\end{CD}$

From this diagram it is clear that the image of the Kummer map lies in the
image of the map
$H^{1}({\mathcal{K}},{X^{[\mathfrak{l}]}}^{\\!\vee}\otimes\mu_{m})\rightarrow
H^{1}({\mathcal{K}},J^{[\mathfrak{l}]}_{n}[m])$.

###  4.6. CM points

Recall we have an embedding $t^{\prime}:K\rightarrow B^{\prime}$ (4.1) which
induces in turn an embedding $t^{\prime}(K^{\times})\subset
t^{\prime}(K^{\times}_{\sigma_{1}})\subset
B^{\prime\times}_{\sigma_{1}}\simeq\mathop{\rm GL}\nolimits_{2}(\mathbf{R})$.
There are two points on
${\mathcal{H}}^{\pm}=\mathbb{P}^{1}(\mathbf{C})-\mathbb{P}^{1}(\mathbf{R})$
fixed by this action and we choose one of them and call it $z^{\prime}$. Then
the set of $\mathfrak{l}$-unramified CM points is defined as

(4.18)
$CM^{\mathfrak{l}-\text{ur}}(M_{\mathcal{U}^{\prime}Y},K)=\\{[z^{\prime},b^{\prime}]:b^{\prime}\in\widehat{B}^{\prime\times},b^{\prime}_{\mathfrak{l}}=1\\}.$

Let $\mathop{\rm rec}\nolimits_{K}:\widehat{K}^{\times}\rightarrow G^{ab}_{K}$
be the geometrically normalized reciprocity map. Then Shimura’s reciprocity
law yields

(4.19) $\mathop{\rm
rec}\nolimits_{K}(a)[z^{\prime},b^{\prime}]=[z^{\prime},t^{\prime}(a)b^{\prime}].$

Since $Y$ contains $F^{\times}_{\mathfrak{l}}$, we have

(4.20)
$\iota_{\mathfrak{l}}:CM^{\mathfrak{l}-\text{ur}}(M_{\mathcal{U}^{\prime}Y},K)\hookrightarrow
M_{\mathcal{U}^{\prime}Y}(K_{\mathfrak{l}}).$

Let $D\in\mathop{\rm
Div}\nolimits^{0}M_{\mathcal{U}^{\prime}Y}(K_{\mathfrak{l}})$ be a divisor
supported on $CM^{\mathfrak{l}-\text{ur}}(M_{\mathcal{U}^{\prime}Y},K)$. The
specialization map can be conveniently calculated by

(4.21)
$r_{{\mathcal{V}}}(\sum_{i}n_{i}[z^{\prime},b^{\prime}_{i}]_{\mathcal{U}^{\prime}Y})=\sum_{i}n_{i}[\varphi_{B,B^{\prime}}^{-1}(b^{\prime}_{i})]_{\mathcal{U}Y}.$

##  5\. Level raising and Euler system of Heegner points

We recall the set up in the beginning of the introduction. Let $f\in
S_{k}(\mathfrak{n},1)$ be a Hilbert modular form of level $\mathfrak{n}$ and
weight $k$. We assume that $f$ is a normalized newform. Then $f$ gives rise to
a morphism $\eta_{f}:{\mathbb{T}}_{\mathfrak{n}}\rightarrow\mathbf{C}$ where
${\mathbb{T}}_{\mathfrak{n}}$ is the Hecke algebra acting on
$S_{k}(\mathfrak{n},1)$. Let $\\{{\bf{a}}_{v}(f)\\}$ be the system of Hecke
eigenvalues attached to $f$. Then we denote by $E_{f}$ the $p$-adic Hecke
field with ring of integers $\mathcal{O}_{f}$ and uniformizer $\varpi$. We set

(5.1) $\displaystyle\alpha_{v}(f)=\text{the unit root of
$X^{2}-{\bf{a}}_{v}(f)X+{\bf{N}}(v)^{k-1}$}\text{ for $v\mid p$},$
$\displaystyle\alpha_{v}(f)={\bf{a}}_{v}(f){\bf{N}}(v)^{\frac{2-k}{2}}\text{
for $v\nmid p$}.$

We define a ring homorphism
$\lambda_{f}:{\mathbb{T}}_{B}(\mathfrak{n}^{+},{\mathfrak{p}}^{n})\otimes\mathcal{O}_{f}={\mathbb{T}}_{B}(\mathfrak{n}^{+},{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}\rightarrow\mathcal{O}_{f}$
by

(5.2) $\displaystyle\lambda_{f}(T_{v})=\alpha_{v}(f),$
$\displaystyle\lambda_{f}(S_{v})=1\text{ for $v\nmid p\mathfrak{n}$},$
$\displaystyle\lambda_{f}(U_{v})=\alpha_{v}(f)\text{ for $v\mid
p\mathfrak{n}$},$ $\displaystyle\lambda_{f}(\langle
a\rangle)=a^{\frac{2-k}{2}}\text{ for
$a\in\mathcal{O}^{\times}_{F,{\mathfrak{p}}}$}.$

###  5.1. Level raising

let $n$ be a positive integer and recall that
$\mathcal{O}_{f,n}=\mathcal{O}_{f}/\varpi^{n}$ and let $\mathfrak{l}$ be an
$n$-admissible prime.

######  Definition 5.1.

An admissible form $\mathcal{D}=(\Delta,g)$ is a pair consisting of a square
free product of primes in $F$ inert in $K$ and an eigenform $g\in
S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f,n})$ on the definite quaternion algebra
$B=B_{\Delta}$ with discriminant $\Delta$ over $F$, of level
$\mathcal{U}=\mathcal{U}_{\mathfrak{n}^{+},{\mathfrak{p}}^{n}}$cf. (3.1) and
which is fixed by the action of $Y$. Then $\mathcal{D}$ is admissible if the
following conditions hold:

  1. (1)

$\mathfrak{n}^{-}\mid\Delta$ and $\Delta/\mathfrak{n}^{-}$ is a product of
$n$-admissible primes.

  2. (2)

$g\,(\mbox{mod }{\varpi})\neq 0.$

  3. (3)

The eigenform $g\equiv f\,(\mbox{mod }{\varpi^{n}})$. This means that
$\lambda_{f}\equiv\lambda_{g}\,(\mbox{mod }{\varpi^{n}})$ where
$\lambda_{g}:{\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}\rightarrow\mathcal{O}_{f,n}$
is the character on the Hecke algebra
${\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ associated to $g$ mod
$\varpi^{n}$.

Let $\mathcal{D}=(\Delta,g)$ be such an admissible form and let
$\mathfrak{l}\nmid\Delta$ be an $n$-admissible prime for $f$ with
$\epsilon_{\mathfrak{l}}\alpha_{\mathfrak{l}}(f)\equiv{\bf{N}}(\mathfrak{l})+1\,(\mbox{mod
}{\varpi^{n}})$. Define
$\lambda_{g}^{[\mathfrak{l}]}:{\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}\rightarrow\mathcal{O}_{f,n}$
by $\lambda^{[\mathfrak{l}]}_{g}(U_{\mathfrak{l}})=\epsilon_{\mathfrak{l}}$.
We define $\mathcal{I}_{g}$ to be the kernel of $\lambda_{g}$ and
$\mathcal{I}^{[\mathfrak{l}]}_{g}$ to be the kernel of
$\lambda^{[\mathfrak{l}]}_{g}$. For $g$ as above, it induces a surjective map
$\psi_{g}:S^{B}_{2}(\mathcal{U}Y,\mathcal{O})/\mathcal{I}_{g}\rightarrow\mathcal{O}_{f,n}$
by $\psi_{g}(h)=\langle g,h\rangle_{\mathcal{U}}$. The following proposition
follows from the multiplicity one result Theorem 9.9. However the multiplicity
one result is proved for Hilbert modular forms with prime to $p$ level, in
order to handle the problem in hand, we use Hida’s theory based on the idea of
[CH15b, Proposition 4.2].

