# Bianchi modular symbols and $p$-adic $L$-functions

JAESUNG KWON  Ulsan National Institute of Science and Technology
Ulsan
Korea<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we construct the integral special $L$-value and integral
valued $p$-adic $L$-function of $p$-ordinary Bianchi Hecke eigenforms. We
obtain the functional equation of the $L$-function of Bianchi modular forms.
By using the functional equation, we compute the approximate functional
equation, which is a fast convergent series of $L$-values. By studying the
fast convergent series of special $L$-values, we prove that the first homology
group of the Bianchi modular manifold is generated by the special Bianchi
modular symbols. As a corollary of the generation result, we have that the
$\mu$-invariant of some isotopic components of the $p$-adic $L$-function of
certain Bianchi Hecke eigenforms vanishes for a positive proportion of
ordinary prime ideals. By studying a $\ell$-adic representation attached to a
Bianchi Hecke eigenform, we show that the density of the set of ordinary prime
ideals is one. Also we obtain the result of residual non-vanishing of the
integral $L$-values.

###### Key words and phrases:

Bianchi modular forms; Bianchi modular symbols; special $L$-values

###### 2020 Mathematics Subject Classification:

11F67, 11F12

###### Contents

  1. 1 Introduction
  2. 2 Cusp forms on GL(2)
  3. 3 Approximate functional equation
  4. 4 Modular symbols and Eichler-Shimura isomorphism
  5. 5 Lattice estimation
  6. 6 Generation of homology groups
  7. 7 Cohomology classes and special $L$-values
  8. 8 $p$-adic $L$-functions and $\mu$-invariants
  9. 9 Density of ordinary primes

##  1\. Introduction

The vanishing of the $\mu$-invariant of the $p$-adic $L$-function of modular
form is a longstanding conjecture in Iwasawa theory, which was introduced by
Greenberg [4]. Kim-Sun [7] gave a proof of the result toward the Greenberg’s
conjecture for elliptic modular forms by studying the distribution of special
modular symbols in the first homology group of the modular curve.

In the present paper, we define an integral valued $p$-adic $L$-function of
Bianchi modular form, which interpolates the integral $L$-values. Also we show
that the $\mu$-invariant of some isotopic components of the $p$-adic
$L$-function of Bianchi modular form is vanishing, which is a generalization
of Kim-Sun [7].

###  1.1. Generation of homology groups by special modular symbols

Let $F$ be an imaginary quadratic field, $h_{F}$ the class number of $F$,
$\\{\mathfrak{a}_{i}\\}_{i=1}^{h_{F}}$ a representative set of the class group
of $F$ such that $\mathfrak{a}_{i}^{-1}$ is integral for each $1\leq i\leq
h_{F}$, $a_{i}\in\mathbb{A}_{F}^{\times}$ an idelic representative of
$\mathfrak{a}_{i}$, and $\mathfrak{N}$ an integral ideal of $F$ such that
$[\mathbb{Z}:\mathfrak{N}\cap\mathbb{Z}]>3$. Let $\mathscr{H}_{3}$ be the
hyperbolic upper half space of real dimension three, $U_{1}(\mathfrak{N})$ the
subgroup of $\operatorname{GL}_{2}(\widehat{\mathcal{O}}_{F})$ defined in the
subsection 1.8, and $\overline{\Gamma_{1}^{i}(\mathfrak{N})}$ the congruence
subgroup of $\operatorname{SL}_{2}(\mathcal{O}_{F})$ defined in the section 4.
Let us define the Bianchi modular manifold $Y_{1}(\mathfrak{N})$ of level
$U_{1}(\mathfrak{N})$ by

$Y_{1}(\mathfrak{N}):=\operatorname{GL}_{2}(F)\backslash\operatorname{GL}_{2}(\mathbb{A}_{F})/\mathbb{C}^{\times}\operatorname{U}_{2}(\mathbb{C})U_{1}(\mathfrak{N})\cong\coprod_{i=1}^{h_{F}}\overline{\Gamma_{1}^{i}(\mathfrak{N})}\backslash\mathscr{H}_{3}=:\coprod_{i}^{h_{F}}Y_{1}^{i}(\mathfrak{N}).$

Let us denote by $X_{1}(\mathfrak{N})$ the Satake compatification of
$Y_{1}(\mathfrak{N})$. For $\alpha,\beta\in\mathbb{P}^{1}(F)$, let us denote
by $\\{\alpha,\beta\\}_{\overline{\Gamma_{1}^{i}(\mathfrak{N})}}$ the image of
the geodesic from $\alpha$ to $\beta$ in $X_{1}^{i}(\mathfrak{N})$. Then we
can consider $\\{\alpha,\beta\\}_{\overline{\Gamma_{1}^{i}(\mathfrak{N})}}$ to
be an element of
$H_{1}(X_{1}^{i}(\mathfrak{N}),\overline{\Gamma_{1}^{i}(\mathfrak{N})}\backslash\mathbb{P}^{1}(F),\mathbb{Z})$.
Then the classes
$\\{\alpha,\beta\\}_{\overline{\Gamma_{1}^{i}(\mathfrak{N})}}$ are called
modular symbols of level $\overline{\Gamma_{1}^{i}(\mathfrak{N})}$ and weight
$(2,2)$.

Let $\mathfrak{l}$ be a prime ideal of $F$ with sufficiently large norm
$N(\mathfrak{l})$, $\mathcal{O}_{F,\mathfrak{l}}$ the $\mathfrak{l}$-adic
completion of the integer ring $\mathcal{O}_{F}$ of $F$, and
$\varpi_{\mathfrak{l}}$ a uniformizer of $\mathcal{O}_{F,\mathfrak{l}}$. In
Section 4, we construct the modular symbol
$\xi_{n,\mathfrak{l}}^{(i)}(a)=\\{\infty,r_{n,\mathfrak{l}}(a)\\}_{\overline{\Gamma_{1}^{i}(\mathfrak{N})}}$
in $X_{1}^{i}(\mathfrak{N})$ corresponds to the finite adele
$a\varpi_{\mathfrak{l}}^{-n}\in\mathbb{A}_{F}$. We consider the submodule
$M_{n,\mathfrak{l}}$ of $H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$, which is
defined by

$M_{n,\mathfrak{l}}:=\bigoplus_{i=1}^{h_{F}}\langle\xi_{n,\mathfrak{l}}^{(i)}(a):a\in\mathcal{O}_{F,\mathfrak{l}}^{\times}\rangle\cap
H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z}).$

We show that $M_{n,\mathfrak{l}}$ is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ for almost all prime ideals
$\mathfrak{l}$:

######  Theorem A (Theorem 6.5 and 6.6).

Let $\mathfrak{l}$ be a prime ideal and $n\geq 22$ an integer. Then we have
the following:

$M_{\mathfrak{l},n}$ is of full rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ for sufficiently large
$N(\mathfrak{l})$.

There is an arithmetic progression $\mathfrak{X}_{n}$ of ideals such that
$M_{n,\mathfrak{p}}\otimes\mathbb{F}_{p}=H_{1}(X_{1}(\mathfrak{N}),\overline{\mathbb{F}}_{q})$
for prime ideals $\mathfrak{p}\in\mathfrak{X}_{n}$ and sufficiently large
prime numbers $q$.

###  1.2. Integral $L$-values and residual non-vanishing

Let $p$ be an odd prime number coprime to
$h_{F}D_{F}|\mathcal{O}_{F}^{\times}|\mathfrak{N}$ such that $F$ does not
contain any primitive $p$-th power roots. Let $f$ be a normalized Bianchi
Hecke eigenform of weight $(2,2)$ and level $U_{1}(\mathfrak{N})$. In Section
7, we define the compact support cohomology class
$\eta_{f,\operatorname{c}}\in
H^{1}_{\operatorname{c}}(Y_{1}(\mathfrak{N}),\overline{\mathbb{Z}}_{p})$
attached to $f$. Let us denote by $\eta_{f,\operatorname{c},i}$ the $i$-th
part of $\eta_{f,\operatorname{c}}$ with respect to the map
$X_{1}^{i}(\mathfrak{N})\hookrightarrow X_{1}(\mathfrak{N})$.

For a finite order Hecke character
$\phi:F^{\times}\backslash\mathbb{A}_{F}^{\times}\rightarrow\mathbb{C}^{\times}$
of prime power conductor $\mathfrak{l}^{n}$, let us define the twisted modular
symbol by

$\Lambda^{(i)}(\phi):=\sum_{r\text{ mod
}\mathfrak{l}^{n}}\phi(a_{i})\phi_{\mathfrak{l}}(r)\xi_{n,\mathfrak{l}}^{(i)}(r)\in
H_{1}(X_{1}^{i}(\mathfrak{N}),\overline{\Gamma_{1}^{i}(\mathfrak{N})}\backslash\mathbb{P}^{1}(F),\mathbb{Z}[\phi]).$

Then we define the integral $L$-value by

$\mathcal{L}_{f}(\phi):=\sum_{i=1}^{h_{F}}\Lambda^{(i)}(\phi)\cap\eta_{f,\operatorname{c},i}\in\overline{\mathbb{Z}}_{p},$

where
$\cap:H_{1}(X_{1}(\mathfrak{N}),\bigcup_{i}\overline{\Gamma_{1}^{i}(\mathfrak{N})}\backslash\mathbb{P}^{1}(F),\overline{\mathbb{Z}}_{p})\times
H^{1}_{\rm{c}}(Y_{1}(\mathfrak{N}),\overline{\mathbb{Z}}_{p})\rightarrow\overline{\mathbb{Z}}_{p}$
is the Lefschetz-Poincaré duality pairing. Also we define the algebraic
$L$-value by

$L_{f}(\phi):=\frac{|D_{F}||\mathcal{O}_{F}^{\times}|G(\phi)L(1,f\otimes\phi)}{8\pi^{2}\Omega_{f}}\in\overline{\mathbb{Q}}.$

Let us define the non-Eisenstein condition introduced in Namikawa [14]:

$(\mathbf{Non\text{-}Eis})\
a_{f}(q\mathcal{O}_{F})-N(q)-1\in\overline{\mathbb{Z}}_{p}^{\times}\text{ for
some prime element }q\equiv 1\ (\text{mod }p\mathfrak{N}).$

Note that this condition is related to the existence of the irreducible
residual Galois representation attached to $f$ mod $p$ (see Scholze [17]).
Also note that in Section 8, if (Non-Eis) holds, then we can see that the
ratio of the integral $L$-value $\mathcal{L}_{f}(\phi)$ and the algebraic
$L$-value $L_{f}(\phi)$ is a $p$-adic unit.

Under the assumption (Non-Eis), Namikawa [14] shows that
$\mathcal{L}_{f}(\phi)$ is a $p$-adic unit for infinitely many Hecke
characters when $h_{F}=1$. By using this result, we obtain the following
result toward the generalization of Stevens’ result [19]:

######  Theorem B (Theorem 7.5).

Assume $(\mathbf{Non\text{-}Eis})$, then $\mathcal{L}_{f}(\phi)$ is a $p$-adic
unit for some finite order Hecke characters $\phi$.

###  1.3. Vanishing of $\mu$-invariant

To discuss Greenberg’s conjecture, we need to define $p$-adic $L$-function for
Bianchi modular forms and its $\mu$-invariant. Namikawa [15] constructed a
two-variable $p$-adic $L$-function for nearly $p$-ordinary cohomological
automorphic forms of $\operatorname{GL}_{2}$ over general number fields, and
proved the non-vanishing of $L$-values for certain weights. Williams [23]
defined a $p$-adic $L$-function of Bianchi modular form by using the
overconvergent modular symbols, and proved the control theorem on the space of
modular symbols.

Let $p$ be an odd prime coprime to
$h_{F}|\mathcal{O}_{F}^{\times}|\mathfrak{N}\mathfrak{a}_{i}^{-1}$ for each
$i$, and $\mathfrak{p}$ a prime ideal of $F$ lying over $p$. Let $\phi$ be a
finite Hecke character of $\mathfrak{p}$-power conductor. Let $f$ be a
normalized $\mathfrak{p}$-ordinary Bianchi Hecke eigenform of weight $(2,2)$
and level $U_{1}(\mathfrak{N})$. Let $\phi$ be a finite order Hecke character
over $F$ of prime power conductor $\mathfrak{p}^{n}$. In Section 8, we
construct an integral $p$-adic $L$-function $L_{\mathfrak{p}}(s,f,\phi)$ by
taking the $p$-adic Mazur-Mellin transformation on the $p$-adic measure
$\nu_{f}$, which is defined by the modular symbols
$\xi^{(i)}_{n,\mathfrak{p}}$ and the cohomology classes $\eta_{f,c,i}$. Then
our $p$-adic $L$-function $L_{\mathfrak{p}}(s,f,\phi)$ interpolates the
integral $L$-value $\mathcal{L}_{f}(\phi)$ (see Proposition 8.2). We suggest
the Bianchi modular version of Greenberg’s conjecture:

######  Conjecture A (Conjecture 8.3).

Assume $(\mathbf{Non\text{-}Eis})$, then
$\mu\big{(}L_{\mathfrak{p}}(s,f,\phi)\big{)}=0$.

Let
$\omega:(\mathcal{O}_{F}/\mathfrak{p})^{\times}\rightarrow\mathcal{O}_{F,\mathfrak{p}}^{\times}$
be the Teichmüller character. By using Theorem A, we obtain the result toward
Conjecture A:

######  Theorem C (Theorem 9.3).

Let $f$ be a normalized Bianchi modular Hecke eigenform of parallel weight
$(2,2)$ with trivial central character and rational Hecke eigenvalues. Assume
that $f$ is neither CM nor the lifting of a classical elliptic modular
newform. Suppose $(\mathbf{Non\text{-}Eis})$, then for a positive proportion
of ordinary prime ideals $\mathfrak{p}$, we have
$\mu\big{(}L_{\mathfrak{p}}(s,f\otimes\psi,\phi\omega^{j})\big{)}=0$ for some
Hilbert characters $\psi$ of $F$ and integers $0\leq j<p-1$.

###  1.4. Skectch of the proof of Theorem A

Let $\mathfrak{l}$ be a prime ideal of $F$ whose norm $\ell$ is coprime to
$\mathfrak{a}_{i}^{-1}\mathfrak{N}$ for each $i$. For an element
$b\in\mathcal{O}_{F,\mathfrak{l}}$ and an open set
$U=a_{0}+\mathfrak{l}^{v}\mathcal{O}_{F,\mathfrak{l}}$, let us introduce the
special modular symbol:

$\Upsilon_{n,\mathfrak{l}}^{(i)}(U,b):=\frac{1}{\ell^{n-v}}\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\mathbf{e}_{F}\Big{(}\frac{-ab}{\varpi_{\mathfrak{l}}^{n}}\Big{)}\cdot\xi^{(i)}_{n,\mathfrak{l}}(a).$

To prove Theorem A, we have to estimate the integral of a Bianchi modular form
along the special modular symbol. By using the arguments of Kim-Sun [7], Kwon-
Sun [9] and Luo-Ramakrishnan [10], the estimation of the integral is given as
follows:

(1.1)
$\langle\Upsilon_{n,\mathfrak{l}}^{(i)}(U,\beta),f\rangle=\frac{a_{f}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}+o(1)\text{
for }\beta\in\mathfrak{a}_{i}^{-1}\text{ and }n,\ell\gg 0,$

where $\langle\cdot,\cdot\rangle$ is the pairing induced by the Lefschetz-
Poincaré duality pairing $\cap$. Note that as $\langle\cdot,\cdot\rangle$ is
non-degenerate, we can change the full-rankness problem into the non-vanishing
one. By using the equation (1.1), we verify the non-vanishing of the pairing.
From this, we deduce Theorem A.

###  1.5. Lattice estimation

To obtain the equation (1.1), we have to compute the estimation on the number
of points in certain lattice in bounded region. To do this, we have to study
the properties of the norm map
$N_{F/\mathbb{Q}}:F^{\times}\rightarrow\mathbb{Q}^{\times}$. If $F$ is
imaginary quadratic, then the norm map is given by the square of the absolute
value, which is easy. But the difficulty arises for the case of general number
fields. For example, if $F=\mathbb{Q}(\sqrt{D})$ is real quadratic, then the
norm form $(a,b)\mapsto N(a+b\sqrt{D})a=a^{2}-b^{2}D$ is not positive
definite.

###  1.6. $\ell$-adic Galois representations and ordinary primes

In Theorem A, we consider the ordinary primes in some arithmetic progression.
Therefore, we have to prove that the density of the ordinary primes is
positive. If a normalized Bianchi Hecke eigenform $f$ of weight $(2,2)$ with
trivial central character is neither CM nor the lifting of a classical
elliptic modular form, then there is a $\ell$-adic Galois representation
$\rho_{f,\ell}$ such that the image of $\rho_{f,\ell}$ is dense in
$\operatorname{GL}_{2}(\mathbb{Q}_{\ell})$ and
$\text{Tr}\big{(}\rho_{f,\ell}(\text{Frob}_{v})\big{)}$ is the Hecke
eigenvalue $a_{f}(v)$ of $T_{v}$ for a set of places $v$ of $F$ of density one
(see Taylor [21]). If the coefficient field of $f$ is rational, then we can
prove the existence of a set of ordinary primes of density one by using the
argument by Serre [18].

###  1.7. Technical Difficulties

To deal with the general class number, we use the adelic language. In Kwon-Sun
[9], we can see the similar techniques.

Consider the module

$M_{n,\mathfrak{l},0}:=\bigg{\langle}\sum_{i=1}^{h_{F}}\xi_{n,\mathfrak{l}}^{(i)}(a):a\in\mathcal{O}_{F,\mathfrak{l}}^{\times}\bigg{\rangle}\cap
H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z}),$

which is a submodule of $M_{n,\mathfrak{l}}$. If we prove that
$M_{n,\mathfrak{l},0}$ is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$, then we can remove the twisting
$\otimes\psi$ in the statement of Theorem C. But the non-trivial class group
is an obstruction for proving the full-rankness of $M_{n,\mathfrak{l},0}$.
Thus we consider a bigger module $M_{n,\mathfrak{l}}$ and the twisting
$\otimes\psi$.

###  1.8. Notations

Let us provide some notations which is used globally in this paper.

  * •

Let $F$ be an imaginary quadratic field, $\mathcal{O}_{F}$ the integer ring of
$F$, $\mathfrak{d}_{F}$ the different ideal of $F$, $d_{F}$ a finite idele of
$\mathfrak{d}_{F}$, $D_{F}$ the discriminant of $F$, and $N$ the norm map of
$F/\mathbb{Q}$.

  * •

For a $\mathbb{Z}$-algebra $A$, let us set
$\widehat{A}:=A\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}$.

  * •

Let $\mathbb{A}_{F}$ be the adele ring of $F$, $\mathbb{A}_{F}^{(\infty)}$ the
ring of finite adeles of $F$, and
$|\cdot|_{\mathbb{A}_{F}}=|\cdot|_{\infty}\times|\cdot|_{\mathbb{A}_{F}^{(\infty)}}$
the idelic norm.

  * •

For a place $v$ of $F$, let us denote by $\mathcal{O}_{F,v}$ the $v$-adic
completion of $\mathcal{O}_{F}$.

  * •

Let us choose a uniformizer $\varpi_{v}$ of $\mathcal{O}_{F,v}$. Let us embed
$\varpi_{v}$ into $\mathbb{A}_{F}^{\times}$ by
$\varpi_{v}\mapsto(1,\cdots,1,\varpi_{v},1,\cdots,1)$. For an ideal
$\mathfrak{a}$ of $F$, define
$\varpi_{\mathfrak{a}}:=\prod_{v|\mathfrak{a}}\varpi_{v}^{\text{ord}_{v}(\mathfrak{a})}$.

  * •

Let $\mathfrak{N}$ be a non-zero integral ideal of $F$ such that
$[\mathbb{Z}:\mathfrak{N}\cap\mathbb{Z}]>3$. Define subgroups
$U_{0}(\mathfrak{N})$ and $U_{1}(\mathfrak{N})$ of
$\operatorname{GL}_{2}(\widehat{\mathcal{O}}_{F})$ by

$\displaystyle U_{0}(\mathfrak{N})$
$\displaystyle:=\bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{GL}_{2}(\widehat{\mathcal{O}}_{F}):c\in\widehat{\mathfrak{N}}\bigg{\\}},$
$\displaystyle U_{1}(\mathfrak{N})$
$\displaystyle:=\bigg{\\{}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
U_{0}(\mathfrak{N}):d\in 1+\widehat{\mathfrak{N}}\bigg{\\}}.$

  * •

Let Cl$(F)$ be the class group of $F$, $h_{F}$ the class number of $F$, and
$H_{F}$ the group of Hilbert characters of $F$. Note that Cl$(F)\cong
F^{\times}\backslash\mathbb{A}_{F}^{\times}/\mathbb{C}^{\times}\widehat{\mathcal{O}}_{F}^{\times}$.

  * •

Without loss of generality, we can choose a representative set
$\\{\mathfrak{a}_{i}\\}_{i=1}^{h_{F}}$ of $\text{Cl}(F)$ such that
$\mathfrak{a}_{1}=\mathcal{O}_{F}$, $\mathfrak{a}_{i}^{-1}$ is integral and
coprime to $\mathfrak{N}$ for each $i$. Let us denote
$a_{i}:=\varpi_{\mathfrak{a}_{i}}$.

