# Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$: Singular terms

Baiqing Zhu

(Date: Jan 2024)

###### Abstract.

For arbitrary level $N$, we relate the generating series of codimension 2
special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of a genus 2
Eisenstein series, especially the singular terms of both sides. On the
analytic side, we use difference formulas of local densities to relate the
singular Fourier coefficients of the genus 2 Eisenstein series to the
nonsingular Fourier coefficients of a genus 1 Eisenstein series. On the
geometric side, we study the reduction of cusps to compute the divisor class
of the Hodge bundle and the heights of special divisors. When $N$ is square-
free, this gives a different proof of the main results in the works of Du and
Yang [DY19] and Sankaran, Shi, and Yang [SSY22].

###### Contents

  1. 1 Introduction
  2. I The Analytic Side
    1. 2 Local densities and difference formulas
    2. 3 Eisenstein series and Whittaker function
    3. 4 Singular Fourier coefficients of Eisenstein series
  3. II The Geometric Side
    1. 5 The geometry of the modular curve $\mathcal{X}_{0}(N)$
    2. 6 Arithmetic intersection theory on $\mathcal{X}_{0}(N)$
    3. 7 Arithmetic special cycles
  4. III Proof of the main results
    1. 8 Heights of arithmetic special divisors
    2. 9 Proof of the arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$

###  1\. Introduction

####  1.1. Background

The classical Siegel-Weil formula relates certain Siegel Eisenstein series to
the arithmetic of quadratic forms, namely it expresses special values of these
series as theta functions – generating series of representation numbers of
quadratic forms. Kudla initiated an influential program to establish the
arithmetic Siegel-Weil formula relating certain Siegel Eisenstein series to
objects in arithmetic geometry.

In this article, we study the case of modular curves. Let $N$ be a positive
integer, the classical modular curve $\mathcal{Y}_{0}(N)_{\mathbb{C}}$ over
$\mathbb{C}$ is defined as the following smooth $1$-dimensional complex curve,

$\mathcal{Y}_{0}(N)_{\mathbb{C}}\coloneqq\textup{GL}_{2}(\mathbb{Q})\backslash\mathbb{H}_{1}^{\pm}\times\textup{GL}_{2}(\mathbb{A}_{f})/\Gamma_{0}(N)(\hat{\mathbb{Z}})\simeq\Gamma_{0}(N)\backslash\mathbb{H}_{1}^{\pm},$

where $\mathbb{H}_{1}^{\pm}=\mathbb{C}\backslash\mathbb{R}$ and
$\mathbb{H}_{1}^{+}=\\{z=x+iy\in\mathbb{C}:x\in\mathbb{R},y\in\mathbb{R}_{>0}\\}$
is the upper half plane. The group $\Gamma_{0}(N)(\hat{\mathbb{Z}})$ is the
following open compact subgroup of $\textup{GL}_{2}(\mathbb{A}_{f})$,

$\Gamma_{0}(N)(\hat{\mathbb{Z}})=\left\\{x=\begin{pmatrix}a&b\\\
Nc&d\end{pmatrix}\in\textup{GL}_{2}(\hat{\mathbb{Z}})\,\,:\,\,a,b,c,d\in\hat{\mathbb{Z}}\right\\},$

and
$\Gamma_{0}(N)=\Gamma_{0}(N)(\hat{\mathbb{Z}})\bigcap\textup{GL}_{2}(\mathbb{Z})$.

The smooth curve $\mathcal{Y}_{0}(N)_{\mathbb{C}}$ is not proper, its
compactification
$\mathcal{X}_{0}(N)_{\mathbb{C}}\coloneqq\mathcal{Y}_{0}(N)_{\mathbb{C}}\cup\\{\textup{cusps}\\}$
is a smooth projective curve over $\mathbb{C}$. Katz, Mazur [KM85] and
Cesnavicius [Ces17] constructed an integral model $\mathcal{X}_{0}(N)$ for the
complex curve $\mathcal{X}_{0}(N)_{\mathbb{C}}$. The model
$\mathcal{X}_{0}(N)$ is a 2-dimensional regular flat Deligne-Mumford stack.
For every pair $(T,\mathsf{y})$ where $T$ is a $2\times 2$ symmetric matrix
with coefficients in $\mathbb{Q}$ and $\mathsf{y}$ is a positive definite
$2\times 2$ symmetric matrix with coefficients in $\mathbb{R}$, we define the
arithmetic special cycle $\widehat{\mathcal{Z}}(T,\mathsf{y})$ on the stack
$\mathcal{X}_{0}(N)$ and study their arithmetic degrees. Finally, we prove
that these arithmetic degrees are identified with the derivatives of Fourier
coefficients of certain Siegel Eisenstein series of genus 2.

When $N$ is an odd, square-free positive integer, the relation has already
been obtained for all the pair $(T,\mathsf{y})$ in the work of Sankaran, Shi,
and Yang [SSY22, Theorem 2.14] by computing both sides explicitly based on the
previous works of Yang [Yn98] and Kudla, Rapoport and Yang [KRY06]. When $T$
is nonsingular, the relation has been obtained for all the level $N$ in the
previous work of the author [Zhu23, Theorem 1.2.1] by establishing difference
formulas on both the analytic and geometric sides. In this article, we give
proof for all the pairs $(T,\mathsf{y})$, and all the levels $N$.

