# An efficient analytical scheme for fuzzy
conformable fractional differential equations
arising in physical sciences

Hadi Eghlimi1, Mohammad Sadegh Asgari2

(1,2Department of Mathematics, Central Tehran Branch,
Islamic Azad University, Tehran, Iran.
<EMAIL_ADDRESS>
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###### Abstract

This article describes the fuzzy conformable fractional derivative which is
based on generalized Hukuhara differentiability. On these topics, we prove a
number of properties concerning this type of differentiability. In addition,
fuzzy conformable Laplace transforms are used to obtain analytical solutions
to the fractional differential equation. Through the use of several practical
examples, such as the fuzzy conformable fractional growth equation, the fuzzy
conformable fractional one-compartment model, and the fuzzy conformable
fractional Newton’s law of cooling, we demonstrate the effectiveness and
efficiency of the approaches.

Keywords: Generalized Hukuhara conformable fractional derivative, fuzzy
conformable Laplace transform, fuzzy conformable fractional growth equation,
one-compartment fuzzy conformable fractional model, fuzzy conformable
fractional Newton’s law of cooling.

2010 AMS Subject Classification: Primary: 26A33; 34A07 Secondary: 34A08.

##  1 Introduction

As fuzzy set theory has developed over the past decades, it has proven to be
an effective tool for modeling systems with uncertainty, providing the models
with a realistic glimpse of reality and enabling them to present a more
comprehensive view. Zadeh introduced fuzzy sets in 1965 [30]. Afterward, the
use of fuzzy sets in modeling increased considerably.

A large number of mathematical, physics, and engineering phenomena are
explained by differential equations. Fuzzy differential equations were first
constructed by Kaleva [20], and Seikkala [28]. There are several approaches to
studying fuzzy differential equations[10, 14, 23, 3, 26]. The first approach
[20] used the Hukuhara derivative (H-derivative) of a fuzzy function. It
should be noted that, for some fuzzy differential equations in this framework,
the diameter of the solution is unbounded as the time $\tau$ increases [13],
which is quite different from crisp differential equations. Consequently, Bede
and Gal introduced weakly generalized differentiability and strongly
generalized differentiability for fuzzy functions in [9]. However, the fuzzy
differential equations expressed by the strongly generalized derivative had no
unique solution. As a result, Stefanini and Bede defined generalized Hukuhara
difference and derivative for interval-valued functions and examined all
conditions for the existence of this kind of difference [11].

Fractional calculus has developed significantly in the past few decades.
Researchers have been using fractional calculus extensively for decades. A
fractional-order differential equation is applied to biological population
models, predator-prey models, infectious disease models, etc. Many of these
models have some uncertainty or ambiguity associated with their measurements
at the beginning. Considering these uncertainties and vagueness, the fuzzy
problem is more closely aligned with current reality and may be able to
express issues with a broader understanding. Researchers combined fuzzy-
yielding algorithms with fractional notions, resulting in a hybrid operator
called fuzzy fractional operator.

Fuzzy fractional calculus and fuzzy fractional differential equations have
emerged as a significant topic; see [2, 17, 18]. The authors in [6] utilized
the results reported in [2] and proved the existence and uniqueness of
fractional differential equations with uncertainty. The generalized Hukuhara
fractional Riemann-Liouville and Caputo gH-differentiability of fuzzy-valued
functions are discussed in [4, 5] . In [25] the authors considered the
solution to the fuzzy fractional initial value problem under Caputo gH-
differentiability using a modified fractional Euler method. A weak version of
the Pontryagin maximum principle for fuzzy fractional optimal control problems
depending on generalized Hukuhara fractional Caputo derivatives is established
in [15].

In recent years, there has been an increase in interest in the use of fuzzy
conformable fractional derivatives. According to our knowledge, all papers
that have used this method have rewritten the fuzzy conformable fractional
differential equation as two crisp fractional differential equations and
solved them using the usual methods. Moreover, the fuzzy conformable
fractional derivative was defined under strongly generalized differentiability
[7]. But the fuzzy conformable fractional differential equation expressed by
the strongly generalized derivative does not have a unique solution. In
comparison, this paper is devoted to developing a new fuzzy conformable
Laplace transform through fuzzy arithmetic. We obtain the fuzzy conformable
Laplace transform of fuzzy derivatives by considering the type of gH-
differentiability. The fuzzy analytical solution of the fuzzy conformable
fractional differential equations can be got without implicitly embedding them
into crisp equations through our method. Our approaches are demonstrated by
solving practical problems, including the fuzzy conformable fractional growth
equation, the fuzzy conformable fractional one-compartment model, and the
fuzzy conformable fractional Newton’s law of cooling. In particular, this
paper deals with the triangular fuzzy solutions of the following fuzzy
conformable fractional differential equation

$\displaystyle\left\\{\begin{array}[]{l}\mathfrak{w}:\mathbb{R}^{+}\rightarrow\Omega,\\\
\mathfrak{F}:\mathbb{R}^{+}\times\Omega\rightarrow\Omega\\\
~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}\mathfrak{w}(\tau)=\mathfrak{F}(\tau,\mathfrak{w}(\tau))\\\
{\mathfrak{w}}(0)=\mathfrak{w}_{0}\end{array}\right.$ (5)

The following will be symbolized in this context: $\tau\in\mathbb{R}^{+}$,
$\mathfrak{w}_{0}\in\Omega$, $\alpha\in(0,1]$, and $\Omega$ denotes the set of
all triangular fuzzy numbers. Here,
$~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))$ is the fuzzy
conformable fractional generalized Hukuhara derivative for $\alpha$ on the
domain of $\mathbb{R}^{+}$.

Now, let’s take a quick look at the contents. Section 2 deals with aspects of
background knowledge in fuzzy mathematics with emphasis on the generalized
Hukuhara differentiability. Fuzzy fractional conformable derivative based on
the concept of generalized Hukuhara difference is defined in Section 3 and
some important properties for this kind of differentiability are proved. The
fuzzy conformable Laplace method is presented in Section 4. Section 5 studies
the fuzzy fractional initial value problem using the concept of conformable
gH-differentiability and the fuzzy analytical triangular solutions for the
fuzzy growth equation, one-compartment fuzzy fractional model and fuzzy
Newton’s law of cooling by considering the type of conformable
differentiability obtained using the conformable Laplace method in Section 5.
The paper ends with Section 6 that presents the conclusions.

##  2 Fundamental Awareness

Fuzzy numbers are a generalization of classical real numbers in the sense that
it does not refer to one single value but rather to a connected set of
possible values, where each possible value has its own weight between 0 and 1.
A fuzzy set $\mathcal{P}:\mathbb{R}\rightarrow[0,1]$ represents a fuzzy number
if it is normal, fuzzy convex, compactly supported and upper semi-continuous
on $\mathbb{R}$. Then $(\mathcal{F},\mathcal{P})$ is called the space of fuzzy
numbers.

Initially, for all $r\in(0,1]$, put
$[\mathcal{P}]^{r}=\\{s\in\mathbb{R}:\mathcal{P}(s)\geq r\\}$ and
$[\mathcal{P}]^{0}=\overline{\\{s\in\mathbb{R}:\mathcal{P}(s)>0\\}}$. Then
$\mathcal{P}\in\mathcal{F}$ iff $[\mathcal{P}]^{r}$ is convex compact in
$\mathbb{R}$ and $[\mathcal{P}]^{1}\neq\phi$ [12]. Indeed, if
$\mathcal{P}\in\mathcal{F}$ then
$[\mathcal{P}]^{r}=\left[\mathcal{P}_{1}(r),\mathcal{P}_{2}(r)\right]$ where,
$\mathcal{P}_{1}(r)=\min\left\\{s\in[\mathcal{P}]^{r}\right\\}$ and
$\mathcal{P}_{2}(r)=\max\left\\{s\in[\mathcal{P}]^{r}\right\\}$.

Let $\mathcal{P},\mathcal{Q}\in\mathcal{F}$. The generalized Hukuhara
difference between in two fuzzy numbers is the fuzzy number $\mathcal{R}$, (if
it exists), such that

$\displaystyle\mathcal{P}\circleddash_{gH}\mathcal{Q}=\mathcal{R}\Longleftrightarrow~{}\mathcal{P}=\mathcal{Q}\oplus\mathcal{R},\quad
or\quad~{}\mathcal{Q}=\mathcal{P}\oplus(-1)\mathcal{R}.$

Throughout of this paper we assume that
$\mathcal{P}\circleddash_{gH}\mathcal{Q}\in\Omega$

######  Definition 2.1.

(See [8]). Let $\mathfrak{w}:\mathbb{J}\rightarrow\mathcal{F}$ be a fuzzy
function and $\tau_{0}\in\mathbb{J}$. If

$\displaystyle\forall\epsilon>0~{}\exists\delta>0~{}\forall\tau\big{(}0<|\tau-\tau_{0}|<\delta\Rightarrow
D(\mathfrak{w}(\tau),L)<\epsilon\big{)},$

where, $D$ is the Hausdorff distance and $\mathbb{J}$ is a bonded domain in
$\mathbb{R}$. Then, we say that $L\in\mathcal{F}$ is limit of $\mathfrak{w}$
in $\tau_{0}$, which is denoted by
$\lim_{\tau\rightarrow\tau_{0}}\mathfrak{w}(\tau)=L$. Also the fuzzy function
$\mathfrak{w}$ is said to be fuzzy continuous if

$\displaystyle\lim_{\tau\rightarrow\tau_{0}}\mathfrak{w}(\tau)=\mathfrak{w}(\tau_{0}).$

A function $\mathfrak{w}:\mathbb{J}\rightarrow\Omega$ is called fuzzy
triangular function and
$\mathfrak{w}(\tau)=\Big{(}\mathfrak{w}_{1}(\tau),\mathfrak{w}_{2}(\tau),\mathfrak{w}_{3}(\tau)\Big{)}$,
for all $\tau\in\mathbb{J}$. Suppose that the notation
$C^{\mathbb{F}}(\mathbb{J},\Omega)$ denotes the set of all fuzzy triangular
functions which are fuzzy continuous on all of $\mathbb{J}$.

######  Theorem 2.1.

( See [8]) Let $\nu,\mathfrak{w}:\mathbb{J}\rightarrow\mathcal{F}$ be two
fuzzy functions. If $\lim_{\tau\rightarrow c}\nu(\tau)=L_{1}$ and
$\lim_{\tau\rightarrow c}\mathfrak{w}(\tau)=L_{2}$, such that
$L_{1},L_{2}\in\mathcal{F}$ then

$\displaystyle\lim_{\tau\rightarrow
c}[\nu(\tau)\ominus_{gH}\mathfrak{w}(\tau)]=L_{1}\ominus_{gH}L_{2}.$

######  Definition 2.2.

(See [12]) The generalized Hukuhara derivative (gH-derivative) of a fuzzy-
valued function $\mathfrak{w}:(a,b)\rightarrow\Omega$ at $\tau_{0}\in(a,b)$ is
defined as

$\mathfrak{w}^{\prime}_{gH}(\tau_{0})=\lim_{\hbar\rightarrow
0}\frac{\mathfrak{w}(\tau_{0}+\hbar)\circleddash_{gH}\mathfrak{w}(\tau_{0})}{\hbar},$

provided that $~{}\mathfrak{w}^{\prime}_{gH}(\tau_{0})\in\mathcal{F}~{}$.
Considering the definition of gH-difference, this derivative has the following
two cases

  * •

Case 1.( $\mathrm{[(i)-gH]}-$differentiability)

$\displaystyle\mathfrak{w}_{i.gH}^{\prime}(\tau)=\Big{(}\mathfrak{w}^{\prime}_{1}(\tau),\mathfrak{w}^{\prime}_{2}(\tau),\mathfrak{w}^{\prime}_{3}(\tau)\Big{)},$

  * •

Case 2.( $\mathrm{[(ii)-gH]}-$differentiability)

$\displaystyle\mathfrak{w}_{ii.gH}^{\prime}(\tau)=\Big{(}\mathfrak{w}^{\prime}_{3}(\tau),\mathfrak{w}^{\prime}_{2}(\tau),\mathfrak{w}^{\prime}_{1}(\tau)\Big{)},$

######  Definition 2.3.

(See [27]) A fuzzy function $\mathfrak{w}$ is piecewise continuous on the
interval $[0,\infty)$ if

1.

$\lim_{\tau\rightarrow 0^{+}}\mathfrak{w}(\tau)=\mathfrak{w}(0^{+}).$

2.

$\mathfrak{w}$ is continuous on every finite interval $(0,b)$ expect possibly
at a finite number of points $\tau_{1},\tau_{2},...,\tau_{n}$ in $(0,b)$ at
which $\mathfrak{w}$ has jump discontinuity.

######  Definition 2.4.

(See [12]) Let $\mathfrak{w}:\mathbb{J}\rightarrow\mathcal{F}$ be a triangular
fuzzy function, then

$\displaystyle\int_{a}^{b}\mathfrak{w}(\tau)\mathrm{d}\tau=\Big{(}\int_{a}^{b}\mathfrak{w}_{1}(\tau)\mathrm{d}\tau,\int_{a}^{b}\mathfrak{w}_{2}(\tau)\mathrm{d}\tau,\int_{a}^{b}\mathfrak{w}_{3}(\tau)\mathrm{d}\tau\Big{)}.$

######  Theorem 2.2.

(See [16])(Integration by part) Consider
$\mathfrak{w}:\mathbb{J}\rightarrow\mathcal{F}$ and $f(\tau)$ be
$gH-$differentiable such that the type of differentiability does not change in
$\mathbb{J}$. If $q(\tau)$ is a differentiable real-valued function such that
$q(\tau)>0$ and $q^{\prime}(\tau)<0$, then

1

. If $\mathfrak{w}(\tau)$ is a $\mathrm{[(i)-gH]}-$differentiable function and

$\displaystyle\int_{a}^{b}q(\tau)\odot\mathfrak{w}^{\prime}_{gH}(\tau)\mathrm{d}\tau$
$\displaystyle=$ $\displaystyle q(b)\odot\mathfrak{w}(b)\ominus
q(a)\odot\mathfrak{w}(a)\oplus\int_{a}^{b}(-q^{\prime}(\tau))\odot\mathfrak{w}(\tau)\mathrm{d}\tau.$

2

. If $\mathfrak{w}(\tau)$ is a $\mathrm{[(ii)-gH]}-$differentiable function
then

$\displaystyle\int_{a}^{b}q(\tau)\odot\mathfrak{w}^{\prime}_{gH}(\tau)\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle(-q(a))\odot\mathfrak{w}(a)\ominus(-q(b))\odot\mathfrak{w}(b)\ominus_{gH}\int_{a}^{b}q^{\prime}(\tau)\odot\mathfrak{w}(\tau)\mathrm{d}\tau.$

##  3 Fractional Conformable Calculus on Fuzzy Functions

This section introduces a new definition of a conformable derivative based on
the generalized Hukuhara differentiability, and several important properties
for this type of differentiability will be expressed and proven.