######  Proposition 5.2.

Assume $(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ , we have an
isomorphism

$\psi_{g}:S^{B}_{2}(\mathcal{U}Y,\mathcal{O})/\mathcal{I}_{g}\rightarrow\mathcal{O}_{f,n}.$

###### Proof.

Let $P_{k}$ be the ideal contained in $\mathcal{I}_{g}$ generated by the set

$\\{\langle
a\rangle-a^{\frac{k-2}{2}}:a\in\mathcal{O}^{\times}_{F,{\mathfrak{p}}}\\}.$

Let $e=\lim_{n\rightarrow\infty}U^{n!}_{{\mathfrak{p}}}$ be Hida’s ordinary
projector. Then by the same computation as in the proof of [Hid88, Theorem
8.1] , we have an isomorphism

$eS^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f,n})[P_{k}]=eS^{B}_{k}(\widehat{R}^{\times}_{{\mathfrak{p}}\mathfrak{n}^{+}}Y,\mathcal{O}_{f,n}).$

Taking the Pontryagin dual yields another isomorphism

$eS^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/(P_{k},\varpi^{n})=eS^{B}_{k}(\widehat{R}^{\times}_{{\mathfrak{p}}\mathfrak{n}^{+}}Y,\mathcal{O}_{f})/(\varpi^{n}).$

Thus there is an action of
$e{\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ on
$eS^{B}_{k}(\widehat{R}^{\times}_{{\mathfrak{p}}\mathfrak{n}^{+}}Y,\mathcal{O}_{f})/(\varpi^{n})$.
Let $e^{\circ}$ be the ordinary projector
$\lim_{n\rightarrow\infty}T^{n!}_{{\mathfrak{p}}}$, then the
${\mathfrak{p}}$-stablization map induces an isomorphism

$e^{\circ}S^{B}_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathcal{O}_{f})_{\mathfrak{m}}\simeq
eS^{B}_{k}(\widehat{R}^{\times}_{{\mathfrak{p}}\mathfrak{n}^{+}}Y,\mathcal{O}_{f})_{\mathfrak{m}}$

by the assumption PO. We define a surjective map
$e{\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}\rightarrow
e_{0}{\mathbb{T}}_{B}(\mathfrak{n}^{+})_{\mathcal{O}_{f}}$ by sending
$U_{{\mathfrak{p}}}$ to $u_{{\mathfrak{p}}}$, a unit root of
$x^{2}-T_{{\mathfrak{p}}}x-{\bf{N}}({\mathfrak{p}})^{k-1}$ in
$e_{0}{\mathbb{T}}_{B}(\mathfrak{n}^{+})_{\mathcal{O}_{f}}$. Hence there is
also an action of
$e{\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}$ on
$e^{\circ}S^{B}_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathcal{O}_{f})$.
Since $U_{{\mathfrak{p}}}-\alpha_{{\mathfrak{p}}}(f)\in\mathfrak{m}$ for the
maximal ideal $\mathfrak{m}$ containing $\mathcal{I}_{g}$, we have the
following sequence of isomorphisms

$S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})_{\mathfrak{m}}/(P_{k},\varpi^{n})\simeq
S^{B}_{k}(\widehat{R}^{\times}_{{\mathfrak{p}}\mathfrak{n}^{+}}Y,\mathcal{O}_{f})_{\mathfrak{m}}/(\varpi^{n})\simeq
S^{B}_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathcal{O}_{f})_{\mathfrak{m}}/(\varpi^{n}).$

We will show later that
$S^{B}_{k}(\widehat{R}^{\times}_{\mathfrak{n}^{+}}Y,\mathcal{O}_{f})_{\mathfrak{m}}$
is a cyclic ${\mathbb{T}}(\mathfrak{n}^{+}Y)_{\mathfrak{m}}$-module assuming
$(\text{CR}^{+})$ and $(\mathfrak{n}^{+}\text{-DT})$ cf. Theorem 9.9. It
follows then $S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/\mathcal{I}_{g}$ is
generated by some $h$ as a
${\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ module. Since
$\psi_{g}$ is surjective, $\psi_{g}(h)\in\mathcal{O}^{\times}_{f,n}$. As the
action of ${\mathbb{T}}_{B}(\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$ is self-
adjoint with respect to the pairing $\langle,\rangle_{\mathcal{U}}$, it
follows that the kernel of the map $\psi_{g}$ is precisely
$\mathcal{I}_{g}S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})$. ∎

Let $B^{\prime}$ be the indefinite quaternion algebra with discriminant
$\mathfrak{l}\Delta$ and let
$M_{n}^{[\mathfrak{l}]}=M_{\mathcal{U}^{\prime}Y}$ be the (quotient) Shimura
curve defined as in section 4.1.1. Let $J^{[\mathfrak{l}]}_{n}$ be the
Jacobian of $M_{n}^{[\mathfrak{l}]}$ and let $\Phi^{[\mathfrak{l}]}$ be the
component group of the Néron model of $J^{[\mathfrak{l}]}_{n}$ over
$\mathcal{O}_{F,\mathfrak{l}^{2}}$.

######  Theorem 5.3.

We have an isomorphism

$\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/\mathcal{I}_{g}\smash{\mathop{\longrightarrow}\limits^{\psi_{g}}}\mathcal{O}_{f,n}.$

###### Proof.

Let $\mathfrak{m}^{[\mathfrak{l}]}$ be the maximal ideal of
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+},{\mathfrak{p}}^{n})={\mathbb{T}}^{[\mathfrak{l}]}$
containing $\mathcal{I}^{[\mathfrak{l}]}_{g}$ and write
$S^{2}=S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})^{\oplus
2}_{\mathfrak{m}^{[\mathfrak{l}]}}$. Then by Proposition 4.1, we have

$(\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}})_{\mathfrak{m}^{[\mathfrak{l}]}}\simeq
S^{2}/(\tilde{U}^{2}_{\mathfrak{l}}-1)\simeq
S^{2}/(\tilde{U}-\epsilon_{\mathfrak{l}}).$

Note it follows from [Nek12, Proposition 1.5.9(1)] that the quotient
$S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})_{0}$
is Eisenstein, while $\mathfrak{m}^{[\mathfrak{l}]}$ is not Eisenstein. Thus

$\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
S^{2}/(\tilde{U}-\epsilon_{\mathfrak{l}})/\mathcal{I}_{g}^{[\mathfrak{l}]}\simeq(S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})^{\oplus
2}/(\tilde{U}_{\mathfrak{l}}-\epsilon_{\mathfrak{l}}))\otimes{\mathbb{T}}^{[\mathfrak{l}]}_{\mathfrak{m}^{[\mathfrak{l}]}}/\mathcal{I}^{[\mathfrak{l}]}_{g}.$

But it is easy to show that

(5.3) $\displaystyle(S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})^{\oplus
2}/(\tilde{U}_{\mathfrak{l}}-\epsilon_{\mathfrak{l}}))\otimes{\mathbb{T}}^{[\mathfrak{l}]}_{\mathfrak{m}^{[\mathfrak{l}]}}/\mathcal{I}_{g}^{[\mathfrak{l}]}$
$\displaystyle\simeq(S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/(\epsilon_{\mathfrak{l}}T_{\mathfrak{l}}-{\bf{N}}(\mathfrak{l})-1))\otimes{\mathbb{T}}_{\mathfrak{m}}/\mathcal{I}_{g}$
$\displaystyle\simeq S^{B}_{2}(\mathcal{U}Y,\mathcal{O}_{f})/\mathcal{I}_{g}.$

This completes the proof in view of Proposition 5.2. ∎

Let
$T_{p}(J^{[\mathfrak{l}]}_{n})=\varprojlim_{m}J^{[\mathfrak{l}]}_{n}[p^{m}](\bar{F})$
be the $p$-adic Tate module of $J^{[\mathfrak{l}]}_{n}$

######  Theorem 5.4.

We have an isomorphism of $G_{F}$-modules

$T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
T_{f,n}.$

###### Proof.

We put $T^{[\mathfrak{l}]}=T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}$.
The proof divides into two parts. First, We prove that
$T^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}\simeq T_{f,1}$. Recall that
the component group is computed by the following exact sequence [Nek12,
1.6.5.1]

$0\smash{\mathop{\longrightarrow}}{\mathcal{X}^{[\mathfrak{l}]}}\smash{\mathop{\longrightarrow}}{{\mathcal{X}^{[\mathfrak{l}]}}^{\\!\vee}}\smash{\mathop{\longrightarrow}}{\Phi^{[\mathfrak{l}]}}\smash{\mathop{\longrightarrow}}0$

where $\mathcal{X}^{[\mathfrak{l}]}$ is the character group of the maximal
torus of the special fiber of the Jacobian $J^{[\mathfrak{l}]}_{n}$. On the
other hand by the theory of $\mathfrak{l}$-adic uniformization of a Jacbian
with totally split toric reduction[Nek12, 1.7.1.1], we have