When the matrix $T$ is singular of rank 1. On the analytic side, we use
difference formulas of local densities to relate the singular Fourier
coefficients of an Eisenstein series of genus 2 to the nonsingular Fourier
coefficients of an Eisenstein series of genus 1. On the geometric side, the
cycle $\widehat{\mathcal{Z}}(T,\mathsf{y})$ is essentially the intersection of
a codimension $1$ cycle and the metrized Hodge line bundle on
$\mathcal{X}_{0}(N)$. We compute this intersection number by investigating the
irreducible components of the special fiber
$\mathcal{X}_{0}(N)_{\mathbb{F}_{p}}\coloneqq\mathcal{X}_{0}(N)\times_{\textup{Spec}\,\mathbb{Z}}\textup{Spec}\,\mathbb{F}_{p}$
of the model $\mathcal{X}_{0}(N)$, and the reduction mod $p$ of the cuspidal
divisor of the curve $\mathcal{X}_{0}(N)$.

When the matrix $T=0$, both the analytic side and geometric side can be
computed explicitly. Here again, the computation of the analytic side is based
on the difference formulas of local densities, while the computation of the
geometric side is based on the intersection of irreducible components of the
special fiber $\mathcal{X}_{0}(N)_{\mathbb{F}_{p}}$.

####  1.2. Summary of the main results

Let $\Delta(N)$ be the following rank 3 quadratic lattice over $\mathbb{Z}$,

(1) $\Delta(N)=\left\\{x=\begin{pmatrix}-Na&b\\\
c&a\end{pmatrix}:\,\,a,b,c\in\mathbb{Z}\right\\}$

equipped with the quadratic form $x\mapsto\textup{det}(x)$.

We use $v$ to denote a place of $\mathbb{Q}$. For every finite place $v$, let
$\delta_{v}(N)=\Delta(N)\otimes_{\mathbb{Z}}\mathbb{Z}_{v}$ be a rank 3
quadratic lattice over $\mathbb{Z}_{v}$. Let $\mathbb{A}$ be the ring of
ad$\grave{\textup{e}}$les over $\mathbb{Q}$. Let
$\mathbb{V}=\\{\mathbb{V}_{v}\\}$ be the incoherent collection of quadratic
spaces of $\mathbb{A}$ of rank 3 nearby $\Delta(N)$ at $\infty$, i.e.,

(2) $\mathbb{V}_{v}=\delta_{v}(N)\otimes\mathbb{Q}_{v}\,\,\textup{if
$v<\infty$, and $\mathbb{V}_{\infty}$ is positive definite.}$

There is a classical incoherent Eisenstein series
$E(\mathsf{z},s,\Delta(N)^{2})$ (cf. §3.1) on the Siegel upper half space of
genus 2,

$\mathbb{H}_{2}=\\{\mathsf{z}=\mathsf{x}+i\mathsf{y}\,\,|\,\,\mathsf{x}\in\textup{Sym}_{2}(\mathbb{R}),\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})_{>0}\\}.$

This is essentially the Siegel Eisenstein series associated to a standard
Siegel-Weil section of the degenerate principal series. The Eisenstein series
here has a meromorphic continuation and a functional equation relating
$s\leftrightarrow-s$. The central value $E(\mathsf{z},0,\Delta(N)^{2})=0$ by
the incoherence. We thus consider its central derivative

$\partial\textup{Eis}(\mathsf{z},\Delta(N)^{2})\coloneqq\frac{\textup{d}}{\textup{d}s}\bigg{|}_{s=0}E(\mathsf{z},s,\Delta(N)^{2}).$

Associated to the standard additive character
$\psi:\mathbb{A}/\mathbb{Q}\rightarrow\mathbb{C}^{\times}$, it has a
decomposition into the central derivatives of the Fourier coefficients

$\partial\textup{Eis}(\mathsf{z},\Delta(N)^{2})=\sum\limits_{T\in\textup{Sym}_{2}(\mathbb{Q})}\partial\textup{Eis}_{T}(\mathsf{z},\Delta(N)^{2})$

On the geometric side, there is a regular integral model of the modular curve
$\mathcal{Y}_{0}(N)_{\mathbb{C}}$ over $\mathbb{Z}$ defined by Katz and Mazur:
for any scheme $S$, the groupoid $\mathcal{Y}_{0}(N)(S)$ consists of objects
$(E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}E^{\prime})$ where $E$,
$E^{\prime}$ are elliptic curves over $S$ and $\pi$ is a cyclic isogeny such
that $\pi^{\vee}\circ\pi=N$. They proved that $\mathcal{Y}_{0}(N)$ is
2-dimensional regular flat Deligne-Mumford stack (cf. [KM85, Theorem 5.1.1]),
but $\mathcal{Y}_{0}(N)$ is not proper. There is a moduli stack
$\mathcal{X}_{0}(N)$ defined in [Ces17] which serves as a “compactification”
of $\mathcal{Y}_{0}(N)$. It is a proper regular flat 2-dimensional Deligne-
Mumford stack that contains $\mathcal{Y}_{0}(N)$ as an open substack, so we
can consider the arithmetic intersection theory on $\mathcal{X}_{0}(N)$
following the lines in [Gil09].

#####  1.2.1. Special cycles of codimension 1

The key concept is that of a special cycle. For every integer $t$, we define a
closed substack of $\mathcal{Y}_{0}(N)$ as follows: For an object
$(E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}E^{\prime})$ of
$\mathcal{Y}_{0}(N)(S)$, the stack $\mathcal{Z}(t,\Delta(N))$ parameterizes
isogenies $j$ between $E$ and $E^{\prime}$ with $j^{\vee}\circ j=t$ and
orthogonal to the cyclic isogeny $\pi$. It can be shown that the stack
$\mathcal{Z}(t,\Delta(N))$ is a generalized Cartier divisor even on the stack
$\mathcal{X}_{0}(N)$ when $t>0$, and it’s empty when $t\leq 0$. For every
positive number $y$, we define the modified special divisor
$\mathcal{Z}^{\ast}(t,y,\Delta(N))$ to be

$\mathcal{Z}^{\ast}(t,y,\Delta(N))=\mathcal{Z}(t,\Delta(N))+g(t,y,\Delta(N))\cdot\mathsf{Cusp}(\mathcal{X}_{0}(N)).$

where $\mathsf{Cusp}(\mathcal{X}_{0}(N))$ is the cuspidal divisor of the stack
$\mathcal{X}_{0}(N)$, and $g(t,y,\Delta(N))$ is a smooth function in $y$
defined in (48), it’s identically $0$ when $t>0$.