######  Definition 3.1.

Let $\mathfrak{w}:[\mathrm{a},\infty)\rightarrow\mathcal{F}$ be a fuzzy
function, $\mathrm{a}\geq 0$ and $\alpha\in(0,1)$. The generalized Hukuhara
conformable fractional derivative of $\mathfrak{w}$ of order $\alpha$ is
defined by

${}_{gH}\mathbf{T}^{\mathrm{a}}_{\alpha}(\mathfrak{w}(\tau))=\lim_{\epsilon\rightarrow
0}\frac{\mathfrak{w}\left(\tau+\epsilon(\tau-\mathrm{a})^{1-\alpha}\right)\ominus_{gH}\mathfrak{w}(\tau)}{\epsilon},\quad\quad\forall\tau\in[a,\infty),$

provided that
${}_{gH}\mathbf{T}^{\mathrm{a}}_{\alpha}(\mathfrak{w}(\tau))\in\mathcal{F}$.
If the generalized Hukuhara conformable fractional derivative of
$\mathfrak{w}$ of order $\alpha$ exists, then we simply say $\mathfrak{w}$ is
$\alpha_{gH}$-differentiable.

######  Theorem 3.1.

If a fuzzy function $\mathfrak{w}:[0,\infty)\rightarrow\mathcal{F}$ is
$\alpha_{gH}$-differentiable at $\tau_{0}>0,~{}\alpha\in(0,1)$, then
$\mathfrak{w}$ is a fuzzy continuous function at $\tau_{0}$.

######  Proof.

We have
$\mathfrak{w}(\tau_{0}+\varepsilon\tau_{0}^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau_{0})=\frac{\mathfrak{w}(\tau_{0}+\varepsilon\tau_{0}^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau_{0})}{\varepsilon}\odot\varepsilon$.
By using Theorem 2.1 we conclude that

$\displaystyle\lim_{\varepsilon\rightarrow
0}[\mathfrak{w}(\tau_{0}+\varepsilon\tau_{0}^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau_{0})]=\lim_{\varepsilon\longrightarrow
0}\frac{\mathfrak{w}(\tau_{0}+\varepsilon\tau_{0}^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau_{0})}{\varepsilon}\odot\lim_{\varepsilon\longrightarrow
0}\varepsilon$

then

$\displaystyle\lim_{\varepsilon\rightarrow
0}[\mathfrak{w}(\tau_{0}+\varepsilon\tau_{0}^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau_{0})]=_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau_{0}))\odot
0$

Now, let $\hbar=\varepsilon\tau_{0}^{1-\alpha}$, therefore

$\displaystyle\lim_{\hbar\rightarrow
0}[\mathfrak{w}(\tau_{0}+\hbar)\ominus_{gH}\mathfrak{w}(\tau_{0})]=0$

Therefore, according to Definition 2.1, it can be concluded that the function
$\mathfrak{w}$ is a fuzzy continuous function. ∎

######  Theorem 3.2.

Let $\alpha\in(0,1)$ and
$\mathfrak{w}:[\mathrm{a},\infty)\rightarrow\mathcal{F}$ be
$\alpha_{gH}$-differentiable fuzzy function at a point $\tau>0$. Then

${}_{gH}\mathbf{T}^{\mathrm{a}}_{\alpha}(\mathfrak{w}(\tau))=(\tau-\mathrm{a})^{1-\alpha}\mathfrak{w}^{\prime}_{gH}(\tau)$

######  Proof.

In Definition 3.1, let $\hbar=\varepsilon(\tau-\mathrm{a})^{1-\alpha}$ and
then $\varepsilon=(\tau-\mathrm{a})^{\alpha-1}\hbar$. Hence

${}_{gH}\mathbf{T}^{\mathrm{a}}_{\alpha}(\mathfrak{w}(\tau))$ $\displaystyle=$
$\displaystyle\lim_{\varepsilon\rightarrow
0}\frac{\mathfrak{w}(\tau+\varepsilon(\tau-\mathrm{a})^{1-\alpha})\ominus_{gH}\mathfrak{w}(\tau)}{\varepsilon}$
$\displaystyle=$ $\displaystyle\lim_{\hbar\rightarrow
0}\frac{\mathfrak{w}(\tau+\hbar)\ominus_{gH}\mathfrak{w}(\tau)}{\hbar(\tau-\mathrm{a})^{\alpha-1}}$
$\displaystyle=$
$\displaystyle(\tau-\mathrm{a})^{1-\alpha}\lim_{\hbar\rightarrow
0}\frac{\mathfrak{w}(\tau+\hbar)\ominus_{gH}\mathfrak{w}(\tau)}{\hbar}$
$\displaystyle=$
$\displaystyle(\tau-\mathrm{a})^{1-\alpha}\mathfrak{w}^{\prime}_{gH}(\tau)$

So the desired result was obtained.∎

######  Definition 3.2.

Let $\alpha\in(0,1)$ and $\mathfrak{w}:[0,\infty)\rightarrow\Omega$ is
$\alpha_{gH}$-differentiable at a point $\tau_{0}>0$. We can say that
$\mathfrak{w}(\tau)$ is

  * $(a)$

$\alpha_{i.gH}$-differentiable function at $\tau_{0}$ if and only if

${}_{i.gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau_{0}))=\Big{(}T_{\alpha}^{0}(\mathfrak{w}_{1}(\tau_{0})),T_{\alpha}^{0}(\mathfrak{w}_{2}(\tau_{0})),T_{\alpha}^{0}(\mathfrak{w}_{3}(\tau_{0}))\Big{)},$
(6)

  * $(b)$

$\alpha_{ii.gH}$-differentiable function at $\tau_{0}$ if and only if

${}_{ii.gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau_{0}))=\Big{(}T_{\alpha}^{0}(\mathfrak{w}_{3}(\tau_{0})),T_{\alpha}^{0}(\mathfrak{w}_{2}(\tau_{0})),T_{\alpha}^{0}(\mathfrak{w}_{1}(\tau_{0}))\Big{)}.$
(7)

Where $T_{\alpha}^{0}(\mathfrak{w}_{i}(\tau_{0}))$ for $i=1,2,3$, are the
conformable fractional derivatives for the real-valued functions
$\mathfrak{w}_{i}(\tau_{0})$, respectively [24].

######  Definition 3.3.

We say that a point $\xi_{0}\in(0,\infty)$, is a switching point for the
differentiability of $\mathfrak{w}$, if in any neighborhood $\mathcal{V}$ of
$\xi_{0}$ there exist points $\xi_{1}<\xi_{0}<\xi_{2}$ such that

Type I.

at $\xi_{1}$ (6) holds while (7) does not hold and at $\xi_{2}$ (7) holds and
(6) does not hold, or

Type II.

at $\xi_{1}$ (7) holds while (6) does not hold and at $\xi_{2}$ (6) holds and
(7) does not hold.

######  Example 3.1.

Consider the fuzzy function $\mathfrak{w}:[0,\pi]\rightarrow\Omega$ defined by

$\displaystyle\mathfrak{w}(\tau)=\Big{(}2.3\sin(\tau),5.6\sin(\tau),9.7\sin(\tau)\Big{)}$

We have the following $\alpha_{gH}$-derivatives of $\mathfrak{w}(\tau)$

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{i.gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))=\Big{(}2.3\tau^{\frac{1}{2}}\cos(\tau),5.6\tau^{\frac{1}{2}}\cos(\tau),9.7\tau^{\frac{1}{2}}\cos(\tau)\Big{)}&\hbox{$\forall\tau\in[0,\frac{\pi}{2}]$}\\\
\\\
~{}_{ii.gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))=\Big{(}9.7\tau^{\frac{1}{2}}\cos(\tau),5.6\tau^{\frac{1}{2}}\cos(\tau),2.3\tau^{\frac{1}{2}}\cos(\tau)\Big{]}&\hbox{$\forall\tau\in[\frac{\pi}{2},\pi]$}\end{array}\right.$

Figure 1: $\mathfrak{w}(\tau)$ (left) and its
${}_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))$-derivative for different
level of r-cut $r\in[0,1]$ (right) of Example 3.1

Therefore, the fuzzy function $\mathfrak{w}(\tau)$ is
$\alpha_{i.gH}-$differentiable function on $\tau\in[0,\frac{\pi}{2}]$. This
function is switched to $\alpha_{ii.gH}-$differentiable at
$\tau=\frac{\pi}{2}$. Hence, the point $\tau=\frac{\pi}{2}$ is a switching
point of Type I for the the differentiability of $\mathfrak{w}$.

######  Theorem 3.3.

Let $\mathfrak{w}:[0,\infty)\rightarrow\Omega$ be a fuzzy function and
$\tau_{0}\in[0,\infty)$.

(a)

$\mathfrak{w}$ is $\mathrm{[(i)-gH]}-$ differentiable at $\tau_{0}$ if and
only if $\mathfrak{w}$ is $\alpha_{i.gH}$-differentiable at $\tau_{0}$.

(b)

$\mathfrak{w}$ is $\mathrm{[(ii)-gH]}-$ differentiable at $\tau_{0}$ if and
only if $\mathfrak{w}$ is $\alpha_{ii.gH}$-differentiable at $\tau_{0}$.

######  Proof.

The desired result can be obtained easily by using Theorem 3.2 and
Definition2.2. ∎

######  Lemma 3.1.

Let $\alpha\in(0,1]$ and $\mathfrak{w}$, $\mathfrak{w}$ are
$\alpha_{gH}-$differentiable at a point $\tau>0$. Then

1.

${}_{i.gH}\textbf{T}_{\alpha}^{0}[\mathfrak{w}(\tau)\oplus\nu(\tau)]=_{i.gH}\textbf{T}_{\alpha}^{0}(\mathfrak{w}(\tau))\oplus_{i.gH}\textbf{T}_{\alpha}^{0}(\nu(\tau))$.

2.

${}_{ii.gH}\textbf{T}_{\alpha}^{0}[\mathfrak{w}(\tau)\oplus\nu(\tau)]=_{ii.gH}\textbf{T}_{\alpha}^{0}(\mathfrak{w}(\tau))\oplus_{ii.gH}\textbf{T}_{\alpha}^{0}(\nu(\tau))$.

######  Proof.

Let $\mathfrak{w}$, $\nu$ are $[\alpha_{i.gH}]$-differentiable functions for
all $\tau>0$, then

${}_{i.gH}\textbf{T}_{\alpha}^{0}(\mathfrak{w}(\tau))$
$\displaystyle\oplus_{i.gH}\textbf{T}_{\alpha}^{0}(\nu(\tau))$
$\displaystyle=\Big{(}T_{\alpha}(\mathfrak{w}_{1}(\tau)),T_{\alpha}(\mathfrak{w}_{2}(\tau)),T_{\alpha}(\mathfrak{w}_{3}(\tau))\Big{)}\oplus\Big{(}T_{\alpha}(\nu_{1}(\tau)),T_{\alpha}(\nu_{2}(\tau)),T_{\alpha}(\nu_{3}(\tau))\Big{)}$
$\displaystyle=\Big{(}T_{\alpha}(\mathfrak{w}_{1}(\tau))+T_{\alpha}(\nu_{1}(\tau)),T_{\alpha}(\mathfrak{w}_{2}(\tau))+T_{\alpha}(\nu_{2}(\tau)),T_{\alpha}(\mathfrak{w}_{3}(\tau))+T_{\alpha}(\nu_{3}(\tau))\Big{)}$
$\displaystyle=\Big{(}T_{\alpha}(\mathfrak{w}_{1}(\tau)+\nu_{1}(\tau)),T_{\alpha}(\mathfrak{w}_{2}(\tau)+\nu_{2}(\tau)),T_{\alpha}(\mathfrak{w}_{3}(\tau)+\nu_{3}(\tau))\Big{)}$
$\displaystyle=_{i.gH}\textbf{T}^{\alpha}_{0}(\mathfrak{w}(\tau)\oplus\nu(\tau))$

In a similar way, we can proof the other case. ∎

######  Definition 3.4.

(See [7]) Assume $\mathfrak{w}\in
C^{\mathbb{F}}([\mathrm{a},\infty),\mathcal{F})$ and $\alpha\in(0,1)$. Then
the fuzzy conformable fractional integral is constructed as

$\displaystyle\mathcal{I}^{\mathrm{a}}_{\alpha}\left(\mathfrak{w}(\tau)\right)=\int_{\mathrm{a}}^{t}\left(\xi-\mathrm{a}\right)^{\alpha-1}\mathfrak{w}(\xi)\mathrm{d}\xi$

######  Remark 1.

If $\mathfrak{w}\in C^{F}([\mathrm{a},\infty),\Omega)$ and
$\mathfrak{w}(\tau)=\Big{(}\mathfrak{w}_{1}(\tau),\mathfrak{w}_{2}(\tau),\mathfrak{w}_{3}(\tau)\Big{)}$
then it is clear that

$\displaystyle\mathcal{I}_{\mathrm{a}}^{\alpha}\left(\mathfrak{w}(\tau)\right)=\Big{(}I_{\mathrm{a}}^{\alpha}\left(\mathfrak{w}_{1}(\tau)\right),I_{\mathrm{a}}^{\alpha}\left(\mathfrak{w}_{2}(\tau)\right),I_{\mathrm{a}}^{\alpha}\left(\mathfrak{w}_{3}(\tau)\right)\Big{)},$

where $I_{\mathrm{a}}^{\alpha}\left(\mathfrak{w}_{i}(\tau)\right)$ for
$i=1,2,3$ are the conformable fractional integral definition for the real-
valued functions $\mathfrak{w}_{i}(\tau)$ [1].

##  4 The Fuzzy Fractional Conformable Laplace Transform

This section will introduce the fuzzy conformable Laplace transform for one-
variable fuzzy-valued functions and prove some important properties for this
transformation.

######  Definition 4.1.

A fuzzy function $\mathfrak{w}$ is said to be fuzzy conformable exponentially
bounded if it satisfies in the following inequality

$\displaystyle D\Big{(}\mathfrak{w}(\tau),0\Big{)}\leq
Me^{c\frac{t^{\alpha}}{\alpha}},$

where $M$ and $c$ are positive real constants and $0<\alpha<1$, for all
sufficiently large $t$.