  * •

Let $\mathbf{e}_{F}:\mathbb{A}_{F}/F\rightarrow\mathbb{C}^{\times}$ be an
additive character defined by

$\mathbf{e}_{F}(x_{\infty}x^{(\infty)}):=\mathbf{e}\big{(}\operatorname{Tr}_{\mathbb{C}/\mathbb{R}}(x_{\infty})\big{)}\cdot\prod_{p}\prod_{\mathfrak{p}|p}\mathbf{e}_{p}\big{(}\operatorname{Tr}_{F_{\mathfrak{p}}/\mathbb{Q}_{p}}(x^{(\infty)})\big{)},$

where $\mathbf{e}(z):=\operatorname{exp}\big{(}{2\pi\sqrt{-1}z}\big{)}$ and
$\mathbf{e}_{p}\big{(}\sum_{j}c_{j}p^{j}\big{)}:=\mathbf{e}\big{(}\sum_{j<0}c_{j}p^{j}\big{)}.$

  * •

We fix an embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{\ell}$ for each
rational primes $\ell$.

##  2\. Cusp forms on GL(2)

In this section, we give the definition of _cuspidal automorphic form_ on
$\operatorname{GL}(2)$ over $F$, which is also called _Bianchi cusp form_.
Note that by Hida [5, Corollary 2.2], all the cusp forms of non-parallel
weights are trivial, thus we only consider the parallel weight.

Let $I_{F}:=\\{\operatorname{id},\operatorname{c}\\}$ be the set of embeddings
of $F$, and $\mathbb{Z}[I_{F}]$ the free $\mathbb{Z}$-module generated by
$I_{F}$. Let us denote
$(k_{\operatorname{id}},k_{\operatorname{c}}):=\sum_{\sigma\in
I_{F}}k_{\sigma}\cdot\sigma\in\mathbb{Z}[I_{F}]$ for $k\geq 2$. Let us denote
$\mathbf{x}:=\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)$, where
$X$ and $Y$ are indeterminates. Let us denote by $L_{d}(R)$ the space of
homogeneous polynomials in two variables $X,Y$ of degree $d\geq 0$ with
coefficients $R$. Note that there is a usual action of GL(2) on $L_{d}(R)$.

######  Definition 2.1 (Hida [5, Section 3]).

Let
$\chi:F^{\times}\backslash\mathbb{A}_{F}^{\times}\rightarrow\mathbb{C}^{\times}$
be a Hecke character of modulus $\mathfrak{N}$ such that
$\chi(z_{\infty})=|z_{\infty}|^{2(2-k)}$ for
$z_{\infty}\in\mathbb{C}^{\times}$. A _(Bianchi) cusp form_ on
$\operatorname{GL}_{2}(\mathbb{A}_{F})$ of weight $(k,k)$ and level
$\mathfrak{N}$ with a central character $\chi$ is a $C^{\infty}$-function
$f:\operatorname{GL}_{2}(\mathbb{A}_{F})\rightarrow L_{2k-2}(\mathbb{C})$ such
that

  1. (1)

$D_{\sigma}f=\big{(}\frac{k^{2}-2k}{2}\big{)}f$ for each $\sigma\in I_{F}$,
where $D_{\sigma}$ is the Casimir operator for $\operatorname{SL}(2)$
corresponding to $\sigma$.

  2. (2)

$f(\gamma
zgh)(\mathbf{x})=\chi(z)\chi_{\mathfrak{N}}(h^{(\infty)})f(g)\big{(}h_{\infty}\mathbf{x}\big{)}$
for $\gamma\in\operatorname{GL}_{2}(F)$, $z\in\mathbb{A}_{F}^{\times}$,
$g\in\operatorname{GL}_{2}(\mathbb{A}_{F})$ and
$h=h_{\infty}h^{(\infty)}\in\operatorname{SU}_{2}(\mathbb{C})U_{0}(\mathfrak{N})$,
where $\chi_{\mathfrak{N}}\big{(}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\big{)}:=\prod_{v|\mathfrak{N}}\chi(d_{v})$.

  3. (3)

$\int_{F\backslash\mathbb{A}_{F}}f\big{(}\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)g\big{)}(\textbf{x})dx=0$ for
$g\in\operatorname{GL}_{2}(\mathbb{A}_{F})$, where $dx$ is a Haar measure on
$F\backslash\mathbb{A}_{F}$.

From now on, we denote by $S_{k}(\mathfrak{N},\chi)$ the space of the
aforementioned cusp forms.

Let us denote by $K_{j}$ the $j$-th modified Bessel function of second kind.
Let $W_{k}:\mathbb{C}^{\times}\rightarrow L_{2k-2}(\mathbb{C})$ be the
_Whittaker function_ , which is defined by

$W_{k}(y_{\infty})(\textbf{x}):=\sum_{m=0}^{2k-2}\left(\begin{smallmatrix}2k-2\\\
m\end{smallmatrix}\right)\bigg{(}\frac{y_{\infty}}{\sqrt{-1}|y_{\infty}|}\bigg{)}^{k-1-m}K_{m+1-k}(4\pi|y_{\infty}|)X^{2k-2-m}Y^{m}.$

Then we have the _Fourier-Whittaker expansion_ of cusp forms:

######  Proposition 2.2 (Hida [5, Theorem 6.1]).

Let $\mathscr{F}$ be the set of fractional ideals of $F$. For each $f\in
S_{k}(\mathfrak{N},\chi)$, there exists a function
$a_{f}:\mathscr{F}\rightarrow\mathbb{C}$ satisfying the following properties:

  1. (1)

$a_{f}(\mathfrak{a})=0$ unless $\mathfrak{a}\in\mathscr{F}$ is integral.

  2. (2)

For $x,y\in\mathbb{A}_{F}$, we have the Fourier-Whittaker expansion of $f$ as
follows:

$f\bigg{(}\begin{pmatrix}y&x\\\
0&1\end{pmatrix}\bigg{)}(\mathbf{x})=|y|_{\mathbb{A}_{F}}\sum_{\xi\in
F^{\times}}a_{f}(\xi y\mathfrak{d}_{F})W_{k}(\xi
y_{\infty})(\mathbf{x})\mathbf{e}_{F}(\xi x),$

where $y_{\infty}\in\mathbb{C}$ is the infinite part of $y$.

##  3\. Approximate functional equation

###  3.1. Integral representation of special $L$-values

Let $f\in S_{2}(\mathfrak{N},\chi)$ be a cusp form and
$u\in\mathbb{A}_{F}^{(\infty)}$ a finite adele. Then we define the _additively
twisted L-function_ $L(s,f,u)$ of $f$ by the following Dirichlet series:

$\displaystyle\frac{1}{|\mathcal{O}_{F}^{\times}|}\sum_{i=1}^{h_{F}}\sum_{\xi\in
F^{\times}}\frac{a_{f}(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})\mathbf{e}_{F}(\xi
a_{i}u)}{N(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})^{s}}\text{, converges
absolutely for }\mathfrak{R}(s)>\frac{k+2}{2}.$

Clearly $L(s,f,0)$ is equal to the usual $L$-function $L(s,f)$ of $f$.

In this section, we obtain an integral representation of $L(s,f,u)$ in a
spirit of Hida [5, Section 7]. Let us denote
$t(u):=\left(\begin{smallmatrix}1&u\\\
0&1\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\mathbb{A}_{F}^{(\infty)})$
and $t_{i}:=\left(\begin{smallmatrix}a_{i}&0\\\ 0&1\end{smallmatrix}\right)$
for each $u\in\mathbb{A}_{F}^{(\infty)}$ and $i$ . We define the $m$-th
component $f_{m}$ of a $L_{2k-2}(\mathbb{C})$-valued function $f$ by

$f(g)(\mathbf{x})=\sum_{m=0}^{2k-2}f_{m}(g)X^{2k-2-m}Y^{m}.$

Let us fix a branch of the square root function on $\mathbb{C}$. For
$x\in\mathbb{C}$ and $y\in\mathbb{C}^{\times}$, define an element
$(x;y)\in\operatorname{SL}_{2}(\mathbb{C})$ by

$(x;y):=\begin{pmatrix}y^{\frac{1}{2}}&xy^{-\frac{1}{2}}\\\
0&y^{-\frac{1}{2}}\end{pmatrix}.$

By Proposition 2.2, we observe that

$\displaystyle
f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}=\left(\begin{smallmatrix}2k-2\\\
k-1\end{smallmatrix}\right)|a_{i}|_{\mathbb{A}_{F}}\sum_{\xi\in
F^{\times}}a_{f}(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})|y|^{k}K_{0}(4\pi|\xi
y|)\mathbf{e}_{F}(\xi a_{i}u).$

So the Mellin transform of the function $y\mapsto
f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}$ on $\mathbb{R}^{\times}_{>0}$ is given by

(3.1)
$\displaystyle\begin{split}&|a_{i}|_{\mathbb{A}_{F}}^{s-1}\int_{0}^{\infty}f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}y^{2s-k}\frac{dy}{y}\\\
=&\left(\begin{smallmatrix}2k-2\\\
k-1\end{smallmatrix}\right)|D_{F}|^{s}\sum_{\xi\in
F^{\times}}\frac{a_{f}(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})\mathbf{e}_{F}(\xi
a_{i}u)}{N(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})^{s}}\int_{0}^{\infty}y^{2s}K_{0}(4\pi
y)\frac{dy}{y}\end{split}$

for $\text{Re}(s)>\frac{k+2}{2}$. By the following integration formula

$\int_{0}^{\infty}y^{s}{K_{j}(ay)}\frac{dy}{y}=\frac{1}{4}\Big{(}\frac{a}{2}\Big{)}^{-s}\Gamma\Big{(}\frac{s+j}{2}\Big{)}\Gamma\Big{(}\frac{s-j}{2}\Big{)}\text{
for }a>0\text{ and }\mathfrak{R}(s\pm j)>0,$

the equation (LABEL:mellin:transform) becomes

$\displaystyle|a_{i}|_{\mathbb{A}_{F}}^{s-1}\int_{0}^{\infty}f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}y^{2s-k}\frac{dy}{y}=\frac{\Gamma_{F,k}(s)}{|\mathcal{O}_{F}^{\times}|}\sum_{\xi\in
F^{\times}}\frac{a_{f}(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})\mathbf{e}_{F}(\xi
a_{i}u)}{N(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})^{s}}$

for $\mathfrak{R}(s)>\frac{k+2}{2}$, where
$\Gamma_{F,k}(s):=\left(\begin{smallmatrix}2k-2\\\
k-1\end{smallmatrix}\right)\frac{|D_{F}|^{s}|\mathcal{O}_{F}^{\times}|}{4(2\pi)^{2s}}\Gamma(s)^{2}.$
Finally, we obtain the integral representation of $L(s,f,u)$:

(3.2)
$\displaystyle\begin{split}\Gamma_{F,k}(s)L(s,f,u)=\sum_{i=1}^{h_{F}}|a_{i}|_{\mathbb{A}_{F}}^{s-1}\int_{0}^{\infty}f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}y^{2s-k}\frac{dy}{y}\end{split}$

for $\mathfrak{R}(s)>\frac{k+2}{2}$.

###  3.2. Functional equations

Let $\omega_{\chi}:=\chi|\cdot|_{\mathbb{A}_{F}}^{k-2}$ be the standard
normalization of $\chi$. Let $\mathfrak{q}$ be an integral ideal of $F$
coprime to $\mathfrak{a}_{i}^{-1}\mathfrak{d}_{F}\mathfrak{N}$ for each $i$.
Let $R_{\mathfrak{q}}$ be the set of elements of
$\prod_{v|\mathfrak{q}}F_{v}^{\times}$ corrensponding to a set of
representatives of $(\mathfrak{q}^{-1}/\mathcal{O}_{F})^{\times}$. Let us
embed $u\in R_{\mathfrak{q}}$ into $\mathbb{A}_{F}^{(\infty)}$ by
$u\mapsto(0,\cdots,0,u,0,\cdots,0)$. Note that we have the following
bijection:

$R_{\mathfrak{q}}\rightarrow(\widehat{\mathcal{O}}_{F}/\widehat{\mathfrak{q}})^{\times},\
u\mapsto\varpi_{\mathfrak{q}}u+\widehat{\mathfrak{q}}.$

For $u\in R_{\mathfrak{q}}$, let us define a _completed L-function_
$\Lambda(s,f,u)$ of $L(s,f,u)$ by

$\Lambda(s,f,u):=|\varpi_{\mathfrak{N}}\varpi_{\mathfrak{q}}^{2}|_{\mathbb{A}_{F}}^{-\frac{s}{2}}\Gamma_{F,k}(s)L(s,f,u).$

Let us define the _Fricke involution_ $W_{\mathfrak{N}}$ by taking the right
translation with the element $\left(\begin{smallmatrix}0&-1\\\
\varpi_{\mathfrak{N}}&0\end{smallmatrix}\right)$ of
$\operatorname{GL}_{2}(\mathbb{A}_{F}^{(\infty)})$:

$W_{\mathfrak{N}}f(g)(\mathbf{x}):=(-1)^{k-1}|\varpi_{\mathfrak{N}}|_{\mathbb{A}_{F}}^{\frac{k}{2}-1}\omega_{\chi}^{-1}\big{(}\det(g)\big{)}f\bigg{(}g\begin{pmatrix}0&-1\\\
\varpi_{\mathfrak{N}}&0\end{pmatrix}\bigg{)}(\mathbf{x}).$

Then $W_{\mathfrak{N}}f$ is a cusp form in
$S_{k}(\mathfrak{N},\overline{\chi})$. Note that for $u\in R_{\mathfrak{q}}$,
we can consider $u\varpi_{\mathfrak{q}}$ to be an element of
$(\mathcal{O}_{F,\mathfrak{q}}/\mathfrak{q}^{2})^{\times}$. For $u\in
R_{\mathfrak{q}}$, let us denote by $u^{\prime}\varpi_{\mathfrak{q}}$ a
representative of the mod $\mathfrak{q}^{2}$ inverse of
$-u\varpi_{\mathfrak{q}}$. Then we obtain a functional equation of
$\Lambda(s,f,u)$ as follows:

######  Proposition 3.1.

For $u\in R_{\mathfrak{q}}$, we have the following functional equation:

(3.3)
$\Lambda(s,f,u)=\omega_{\chi}(\varpi_{\mathfrak{q}})\Lambda(k-s,W_{\mathfrak{N}}f,u^{\prime}).$

Also we have the anlaytic continuation of $\Lambda(s,f,u)$ to the entire
$s\in\mathbb{C}$.

###### Proof.

Let us denote $\mathfrak{Q}=\mathfrak{N}\mathfrak{q}^{2}$. Then there exists a
$h_{F}$-permutation $\sigma$ and elements $\xi_{i}\in F^{\times}$ such that
$\mathfrak{a}_{i}^{-1}\mathfrak{Q}=\xi_{i}\mathfrak{a}_{\sigma(i)}$ for each
$i$. For $\xi\in F^{\times}$, let us denote
$w_{\xi}:=\xi^{(\infty)}\varpi_{\xi\mathcal{O}_{F}}^{-1}$. Then we obtain

$w_{\xi_{i}}\in\widehat{\mathcal{O}}_{F}^{\times},\
(w_{\xi_{i}})_{\mathfrak{q}}=1,\
a_{i}^{-1}\varpi_{\mathfrak{Q}}=a_{\sigma(i)}\xi_{i}^{(\infty)}w_{\xi_{i}}^{-1},\text{
and
}|\xi_{i}|=|a_{i}^{2}a_{\sigma(i)}^{2}\varpi_{\mathfrak{Q}}^{-2}|_{\mathbb{A}_{F}}^{1/4}.$

Let us denote by $f_{(i),j}^{u}$ the function $y\mapsto
f_{j}\big{(}t_{i}(0;y)t(u)\big{)}$ on $\mathbb{R}_{>0}^{\times}$. Then by the
equation (3.2), we obtain

(3.4)
$\displaystyle\begin{split}\Lambda(s,f,u)=\sum_{i=1}^{h_{F}}\frac{|\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{-\frac{s}{2}}}{|a_{i}|_{\mathbb{A}_{F}}^{1-s}}\bigg{(}\int_{|a_{i}^{-2}\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1/4}}^{\infty}+\int_{0}^{|a_{i}^{-2}\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1/4}}\bigg{)}f_{(i),k-1}^{u}(y)y^{2s-k}\frac{dy}{y}.\end{split}$

Let us set $\gamma_{i}=\left(\begin{smallmatrix}0&-1\\\
\xi_{i}&0\end{smallmatrix}\right)\in\operatorname{GL}_{2}(F)$. Let
$b(u)\in\widehat{\mathcal{O}}_{F}$ be the element such that
$b(u)_{\mathfrak{q}}\varpi_{\mathfrak{q}}^{2}-(u\varpi_{\mathfrak{q}})(u^{\prime}\varpi_{\mathfrak{q}})=1$
and $b(u)_{v}=1$ for $v$ coprime to $\mathfrak{q}$. Under the isomorphism
$\operatorname{GL}_{2}(\mathbb{A}_{F})\cong\operatorname{GL}_{2}(\mathbb{C})\times\operatorname{GL}_{2}(\mathbb{A}_{F}^{(\infty)})$,
$\gamma_{i}t_{i}(0;y)t(u)$ can be decomposed into

$\displaystyle\gamma_{i}t_{i}(0;y)t(u)=\big{(}\xi_{i}^{1/2}(0;|\xi_{i}|^{-1}y^{-1})U_{i},\varpi_{\mathfrak{q}}a_{\sigma(i)}^{-1}t_{\sigma(i)}t(u^{\prime})A_{\mathfrak{N}}B_{u}C_{i}\big{)}$

for each $y\in\mathbb{C}^{\times}$ and $i$, where
$U_{i}=(0;|\xi_{i}|\xi_{i}^{-1})\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)\in\operatorname{SU}_{2}(\mathbb{C})$,
$A_{\mathfrak{N}}=\left(\begin{smallmatrix}0&-1\\\
\varpi_{\mathfrak{N}}&0\end{smallmatrix}\right)$,

$B_{u}=\begin{pmatrix}\varpi_{\mathfrak{q}}&u\varpi_{\mathfrak{q}}\\\
u^{\prime}\varpi_{\mathfrak{q}}&b(u)\varpi_{\mathfrak{q}}\end{pmatrix}\in
U_{1}(\mathfrak{N}),\text{ and }C_{i}=\begin{pmatrix}w_{\xi_{i}}&0\\\
0&1\end{pmatrix}\in U_{1}(\mathfrak{N}).$

Note that $\operatorname{GL}_{2}(F)$ acts on
$\operatorname{GL}_{2}(\mathbb{A}_{F})\cong\operatorname{GL}_{2}(\mathbb{C})\times\operatorname{GL}_{2}(\mathbb{A}_{F}^{(\infty)})$
diagonally. By using the above equations and the properties of $f$, we have

$\displaystyle f_{(i),k-1}^{u}(y)$
$\displaystyle=f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}=f_{k-1}\big{(}\gamma_{i}t_{i}(0;y)t(u)\big{)}$
$\displaystyle=|\xi_{i}|^{2-k}\chi_{\mathfrak{N}}(B_{u}C_{i})\chi(\varpi_{\mathfrak{q}}a_{\sigma(i)}^{-1})\Big{(}f\big{(}t_{\sigma(i)}(0;|\xi_{i}|^{-1}y^{-1})t(u^{\prime})A_{\mathfrak{N}}\big{)}\big{(}U_{i}\mathbf{x}\big{)}\Big{)}_{k-1}$
$\displaystyle=|\xi_{i}|^{2-k}|a_{\sigma(i)}|_{\mathbb{A}_{F}}^{k-2}|\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1-\frac{k}{2}}\omega_{\chi}(\varpi_{\mathfrak{q}})(W_{\mathfrak{N}}f)^{u^{\prime}}_{(\sigma(i)),k-1}(|\xi_{i}|^{-1}y^{-1}).$

By using the above equations, we can write the second term of the integral
(3.4) as

$\displaystyle\sum_{i=1}^{h_{F}}\frac{\omega_{\chi}(\varpi_{\mathfrak{q}})|\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{\frac{s-k}{2}}}{|a_{\sigma(i)}|^{1-k+s}_{\mathbb{A}_{F}}}\int_{0}^{|a_{i}^{-2}\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1/4}}(W_{\mathfrak{N}}f)^{u^{\prime}}_{(\sigma(i)),k-1}\big{(}|\xi_{i}|^{-1}y^{-1}\big{)}(|\xi_{i}|y)^{2s-k}\frac{dy}{y}.$

By applying the change of variable to the above equation, we can rewrite the
equation (3.4) as follows:

(3.5)
$\displaystyle\begin{split}\Lambda(s,f,u)=&\sum_{i=1}^{h_{F}}\bigg{(}\int_{|a_{i}^{-2}\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1/4}}^{\infty}r_{i}(s)f^{u}_{(i),k-1}(y)y^{2s-k}\frac{dy}{y}\\\
&+\int_{|a_{i}^{-2}\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{1/4}}^{\infty}\omega_{\chi}(\varpi_{\mathfrak{q}})r_{i}(k-s)(W_{\mathfrak{N}}f)^{u^{\prime}}_{(i),k-1}(y)y^{k-2s}\frac{dy}{y}\bigg{)},\end{split}$

where
$r_{i}(s)=|\varpi_{\mathfrak{Q}}|_{\mathbb{A}_{F}}^{-\frac{s}{2}}|a_{i}|_{\mathbb{A}_{F}}^{s-1}$.
Note that $W_{\mathfrak{N}}f\in S_{2}(\mathfrak{N},\overline{\chi})$,
$W_{\mathfrak{N}}^{2}f=f$ for $f\in S_{2}(\mathfrak{N},\chi)$, and
$(u^{\prime})^{\prime}=u$. By the straightforward computation, we can obtain
the desired funtional equation from the integral representation (3.5). Also
the equation (3.5) implies that $\Lambda(s,f,u)$ has the analytic continuation
to the entire $s\in\mathbb{C}$. ∎

By using our functional equation $(\ref{func:eq})$, we obtain an approximate
functional equation of $\Lambda(s,f,u)$ for $u\in R_{\mathfrak{q}}$ in a
spirit of Luo-Ramakrishnan [10]. Let us fix an infinitely differentiable
function $\Phi$ on $\mathbb{R}_{>0}^{\times}$ with compact support such that
$\int_{0}^{\infty}\Phi(y)\frac{dy}{y}=1$. Define functions

$\displaystyle V_{1,s}(x)$ $\displaystyle:=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\kappa(t)\Gamma_{F,k}(s+t)x^{-t}\frac{dt}{t},$
$\displaystyle V_{2,s}(x)$ $\displaystyle:=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\kappa(-t)\Gamma_{F,k}(s+t)x^{-t}\frac{dt}{t},$

where $\kappa(t)=\int_{0}^{\infty}\Phi(y)y^{t}\frac{dy}{y}$. By shifting the
contour, one can show that for $s\neq 1/2$, $V_{1,s}$ and $V_{2,s}$ satisfy
following:

(3.6)
$\displaystyle\begin{split}V_{j,s}(x)&=O(\Gamma_{F,k}(\mathfrak{R}(s)+j)x^{-j})\text{
for all }j\geq 1\text{ as }x\rightarrow\infty,\\\
V_{j,s}(x)&=\Gamma_{F,k}(s)+O\Big{(}\Gamma_{F,k}\Big{(}\mathfrak{R}(s)-\frac{1}{2}\Big{)}x^{\frac{1}{2}}\Big{)}\text{
as }x\rightarrow 0,\end{split}$

where the implicit constants depend only on $j$ and $\Phi$. Let $u$ be an
element of $R_{\mathfrak{q}}$. Let us recall that $\mathfrak{q}$ is coprime to
$\mathfrak{a}_{i}^{-1}\mathfrak{N}$ for each $i$. For
$\mathfrak{R}(s)>\frac{k+2}{2}$ and $y>0$, we obtain

(3.7) $\displaystyle\begin{split}&\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\kappa\big{(}(-1)^{j-1}t\big{)}\Gamma_{F,k}(s+t)L(s+t,f,u)y^{t}\frac{dt}{t}\\\
=&\frac{1}{|\mathcal{O}_{F}^{\times}|}\sum_{i=1}^{h_{F}}\sum_{\xi\in
F^{\times}}\frac{a_{f}(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})\mathbf{e}_{F}(\xi
a_{i}u)}{N(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})^{s}}V_{j,s}\Big{(}\frac{N(\xi\mathfrak{a}_{i}\mathfrak{d}_{F})}{y}\Big{)}.\end{split}$

by the definition of $L(s,f,u)$. From the Residue theorem, we have

$\displaystyle\Gamma_{F,k}(s)L(s,f,u)$ $\displaystyle=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\kappa(t)\Gamma_{F,k}(s+t)L(s+t,f,u)y^{t}\frac{dt}{t}$
$\displaystyle+\frac{1}{2\pi
i}\int_{-2+i\infty}^{-2-i\infty}\kappa(t)\Gamma_{F,k}(s+t)L(s+t,f,u)y^{t}\frac{dt}{t}.$

Putting the equations (3.3), (LABEL:series:exp) in the above equation and
applying Proposition 2.2.(1), we obtain

(3.8)
$\displaystyle\begin{split}\Lambda(s,f,u)&=\sum_{i=1}^{h_{F}}\bigg{(}N(\mathfrak{N}\mathfrak{q}^{2})^{\frac{s}{2}}\sum_{0\neq\alpha\in\mathfrak{a}_{i}^{-1}}\frac{a_{f}(\alpha\mathfrak{a}_{i})\mathbf{e}_{F}(\alpha
u)}{N(\alpha\mathfrak{a}_{i})^{s}|\mathcal{O}_{F}^{\times}|}V_{1,s}\Big{(}\frac{N(\alpha\mathfrak{a}_{i})}{y}\Big{)}\\\
+\omega_{\chi}(\varpi_{\mathfrak{q}})&N(\mathfrak{N}\mathfrak{q}^{2})^{\frac{k-s}{2}}\sum_{0\neq\alpha\in\mathfrak{a}_{i}^{-1}}\frac{a_{{W_{\mathfrak{N}}}f}(\alpha\mathfrak{a}_{i})\mathbf{e}_{F}(\alpha
u^{\prime})}{N(\alpha\mathfrak{a}_{i})^{k-s}|\mathcal{O}_{F}^{\times}|}V_{2,k-s}\Big{(}\frac{N(\alpha\mathfrak{a}_{i})y}{N(\mathfrak{N}\mathfrak{q}^{2})}\Big{)}\bigg{)}.\end{split}$

Note that $\mathbf{e}_{F}(\alpha u)=\mathbf{e}_{F}(\alpha a_{i}d_{F}^{-1}u)$
as $\mathfrak{q}$ is coprime to $\mathfrak{a}_{i}^{-1}\mathfrak{d}_{F}$ for
each $i$.

###  3.3. Twisted cusp forms

Let $f\in S_{k}(\mathfrak{N},\chi)$ be a cusp form. For a finite order Hecke
character
$\phi:F^{\times}\backslash\mathbb{A}_{F}^{\times}\rightarrow\mathbb{C}^{\times}$
of conductor $\mathfrak{q}$, the _twisted cusp form_
$f\otimes\phi:\text{GL}_{2}(\mathbb{A}_{F})\rightarrow L(2k-2,\mathbb{C})$ is
defined by

(3.9)
$f\otimes\phi(g)(\mathbf{x}):=G(\phi)^{-1}\phi\big{(}\det(g)\big{)}\sum_{u\in
R_{\mathfrak{q}}}\phi(\varpi_{\mathfrak{q}}u)f\big{(}gt(u)\big{)}(\mathbf{x}),$

where $G(\phi)$ is the Gauss sum of $\phi$ (see Hida [5, Section 6]). Then we
have $f\otimes\phi\in S_{k}(\mathfrak{N}\cap\mathfrak{q}^{2},\chi\phi^{2})$
and $a_{f\otimes\phi}(\mathfrak{a})=a_{f}(\mathfrak{a})\phi(\mathfrak{a})$,
where

$\phi(\mathfrak{a}):=\left\\{\begin{array}[]{ll}\phi(\varpi_{\mathfrak{a}})&\textrm{if
$\mathfrak{a}$ is coprime to $\mathfrak{f}(\phi)$},\\\
0&\textrm{otherwise}.\end{array}\right.$

From the equation (3.2), the special value $L(\frac{k}{2},f\otimes\phi)$ of
the $L$-function of $f\otimes\phi$ is given by

(3.10)
$\Gamma_{F,k}\Big{(}\frac{k}{2}\Big{)}L\Big{(}\frac{k}{2},f\otimes\phi\Big{)}=\sum_{i=1}^{h_{F}}\sum_{u\in
R_{\mathfrak{q}}}\frac{\phi(a_{i})\phi(\varpi_{\mathfrak{q}}u)}{G(\phi)N(\mathfrak{a}_{i})^{\frac{k}{2}-1}}\int_{0}^{\infty}f_{k-1}\big{(}t_{i}(0;y)t(u)\big{)}\frac{dy}{y}.$

##  4\. Modular symbols and Eichler-Shimura isomorphism

In this section, we will give the definition and the properties of modular
symbols. Also we discuss the homology and cohomology groups of Bianchi modular
manifolds. To associate the cusp forms and the cohomology classes, we
introduce the generalized Eichler-Shimura isomorphism.

###  4.1. Homology and Cohomology of Bianchi modular manifolds

We will introduce the Bianchi modular manifold and study its homology and
cohomology. For the case of classical modular curves, see Ash-Stevens [1] and
Manin [11].

Let us define the hyperbolic upper half space by
$\mathscr{H}_{3}:=\mathbb{C}\times\mathbb{R}_{>0}$. Then there is a left
action of $\operatorname{SL}_{2}(\mathbb{C})$ on $\mathscr{H}_{3}$, which is
defined by

$\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot(x,y):=\bigg{(}\frac{(ax+b)\overline{(cx+d)}+a\overline{c}y^{2}}{|cx+d|^{2}+|c|^{2}y^{2}},\frac{y}{|cx+d|^{2}+|c|^{2}y^{2}}\bigg{)}.$

Note that the element $(x;y)\in\operatorname{SL}_{2}(\mathbb{C})$ for
$x\in\mathbb{C}$ and $y\in\mathbb{R}_{>0}^{\times}$ defined in the Section 3
corresponds to $(x,y)\in\mathscr{H}_{3}$ under the following bijection:

$\operatorname{SL}_{2}(\mathbb{C})/\operatorname{SU}_{2}(\mathbb{C})\rightarrow\mathscr{H}_{3},\
g\operatorname{SU}_{2}(\mathbb{C})\mapsto g\cdot(0,1).$

Let $R$ be a Dedekind domain, $\Gamma$ a congruence subgroup of
$\operatorname{SL}_{2}(\mathcal{O}_{F})$ such that the orders of the torsion
elements of $\Gamma$ are invertible in $R$. Let us define the Bianchi modular
manifold $Y_{\Gamma}$ of $\Gamma$ by
$Y_{\Gamma}:=\Gamma\backslash\mathscr{H}_{3}$. Let us denote
$\overline{\mathscr{H}_{3}}$ by the space obtained by attaching
$\mathbb{P}^{1}(F)$ to $\mathscr{H}_{3}$ via the map $\alpha\mapsto(\alpha,0)$
and $\infty\mapsto(0,\infty)$. Let us set
$X_{\Gamma}:=\Gamma\backslash\overline{\mathscr{H}_{3}}$. Then $X_{\Gamma}$ is
the Satake compactifiation of $Y_{\Gamma}$. Let us denote the set of cusps of
$X_{\Gamma}$ by $C_{\Gamma}:=\Gamma\backslash\mathbb{P}^{1}(F)$. From the
relative homology sequence of a pair $(X_{\Gamma},C_{\Gamma})$, we can obtain
the following exact sequence:

(4.1) $0\rightarrow H_{1}(X_{\Gamma},R)\rightarrow
H_{1}(X_{\Gamma},C_{\Gamma},R)\rightarrow H_{0}(C_{\Gamma},R)\rightarrow
H_{0}(X_{\Gamma},R)\rightarrow 0.$

Note that $H_{1}(C_{\Gamma},R)=H_{0}(X_{\Gamma},C_{\Gamma},R)=0$.

Let $Y^{*}_{\Gamma}$ be the Borel-Serre compactification of $Y_{\Gamma}$ (see
Borel-Serre [2]). Note that the boundary $\partial Y_{\Gamma}^{*}$ of
$Y_{\Gamma}^{*}$ is a disjoint union of $|C_{\Gamma}|$ $2$-tori. Also note
that there is a homotopy equivalence $Y_{\Gamma}\cong Y_{\Gamma}^{*}$. Then we
have the following boundary exact sequence:

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Breaking the sequence (4.2), we obtain the following exact sequence:

(4.3) $\displaystyle 0\rightarrow H^{0}(Y_{\Gamma},R)\rightarrow
H^{0}(\partial Y_{\Gamma}^{*},R)\rightarrow
H^{1}_{\operatorname{c}}(Y_{\Gamma},R)\rightarrow
H^{1}_{\text{par}}(Y_{\Gamma},R)\rightarrow 0,$

where $H^{1}_{\text{par}}(Y_{\Gamma},R)$ is the image of the natural map
$H^{1}_{\operatorname{c}}(Y_{\Gamma},R)\rightarrow H^{1}(Y_{\Gamma},R)$. Note
that $H^{0}(Y_{\Gamma},R)\cong R$ and $H^{0}(\partial
Y_{\Gamma},R)\cong\oplus_{s}R^{\Gamma_{s}}$, where $s$ runs over $C_{\Gamma}$.

We have the isomorphism $H^{1}_{\text{par}}(Y_{\Gamma},R)\cong
H^{1}(X_{\Gamma},R)$ induced by the injection $Y^{*}_{\Gamma}\rightarrow
X_{\Gamma}$. Also we have $H^{2}(Y_{\Gamma},R)\cong
H_{1}(X_{\Gamma},C_{\Gamma},R)$ by the Lefschetz-Poincaré duality. Then from
the Poincaré duality, we can induce non-degenerate $R$-bilinear pairings as
follows:

(4.4) $\displaystyle\cap:$ $\displaystyle H_{1}(X_{\Gamma},R)^{\prime}\times
H^{1}_{\text{par}}(Y_{\Gamma},R)^{\prime}\rightarrow R,$ (4.5)
$\displaystyle\cap:$ $\displaystyle
H_{1}(X_{\Gamma},C_{\Gamma},R)^{\prime}\times
H^{1}_{\operatorname{c}}(Y_{\Gamma},R)^{\prime}\rightarrow R,$

where $H^{\prime}$ is the torsion free part of a finitely generated $R$-module
$H$.

######  Remark 4.1.

Note that there is a Hecke equivariant section $\iota$ of the map

$H^{1}_{\operatorname{c}}(Y_{\Gamma},\mathbb{C})\rightarrow
H^{1}_{\text{par}}(Y_{\Gamma},\mathbb{C})\cong H^{1}(X_{\Gamma},\mathbb{C})$

(see Hida [6, Section 2]). Also note that for a parabolic cohomology class
$\delta\in H^{1}_{\text{par}}(Y_{\Gamma},\mathbb{C})$, $\delta-\iota(\delta)$
is an exact form. Thus we have

(4.6) $c\cap\delta=c\cap\iota(\delta)$

for each homology class $c\in H_{1}(X_{\Gamma},\mathbb{Z})$.

On the other hand, choose a relative homology class $\xi\in
H_{1}(X_{\Gamma},C_{\Gamma},\mathbb{Z})$. By the generalized Manin-Drinfeld
theorem (Kurčanov [8, Lemma 2, Theorem 2]), $N_{\xi}\xi\in
H_{1}(X_{\Gamma},\mathbb{Z})$ for some non-zero integer $N_{\xi}$. Thus we
have $N_{\xi}\xi\cap\delta=N_{\xi}\xi\cap\iota(\delta)$. Therefore, if
$R=\mathbb{C}$, then the pairings (4.4) and (4.5) are compatible under the map
$\iota$ and the sequence (4.1).

###  4.2. Modular symbols of weight (2,2)

Let $\Gamma$ be a torsion free congruence subgroup of
$\operatorname{SL}_{2}(\mathcal{O}_{F})$. For
$\alpha,\beta\in\mathbb{P}^{1}(F)$, let $\\{\alpha,\beta\\}$ be the geodesic
from $\alpha$ to $\beta$ in $\overline{\mathscr{H}_{3}}$. Let
$\\{\alpha,\beta\\}_{\Gamma}$ be the image of $\\{\alpha,\beta\\}$ in
$X_{\Gamma}$. Then we can consider $\\{\alpha,\beta\\}_{\Gamma}$ to be an
element of $H_{1}(X_{\Gamma},C_{\Gamma},\mathbb{Z})$. The classes
$\\{\alpha,\beta\\}_{\Gamma}$ are called _modular symbols_ of level $\Gamma$
and weight $(2,2)$. We have the following relations on the modular symbols:

  * (1)

$\\{\alpha,\alpha\\}_{\Gamma}=0$ in $H_{1}(X_{\Gamma},C_{\Gamma},\mathbb{Z})$
for $\alpha\in\mathbb{P}^{1}(F)$.

  * (2)

$\\{\alpha_{1},\alpha_{2}\\}_{\Gamma}=\\{\alpha_{1},\alpha_{3}\\}_{\Gamma}+\\{\alpha_{3},\alpha_{2}\\}_{\Gamma}$
for $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{P}^{1}(F)$.

  * (3)

$\\{\alpha,\beta\\}_{\Gamma}=\\{\gamma\alpha,\gamma\beta\\}_{\Gamma}$ for
$\alpha,\beta\in\mathbb{P}^{1}(F)$ and $\gamma\in\Gamma$.

By the definition of the relative homology group,
$H_{1}(X_{\Gamma},C_{\Gamma},\mathbb{Z})$ is generated by the symbols
$\\{\alpha,\beta\\}_{\Gamma}$. From the exact sequence (4.1), we have
$\ker(\partial)=H_{1}(X_{\Gamma},\mathbb{Z})$, where $\partial$ is the
boundary map defined by

$\partial:H_{1}(X_{\Gamma},C_{\Gamma},\mathbb{Z})\rightarrow
H_{0}(C_{\Gamma},\mathbb{Z}),\
\\{\alpha,\beta\\}_{\Gamma}\mapsto\\{\alpha\\}_{\Gamma}-\\{\beta\\}_{\Gamma}.$

Note that we have the following map:

$\text{pr}_{\Gamma}:\Gamma\rightarrow H_{1}(X_{\Gamma},\mathbb{Z}),\
\gamma\mapsto\\{0,\gamma\cdot 0\\}_{\Gamma}.$

Then $\text{pr}_{\Gamma}$ is a surjective group homomorphism, which is trivial
on the parabolic subgroups $\Gamma_{s}$ of $\Gamma$.

###  4.3. Special modular symbols

For each $1\leq i\leq h_{F}$, let us denote

$\Gamma_{1}^{i}(\mathfrak{N}):=\operatorname{GL}_{2}(F)\cap
t_{i}\operatorname{GL}_{2}(\mathbb{C})U_{1}(\mathfrak{N})t_{i}^{-1},\
\overline{\Gamma_{1}^{i}(\mathfrak{N})}:=\operatorname{SL}_{2}(F)\cap\Gamma_{1}^{i}(\mathfrak{N}),$

which are congruence subgroups of $\operatorname{GL}_{2}(\mathcal{O}_{F})$ and
$\operatorname{SL}_{2}(\mathcal{O}_{F})$, respectively. Since
$[\mathbb{Z}:\mathfrak{N}\cap\mathbb{Z}]>3$, we observe that
$\overline{\Gamma_{1}^{i}(\mathfrak{N})}$ has no torsion element (see Urban
[22, Lemme 2.3.1]). Therefore, for
$\Gamma=\overline{\Gamma_{1}^{i}(\mathfrak{N})}$, $Y_{\Gamma}$ is a manifold,
not an orbifold. Let us denote $Y_{1}^{i}(\mathfrak{N}):=Y_{\Gamma}$ and
$X_{1}^{i}(\mathfrak{N}):=X_{\Gamma}$ for
$\Gamma=\overline{\Gamma_{1}^{i}(\mathfrak{N})}$. Let us define the Bianchi
modular manifold $Y_{1}(\mathfrak{N})$ of level $U_{1}(\mathfrak{N})$ by

$Y_{1}(\mathfrak{N}):=\operatorname{GL}_{2}(F)\backslash\operatorname{GL}_{2}(\mathbb{A}_{F})/\mathbb{C}^{\times}\operatorname{U}_{2}(\mathbb{C})U_{1}(\mathfrak{N}).$

Let us recall that $\mathfrak{a}_{i}^{-1}$ is coprime to $\mathfrak{N}$ for
each $i$. Then by the strong approximation theorem on $\operatorname{GL}(2)$,
we obtain the following isomorphism:

$Y_{1}(\mathfrak{N})\rightarrow\coprod_{i=1}^{h_{F}}Y_{1}^{i}(\mathfrak{N}),\
g=\gamma
t_{j}g^{\prime}h\mapsto\overline{\Gamma_{1}^{j}(\mathfrak{N})}\frac{g^{\prime}}{\det(g^{\prime})}\operatorname{SU}_{2}(\mathbb{C})\in
Y_{1}^{j}(\mathfrak{N}),$

where $\gamma\in\operatorname{GL}_{2}(F)$,
$g^{\prime}\in\operatorname{GL}_{2}(\mathbb{C})$, $h\in U_{1}(\mathfrak{N})$
and $1\leq j\leq h_{F}$. Note that

$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})\cong\bigoplus_{i=1}^{h_{F}}H_{1}(X_{1}^{i}(\mathfrak{N}),\mathbb{Z}).$

Let $\mathfrak{q}$ be an integral ideal of $F$. For $u\in R_{\mathfrak{q}}$
and $i$, consider the following set:

$\\{t_{i}(0;y)t(u):y\in\mathbb{R}^{\times}_{>0}\\}\subset\operatorname{GL}_{2}(\mathbb{A}_{F}).$

Then we have the following proposition:

######  Proposition 4.2.

Assume that $\mathfrak{q}$ is coprime to $\mathfrak{a}_{i}^{-1}$. Then for
each $u\in R_{\mathfrak{q}}$, there is an element
$\alpha_{u}=\left(\begin{smallmatrix}1&r_{u}\\\
0&1\end{smallmatrix}\right)\in\operatorname{SL}_{2}(F)$ and
$c^{(i)}_{u}\in\widehat{\mathcal{O}}_{F}$ such that
$(c^{(i)}_{u})_{\mathfrak{q}}=0$ and

$\alpha_{u}t_{i}gt(u)=t_{i}\alpha_{u,\infty}gt(c^{(i)}_{u})$

in $\operatorname{GL}_{2}(\mathbb{A}_{F})$ for any
$g\in\operatorname{GL}_{2}(\mathbb{C})$.