For every pair $(t,y)$ such that $t$ is a nonzero integer and $y>0$, a green
function $\mathfrak{g}(t,y,\Delta(N))$ of the divisor
$\mathcal{Z}^{\ast}(t,y,\Delta(N))$ is constructed by Kudla [Kud97, (12.21)]
(see also [DY19, §5]). We will recall the construction in $\S$7.1.2. Finally,
we define the following element in the codimension 1 arithmetic Chow group
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$ of $\mathcal{X}_{0}(N)$
(Definition 7.1.3):

(3)
$\widehat{\mathcal{Z}}(t,y)=(\mathcal{Z}^{\ast}(t,y,\Delta(N)),\mathfrak{g}(t,y,\Delta(N))).$

These elements $\widehat{\mathcal{Z}}(t,y)$ are invariant under the Atkin-
Lehner involution $W_{N}$ of the stack $\mathcal{X}_{0}(N)$, i.e.,
$W_{N}^{\ast}\widehat{\mathcal{Z}}(t,y)=\widehat{\mathcal{Z}}(t,y)$ (Lemma
7.1.4).

When $t=0$, the definition of $\widehat{\mathcal{Z}}(0,y)$ is slightly
different from (3). Recall that there is a metrized Hodge line bundle
$\widehat{\omega}_{N}$ on the stack $\mathcal{X}_{0}(N)$ (Example 6.2.2),
however, this bundle is not invariant under the Atkin-Lehner involution
$W_{N}$ (Corollary 6.4.4). Therefore we consider the following element

$\widehat{\omega}=-\widehat{\omega}_{N}-W_{N}^{\ast}\widehat{\omega}_{N},$

the element $\widehat{\omega}$ is clearly invariant under the Atkin-Lehner
involution. We define

(4)
$\widehat{\mathcal{Z}}(0,y,\Delta(N))=\widehat{\omega}+(\mathcal{Z}^{\ast}(0,y,\Delta(N)),\mathfrak{g}(0,y,\Delta(N)))-(0,\log
y).$

Now for any rational number $t$ which is not an integer, we simply define
$\widehat{\mathcal{Z}}(t,y)=0\in\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$.
Let $\tau=x+iy\in\mathbb{H}_{1}^{+}$, we consider the following generating
series with coefficients in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$,

$\widehat{\phi}_{1}(\tau)=\sum\limits_{t\in\mathbb{Q}}\widehat{\mathcal{Z}}(t,y,\Delta(N))\cdot
q^{t}$

where $q^{t}=e^{2\pi i\,t\tau}$.

######  Theorem 1.2.1 (Theorem 8.2.5).

Let $N$ be a positive integer. The generating series $\widehat{\phi}_{1}$ is a
nonholomorphic Siegel modular form of genus 1 and weight $\frac{3}{2}$ with
values in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$.

######  Remark 1.2.2.

When $N$ is square-free, the above theorem has been proved by Du and Yang
[DY19, Theorem 1.1]. Similar results for Shimura curves are proved by Kudla,
Rapoport, and Yang [KRY06, Theorem A].

#####  1.2.2. Special cycles of codimension 2

We are mainly interested in the elements in the codimension 2 arithmetic Chow
group $\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$.

Let $T\in\textup{Sym}_{2}(\mathbb{Q})$ be a symmetric matrix. If
$T=\mathbf{0}_{2}$, for every positive definite matrix
$\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})$, we define

$\widehat{\mathcal{Z}}(\mathbf{0}_{2},\mathsf{y})=\widehat{\omega}\cdot\widehat{\omega}+(0,\log\textup{det}\mathsf{y}\cdot[\Omega])\in\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N)).$

If the rank of $T$ is 1, there exists a matrix
$g\in\textup{GL}_{2}(\mathbb{Z})$ such that
${{}^{t}g}Tg=\textup{diag}\\{0,t\\}$ for some nonzero rational number $t$, let
$\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a symmetric matrix with
$y_{1},y_{2}>0$, we define the following element in
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$

$\widehat{\mathcal{Z}}(T,\mathsf{y})=\widehat{\mathcal{Z}}(t,y_{2},\Delta(N))\cdot\widehat{\omega}-\left(0,\log
y_{1}\cdot\delta_{\mathcal{Z}^{\ast}(t,y_{2},\Delta(N))_{\mathbb{C}}}\right).$

If $T$ is nonsingular, a detailed definition of the element
$\widehat{\mathcal{Z}}(T,\mathsf{y})$ can be found in [Zhu23, (16)]. Now for
any element $\mathsf{z}=\mathsf{x}+i\mathsf{y}\in\mathbb{H}_{2}$, we consider
the following generating series with coefficients in
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$,

$\widehat{\phi}_{2}(\mathsf{z})=\sum\limits_{T\in\textup{Sym}_{2}(\mathbb{Q})}\widehat{\mathcal{Z}}(T,\mathsf{y})\cdot
q^{T}$

where $q^{T}=e^{2\pi i\,\textup{tr}\,(T\mathsf{z})}$.