######  Definition 4.2.

Let $\mathfrak{w}:[\tau_{0},\infty)\rightarrow\mathcal{F}$ be a fuzzy
function. The fuzzy fractional conformable Laplace transform of order $\alpha$
of fuzzy function $\mathfrak{w}(\tau)$ is defined as follows

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)=\textbf{W}^{\tau_{0}}_{\alpha}(s)=\int_{\tau_{0}}^{\infty}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\mathfrak{w}(\tau)\mathrm{d}\tau,$

whenever the limit exist.

######  Remark 2.

Consider the fuzzy function
$\mathfrak{w}(\tau)=\Big{(}\mathfrak{w}_{1}(\tau),\mathfrak{w}_{2}(\tau),\mathfrak{w}_{3}(\tau)\Big{)}$,
then by attention to Definition 2.4, the fuzzy fractional conformable Laplace
transform of this function is denoted by

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)$
$\displaystyle=$
$\displaystyle\Big{(}L_{\alpha}^{\tau_{0}}\\{\mathfrak{w}_{1}(\tau)\\},~{}L_{\alpha}^{\tau_{0}}\\{\mathfrak{w}_{2}(\tau)\\},L_{\alpha}^{\tau_{0}}\\{\mathfrak{w}_{3}(\tau)\\}\Big{)},$

where $L_{\alpha}^{\tau_{0}}$ is the definition of classical fractional
conformable Laplace transform for the real-valued functions
$\mathfrak{w}_{i}(\tau)$ [1, 24], and

$\displaystyle
L_{\alpha}^{\tau_{0}}\\{\mathfrak{w}_{i}(\tau)\\}=\int_{\tau_{0}}^{\infty}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}v_{i}(\tau)d\tau=\lim_{\mathcal{R}\rightarrow\infty}\int_{\tau_{0}}^{\mathcal{R}}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}v_{i}(\tau)d\tau,\quad
i=1,2,3.$

######  Definition 4.3.

Let $\mathfrak{w}(\tau)$ is a fuzzy function. If
$\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)=\textbf{W}^{\tau_{0}}_{\alpha}(s)$
then $\\{\mathfrak{L}_{\alpha}^{\tau_{0}}\\}^{-1}$ is used to denote the
inverse fuzzy fractional conformable Laplace transform of
$\textbf{W}^{\tau_{0}}_{\alpha}(s)$ and we have

$\displaystyle\\{\mathfrak{L}_{\alpha}^{\tau_{0}}\\}^{-1}[\textbf{W}^{\tau_{0}}_{\alpha}(s)]=\mathfrak{w}(\tau),\quad\quad\tau\geq
0,$

which maps the fuzzy fractional conformable Laplace transform of a fuzzy
function back to the original function.

######  Lemma 4.1.

Let $\mathfrak{w}$ be piecewise continuous on $[\tau_{0},\infty)$ and fuzzy
conformable exponentially bounded. Then the fuzzy conformable Laplace
transform of $\mathfrak{w}(\tau)$ exists for all s provided $Re(s)>c$.

######  Proof.

$\mathfrak{w}$ is conformable exponentially bounded function, so there exists
$\tau_{0},M_{1}$ and $c$ such that

$\displaystyle D(\mathfrak{w}(\tau),0)\leq
M_{1}e^{c\frac{\tau^{\alpha}}{\alpha}},$

for all $\tau\geq\tau_{0}$. Furthermore, $\mathfrak{w}$ is piecewise
continuous on $[0,\tau_{0}]$ and hence bounded there, so

$\displaystyle D(\mathfrak{w}(\tau),0)\leq
M_{2},\quad\forall\tau\in[0,\tau_{0}].$

This means that, a constant $\mathcal{M}$ can be chosen sufficiently large so
that
$D(\mathfrak{w}(\tau),0)\leq\mathcal{M}e^{c\frac{\tau^{\alpha}}{\alpha}}$.
Now, let
$\mathfrak{L}_{\alpha}^{0}\\{\mathfrak{w}(\tau)\\}=\textbf{W}_{\alpha}^{0}(s)$,
therefore,

$\displaystyle
D\Big{(}\int_{0}^{\mathcal{R}}\tau^{\alpha-1}e^{-s\frac{\tau^{\alpha}}{\alpha}}\mathfrak{w}(\tau)\mathrm{d}\tau,0\Big{)}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\mathcal{R}}\tau^{\alpha-1}e^{-s\frac{\tau^{\alpha}}{\alpha}}D\Big{(}\mathfrak{w}(\tau),0\Big{)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\mathcal{M}\int_{0}^{\mathcal{R}}\tau^{\alpha-1}e^{(c-s)\frac{\tau^{\alpha}}{\alpha}}\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\frac{\mathcal{M}}{s-c}-\frac{e^{(c-s)\frac{\mathcal{R}^{\alpha}}{\alpha}}}{s-c}$

Letting $\mathcal{R}\rightarrow\infty$, so when $Re(s)>c$ we have

$\displaystyle D\Big{(}\textbf{W}_{\alpha}^{0}(s),0\Big{)}$
$\displaystyle\leq$ $\displaystyle\frac{\mathcal{M}}{s-c}.$

The proof is complete. ∎

######  Lemma 4.2.

Let $\mathfrak{w}:[\tau_{0},\infty)\rightarrow\mathbb{R}_{F}$ be a fuzzy
function such that
$\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)=\textbf{W}_{\alpha}^{\tau_{0}}(s)$
exists. Then

$\displaystyle\textbf{W}_{\alpha}^{\tau_{0}}(s)=\mathcal{L}[\mathfrak{w}(\tau_{0}+(\alpha\tau)^{\frac{1}{\alpha}}],\quad\quad
0<\alpha\leq 1$

where
$\mathcal{L}[\nu(\tau)]=\int_{0}^{\infty}e^{-s\tau}\nu(\tau)\mathrm{d}\tau$ is
the fuzzy Laplace transform for the fuzzy function $\nu(\tau)$ [27].

######  Proof.

Let $u=\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}$, so the proof is clear. ∎

######  Lemma 4.3.

Consider $\mathfrak{w}(\tau)$ and $\mathfrak{w}(\tau)$ are two fuzzy
functions. Let $\mathrm{a}$, $\mathrm{b}$ are two real constant such that
$\mathrm{a},\mathrm{b}>0$ (or $\mathrm{a},\mathrm{b}<0$). If the fuzzy
conformable Laplace transform $\mathfrak{w}$ and $\mathfrak{w}$ exist hence
the fuzzy conformable Laplace transform of
$\mathrm{a}\mathfrak{w}(\tau)\oplus\mathrm{b}\mathfrak{w}(\tau)$ and
$\mathrm{a}\mathfrak{w}(\tau)\circleddash_{gH}\mathrm{b}g\mathfrak{w}(\tau)$
exist, and

  1. 1.

$\mathrm{a}\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)\ominus_{gH}\mathrm{b}\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\nu(\tau)\\}(s)=\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathrm{a}\mathfrak{w}(\tau)\ominus_{gH}\mathrm{b}\nu(\tau)\\}(s)$.

  2. 2.

$\mathrm{a}\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)\oplus\mathrm{b}\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\nu(\tau)\\}(s)=\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathrm{a}\mathfrak{w}(\tau)\oplus\mathrm{b}\nu(\tau)\\}(s)$

######  Proof.

The proof of this Lemma is clear with respect to Lemma 4.2 and Lemma 3.1 in
[22].∎

######  Theorem 4.1.

Let us consider $\tau_{0}\in\mathbb{R}$, $0<\alpha\leq 1$ and
$\mathfrak{w}:[\tau_{0},\infty)\rightarrow\mathbb{R}_{F}$ be
$\alpha_{gH}$-differentiable fuzzy function provided that the type of
$\alpha_{gH}$-differentiability doesn’t change in interval
$[\tau_{0},\infty)$. Then

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\Big{\\{}~{}_{i.gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\Big{\\}}(s)=s\textbf{W}_{\alpha}^{\tau_{0}}(s)\ominus\mathfrak{w}(\tau_{0}),$
$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\Big{\\{}~{}_{ii.gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\Big{\\}}(s)=(-1)\mathfrak{w}(\tau_{0})\ominus_{gH}(-1)s\textbf{W}_{\alpha}^{\tau_{0}}(s),$

where
$\textbf{W}_{\alpha}^{\tau_{0}}(s)=\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)$.

######  Proof.

First, let $\mathfrak{w}(\tau)$ is a $\alpha_{i.gH}-$differentiable triangular
fuzzy function. According to the definition of conformable Laplace transform
for a fuzzy function, we have

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=\lim_{\mathcal{R}\rightarrow\infty}\int_{\tau_{0}}^{\mathcal{R}}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}~{}_{gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\mathrm{d}\tau,$

provided that the limit exists. Using Theorem 3.2, we conclude that

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=\lim_{\mathcal{R}\rightarrow\infty}\int_{\tau_{0}}^{\mathcal{R}}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\mathfrak{w}^{\prime}_{gH}(\tau)\mathrm{d}\tau,$

Now, let $\mathfrak{w}$ be a $\alpha_{i.gH}-$ differentiable function. Then
Theorem 3.3 and Theorem 2.2 result

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{i.gH}$
$\displaystyle\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=\lim_{\mathcal{R}\rightarrow\infty}\int_{\tau_{0}}^{\mathcal{R}}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\mathfrak{w}^{\prime}_{i.gH}(\tau)\mathrm{d}\tau$
$\displaystyle=\lim_{\mathcal{R}\rightarrow\infty}\bigg{[}\Big{(}e^{-s\frac{(\mathcal{R}-\tau_{0})^{\alpha}}{\alpha}}\odot\mathfrak{w}(\mathcal{R})\ominus\mathfrak{w}(\tau_{0})\Big{)}\oplus
s\int_{0}^{\mathcal{R}}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\odot\mathfrak{w}(\tau)\mathrm{d}\tau\bigg{]}.$

Therefore, we have

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{i.gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=s\textbf{W}_{\alpha}^{\tau_{0}}(s)\ominus\mathfrak{w}(\tau_{0}).$

Now, let $\mathfrak{w}$ be a $\alpha_{ii.gH}-$ differentiable function, then
Theorem 3.3 and Theorem 2.2 result

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}$
$\displaystyle\\{~{}_{ii.gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=\lim_{\mathcal{R}\rightarrow\infty}\int_{\tau_{0}}^{\mathcal{R}}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\mathfrak{w}^{\prime}_{ii.gH}(\tau)\mathrm{d}\tau$
$\displaystyle=\lim_{\mathcal{R}\rightarrow\infty}\bigg{[}\Big{(}(-1)\mathfrak{w}(\tau_{0})\ominus(-1)e^{-s\frac{(\mathcal{R}-\tau_{0})^{\alpha}}{\alpha}}\odot\mathfrak{w}(\mathcal{R})\Big{)}\ominus_{gH}(-1)s\int_{0}^{\mathcal{R}}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\odot\mathfrak{w}(\tau)\mathrm{d}\tau\bigg{]}$
$\displaystyle=(-1)\mathfrak{w}(\tau_{0})\ominus_{gH}(-1)\lim_{\mathcal{R}\rightarrow\infty}s\int_{0}^{\mathcal{R}}(\tau-\tau_{0})^{\alpha-1}e^{-s\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\odot\mathfrak{w}(\tau)\mathrm{d}\tau.$

Hence

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{ii.gH}\textbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=(-1)\mathfrak{w}(\tau_{0})\ominus_{gH}(-1)s\textbf{W}_{\alpha}^{\tau_{0}}(s).$

∎

##  5 The Fuzzy Conformable Fractional Initial Value Problem

Consider the following fuzzy conformable fractional initial value problem

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))=\mathfrak{F}(\tau,\mathfrak{w}(\tau)),&\hbox{}\\\
\\\
\mathfrak{w}(\tau_{0})=\Big{(}\mathfrak{w}_{1}(\tau_{0}),\mathfrak{w}_{2}(\tau_{0}),\mathfrak{w}_{3}(\tau_{0})\Big{)},&\hbox{}\end{array}\right.,$
(12)

where $\mathfrak{w}(\tau_{0})$ is a fuzzy triangular continuous function.
There exists at most one solution for the problem (12) [7].

The fuzzy conformable fractional initial value problem (12) can be solved
using the fuzzy Laplace transform method and based on the following steps

i.

Take the fuzzy Laplace transform of this fuzzy initial value problem using all
theorems and properties in Section 4 as necessary.

ii.

Put initial conditions into the resulting equation.

iii.

Solve for the output variable.

iv.

Use the fuzzy inverse Laplace transform.

In the following, we will obtain an analytical triangular fuzzy solution for
several practical examples such as the fuzzy conformable fractional Growth
equation, the one-compartment fuzzy conformable fractional model and the fuzzy
conformable fractional Newton’s law of cooling by using the fuzzy conformable
Laplace transform.

###  5.1 The Fuzzy Growth Equation

The growth of microorganisms, such as bacteria, fungi, and algae, is governed
by differential equations or systems of differential equations. Despite the
fact that the conditions in the laboratory may vary over time, actual
conditions in the real world may differ from laboratory conditions. Thus,
getting a precise figure for the colony’s population is virtually impossible.
If $\mathfrak{w}(\tau)$ represents the population at any given time, then
$\tau$ would represent the population at that time. In addition, let
$\mathfrak{w}_{0}$ be the fuzzy initial population at time $\tau_{0}$, that
is, $\mathfrak{w}(\tau_{0})=\mathfrak{w}_{0}$. Then if the population grows
exponentially

$\displaystyle\text{(Rate of change of population at time }\tau)=\kappa(\text{
Current population at time }\tau)$

In mathematical terms, this can be written as

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))=\kappa\odot\mathfrak{w}(\tau),&\hbox{$0<\alpha<1$,}\\\
\\\ \mathfrak{w}(\tau_{0})=\mathfrak{w}_{0},&\hbox{}\end{array}\right.$ (16)

The value $\kappa$ is known as the relative growth rate and is a positive real
constant and
$\mathfrak{w}_{0}=\Big{(}\mathfrak{w}_{0_{1}},\mathfrak{w}_{0_{2}},\mathfrak{w}_{0_{3}}\Big{)}$
is a triangular fuzzy constant.

######  Remark 3.

It is easy to investigate that $\kappa$ if in Eq.(16) is a positive real
constant then the fuzzy solution of this equation is a
$\alpha_{i.gH}$-differentiable function and if $\kappa$ is a negative real
constant then Eq.(16) has $\alpha_{ii.gH}$-differentiable fuzzy solution.