$0\smash{\mathop{\longrightarrow}}{{\mathcal{X}^{[\mathfrak{l}]}}^{\\!\vee}\otimes\mu_{p}}\smash{\mathop{\longrightarrow}}{J^{[\mathfrak{l}]}_{n}[p]}\smash{\mathop{\longrightarrow}}{\mathcal{X}^{[\mathfrak{l}]}/p}\smash{\mathop{\longrightarrow}}0.$

Taking the above exact sequence modulo $\mathfrak{m}^{[\mathfrak{l}]}$, we
obtain

$0\smash{\mathop{\longrightarrow}}{({\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}}^{\\!\vee}/\mathfrak{m}^{[\mathfrak{l}]}/\mathcal{Y})\otimes\mu_{p}}\smash{\mathop{\longrightarrow}}{J^{[\mathfrak{l}]}_{n}[p]_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]}}\smash{\mathop{\longrightarrow}}{\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]}}\smash{\mathop{\longrightarrow}}0$

for some submodule $\mathcal{Y}$. It induces a long exact sequence

$\mathcal{X}^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}\smash{\mathop{\longrightarrow}}H^{1}(F_{\mathfrak{l}^{2}},({\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}}^{\\!\vee}/\mathfrak{m}^{[\mathfrak{l}]}/\mathcal{Y})\otimes\mu_{p})\smash{\mathop{\longrightarrow}}H^{1}(F_{\mathfrak{l}^{2}},J^{[\mathfrak{l}]}_{n}[p]_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})\smash{\mathop{\longrightarrow}}H^{1}(F_{\mathfrak{l}^{2}},\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]}).$

We analyze some terms in this exact sequence as follows

  1. (1)

$H^{1}(F_{\mathfrak{l}^{2}},({\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}}^{\\!\vee}/\mathfrak{m}^{[\mathfrak{l}]}/\mathcal{Y})\otimes\mu_{p})=({\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}}^{\\!\vee}/\mathfrak{m}^{[\mathfrak{l}]}/\mathcal{Y})\otimes
F_{\mathfrak{l}^{2}}^{\times}/(F_{\mathfrak{l}^{2}}^{\times})^{p}={\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}}^{\\!\vee}/\mathfrak{m}^{[\mathfrak{l}]}/\mathcal{Y}$
by Kummer theory and the assumption that $\mathfrak{l}$ is $n$-admissible;

  2. (2)

$H^{1}(F_{\mathfrak{l}^{2}},\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})=H^{1}_{{\rm
fin}}(F_{\mathfrak{l}^{2}},\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})$
by the inflation-restriction exact sequence and local class field theory.

Combining these two facts we obtain an exact sequence

(5.4)
$0\smash{\mathop{\longrightarrow}}{\overline{\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]}}}\smash{\mathop{\longrightarrow}}{H^{1}(F_{\mathfrak{l}^{2}},T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})}\smash{\mathop{\longrightarrow}}{H^{1}_{{\rm
fin}}(F_{\mathfrak{l}^{2}},\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})}$

where
$\overline{\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}}$
is a quotient of
$\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}$.

Now by the main result in [BLR91], we see that

$H^{1}(F_{\mathfrak{l}^{2}},T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathfrak{m}^{[\mathfrak{l}]})=H^{1}(F_{\mathfrak{l}^{2}},T_{f,1})^{s}$

for some $s\geq 1$. Each $H^{1}(F_{\mathfrak{l}^{2}},T_{f,1})$ splits into
unramified classes and ordinary classes by Lemma 2.7 and both are of rank one.
Since
$\overline{\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}}$
is at most one dimensional by Theorem 5.3, it follows that $s=1$ and
$\overline{\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/}\mathfrak{m}^{[\mathfrak{l}]}=\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}$.
We get the result of the first part.

In the second part we prove that
$T^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq T_{f,n}$. By the
first part and Nakayama’s lemma, we can find generators $e_{1},e_{2}\in
T^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}$ such that their
reductions generate $T^{[\mathfrak{l}]}/\mathfrak{m}^{[\mathfrak{l}]}\simeq
T_{f,1}$. We will prove that the exponent of
$T^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}$ is $\varpi^{n}$ in the
next lemma. By the irreducibility of the $T_{f,1}$, we prove that
$e_{1},e_{2}$ have the same exponent and is $\varpi^{n}$. The result follows.
∎

######  Lemma 5.5.

The exponent of
$T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$
is $\varpi^{n}$.

###### Proof.

First notice that
$T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$
is naturally a
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$-module,
hence its exponent is at most $\varpi^{n}$. Then by the proof of [BD05, Lemma
5.16], we can find $n^{\prime}$ large enough such that
$J^{[\mathfrak{l}]}_{n}[{p}^{n^{\prime}}]_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$
surjects onto
$\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$. By
Theorem 5.3,
$\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$ has
exponent $\varpi^{n}$. It follows that the exponent of
$T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}$
is exactly $\varpi^{n}$. ∎

###  5.2. Reduction and residue map

Now let
$\tilde{r}_{\mathfrak{l}}:J^{[\mathfrak{l}]}_{n}(K_{m})\smash{\mathop{\longrightarrow}}{}{}\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}\otimes\mathcal{O}_{f,n}[\Gamma_{m}]$
be the reduction map given by

(5.5)
$\tilde{r}_{\mathfrak{l}}(D)=\sum_{\sigma\in\Gamma_{m}}r_{\mathfrak{l}}(\iota_{\mathfrak{l}}(D^{\sigma}))[\sigma]$

where $\iota_{\mathfrak{l}}$ is defined as in (4.20).

######  Proposition 5.6.

We have an isomorphism

$\psi_{g}:\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
H^{1}_{{\rm sing}}(K_{\mathfrak{l}},T_{f,n}).$

###### Proof.

By the proof of Theorem 5.4, and in particular replacing
$\mathfrak{m}^{[\mathfrak{l}]}$ by $\mathcal{I}^{[\mathfrak{l}]}_{g}$ in the
exact sequence (5.4), we obtain an exact sequence

(5.6)
$0\smash{\mathop{\longrightarrow}}{\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}}\smash{\mathop{\longrightarrow}}{H^{1}(K_{\mathfrak{l}},T_{f,n})}\smash{\mathop{\longrightarrow}}{H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},\mathcal{X}^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g})}.$

Since
$\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}$ is
of rank one and $H^{1}(K_{\mathfrak{l}},T_{f,n})=H^{1}_{{\rm
fin}}(K_{\mathfrak{l}},T_{f,n})\oplus H^{1}_{{\rm
sing}}(K_{\mathfrak{l}},T_{f,n})$ as $\mathfrak{l}$ is $n$-admissible, it
follows that
$\Phi_{\mathcal{O}_{f}}^{[\mathfrak{l}]}/\mathcal{I}^{[\mathfrak{l}]}_{g}\simeq
H^{1}_{{\rm sing}}(K_{\mathfrak{l}},T_{f,n})$. ∎

Let
$\partial_{\mathfrak{l}}:H^{1}(K_{m},T_{f,n})\smash{\mathop{\longrightarrow}}H^{1}_{{\rm
sing}}(K_{m,\mathfrak{l}},T_{f,n})$ be the residue map defined in (2.1). The
following theorem gives a relation between the reduction map and the residue
map.