###### Proof.

Choose an element $r_{u}$ of $(\mathfrak{q}^{-1}/\mathcal{O}_{F})^{\times}$
corresponding to the element
$-\varpi_{\mathfrak{q}}u\in(\widehat{\mathcal{O}}_{F}/\widehat{\mathfrak{q}})^{\times}$
under the following well-defined bijection:

$(\mathfrak{q}^{-1}/\mathcal{O}_{F})^{\times}\rightarrow(\widehat{\mathcal{O}}_{F}/\widehat{\mathfrak{q}})^{\times},\
r+\mathcal{O}_{F}\mapsto\varpi_{\mathfrak{q}}r_{\mathfrak{q}}+\widehat{\mathfrak{q}}.$

Then we have the equation as follows:

$\begin{pmatrix}1&r_{u}\\\
0&1\end{pmatrix}t_{i}(0;y)t(u)=t_{i}\begin{pmatrix}1&r_{u}\\\
0&1\end{pmatrix}_{\infty}(0;y)\begin{pmatrix}1&u+r_{u}^{(\infty)}a_{i}^{-1}\\\
0&1\end{pmatrix}.$

Note that $(u+r_{u}^{(\infty)}a_{i}^{-1})_{v}=0$ for each $v|\mathfrak{q}$,
and $(u+r_{u}^{(\infty)}a_{i}^{-1})_{v}\in\mathcal{O}_{v}$ for each $v$
coprime to $\mathfrak{q}$. So we conclude the proof. ∎

######  Remark 4.3.

Note that we can observe that $\alpha_{u}$ and $r_{u}$ does not depend on $i$.
For $u\in R_{\mathfrak{q}}$, the set
$\\{t_{i}(0;y)t(u):y\in\mathbb{R}^{\times}_{>0}\\}$ represents the homology
class
$\\{\infty,r_{u}\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\alpha_{u,\infty}\\{\infty,0\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}$
by the Proposition 4.2. So from now on, let us denote by $\xi^{(i)}_{u}$ the
class $\\{\infty,r_{u}\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}$ for $u\in
R_{\mathfrak{q}}$ if $\mathfrak{q}$ is coprime to $\mathfrak{a}_{i}^{-1}$.

Let $\mathfrak{l}$ be a prime ideal of $F$ coprime to $\mathfrak{a}_{i}^{-1}$
for each $i$. For an integer $n>0$ and an element
$a\in\mathcal{O}_{F,\mathfrak{l}}^{\times}$, let us define the homology class
$\xi^{(i)}_{\mathfrak{l},n}(a)$ by

$\xi^{(i)}_{n,\mathfrak{l}}(a):=\xi_{a/\varpi_{\mathfrak{l}}^{n}}^{(i)}\in
H_{1}(X_{1}^{i}(\mathfrak{N}),C_{1}^{i}(\mathfrak{N}),\mathbb{Z}),$

or simply denote by $\xi^{(i)}_{n}(a)$ if there is no ambiguity on
$\mathfrak{l}$. We consider the submodule $M_{n,\mathfrak{l}}$ of
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$, which is defined by

$M_{n,\mathfrak{l}}:=\bigoplus_{i=1}^{h_{F}}\langle\xi_{n}^{(i)}(a):a\in\mathcal{O}_{F,\mathfrak{l}}^{\times}\rangle\cap
H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z}).$

Note that $H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ is a finite generated
abelian group since $X_{1}(\mathfrak{N})$ is a finite dimensional compact
manifold. In section 6, we will show that $M_{n,\mathfrak{l}}$ is of full-rank
in $H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ for prime ideals $\mathfrak{l}$ of
sufficiently large norm $N(\mathfrak{l})$.

###  4.4. Generalized Eichler-Shimura isomorphism

Let us follow Hida [5] to define the generalized Eichler-Shimura map. Let
$S_{2}\big{(}\Gamma_{1}^{i}(\mathfrak{N})\big{)}$ be the space of cusp forms
on $\operatorname{GL}_{2}(\mathbb{C})$ of weight $(2,2)$ and level
$\Gamma_{1}^{i}(\mathfrak{N})$ (for the definition, see Hida [5, Section 2]).
For $f\in S_{2}\big{(}\Gamma_{1}^{i}(\mathfrak{N})\big{)}$, define a function
$f^{\prime}$ on $\operatorname{SL}_{2}(\mathbb{C})$ by

$f^{\prime}(g,\mathbf{a}):=\begin{pmatrix}f_{0}(g)&f_{1}(g)&f_{2}(g)\end{pmatrix}\Psi\big{(}{}^{t}j\big{(}g,(0,1)\big{)}\mathbf{a}\big{)},$

where $\mathbf{a}=\left(\begin{smallmatrix}A\\\ B\end{smallmatrix}\right)$,
$\Psi(\mathbf{a})={}^{t}\begin{pmatrix}A^{2}&AB&B^{2}\end{pmatrix}$ and

$j\bigg{(}\begin{pmatrix}a&b\\\
c&d\end{pmatrix},(x,y)\bigg{)}:=\begin{pmatrix}cx+d&-cy\\\
\overline{c}y&\overline{cx+d}\end{pmatrix}$

for $(x,y)\in\mathscr{H}_{3}$. Let us define a differential $1$-form
$\delta^{(i)}(f)$ on $Y_{1}^{i}(\mathfrak{N})$ by

$\delta^{(i)}(f)\big{(}(x,y)\big{)}:=f^{\prime}\big{(}(x;y),\mathbf{a}\big{)}|_{A^{2}=dx,\
AB=-dy,\ B^{2}=-d\overline{x}},$

where $x,y,\overline{x}$ are the local coordinates on $Y_{\Gamma}$. Note that
we have following:

$\overline{\Gamma_{1}^{i}(\mathfrak{N})}\backslash\operatorname{SL}_{2}(\mathbb{C})/\operatorname{SU}_{2}(\mathbb{C})\cong
Y_{1}^{i}(\mathfrak{N}),\
\overline{\Gamma_{1}^{i}(\mathfrak{N})}g\operatorname{SU}_{2}(\mathbb{C})\mapsto\overline{\Gamma_{1}^{i}(\mathfrak{N})}\big{(}g\cdot(0,1)\big{)}.$

Then we obtain the _generalized Eichler-Shimura isomorphism_ :

######  Proposition 4.4.

For $1\leq i\leq h_{F}$, the map $\delta^{(i)}$ is a Hecke equivariant
isomorphism of $\mathbb{C}$-vector spaces:

$\delta^{(i)}:S_{2}\big{(}\Gamma_{1}^{i}(\mathfrak{N})\big{)}\rightarrow
H^{1}_{\operatorname{par}}(Y_{1}^{i}(\mathfrak{N}),\mathbb{C})\cong
H^{1}(Y_{1}^{i}(\mathfrak{N}),\mathbb{C}).$

###### Proof.

By Hida [5, Corollary 2.2, Proposition 3.1], we are done. ∎

Note that for $(x,y)\in Y_{1}^{i}(\mathfrak{N})$, we have

(4.7)
$\delta^{(i)}(f)\big{(}(x,y)\big{)}=f_{0}\big{(}(x;y)\big{)}\frac{dx}{y}-f_{1}\big{(}(x;y)\big{)}\frac{dy}{y}-f_{2}\big{(}(x;y)\big{)}\frac{d\overline{x}}{y}.$

Let us denote by $S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$ the space of cusp
forms on $\operatorname{GL}_{2}(\mathbb{A}_{F})$ of weight $(2,2)$ and level
$U_{1}(\mathfrak{N})$ (for the definition, see Hida [5, Section 3]). For $f\in
S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$, let $f_{(i)}$ be the function on
$\operatorname{GL}_{2}(\mathbb{C})$ defined by $f_{(i)}(g):=f(t_{i}g)$. Then
we have the following isomorphism:

$S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}\cong\bigoplus_{i=1}^{h_{F}}S_{2}\big{(}\Gamma_{1}^{i}(\mathfrak{N})\big{)},\
f\mapsto(f_{(i)})_{i=1}^{h_{F}}.$

Then for a cusp form $f\in S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$, we have

(4.8)
$\xi_{u}^{(i)}\cap\iota\big{(}\delta^{(i)}(f_{(i)})\big{)}=\int_{0}^{\infty}f_{1}\big{(}t_{i}(r_{u};y)\big{)}\frac{dy}{y}=\int_{0}^{\infty}f_{1}\big{(}t_{i}(0;y)t(u)\big{)}\frac{dy}{y}$

by Proposition 4.2 and the equation (4.7).

##  5\. Lattice estimation

In this section, we estimate the number of the elements in the lattices of
bounded norm. Also we find the elements of the smallest norm in certain
lattices.

Let $n>0$ be an integer and $\mathfrak{l}$ a prime ideal of $F$. Then we have
the following lemma:

######  Lemma 5.1.

For each $\beta\in\mathcal{O}_{F}$, we have

$\displaystyle\\#\\{\alpha\in\beta+\mathfrak{l}^{n}:\ N(\alpha)\leq
x\\}\ll_{F}\max\Big{\\{}\frac{x}{N(\mathfrak{l})^{n}},1\Big{\\}}.$

###### Proof.

By Kwon-Sun [9, Proposition 5.2], we can conclude the proof since the size of
the unit group $\mathcal{O}_{F}^{\times}$ is finite. Note that [9, Proposition
5.2] comes from Rohrlich [16], which is about the coherent cone decomposition
of lattices in the Minkowski space of $F$. ∎

For $\beta\in\mathcal{O}_{F}$, let $[\beta]_{n,\mathfrak{l}}$ be the set
consists of the elements $\alpha\in\beta+\mathfrak{l}^{n}$ such that

$N(\alpha)=\min_{\alpha_{1}\in\beta+\mathfrak{l}^{n}}N(\alpha_{1}).$

We show that $[\beta]_{n,\mathfrak{l}}$ is a singleton set for sufficiently
large $N(\mathfrak{l})^{n}$:

######  Lemma 5.2.

Let $\beta\in\mathcal{O}_{F}$ such that $4N(\beta)<N(\mathfrak{l})^{n}$. Then
we have $[\beta]_{n,\mathfrak{l}}=\\{\beta\\}$.

###### Proof.

By the definition, the set $[\beta]_{n,\mathfrak{l}}$ is non-empty. Thus we
can choose an element $\beta_{1}\in[\beta]_{n,\mathfrak{l}}$. Let us assume
that $\beta_{1}\neq\beta$. Note that we have
$0\neq\beta_{1}-\beta\in\mathfrak{l}^{n}$ and $N(\beta_{1})\leq N(\beta)$ by
the definition of $[\beta]_{n,\mathfrak{l}}$. Then by the triangle inequality,
we obtain

$N(\mathfrak{l})^{n}\leq
N(\beta-\beta_{1})\leq(N(\beta)^{\frac{1}{2}}+N(\beta_{1})^{\frac{1}{2}})^{2}\leq
4N(\beta)<N(\mathfrak{l})^{n},$

which is a contradiction. Therefore, $[\beta]_{n,\mathfrak{l}}=\\{\beta\\}$,
which concludes the proof. ∎

##  6\. Generation of homology groups

###  6.1. Estimation of integrals along special modular symbols

Let $n>0$ be an integer and $\mathfrak{l}$ a prime ideal of $F$ whose norm
$N(\mathfrak{l})$ is coprime to $\mathfrak{a}_{i}^{-1}\mathfrak{N}$ for each
$i$. Let us denote $\ell:=N(\mathfrak{l})$. In this section, we prove that the
module $M_{n,\mathfrak{l}}$ defined in Section 4 is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ for sufficiently large $\ell^{n}$.
Before proving the generation result, we need the following lemma about the
estimation on the Kloosterman-like exponential sum:

######  Lemma 6.1.

Le $v>0$ be an integer such that $v\leq\lceil\frac{n}{2}\rceil$. For
$b,c\in\mathcal{O}_{F,\mathfrak{l}}$ with $v\leq
n-\lceil\frac{n}{2}\rceil-\min\\{v_{\mathfrak{l}}(b),v_{\mathfrak{l}}(c)\\}$,
we have

(6.1) $\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\mathbf{e}_{F}\bigg{(}\frac{ab+a^{\prime}c}{\varpi_{\mathfrak{l}}^{n}}\bigg{)}\leq
2\ell^{\lceil\frac{n}{2}\rceil+\min\\{v_{\mathfrak{l}}(b),v_{\mathfrak{l}}(c)\\}},$

where $a^{\prime}$ is the mod $\mathfrak{l}^{n}$ inverse of $a$.

###### Proof.

Let us denote $\Gamma_{n}=1+\mathfrak{l}^{n}\mathcal{O}_{F,\mathfrak{l}}$,
$Q(j)=a_{0}bj+a_{0}^{\prime}cj^{\prime}$, $h=\lceil\frac{n}{2}\rceil$, and
$w=\min\\{v_{\mathfrak{l}}(b),v_{\mathfrak{l}}(c)\\}$. Let us set
$d\in\mathcal{O}_{F,\mathfrak{l}}$ and $r\in\Gamma_{v}$. As the map
$a\mapsto\operatorname{\textbf{e}}_{F}\big{(}\frac{a}{\varpi_{\mathfrak{l}}^{n}}\big{)}$
is an additive character on $\mathcal{O}_{F,\mathfrak{l}}$, we have

$\displaystyle Q\big{(}r(1+\varpi_{\mathfrak{l}}^{h}d)\big{)}$
$\displaystyle\equiv
a_{0}br+a_{0}br\varpi_{\mathfrak{l}}^{h}d+a_{0}^{\prime}cr^{\prime}\big{(}1-\varpi_{\mathfrak{l}}^{h}d+O(\varpi_{\mathfrak{l}}^{2h})\big{)}$
$\displaystyle\equiv Q(r)+\varpi_{\mathfrak{l}}^{h}d(a_{0}br-
a_{0}^{\prime}cr^{\prime})\ (\text{mod }\mathfrak{l}^{n}).$

Note that we have the following group isomorphism:

$\displaystyle\begin{split}r(1+\varpi_{\mathfrak{l}}^{h}d)\in\Gamma_{v}/\Gamma_{n}\cong\Gamma_{v}/\Gamma_{h}\times\Gamma_{h}/\Gamma_{n}\cong\Gamma_{v}/\Gamma_{h}\times\mathcal{O}_{F,\mathfrak{l}}/\mathfrak{l}^{n-h}\mathcal{O}_{F,\mathfrak{l}}\ni(r,d).\end{split}$

From the above equations, we can rewrite (6.1) as

(6.2)
$\displaystyle\begin{split}\sum_{r\in\Gamma_{v}/\Gamma_{h}}\textbf{e}_{F}\bigg{(}\frac{Q(r)}{\varpi_{\mathfrak{l}}^{n}}\bigg{)}\sum_{d\
\text{mod }\mathfrak{l}^{n-h}}\textbf{e}_{F}\bigg{(}\frac{(a_{0}br-
a_{0}^{\prime}cr^{\prime})d}{\varpi_{\mathfrak{l}}^{n-h}}\bigg{)}.\end{split}$

By the orthogonality of characters of finite group, the above equation becomes

(6.3) $\displaystyle\begin{split}\ell^{n-h}\sum_{a_{0}br\equiv
a_{0}^{\prime}cr^{\prime}\ (\text{mod
}\mathfrak{l}^{n-h})}\textbf{e}_{F}\bigg{(}\frac{Q(r)}{\varpi_{\mathfrak{l}}^{n}}\bigg{)},\end{split}$

where $r$ runs over $\Gamma_{v}/\Gamma_{h}$. As $v\leq n-h-w$, we can observe
that

$\\{r\in\Gamma_{v}/\Gamma_{h}:\ a_{0}br\equiv a_{0}^{\prime}cr^{\prime}\
(\text{mod
}\mathfrak{l}^{n-h})\\}\leq\frac{2|\Gamma_{1}/\Gamma_{h}|}{\ell^{n-h-w}}=2\ell^{2h-n+w}.$

Hence the absolute value of (6.3) is bounded by $2\ell^{h+w}.$ So we are done.
∎

We have the following lemma about the Fourier-Whittaker coefficients of the
cusp form $W_{\mathfrak{N}}(f\otimes\psi)$:

######  Lemma 6.2.

For $f\in S_{2}(\mathfrak{N},\chi)$, we have
$a_{W_{\mathfrak{N}}(f\otimes\psi)}(\xi\mathfrak{a}_{i})=\psi(a_{j})a_{W_{\mathfrak{N}}f}(\xi\mathfrak{a}_{i})$,
where $j$ is the integer such that
$\mathfrak{a}_{j}=\mathfrak{a}_{i}^{-1}\mathfrak{N}$ in
$\operatorname{Cl}(F)$.

###### Proof.

Note that $f\otimes\psi\in S_{2}(\mathfrak{N},\chi\psi^{2})$. By the
definition of $W_{\mathfrak{N}}f$ and $f\otimes\psi$, we have

$W_{\mathfrak{N}}(f\otimes\psi)(g)=\psi^{-1}\big{(}\det(g)\big{)}\psi(\varpi_{\mathfrak{N}})W_{\mathfrak{N}}f(g).$

Hence we obtain
$a_{W_{\mathfrak{N}}(f\otimes\psi)}(\xi\mathfrak{a}_{i})=\psi(a_{i}^{-1}\varpi_{\mathfrak{N}})a_{W_{\mathfrak{N}}f}(\xi\mathfrak{a}_{i})$
for $\xi\in F^{\times}$ by the Proposition 2.2 and the equation (3.2). So we
conclude the proof. ∎

Let us recall that $H_{F}$ is the group of Hilbert characters of $F$. For
$f\in S_{2}(\mathfrak{N},\chi)$, we obtain

(6.4) $\displaystyle\frac{1}{h_{F}}\sum_{\psi\in
H_{F}}\psi(a_{i})^{-1}\Gamma_{F,2}(1)L\Big{(}1,f\otimes\psi,\frac{a}{\varpi_{\mathfrak{l}}^{n}}\Big{)}=\xi^{(i)}_{n,\mathfrak{l}}(a)\cap\delta^{(i)}(f_{(i)})$

by using the equations (3.10), (4.8) and the orthogonality of characters.

Note that $S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$ is a finite dimensional
complex vector space. From the pairing (4.4) and the generalized Eichler-
Shimura isomorphism, we induce the following non-degenerate pairing:

$\displaystyle\langle\cdot,\cdot\rangle:H_{1}(X_{1}(\mathfrak{N}),\mathbb{C})\times
S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}\rightarrow\mathbb{C},\ \langle
c,f\rangle\mapsto\sum_{i=1}^{h_{F}}c_{i}\cap\delta^{(i)}(f_{(i)}),$

where $c_{i}$ is the $i$-th component of a homology class $c$ under the
isomorphism
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{C})\cong\bigoplus_{i=1}^{h_{F}}H_{1}(X_{1}^{i}(\mathfrak{N}),\mathbb{C})$.

For an open subset
$U_{v}(a_{0}):=a_{0}+\mathfrak{l}^{v}\mathcal{O}_{F,\mathfrak{l}}$ of
$\mathcal{O}_{F,\mathfrak{l}}^{\times}$ and an element
$b\in\mathcal{O}_{F,\mathfrak{l}}$, let us define a special modular symbol
$\Upsilon_{n,\mathfrak{l}}^{(i)}\big{(}U_{v}(a_{0}),b\big{)}$ by

$\Upsilon_{n,\mathfrak{l}}^{(i)}\big{(}U_{v}(a_{0}),b\big{)}:=\frac{1}{\ell^{n-v}}\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\mathbf{e}_{F}\Big{(}\frac{-ab}{\varpi_{\mathfrak{l}}^{n}}\Big{)}\cdot\xi^{(i)}_{n,\mathfrak{l}}(a),$

where $a-a_{0}$ runs over
$\mathfrak{l}^{v}\mathcal{O}_{F,\mathfrak{l}}/\mathfrak{l}^{n}\mathcal{O}_{F,\mathfrak{l}}$.
By the generalized Manin-Drinfeld theorem, we have
$\Upsilon_{n,\mathfrak{l}}^{(i)}\big{(}U_{v}(a_{0}),b\big{)}\in
H_{1}(X_{1}^{i}(\mathfrak{N}),\mathbb{C})$. If there is no ambiguity on
$\mathfrak{l}$, we just denote $\Upsilon_{n,\mathfrak{l}}^{(i)}$ by
$\Upsilon_{n}^{(i)}$. Then we have the following estimation:

######  Proposition 6.3.

Let $\beta$ be a non-zero element of $\mathfrak{a}_{i}^{-1}$ and $f\in
S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$. For
$n>\max\\{2v+2v_{\mathfrak{l}}(\beta),v+\operatorname{log}_{\ell}\big{(}4N(\beta)\big{)}\\}$
and $\epsilon>0$, we have

$\displaystyle\begin{split}\langle\Upsilon_{n,\mathfrak{l}}^{(i)}(U_{v}(a_{0}),\beta),f\rangle=\frac{a_{f}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}\big{(}1+O(\ell^{-\frac{n}{6}})\big{)}+O\Big{(}\frac{\ell^{v}}{\ell^{(\frac{3}{16}-\varepsilon)n}}\Big{)}+O\bigg{(}\frac{\ell^{v+v_{\mathfrak{l}}(\beta)+1}}{\ell^{(\frac{3}{32}-\varepsilon)n}}\bigg{)},\end{split}$

where the implicit constant does not depend on $\beta$, $n$, and $\ell$.