In the following, we will obtain the fuzzy analytical triangular solution of
problem (16) with the fuzzy initial condition $\mathfrak{w}(\tau_{0})$ by the
fuzzy conformable Laplace transform. The fuzzy conformable Laplace transform
is applied to the problem (16). We had assumed that $\kappa$ is a positive
real constant, hence according to Remark 3 and Theorem 4.1 and the fuzzy
initial condition $\mathfrak{w}(\tau_{0})$, we have

$\displaystyle\mathfrak{L}_{\alpha}^{\tau_{0}}\\{~{}_{gH}\mathbf{T}^{t_{0}}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=\kappa\odot\mathfrak{L}_{\alpha}^{\tau_{0}}\\{\mathfrak{w}(\tau)\\}(s)\quad\Rightarrow\quad
s\textbf{W}_{\alpha}^{\tau_{0}}(s)\ominus\mathfrak{w}_{0}=\kappa\odot\textbf{W}_{\alpha}^{\tau_{0}}(s)$

We can rearrange the above equation by using some basic rules of fuzzy
arithmetic as follows

$\displaystyle(s-\kappa)\odot\textbf{W}_{\alpha}^{\tau_{0}}(s)$
$\displaystyle=$
$\displaystyle\mathfrak{w}_{0},\quad\quad\Rightarrow\quad\textbf{W}_{\alpha}^{\tau_{0}}(s)=\frac{\mathfrak{w}_{0}}{(s-\kappa)}$

The fuzzy inverse Laplace transform is applied to the above equation

$\displaystyle\textbf{W}_{\alpha}^{\tau_{0}}(s)$ $\displaystyle=$
$\displaystyle\Big{(}\\{L^{\tau_{0}}_{\alpha}\\}^{-1}\\{\frac{\mathfrak{w}_{0_{1}}}{(s-\kappa)}\\},\\{L^{\tau_{0}}_{\alpha}\\}^{-1}\\{\frac{\mathfrak{w}_{0_{2}}}{(s-\kappa)}\\},\\{L^{\tau_{0}}_{\alpha}\\}^{-1}\\{\frac{\mathfrak{w}_{0_{3}}}{(s-\kappa)}\\}\Big{)}$

Therefore, the fuzzy analytical triangular solution of problem (16) with the
fuzzy initial condition $\mathfrak{w}(\tau_{0})$ is

$\displaystyle\mathfrak{w}(\tau)$ $\displaystyle=$
$\displaystyle\Big{(}\mathfrak{w}_{0_{1}}e^{\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},\mathfrak{w}_{0_{2}}e^{\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},\mathfrak{w}_{0_{3}}e^{\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\Big{)}$

######  Example 5.1.

In the refrigerator, you have an old jar of yogurt that is growing bacteria.
The number of bacteria in the yogurt jar will be denoted by
$\mathfrak{w}(\tau)$ in this problem. Let the initial fuzzy population of
bacteria at time zero equal $\mathfrak{w}(0)=\Big{(}516,540,598\Big{)}$ and
$\kappa=\frac{1}{30}$.

As bacteria number changes over time $\tau$, the fuzzy initial valued problem
describes the changing number of bacteria as being

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))=\kappa\odot\mathfrak{w}(\tau),&\hbox{}\\\
\\\ \mathfrak{w}(0)=\Big{(}516,540,598\Big{)},&\hbox{}\end{array}\right.$

Now, suppose that $\alpha=\frac{1}{5}$, so the number of bacteria over time
$\tau$ is obtained by using the fuzzy conformable Laplace transform as follows

$\displaystyle\mathfrak{w}(\tau)=\Big{(}516e^{\frac{1}{6}\tau^{\frac{1}{5}}},540e^{\frac{1}{6}\tau^{\frac{1}{5}}},598e^{\frac{1}{6}\tau^{\frac{1}{5}}}\Big{)}$

This exact solution for various $\tau$ in $[0,1]$ have been reported in
Figures 2.

Figure 2: Graphs of fuzzy exact solution of Example 5.1

The r-cut of this solution, $\mathfrak{w}(\tau)$ and
$~{}_{gH}\mathbf{T}^{0}_{\frac{1}{5}}(\mathfrak{w}(\tau))$ of this solution
for $0\leq r\leq 1$ are showed in Figures 3. As you can see , the position of
lower cut(blue) and upper cut(red) isn’t changed. It shows that
$\mathfrak{w}(\tau)$ is $\alpha_{i.gH}-$differentiable.

(a).$r-$cut of $\mathfrak{w}(\tau)$

(b).$r$-cut of $~{}_{gH}\mathbf{T}^{0}_{\frac{1}{5}}(\mathfrak{w}(\tau))$

Figure 3: r-cut representation of different functions for Example 5.1 for all
$r\in[0,1]$.

###  5.2 The One-compartment Fuzzy Fractional Model

The one-compartment open model refers to the rate and extent of distribution
of a drug to different tissues, and the rate of elimination of the drug. The
rate of drug movement between compartments is described by the first order
kinetics. If $\mathfrak{w}(\tau)$ denotes the concentration of drug in the
compartment at time $\tau$, then the rate of change of $\mathfrak{w}(\tau)$ is

$\displaystyle~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))=(-1)\kappa\odot\mathfrak{w}(\tau),\quad
0<\alpha<1,$

where $\kappa$ is the elimination rate constant [22].

Now, by the fuzzy conformable Laplace transform we have

$\displaystyle\mathfrak{L}_{\alpha}^{0}\\{~{}_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=(-1)\kappa\mathfrak{L}_{\alpha}^{0}\\{\mathfrak{w}(\tau)\\}(s)$

Let $\mathfrak{w}_{0}$ represents the fuzzy initial amount of a drug. Remark 3
and Theorem 3.2 yield to

$\displaystyle(-1)\mathfrak{w}_{0}\ominus_{gH}(-1)s\textbf{W}_{\alpha}^{\tau_{0}}(s)=(-1)\kappa\textbf{W}_{\alpha}^{\tau_{0}}(s),\quad\Rightarrow\quad\textbf{W}_{\alpha}^{\tau_{0}}(s)=\frac{\mathfrak{w}_{0}}{s+\kappa}$

By applying the inverse conformable Laplace transform we obtain

$\displaystyle\mathfrak{w}(\tau)=\Big{(}\mathfrak{w}_{0_{1}}e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},\mathfrak{w}_{0_{2}}e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},\mathfrak{w}_{0_{3}}e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\Big{)}$

######  Example 5.2.

Assume that $\Big{(}3.5,4.3,5.1\Big{)}$ mg of a highly hydrophilic drug (like
Aminoglycosides) is injected into the body at time $\tau_{0}$. The fuzzy
initial valued problem that describes the amount of this drug in the tissues
is

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))=(-1)\kappa\odot\mathfrak{w}(\tau)\\\
\mathfrak{w}(\tau_{0})=\Big{(}3.97,4.3,5.1\Big{)}\end{array}\right.$

Hence, the amount of this drug in the tissues, $\mathfrak{w}(\tau)$, is

$\displaystyle\mathfrak{w}(\tau)=\Big{(}3.97e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},4.3e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}},5.1e^{-\kappa\frac{(\tau-\tau_{0})^{\alpha}}{\alpha}}\Big{)}$

###  5.3 The Fuzzy Newton Fractional Conformable Cooling Law

The temperature difference in any situation results from energy flow into a
system or energy flow from a system to the surroundings. The former leads to
heating, whereas the latter leads to cooling. Newton’s law of cooling states
that the rate of change temperature is proportional to the difference between
the body’s temperature and the surrounding medium’s temperature.

Consider a body in the temperature $\mathfrak{w}$ placed in a medium of
temperature $\mathcal{M}$ medium, which is considered a fuzzy constant. This
law can be written in the form.

$\displaystyle~{}_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))=(-1)\kappa\odot(\mathfrak{w}\ominus\mathcal{M})\quad
0<\alpha<1,$ (19)

where $\kappa$ is a real positive constant of the cooling coefficient.
Consider that the initial temperature of the body
$\mathfrak{w}(0)=\mathfrak{w}_{0}$ is a fuzzy constant.

By Remark 3, the Eq.(19) has a $\alpha_{ii.gH}-$ differential solution.
Applying the fuzzy conformable Laplace transform to problem (19) and using
Lemma 4.3 we have

$\displaystyle\mathfrak{L}_{\alpha}^{0}\\{~{}_{gH}\mathbf{T}^{0}_{\alpha}(\mathfrak{w}(\tau))\\}(s)=(-1)\kappa\odot\Big{(}\mathfrak{L}_{\alpha}^{0}\\{\mathfrak{w}(\tau)\\}(s)\ominus\mathfrak{L}_{\alpha}^{0}\\{\mathcal{M}\\}(s)\Big{)}$

So the body’s temperature is

$\displaystyle\mathfrak{w}(\tau)=(\mathfrak{w}_{0}\ominus\mathcal{M})e^{-\kappa\frac{\tau^{\alpha}}{\alpha}}\oplus\mathcal{M}$

######  Example 5.3.

At midnight the furnace fails inside a building such that the outside
temperature at a constant $\Big{(}6.8,7,7.85\Big{)}^{\circ}$ F and the inside
temperature at $\Big{(}59.1,70,80.6\Big{)}^{\circ}$ F. The fuzzy initial
valued problem that describes the temperature inside the building is

$\displaystyle\left\\{\begin{array}[]{ll}~{}_{gH}\mathbf{T}^{\tau_{0}}_{\alpha}(\mathfrak{w}(\tau))=(-1)\kappa\odot\bigg{(}\mathfrak{w}\ominus\Big{(}6.8,7,7.85\Big{)}\bigg{)}\\\
\mathfrak{w}(0)=\Big{(}59.1,70,80.6\Big{)}\end{array}\right.$

Let $\alpha=\frac{1}{2}$ and $\kappa=\frac{1}{20}$. Then by using the method
described above, the temperature inside the building decrease

$\displaystyle\mathfrak{w}(\tau)=\Big{(}52.6,63,72.75\Big{)}e^{-\frac{1}{10}\tau^{\frac{1}{2}}}\oplus\Big{(}6.8,7,7.85\Big{)}$

The r-cut of this solution, $\mathfrak{w}(\tau)$ and
$~{}_{gH}\mathbf{T}^{0}_{\frac{1}{2}}(\mathfrak{w}(\tau))$ of this solution
for $0\leq r\leq 1$ are showed in Figures 4. As you can see , the position of
lower cut(blue) and upper cut(red) is changed. It shows that
$\mathfrak{w}(\tau)$ is $\alpha_{ii.gH}-$differentiable.

(a).$r-$cut of $\mathfrak{w}(\tau)$

(b).$r$-cut of $~{}_{gH}\mathbf{T}^{0}_{\frac{1}{2}}(\mathfrak{w}(\tau))$

Figure 4: r-cut representation of different functions for Example 5.3 for all
$r\in[0,1]$.

##  6 Conclusion

This study discusses the fuzzy conformable fractional order initial problem
within the context of conformable generalized Hukuhara differentiability. A
fuzzy conformable fractional derivative based on generalized Hukuhara
differentiability was introduced, and a number of associated properties were
shown on the topics. As a result, the fuzzy conformable Laplace transform was
employed to determine the analytical solutions for the fractional differential
equation. A number of practical applications such as fuzzy Newton’s law of
cooling, fuzzy growth equation, and one-compartment fuzzy fractional model,
have been used to demonstrate the effectiveness and efficiency of the
approaches. According to the results, the fuzzy conformable Laplace transform
is an effective and convenient tool for solving fuzzy conformable fractional
differential equations. We have obtained interesting results here that could
be used to conduct future studies on fuzzy fractional partial differential
equations under conformable gH-differentiability.

## Acknowledgements

This work was partially supported by the Central Tehran Branch of Islamic Azad
University.

### Compliance with Ethical Standards

### Conflict of interest

All authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals
performed by any of the authors.

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#  $p$-adic measures for reciprocals of $L$-functions of totally real number
fields

Razan Taha

###### Abstract.

We construct $p$-adic measures which interpolate the special values of
reciprocals of $p$-adic $L$-functions of totally real number fields $K$ at
negative integers. These measures are defined by analyzing the non-constant
term of partial Eisentein series of the Hilbert modular group of $K$.

###### Contents

  1. 1 Introduction
  2. 2 Hilbert Modular Forms
    1. 2.1 Structure of the Hilbert Modular Group
    2. 2.2 Modular Forms on Congruence Subgroups
    3. 2.3 Eisenstein Series
  3. 3 Fourier Coefficients of Certain Eisenstein Series
  4. 4 Constructing a p-adic Measure
    1. 4.1 p-adic Measures on Galois Groups
    2. 4.2 The Main Construction
  5. 5 p-adic Measures of Totally Real Fields
    1. 5.1 Invertibility of Measures in the Iwasawa Algebra
    2. 5.2 Boundedness of $\lambda$

##  1\. Introduction

Let $K$ be a totally real number field of finite degree over $\mathbb{Q}$ and
let $K(p^{\infty})$ be the maximal abelian unramified-outside-$p$ extension of
$K.$ The goal of this paper is to construct a $p-$adic measure $\lambda$ on
the Galois group $G\left(K(p^{\infty})/K\right)$ from the non-constant terms
of the Fourier expansion of holomorphic Eisenstein series on
$SL_{2}(\mathcal{O}),$ where $\mathcal{O}$ is the ring of integers of $K$.
Moreover, the Mellin transform of $\lambda$ turns out to be the reciprocal of
the $p-$adic $L-$function of the totally real field $K$. This $L$-function is
the $p$-adic analogue of the classical complex valued Hecke $L$-function and
has several different constructions which we will briefly describe.

The study of $p$-adic $L$-functions starts with the fact that the values of
the Riemann zeta function at negative integers are rational and can be
expressed in terms of Bernoulli numbers as

$\zeta(1-k)=(-1)^{k+1}\frac{B_{k}}{k},\,\,\,\forall k>0.$

Kummer [Kum51] proved that these values satisfy special congruences modulo
powers of a prime number p: given two even integers $k$ and $k^{\prime}$ which
are not divisible by $p-1$ and for all $d>0,$ if $k\equiv k^{\prime}\bmod
p^{d-1}(p-1)$ then

(1.1) $(1-p^{k-1})\zeta(1-k)\equiv(1-p^{k^{\prime}-1})\zeta(1-k^{\prime})\bmod
p^{d}.$

Kubota and Leopoldt [KL64] were the first to interpret the Kummer congruences
as a $p$-adic function which interpolates the values of the Riemann zeta
function at negative integers. They proved that there exists a continuous
function $L_{p}$ on $\mathbb{Z}_{p}$ such that for all positive integers $k$
which are congruent to $0\bmod p-1,$

(1.2) $L_{p}(\chi,1-k)=L(\chi,1-k)(1-\chi(p)p^{k-1})$

where the right hand side of (1.2) is the Dirichlet $L$-function with the
Euler factor at $p$ removed.