There is an isomorphism
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))\simeq\mathbb{C}$ given by the
arithmetic degree map
$\widehat{\textup{deg}}:\widehat{\textup{CH}}^{2}_{\mathbb{C}}(\mathcal{X}_{0}(N))\rightarrow\mathbb{C}$
constructed by Kudla, Rapoport and Yang [KRY06, $\S$2.4] (see also (35)). Our
main result is the following

######  Theorem 1.2.3 (Theorem 7.2.5).

Let $N$ be a positive integer. The generating series $\widehat{\phi}_{2}$ is a
nonholomorphic Siegel modular form of genus 2 and weight $\frac{3}{2}$. More
precisely, under the isomorphism
$\widehat{\textup{deg}}:\widehat{\textup{CH}}^{2}_{\mathbb{C}}(\mathcal{X}_{0}(N))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C}$

$\widehat{\phi}_{2}(\mathsf{z})=\frac{\psi(N)}{24}\cdot\partial\textup{Eis}(\mathsf{z},\Delta(N)^{2}),$

here $\psi(N)=N\cdot\prod\limits_{p|N}(1+p^{-1})$.

######  Remark 1.2.4.

When $N$ is square-free, the above theorem has been proved by Sankaran, Shi,
and Yang [SSY22]. Similar results for Shimura curves are proved by Kudla,
Rapoport, and Yang [KRY06, Theorem B].

####  1.3. Strategy of the proof of main results

In this section, we mainly explain the proof of Theorem 1.2.3 in this section,
while Theorem 1.2.1 will come as a byproduct.

We will prove Theorem 1.2.3 term-by-term, i.e., for any symmetric matrix
$T\in\textup{Sym}_{2}(\mathbb{Q})$, we prove that for any
$\mathsf{z}=\mathsf{x}+i\mathsf{y}\in\mathbb{H}_{2}$,

(5) $\widehat{\textup{deg}}\,\widehat{\mathcal{Z}}(T,\mathsf{y})\cdot
q^{T}=\frac{\psi(N)}{24}\cdot\partial\textup{Eis}_{T}(\mathsf{z},\Delta(N)^{2}).$

When $T$ is nonsingular, this has been proved in [Zhu23, Theorem 1.2.1]. In
this article, we will focus on the case that $T$ is singular. In the following
we refer to $\widehat{\textup{deg}}\,\widehat{\mathcal{Z}}(T,\mathsf{y})$ as
the geometric side and $\partial\textup{Eis}_{T}(\mathsf{z},\Delta(N)^{2})$
the analytic side.

#####  1.3.1. The singular coefficients of Eisenstein series

Our primary goals in the analytic side are: (a). When the rank of $T$ is 1, we
relate the singular Fourier coefficients of the Eisenstein series
$E_{T}(\mathsf{z},s,\Delta(N)^{2})$ to nonsingular Fourier coefficients of
another Eisenstein series of lower genus. (b). When $T=0$, we compute the
exact value of
$\partial\textup{Eis}_{\mathbf{0}_{2}}(\mathsf{z},\Delta(N)^{2})$.

Before explaining the main idea, we give some necessary definitions. For any
two integral quadratic $\mathbb{Z}_{p}$-lattices $L$ and $M$. Let
$\textup{Rep}_{M,L}$ be the scheme of integral representations, an
$\mathbb{Z}_{p}$-scheme such that for any $\mathbb{Z}_{p}$-algebra $R$,
$\textup{Rep}_{M,L}(R)=\textup{QHom}(L\otimes_{\mathbb{Z}_{p}}R,M\otimes_{\mathbb{Z}_{p}}R)$,
where QHom denotes the set of quadratic module homomorphisms. The local
density of integral representations is defined to be

$\textup{Den}(M,L)=\lim\limits_{d\rightarrow\infty}\frac{\\#\textup{Rep}_{M,L}(\mathbb{Z}_{p}/p^{d})}{p^{d\cdot\textup{dim}(\textup{Rep}_{M,L})_{\mathbb{Q}_{p}}}}.$

Let $H_{2}^{+}=\mathbb{Z}_{p}^{2}$ be the rank 2 quadratic
$\mathbb{Z}_{p}$-lattice equipped with the quadratic form
$q_{H_{2}^{+}}(x,y)=xy$. For any positive integer $k$, let
$H_{2k}^{+}=\left(H_{2}^{+}\right)^{\obot k}$ be the rank $2k$ quadratic
lattice obtained by orthogonal direct sum of $k$ copies of $H_{2}^{+}$.

Let’s explain the idea shortly. As is well-known, the Fourier coefficients of
Eisenstein series are (linear combinations) of Whittaker functions which are
essentially products of local representation densities of quadratic lattices.
Our main tool is the difference formula of local representation densities of
quadratic lattices proved in [Zhu23, §7.2] (see also Theorem 2.2.5) which
roughly says that for any number $l\in\mathbb{Z}_{p}$ and any quadratic
lattice $M$, the following two local densities are related

(6) $\textup{Den}(\langle-l\rangle\obot
H_{2k+2}^{+},M)\leftrightarrow\textup{Den}(H_{2k+4}^{+},M\obot\langle
l\rangle).$

This relation has two applications:
$(a)$. Let $l=N$, the quadratic lattice $\langle-l\rangle\obot H_{2k+2}^{+}$
is isometric to $\delta_{p}(N)\obot H_{2k}^{+}$. The relation (6) reduces the
computation of a local density function with level structure to that of an
unramified local density function, which has very explicit expression and
functional equation by the works of Cho and Yamauchi [CY20, (3.4)], Ikeda
[Ike17, Theorem 4.1], see also the works of Li and Zhang [LZ22, $\S$3].
$(b)$. Taking the limit $\nu_{p}(l)\rightarrow\infty$, the right-hand side of
(6) converges to the Whittaker function with singular coefficient, while the
left-hand side of (6) converges to $\textup{Den}(H_{2k+2}^{+},M)$ (Lemma
2.2.4), i.e., we have the following relation,

(7)
$\textup{Den}(H_{2k+2}^{+},M)=\lim\limits_{\nu_{p}(l)\rightarrow\infty}\textup{Den}(H_{2k+4}^{+},M\obot\langle
l\rangle).$