######  Theorem 5.7.

The following diagram commutes

(5.7)
$\begin{CD}J^{[\mathfrak{l}]}_{n}(K_{m})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}@>{\text{Kum}}>{}>H^{1}(K_{m},T_{f,n})\\\
@V{}V{\tilde{r}_{\mathfrak{l}}}V@V{}V{\partial_{\mathfrak{l}}}V\\\
\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\otimes\mathcal{O}_{f,n}[\Gamma_{m}]@>{\psi_{g}}>{\simeq}>H^{1}_{{\rm
sing}}(K_{m,\mathfrak{l}},T_{f,n}).\\\ \end{CD}$

###### Proof.

This is a formal consequence of the previous proposition. It is important to
note that the proof for the exact sequence (5.4) shows that the natural Kummer
map in fact factors through the specialization map $\tilde{r}_{\mathfrak{l}}$.
∎

######  Remark.

We can take a limit in the above theorem to get the following diagram

(5.8)
$\begin{CD}J^{[\mathfrak{l}]}_{n}(K_{\infty})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}@>{\text{Kum}}>{}>H^{1}(K_{\infty},T_{f,n})\\\
@V{}V{\tilde{r}_{\mathfrak{l}}}V@V{}V{\partial_{\mathfrak{l}}}V\\\
\Phi^{[\mathfrak{l}]}_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}\otimes\mathcal{O}_{f,n}[[\Gamma_{\infty}]]@>{\psi_{g}}>{\simeq}>H^{1}_{{\rm
sing}}(K_{\infty,\mathfrak{l}},T_{f,n}).\\\ \end{CD}$

###  5.3. Gross points

Let
$\mathfrak{n}^{+}\mathcal{O}_{K}=\mathfrak{N}^{+}\overline{\mathfrak{N}^{+}}$,
we define an element $\zeta_{v}\in\mathop{\rm GL}\nolimits_{2}(F_{v})$ for
each $v\mid\mathfrak{n}^{+}$ by

(5.9)
$\zeta_{v}=\delta^{-1}\begin{pmatrix}{\boldsymbol{\theta}}&{\bar{\boldsymbol{\theta}}}\\\
{1}&{1}\end{pmatrix}\in\mathop{\rm GL}\nolimits_{2}(K_{w})=\mathop{\rm
GL}\nolimits_{2}(F_{v})\text{ if $v=w\bar{w}$ in $K$.}$

Let $m$ be a positive integer, we define
$\zeta_{{\mathfrak{p}}}^{(m)}\in\mathop{\rm
GL}\nolimits_{2}(F_{{\mathfrak{p}}})$ by

(5.10)
$\displaystyle\zeta_{{\mathfrak{p}}}^{(m)}=\begin{pmatrix}{\boldsymbol{\theta}}&{-1}\\\
{1}&{0}\end{pmatrix}\begin{pmatrix}{\varpi_{{\mathfrak{p}}}^{m}}&{0}\\\
{0}&{1}\end{pmatrix}\in\mathop{\rm
GL}\nolimits_{2}(K_{\mathfrak{P}})=\mathop{\rm
GL}\nolimits_{2}(F_{{\mathfrak{p}}})\text{ if
${\mathfrak{p}}=\mathfrak{P}\bar{\mathfrak{P}}$},$
$\displaystyle\zeta_{{\mathfrak{p}}}^{(m)}=\begin{pmatrix}{0}&{1}\\\
{-1}&{0}\end{pmatrix}\begin{pmatrix}{\varpi^{m}_{{\mathfrak{p}}}}&{0}\\\
{0}&{1}\end{pmatrix}\text{if ${\mathfrak{p}}$ is inert}.$

We set
$\zeta^{(m)}=\zeta_{{\mathfrak{p}}}^{(m)}\prod_{v\mid\mathfrak{n}^{+}}\zeta_{v}\in\widehat{B}^{\times}_{({\mathfrak{p}}\mathfrak{n}^{+})}\hookrightarrow\widehat{B}^{\times}$.
Let
${\mathcal{R}}_{m}=\mathcal{O}_{F}+\varpi_{{\mathfrak{p}}}^{m}\mathcal{O}_{K}$
be the order of $K$ of level $\varpi^{m}_{{\mathfrak{p}}}$. It is almost
immediately verified that
$(\zeta^{(m)})^{-1}\widehat{{\mathcal{R}}}^{\times}_{m}\zeta^{(m)}\subset\mathcal{U}_{\mathfrak{n}^{+},{\mathfrak{p}}^{n}}$
if $m\geq n$. Recall
$\mathcal{U}=\mathcal{U}_{\mathfrak{n}^{+},{\mathfrak{p}}^{n}}$, we define a
map

(5.11) $\displaystyle
x_{m}:\text{Pic}({\mathcal{R}}_{m}Y)=K^{\times}\backslash\widehat{K}^{\times}/\widehat{{\mathcal{R}}}^{\times}_{m}Y\smash{\mathop{\longrightarrow}}X_{B}(\mathcal{U}Y),$
$\displaystyle
K^{\times}a\widehat{{\mathcal{R}}}^{\times}_{m}Y\smash{\mathop{\longrightarrow}}x_{m}(a)=[a\zeta^{(m)}]_{\mathcal{U}Y}.$

The set of points of the form $x_{m}(a)$ for
$a\in\text{Pic}({\mathcal{R}}_{m}Y)$ are referred as Gross points of conductor
${\mathfrak{p}}^{m}$ on the definite Shimura set $X_{B}(\mathcal{U}Y)$.

###  5.4. Construction of the Euler system
$\kappa_{\mathcal{D}}(\mathfrak{l})$

Given an admissible form $\mathcal{D}=(\Delta,g)$ and an admissible prime
$\mathfrak{l}\nmid\Delta$, in this section we will define a cohomology class
$\kappa_{\mathcal{D}}(\mathfrak{l})=\\{\kappa_{\mathcal{D}}(\mathfrak{l})_{m}\\}_{m}\in\widehat{\text{Sel}}_{\Delta\mathfrak{l}}(K_{\infty},T_{f,n})$.
See [Lon12, section 7.4.2] for similar construction of the Euler system.

####  5.4.1. Heegner points on Shimura curves and its Jacobian

Let
$\mathcal{U}^{\prime}=\varphi_{B,B^{\prime}}(\mathcal{U}^{(\mathfrak{l})})\mathcal{O}^{\times}_{B^{\prime}_{\mathfrak{l}}}$
and let $M^{[\mathfrak{l}]}_{n}=M_{\mathcal{U}^{\prime}Y}$ be the quotient
Shimura curve. For each $a\in\widehat{K}^{\times}$ and $m\geq n$, we define a
Heegner point $P_{m}(a)$ by

$P_{m}(a)=[z^{\prime},\varphi_{B,B^{\prime}}(a^{(\mathfrak{l})}\zeta^{(m)}\tau_{n})]_{\mathcal{U}^{\prime}Y}\in
M^{[\mathfrak{l}]}_{n}(\mathbf{C})$

where $\tau_{n}$ is the Atkin-Lehner involution (3.4). This point is defined
on a subfield of the ring class field $H_{m}$ and the Galois action of
$G_{m}=\mathop{\rm Gal}\nolimits(H_{m}/K)$ is given by Shimura’s reciprocity
law (4.19). We choose an auciliary prime $\mathfrak{q}\nmid
p\mathfrak{l}\mathfrak{n}^{+}\Delta$ such that
$1+{\bf{N}}(\mathfrak{q})-\alpha_{\mathfrak{q}}(f)\in\mathcal{O}^{\times}_{f}$.
We define a map

(5.12) $\displaystyle\xi_{\mathfrak{q}}:\mathop{\rm Div}\nolimits
M^{[\mathfrak{l}]}_{n}(H_{m})\smash{\mathop{\longrightarrow}}J^{[\mathfrak{l}]}_{n}(H_{m})_{\mathcal{O}_{f}};$
$\displaystyle\xi_{\mathfrak{q}}(P)=\frac{1}{(1+{\bf{N}}(\mathfrak{q})-\alpha_{\mathfrak{q}}(f))}{\rm
cl}((1+{\bf{N}}(\mathfrak{q})-T_{\mathfrak{q}})P).$

Put $P_{m}=P_{m}(1)$. We define the Heegner divisor over the anticyclotomic
tower by

$D_{m}=\sum_{\sigma\in\mathop{\rm
Gal}\nolimits(H_{m}/K_{m})}\xi_{\mathfrak{q}}(P^{\sigma}_{m})\in
J^{[\mathfrak{l}]}_{n}(K_{m})_{\mathcal{O}_{f}}.$

####  5.4.2. Definition of $\kappa_{D}(\mathfrak{l})$

Let
$\text{Kum}:J^{[\mathfrak{l}]}_{n}(K_{m})_{\mathcal{O}_{f}}\smash{\mathop{\longrightarrow}}H^{1}(K_{m},T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}})$
be the Kummer map. We define the cohomology class
$\kappa_{\mathcal{D}}(\mathfrak{l})_{m}$ as:

(5.13)
$\kappa_{\mathcal{D}}(\mathfrak{l})_{m}:=\frac{1}{\alpha^{m}_{{\mathfrak{p}}}(f)}\text{Kum}(D_{m})\,(\mbox{mod
}{\mathcal{I}^{[\mathfrak{l}]}_{g}})\in
H^{1}(K_{m},T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}).$

Note we have
$H^{1}(K_{m},T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}/\mathcal{I}^{[\mathfrak{l}]}_{g}){\simeq}H^{1}(K_{m},T_{f,n})$
by Theorem 5.4. Thus we treat $\kappa_{\mathcal{D}}(\mathfrak{l})_{m}$ as an
element in $H^{1}(K_{m},T_{f,n})$.