For a totally real field $K,$ the Hecke $L$-function attached to a character
$\chi$ with conductor $\mathfrak{n}$ is given by

(1.3)
$L(s,\chi)=\sum_{(\mathfrak{a},\mathfrak{n})=1}\frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^{s}}=\sum_{[\mathfrak{a}]}\chi([\mathfrak{a}])\sum_{\mathfrak{b}\in[\mathfrak{a}]}\frac{1}{N(\mathfrak{b})^{s}}$

where $[\mathfrak{a}]$ denotes the class of the ideal $\mathfrak{a}.$ Siegel
[Sie70] proved that the values of the partial $L$-function
$\sum_{\mathfrak{b}\in[\mathfrak{a}]}N(\mathfrak{b})^{k-1}$ are rational,
which led to several different constructions of the $p$-adic $L$-function for
a totally real field. Serre [Ser73] constructed $p$-adic modular forms and
obtained the $p$-adic $L$-function as the constant term of a $p$-adic
Eisenstein series. Shintani [Shi76] proved an elementary formula for this
rational number in terms of certain generalized Bernoulli numbers, which
Cassou-Nogues [CN79] then used to define the p-adic L-function in terms of
generalized Bernoulli numbers. Coates and Sinnott [CS74] equivalently proved
the generalized Kummer congruences for real quadratic fields and Deligne and
Ribet [DR80] proved them in general for any totally real field.

One long term goal of this project is to study $p$-adic $L$-functions using
methods inspired by the Langlands program. In the complex setting, the
Langlands-Shahidi method establishes the analytic continuation and functional
equations of many $L$-functions by studying the non-constant terms of the
Fourier expansion of Eisenstein series on reductive groups. In particular, the
denominators of these Fourier coefficients contain suitable $L$-functions.
This inspired Gelbart, Miller, Panchishkin, and Shahidi to introduce an analog
of parts of the Langlands-Shahidi method to the $p$-adic setting in [GMPS14].
They proved the existence of a $p$-adic measure $\mu^{*}$ which can be
expressed in terms of the non-zero Fourier coefficients of classical
holomorphic Eisenstein series such that its Mellin transform is the reciprocal
of the $p$-adic Riemann $\zeta$-function.

In this paper, we generalize the result of [GMPS14] to the case of totally
real number fields. We now state our main theorem:

######  Theorem 1.1.

Let $p\in\mathbb{Q}$ be an odd prime number such that $p$ does not divide
$[K:\mathbb{Q}]$ or the class number of $K(e^{2\pi i/p})$. Assume further that
no prime $\wp$ of the field $K(e^{2\pi i/p}+e^{-2\pi i/p})$ lying above $p$
splits in $K(e^{2\pi i/p}).$ Let $\mathfrak{p}$ be a prime ideal of
$\mathcal{O}$ lying above $p$ and $h^{+}$ the strict class number of $K.$

  1. (1)

There exists a $p$-adic measure $\lambda$ on $G\left(K(p^{\infty})/K\right)$
whose Mellin transform is the reciprocal of the $p$-adic $L$-function of the
totally real field $K$

$\int_{G\left(K(p^{\infty})/K\right)}N(x)^{k-1}\,d\lambda=h^{+}(1-N(\mathfrak{p})^{k-1})^{-1}\zeta(1-k)^{-1}$

for all even positive integers $k.$

  2. (2)

For any non-trivial Hecke character of finite type $\chi\mod\mathfrak{p}^{m}$
with $m>0$ , we have

$\int_{G\left(K(p^{\infty})/K\right)}\chi(x)N(x)^{k-1}\,d\lambda=h^{+}L(1-k,\chi)^{-1}$

for any positive integer $k$ satisfying the parity condition
$\chi(-1)=(-1)^{k}$.

Moreover, the measure $\lambda$ can be expressed in terms of Fourier
coefficients of the Eisenstein series defined by $\chi$ on the Hilbert upper
half plane.

######  Remark 1.2.

The conditions on $p$ are those that ensure that Kummer’s criterion holds for
$K$ ([Gre73]). Greenberg mentions in the introduction to loc. cit. that the
condition $p\nmid[K:\mathbb{Q}]$ may be relaxed. However, the other conditions
are the regularity conditions on $p$ and cannot be removed.

The paper is organized as follows: In Section 2, we give a brief overview of
Hilbert modular forms in the classical setting and we define holomorphic
Hilbert Eisenstein series. In Section 3, we review the Fourier expansion of
holomorphic Eisenstein series of weight $k$ and level $\mathfrak{n}.$ In
Section 4, we use the non-constant terms of this Fourier expansion to
construct a $p$-adic distribution $\lambda.$ This is the main construction of
the paper. Moreover, we show that $\lambda$ satisfies the properties given in
Theorem 1.1. In Section 5, we prove that $\lambda$ is in fact a $p$-adic
measure.

Acknowledgement. The author would like to thank Andrei Jorza, Ellen Eischen,
and A. Raghuram for their helpful conversations. She would also like to thank
Romyar Sharifi for his explanation of the structure of the Iwasawa algebra.
This work is partially supported by Freydoon Shahidi’s NSF grants DMS-1500759
and DMS-1801273 as a part of the thesis written under him at Purdue
University.

##  2\. Hilbert Modular Forms

In this section, we set up the main definitions and results that we need from
the theory of Hilbert modular forms. A thorough treatment of the subject can
be found in [Fre90], for example.

###  2.1. Structure of the Hilbert Modular Group

Let $K$ be a totally real number field of degree $r$ over $\mathbb{Q}.$ Let
$\mathcal{O}$ denote the ring of integers of $K,$ $\mathfrak{d}$ its different
ideal, $D_{K}$ its discriminant, and $h$ its class number. Let
$N:K\rightarrow\mathbb{Q}$ denote the norm map on $K$.

We fix an ordering of the $r$ embeddings $K\hookrightarrow\mathbb{R}.$ For any
$\alpha\in K,$ we denote by $\alpha^{(1)},\cdots,\alpha^{(r)}$ the conjugates
of $\alpha.$ We say $\alpha$ is totally positive, denoted $\alpha\gg 0,$ if
$\alpha^{(i)}\geq 0$ for all $1\leq i\leq r.$

The Hilbert modular group $SL_{2}(\mathcal{O})$ acts discontinuously on the
product of $r$ copies of the upper half plane $\mathfrak{h}$ by Mobius
transformations componentwise: for any $A=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL_{2}(\mathcal{O})$ and any
$z=(z_{1},\cdots,z_{r})\in\mathfrak{h}^{r},$

(2.1) $A\cdot
z=\left(\frac{a^{(1)}z_{1}+b^{(1)}}{c^{(1)}z_{1}+d^{(1)}},\cdots,\frac{a^{(r)}z_{r}+b^{(r)}}{c^{(r)}z_{r}+d^{(r)}}\right).$

This produces an infinite number of cusps but only a finite number of
inequivalent cusps.

######  Theorem 2.1.

([Fre90] pg. 37 Corollary $3.5_{1}$) The equivalence classes of cusps $\kappa$
under the action of $SL_{2}(\mathcal{O})$ are in one to one correspondence
with the ideal classes of $K.$ In particular, the number of inequivalent cusps
is $h$.

The correspondence is easy to describe: the cusp represented by
$(a:b)\in\mathbb{P}^{1}(K)$ is mapped to the class of the ideal $\langle
a,b\rangle.$ This implies that the cusp $\kappa=\infty$ which is represented
by $(1:0)\in\mathbb{P}^{1}(K)$ corresponds to the trivial ideal class
consisting of all principal ideals of $K$.

###  2.2. Modular Forms on Congruence Subgroups

For
$k=(k_{1},\cdots,k_{r})\in\mathbb{Z}^{r},\,z=(z_{1},\cdots,z_{r})\in\mathbb{C}^{r},$
and $c,d\in K,$ we write

(2.2) $\displaystyle Tr(k)=\sum_{i=1}^{r}k_{i}\qquad$ $\displaystyle\qquad
Tr(z)=\sum_{i=1}^{r}z_{i}$ (2.3) $\displaystyle(cz+d)^{k}$
$\displaystyle=\prod_{i=1}^{r}(c^{(i)}z_{i}+d^{(i)})^{k_{i}}$

The slash operator $\mid_{k}$ is defined for a complex valued function $f$ on
$\mathfrak{h}^{r}$ and an element $\begin{pmatrix}*&*\\\ c&d\end{pmatrix}\in
SL_{2}(\mathcal{O})$ by

(2.4) $(f\mid_{k}\begin{pmatrix}*&*\\\
c&d\end{pmatrix})(z)=(cz+d)^{-k}f(\begin{pmatrix}*&*\\\ c&d\end{pmatrix}\cdot
z).$

Given a congruence subgroup $\Gamma\subset SL_{2}(\mathcal{O}),$ we say that
$f$ is a Hilbert modular form of weight $k=(k_{1},\cdots,k_{r})$ and level
$\Gamma$ if $f\mid_{k}\gamma=f$ for all $\gamma\in\Gamma.$ We will only be
interested in the case when $k_{1}=k_{2}=\cdots=k_{r}=k$ and we will say the
Hilbert modular form is of (parallel) weight $k$. One basic example of Hilbert
modular forms is the Eisenstein series attached to a character of $K^{*}$. We
denote the space of all the Hilbert modular forms of weight $k$ and level
$\Gamma$ by $\mathcal{M}_{k}(\Gamma).$

We will be most interested in the congruence subgroups

(2.5) $\displaystyle\Gamma_{0}(\mathfrak{n})$
$\displaystyle=\left\\{\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathcal{O})\,\middle|\,\gamma\equiv\begin{pmatrix}*&*\\\
0&*\end{pmatrix}\bmod\mathfrak{n}\right\\}$ (2.6)
$\displaystyle\Gamma_{1}(\mathfrak{n})$
$\displaystyle=\left\\{\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathcal{O})\,\middle|\,\gamma\equiv\begin{pmatrix}1&*\\\
0&1\end{pmatrix}\bmod\mathfrak{n}\right\\}$

of $SL_{2}(\mathcal{O}),$ where $\mathfrak{n}$ is an integral ideal of $K.$

If $f\in\mathcal{M}_{k}(\Gamma_{0}(\mathfrak{n}))$ or
$f\in\mathcal{M}_{k}(\Gamma_{1}(\mathfrak{n})),$ it has a Fourier expansion at
the cusp $\infty$ of the form

(2.7) $f(z)=\sum_{\begin{subarray}{c}\nu\equiv
0\bmod(\mathfrak{d}\mathfrak{n})^{-1}\\\ \nu\gg 0\end{subarray}}a(\nu)\exp(\nu
z)$

where $\exp(\nu z)=e^{2\pi iTr(\nu z)}.$

For any general cusp $\kappa,$ there is a matrix
$A=\begin{pmatrix}\alpha_{1}&\alpha_{2}\\\ \alpha_{3}&\alpha_{4}\end{pmatrix}$
of determinant 1 such that $A\kappa=\infty$ ([Kli62] pg. 181). Moreover, $A$
can be chosen so that

(2.8)
$\kappa=-\frac{\alpha_{4}}{\alpha_{3}},\quad\mathfrak{b}=\langle\alpha_{3},\alpha_{4}\rangle,\quad\mathfrak{b}^{-1}=\langle\alpha_{1},\alpha_{2}\rangle,\quad\gcd(\mathfrak{n},\mathfrak{b})=1.$

In this case, $f$ admits a Fourier expansion at $\kappa=A^{-1}\infty$ of the
form

(2.9)
$(\alpha_{3}A^{-1}z+\alpha_{4})^{k}f(A^{-1}z)=\sum_{\begin{subarray}{c}\nu\equiv
0\bmod\mathfrak{b}^{2}(\mathfrak{d}\mathfrak{n})^{-1}\\\ \nu\gg
0\end{subarray}}a_{A}(\nu)\exp(\nu z).$

###  2.3. Eisenstein Series

Recall that $\mathfrak{n}$ is an integral ideal of $K.$ A grossencharacter
$\chi$ is said to be of finite type with conductor $\mathfrak{n},$ denoted
$\chi\bmod\mathfrak{n},$ whenever $\chi$ is a character
$\left(\mathcal{O}/\mathfrak{n}\right)^{*}\to\mathbb{C}^{*}$.