When the rank of $T$ is 1, we assume $T=\textup{diag}\\{0,t\\}$ with $t\neq 0$
for simplicity, let $\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a positive
definite symmetric matrix, then there is a well-known relation (see [GS19,
Lemma 5.4])

$E_{T}(i\mathsf{y},s,\Delta(N)^{2})=y_{1}^{s/2}y_{2}^{-3/4}W_{t}(g_{iy_{2}},s+\frac{1}{2},\Delta(N))+(y_{1}y_{2})^{-3/4}W_{T}(g_{i\mathsf{y}},s,\Delta(N)^{2}),$

where both of the terms $W_{t}(g_{iy_{2}},s+\frac{1}{2},\Delta(N))$ and
$W_{T}(g_{i\mathsf{y}},s,\Delta(N)^{2})$ are product of local Whittaker
functions (see (13) and (14) for the precise definition).

Since $t\neq 0$, the term $W_{t}(g_{iy_{2}},s+\frac{1}{2},\Delta(N))$ is
already the nonsingular Fourier coefficient of an Eisenstein series of the
lower genus. Let’s now consider the term
$W_{T}(g_{i\mathsf{y}},s,\Delta(N)^{2})$, which is the product of local
Whittaker functions $W_{T,v}$ over all the places $v$ of $\mathbb{Q}$. The
relation between $W_{T,v}$ and $W_{t,v}$ is sketched in the following way: Let
$p$ be a finite prime, for any integer $m$, let
$T_{m}=\textup{diag}\\{p^{m},t\\}$ be a $2\times 2$ nonsingular matrix. By the
well-known relation between Whittaker functions and local densities
(Proposition 3.2.1), we have

$W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})=\lim\limits_{m\rightarrow\infty}W_{T_{m},p}(1,k,1_{\delta_{p}(N)^{2}})\leftrightarrow\textup{Den}(\delta_{p}(N)\obot
H_{2k}^{+},\langle t\rangle\obot\langle p^{m}\rangle).$

Applying $(a)$ of (6) twice,

(8) $\textup{Den}(\delta_{p}(N)\obot H_{2k}^{+},\langle t\rangle\obot\langle
p^{m}\rangle)\stackrel{{\scriptstyle
l=N}}{{\leftrightarrow}}\textup{Den}(H_{2k+4}^{+},\langle t\rangle\obot\langle
p^{m}\rangle\obot\langle N\rangle)\stackrel{{\scriptstyle
l=p^{m}}}{{\leftrightarrow}}\textup{Den}(\langle-p^{m}\rangle\obot
H_{2k+2}^{+},\langle t\rangle\obot\langle N\rangle).$

Taking limit $m\rightarrow\infty$, the formula (7) tells us that

(9) $\lim_{m\rightarrow\infty}\textup{Den}(\langle-p^{m}\rangle\obot
H_{2k+2}^{+},\langle t\rangle\obot\langle
N\rangle)=\textup{Den}(H_{2k+2}^{+},\langle t\rangle\obot\langle N\rangle).$

Applying $(a)$ again, we get

(10) $\textup{Den}(H_{2k+2}^{+},\langle t\rangle\obot\langle
N\rangle)\stackrel{{\scriptstyle
l=N}}{{\leftrightarrow}}\textup{Den}(\delta_{p}(N)\obot H_{2k-2}^{+},\langle
t\rangle).$

The last term $\textup{Den}(\delta_{p}(N)\obot H_{2k-2}^{+},\langle t\rangle)$
is clearly related to $W_{t,p}(1,k-\frac{1}{2},1_{\delta_{p}(N)})$ by
Proposition 3.2.1. Therefore we have successfully built the bridge from
$W_{T,v}$ and $W_{t,v}$ for a finite prime $v$.

However, we warn the reader that all the symbols “$\leftrightarrow$” are not
exact equal, it just means that there are some relations between the two sides
of the symbol. Detailed calculations based on the principles above are given
in $\S$4. Our final result is the following proposition.

######  Proposition 1.3.1 (Proposition 4.2.4).

Let $T$ be a $2\times 2$ matrix of rank 1 which can be diagonalized to
$\textup{diag}\\{0,t\\}$ with $t\neq 0$. Let
$\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a positive definite symmetric
matrix, then for any complex number $k$, we have

$\displaystyle E_{T}(i\mathsf{y},k,\Delta(N)^{2})$
$\displaystyle=y_{1}^{k/2}E_{t}(iy_{2},\frac{1}{2}+k,\Delta(N))$
$\displaystyle+y_{1}^{-k/2}\cdot\frac{k-1}{k+1}\cdot
N^{-k}\cdot\frac{\Lambda(2-2k)}{\Lambda(2+2k)}\cdot\prod\limits_{p|N}\beta_{p}(k)\cdot
E_{t}(iy_{2},\frac{1}{2}-k,\Delta(N)),$

where $\beta_{p}(s)$ is a rational function in $p^{-s}$.

######  Remark 1.3.2.

The same formula in Proposition 1.3.1 has been proved when $N$ is squarefree
in [SSY22, Lemma 4.13] by explicit computations of local densities. Our proof
is more conceptual and can be easily generalized to higher dimensions by the
principles displayed in (8), (9) and (10).

When $T=\mathbf{0}_{2}$, all the local Whittaker functions
$W_{T,v}(1_{v},s,1_{\delta_{v}(N)^{2}})$ can be computed explicitly by
combining Wedhorn’s computations [Wed07, §2.11] and the difference formula.

#####  1.3.2. Vertical components of the Hodge line bundle

The primary goal on the geometric side is to compute the height pairings of
(metrized) special divisors $\widehat{\mathcal{Z}}(t,y,\Delta(N))$ and the
modified metrized Hodge line bundle $\widehat{\omega}$, and the self-
intersection numbers of the metrized line bundle $\widehat{\omega}$.