######  Lemma 5.8 (Norm compatibility).

$\text{Cor}_{K_{m+1}/K_{m}}(D_{m+1})=U_{{\mathfrak{p}}}D_{m}.$

###### Proof.

We need to calculate $\text{Cor}_{K_{m+1}/K_{m}}(P_{m+1})$:

(5.14) $\displaystyle\text{Cor}_{K_{m+1}/K_{m}}P_{m+1}$
$\displaystyle=\sum_{\sigma\in\mathop{\rm Gal}\nolimits(K_{m+1}/K_{m})}\sigma
P_{m+1}$
$\displaystyle=\sum_{y\in\widehat{{\mathcal{R}}}^{\times}_{m}/\widehat{{\mathcal{R}}}^{\times}_{m+1}}\mathop{\rm
rec}\nolimits{y}(P_{m+1})$
$\displaystyle=\sum_{y\in\widehat{{\mathcal{R}}}^{\times}_{m}/\widehat{{\mathcal{R}}}^{\times}_{m+1}}[z^{\prime},t^{\prime}(y)\varphi_{B,B^{\prime}}(\zeta^{(m+1)}\tau_{n})]_{\mathcal{U}^{\prime}Y}.$

Note that $\mathop{\rm
Gal}\nolimits(K_{m+1}/K_{m})=\\{[1+\varpi_{{\mathfrak{p}}}^{m}u\boldsymbol{\theta}]_{m+1}:u\in\mathcal{O}_{F,{\mathfrak{p}}}/{\mathfrak{p}}\\}$
and for each $u$,
$t^{\prime}([1+\varpi_{{\mathfrak{p}}}^{m}u\boldsymbol{\theta}]_{m+1})=\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}&{-\varpi_{{\mathfrak{p}}}^{m}uN(\boldsymbol{\theta})}\\\
{\varpi_{{\mathfrak{p}}}^{m}u}&{1}\end{pmatrix}$.

In the case when ${\mathfrak{p}}$ is split in $K$, we have
$\zeta^{(m+1)}=\begin{pmatrix}{\boldsymbol{\theta}}&{-1}\\\
{1}&{0}\end{pmatrix}\begin{pmatrix}{\varpi^{m+1}_{{\mathfrak{p}}}}&{0}\\\
{0}&{1}\end{pmatrix}$ we compute

(5.15)
$\displaystyle(\zeta^{(m+1)})^{-1}\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}&{-\varpi_{{\mathfrak{p}}}^{m}uN(\boldsymbol{\theta})}\\\
{\varpi_{{\mathfrak{p}}}^{m}u}&{1}\end{pmatrix}\zeta^{(m+1)}$
$\displaystyle=\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta}}&{-\frac{u}{\varpi_{{\mathfrak{p}}}}}\\\
{0}&{1+\varpi^{m}_{{\mathfrak{p}}}u\bar{\boldsymbol{\theta}}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}{1}&{-\frac{u}{\varpi_{{\mathfrak{p}}}(1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta})}}\\\
{0}&{1}\end{pmatrix}\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta}}&{0}\\\
{0}&{1+\varpi^{m}_{{\mathfrak{p}}}u\bar{\boldsymbol{\theta}}}\end{pmatrix}.$

Since
$\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta}}&{0}\\\
{0}&{1+\varpi^{m}_{{\mathfrak{p}}}u\bar{\boldsymbol{\theta}}}\end{pmatrix}\in\mathcal{U}Y$,
we can ignore it.

Note $\zeta^{(m+1)}=\zeta^{(m)}\begin{pmatrix}{\varpi_{{\mathfrak{p}}}}&{0}\\\
{0}&{1}\end{pmatrix}$, and we get

(5.16)
$\displaystyle\zeta^{(m+1)}(\zeta^{(m+1)})^{-1}\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}&{-\varpi_{{\mathfrak{p}}}^{m}uN(\boldsymbol{\theta})}\\\
{\varpi_{{\mathfrak{p}}}^{m}u}&{1}\end{pmatrix}\zeta^{(m+1)}$
$\displaystyle=\zeta^{(m)}\begin{pmatrix}{\varpi_{{\mathfrak{p}}}}&{0}\\\
{0}&{1}\end{pmatrix}\begin{pmatrix}{1}&{-\frac{u}{\varpi_{{\mathfrak{p}}}(1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta})}}\\\
{0}&{1}\end{pmatrix}$
$\displaystyle=\zeta^{(m)}\begin{pmatrix}{\varpi_{{\mathfrak{p}}}}&{-\frac{u}{(1+\varpi^{m}_{{\mathfrak{p}}}u\boldsymbol{\theta})}}\\\
{0}&{1}\end{pmatrix}.$

In the case when ${\mathfrak{p}}$ is inert in $K$, we have
$\zeta^{(m+1)}=\begin{pmatrix}{0}&{1}\\\
{-1}&{0}\end{pmatrix}\begin{pmatrix}{\varpi^{m+1}_{{\mathfrak{p}}}}&{0}\\\
{0}&{1}\end{pmatrix}$, and we do exactly the same computation

(5.17)
$\displaystyle(\zeta^{(m+1)})^{-1}\begin{pmatrix}{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}&{-\varpi_{{\mathfrak{p}}}^{m}uN(\boldsymbol{\theta})}\\\
{\varpi_{{\mathfrak{p}}}^{m}u}&{1}\end{pmatrix}\zeta^{(m+1)}$
$\displaystyle=\begin{pmatrix}{1}&{\frac{u}{-\varpi_{{\mathfrak{p}}}}}\\\
{\varpi^{2m+1}_{{\mathfrak{p}}}uN(\boldsymbol{\theta})}&{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}{1}&{-\frac{u}{\varpi_{{\mathfrak{p}}}}}\\\
{0}&{1}\end{pmatrix}\begin{pmatrix}{1+\varpi^{2m}_{{\mathfrak{p}}}u^{2}N(\boldsymbol{\theta})}&{\varpi^{m-1}_{{\mathfrak{p}}}u^{2}T(\boldsymbol{\theta})}\\\
{\varpi^{2m+1}_{{\mathfrak{p}}}uN(\boldsymbol{\theta})}&{1+\varpi^{m}_{{\mathfrak{p}}}uT(\boldsymbol{\theta})}\end{pmatrix}.$

Again the last matrix is in $\mathcal{U}Y$ and hence can be ignored. The rest
of the computation is the same as in the previous case. ∎

######  Remark.

We remark that the Hecke action on $J^{[\mathfrak{l}]}_{n}$ and
$\mathbf{Z}[X_{B}(\mathcal{U}Y)]$ is with respect to the Picard functoriality.
The additional $\tau_{n}$ is a modification due to the fact that the action of
the Hecke correspondence induces dual actions on the Jacobian under Picard
functoriality and on the space of modular forms.

By the above lemma, we can define the cohomology class
$\kappa_{\mathcal{D}}(\mathfrak{l})=\\{\kappa_{\mathcal{D}}(\mathfrak{l})_{m}\\}_{m}\in\widehat{H}^{1}(K_{\infty},T_{f,n})$.
These elements are in fact in the Selmer group defined in Definition 2.3 as
shown by the following propositon.

######  Proposition 5.9.

The cohomology class $\kappa_{\mathcal{D}}(\mathfrak{l})$ is in the Selmer
group $\widehat{\text{Sel}}_{\Delta\mathfrak{l}}(K_{\infty},T_{f,n}).$

###### Proof.

For each $m\geq n$, we need to show that

  1. (1)

$\partial_{v}(\kappa_{\mathcal{D}}(\mathfrak{l}))=0$ for
$v\nmid\Delta\mathfrak{l}p$;

  2. (2)

${\rm
res}_{v}(\kappa_{\mathcal{D}}(\mathfrak{l}))\in{H^{1}_{\text{ord}}}(K_{\infty},T_{f,n})$
for $v\mid\Delta\mathfrak{l}p$.