Given a grossencharacter $\chi\bmod\mathfrak{n}$ of finite type, an integer
$k\geq 3,$ and a cusp $\kappa$ of $\Gamma_{0}(\mathfrak{n}),$ the weight $k$
holomorphic Eisenstein series for $\Gamma_{0}(\mathfrak{n})$ at $\kappa$ is
given by

(2.10)
$E_{k,\mathfrak{n}}(\chi,z,\kappa)=\sum_{\gamma\in\Gamma_{0}(\mathfrak{n})_{\kappa}\backslash\Gamma_{0}(\mathfrak{n})}\text{Im}(\gamma
z)^{k}$

for any $z\in\mathfrak{h}^{r}.$ Here, for any congruence subgroup $\Gamma,$
$\Gamma_{\kappa}$ denotes the stabilizer of $\kappa$ inside $\Gamma.$

If $E_{k,\mathfrak{n}}^{\Gamma_{1}}(\chi,z,\kappa_{1})$ denotes the weight $k$
holomorphic Eisenstein series for $\Gamma_{1}(\mathfrak{n})$ at a cusp
$\kappa_{1}$ lying above $\kappa,$ we have

(2.11)
$E_{k,\mathfrak{n}}(\chi,z,\kappa)=\sum_{\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma_{1}(\mathfrak{n})\backslash\Gamma_{0}(\mathfrak{n})}\chi(\langle
d\rangle)E_{k,\mathfrak{n}}^{\Gamma_{1}}(\chi,z,\kappa_{1}).$

We can describe the Eisenstein series explicitly at the cusp $\kappa=\infty.$
This cusp corresponds to the trivial ideal class so all the ideals are
principal. Two matrices in $SL_{2}(\mathcal{O})$ are left-equivalent under
$\Gamma_{\infty}$ if and only if they have the same bottom row
$(c,d)\in\mathcal{O}\times\mathcal{O}.$ Fix a principal ideal $\mathfrak{a}$
which is not necessarily integral. A tuple $(c,d)$ is the bottom row of a
matrix in $\Gamma_{0}(\mathfrak{n})$ or $\Gamma_{1}(\mathfrak{n})$ precisely
when $c,d\in\mathfrak{a},$
$\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1,$ and the
tuple satisfies the congruence condition (2.5) and (2.6) of the respective
subgroup ([BMP03] pgs. 706 Equation (46)). Since
$\mathfrak{a}=\langle\alpha\rangle$ is principal, every element
$a\in\mathfrak{a}$ can be written as $a=a^{*}\alpha,$ and we use the notation
$\frac{a}{\mathfrak{a}}$ to denote $a^{*}.$ Choose a set of representatives
$a_{2}\in\mathfrak{a}$ which runs over all the congruence classes modulo
$\mathfrak{na}.$ Now the sum in Equation (2.10) at the cusp $\kappa=\infty$
can be written as

(2.12) $\displaystyle E_{k,\mathfrak{n}}(\chi,z)\coloneqq
E_{k,\mathfrak{n}}(\chi,z,\infty)$
$\displaystyle=\sum_{a_{2}\in\left(\mathfrak{a}/\mathfrak{na}\right)^{\times}}\chi(\langle
a_{2}\rangle)\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\
\mathfrak{n}|c\\\ d\equiv
a_{2}\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}$
(2.13)
$\displaystyle=\sum_{a_{2}\in\left(\mathfrak{a}/\mathfrak{na}\right)^{\times}}\chi(\langle
a_{2}\rangle)\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\ d\equiv
a_{2}\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(\mathfrak{n}cz+d)^{k}}.$

The notation $\\{c,d\\}$ means that two tuples $(c,d)$ and
$(c^{\prime},d^{\prime})$ are identified together if there exists
$\varepsilon\equiv 1\bmod\mathfrak{n}$ such that $c=\varepsilon c^{\prime}$
and $d=\varepsilon d^{\prime}.$

This definition does not depend on the ideal $\mathfrak{a}$ that was chosen.
Given another ideal $\mathfrak{a}^{\prime}$ in the same ideal class as
$\mathfrak{a},$ there exists an element $\alpha\in K^{*}$ such that
$\mathfrak{a}^{\prime}=\alpha\mathfrak{a}.$ This implies that the numerator
and denominator of the inner sum in (2.12) would both be multiplied by
$N(\alpha)^{k},$ and hence the expression is invariant ([CS17] pg. 662).

We now want to describe the Eisenstein series at a cusp $\kappa\neq\infty,$
which corresponds to a (fixed) non-trivial ideal class. The idea is to
translate the cusp $\kappa$ to $\infty$ where an explicit expression of the
Eisenstein series is easy to describe.

By Equation (2.9), the Fourier expansion of
$E_{k,\mathfrak{n}}(\chi,z,\kappa)$ at $\kappa$ is equal to the Fourier
expansion of
$E_{k,\mathfrak{n}}(\chi,z)|_{k}\,A^{-1}=(-\alpha_{3}z+\alpha_{1})^{-k}E_{k,\mathfrak{n}}(\chi,A^{-1}z)$
at $\infty.$ Since $\det A=1,$ a direct computation shows that
$(-\alpha_{3}z+\alpha_{1})=(\alpha_{3}A^{-1}z+\alpha_{4})^{-1}.$ Hence, we
have

(2.14)
$E_{k,\mathfrak{n}}(\chi,z,\kappa)=(\alpha_{3}A^{-1}z+\alpha_{4})^{k}\sum_{a_{2}\in\left(\mathfrak{a}/\mathfrak{na}\right)^{\times}}\chi(\langle
a_{2}\rangle)\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\ d\equiv
a_{2}\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(\mathfrak{n}cA^{-1}z+d)^{k}}.$

##  3\. Fourier Coefficients of Certain Eisenstein Series

We are interested in calculating the non-constant Fourier coefficients of
(2.14). Indeed, this will be the main calculation needed to construct our
$p$-adic measure $\lambda.$ We start by defining a more general Eisenstein
series by

(3.1)
$G_{k}(z,a_{1},a_{2},\mathfrak{n},\mathfrak{a})=\sum_{\begin{subarray}{c}\\{c,d\\}^{+}\in\mathfrak{a}\times\mathfrak{a}\\\
(c,d)\equiv(a_{1},a_{2})\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}.$

The notation $\\{c,d\\}^{+}$ means that two tuples $(c,d)$ and
$(c^{\prime},d^{\prime})$ are identified together if there exists a totally
positive $\varepsilon\equiv 1\bmod\mathfrak{n}$ such that $c=\varepsilon
c^{\prime}$ and $d=\varepsilon d^{\prime}.$

######  Lemma 3.1.

We have

$\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\ d\equiv
a_{2}\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}=\sum_{a_{1}\in\mathfrak{a}/\mathfrak{a}\mathfrak{n}}\sum_{i=1}^{h(\mathfrak{n})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}G_{k}(z,\tau a_{1},\tau
a_{2},\mathfrak{n},\mathfrak{a}\mathfrak{t})$

where $\mu(\mathfrak{m})$ denotes the Mobius function for any integral ideal
$\mathfrak{m}$ given by

(3.2) $\mu(\mathfrak{m})=\begin{cases}1&\text{if }\mathfrak{m}=\mathcal{O}\\\
(-1)^{r}&\text{if }\mathfrak{m}=\mathfrak{p}_{1}\cdots\mathfrak{p}_{r}\\\
0&\text{if }\mathfrak{p}^{2}|\mathfrak{m}\text{ for some prime ideal
}\mathfrak{p}\end{cases}$

and $h(\mathfrak{n})$ denotes the strict ideal class number of
$K\bmod\mathfrak{n}.$

###### Proof.

Recall the following property of the Mobius function:

(3.3)
$\sum_{\mathfrak{t}|\mathfrak{n}}\mu(\mathfrak{t})=\begin{cases}1&\text{if
}\mathfrak{n}=\mathcal{O}\\\ 0&\text{otherwise}\end{cases}.$

We can use this property to remove the coprimeness condition. Recall that
$\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\in\mathcal{O},$ which implies
that the ideal
$\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)$ is an
integral ideal. We now have

(3.4)
$\displaystyle\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\
(c,d)\equiv(a_{1},a_{2})\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}$
$\displaystyle=\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
(c,d)\equiv(a_{1},a_{2})\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}\sum_{\mathfrak{t}|\gcd(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}})}\mu(\mathfrak{t})$
(3.5)
$\displaystyle=\sum_{\gcd(\mathfrak{t},\mathfrak{n})=1}\mu(\mathfrak{t})\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
(c,d)\equiv(a_{1},a_{2})\bmod\mathfrak{n}\mathfrak{a}\\\
(c,d)\equiv(0,0)\bmod\mathfrak{a}\mathfrak{t}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}.$

Let $C_{1},\cdots,C_{h(\mathfrak{n})}$ denote the strict ideal classes of $K$
modulo $\mathfrak{n}.$ Choose integral ideals
$\mathfrak{t}_{1},\cdots,\mathfrak{t}_{h(\mathfrak{n})}$ such that
$\mathfrak{t}_{i}\in C_{i}^{-1}$ for all $i\in\\{1,\cdots h(\mathfrak{n})\\}$.
For each $\mathfrak{t}\in C_{i},$ let $(\tau)=\mathfrak{t}\mathfrak{t}_{i}$
where $\tau\equiv 1\bmod\mathfrak{n}.$ Then the previous sum is equal to

(3.6)
$\displaystyle\sum_{i=1}^{h(\mathfrak{n})}\sum_{\begin{subarray}{c}\mathfrak{t}\in
C_{i}\\\
\gcd(\mathfrak{t},\mathfrak{n})=1\end{subarray}}\mu(\mathfrak{t})\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
(c,d)\equiv(\tau a_{1},\tau
a_{2})\bmod\mathfrak{n}\mathfrak{a}\mathfrak{t}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(cz+d)^{k}}$
(3.7) $\displaystyle=$
$\displaystyle\sum_{i=1}^{h(\mathfrak{n})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}G_{k}(z,\tau a_{1},\tau
a_{2},\mathfrak{n},\mathfrak{a}\mathfrak{t}).$

Summing up over the equivalence classes of $a_{1}$ modulo $\mathfrak{na},$ we
get the desired result. ∎

The Fourier expansion of the Eisenstein series in Equation (3.1) has been
computed by Klingen ([Kli62], pg. 181, 182) as follows:

######  Theorem 3.2.

The Fourier expansion of $G_{k}(z,a_{1},a_{2},\mathfrak{n},\mathfrak{a})$ at a
general cusp $\kappa$ is given by

(3.8)
$\delta\left(\frac{(a_{1}^{*})}{\mathfrak{n}\mathfrak{ab}}\right)N\left(\mathfrak{a}\right)^{k}\sum_{\begin{subarray}{c}d\equiv
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{ab}^{-1}\\\
\\{d\\}^{+}\end{subarray}}\frac{\text{sgn}^{k}N(d)}{|N(d)|^{k}}\\\
+\frac{(-2\pi
i)^{kr}N(\mathfrak{a})^{k-1}N(\mathfrak{b})}{(k-1)!^{r}N(\mathfrak{n})|D|^{1/2}}\sum_{\begin{subarray}{c}c\equiv
a_{1}^{*}\bmod\mathfrak{n}\mathfrak{ab}\\\ \nu\equiv
0\bmod\mathfrak{b}/\mathfrak{n}\mathfrak{a}\mathfrak{d}\\\ c\nu\gg
0,\\{c\\}^{+}\end{subarray}}\text{sgn}N(\nu)N(\nu)^{k-1}\exp{(a_{2}^{*}\nu+c\nu
z)}$

where $a_{1}^{*}=\alpha_{4}a_{1}-\alpha_{1}a_{2}\in\mathfrak{ab}$ and
$a_{2}^{*}=\alpha_{1}a_{2}-\alpha_{2}a_{1}\in\mathfrak{ab}^{-1}$.

We use this Fourier expansion to prove the following:

######  Theorem 3.3.

The Fourier expansion of the partial Eisenstein series

$(\alpha_{3}A^{-1}z+\alpha_{4})^{k}\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\ d\equiv
a_{2}\bmod\mathfrak{n}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(\mathfrak{n}cA^{-1}z+d)^{k}}$

at the cusp $\infty$ is given by

$\sum_{l\equiv
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{ab}^{-1}}\frac{N(\mathfrak{a})^{k}}{N(l)^{k}}\sum_{\begin{subarray}{c}\mathfrak{t}\in
K\\\ \mathfrak{t}\mid\langle l\rangle\\\
\mathfrak{n}\nmid\mathfrak{t}\end{subarray}}\mu(\mathfrak{t})\\\ +\frac{(-2\pi
i)^{kr}N(\mathfrak{b})^{k}}{(k-1)!^{r}N(\mathfrak{n})^{k}|D|^{\frac{2k-1}{2}}}\sum_{i=1}^{h(\mathfrak{n})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\begin{subarray}{c}\nu\in\mathcal{O}\\\
c\nu\gg
0,\\{c\\}^{+}\end{subarray}}\text{sgn}N(\nu)N(\nu)^{k-1}\exp\left(\frac{\mathfrak{t}_{i}a_{2}^{\prime}\nu\mathfrak{b}}{\mathfrak{n}\mathfrak{d}}+\frac{c\nu\mathfrak{b}^{2}z}{\mathfrak{d}}\right).$

###### Proof.

Lemma 3.1 implies that we need to calculate the Fourier expansion of

(3.9)
$(\alpha_{3}A^{-1}z+\alpha_{4})^{k}\sum_{a_{1}\in\mathfrak{a}/\mathfrak{a}\mathfrak{n}}\sum_{i=1}^{h(\mathfrak{\mathfrak{n}})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}G_{k}(\mathfrak{n}A^{-1}z,\tau
a_{1},\tau a_{2},\mathfrak{n},\mathfrak{a}\mathfrak{t}).$

Theorem 3.2 along with Equation (2.9) now show that the constant term of (3.9)
is non-zero only when $a_{1}^{*}\equiv 0\bmod\mathfrak{n}\mathfrak{atb}.$ This
implies that the contribution from the terms of the outermost sum are all zero
except for one equivalence class of $a_{1}.$ The nonzero term remaining is

(3.10) $\sum_{i=1}^{h(\mathfrak{\mathfrak{n}})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}N\left(\mathfrak{at}\right)^{k}\sum_{\begin{subarray}{c}d\equiv\tau
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{atb}^{-1}\\\
\\{d\\}^{+}\end{subarray}}\frac{\text{sgn}^{k}N(d)}{|N(d)|^{k}}.$

The ideals $\mathfrak{n}$ and $\mathfrak{t}$ being coprime, we can rewrite
$d\equiv\tau a_{2}^{*}\bmod\mathfrak{n}\mathfrak{atb}^{-1}$ as two
equivalences modulo $\mathfrak{n}\mathfrak{ab}^{-1}$ and
$\mathfrak{atb}^{-1}$. We now obtain the constant term to be

(3.11) $\sum_{i=1}^{h(\mathfrak{\mathfrak{n}})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\begin{subarray}{c}d\equiv
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{ab}^{-1}\\\ d\equiv
0\bmod\mathfrak{ab}^{-1}\mathfrak{t}\\\
\\{d\\}^{+}\end{subarray}}N\left(\mathfrak{at}\right)^{k}\frac{\text{sgn}^{k}N(d)}{|N(d)|^{k}}.$

Switching the order of summation and writing $\langle l\rangle$ for the ideal
generated by $l,$ the constant term becomes

(3.12) $\displaystyle\sum_{\begin{subarray}{c}\mathfrak{t}\in K\\\
\mathfrak{p}\nmid\mathfrak{t}\end{subarray}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\begin{subarray}{c}\mathfrak{t}d\equiv
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{ab}^{-1}\\\
\\{d\\}^{+}\end{subarray}}N\left(\mathfrak{a}\right)^{k}\frac{1}{N(d)^{k}}$
(3.13) $\displaystyle=\sum_{l\equiv
a_{2}^{*}\bmod\mathfrak{n}\mathfrak{ab}^{-1}}\frac{N(\mathfrak{a})^{k}}{N(l)^{k}}\sum_{\begin{subarray}{c}\mathfrak{t}\in
K\\\ \mathfrak{t}\mid\langle l\rangle\\\
\mathfrak{n}\nmid\mathfrak{t}\end{subarray}}\mu(\mathfrak{t}).$