The main difficulty comes from the fact that the bundle $\widehat{\omega}$
contains vertical components. These components should be a linear combination
of irreducible components of the reduction mod $p$ of the stack
$\mathcal{X}_{0}(N)$ for prime numbers $p$. We need to know the explicit
multiplicities of these irreducible components appearing in $\widehat{\omega}$
and their intersections.

We first study the special fiber of the stack $\mathcal{X}_{0}(N)$. Let $p$ be
a prime number, by the works of Katz and Mazur, if $n=\nu_{p}(N)\geq 0$ is the
$p$-adic valuation of the number $N$, then
$\mathcal{X}_{0}(N)_{p}\coloneqq\mathcal{X}_{0}(N)\times_{\textup{Spec}\,\mathbb{Z}}\textup{Spec}\,\mathbb{F}_{p}$
has $n+1$ irreducible components $\mathcal{X}_{p}^{a}(N)$ and they meet each
other at every supersingular point, where the index $a$ satisfies that $-n\leq
a\leq n$ and has the same parity with $n$. Moreover, every component
$\mathcal{X}_{p}^{a}(N)$ has two natural morphisms to the stack
$\mathcal{X}_{0}(Np^{-n})_{p}$, one of them is an isomorphism while the other
one is finite flat of degree $p^{|a|}$ (see Theorem 5.4.2). Later in $\S$6.3,
we compute the intersection numbers between these irreducible components based
on the explicit local equations of these components at supersingular points
obtained by Katz and Mazur [KM85, Theorem 13.4.7] (see also [Zhu23, Corollary
6.2.7]).

After that, in $\S$6.4 we use an explicit rational section $\Delta_{N}$ of the
bundle $\widehat{\omega}_{N}^{\otimes 12\varphi(N)}$ to give an explicit
expression of this bundle in the Chow group
$\textup{CH}^{1}(\mathcal{X}_{0}(N))$ by computing the element
$\textup{div}(\Delta_{N})\in\textup{CH}^{1}(\mathcal{X}_{0}(N))$ (Theorem 45),
this idea originates from the work of Du and Yang [DY19], where they consider
the case that $N$ is squarefree.

Our idea of computing the element $\textup{div}(\Delta_{N})$ for general $N$
comes from the observation that there is a correspondence between cusps and
the special fibers of the stack $\mathcal{X}_{0}(N)$. Let’s explain this in
the specific case that $N=p^{n}$ for some integer $n\geq 0$. There is a
cuspidal divisor $\mathsf{Cusp}(\mathcal{X}_{0}(p^{n}))$ on the stack
$\mathcal{X}_{0}(p^{n})$, it is a disjoint union of $n+1$ connected components
(Proposition 5.3.1),

$\mathsf{Cusp}(\mathcal{X}_{0}(p^{n}))=\coprod\limits_{\begin{subarray}{c}-n\leq
a\leq n\\\ a\equiv n\,\textup{mod}\,2\end{subarray}}\mathsf{C}^{a}(p^{n}).$

Note that the index set is exactly the same as the index set for the
irreducible components of the stack $\mathcal{X}_{0}(p^{n})_{p}$! Actually we
will prove that the connected component $\mathsf{C}^{a}(p^{n})$ pulls back to
the cusp of the curve $\mathcal{X}_{p}^{a}(p^{n})$ (Proposition 5.4.5). Then
we pick a specific cusp lying in the component $\mathsf{C}^{a}(p^{n})$ and
consider the local expansion of the section $\Delta_{p^{n}}$ around this
point, the multiplicity of $p$ appearing in the local expansion gives the
multiplicity of the vertical component of $\mathcal{X}_{p}^{a}(p^{n})$ in the
element $\textup{div}(\Delta_{p^{n}})$. The final result is

$\textup{div}(\Delta_{N})=\psi(N)\varphi(N)P_{\infty}(N)+\sum\limits_{p|N}f_{p}(N),$

where $P_{\infty}(N)$ is the connected component of the cuspidal divisor
containing the cusp $\infty$, and for any $p|N$, $f_{p}(N)$ is the following
vertical divisor,

$f_{p}(N)=12p^{n-1}\varphi(N_{p})\sum\limits_{\begin{subarray}{c}-n\leq a<n\\\
a\equiv n\,\textup{mod
2}\end{subarray}}\left(\frac{1-p}{2}(n-a)-1\right)\varphi(p^{(n-|a|)/2})\cdot\mathcal{X}_{p}^{a}(N).$

This expression is the main tool to compute the self intersection of the
modified Hodge bundle $\widehat{\omega}$ and the intersection pairing
$\widehat{\mathcal{Z}}(t,y,\Delta(N))\cdot\widehat{\omega}$.

####  1.4. Acknowledgement

The author is grateful to Professor Chao Li for his careful reading of the
original manuscript and many helpful comments. The author is supported by the
Department of Mathematics at Columbia University in the city of New York.

##  Part I The Analytic Side

###  2\. Local densities and difference formulas

####  2.1. Notations on quadratic lattices

Let $p$ be a prime number. Let $F$ be a nonarchimedean local field of residue
characteristic $p$, with ring of integers $\mathcal{O}_{F}$, residue field
$\kappa=\mathbb{F}_{q}$ of size $q$, and uniformizer $\pi$. Let
$\nu_{\pi}:F\rightarrow\mathbb{Z}\cup\\{\infty\\}$ be the valuation on $F$ and
$|\cdot|:F\rightarrow\mathbb{R}_{\geq 0}$ be the normalized absolute value on
$F$. Let $(\cdot,\cdot)_{F}$ be the Hilbert symbol on the local field $F$.