For the first point, if $v\nmid\mathfrak{n}^{+}$, the result follows from the
fact that $J^{[\mathfrak{l}]}_{n}$ has good reduction at such $v$. On the
other hand if $v\mid\mathfrak{n}^{+}$, then
$\widehat{H}^{1}(K_{\infty,v},T_{f,n})=\widehat{H}_{{\rm
fin}}^{1}(K_{\infty,v},T_{f,n})$ by Lemma 2.6 and this shows the claim.

For the second point if $v\mid\mathfrak{l}\Delta$, then the result follows
from our previous discussion cf. Section 4.5. Note the Kummer image belongs to
the ordinary part of the cohomology group as the Jacobian has purely toric
reduction at these primes. See (4.17) and the discussion after the diagram.

Finally it remains to prove the case for $v\mid{\mathfrak{p}}$. Let
$T=T_{p}(J^{[\mathfrak{l}]}_{n})_{\mathcal{O}_{f}}$,
$\mathcal{I}=\mathcal{I}^{[\mathfrak{l}]}_{g}$ and
${\mathbb{T}}^{[\mathfrak{l}]}={\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})$,
then $T_{\mathfrak{m}^{[\mathfrak{l}]}}$ is a direct summand of $T$ and
$T_{\mathfrak{m}^{[\mathfrak{l}]}}/\mathcal{I}\simeq T_{f,n}$. Notice that
$V_{\mathfrak{m}^{[\mathfrak{l}]}}=T_{\mathfrak{m}^{[\mathfrak{l}]}}\otimes
E_{f}$ is a direct sum of representations $\rho_{h}\otimes\epsilon$ for $h$ a
$p$-ordinary Hilbert new form of weight $2$ and central character
$\epsilon^{-2}$ such that
$\rho_{h}\otimes\epsilon\simeq\rho^{*}_{f}\,\mbox{mod }{\varpi}$. As before we
have an exact sequence

$0\smash{\mathop{\longrightarrow}}{F^{+}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}}\smash{\mathop{\longrightarrow}}{V_{\mathfrak{m}^{[\mathfrak{l}]}}}\smash{\mathop{\longrightarrow}}{F^{-}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}}\smash{\mathop{\longrightarrow}}0.$

The inertia group at $v$ acts on $F^{+}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}$
by $\epsilon_{{\mathbb{T}}}\epsilon$ and on
$F^{-}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}$ via
$\epsilon_{{\mathbb{T}}}^{-1}$. Here
$\epsilon_{{\mathbb{T}}}:G_{F_{v}}\rightarrow{\mathbb{T}}^{[\mathfrak{l}]}$ is
given by $\epsilon_{{\mathbb{T}}}(\sigma)=\langle\epsilon(\sigma)\rangle$. We
set
$F^{+}_{v}T_{\mathfrak{m}^{[\mathfrak{l}]}}=T_{\mathfrak{m}^{[\mathfrak{l}]}}\cap
F^{+}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}$, then
$F^{+}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}/\mathcal{I}\simeq
F^{+}_{v}T_{f,n}$ as Galois modules. We have the following commutative diagram
cf. [Nek06, Proposition 12.5.8]

(5.18)
$\begin{CD}H^{1}(K_{m,v},T_{\mathfrak{m}^{[\mathfrak{l}]}})@>{\alpha}>{}>H^{1}(K_{m,v},F^{-}_{v}T_{\mathfrak{m}^{[\mathfrak{l}]}})\\\
@V{}V{\beta}V@V{}V{\beta^{\prime}}V\\\
H^{1}(K_{m,v},V_{\mathfrak{m}^{[\mathfrak{l}]}})@>{\alpha^{\prime}}>{}>H^{1}(K_{m,v},F^{-}_{v}V_{\mathfrak{m}^{[\mathfrak{l}]}}).\\\
\end{CD}$

Now let $\text{Kum}(D_{m})_{\mathfrak{m}^{[\mathfrak{l}]}}$ be the image of
$\text{Kum}(D_{m})$ in
$H^{1}(K_{m,{\mathfrak{p}}},T_{\mathfrak{m}^{[\mathfrak{l}]}})$, it suffices
to show that $\alpha(\text{Kum}(D_{m})_{\mathfrak{m}^{[\mathfrak{l}]}})=0$.
This follows from the fact that
$\alpha^{\prime}\circ\beta(\text{Kum}(D_{m})_{\mathfrak{m}^{[\mathfrak{l}]}})=0$
by [BK90, Example 3.11] and that $\beta^{\prime}$ is injective by Lemma 2.9. ∎

.

##  6\. Explicit reciprocity laws

In this chapter we explain the relationship between Heegner point and the
theta element. While the theta element interpolates $L$-values, Heegner points
give rise to elements in the Selmer group. The relationship allows us to bound
the Selmer group in terms of $L$-values. The coexistence of Heegner point and
theta element is a reflection of the parity of the Selmer group and is a
special case of the so called bipartite Euler system [How06].

###  6.1. The first explicit reciprocity law

Let $\mathcal{D}=(\Delta,g)$ be an admissible form. Define

$\Theta_{m}(\mathcal{D})=\frac{1}{\alpha_{{\mathfrak{p}}}^{m}}\sum_{[a]\in
G_{m}}g(x_{m}(a))[a]_{m}\in\mathcal{O}_{f,n}[G_{m}]$

where $[a]_{m}=\mathop{\rm rec}\nolimits_{K}(a)\mid_{G_{m}}\in G_{m}$ is the
map induced by the reciprocity law normalized geometrically. These elements
are norm compatible in the following sense.

######  Lemma 6.1.

If we denote the quotient map by
$\pi_{m+1,m}:G_{m+1}\smash{\mathop{\longrightarrow}}G_{m}$, then

$\pi_{m+1,m}(\Theta_{m+1}(\mathcal{D}))=\Theta_{m}(\mathcal{D}).$

###### Proof.

This follows from the exact same computation in Lemma 5.8 and the fact that
$g$ is a $U_{{\mathfrak{p}}}$-eigenform. ∎

Let $\pi_{m}:G_{m}\rightarrow\Gamma_{m}=\mathop{\rm Gal}\nolimits(K_{m}/K)$ be
the natural map and let

$\theta_{m}(\mathcal{D})=\pi_{m}(\Theta_{m}(\mathcal{D}))\in\mathcal{O}_{f,n}[\Gamma_{m}],$
$\theta_{\infty}(\mathcal{D})=(\theta_{m}(\mathcal{D}))_{m}\in\mathcal{O}_{f,n}[[\Gamma]].$

The following theorem is referred to as the first reciprocity law which was a
terminology used first in [BD05]. For proofs of the first reciprocity in other
setups, see [BD05, Theorem 4.1], [Lon12, Theorem 7.22].

######  Theorem 6.2 (First reciprocity law).

Let $m\geq n\geq 0$ and $\mathfrak{l}$ be an $n$-admissible prime for $f$. For
an $n$-admissible form $\mathcal{D}=(\Delta,g)$, we have

$\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}}(\mathfrak{l})_{m})=\theta_{m}(\mathcal{D})\in\mathcal{O}_{f,n}[\Gamma_{m}]$

and in paticular we have

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

###### Proof.

By the commutativity of the diagram in Theorem 5.7 and the definition of
$\tilde{r}_{\mathfrak{l}}$, we obtain

(6.1)
$\displaystyle\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}}(\mathfrak{l})_{m})=\sum_{\sigma\in\Gamma_{m}}\psi_{g}(r_{\mathfrak{l}}(D^{\sigma}_{m}))[\sigma].$

By the commutativity of the diagram (4.10) and equation (4.21), we have

(6.2) $r_{\mathfrak{l}}(D_{m}^{\sigma})=\sum_{[b]_{m}\in\mathop{\rm
Gal}\nolimits(H_{m}/K_{m})}x_{m}(ub)\tau_{n}$

where $u$ is given by $[u]_{m}=\sigma\in\mathop{\rm Gal}\nolimits(K_{m}/K)$
and $\tau_{n}$ is the Atkin-Lehner involution given in (3.4).