We consider next the non-constant terms. Note that as $a_{1}$ varies over the
equivalence classes of $\mathfrak{a}/\mathfrak{an},$ $a_{1}^{*}$ varies over
the equivalence classes of $\mathfrak{ab}/\mathfrak{abn}.$ It is clear then
that the non-constant terms are

(3.14) $\frac{(-2\pi
i)^{kr}N(\mathfrak{b})}{(k-1)!^{r}N(\mathfrak{n})|D|^{1/2}}\sum_{a_{1}^{*}\in\mathfrak{ab}/\mathfrak{ab}\mathfrak{n}}\sum_{i=1}^{h(\mathfrak{n})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}N(\mathfrak{at})^{k-1}\\\
\sum_{\begin{subarray}{c}c\equiv\tau
a_{1}^{*}\bmod\mathfrak{n}\mathfrak{ab}\mathfrak{t}\\\ \nu\equiv
0\bmod\mathfrak{b}/\mathfrak{n}\mathfrak{a}\mathfrak{t}\mathfrak{d}\\\ c\nu\gg
0,\\{c\\}^{+}\end{subarray}}\text{sgn}N(v)N(\nu)^{k-1}\exp(\tau
a_{2}^{*}\nu+c\nu\mathfrak{n}z).$

As before, we can split the equivalence modulo
$\mathfrak{n}\mathfrak{ab}\mathfrak{t}$ into two equivalences modulo
$\mathfrak{n}\mathfrak{ab}$ and $\mathfrak{ab}\mathfrak{t}.$ Moreover, we make
a change of variables
$\nu\to\frac{\nu\mathfrak{b}}{\mathfrak{n}\mathfrak{adt}}$ and
$c\to\mathfrak{abt}c.$ Since $N(\mathfrak{d})=|D|,$ we get

(3.15) $\frac{(-2\pi
i)^{kr}N(\mathfrak{b})^{k}}{(k-1)!^{r}N(\mathfrak{n})^{k}|D|^{\frac{2k-1}{2}}}\sum_{i=1}^{h(\mathfrak{n})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\begin{subarray}{c}\nu\in\mathcal{O}\\\
c\nu\gg
0,\\{c\\}^{+}\end{subarray}}\text{sgn}N(\nu)N(\nu)^{k-1}\exp\left(\frac{\mathfrak{t}_{i}a_{2}^{\prime}\nu\mathfrak{b}}{\mathfrak{n}\mathfrak{d}}+\frac{c\nu\mathfrak{b}^{2}z}{\mathfrak{d}}\right).$

∎

##  4\. Constructing a p-adic Measure

###  4.1. p-adic Measures on Galois Groups

Let $\mathfrak{b}_{1},\cdots,\mathfrak{b}_{h}$ represent the strict ideal
classes of $K,$ where each $\mathfrak{b}_{j}$ is the ideal generated by the
lower row of the matrix $A_{j}$ associated with the distinct cusps
$\kappa_{j}.$ By construction, all the ideals $\mathfrak{b}_{j}$ are
relatively prime to $\mathfrak{p}.$

We can decompose $G=G\left(K(p^{\infty})/K\right)$ as
$\sqcup_{j}\mathfrak{b}_{j}^{-1}(\mathcal{O}_{\mathfrak{p}}^{*}/U),$ where $U$
is the closure in $\mathcal{O}_{\mathfrak{p}}^{*}$ of the subgroup of all
totally positive units in $\mathcal{O}^{*}$ ([Hid93], Section 3.9). Then for
any continuous function $\phi$ on $G$, we can define a continuous function
$\phi_{j}$ on $\mathcal{O}_{\mathfrak{p}}$ by

(4.1) $\phi_{j}(x)=\begin{cases}\phi(\mathfrak{b}_{j}^{-1}x)&\text{ if
}x\in\mathcal{O}_{\mathfrak{p}}^{*}\\\ 0&\text{ otherwise.}\end{cases}$

We then define p-adic measures on $G$ by setting

(4.2)
$\int_{G}\phi\,d\mu=\sum_{j}\int_{\mathcal{O}_{\mathfrak{p}}}\phi_{j}d\mu_{j}.$

###  4.2. The Main Construction

The structure of the Galois group $G$ allows us to define a distribution
$\lambda$ on $G$ by defining it on each piece
$\mathfrak{b}_{j}^{-1}(\mathcal{O}_{\mathfrak{p}}^{*}/U).$ We will denote the
restriction of $\lambda$ to each
$\mathfrak{b}_{j}^{-1}(\mathcal{O}_{\mathfrak{p}}^{*}/U)$ by $\lambda_{j}.$

Fix a strict ideal class $\mathfrak{b}_{j}.$ Define a map

(4.3)
$\varepsilon_{k,\mathfrak{p}^{m},\mathfrak{b}_{j}}(\mathfrak{a}):\mathcal{O}/\mathfrak{p}^{m}\mathcal{O}\to\mathcal{(}M_{k}(\Gamma_{1}(\mathfrak{p}^{m})))$

by

(4.4)
$\varepsilon_{k,\mathfrak{p}^{m},\mathfrak{b}_{j}}(\mathfrak{a})\coloneqq(\alpha_{3}A^{-1}z+\alpha_{4})^{k}\sum_{\begin{subarray}{c}\\{c,d\\}\in\mathfrak{a}\times\mathfrak{a}\\\
\gcd\left(\frac{c}{\mathfrak{a}},\frac{d}{\mathfrak{a}}\right)=1\\\ d\equiv
a_{2}\bmod\mathfrak{p}^{m}\mathfrak{a}\end{subarray}}\frac{N(\mathfrak{a})^{k}}{(\mathfrak{p}^{m}cA^{-1}z+d)^{k}}.$

Let $\mathcal{C}_{n}$ denote the map which sends a holomorphic Hilbert modular
form to the coefficient of $e^{2\pi iTr(nz)}$ in its Fourier expansion.
Theorem 3.3 states that

(4.5)
$\mathcal{C}_{\mathfrak{p}^{m}}(\varepsilon_{k,\mathfrak{p}^{m},\mathfrak{b}_{j}})=\frac{(-2\pi
i)^{kr}N(\mathfrak{b}_{j})^{k}}{(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\sum_{u=0}^{m}N(\mathfrak{p})^{u(k-1)-mk}\\\
\sum_{i=1}^{h(\mathfrak{p}^{m})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\exp\left(\frac{\mathfrak{t}_{i}a_{2}^{\prime}\mathfrak{p}^{u}\mathfrak{b}_{j}}{\mathfrak{p}^{m}\mathfrak{d}}\right).$

Define a distribution on
$\mathfrak{b}_{j}^{-1}(\mathcal{O}_{\mathfrak{p}}^{*}/U)$ by

(4.6)
$\lambda_{j}(\mathfrak{a}+\mathfrak{p}^{m}\mathcal{O}_{\mathfrak{p}})=\frac{1}{2^{r}N(\mathfrak{b}_{j})^{k}}\mathcal{C}_{\mathfrak{p}^{m}}(\varepsilon_{k,\mathfrak{p}^{m},\mathfrak{b}_{j}})+\gamma(k)\lambda_{Haar}$

where

(4.7)
$\gamma(k)=\begin{cases}\frac{N(\mathfrak{p})^{2k-1}}{N(\mathfrak{p})^{k}-1}\frac{(1-N(\mathfrak{p})^{k-1})^{-1}}{\zeta(1-k)}&\text{
for k even}\\\ 0&\text{ otherwise}\end{cases}.$

Note that since $a_{2}^{\prime}\in\mathfrak{a}\mathfrak{b}_{j}^{-1},$ the term
inside the exponential function, and hence the definition of $\lambda,$ does
not depend on the ideals $\mathfrak{b}_{j}.$ Henceforth we will write
$\mathfrak{a}$ instead of $a_{2}^{\prime}\mathfrak{b}_{j}.$

######  Theorem 4.1.

Let $\chi$ be a Hecke character of finite type with conductor
$\mathfrak{p}^{m}$ for some $m\geq 1.$ Let $k$ be an integer of the same
parity as $\chi.$ Then we have

(4.8)
$\int_{G}\chi^{-1}(x)\,d\lambda(x)=\frac{h^{+}\left(1-\chi(\mathfrak{p})N(\mathfrak{p})^{k-1}\right)^{-1}}{L(1-k,\chi)}$

where $h^{+}$ denotes the strict class number of $K$.

###### Proof.

First, we consider the case when $\chi$ is the trivial character and k is
even. By finite additivity, we have

(4.9) $\displaystyle\lambda_{j}(\mathcal{O}_{\mathfrak{p}}^{*})$
$\displaystyle=\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\lambda_{j}(\mathfrak{a}+\mathfrak{p}\mathcal{O}_{\mathfrak{p}})$
(4.10)
$\displaystyle=\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\frac{1}{2^{r}}\mathcal{C}_{\mathfrak{p}}(\varepsilon_{k,\mathfrak{p},\mathfrak{b}_{j}})+\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\gamma(k)\lambda_{Haar}.$

We calculate each sum separately and start with the first sum. Since the sum
of roots of unity modulo $\mathfrak{p}$ is 0, we have

(4.11)
$\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\exp\left(\frac{\mathfrak{t}_{i}\mathfrak{a}}{\mathfrak{p}\mathfrak{d}}\right)=-1.$

Moreover,
$\exp\left(\frac{\mathfrak{t}_{i}\mathfrak{a}}{\mathfrak{d}}\right)=e^{2\pi
iTr(\mathfrak{t}_{i}\mathfrak{a}/\mathfrak{d})}=1$ for our choice of
$\mathfrak{t}_{i}$ and $\mathfrak{a},$ so

(4.12)
$\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\exp\left(\frac{\mathfrak{t}_{i}\mathfrak{a}}{\mathfrak{d}}\right)=N(\mathfrak{p})-1.$

These comments directly imply that the first sum is equal to

(4.13) $\displaystyle\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\sum_{\begin{subarray}{c}\mathfrak{t}\in
K\\\
\mathfrak{p}\nmid\mathfrak{t}\end{subarray}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{u=0}^{1}N(\mathfrak{p})^{u(k-1)-k}\sum_{\mathfrak{a}\in\mathcal{O}/\mathfrak{p}\mathcal{O}}\exp\left(\frac{\mathfrak{t}_{i}\mathfrak{p}^{u}\mathfrak{a}}{\mathfrak{p}\mathfrak{d}}\right)$
(4.14) $\displaystyle=\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\sum_{\begin{subarray}{c}\mathfrak{t}\in
K\\\
\mathfrak{p}\nmid\mathfrak{t}\end{subarray}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\left(-N(\mathfrak{p})^{-k}+\frac{N(\mathfrak{p})-1}{N(\mathfrak{p})}\right).$

The term $\sum_{\begin{subarray}{c}\mathfrak{t}\in K\\\
\mathfrak{p}\nmid\mathfrak{t}\end{subarray}}\mu(\mathfrak{t})N(\mathfrak{t})^{-k}$
is the reciprocal of the Dedekind zeta function with the Euler factor at
$\mathfrak{p}$ removed, so the previous sum is equal to

(4.15) $\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\frac{(1-N(\mathfrak{p})^{-k})^{-1}}{\zeta(k)}\left(-N(\mathfrak{p})^{-k}+\frac{N(\mathfrak{p})-1}{N(\mathfrak{p})}\right).$

The functional equation of the Dedekind zeta function is given by

(4.16) $\frac{1}{\zeta(1-k)}=\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\frac{1}{\zeta(k)}.$

It directly implies the first sum is equal to

(4.17)
$\frac{(1-N(\mathfrak{p})^{-k})^{-1}}{\zeta(1-k)}\left(\frac{N(\mathfrak{p})^{k}-N(\mathfrak{p})^{k-1}-1}{N(\mathfrak{p})^{k}}\right).$

The second sum is

(4.18)
$\frac{N(\mathfrak{p})^{2k-1}}{N(\mathfrak{p})^{k}-1}\frac{(1-N(\mathfrak{p})^{k-1})^{-1}}{\zeta(1-k)}\frac{N(\mathfrak{p})-1}{N(\mathfrak{p})}.$

Adding Equations (4.17) and (4.18) gives

(4.19)
$\lambda_{j}(\mathcal{O}_{\mathfrak{p}}^{*})=\frac{(1-N(\mathfrak{p})^{-k})^{-1}}{\zeta(1-k)}.$

Summing up over $j$ equivalence classes we get the required result.