A quadratic lattice $(L,q_{L})$ is a finite free $\mathcal{O}_{F}$-module
equipped with a quadratic form $q_{L}:L\rightarrow F$. The quadratic form
$q_{L}$ also induces a symmetric bilinear form $L\times
L\stackrel{{\scriptstyle(\cdot,\cdot)}}{{\longrightarrow}}F$ by
$(x,y)=q_{L}(x+y)-q_{L}(x)-q_{L}(y)$. Let $L^{\vee}=\\{x\in
L\otimes_{\mathcal{O}_{F}}F:(x,L)\subset\mathcal{O}_{F}\\}$. We say a
quadratic lattice is integral if $q_{L}(x)\in\mathcal{O}_{F}$ for all $x\in
L$, is self-dual if it is integral and $L$ = $L^{\vee}$. There are three
invariants associated to a quadratic lattice: the rank of $L$ over
$\mathcal{O}_{F}$, the discriminant $\chi(L)\in\\{0,\pm 1\\}$ and the Hasse
invariant $\epsilon(L)\in\\{\pm 1\\}$. We refer the readers to [Zhu23, §2] and
[LZ22, §2] for details about the last two invariants.

When $p$ is odd, we use $H_{k}^{\varepsilon}$ to denote the unique (up to
isometry) self-dual lattice of rank $k$ and discriminant $\chi(L)=\varepsilon$
. When $p=2$, let $H_{2n}^{+}=(H_{2}^{+})^{\obot n}$ be a self-dual lattice of
rank $2n$, where the quadratic form on $H_{2}^{+}=\mathcal{O}_{F}^{2}$ is
given by $(x,y)\in\mathcal{O}_{F}^{2}\mapsto xy$.

######  Example 2.1.1.

Let $N\in\mathcal{O}_{F}$. Let $\delta_{F}(N)$ be the following rank 3
quadratic lattice over $\mathcal{O}_{F}$,

$\delta_{F}(N)=\left\\{x=\begin{pmatrix}-Na&b\\\
c&a\end{pmatrix}:\,\,a,b,c\in\mathcal{O}_{F}\right\\}.$

equipped with the quadratic form induced by $x\mapsto\textup{det}(x)$. Under
the following basis of $\delta_{F}(N)$,

$e_{1}=\begin{pmatrix}-N&\\\ &1\end{pmatrix},\,\,e_{2}=\begin{pmatrix}\,&1\\\
\,&\end{pmatrix},\,\,e_{3}=\begin{pmatrix}\,&\,\\\ 1&\,\end{pmatrix}.$

the quadratic form can be represented by the following symmetric matrix,

$T=\begin{pmatrix}-N&0&0\\\ 0&0&-\frac{1}{2}\\\
0&-\frac{1}{2}&0\end{pmatrix}.$

therefore $\chi(\delta_{F}(N))=\left(\frac{-N}{p}\right)$, where
$\left(\frac{\cdot}{p}\right)$ is the extended quadratic residue symbol (see
[Zhu23, §2.1]), and $\epsilon(\delta_{F}(N))=(N,-1)_{F}$. Moreover,

$\delta_{F}(N)^{\vee}=\left\\{x=\begin{pmatrix}-Na&b\\\
c&a\end{pmatrix}:\,\,a\in\frac{1}{2N}\mathcal{O}_{F},b,c\in\mathcal{O}_{F}\right\\}.$

therefore $\delta_{F}(N)^{\vee}/\delta_{F}(N)\simeq\mathcal{O}_{F}/2N$.

Throughout this article, we will mainly focus on the case that
$F=\mathbb{Q}_{p}$. In this case, we simply use $\delta_{p}(N)$ to denote the
lattice $\delta_{\mathbb{Q}_{p}}(N)$ (as we did in the introduction).

####  2.2. Local densities of quadratic lattices

######  Definition 2.2.1.

Let $L,M$ be two quadratic $\mathcal{O}_{F}$-lattices. Let
$\textup{Rep}_{M,L}$ be the scheme of integral representations, an
$\mathcal{O}_{F}$-scheme such that for any $\mathcal{O}_{F}$-algebra $R$,

$\textup{Rep}_{M,L}(R)=\textup{QHom}(L\otimes_{\mathcal{O}_{F}}R,M\otimes_{\mathcal{O}_{F}}R),$

where QHom denotes the set of injective module homomorphisms which preserve
the quadratic forms. The local density of integral representations is defined
to be

$\textup{Den}(M,L)=\lim\limits_{d\rightarrow\infty}\frac{\\#\textup{Rep}_{M,L}(\mathcal{O}_{F}/\pi^{d})}{q^{d\cdot\textup{dim}(\textup{Rep}_{M,L})_{F}}}.$

######  Definition 2.2.2.

Let $L,M$ be two quadratic $\mathcal{O}_{F}$-lattices. Let
$\textup{PRep}_{M,L}$ be the $\mathcal{O}_{F}$-scheme of primitive integral
representations such that for any $\mathcal{O}_{F}$-algebra $R$,

$\textup{PRep}_{M,L}(R)=\\{\phi\in\textup{Rep}_{M,L}(R):\textup{$\phi$ is an
isomorphism between $L_{R}$ and a direct summand of $M_{R}$}\\}.$

where $L_{R}$ (resp. $M_{R}$) is $L\otimes_{\mathcal{O}_{F}}R$ (resp.
$M\otimes_{\mathcal{O}_{F}}R$). The primitive local density is defined to be

$\textup{Pden}(M,L)=\lim\limits_{d\rightarrow\infty}\frac{\\#\textup{PRep}_{M,L}(\mathcal{O}_{F}/\pi^{d})}{q^{d\cdot\textup{dim}(\textup{Rep}_{M,L})_{F}}}.$

For any $N\in F$, let $(\langle N\rangle,q_{\langle N\rangle})$ be the rank 1
$\mathcal{O}_{F}$-lattice with an $\mathcal{O}_{F}$ generator $l_{N}$ such
that $q_{\langle N\rangle}(l_{N})=N$. For $a_{1},a_{2},\cdots,a_{n}\in F$, we
define $\langle a_{1},a_{2},\cdots,a_{n}\rangle\coloneqq\langle
a_{1}\rangle\obot\langle a_{2}\rangle\obot\cdots\obot\langle a_{n}\rangle$.