###### Proof.

Let $f_{\chi}$ be the $\chi$-component of $f$ with respect to the isomorphism
$S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}\cong\bigoplus_{\chi}S_{2}(\mathfrak{N},\chi)$.
Then we have $\langle-,f\rangle=\sum_{\chi}\langle-,f_{\chi}\rangle$. Note
that the size of the sum $\sum_{\chi}$ depends only on $h_{F}$ and
$N(\mathfrak{N})$. So we can assume that $f\in S_{2}(\mathfrak{N},\chi)$.

For the simplicity, let us denote $U=U_{v}(a_{0})$. By the definition of the
special modular symbols and the equation (6.4), we obtain

(6.5)
$\displaystyle\begin{split}\langle\Upsilon_{n,\mathfrak{l}}^{(i)}\big{(}U,\beta\big{)},f\rangle=\frac{1}{\ell^{n-v}}\sum_{a\equiv
a_{0}(\mathfrak{l}^{n})}\mathbf{e}_{F}\Big{(}\frac{-a\beta}{\varpi_{\mathfrak{l}}^{n}}\Big{)}\sum_{\psi\in
H_{F}}\frac{\Gamma_{F,2}(1)}{h_{F}\psi(a_{i})}L\Big{(}1,f\otimes\psi,\frac{a}{\varpi_{\mathfrak{l}}^{n}}\Big{)}.\end{split}$

By the equation (3.8) for $\mathfrak{q}=\mathfrak{l}^{n}$, Lemma 6.2, and the
orthogonality of characters, we have the following equation of the
$L$-function:

(6.6) $\displaystyle\begin{split}\sum_{\psi\in
H_{F}}\frac{\Gamma_{F,2}(1)}{h_{F}\psi(a_{i})}L\Big{(}1,f\otimes\psi,\frac{a}{\varpi_{\mathfrak{l}}^{n}}\Big{)}=\sum_{0\neq\alpha\in\mathfrak{a}_{i}^{-1}}\frac{a_{f}(\alpha\mathfrak{a}_{i})\textbf{e}_{F}\big{(}\frac{\alpha
a}{\varpi_{\mathfrak{l}}^{n}}\big{)}}{N(\alpha\mathfrak{a}_{i})|\mathcal{O}_{F}^{\times}|}V_{1,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{i})}{y}\bigg{)}&\\\
+\chi(\varpi_{\mathfrak{l}})^{n}\sum_{0\neq\alpha\in\mathfrak{a}_{j}^{-1}}\frac{a_{W_{\mathfrak{N}}f}(\alpha\mathfrak{a}_{j})\textbf{e}_{F}\big{(}\frac{\alpha
a^{\prime}}{\varpi_{\mathfrak{l}}^{n}}\big{)}}{N(\alpha\mathfrak{a}_{j})|\mathcal{O}_{F}^{\times}|}V_{2,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{j})y}{N(\mathfrak{N})\ell^{2n}}\bigg{)}&,\end{split}$

where $a^{\prime}\in\mathcal{O}_{F,\mathfrak{l}}$ is the mod
$\mathfrak{l}^{n}$ inverse of $-a$, and $j$ is the integer such that
$\mathfrak{a}_{j}=\mathfrak{a}_{i}^{-1}\mathfrak{N}$ in $\text{Cl}(F)$. By
using the equation (6.6), we split the equation (6.5) into the two terms:

(6.7) $\displaystyle\frac{1}{\ell^{n-v}}$ $\displaystyle\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\sum_{0\neq\alpha\in\mathfrak{a}_{i}^{-1}}\frac{a_{f}(\alpha\mathfrak{a}_{i})\textbf{e}_{F}\big{(}\frac{a(\alpha-\beta)}{\varpi_{\mathfrak{l}}^{n}}\big{)}}{N(\alpha\mathfrak{a}_{i})|\mathcal{O}_{F}^{\times}|}V_{1,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{i})}{y}\bigg{)}$
(6.8) $\displaystyle+\frac{\chi(\varpi_{\mathfrak{l}})^{n}}{\ell^{n-v}}$
$\displaystyle\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\sum_{0\neq\alpha\in\mathfrak{a}_{j}^{-1}}\frac{a_{W_{\mathfrak{N}}f}(\alpha\mathfrak{a}_{j})\textbf{e}_{F}\big{(}\frac{a^{\prime}\alpha-a\beta}{\varpi_{\mathfrak{l}}^{n}}\big{)}}{N(\alpha\mathfrak{a}_{j})|\mathcal{O}_{F}^{\times}|}V_{2,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{j})y}{N(\mathfrak{N})\ell^{2n}}\bigg{)}.$

Note that by the orthogonality of characters, we observe that

$\frac{1}{\ell^{n-v}}\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\textbf{e}_{F}\Big{(}\frac{ar}{\varpi_{\mathfrak{l}}^{n}}\Big{)}=\left\\{\begin{array}[]{ll}\textbf{e}_{F}\big{(}\frac{a_{0}r}{\varpi_{\mathfrak{l}}^{n}}\big{)}&\textrm{if
$v_{\mathfrak{l}}(r)\geq n-v$,}\\\ 0&\textrm{otherwise}.\end{array}\right.$

By Lemma 5.2, we can write (6.7) as

(6.9)
$\displaystyle\frac{a_{f}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})|\mathcal{O}_{F}^{\times}|}V_{1,1}\bigg{(}\frac{N(\beta\mathfrak{a}_{i})}{y}\bigg{)}+\sum_{\alpha\in
X^{(i)}_{\beta,n}}\frac{a_{f}(\alpha\mathfrak{a}_{i})\textbf{e}_{F}\big{(}\frac{a_{0}(\alpha-\beta)}{\varpi_{\mathfrak{l}}^{n}}\big{)}}{N(\alpha\mathfrak{a}_{i})|\mathcal{O}_{F}^{\times}|}V_{1,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{i})}{y}\bigg{)}$

where $X^{(i)}_{\beta,n}$ is the set defined by
$\\{\alpha\in(\beta+\mathfrak{l}^{n-v})\cap\mathfrak{a}_{i}^{-1}:N(\alpha)>N(\beta)\\}$.
Note that if $\alpha\in X^{(i)}_{\beta,n}(y)$, then we have
$4^{-1}\ell^{n-v}<(\ell^{\frac{n-v}{2}}-N(\beta)^{1/2})^{2}\leq N(\alpha)$ by
the triangle inequality, Let $0\leq\theta<\frac{1}{2}$ be a number such that
$|a_{f}(\mathfrak{a})|\ll_{\varepsilon}N(\mathfrak{a})^{\frac{1}{2}+\theta+\epsilon}$
for any ideals $\mathfrak{a}$ of $F$. Then by using the equation (3.6), the
estimation of the equation (6.9) is given by

$\frac{a_{f}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}\bigg{(}1+O\bigg{(}\frac{\ell^{\frac{n-v}{2}}}{y^{1/2}}\bigg{)}\bigg{)}$

for $y\geq N(\mathfrak{a}_{i})\ell^{n-v}$, where the implicit constants
depends only on $D_{F}$ and $N(\mathfrak{a}_{i})$. We split the second term of
(6.9) into the two parts:

$\text{2nd term of }(\ref{approx:func:eq:2:1st:char})=(*)+(**),\text{ where
}(*)=\sum_{N(\alpha\mathfrak{a}_{i})\leq y},\
(**)=\sum_{N(\alpha\mathfrak{a}_{i})>y}.$

For $a\in\mathcal{O}_{F,\mathfrak{l}}$, define functions $u_{a,n}^{(i)}$ and
$U_{a,n}^{(i)}$ by

$u_{a,n}^{(i)}(\alpha)=\left\\{\begin{array}[]{ll}1&\textrm{if
$\alpha\in(a+\mathfrak{l}^{n-v})\cap\mathfrak{a}_{i}^{-1}$},\\\
0&\textrm{otherwise},\end{array}\right.$

and $U_{a,n}^{(i)}(y):=\sum_{N(\alpha)\leq
y}u_{a,n}^{(i)}(\alpha)=\\#\\{\alpha\in(a+\mathfrak{l}^{n-v})\cap\mathfrak{a}_{i}^{-1}:\
N(\alpha)\leq y\\}.$ By using the Abel summation formula, the bound on
$a_{f}(\mathfrak{a})$, the equation (3.6), Lemma 5.1 and Lemma 5.2, the
estimation of $(*)$ is given by

$\displaystyle(*)\ll_{D_{F},N(\mathfrak{a}_{i}),\varepsilon}$
$\displaystyle\sum_{4^{-1}\ell^{n-v}<N(\alpha)\leq
yN(\mathfrak{a}_{i}^{-1})}\frac{u_{\beta,n}^{(i)}(\alpha)}{N(\alpha)^{\frac{1}{2}-\theta-\epsilon}}$
$\displaystyle\ll_{N(\mathfrak{a}_{i}),\varepsilon,\theta}$
$\displaystyle\frac{U_{\beta,n}^{(i)}(yN(\mathfrak{a}_{i}^{-1}))}{y^{\frac{1}{2}-\theta-\epsilon}}+\frac{U_{\beta,n}^{(i)}(4^{-1}\ell^{n-v})}{(4^{-1}\ell^{n-v})^{\frac{1}{2}-\theta-\epsilon}}$
$\displaystyle+$
$\displaystyle\int_{4^{-1}\ell^{n-v}}^{yN(\mathfrak{a}_{i}^{-1})}\frac{U_{\beta,n}^{(i)}(t)}{t^{\frac{3}{2}-\theta-\epsilon}}dt\ll_{F,N(\mathfrak{a}_{i}),\varepsilon,\theta}\frac{y^{\frac{1}{2}+\theta+\epsilon}}{\ell^{n-v}}+\ell^{(\theta+\epsilon-\frac{1}{2})(n-v)}$

for $y\geq N(\mathfrak{a}_{i})\ell^{n-v}$. Similarly as above, the estimation
of $(**)$ is given by

$\displaystyle(**)\ll_{F,N(\mathfrak{a}_{i}),\varepsilon}$
$\displaystyle\sum_{N(\alpha\mathfrak{a}_{i})>y}\frac{u_{\beta,n}^{(i)}(\alpha)y}{N(\alpha)^{\frac{3}{2}-\theta-\epsilon}}\ll_{N(\mathfrak{a}_{i}),\varepsilon,\theta}\frac{U_{\beta,n}^{(i)}(yN(\mathfrak{a}_{i})^{-1})y}{y^{\frac{3}{2}-\theta-\epsilon}}$
$\displaystyle+$
$\displaystyle\int^{\infty}_{yN(\mathfrak{a}_{i})^{-1}}\frac{U_{\beta,n}^{(i)}(t)y}{t^{\frac{5}{2}-\theta-\epsilon}}dt\ll_{F,N(\mathfrak{a}_{i}),\varepsilon,\theta}\frac{y^{\frac{1}{2}+\theta+\epsilon}}{\ell^{n-v}}$

for $y\geq N(\mathfrak{a}_{i})\ell^{n-v}$. In sum, the estimation of the
equation (6.9) is given by

$\displaystyle\frac{a_{f_{\nu}}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}\bigg{(}1+O\bigg{(}\frac{\ell^{\frac{n-v}{2}}}{y^{1/2}}\bigg{)}\bigg{)}+O\bigg{(}\frac{y^{\frac{1}{2}+\theta+\epsilon}}{\ell^{n-v}}\bigg{)}+O(\ell^{(\theta+\epsilon-\frac{1}{2})(n-v)})$

for $y\geq N(\mathfrak{a}_{i})\ell^{n-v}$, where the implicit constants
depends only on $F,N(\mathfrak{a}_{i}),\varepsilon,\theta$.

On the other hand, we can write the term (6.8) by

(6.10)
$\chi(\varpi_{\mathfrak{l}})^{n}\sum_{0\neq\alpha\in\mathfrak{a}_{j}^{-1}}\frac{a_{W_{\mathfrak{N}}f}(\alpha\mathfrak{a}_{j})\text{Kl}_{n}(\alpha)}{N(\alpha\mathfrak{a}_{j})|\mathcal{O}_{F}^{\times}|}V_{2,1}\bigg{(}\frac{N(\alpha\mathfrak{a}_{j})y}{N(\mathfrak{N})\ell^{2n}}\bigg{)},$

where $\text{Kl}_{n}(\alpha)$ is a number defined by

$\text{Kl}_{n}(\alpha):=\frac{1}{\ell^{n-v}}\sum_{a\equiv
a_{0}(\mathfrak{l}^{v})}\textbf{e}_{F}\bigg{(}\frac{-a\beta+a^{\prime}\alpha}{\varpi_{\mathfrak{l}}^{n}}\bigg{)}.$

We split the term (6.10) into the two parts:

$(\ref{approx:func:eq:2:2nd:char})=I+II,\text{ where
}I=\sum_{N(\alpha\mathfrak{a}_{j})\leq N(\mathfrak{N})\ell^{2n}/y},\
II=\sum_{N(\alpha\mathfrak{a}_{j})>N(\mathfrak{N})\ell^{2n}/y}.$

Since $n>2v+2v_{\mathfrak{l}}(\beta)$, we have
$\text{Kl}_{n}(\alpha)\leq\ell^{-\frac{n}{2}+v+v_{\mathfrak{l}}(\beta)+1}$ by
Lemma 6.1. Note that we have
$a_{W_{\mathfrak{N}}f}(\mathfrak{a})\ll_{\varepsilon}N(\mathfrak{a})^{\frac{1}{2}+\theta+\varepsilon}$.
Therefore, by the equation (3.6), the estimations of $I$ and $II$ are given by

$\displaystyle I$
$\displaystyle\ll_{D_{F},N(\mathfrak{N}),\varepsilon}\frac{\ell^{v+v_{\mathfrak{l}}(\beta)+1}}{\ell^{\frac{n}{2}}}\sum_{N(\alpha)\leq
N(\mathfrak{N}\mathfrak{a}_{j}^{-1})\ell^{2n}/y}\frac{1}{N(\alpha)^{\frac{1}{2}-\theta-\varepsilon}}$
$\displaystyle\ll_{N(\mathfrak{a}_{j}),N(\mathfrak{N}),\theta,\varepsilon}\frac{\ell^{(\frac{1}{2}+2\theta+2\varepsilon)n+v+v_{\mathfrak{l}}(\beta)+1}}{y^{\frac{1}{2}+\theta+\varepsilon}},$
$\displaystyle II$
$\displaystyle\ll_{D_{F},N(\mathfrak{N}),\varepsilon}\ell^{v+v_{\mathfrak{l}}(\beta)+1+\frac{3n}{2}}\sum_{N(\alpha)>N(\mathfrak{N}\mathfrak{a}_{j}^{-1})\ell^{2n}/y}\frac{1}{N(\alpha)^{\frac{3}{2}-\theta-\varepsilon}y}$
$\displaystyle\ll_{N(\mathfrak{a}_{j}),N(\mathfrak{N}),\theta,\varepsilon}\frac{\ell^{(\frac{1}{2}+2\theta+2\varepsilon)n+v+v_{\mathfrak{l}}(\beta)+1}}{y^{\frac{1}{2}+\theta+\varepsilon}}.$

Hence we have

$\displaystyle(\ref{approx:func:eq:2:2nd:char})=I+II=O\bigg{(}\frac{\ell^{(\frac{1}{2}+2\theta+2\varepsilon)n+v+v_{\mathfrak{l}}(\beta)+1}}{y^{\frac{1}{2}+\theta+\varepsilon}}\bigg{)},$

where the implicit constant depends only on
$D_{F},N(\mathfrak{a}_{j}),N(\mathfrak{N}),\varepsilon$ and $\theta$.

In sum, we have

$\displaystyle\begin{split}\langle\Upsilon_{n,\mathfrak{l}}^{(i)}(U_{v}(a_{0}),\beta),f\rangle=&\frac{a_{f_{\nu}}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}\bigg{(}1+O\bigg{(}\frac{\ell^{\frac{n-v}{2}}}{y^{1/2}}\bigg{)}\bigg{)}+O\bigg{(}\frac{y^{\frac{1}{2}+\theta+\epsilon}}{\ell^{n-v}}\bigg{)}\\\
&+O(\ell^{(\theta+\epsilon-\frac{1}{2})(n-v)})+O\bigg{(}\frac{\ell^{(\frac{1}{2}+2\theta+2\varepsilon)n+v+v_{\mathfrak{l}}(\beta)+1}}{y^{\frac{1}{2}+\theta+\varepsilon}}\bigg{)}\end{split}$

for $y\geq N(\mathfrak{a}_{i})\ell^{n-v}$, where the implicit constant depends
only on $F$, $N(\mathfrak{a}_{i})$, $N(\mathfrak{a}_{j})$, $N(\mathfrak{N})$,
$\varepsilon$ and $\theta$. Note that $\ell$ is coprime to
$\mathfrak{a}_{i}^{-1}\mathfrak{N}$ for each $i$, so the implicit constant
does not depend on $\ell$. By Nakasuji [13, Corollary 1.1], we can choose
$\theta=\frac{7}{64}$. By letting $y=\ell^{\frac{4}{3}(n-v)}$, we conclude the
proof. ∎

###  6.2. Full-rankness results

From the estimation in the previous subsection, we can prove the full-rankness
of $M_{n,\mathfrak{l}}$ in $H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$. We need
the following fundamental lemma:

######  Lemma 6.4.

Let $X$ and $Y$ be finite dimensional complex vector spaces with a non-
degenerate pairing $\langle\cdot,\cdot\rangle:X\times Y\rightarrow\mathbb{C}$.
For any sequence of proper subspaces $\\{X_{m}\\}_{m}$ of $X$, there is a non-
zero vector $y\in Y$ and a sequence $\\{m_{j}\\}$ in $\mathbb{Z}_{>0}$ such
that $\lim_{j\rightarrow\infty}\langle u_{m_{j}},y\rangle=0$ for any sequence
$\\{u_{m}\\}_{m>0}$ of vectors $u_{m}\in X_{m}$ of bounded norm.

###### Proof.

Let us choose $m>0$. Since $X_{m}$ is proper in $X$, there is a non-zero
element $y_{m}\in Y$ such that $|y_{m}|_{Y}=1$ and $\langle x,y_{m}\rangle=0$
for all $x\in X_{m}$. Since the unit ball of $Y$ is compact, there is a
sequence $\\{m_{j}\\}_{j}$ in $\mathbb{Z}_{>0}$ and a vector $y$ in the unit
ball of $Y$ such that $y_{m_{j}}$ converges to $y$. Then by applying the
Cauchy-Schwarz inequality, we have

$|\langle u_{m_{j}},y\rangle|=|\langle
u_{m_{j}},y-y_{m_{j}}\rangle|\leq|u_{m_{j}}|\cdot|y-y_{m_{j}}|$

for any bounded sequence $\\{u_{m}\\}_{m}$ in $X$. By taking the limit
$j\rightarrow\infty$, we finish the proof. ∎

######  Theorem 6.5.

For $n\geq 22$, the module $M_{n,\mathfrak{l}}$ is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ for the prime ideals $\mathfrak{l}$ of
sufficiently large norm.

###### Proof.

Let us set $n\geq 22$. We need to show that
$M_{n,\mathfrak{l}}\otimes_{\mathbb{Z}}\mathbb{C}=H_{1}(X_{1}(\mathfrak{N}),\mathbb{C})$
for the prime ideals $\mathfrak{l}$ of sufficienly large norm. Let us assume
the contrary, i,e., assume that there is a sequence
$\\{\mathfrak{l}_{m}\\}_{m}$ of prime ideals such that
$M_{n,\mathfrak{l}_{m}}\otimes_{\mathbb{Z}}\mathbb{C}$ is proper in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{C})$ and
$N(\mathfrak{l}_{m})\rightarrow\infty$ as $m\rightarrow\infty$. Note that
without loss of generality, we can discard the prime ideals whose norm divides
$\mathfrak{a}_{i}^{-1}\mathfrak{N}$ for some $i$ from the sequence
$\\{\mathfrak{l}_{m}\\}_{m}$ . By Lemma 6.4, there is a non-zero element $f\in
S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$ of bounded norm and a sequence
$\\{m_{j}\\}$ in $\mathbb{Z}_{>0}$ such that for any bounded sequence
$\\{u_{m}\\}$ in $M_{n,\mathfrak{l}_{m}}\otimes_{\mathbb{Z}}\mathbb{C}$, we
have

(6.11) $\lim_{j\rightarrow\infty}\langle u_{m_{j}},f\rangle=0.$

Since $f$ is non-zero, there is a non-zero integral ideal
$\mathfrak{a}=\beta\mathfrak{a}_{i}$ such that
$a_{f}(\beta\mathfrak{a}_{i})\neq 0$.