Next we consider the case when $\chi$ is a non-trivial Hecke character of
finite type with conductor $\mathfrak{p}^{m}$ and $k$ is an integer of the
same parity as $\chi.$ Since $\chi$ is orthogonal to Haar measure, the
integral of $\chi$ against $\lambda_{Haar}$ is zero. The integral of $\chi$
against $\lambda_{j}$ is

(4.20)
$\int_{\mathcal{O}_{\mathfrak{p}}^{*}}\chi^{-1}(x)\,d\lambda_{j}(x)=\frac{(-2\pi
i)^{kr}}{(k-1)!^{r}|D|^{\frac{2k-1}{2}}}\sum_{u=0}^{m}N(\mathfrak{p})^{u(k-1)-mk}\\\
\sum_{i=1}^{h(\mathfrak{p}^{m})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\mathfrak{a}\in\left(\mathcal{O}/\mathfrak{p}^{m}\right)^{\times}}\chi(\mathfrak{a})\exp\left(\frac{\mathfrak{t}_{i}\overline{\mathfrak{a}}\mathfrak{p}^{u}}{\mathfrak{p}^{m}\mathfrak{d}}\right).$

We define the Gauss sum

(4.21)
$\tau(\chi)=\sum_{\mathfrak{x}\in\mathcal{O}/\mathfrak{p}^{m}}\chi(\mathfrak{x})e^{2\pi
iTr(\mathfrak{x}/\mathfrak{p}^{m}\mathfrak{d})}$

If $\chi$ is primitive and $x\in\mathcal{O},$ then

(4.22)
$\sum_{\mathfrak{x}\in\mathcal{O}/\mathfrak{p}^{m}}\chi(\mathfrak{x})e^{2\pi
iTr(\mathfrak{x}x/\mathfrak{d})}=\begin{cases}\overline{\chi}(x)\tau(\chi)&\text{
if }\gcd(x,\mathfrak{p}^{m})=1\\\ 0&\text{ otherwise}\end{cases}.$

Then the term with $u=0$ is the only non-zero term in Equation (4.20), so we
have

(4.23) $\displaystyle\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}N(\mathfrak{p}^{m})^{k}|D|^{\frac{2k-1}{2}}}\sum_{i=1}^{h(\mathfrak{p}^{m})}\sum_{\mathfrak{t}\in
C_{i}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\sum_{\mathfrak{a}\in\left(\mathcal{O}/\mathfrak{p}^{m}\right)^{\times}}\chi(\mathfrak{a})\exp\left(\frac{\mathfrak{t}_{i}\overline{\mathfrak{a}}}{\mathfrak{p}^{m}\mathfrak{d}}\right)$
(4.24) $\displaystyle=\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}N(\mathfrak{p}^{m})^{k}|D|^{\frac{2k-1}{2}}}\sum_{\begin{subarray}{c}\mathfrak{t}\in
K\\\
\mathfrak{p}\nmid\mathfrak{t}\end{subarray}}\frac{\mu(\mathfrak{t})}{N(\mathfrak{t})^{k}}\chi(\mathfrak{t})^{-1}\tau(\overline{\chi})$
(4.25) $\displaystyle=\frac{(-2\pi
i)^{kr}}{2^{r}(k-1)!^{r}N(\mathfrak{p}^{m})^{k}|D|^{\frac{2k-1}{2}}}\tau(\overline{\chi})\frac{1}{L(k,\overline{\chi})}.$

The functional equation of the Hecke $L$-function of a totally real field

(4.26) $L(k,\chi)=\frac{(2\pi
i)^{kr}}{2^{r}(k-1)!^{r}|D|^{\frac{2k-1}{2}}N(\mathfrak{p}^{m})^{k-1}\tau(\chi^{-1})}L(1-k,\chi^{-1})$

along with the relation
$\tau(\chi)\tau(\overline{\chi})=\chi(-1)N(\mathfrak{p}^{m})$ gives the
integral against $\lambda_{j}$ to be $L(1-k,\chi)^{-1}.$ Summing over the
strict ideal classes of $K$ proves the result. ∎

##  5\. p-adic Measures of Totally Real Fields

It remains to show that the distribution $\lambda$ is bounded. It is desirable
to prove the boundedness directly by properties of the Eisenstein series.
Instead, we show that $p$-adic $L$-function is invertible as an element of the
Iwasawa algebra. We first introduce the $p$-adic $L$-function of totally real
fields and follow the construction of Deligne and Ribet ([DR80]).

Let $\mathfrak{n}$ be a non-zero integral ideal of $K$ and let
$G_{\mathfrak{n}}$ denote the strict ideal class group of $K$ modulo
$\mathfrak{n}$ defined by:

(5.1) $\left\\{\text{prime-to-}\mathfrak{n}\text{ fractional ideals of
}K\right\\}/\left\\{\langle\alpha\rangle\mid\alpha\gg 0\text{ and }\alpha\in
1+\mathfrak{n}\mathfrak{b}^{-1}\text{ for some ideal }\mathfrak{b}\right\\}.$

By Class Field Theory, we can identify $\varprojlim_{n}G_{\mathfrak{n}p^{n}}$
with the Galois group $G=G\left(K(\mathfrak{n}p^{\infty})/K\right)$ of the
largest abelian extension of $K$ which is unramified at all the finite places
of $K$ prime to $\mathfrak{n}p.$

Let $\chi$ be any $\mathbb{C}$-valued locally constant character of
$G_{\mathfrak{n}}.$ We extend the norm map $N:K\rightarrow\mathbb{Q}$ to the
unique continuous norm map $G\rightarrow\mathbb{Z}_{p}^{*}$ which we again
denote by $N$. Siegel ([Sie70]) showed that the values of
$L(s,\chi)=\sum_{\gcd(\mathfrak{a},\mathfrak{n})=1}\chi(\mathfrak{a})N(\mathfrak{a})^{-s}$
at negative integers are rational whenever the values of $\chi$ are rational.

We now assume that all the primes lying above $p$ divide $\mathfrak{n}.$ Any
locally constant function $\chi:G\to\mathbb{Q}_{p}$ factorizes through
$G_{\mathfrak{n}^{\prime}}$ for some $\mathfrak{n}\mid\mathfrak{n}^{\prime},$
so we can define the values $L(1-k,\chi)\in\mathbb{Q}_{p}$ unambiguously. In
particular, for any $c\in G,k\geq 1,$ and
$\chi:G_{\mathfrak{n}}\to\mathbb{Z}_{p}$ the values

(5.2) $\Delta_{c}(1-k,\chi)=L(1-k,\chi)-N(c)^{k}L(1-k,\chi_{c}),$

where $\chi_{c}$ is the twist of $\chi$ by $c\in G_{\mathfrak{n}},$ are well
defined independently of the choice of $\mathfrak{n}$ and define a $p$-adic
measure. Explicitly, the following $p$-adic $L$-function exists:

######  Theorem 5.1.

([DR80] Theorem 8.2) Let $\mathfrak{c}$ be an ideal of $K$ which is relatively
prime to $p$. Then there exists a $\mathbb{Z}_{p}$-valued measure
$\mu_{1,\mathfrak{c}}$ on $G$ such that for every integer $k\geq 0$ and every
locally constant character $\chi,$ we have

(5.3)
$\int_{G}\chi(x)N(x)^{k-1}\,d\mu_{1,\mathfrak{c}}=(1-\chi(\mathfrak{c})N(\mathfrak{c})^{k}))L(1-k,\chi).$

###  5.1. Invertibility of Measures in the Iwasawa Algebra

We begin by describing the $p$-adic measures on $G$ as elements of the Iwasawa
algebra $\Lambda(G).$ We refer the reader to [CS06] for more details.

Let $\mu_{p}$ denote the group of all $p^{th}$ roots of unity and let
$K^{cyc}$ be the unique $\mathbb{Z}_{p}$ extension of $K$ in $K(p^{\infty})$.
The group $G=Gal(K(p^{\infty})/K)$ decomposes as $\Delta\times\Gamma,$ where
$\Delta=Gal(K(\mu_{p})/K)$ and $\Gamma=Gal(K^{cyc}/K)\cong\mathbb{Z}_{p}.$

Let $\omega(x)=\lim_{n\to\infty}x^{p^{n}}$ be the Teichmuller character
defined over $\mathbb{Z}_{p}.$ $\omega\circ N$ defines a character of order
$q=[(K(\mu_{p}):K]$ on integral ideals of $K$ which we will again denote by
$\omega.$ An arbitrary element $x\in G$ factors as
$\omega(x)\cdot\omega(x)^{-1}x\in\Delta\times\Gamma.$

Any measure $\nu$ on $\Gamma$ can be extended to the rest of $G$ by the
relation

(5.4) $\nu(aU)=\nu(U)$

for any $a\in\Delta$ and compact open $U\subset\Gamma.$ Since each
$\omega^{i}$ is a continuous function on $G,$ the product
$(\omega^{i}\nu)(aU)$ is a measure on $G$ satisfying

(5.5) $(\omega^{i}\nu)(aU)=a^{i}(\omega^{i}\nu)(U).$

On the other hand, every measure $\mu$ on $G$ can be decomposed as a sum of
measures of the form $\omega^{i}\nu_{i}:$

(5.6) $\mu=\frac{1}{q}\sum_{i=1}^{q}\omega^{i}\nu_{i}$

by setting $\omega^{i}\nu_{i}(U)=\sum_{a\in\Delta}\omega(a)^{-i}\mu(aU).$

For any continuous homomorphism $\chi:G\rightarrow\mathbb{C}_{p}^{*},$ let
$\chi|_{\Gamma}$ denote the restriction of $\chi$ to $\Gamma.$ The values of
$\chi(\omega(x))$ are $q^{th}$ roots of unity in $\mathbb{C}_{p}^{*},$ and
hence $\chi|_{\Gamma}$ has the form $x\to x^{-j}$ for some $j\bmod q.$ Since
$\chi$ transforms under $\Delta$ by $\omega^{-j},$ the integral of $\chi$
against $\mu$ only involves the term for $i=j$ and equals

(5.7)
$\int_{G}\chi(x)\,d\mu=\frac{1}{q}\int_{G}\chi(x)\,d(\omega^{j}\nu_{j})=\int_{\Gamma}\chi|_{\Gamma}\,d\nu_{j}$

reducing the integration of the character $\chi$ on $G$ to that of
$\chi|_{\Gamma}$ on $\Gamma.$

Now assume that $\nu_{i}\equiv 0$ whenever $i$ is even and that
$\nu_{i}|_{\Gamma}$ is a unit in the Iwasawa algebra $\Lambda(G)$ whenever $i$
is odd. Since $\Lambda(G)\cong\mathcal{O}[[T]]$ has a formal power structure
([CN79] Section 4), the invertibility condition is equivalent to the constant
term

(5.8)
$(\omega^{i}\nu_{i})(\Gamma)=\sum_{a\in\Delta}\omega(a)^{-i}\mu(a\Gamma)=\int_{G}\omega(x)^{-i}\,d\mu$

being a $p$-adic unit. In this case, for each odd value of $i\bmod q$, there
exists an inverse measure $\nu_{i}^{-1}$ such that $\nu_{i}\ast\nu_{i}^{-1}$
is equal to the $\delta$-distribution $\delta_{1}$ at the identity by
Iwasawa’s isomorphism theorem, or in other words

(5.9)
$\int_{\Gamma}f(x)\,d(\nu_{i}\ast\nu_{i}^{-1})(x)=\int_{\Gamma}\int_{\Gamma}f(xy)\,d\nu_{i}(x)\,d\nu_{i}^{-1}(y)=f(1)$

for any continuous $\mathbb{C}_{p}$-valued function on $\Gamma.$ In
particular, when $i$ is odd,

(5.10)
$\int_{\Gamma}\chi|_{\Gamma}\,d\nu_{i}^{-1}=\left(\int_{\Gamma}\chi|_{\Gamma}\,d\nu_{i}\right)^{-1}.$

We can then define an inverse measure $\mu^{-1}$ on $G$ by

(5.11) $\mu^{-1}=\frac{1}{q}\sum_{\begin{subarray}{c}1\leq i\leq q\\\ i\text{
odd}\end{subarray}}\omega^{i}\nu_{i}^{-1}.$

Moreover, evaluating the inverse measure $\mu^{-1}$ against odd characters
$\chi,$

(5.12)
$\int_{G}\chi(x)\,d\mu^{-1}=\int_{\Gamma}\chi|_{\Gamma}\,d\nu_{j}^{-1}=\left(\int_{\Gamma}\chi|_{\Gamma}\,d\nu_{j}\right)^{-1}=\left(\int_{G}\chi(x)\,d\mu\right)^{-1},$

we see that the integrals of $\mu$ and $\mu^{-1}$ are reciprocals of each
other.

###  5.2. Boundedness of $\lambda$

We will prove that $\lambda$ is a bounded measure by showing that the $p$-adic
measure on totally real fields $\mu_{1,\mathfrak{c}}$ is a unit in the Iwasawa
algebra and that the regularization $\mu^{*}$ of its inverse
$\mu_{1,\mathfrak{c}}^{-1}$ is equal as a distribution to a multiple of
$\lambda$.

We start by considering the $p$-adic measure on totally real fields
$\mu_{1,\mathfrak{c}}$. By Equation (5.8), $\nu_{i}$ is invertible whenever
the integral

(5.13) $\int_{G}\omega^{-i}(x)\,d\mu_{1,\mathfrak{c}}$

is a $p$-adic unit. Using the property

(5.14)
$\int_{G}\chi(x)N(x)^{n-1}\,d\mu_{1,\mathfrak{c}}(x)=\left(1-\chi(\mathfrak{c})N(\mathfrak{c})^{n}\right)L(1-n,\chi),$

the integral in (5.13) is equal to
$\left(1-\omega^{-i}(\mathfrak{c})N(\mathfrak{c})\right)L(0,\omega^{-i})$.
Kummer’s criterion in the case of totally real fields ([Gre73]) implies that
$p$ divides $L(0,\omega^{-i})$ for any odd $1\leq i\leq q$ if and only if $p$
divides the class number of $K(e^{2\pi i/p}).$ Since $p$ is regular by
assumption, this does not happen, so the $\nu_{i}$ are $p$-adic units.

Therefore, we conclude that the measure $\mu_{1,\mathfrak{c}}^{-1}$ satisfies
the property

(5.15)
$\int_{G}\chi(x)N(x)^{k-1}\,d\mu_{1,\mathfrak{c}}^{-1}=(1-\chi(\mathfrak{c})N(\mathfrak{c})^{k})^{-1}L(1-k,\chi)^{-1}$

for any Hecke character $\chi$ and non-negative integer $k$ of the same
parity. Moreover, by direct computation, the regularization

(5.16)
$\mu^{*}(U)=\mu_{1,\mathfrak{c}}^{-1}(U)-N(\mathfrak{c})\mu_{1,\mathfrak{c}}^{-1}(\mathfrak{c}U)$

satisfies

(5.17) $\int_{G}\chi(x)N(x)^{k-1}\,d\mu^{*}=L(1-k,\chi)^{-1}.$

We are now ready to prove that $\lambda$ is a measure.

######  Theorem 5.2.

The distribution $\lambda$ defined in Equation 4.6 is a $p$-adic measure.

###### Proof.

Define a distribution $\overline{\lambda}$ on $G$ by
$\overline{\lambda}(U)=\lambda(U^{-1}).$ The integral in Equation (4.8) is
then equal to $\int_{G}\chi(x)\,d\overline{\lambda}.$ Moreover, since
$\mu^{*}$ is a measure, so is $N(x)^{k-1}\mu^{*}$ for any $k.$ Comparing
Equations (4.8) and (5.17), we see that
$\overline{\lambda}-h^{+}N(x)^{k-1}\mu^{*}$ vanishes when integrated against
any Hecke character of finite type whenever $k\geq 3$. Since these span the
space of locally constant functions on $G,$ the distribution
$\overline{\lambda}-h^{+}N(x)^{k-1}\mu^{*}$ is identically zero. This implies
that $\overline{\lambda}$ is equal to $h^{+}N(x)^{k-1}\mu^{*}$ as
distributions and so it is bounded. This implies that $\lambda$ is a measure.
∎

## References

  * [BMP03] R. W. Bruggeman, R. J. Miatello, and I. Pacharoni, _Density results for automorphic forms on Hilbert modular groups_ , Geom. Funct. Anal. 13 (2003), no. 4, 681–719.
  * [CN79] Pierrette Cassou-Noguès, _Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques_, Invent. Math. 51 (1979), no. 1, 29–59.
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