Hence we conclude that

(6.3)
$\displaystyle\partial_{\mathfrak{l}}(\kappa_{\mathcal{D}}(\mathfrak{l})_{m})=$
$\displaystyle\sum_{[u]_{m}\in\mathop{\rm
Gal}\nolimits(K_{m}/K)}\sum_{[b]_{m}\in\mathop{\rm
Gal}\nolimits(H_{m}/K_{m})}\langle
g,x_{m}(ub)\tau_{n}\rangle_{\mathcal{U}Y}[u]_{m}$
$\displaystyle=\sum_{a\in\mathop{\rm
Gal}\nolimits(H_{m}/K)}g(x_{m}(a))\pi_{m}([a]_{m}).$

∎

######  Remark.

In the above proof, especially the last line, we identify the Gross point
$x_{m}(a)$ as an element in $S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})$ in the
following manner. We have a sequence of isomorpshims

$\mathbf{Z}[X_{B}(\mathcal{U}Y)]\simeq\mathop{\rm
Hom}\nolimits_{\mathbf{Z}}(S^{B}_{2}(\mathcal{U}Y,\mathbf{Z}),\mathbf{Z})\simeq
S^{B}_{2}(\mathcal{U}Y,\mathbf{Z})$

where the first isomorphism is given by sending $x\in X_{B}(\mathcal{U}Y)$ to
the evaluation function of $f\rightarrow f(x)$ and the second isomorphism is
given by $\langle,\rangle_{\mathcal{U}Y}$. These isomorphisms are all "Hecke
compatible".

###  6.2. Ihara’s lemma for Shimura curves

Let $\mathcal{U}$ be an open compact subgroup of $\widehat{B}^{\prime\times}$
and $M_{\mathcal{U}}$ be the Shimura curve of level $\mathcal{U}$. For any
ring $A$, let ${\mathcal{F}}_{k}(A)$ be the local system associated to
$L_{k}(A)$ over $M_{\mathcal{U}}$ and we set
${\mathcal{L}}_{k}(\mathcal{U},A)=H_{et}^{1}(M_{\mathcal{U}},{\mathcal{F}}_{k}(A))$.
If $v\nmid\Delta_{B}p\mathfrak{n}^{+}$ is a prime in $F$, we denote by
$\mathcal{U}(v)$ the open compact subgroup of $\widehat{B}^{\prime\times}$
obtained by adding a level at $v$ to $\mathcal{U}$. Let
$\mathbb{F}=\mathcal{O}_{f,1}$ Then the following assumption is usually known
as the "Ihara’s lemma":

######  Assumption 6.3.

If $\mathcal{U}$ is maximal at $v$ and places above $p$ and $2\leq k<p+1$,
then we have an injective map

$1+\eta_{v}^{*}:{\mathcal{L}}_{k}(\mathcal{U},{\mathbb{F}})^{2}_{\mathfrak{m}}\rightarrow{\mathcal{L}}_{k}(\mathcal{U}(v),{\mathbb{F}})_{\mathfrak{m}}.$

Here $\eta_{v}$ is the natural degeneracy map from $M_{\mathcal{U}(v)}$ to
$M_{\mathcal{U}}$ and $\mathfrak{m}$ is a non-Eisenstein maximal ideal in the
spherical Hecke algebra of level $\mathcal{U}(v)$.

######  Remark.

A proof of the Ihara’s lemma is contained in [Che11] following the method of
Diamond-Taylor [DT94]. It was never published and we treat this as an
assumption. Note that recently, Manning and Shotton provide another proof of
the Ihara’s lemma via patching in [MS19] and if we assume that the residual
representation has large image we can apply their results.

###  6.3. The second explicit reciprocity law

Recall we have a homomorphism

$\lambda^{[\mathfrak{l}]}_{g}:{\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}\rightarrow\mathcal{O}_{f,n}$

whose kernel is denoted by $\mathcal{I}^{[\mathfrak{l}]}_{g}$. Let
$\mathfrak{m}$ be the maximal ideal containing
$\mathcal{I}^{[\mathfrak{l}]}_{g}$ in
${\mathbb{T}}_{B}(\mathfrak{l}\mathfrak{n}^{+}Y,{\mathfrak{p}}^{n})_{\mathcal{O}_{f}}$.
Also denote $\widehat{R^{\prime}}_{{\mathfrak{c}}}^{\times}$ the Eichler order
of level ${\mathfrak{c}}$ for some integral ideal ${\mathfrak{c}}$ in $F$.
Note we can not apply the Ihara’s lemma directly as our open compact group
$\mathcal{U}^{\prime}=\mathcal{U}_{\mathfrak{n}^{+},{\mathfrak{p}}^{n}}$ is
not maximal at ${\mathfrak{p}}$. To remedy, we use the following proposition
which again uses Hida theory.

######  Proposition 6.4.

Assume PO and the "Ihara’s lemma", we have

$1+\eta_{v}^{*}:{\mathcal{L}}_{2}(\mathcal{U}^{\prime},\mathbb{F})_{\mathfrak{m}}^{2}[P_{k}]\rightarrow{\mathcal{L}}_{2}(\mathcal{U}^{\prime}(v),\mathbb{F})_{\mathfrak{m}}[P_{k}]$

is injective.

###### Proof.

Using [Hid88, Theorem 8.1, Corollary 8.2], we have the following isomorphisms

${\mathcal{L}}_{2}(\mathcal{U}^{\prime},\mathbb{F})_{\mathfrak{m}}[P_{k}]\simeq{\mathcal{L}}_{k}(\widehat{R^{\prime}}_{\mathfrak{n}^{+}{\mathfrak{p}}}^{\times},\mathbb{F})_{\mathfrak{m}},$
${\mathcal{L}}_{2}(\mathcal{U}^{\prime}(v),\mathbb{F})_{\mathfrak{m}}[P_{k}]\simeq{\mathcal{L}}_{k}(\widehat{R^{\prime}}_{\mathfrak{n}^{+}v{\mathfrak{p}}}^{\times},\mathbb{F})_{\mathfrak{m}}.$

Now let $\mathcal{U}$ be either
$\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}}$ or
$\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}v}$ and consider the injective
map

${\mathcal{L}}_{k}(\mathcal{U},\mathbf{C}_{p})_{\mathfrak{m}}\rightarrow{\mathcal{L}}_{k}(\mathcal{U}({\mathfrak{p}}),\mathbf{C}_{p})_{\mathfrak{m}}.$

By the Eichler-Shimura relation, the cokernel of this map is is two copies of
the ${\mathfrak{p}}$-new part of the ordinary Hilbert modular forms on
$B^{\prime}$ and hence is killed by
$U_{{\mathfrak{p}}}^{2}-{\bf{N}}({\mathfrak{p}})^{k-2}$. Thus the cokernel of
the map

${\mathcal{L}}_{k}(\mathcal{U},\mathcal{O}_{f})_{\mathfrak{m}}\rightarrow{\mathcal{L}}_{k}(\mathcal{U}({\mathfrak{p}}),\mathcal{O}_{f})_{\mathfrak{m}}$

is $\mathcal{O}_{f}$-torsion under the assumption PO. By following the same
argument as in [DT94, Lemma 4],
${\mathcal{L}}_{k}(\mathcal{U}({\mathfrak{p}}),\mathcal{O}_{f})_{\mathfrak{m}}$
is torsion free. Therefore
${\mathcal{L}}_{k}(\mathcal{U},\mathcal{O}_{f})_{\mathfrak{m}}\rightarrow{\mathcal{L}}_{k}(\mathcal{U}({\mathfrak{p}}),\mathcal{O}_{f})_{\mathfrak{m}}$
is an isomorphism and so is its reduction modulo $\varpi$. Thus we have proved
the following isomorphisms:

${\mathcal{L}}_{k}(\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}{\mathfrak{p}}},\mathbb{F})_{\mathfrak{m}}\simeq{\mathcal{L}}_{k}(\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}},\mathbb{F})_{\mathfrak{m}},$
${\mathcal{L}}_{k}(\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}v{\mathfrak{p}}},\mathbb{F})_{\mathfrak{m}}\simeq{\mathcal{L}}_{k}(\widehat{R^{\prime}}^{\times}_{\mathfrak{n}^{+}v},\mathbb{F})_{\mathfrak{m}}$

under the PO assumption. Therefore the conclusion follows from "Ihara’s
lemma". ∎