Let us consider a sequence
$\\{\Upsilon_{n}^{(i)}(U_{\mathfrak{l}_{m}},\beta)\\}_{m}$, where
$U_{\mathfrak{l}}=1+\mathfrak{l}\mathcal{O}^{\times}_{F,\mathfrak{l}}$. Note
that $\Upsilon_{n}^{(i)}(U_{\mathfrak{l}_{m}},\beta)\in
M_{n,\mathfrak{l}_{m}}\otimes_{\mathbb{Z}}\mathbb{C}$ and
$v_{\mathfrak{l}_{m}}(\beta)=0$ for sufficiently large $m$. From Proposition
6.3, we obtain

(6.12)
$\langle\Upsilon_{n}^{(i)}(U_{\mathfrak{l}_{m}},\beta),g\rangle=\frac{a_{g}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}+o(1)$

as $m\rightarrow\infty$ for any $g\in S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}$,
where the implicit constant does not depend on $\\{\mathfrak{l}_{m}\\}$. So
the sequence $\\{\Upsilon_{n}^{(i)}(U_{\mathfrak{l}_{m}},\beta)\\}_{m}$ is
bounded. By Proposition 6.3 and the equation (6.11), we obtain

$\lim_{m\rightarrow\infty}\langle\Upsilon_{n}^{(i)}(U_{\mathfrak{l}_{m}},\beta),f\rangle=\frac{a_{f}(\beta\mathfrak{a}_{i})}{N(\beta\mathfrak{a}_{i})}=0,$

which is a contradiction. ∎

From the above result, we obtain the full-rankness theorem of horizontal
direction, which has a key role to prove the result toward Greenberg’s
conjecure of Bianchi modular version:

######  Theorem 6.6 (Horizontal direction).

For an integer $n\geq 22$, there is a prime ideal
$\mathfrak{p}_{n}=\alpha_{n}\mathcal{O}_{F}$ whose norm is coprime to
$\mathfrak{N}\prod_{i=1}^{h_{F}}\mathfrak{a}_{i}^{-1}$ such that
$M_{n,\mathfrak{p}_{n}}$ is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$. And there is an arithmetic
progression
$\mathfrak{X}_{n}\subset\big{\\{}\alpha\mathcal{O}_{F}:\alpha\in\alpha_{n}+\mathfrak{N}\prod_{i=1}^{h_{F}}\mathfrak{a}_{i}^{-1}\big{\\}}$
of ideals such that $M_{n,\mathfrak{p}_{n}}\subset M_{n,\mathfrak{p}}$ for the
prime ideals $\mathfrak{p}$ in $\mathfrak{X}_{n}$.

###### Proof.

Let us choose an integer $n\geq 22$. By Theorem 6.5, there is a prime ideal
$\mathfrak{p}_{n}=\alpha_{n}\mathcal{O}_{F}$ of sufficiently large norm so
that $N(\mathfrak{p}_{n})$ is coprime to
$\mathfrak{N}\prod_{i=1}^{h_{F}}\mathfrak{a}_{i}^{-1}$ and
$M_{n,\mathfrak{p}_{n}}$ is of full-rank in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$. Choose a generator $\xi$ of the
module $M_{n,\mathfrak{p}_{n}}$. Note that we have
$\xi=\xi^{(i)}_{n,\mathfrak{p}_{n}}(\beta)$ for some
$\beta\in(\mathcal{O}_{F}/\mathfrak{p}_{n}^{n})^{\times}$. From Proposition
4.2 and our assumption on $\mathfrak{p}_{n}$, we obtain

$\xi=\xi^{(i)}_{n,\mathfrak{p}_{n}}(\beta)=\\{\infty,\beta\alpha_{n}^{-n}\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}.$

Let us set
$B=\prod_{\beta\in(\mathcal{O}_{F}/\mathfrak{p}_{n}^{n})^{\times}}\beta$ and
$\mathfrak{C}=\prod_{i=1}^{h_{F}}\mathfrak{a}_{i}^{-1}$. Let us define an
arithmetic progression $\mathfrak{X}_{n}$ of ideals by
$\mathfrak{X}_{n}:=\\{\alpha\mathcal{O}_{F}:\alpha\in\alpha_{n}+B\mathfrak{C}\mathfrak{N},\
N(\mathfrak{p})\geq N(\mathfrak{p}_{n})\\}$. Let us choose a prime ideal
$\mathfrak{p}=\alpha\mathcal{O}_{F}\in\mathfrak{X}_{n}$. Then we observe that
$\mathfrak{p}$ is coprime to $\mathfrak{a}_{i}^{-1}$ for each $i$ so that the
symbols $\xi_{n,\mathfrak{p}}^{(i)}(a)$ and modules $M_{n,\mathfrak{p}}$ are
well defined. Also we have
$\xi^{(i)}_{n,\mathfrak{p}}(\beta)=\\{\infty,\beta\alpha^{-n}\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}$
and $\alpha^{n}=\alpha_{n}^{n}+BCN$ for some $C\in\mathfrak{C}$ and
$N\in\mathfrak{N}$.

Consider the action of $\operatorname{SL}_{2}(F)$ on $Y_{1}(\mathfrak{N})$. If
$c\in\mathfrak{a}_{i}^{-1}\mathfrak{N}$, then
$\gamma_{c}=\left(\begin{smallmatrix}1&0\\\
c&1\end{smallmatrix}\right)\in\operatorname{GL}_{2}(F)$ and
$\gamma_{c,\infty}\in\Gamma_{1}^{i}(\mathfrak{N})$. For $b\in F$ and
$y\in\mathbb{R}_{>0}^{\times}$, we have

$t_{i}(b;y)=\gamma_{c}t_{i}(b;y)=t_{i}\gamma_{c,\infty}(b;y)\begin{pmatrix}1&0\\\
a_{i}c^{(\infty)}&1\end{pmatrix}=t_{i}\begin{pmatrix}1&0\\\
c&1\end{pmatrix}_{\infty}(b;y)$

in $Y_{1}(\mathfrak{N})$. Note that
$\\{\infty,b\\}_{\Gamma_{1}^{i}(\mathfrak{N})}$ is represented by the set
$\\{t_{i}(b;y):y\in\mathbb{R}_{>0}^{\times}\\}$. Also note that the action of
$\operatorname{SL}_{2}(\mathbb{C})$ on $Y_{1}^{i}(\mathfrak{N})$ preserve the
metric, and

$\begin{pmatrix}1&0\\\
c&1\end{pmatrix}_{\infty}(b;y)\mapsto\bigg{(}\frac{\overline{c}y^{2}+b\overline{(bc+1)}}{|c|^{2}y^{2}+|bc+1|^{2}},\frac{y}{|c|^{2}y^{2}+|bc+1|^{2}}\bigg{)}$

under the bijection
$\operatorname{SL}_{2}(\mathbb{C})/\operatorname{SU}_{2}(\mathbb{C})\rightarrow\mathscr{H}_{3}$.
Thus we have

$\\{\infty,b\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\Big{\\{}\frac{1}{c},\frac{b}{bc+1}\Big{\\}}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\gamma_{c}\\{\infty,0\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}+\\{0,\infty\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}+\Big{\\{}\infty,\frac{b}{bc+1}\Big{\\}}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}.$

Let us set $b=\beta\alpha_{n}^{-n}\in F^{\times}$ and
$c=\beta^{-1}BCN\in\mathfrak{N}\prod_{i=1}\mathfrak{a}_{i}^{-1}$. Note that
$\gamma_{c}\in\operatorname{GL}_{2}(F)$ and
$\gamma_{c,\infty}\in\bigcap_{i=1}^{h_{F}}\Gamma_{1}^{i}(\mathfrak{N})$. Then
by the above equations, we obtain

$\xi=\xi_{n,\mathfrak{p}_{n}}^{(i)}(\beta)=\\{\infty,b\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\\{\infty,\beta\alpha^{-n}\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\xi_{n,\mathfrak{p}}^{(i)}(\beta),$

which is a generator of $M_{n,\mathfrak{p}}$. Thus $M_{n,\mathfrak{p}_{n}}$ is
a submodule of $M_{n,\mathfrak{p}}$, which finishes the proof. ∎

######  Remark 6.7.

Let us follow the notations in Theorem 6.6. Let us define a positive integer
as follows:

$\mathfrak{h}_{n}:=[H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z}):M_{n,\mathfrak{p}_{n}}].$

Then by Theorem 6.6, the indices of $M_{n,\mathfrak{p}}$ in
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})$ divide $\mathfrak{h}_{n}$ for the
prime ideals $\mathfrak{p}\in\mathfrak{X}_{n}$. Let $q$ be an odd prime number
does not divide $\mathfrak{h}_{n}$. Then we have

$M_{n,\mathfrak{p}}\otimes_{\mathbb{Z}}\overline{\mathbb{F}}_{q}=H_{1}(X_{1}(\mathfrak{N}),\overline{\mathbb{F}}_{q})$

for the prime ideals $\mathfrak{p}\in\mathfrak{X}_{n}$ by Theorem 6.6 and the
universal coefficient theorem.

Let $f\in S_{2}(\mathfrak{N},\chi)$ be a normalized Hecke eigenform and
$\mathfrak{p}$ a prime ideal coprime to $\mathfrak{a}_{i}^{-1}$ for each $i$.
For an open subset $Z\subset\mathcal{O}_{F,\mathfrak{p}}$, define a module
$M^{f}_{n,\mathfrak{p}}(Z)$ by

$M^{f}_{n,\mathfrak{p}}(Z):=\langle
a_{f}(\mathfrak{p})\xi_{n}(r)-\chi(\varpi_{\mathfrak{p}})\xi_{n-1}(r):\ r\in
Z\rangle\cap H_{1}(X_{1}(\mathfrak{N}),\overline{\mathbb{Z}}),$

which depends on $a_{f}(\mathfrak{p})$. We denote
$M^{f}_{\mathfrak{p},n}(\mathcal{O}_{F,\mathfrak{p}})$ by
$M^{f}_{\mathfrak{p},n}$. Then we have a variation of Theorem 6.6:

######  Proposition 6.8.

Let us follow the notations in Theorem 6.6 and Remark 6.7. Let $\mathfrak{p}$
be an $f$-ordinary prime ideal $\mathfrak{p}\in\mathfrak{X}_{n}$ coprime to
$\mathfrak{h}_{n}$. Let us denote $p$ the odd rational prime lying below
$\mathfrak{p}$. Then we have
$M^{f}_{n,\mathfrak{p}}\otimes_{\mathbb{Z}}\mathbb{F}_{p}=H_{1}(X_{1}(\mathfrak{N}),\overline{\mathbb{F}}_{p})$.

###### Proof.

For each open subset $Z\subset\mathcal{O}_{F,\mathfrak{p}}^{\times}$, let us
denote $M_{n,\mathfrak{p}}^{(i)}(Z):=\langle\xi_{n,\mathfrak{p}}^{(i)}(r):r\in
Z\rangle\cap H_{1}(X_{1}^{i}(\mathfrak{N}),\mathbb{Z})$ and
$M^{f,(i)}_{n,\mathfrak{p}}(Z):=M^{f}_{n,\mathfrak{p}}(Z)\cap
H_{1}(X_{1}^{i}(\mathfrak{N}),\overline{\mathbb{Z}})$. For
$a\in(\mathcal{O}_{F}/\mathfrak{p}^{n-1})^{\times}$, let us choose an element
$\xi\in
M^{(i)}_{n,\mathfrak{p}}(a+\mathfrak{p}^{n-1}\mathcal{O}_{F,\mathfrak{p}})$.
Then
$\xi=\sum_{b\in\mathfrak{p}^{n-1}/\mathfrak{p}^{n}}N_{b}\xi^{(i)}_{n,\mathfrak{p}}(a+b)$,
where $N_{b}$ is a non-negative integer. Then by the proof of Theorem 6.6, we
have

$\xi^{(i)}_{n,\mathfrak{p}}(a+b)=\Big{\\{}\infty,\frac{a+b}{\alpha^{n}}\Big{\\}}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}},$

where $\alpha$ is a generator of an ideal $\mathfrak{p}$ of $F$. Taking the
boundary map $\partial$ on the symbol $\xi$, we obtain

$\sum_{b}N_{b}\Big{\\{}\frac{a+b}{\alpha^{n}}\Big{\\}}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}=\sum_{b}N_{b}\\{\infty\\}_{\overline{\Gamma^{i}_{1}(\mathfrak{N})}}.$

Note that the above cusps are not equivalent by the definition of
$\Gamma_{1}^{i}(\mathfrak{N})$ and $\mathfrak{X}_{n}$. Thus we have
$\sum_{b}N_{b}=0$. From this relation, we obtain

$a_{f}(\mathfrak{p})\xi=a_{f}(\mathfrak{p})\sum_{b}N_{r}\xi^{(i)}_{n,\mathfrak{p}}(a+b)=\sum_{b}N_{b}\big{(}a_{f}(\mathfrak{p})\xi^{(i)}_{n,\mathfrak{p}}(a+b)-\chi(\varpi_{\mathfrak{p}})\xi^{(i)}_{n-1,\mathfrak{p}}(a)\big{)},$

which implies that

$M^{(i)}_{n,\mathfrak{p}}(a+\mathfrak{p}^{n-1}\mathcal{O}_{F,\mathfrak{p}})\otimes_{\mathbb{Z}}\overline{\mathbb{F}}_{p}\subset
M^{f,(i)}_{n,\mathfrak{p}}(a+\mathfrak{p}^{n-1}\mathcal{O}_{F,\mathfrak{p}})\otimes_{\mathbb{Z}}\mathbb{F}_{p}\subset
H_{1}(X^{i}_{1}(\mathfrak{N}),\overline{\mathbb{F}}_{p}).$

By using Remark 6.7 and summing the spaces over $a$ and $i$, we conclude the
proof. ∎

##  7\. Cohomology classes and special $L$-values

In this section, we define integral $L$-values of Hecke eigenforms and study
their residual non-vanishing. Also we compare the integral $L$-values and
algebraic $L$-values.

###  7.1. Special $L$-values

In this section, we follow Namikawa [14, Section 2] to define the cohomology
classes of integral coefficients and the integral $L$-values.

Let $p$ be a odd prime number coprime to $h_{F}D_{F}$ and $\mathfrak{p}$ a
prime ideal of $F$ lying above $p$. Let $f\in S_{k}(\mathfrak{N},\chi)$ be a
normalized Hecke eigenform, and
$\mathcal{O}=\mathbb{Z}_{p}[\\{a_{f}(\mathfrak{l}):\mathfrak{l}\\}]$. For a
cohomology class $\delta\in H_{*}^{1}(Y_{1}(\mathfrak{N}),R)$, let us denote
by $\delta_{i}$ the image of $\delta$ under the map
$H_{*}^{1}(Y_{1}(\mathfrak{N}),\mathcal{O})\rightarrow
H_{*}^{1}(Y_{1}^{i}(\mathfrak{N}),R)$, where
$*\in\\{\operatorname{c},\text{par}\\}$ and $R=\mathcal{O}$,
$\overline{\mathbb{F}}_{p}$ or $\mathbb{C}$. Let
$\delta:S_{2}\big{(}U_{1}(\mathfrak{N})\big{)}\rightarrow
H^{1}_{\text{par}}(Y_{1}(\mathfrak{N}),\mathbb{C})$ be the generalized
Eichler-Shimura isomorphism defined by
$\delta(f):=\sum_{i=1}^{h_{F}}\delta^{(i)}(f_{(i)})$.

Let us fix an isomorphism $\mathbb{C}\cong\mathbb{C}_{p}$, then we have the
scalar extension map $H^{1}(Y_{1}(\mathfrak{N}),\mathcal{O})\rightarrow
H^{1}(Y_{1}(\mathfrak{N}),\mathbb{C})$. Let us recall that there is a section
$\iota$ of the map
$H^{1}_{\operatorname{c}}(Y_{1}(\mathfrak{N}),\mathbb{C})\rightarrow
H^{1}_{\text{par}}(Y_{1}(\mathfrak{N}),\mathbb{C})$. Define the _cuspidal
cohomology group_ of coefficient $\mathcal{O}$ by

$H^{1}_{\text{cusp}}(Y_{1}(\mathfrak{N}),\mathcal{O}):=\operatorname{Im}(\iota)\cap
H^{1}_{\operatorname{c}}(Y_{1}(\mathfrak{N}),\mathcal{O}),$

where the intersection is defined via the scalar extension map. Then the
$f$-eigenspace $H^{1}_{\text{cusp}}(Y_{1}(\mathfrak{N}),\mathcal{O})[f]$ is a
free $\mathcal{O}$-module of rank one. Fix a $\mathcal{O}$-free generator
$\eta_{f,{\operatorname{c}}}$ of
$H^{1}_{\text{cusp}}(Y_{1}(\mathfrak{N}),\mathcal{O})[f]$, which is determined
up to multiplication by $\mathcal{O}^{\times}$. We define the _Hida’s
canonical period_ $\Omega_{f,{\operatorname{c}}}\in\mathbb{C}^{\times}$ by the
number which satisfies
$\iota\big{(}\delta(f)\big{)}=\Omega_{f,{\operatorname{c}}}\eta_{f,{\operatorname{c}}}$.

For a finite order Hecke character $\phi$ whose conductor $\mathfrak{f}(\phi)$
is coprime to $\mathfrak{a}_{i}^{-1}$, let us define a _twisted modular
symbol_ $\Lambda^{(i)}(\phi)$ by

$\Lambda^{(i)}(\phi):=\sum_{u\in
R_{\mathfrak{f}(\phi)}}\phi(a_{i})\phi(\varpi_{\mathfrak{f}(\phi)}u)\xi_{u}^{(i)}\in
H_{1}(X_{1}^{i}(\mathfrak{N}),C_{1}^{i}(\mathfrak{N}),\mathbb{Z}[\phi]).$

######  Remark 7.1.

By Namikawa [14, Lemma 2.2], a relative homology classes $\Lambda^{(1)}(\phi)$
can be lifted to a homology class when $h_{F}=1$.

When $\mathfrak{f}(\phi)$ is coprime to $\mathfrak{a}_{i}^{-1}$ for each $i$,
we define the _integral L-value_ $\mathcal{L}_{f}(\phi)$ by

$\mathcal{L}_{f}(\phi):=\sum_{i=1}^{h_{F}}\Lambda^{(i)}(\phi)\cap\eta_{f,\operatorname{c},i}\in\mathcal{O}[\phi].$

Then by the equation (3.10) and (4.8), we have

$\mathcal{L}_{f}(\phi)=\sum_{i=1}^{h_{F}}\Lambda^{(i)}(\phi)\cap\frac{\iota\big{(}\delta^{(i)}(f_{(i)})\big{)}}{\Omega_{f,\operatorname{c}}}=\frac{|D_{F}||\mathcal{O}_{F}^{\times}|G(\phi)L(1,f\otimes\phi)}{8\pi^{2}\Omega_{f,{\operatorname{c}}}}\in\mathcal{O}[\phi].$

Similar with the case of $H_{c}^{1}$, the $f$-eigenspace
$H^{1}_{\text{par}}(Y_{1}(\mathfrak{N}),\mathcal{O})^{\prime}[f]$ is free
$\mathcal{O}$-module of rank one by Hida [5, Section 8]. Fix a free generator
$\eta_{f}$ of
$H^{1}_{\text{par}}(Y_{1}(\mathfrak{N}),\mathcal{O})^{\prime}[f]$, which is
determined up to multiplication by $\mathcal{O}^{\times}$. We define the
_period_ $\Omega_{f}\in\mathbb{C}^{\times}$ by a complex number such that
$\delta(f)=\Omega_{f}\eta_{f}$. Note that we have the _algebraic L-value_
$L_{f}(\phi)$ of $f$ by Hida [5, Theorem 8.1]:

$L_{f}(\phi):=\frac{|D_{F}||\mathcal{O}_{F}^{\times}|G(\phi)L(1,f\otimes\phi)}{8\pi^{2}\Omega_{f}}\in\overline{\mathbb{Q}}.$

To compare the $L$-values $\mathcal{L}_{f}(\phi)$ and $L_{f}(\phi)$, we have
to compare the periods $\Omega_{f,\operatorname{c}}$ and $\Omega_{f}$. We have
the following lemma about the Hecke action on the cohomology classes:

######  Lemma 7.2.

Let $j$ be the integer such that
$\mathfrak{a}_{i}\mathfrak{p}^{-1}=\mathfrak{a}_{j}$ in
$\operatorname{Cl}(F)$. Then we have

$T_{\mathfrak{p}}\eta_{f,i}=a_{f}(\mathfrak{p})\eta_{f,j},\
T_{\mathfrak{p}}\eta_{f,\operatorname{c},i}=a_{f}(\mathfrak{p})\eta_{f,\operatorname{c},j}.$

###### Proof.

Let us set $\pi_{i}(f):=h_{F}^{-1}\sum_{\psi\in
H_{F}}\psi^{-1}(a_{i})f\otimes\psi.$ By the definition of $f\otimes\psi$, we
have

$\delta\big{(}\pi_{i}(f)\big{)}=\frac{1}{h_{F}}\sum_{\psi\in
H_{F}}\sum_{j=1}^{h_{F}}\psi^{-1}(a_{i})\delta^{(j)}\big{(}(f\otimes\psi)_{(j)}\big{)}=\delta^{(i)}(f_{(i)}).$

Also we have

$T_{\mathfrak{p}}\pi_{i}(f)=\frac{1}{h_{F}}\sum_{\psi\in
H_{F}}a_{f}(\mathfrak{p})\psi^{-1}(\varpi_{\mathfrak{p}}^{-1}a_{i})f\otimes\psi=a_{f}(\mathfrak{p})\pi_{j}(f).$

From the above equations, we have

$T_{\mathfrak{p}}\eta_{f,\operatorname{c},i}=T_{\mathfrak{p}}\frac{\iota\big{(}\delta^{(i)}(f_{(i)})\big{)}}{\Omega_{f,\operatorname{c}}}=\frac{\iota\big{(}T_{\mathfrak{p}}\delta(\pi_{i}(f))\big{)}}{\Omega_{f,\operatorname{c}}}=a_{f}(\mathfrak{p})\eta_{f,\operatorname{c},j}.$

Note that we can do the same process for $\eta_{f}$, so we are done. ∎

Let us introduce the _non-Eisenstein condition_ :

$(\mathbf{Non\text{-}Eis})\
a_{f}(q\mathcal{O}_{F})-N(q)-1\in\mathcal{O}^{\times}\text{ for some prime
element }q\equiv 1\ (\text{mod }p\mathfrak{N}).$

Then this condition implies following:

######  Lemma 7.3.

Assume $(\mathbf{Non\text{-}Eis})$, then
$\Omega_{f}/\Omega_{f,{\operatorname{c}}}\in\mathcal{O}^{\times}$.

###### Proof.

We follow the proof of Namikawa [14, Proposition 2.6]. By our assumption,
there is a prime element $q\equiv 1\ (\text{mod }p\mathfrak{N})$ of $F$ such
that $\epsilon_{q}:=a_{f}(q\mathcal{O}_{F})-N(q)-1\in\mathcal{O}^{\times}$.
Let us denote by $\mathcal{E}_{q}$ the operator $T_{q\mathcal{O}_{F}}-N(q)-1$.

From the sequence (4.3), we obtain a Hecke-equivariant exact sequence:

${H^{0}(\partial
Y_{1}(\mathfrak{N})^{*},\mathcal{O})}$${H^{1}_{\operatorname{c}}(Y_{1}(\mathfrak{N}),\mathcal{O})}$${H^{1}_{\text{par}}(Y_{1}(\mathfrak{N}),\mathcal{O})}$${0.}$$\scriptstyle{{\operatorname{c}}}$par

We can consider $\eta_{f}$ to be the element of
$H_{\text{par}}^{1}(Y_{1}(\mathfrak{N}),\mathcal{O})$ or
$H_{\text{par}}^{1}(Y_{1}(\mathfrak{N}),\mathbb{C})$. Since the map par is
Hecke-equivariant, $\text{par}(\eta_{f,{\operatorname{c}}})=\alpha\eta_{f}$
for some $\alpha\in\mathcal{O}$ by the definition of $\eta_{f}$. Choose a
class $\eta\in H^{1}_{\operatorname{c}}(Y_{1}(\mathfrak{N}),\mathcal{O})$ such
that $\text{par}(\eta)=\eta_{f}$. Then we have
$\mathcal{E}_{q}(\eta_{f,{\operatorname{c}},1}-\alpha\eta_{1})=0$ by Namikawa
[14, Lemma 2.3]. Since $\eta_{f,{\operatorname{c}}}$ is an element of the
eigenspace, we obtain
$\alpha\mathcal{E}_{q}\eta_{1}=\epsilon_{q}\eta_{f,{\operatorname{c}},1}$ by
Lemma 7.2, which is a $\mathcal{O}$-genetator of
$H_{\operatorname{cusp}}^{1}(Y_{1}^{1}(\mathfrak{N}),\mathcal{O})[f]$. As
$\mathcal{E}_{q}\eta_{1}\in
H_{\operatorname{c}}^{1}(Y_{1}^{1}(\mathfrak{N}),\mathcal{O})$, we have
$\mathcal{E}_{q}\eta_{1}=\beta\eta_{f,{\operatorname{c}},1}$ for some
$\beta\in\mathcal{O}$. Taking the map par to the above equation, we obtain
$\epsilon_{q}\eta_{f,1}=\text{par}(\mathcal{E}_{q}\eta_{1})=\alpha\beta\eta_{f,1}$.
Note that $\eta_{f,1}$ is non-trivial since $f$ is an eigenform. Thus $\alpha$
is a $p$-adic unit and

$\frac{\Omega_{f}}{\Omega_{f,{\operatorname{c}}}}\eta_{f}=\text{par}(\eta_{f,{\operatorname{c}}})=\alpha\eta_{f}.$

Therefore, the ratio $\Omega_{f}/\Omega_{f,{\operatorname{c}}}$ is a $p$-adic
unit. ∎

From Lemma 7.3, the ratio of $\mathcal{L}_{f}(\phi)$ and $L_{f}(\phi)$ is a
$p$-adic unit under the assumption (Non-Eis). Also we can compare the pairings
(4.4) and (4.5) for $R=\mathcal{O}$.

###  7.2. Residual non-vanishing

In this subsection, we add assumption on $p$ that $F$ does not contain the
primitive $p$-th power roots and $p$ is coprime to
$|\mathcal{O}_{F}^{\times}|\mathfrak{N}$. Let $\mathfrak{M}_{p}$ be the
conductor of the extension $F(\zeta_{p})/F$ and
$\mathfrak{M}:=\mathfrak{M}_{p}\mathfrak{N}$. Note that by the straightforward
computation, we have

$\overline{\Gamma_{1}^{i}(\mathfrak{N})}=\bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL}_{2}(\mathcal{O}_{F}):a,d\in
1+\mathfrak{N},\ b\in\mathfrak{a}_{i},\ c\in\mathfrak{N}\mathfrak{a}_{i}^{-1}\
\bigg{\\}}.$

Let us denote $S(\mathfrak{M}_{p},\mathfrak{N})$ the set of pairs $(b,d)$ of
elements $b,d\in F$ such that $b$ is coprime to $d\mathfrak{M}_{p}$ and
$d\equiv 1\ (\text{mod }\mathfrak{M})$. From Namikawa [14, Lemma 3.2], for any
$(b,d)\in S(\mathfrak{M}_{p},\mathfrak{N})$, there is an infinite set
$P_{b,d}$ of prime elements of $F$ such that for each $\pi\in P_{b,d}$, we
have

  1. (1)

$\big{\\{}0,\frac{b}{d}\big{\\}}_{\overline{\Gamma_{1}^{1}(\mathfrak{N})}}=\big{\\{}0,\frac{b}{\pi}\big{\\}}_{\overline{\Gamma_{1}^{1}(\mathfrak{N})}}$.

  2. (2)

$\pi\in 1+\mathfrak{N}$.

  3. (3)

$N(\pi)-1$ is coprime to $p$.

  4. (4)

$N(\pi)-1\neq|\mathcal{O}_{F}^{\times}|$.

Let us denote by $\mathfrak{Y}$ the set of finite order Hecke characters over
$F$ such that if $\psi\in\mathfrak{Y}$, then the conductor of $\psi$ is
$\pi\mathcal{O}_{F}$ for some $\pi\in\bigcup_{(b,d)\in
S(\mathfrak{M}_{p},\mathfrak{N})}P_{b,d}$. Note that we have the following
conjecture on the generalization of the result of Stevens [19]:

######  Conjecture 7.4.

Assume $(\mathbf{Non\text{-}Eis})$. Then there exist infinitley many Hecke
characters $\phi$ such that $\mathcal{L}_{f}(\phi)$ is a $p$-adic unit.

By using the result of Namikawa [14], we prove the following partial result
toward the conjecture 7.4:

######  Theorem 7.5.

Assume $(\mathbf{Non\text{-}Eis})$, then $\mathcal{L}_{f}(\phi)$ is a $p$-adic
unit for some Hecke characters $\phi\in\mathfrak{Y}$.

###### Proof.

Let us recall that
$\operatorname{pr}^{1}:\overline{\Gamma^{1}_{1}(\mathfrak{N})}\rightarrow
H_{1}(X_{1}^{1}(\mathfrak{N}),\mathbb{Z})$ is a surjective homomorphism
defined by $\gamma\mapsto\\{0,\gamma\cdot
0\\}_{\overline{\Gamma^{1}_{1}(\mathfrak{N})}}$, which is trivial for the
parabolic elements. First, let us assume that
$\Lambda^{(1)}(\phi)\cap\eta_{f,c,1}=0$ in $\overline{\mathbb{F}}_{p}$ for any
$\phi\in\mathfrak{Y}$. Choose $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{1}^{1}(\mathfrak{M})$. Then we can
modify $\gamma$ by using the parabolic element
$\left(\begin{smallmatrix}1&t-b\\\ 0&1\end{smallmatrix}\right)$ of
$\Gamma_{1}^{1}(\mathfrak{M})$:

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\begin{pmatrix}1&t-b\\\
0&1\end{pmatrix}=\begin{pmatrix}a&b+a(t-b)\\\
c&d+c(t-b)\end{pmatrix}\in\Gamma_{1}^{1}(\mathfrak{M}),$

where $t$ is an element of $\mathcal{O}_{F}$ coprime to $\mathfrak{M}_{p}$.
Then $b+a(t-b)\in t+\mathfrak{M}_{p}$, thus we can assume that $b$ is coprime
to $\mathfrak{M}_{p}$. Let us choose $\pi\in P_{b,d}$. Let $C_{\pi}$ be the
set of finite order Hecke characters whose conductor dividing
$\pi\mathcal{O}_{F}$. Then $|C_{\pi}|$ divides $h_{F}(N(\pi)-1)$, thus
$|C_{\pi}|$ is a $p$-adic unit. By the orthogonality of characters, we have

$\displaystyle\operatorname{pr}^{1}(\gamma)=\Big{\\{}0,\frac{b}{\pi}\Big{\\}}_{\overline{\Gamma^{1}_{1}(\mathfrak{N})}}=\frac{1}{|C_{\pi}|}\sum_{\mathbf{1}\neq\phi\in
C_{\pi}}\overline{\phi}(b_{\pi\mathcal{O}_{F}})\Lambda^{(1)}(\phi)+\frac{1}{|C_{\pi}|}\sum_{u\in
R_{\pi\mathcal{O}_{F}}}\xi_{u}^{(1)}$

in $H_{1}(X_{1}^{1}(\mathfrak{N}),C_{1}^{1}(\mathfrak{N}),\mathcal{O}[\phi])$.
Note that $C_{\pi}\subset\mathfrak{Y}$ by the definition. Therefore, by the
equation (4.6), Lemma 7.3 and our assumption, we have

$\operatorname{pr}^{1}(\gamma)\cap\eta_{f,1}=\Big{\\{}0,\frac{b}{\pi}\Big{\\}}_{\overline{\Gamma^{1}_{1}(\mathfrak{N})}}\cap\frac{\Omega_{f,\operatorname{c}}}{\Omega_{f}}\eta_{f,{\operatorname{c}},1}=\frac{1}{|C_{\pi}|}\sum_{u\in
R_{\pi\mathcal{O}_{F}}}\xi_{u}^{(1)}\cap\frac{\Omega_{f,\operatorname{c}}}{\Omega_{f}}\eta_{f,{\operatorname{c}},1}$

in $\overline{\mathbb{F}}_{p}$, which is independent on $b$. Note that
$\left(\begin{smallmatrix}1&1\\\
\pi-1&\pi\end{smallmatrix}\right)\in\Gamma_{1}^{1}(\mathfrak{N})$, thus we
have

$\operatorname{pr}^{1}(\gamma)\cap\eta_{f,1}=\Big{\\{}0,\frac{1}{\pi}\Big{\\}}_{\overline{\Gamma^{1}_{1}(\mathfrak{N})}}\cap\frac{\Omega_{f,\operatorname{c}}}{\Omega_{f}}\eta_{f,{\operatorname{c}},1}=\operatorname{pr}^{1}\Big{(}\left(\begin{smallmatrix}1&0\\\
\pi-1&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\Big{)}\cap\frac{\Omega_{f,\operatorname{c}}}{\Omega_{f}}\eta_{f,{\operatorname{c}},1}=0$

in $\overline{\mathbb{F}}_{p}$ since $\left(\begin{smallmatrix}1&0\\\
\pi-1&1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)$ are parabolic elements of
$\Gamma_{1}^{1}(\mathfrak{N})$. By the property of the map pr1 and Namikawa
[14, Corollary A.2], we have $c_{1}\cap\eta_{f,1}=0$ in
$\overline{\mathbb{F}}_{p}$ for each $c_{1}\in
H_{1}(X_{1}^{1}(\mathfrak{N}),\mathbb{Z})$. Then from the extension of
coefficients
$H_{1}(X_{1}(\mathfrak{N}),\mathbb{Z})\otimes\overline{\mathbb{F}}_{p}\cong
H_{1}(X_{1}(\mathfrak{N}),\overline{\mathbb{F}}_{p})$ and the Poincaré
duality, we conclude that $\eta_{f,1}$ is trivial, which is a contradiction
since $f$ is an eigenform. Therefore,
$\Lambda^{(1)}(\phi)\cap\eta_{f,{\operatorname{c}},1}$ is a $p$-adic unit for
some $\phi\in\mathfrak{Y}$.

Let us assume the contrary of Theorem 7.5, i.e., assume that
$\mathcal{L}_{f}(\phi)=0$ in $\overline{\mathbb{F}}_{p}$ for any
$\phi\in\mathfrak{Y}$. Note that the conductor of $\psi\phi$ and $\phi$ are
same for each $\psi\in H_{F}$, thus we have $\psi\phi\in\mathfrak{Y}$ for
$\psi\in H_{F}$ and $\phi\in\mathfrak{Y}$. Therefore, we have

$\displaystyle 0$ $\displaystyle=\frac{1}{h_{F}}\sum_{\psi\in
H_{F}}\mathcal{L}_{f}(\psi\phi)=\sum_{i=1}^{h_{F}}\bigg{(}\frac{1}{h_{F}}\sum_{\psi\in
H_{F}}\psi(a_{i})\bigg{)}\sum_{u\in
R_{\mathfrak{f}(\phi)}}\phi(a_{i})\phi(\varpi_{\mathfrak{f}(\phi)}u)\xi_{u}^{(i)}\cap\eta_{f,{\operatorname{c}},i}$
$\displaystyle=\sum_{u\in
R_{\mathfrak{f}(\phi)}}\phi(a_{1})\phi(\varpi_{\mathfrak{f}(\phi)}u)\xi_{u}^{(1)}\cap\eta_{f,{\operatorname{c}},1}=\Lambda^{(1)}(\phi)\cap\eta_{f,{\operatorname{c}},1}$

in $\overline{\mathbb{F}}_{p}$ for any $\phi\in\mathfrak{Y}$, which is a
contradiction. ∎

##  8\. $p$-adic $L$-functions and $\mu$-invariants

In this section, we define the Bianchi modular version of Mazur-Tate-
Teitelbaum $p$-adic $L$-function (see [12]). Also we prove the partial result
toward the Bianchi modular version of Greenberg’s conjecture.

###  8.1. $p$-adic integral

Let $m>0$ be an integer. Let us fix a normalized eigenform $f\in
S_{2}(\mathfrak{N},\chi)$. Let $p$ be a odd prime number coprime to the
following ideal

$h_{F}D_{F}|\mathcal{O}_{F}^{\times}|\mathfrak{h}_{n}\mathfrak{N}\prod_{i=1}^{h_{F}}\mathfrak{a}_{i}^{-1},$

where $\mathfrak{h}_{n}$ is the integer defined in the Remark 6.7. Let
$\mathfrak{p}$ be an ordinary prime ideal lying above $p$. Let us denote
$W_{\mathfrak{p}}$ the torsion subgroup of
$\mathcal{O}_{F,\mathfrak{p}}^{\times}$. Let $\alpha_{f,\mathfrak{p}}$ be the
$p$-adic unit root of the equation
$x^{2}-a_{f}(\mathfrak{p})x+\chi(\varpi_{\mathfrak{p}})N(\mathfrak{p})=0$. Let
us set $\phi$ by a finite order Hecke character of $\mathfrak{p}$-power
conductor. Let
$\mathcal{O}:=\mathbb{Z}_{p}[\alpha_{f,\mathfrak{p}},\mu_{h_{F}},\phi,\chi]\mathbb{Z}_{p}[\\{a_{f}(\mathfrak{l}):\mathfrak{l}\\}]$.
Let $\varpi$ be a uniformizer of $\mathcal{O}$. Define a function $\nu_{f}$ on
the set of the basic open subsets of $\mathcal{O}_{F,\mathfrak{p}}$ as
follows:

$\nu_{f}(a+\mathfrak{p}^{m}\mathcal{O}_{F,\mathfrak{p}}):=\alpha_{f,\mathfrak{p}}^{-m}\sum_{i=1}^{h_{F}}\big{(}\xi^{(i)}_{m}(a)-\chi(\varpi_{\mathfrak{p}})\alpha_{\mathfrak{p}}^{-1}\xi^{(i)}_{m-1}(a)\big{)}\cap\eta_{f,{\operatorname{c}},i}\in\mathcal{O}.$

Then we have following:

######  Proposition 8.1.

$\nu_{f}$ is a $p$-adic measure on $\mathcal{O}_{F,\mathfrak{p}}$.

###### Proof.

Note that there are $h_{F}$-permutations $\sigma$ and $\tau$ such that

$\alpha_{i}\mathfrak{a}_{i}\mathfrak{p}^{-1}=\mathfrak{a}_{\sigma(i)},\
\beta_{i}\mathfrak{a}_{i}\mathfrak{p}=\mathfrak{a}_{\tau(i)}.$

Note that $\tau=\sigma^{-1}$ and $\beta_{\sigma(i)}=\alpha_{i}^{-1}$. Let us
denote by $r(\mathfrak{p})$ the matrix $\left(\begin{smallmatrix}1&0\\\
0&\varpi_{\mathfrak{p}}\end{smallmatrix}\right)$. For $\alpha\in F^{\times}$,
let us denote $\gamma_{\alpha}:=\left(\begin{smallmatrix}\alpha&0\\\
0&1\end{smallmatrix}\right)$ and
$w_{\alpha}:=\alpha^{(\infty)}\varpi_{\alpha\mathcal{O}_{F}}^{-1}\in\widehat{\mathcal{O}}_{F}^{\times}$.
As $(a_{i})_{\mathfrak{p}}=1$ for each $i$, we have
$(w_{\alpha_{i}})_{\mathfrak{p}}=(w_{\beta_{i}})_{\mathfrak{p}}=1$. Then by
Proposition 4.2, we obtain that

$\displaystyle\gamma_{\alpha_{i}}t_{i}(0;y)t\Big{(}\frac{a}{\varpi_{\mathfrak{p}}^{m}}\Big{)}r(\mathfrak{p})=\alpha_{i,\infty}^{\frac{1}{2}}\varpi_{\mathfrak{p}}t_{\sigma(i)}(0;|\alpha_{i}|y)\Big{(}0;\frac{\alpha_{i}}{|\alpha_{i}|}\Big{)}t\Big{(}\frac{a}{\varpi_{\mathfrak{p}}^{m-1}}\Big{)}\begin{pmatrix}w_{\alpha_{i}}&0\\\
0&1\end{pmatrix},$
$\displaystyle\gamma_{\beta_{i}}t_{i}(0;y)t\Big{(}\frac{a}{\varpi_{\mathfrak{p}}^{m}}\Big{)}\begin{pmatrix}\varpi_{\mathfrak{p}}&c\\\
0&1\end{pmatrix}=\beta_{i,\infty}^{\frac{1}{2}}t_{\tau(i)}(0;|\beta_{i}|y)\Big{(}0;\frac{\beta_{i}}{|\beta_{i}|}\Big{)}t\Big{(}\frac{a+\varpi_{\mathfrak{p}}^{m}c}{\varpi_{\mathfrak{p}}^{m+1}}\Big{)}\begin{pmatrix}w_{\beta_{i}}&0\\\
0&1\end{pmatrix}$

in $\operatorname{GL}_{2}(\mathbb{A}_{F})$ for each
$c\in\mathcal{O}_{F,\mathfrak{p}}$. Note that
$\left(\begin{smallmatrix}w_{\alpha_{i}}&0\\\ 0&1\end{smallmatrix}\right)$,
$\left(\begin{smallmatrix}w_{\beta_{i}}&0\\\ 0&1\end{smallmatrix}\right)\in
U_{1}(\mathfrak{N})$. From the definition of the Hecke operator
$T_{\mathfrak{p}}$ (Hida [5, Section 4]), $T_{\mathfrak{p}}(f)$ can be written
as

$T_{\mathfrak{p}}(f)(g):=f\big{(}gr(\mathfrak{p})\big{)}+\sum_{c\text{ mod
}{\mathfrak{p}}}f\bigg{(}g\begin{pmatrix}\varpi_{\mathfrak{p}}&c\\\
0&1\end{pmatrix}\bigg{)}.$