Let $\mathfrak{l}_{1},\mathfrak{l}_{2}$ be two $n$-admissible primes such that
$\mathfrak{l}_{1}\mathfrak{l}_{2}\nmid\Delta_{B}$. We denote the underlying
rational prime in $\mathbf{Q}$ of $\mathfrak{l}_{2}$ by $l_{2}$. Let
${\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}$ be the integral model of
$M^{[\mathfrak{l}_{1}]}_{n}$ over $\mathcal{O}_{F,\mathfrak{l}_{2}}$ studied
by Carayol [Car83] and also in Zhang [Zha01]. We review the definition of this
model in preparation for the next theorem. Let $F^{\prime}$ be an auxiliary CM
field over $F$ such that $\mathfrak{l}_{2}$ is split in $F^{\prime}$. We set
$D=B^{\prime}\otimes F^{\prime}$ and denote
$\upsilon(\mathcal{U}^{\prime})={N}_{D}(U^{\prime}\widehat{F^{\prime}}^{\times})\cap\mathcal{O}^{\times}_{F}$.
We fix an embedding $\iota_{\mathfrak{l}_{2}}:F^{\prime}\hookrightarrow
F_{\mathfrak{l}_{2}}$. For an $\mathcal{O}_{D,l_{2}}$ module $\Lambda$, there
is an natural decomposition
$\Lambda=(\oplus_{i=1}^{r}\Lambda_{1}^{i})\oplus(\oplus_{i=1}^{r}\Lambda_{2}^{i})$
compatible with respect to the natural decomposition
$\mathcal{O}_{D,l_{2}}=(\oplus_{i=1}^{r}\mathcal{O}_{D,1}^{i})\oplus(\oplus_{i=1}^{r}\mathcal{O}_{D,2}^{i})$
where we make a choice such that $\mathcal{O}_{D,\mathfrak{l}_{2}}$
corresponds to $\mathcal{O}_{D,2}^{1}$. Note that as $D$ splits at
$\mathfrak{l}_{2}$, we can set $\Lambda^{1,1}_{2}=\begin{pmatrix}{1}&{0}\\\
{0}&{0}\end{pmatrix}\Lambda^{1}_{2}$. Let $\widehat{V}$ be the profinite
completion of an $\mathcal{O}_{D}$-module $V$ and $\widehat{V}^{(l_{2})}$ be
the subset of $\widehat{V}$ with trivial $l_{2}$ component. We further denote
by $V^{(\mathfrak{l}_{2})}_{2}=\oplus^{r}_{i=2}V_{l_{2},2}^{i}$. These
notations in particular applies to the case when $V=\mathcal{O}_{D}$ viewed as
an $\mathcal{O}_{D}$-module via the natural left multiplication,
$V=H_{1}(A,{\mathbf{Z}})$ or $\mathop{\rm Lie}\nolimits(A)$ for an abelian
variety $A$ with $\mathcal{O}_{D}$ action. To define the integral model
${\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}$, one first constructs an auxiliary
integral model ${\tilde{\mathcal{M}}}^{[\mathfrak{l}_{1}]}_{n}$ for another
Shimura curve ${\tilde{M}^{[\mathfrak{l}_{1}]}}_{n}$ which admits a finite map
$f:M^{[\mathfrak{l}_{1}]}_{n}\rightarrow\tilde{M}^{[\mathfrak{l}_{1}]}_{n}$.
Then one takes ${\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}$ to be the
normalization of $\tilde{\mathcal{M}}^{[\mathfrak{l}]}_{n}$.

Let $x\in{\tilde{\mathcal{M}}}^{[\mathfrak{l}_{1}]}_{n}(S)$ for a scheme $S$
over $\mathcal{O}_{F,\mathfrak{l}_{2}}$. Then
$x=(A/S,\iota,\bar{\theta},\bar{\kappa}^{[\mathfrak{l}_{2}]})$ where

  * •

$A/S$ is an abelian scheme of dimension $4d$ with an action
$\iota:\mathcal{O}_{D}\rightarrow\mathop{\rm End}\nolimits{A}$ satisfying
certain conditions on $\mathop{\rm Lie}\nolimits(A)$: $\mathop{\rm
Lie}\nolimits(A)^{1}_{2}$ is a locally free $\mathcal{O}_{S}$-module of rank
$2$ and $\mathop{\rm Lie}\nolimits(A)^{\mathfrak{l}_{2}}_{l_{2}}=0$ cf.
[Zha01, Proposition 1,22];

  * •

$\bar{\theta}:A\rightarrow{A}^{\\!\vee}$ is an
$\upsilon(\mathcal{U}^{\prime})$-isomorphism class of polarizations of $A$
with prime to $l_{2}$ degree;

  * •

$\bar{\kappa}^{[\mathfrak{l}_{2}]}:{T}^{(\mathfrak{l}_{2})}_{l_{2}}(A)\oplus\widehat{T}^{(l_{2})}(A)\rightarrow
W^{(\mathfrak{l}_{2})}_{l_{2}}\oplus\widehat{W}^{(l_{2})}$ is an isomorphism
up to the action of $\mathcal{U}^{\prime}Y$.

Let $A$ be the universal abelian scheme over
${\tilde{\mathcal{M}}}^{[\mathfrak{l}_{1}]}_{n}$ and $A_{l_{2}^{\infty}}$ be
the $l_{2}^{\infty}$-divisible group of $A$, then
$(A_{l_{2}^{\infty}})^{1,1}_{2}$ is a divisible
$\mathcal{O}_{F,{\mathfrak{l}_{2}}}$-module of height $2$ and dimension $1$.
We denote by $\mathbf{E}_{\infty}$ the pullback of
$(A_{l_{2}^{\infty}})^{1,1}_{2}$ to $\mathcal{M}^{[\mathfrak{l}_{1}]}_{n}$.
Let $x$ be a geometric point of ${\mathcal{M}}^{[\mathfrak{l}_{2}]}_{n}$, then
the fiber $\mathbf{E}_{\infty,x}$ is either of the form
$F_{\mathfrak{l}_{2}}/\mathcal{O}_{F,\mathfrak{l}_{2}}\times\Sigma_{1}$ or
$\Sigma_{2}$ where $\Sigma_{1}$ is the formal
$\mathcal{O}_{F,\mathfrak{l}_{2}}$ module of height $1$ and $\Sigma_{2}$ is
the formal $\mathcal{O}_{F,\mathfrak{l}_{2}}$ module of height $2$. If the
first case happens, then we say $x$ is ordinary and if the second case happens
then we say $x$ is supersingular. See [Car83, Section 0.8].

We fix an isomorphism
$\varphi_{B^{\prime\prime},B^{\prime}}:{\widehat{B}^{\prime\prime(\mathfrak{l}_{2})\times}}\rightarrow{\widehat{B}^{\prime(\mathfrak{l}_{2})\times}}$
and put
$\mathcal{U}^{\prime\prime}=\varphi^{-1}_{B^{\prime\prime},B^{\prime}}(\mathcal{U}^{\prime(\mathfrak{l}_{2})})\mathcal{O}^{\times}_{B^{\prime\prime},\mathfrak{l}_{2}}$.
By [Car83, Section 11.2], we have an identification

$\gamma:X_{B^{\prime\prime}}(\mathcal{U}^{\prime\prime}Y)\smash{\mathop{\longrightarrow}}{\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}(\mathbb{F}_{\mathfrak{l}^{2}_{2}})^{ss}$

where the righthand side is the supersingular locus of the special fiber of
${\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}$ and
$\mathbb{F}_{\mathfrak{l}^{2}_{2}}$ is the residue field of
$\mathcal{O}_{K,\mathfrak{l}_{2}}$.

######  Lemma 6.5.

We have
$\text{red}_{\mathfrak{l}_{2}}(P_{m}(a))\in{\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}(\mathbb{F}_{\mathfrak{l}^{2}_{2}})^{ss}$
and moreover
$\text{red}_{\mathfrak{l}_{2}}(P_{m}(a))=\gamma(x_{m}(a)\tau_{n})$.

###### Proof.

The first part is proved by observing that for a geometric point $x$
underlying $\text{red}_{\mathfrak{l}_{2}}(P_{m}(a))$ the formal
$\mathcal{O}_{F,\mathfrak{l}_{2}}$ module $\mathbf{E}_{\infty,x}$ admitts an
$F_{\mathfrak{l}_{2}}$ linear action of $K_{\mathfrak{l}_{2}}$. Since
$\mathfrak{l}_{2}$ is inert, it follows that the connected part
$\mathbf{E}^{0}_{\infty,x}$ of $\mathbf{E}_{\infty,x}$ has to be $\Sigma_{2}$.

By [Car83, Section 11.2], the action of $\widehat{B}^{\prime\prime\times}$ is
transitive on
${\mathcal{M}}^{[\mathfrak{l}_{1}]}_{n}(\mathbb{F}_{\mathfrak{l}^{2}_{2}})^{ss}$
and so $\gamma$ can be chosen such that