Together with the equation (4.8) and the above equations, we obtain

$\displaystyle
a_{f}(\mathfrak{p})\xi_{m}^{(i)}(a)\cap\iota\big{(}\delta^{(i)}(f_{(i)})\big{)}=$
$\displaystyle\chi(\varpi_{\mathfrak{p}})\xi_{m-1}^{(\sigma(i))}(a)\cap\iota\big{(}\delta^{(\sigma(i))}(f_{(\sigma(i))})\big{)}$
$\displaystyle+\sum_{c\text{ mod
}{\mathfrak{p}}}\xi_{m+1}^{(\tau(i))}(a+\varpi_{\mathfrak{p}}^{m}c)\cap\iota\big{(}\delta^{(\tau(i))}(f_{(\tau(i))})\big{)}.$

From the above equation, we obtain

$\sum_{b\equiv a\text{ mod
}\mathfrak{p}^{m}}\nu_{f}(b+\mathfrak{p}^{m+1}\mathcal{O}_{F,\mathfrak{p}})=\nu_{f}(a+\mathfrak{p}^{m}\mathcal{O}_{F,\mathfrak{p}}),$

which implies that $\nu_{f}$ is a $p$-adic distribution on
$\mathcal{O}_{F,\mathfrak{p}}$. ∎

Let $\langle\cdot\rangle:\mathcal{O}_{F,\mathfrak{p}}^{\times}\rightarrow
1+\mathfrak{p}\mathcal{O}_{F,\mathfrak{p}}$ be the projection map. Let us
define the $p$-_adic L-function_ $L_{\mathfrak{p}}(s,f,\phi)$ of $f$ by the
Mazur-Mellin transformation of $\nu_{f}$:

$L_{\mathfrak{p}}(s,f,\phi):=\int_{\mathcal{O}_{F,\mathfrak{p}}^{\times}}\langle
x\rangle^{-s}\phi(x)d\nu_{f}(x).$

Then our $p$-adic $L$-function interpolates the integral $L$-values
$\mathcal{L}_{f}(\phi)$:

######  Proposition 8.2.

If the Hilbert character part of $\phi$ is trivial and the conductor of $\phi$
is $\mathfrak{p}^{m}$ for $m\geq 1$, then we have

$L_{\mathfrak{p}}(0,f,\phi)=\alpha_{\mathfrak{p}}^{-m}\mathcal{L}_{f}(\phi).$

###### Proof.

From our assumption, the value $L_{\mathfrak{p}}(0,f,\phi)$ is given by

$\alpha_{f,\mathfrak{p}}^{-m}\mathcal{L}_{f}(\phi)-\alpha_{f,\mathfrak{p}}^{-m-1}\chi(\varpi_{\mathfrak{p}})\sum_{i=1}^{h_{F}}\sum_{a\in(\mathcal{O}_{F}/\mathfrak{p}^{m})^{\times}}\phi(a)\xi_{m-1}^{(i)}(a)\cap\eta_{f,\operatorname{c},i}.$

Then the latter term of the above equation can be written as follows:

$-\alpha_{f,\mathfrak{p}}^{-m-1}\chi(\varpi_{\mathfrak{p}})\sum_{i=1}^{h_{F}}\sum_{b\in(\mathcal{O}_{F}/\mathfrak{p}^{m-1})^{\times}}\sum_{d\text{
mod
}\mathfrak{p}}\phi(b)\phi(1+b^{-1}d\varpi_{\mathfrak{p}}^{m-1})\xi_{m-1,\mathfrak{N}}^{(i)}(b)\cap\eta_{f,\operatorname{c},i}.$

By the orthogonality of characters, the above equation is equal to zero.
Therefore, we obtain the desired result. ∎

Let $\gamma$ be a topological generator of
$1+\mathfrak{p}\mathcal{O}_{F,\mathfrak{p}}$, then we have the isomorphism
$(y\mapsto\gamma^{y}):\mathcal{O}_{F,\mathfrak{p}}\cong
1+\mathfrak{p}\mathcal{O}_{F,\mathfrak{p}}$. By the decomposition
$\mathcal{O}_{F,\mathfrak{p}}^{\times}=W_{\mathfrak{p}}\times(1+\mathfrak{p}\mathcal{O}_{F,\mathfrak{p}})$,
the $p$-adic Iwasawa power series $L_{\mathfrak{p}}(T;f,\phi)$ of
$L_{\mathfrak{p}}(s,f,\phi)$ is defined by

$L_{\mathfrak{p}}(T;f,\phi):=\int_{\mathcal{O}_{F,\mathfrak{p}}}T^{y}\phi(\gamma^{y})\sum_{\kappa\in
W_{\mathfrak{p}}}\phi(\kappa)d\nu_{f}(\kappa\gamma^{y})\in\mathcal{O}[[T-1]].$

Then we have
$L_{\mathfrak{p}}(\gamma^{-s};f,\phi)=L_{\mathfrak{p}}(s,f,\phi)$. By the
Weierstrass preparation theorem, there exists a product
$P(T)\in\mathcal{O}[[T-1]]$ of distinguished polynomials, a unit element
$U(T)\in\mathcal{O}[[T-1]]^{\times}$ and an integer $\mu$ such that
$L_{\mathfrak{p}}(T;f,\phi)=\pi^{\mu}P(T)U(T)$. Let us define the
$\mu$-_invariant_ $\mu\big{(}L_{\mathfrak{p}}(s,f,\phi)\big{)}$ of
$L_{\mathfrak{p}}(s,f,\phi)$ by $\mu$.

###  8.2. Results toward the vanishing of $\mu$-invariant

By using the results in the previous subsection, we can suggest the Bianchi
modular version of Greenberg conjecture :

######  Conjecture 8.3.

Assume $(\mathbf{Non\text{-}Eis})$, then
$\mu\big{(}L_{\mathfrak{p}}(s,f,\phi)\big{)}=0$.

In the main theorem, we discuss the $\mu$-invariant of
$L_{\mathfrak{p}}(s,f\otimes\psi,\phi)$ for $\psi\in H_{F}$ instead of
$\mu\big{(}L_{\mathfrak{p}}(s,f,\phi)\big{)}$ due to the technical issues. To
achieve this, we need to compare the periods of $f$ and $f\otimes\psi$:

######  Proposition 8.4.

Let $\alpha_{f,\psi,\operatorname{c}}\in\mathbb{C}_{p}$ be the ratio of the
periods:

$\alpha_{f,\psi,\operatorname{c}}:=\frac{\Omega_{f\otimes\psi,\operatorname{c}}}{\Omega_{f,\operatorname{c}}}$

Then $\alpha_{f,\psi,\operatorname{c}}\in\mathcal{O}^{\times}$ for any
$\psi\in H_{F}$.

###### Proof.

Let us choose $\psi\in H_{F}$. By the definition,
$\eta_{f\otimes\psi,\operatorname{c}}$ is a $\mathcal{O}$-generator of
$H^{1}_{\text{cusp}}(X_{1}(\mathfrak{N}),\mathcal{O})[f\otimes\psi]$. Note
that $f\otimes\psi$ is a normalized Hecke eigenform. Let us define the
cohomology classes $\eta_{f,\psi,\operatorname{c}}$ by

$\eta_{f,\psi,\operatorname{c}}:=\sum_{i=1}^{h_{F}}\psi(a_{i})\eta_{f,\operatorname{c},i}.$

We observe that the class $\eta_{f,\psi,\operatorname{c}}$ is an element in
$H_{\text{cusp}}^{1}(X_{1}(\mathfrak{N}),\mathcal{O})$. By Lemma 7.2, we
obtain
$T_{\mathfrak{q}}\eta_{f,\psi,\operatorname{c}}=a_{f}(\mathfrak{q})\psi(\varpi_{\mathfrak{q}})\eta_{f,\psi,\operatorname{c}}$
for almost all prime ideals $\mathfrak{q}$. Thus by the strong multiplicity
one theorem, we have

$\eta_{f,\psi,\operatorname{c}}\in
H_{\text{cusp}}^{1}(X_{1}(\mathfrak{N}),\mathcal{O})[f\otimes\psi],$

which implies that
$\eta_{f,\psi,\operatorname{c}}=\beta_{f,\psi,\operatorname{c}}\eta_{f\otimes\psi,\operatorname{c}}$
for some $\beta_{f,\psi,\operatorname{c}}\in\mathcal{O}$.

On the other hand, by the definition of $f\otimes\psi$, we have

$\alpha_{f,\psi,\operatorname{c}}\eta_{f\otimes\psi,\operatorname{c}}=\frac{\iota\big{(}\delta(f\otimes\psi)\big{)}}{\Omega_{f,\operatorname{c}}}=\sum_{i=1}^{h_{F}}\psi(a_{i})\frac{\iota\big{(}\delta^{(i)}(f_{(i)})\big{)}}{\Omega_{f,\operatorname{c}}}=\eta_{f,\psi,\operatorname{c}},$

which implies that
$\alpha_{f,\psi,\operatorname{c}}=\beta_{f,\psi,\operatorname{c}}\in\mathcal{O}.$
Since $(f\otimes\psi)\otimes\psi^{-1}=f$, we have
$\alpha_{f,\psi,\operatorname{c}}^{-1}=\alpha_{f\otimes\psi,\psi^{-1},\operatorname{c}}\in\mathcal{O}$.
Therefore, we conclude that $\alpha_{f,\psi,\operatorname{c}}$ is a $p$-adic
unit. ∎

Let $n\geq\max\\{22,m\\}$ be an integer, where $m$ is the integer such that
$\mathfrak{p}^{m}$ is the conductor of $\phi$. Let $\mathfrak{X}_{n}$ be the
arithmetic progression defined in Theorem 6.6. Let
$\omega:W_{\mathfrak{p}}\rightarrow\mathcal{O}_{F,\mathfrak{p}}^{\times}$ be
the Teichmüller character. Note that we can consider $\omega$ to be a Hecke
character of conductor $\mathfrak{p}$. Using the results on the full-rankness
and the periods, we obtain the following result toward Conjecture 8.3:

######  Theorem 8.5.

Assume $(\mathbf{Non}$-$\mathbf{Eis})$. If $\mathfrak{p}\in\mathfrak{X}_{n}$,
then we have
$\mu\big{(}L_{\mathfrak{p}}(s,f\otimes\psi,\phi\omega^{j})\big{)}=0$ for some
$\psi\in H_{F}$ and $0\leq j<p-1$.

###### Proof.

We observe that $f\otimes\psi$ is a $\mathfrak{p}$-ordinary normalized
eigenform for any $\psi\in H_{F}$. Assume the contrary, in other words,
$\mu\big{(}L_{\mathfrak{p}}(s,f\otimes\psi,\phi\omega^{j})\big{)}>0$ for any
$\psi\in H_{F}$ and integers $0\leq j<p-1$. Then for each sufficiently small
open subset $U$ of $\mathcal{O}_{F,\mathfrak{p}}$, we have

(8.1) $\int_{U}\phi\omega^{j}(\gamma^{y})\sum_{\kappa\in
W_{\mathfrak{p}}}\phi\omega^{j}(\kappa)d\nu_{f\otimes\psi}(\kappa\gamma^{y})\equiv
0\ (\text{mod }\varpi).$

Thus for each $a\equiv 1\ (\text{mod }\mathfrak{p})$, we obtain

(8.2) $\sum_{\kappa\in
W_{\mathfrak{p}}}\phi\omega^{j}(\kappa)\nu_{f\otimes\psi}(a\kappa+\mathfrak{p}^{n}\mathcal{O}_{F,\mathfrak{p}})\equiv
0\ (\text{mod }\varpi).$

Note that we have $\alpha_{f\otimes\psi,\mathfrak{p}}=\alpha_{f,\mathfrak{p}}$
for each $\psi\in H_{F}$ as $\psi(\varpi_{\mathfrak{p}})=1$. By using the
congruence $a_{f\otimes\psi}(\mathfrak{p})\equiv\alpha_{f,\mathfrak{p}}\
(\text{mod }\varpi)$, we can write the equation (8.2) by

(8.3) $\sum_{i=1}^{h_{F}}\sum_{\kappa\in
W_{\mathfrak{p}}}\phi\omega^{j}(\kappa)\big{(}a_{f}(\mathfrak{p})\xi^{(i)}_{n}(a\kappa)-\chi(\varpi_{\mathfrak{p}})\xi^{(i)}_{n-1}(a\kappa)\big{)}\cap\eta_{f\otimes\psi,{\operatorname{c}},i}\equiv
0\ (\text{mod }\varpi).$

From Proposition 8.4, we can replace the period $\Omega_{f\otimes\psi,c}$ by
$\Omega_{f,c}$, thus we can replace $\eta_{f\otimes\psi,c,i}$ by
$\psi(a_{i})\eta_{f,c,i}$. By applying the following operator

$\frac{1}{h_{F}(p-1)}\sum_{\psi\in
H_{F}}\sum_{j^{\prime}=0}^{p-2}\omega^{-j^{\prime}}(\nu)$

for $\nu\in W_{\mathfrak{p}}$ on the equation (8.3), we obtain

(8.4)
$\big{(}a_{f}(\mathfrak{p})\xi^{(1)}_{n}(a\nu)-\chi(\varpi_{\mathfrak{p}})\xi^{(1)}_{n-1}(a\nu)\big{)}\cap\eta_{f,\operatorname{c},1}\equiv
0\ (\text{mod }\varpi)$

for each $a\equiv 1\ (\text{mod }\mathfrak{p})$ and $\nu\in W_{\mathfrak{p}}$.
Hence for each $c\in
H_{1}(X_{1}^{1}(\mathfrak{N}),\overline{\mathbb{F}}_{p})$, we have

$c\cap\eta_{f,1}=\sum_{a,\nu}c_{a,\nu}\big{(}a_{f}(\mathfrak{p})\xi^{(1)}_{n}(a\nu)-\chi(\varpi_{\mathfrak{p}})\xi^{(1)}_{n-1}(a\nu)\big{)}\cap\frac{\Omega_{f,\operatorname{c}}}{\Omega_{f}}\eta_{f,\operatorname{c},1}\equiv
0\ (\text{mod }\varpi)$

due to the equations (4.6), (8.4), Proposition 6.8 and Lemma 7.3. Note that
the pairing $\cap$ which is defined in the equation (4.4) is compatible for
$R=\mathcal{O}$ and $R=\overline{\mathbb{F}}_{p}$ under the projection map
$(\text{mod }\pi):\mathcal{O}\rightarrow\overline{\mathbb{F}}_{p}$ by the
assumption on $\mathfrak{p}$. Therefore, it is a contradiction since $f$ is
normalized eigenform, $\eta_{f}$ is chosen to be $p$-optimal, and the pairing
$\cap$ is non-degenerate. ∎

######  Remark 8.6.

Note that if there is no ordinary prime ideals $\mathfrak{p}$ in
$\mathfrak{X}_{n}$, then our main theorem is vacuously true. Therefore, we
should verify that the set of ordinary prime ideals has positive density.

##  9\. Density of ordinary primes

Let $\ell$ be a rational prime. By using the result of Serre [18] and the
prime number theorem, the set of ordinary rational primes has density one for
any classical elliptic modular forms with no CM. In this section, we prove the
similar result.

Let $f\in
S_{2}\big{(}U_{0}(\mathfrak{N})\big{)}=S_{2}(\mathfrak{N},\mathbf{1})$ be a
normalized Hecke eigenform, which is neither CM nor the lifting of classical
elliptic modular newform. For simplicity, we call such $f$ _Genuine newform_.
Let us denote by $\mathbb{Q}_{f}$ the coefficient field of $f$ By Friedberg-
Hoffstein [3, Theorem A, Theorem B] and Taylor [21, Theorem 2, Corollary 4],
we obtain a continuous irreducible representation

$\rho_{f,\ell}:G_{F}\rightarrow\operatorname{GL}_{2}(\mathbb{Q}_{\ell})$

such that if $v$ is an unramified place of $F$ which does not divide
$\mathfrak{N}\ell$, then

  1. (1)

$\rho_{f,\ell}$ is unramified at $v$,

  2. (2)

Im$(\rho_{f,\ell}\oplus\rho^{\operatorname{c}}_{f,\ell})$ is dense in
$\\{(a,b)\in\operatorname{GL}_{2}^{2}:\det(a)=\det(b)\\}$, and

  3. (3)

either

$\text{char}\big{(}\rho_{f,\ell}(\text{Frob}_{v})\big{)}(X)=X^{2}-a_{f}(v)X+N(v)$

or $a_{f}(v^{\operatorname{c}})=0$ and

$\text{char}\big{(}\rho_{f,\ell}(\text{Frob}_{v})\big{)}(X)=X^{2}+N(v),$

where the first possibility occurs outside a set $\\{v\\}_{v}$ of density
zero.

From this, we have the following proposition:

######  Proposition 9.1.

For $x>0$, the number of places $v$ of $F$ such that
$\operatorname{Tr}\big{(}\rho_{f,\ell}(\operatorname{Frob}_{v})\big{)}=0$ and
$N(v)<x$ is given by

$O\bigg{(}\frac{x}{\operatorname{log}(x)^{\frac{3}{2}-\varepsilon}}\bigg{)}$

for any $\varepsilon>0$.

###### Proof.

Let
$G:=\text{Im}(\rho_{f,\ell})/\big{(}\mathbb{Q}_{\ell}^{\times}\cap\text{Im}(\rho_{f,\ell})\big{)}$,
$C$ the image of the set $\\{s\in\text{Im}(\rho_{f,\ell}):\text{Tr}(s)=0\\}$
in $G$. Note that Im$(\rho_{f,\ell})$ is dense in $\operatorname{GL}_{2}$,
thus $\dim_{\mathbb{Q}_{\ell}}(G)=3$ and $\dim_{\mathbb{Q}_{\ell}}(C)\leq 2$.
By the argument of Serre [18, Théorème 15], we have

$\inf_{s\in C}\dim_{\mathbb{Q}_{\ell}}(\text{Ad}(s)\rm{Id})=2,$

where Ad is the adjoint representation of the Lie algebra $\mathfrak{g}$ of
$G$ induced from $\rho_{f,\ell}$. Applying [18, Théorème 12] for a pair
$(G,C)$, we obtain the desired estimation. ∎

From this, we obtain the result toward Conjecture 8.3:

######  Theorem 9.2.

If $\mathbb{Q}_{f}=\mathbb{Q}$, then $f$ is $\mathfrak{l}$-ordinary for a set
$\\{\mathfrak{l}\\}_{\mathfrak{l}}$ of prime ideals of $F$ of density one.

###### Proof.

Let $\mathfrak{l}$ be a non-ordinary prime ideal of $F$ lying above a rational
prime $\ell$ such that $N(\mathfrak{l})>2^{\frac{64}{25}}$. Then we obtain
$a_{f}(\mathfrak{l})\in N(\mathfrak{l})\mathbb{Z}$. Assume that
$a_{f}(\mathfrak{l})\neq 0$, then we have

$N(\mathfrak{l})\leq|a_{f}(\mathfrak{l})|\leq
2N(\mathfrak{l})^{\frac{39}{64}}$

by Nakasuji [13, Corollary 1.1], which is a contradiction. Therefore, we have
$a_{f}(\mathfrak{l})=0$ if $\mathfrak{l}$ is non-ordinary and
$N(\mathfrak{l})>2^{\frac{64}{25}}$. The converse direction is clear.
Therefore, if $N(\mathfrak{l})$ is sufficiently large, then
$a_{f}(\mathfrak{l})$ is zero if and only if $\mathfrak{l}$ is non-ordinary.

On the other hand, by the prime number theorem and Proposition 9.1, the
density of prime ideals $\mathfrak{l}$ of $F$ such that
$\text{Tr}\big{(}\rho_{f,\ell}(\text{Frob}_{\mathfrak{l}})\big{)}=0$ is given
by

$\lim_{x\rightarrow\infty}\operatorname{log}(x)^{\varepsilon-\frac{1}{2}}=0.$

Note that the equation
$\text{Tr}\big{(}\rho_{f,\ell}(\text{Frob}_{\mathfrak{l}})\big{)}=a_{f}(\mathfrak{l})$
holds for a set of prime ideals $\\{\mathfrak{l}\\}_{\mathfrak{l}}$ of density
one. Hence we obtain the desired result. ∎

From Theorem 8.5 and Theorem 9.2, we obtain the main result of the present
paper:

######  Theorem 9.3.

Let $f\in S_{2}\big{(}U_{0}(\mathfrak{N})\big{)}$ be a Genuine newform such
that $\mathbb{Q}_{f}=\mathbb{Q}$. Let $\phi$ be a finite order Hecke character
of $\mathfrak{p}$-power conductor. Assume $(\mathbf{Non}$-$\mathbf{Eis})$,
then for a positive proportion of ordinary primes $\mathfrak{p}$, we have
$\mu\big{(}L_{\mathfrak{p}}(s,f\otimes\psi,\phi\omega^{j})\big{)}=0$ for some
$\psi\in H_{F}$ and $0\leq j<p-1$.

## Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government(MSIT)(No. 2019R1A2C108860913). The author
would like to thank his advisor Hae-Sang Sun for helpful discussions,
comments, and suggesting the research topic. The author also would like to
thank Ashay Burungale, Ming-Lun Hsieh, Chan-Ho Kim, Kenichi Namikawa and Chol
Park and for helpful discussions and comments. The author plans to generalize
the Hecke field generation results by Kim-Sun [7] and Sun [20] to the case of
Bianchi modular forms.

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