######  Example 2.2.3.

When $p$ odd, it has been calculated explicitly that for any
$N\in\mathcal{O}_{F}$ (cf. [LZ22, (3.3.2.1)]),

(11) $\textup{Pden}(H_{k}^{\varepsilon},\langle
N\rangle)=\begin{cases}1-q^{1-k},&\textup{when $k$ is odd and $\pi\,|\,N$;}\\\
1+\varepsilon\chi_{F}(N)q^{(1-k)/2},&\textup{when $k$ is odd and $\pi\nmid
N$;}\\\ (1-\varepsilon q^{-k/2})(1+\varepsilon q^{1-k/2}),&\textup{when $k$ is
even and $\pi\,|\,N$;}\\\ 1-\varepsilon q^{-k/2},&\textup{when $k$ is even and
$\pi\nmid N$.}\end{cases}$

When $p=2$, the same formula makes sense and holds true only in the case that
$k$ is even and $\varepsilon=+1$.

######  Lemma 2.2.4.

Let $L,M$ be two quadratic $\mathcal{O}_{F}$-lattices such that the quadratic
form on $L$ is nondegenerate, then there exists a positive number $m$ such
that for any $a\in\mathcal{O}_{F}$ with $\nu_{\pi}(a)\geq m$, the following
identity holds,

$\textup{Den}(\langle a\rangle\obot M,L)=\textup{Den}(M,L).$

###### Proof.

We assume that the rank of $L$ is $n$ and the rank of $M$ is $m$. Let
$\\{l_{i}\\}_{i=1}^{n}$ be an $\mathcal{O}_{F}$-basis of $L$, and let
$T=(t_{ij})_{1\leq i,j\leq n}$ be the moment matrix of the this basis under
the quadratic form on $L$. For any positive integer $d$, we have

$\displaystyle\textup{Rep}_{\langle a\rangle\obot
M,L}(\mathcal{O}_{F}/\pi^{d})=\\{(\overline{m}_{1}^{\prime},\cdots,\overline{m}_{n}^{\prime})\in(\langle
a\rangle/\pi^{d}\langle a\rangle\obot
M/\pi^{d}M)^{n}:(\overline{m}_{i}^{\prime},\overline{m}_{j}^{\prime})\equiv
2t_{ij}\,\textup{mod}\,2\pi^{d}\,\,\textup{for}\,i,j\geq 1.\\}.$

For any element
$(\overline{m}_{1}^{\prime},\cdots,\overline{m}_{n}^{\prime})\in(\langle
a\rangle/\pi^{d}\langle a\rangle\obot M/\pi^{d}M)^{n}$ in the set
$\textup{Rep}_{\langle a\rangle\obot M,L}(\mathcal{O}_{F}/\pi^{d})$, let
$(\overline{m}_{1}^{\prime\prime},\cdots,\overline{m}_{n}^{\prime\prime})\in(M/\pi^{d}M)^{n}$
be the $M/\pi^{d}M$-component of this element. Let
$(m_{1}^{\prime\prime},\cdots,m_{n}^{\prime\prime})\in M^{n}$ be an arbitrary
lift of the element
$(\overline{m}_{1}^{\prime\prime},\cdots,\overline{m}_{n}^{\prime\prime})$ to
$M$, when $\nu_{\pi}(a)$ and $d$ are sufficiently large, the sublattice
spanned by $(m_{1}^{\prime\prime},\cdots,m_{n}^{\prime\prime})$ inside $M$ is
isometric to $L$ due to the local constancy of equivalent class of
nondegenerate quadratic lattice (see the work of Durfee [Dur44]).

Therefore when $\nu_{\pi}(a)$ and $d$ are sufficiently large, we have a
natural map by taking the $M/\pi^{d}M$-component,

$\textup{Rep}_{\langle a\rangle\obot
M,L}(\mathcal{O}_{F}/\pi^{d})\longrightarrow\textup{Rep}_{M,L}(\mathcal{O}_{F}/\pi^{d}).$

hence $\\#\textup{Rep}_{\langle a\rangle\obot
M,L}(\mathcal{O}_{F}/\pi^{d})=q^{nd}\cdot\\#\textup{Rep}_{M,L}(\mathcal{O}_{F}/\pi^{d})$.

Notice that $\textup{dim}(\textup{Rep}_{M,L}=mn-\frac{n(n+1)}{2}$ and
$\textup{dim}(\textup{Rep}_{\langle a\rangle\obot
M,L}=(m+1)n-\frac{n(n+1)}{2}$, hence when $\nu_{\pi}(a)$ and $d$ are
sufficiently large,

$\frac{\\#\textup{Rep}_{\langle a\rangle\obot
M,L}(\mathcal{O}_{F}/\pi^{d})}{q^{d\cdot\textup{dim}(\textup{Rep}_{\langle
a\rangle\obot
M,L})}}=\frac{\\#\textup{Rep}_{M,L}(\mathcal{O}_{F}/\pi^{d})}{q^{d\cdot\textup{dim}(\textup{Rep}_{M,L})}}.$

Taking limit $d\rightarrow\infty$, we get $\textup{Den}(\langle a\rangle\obot
M,L)=\textup{Den}(M,L)$ under the assumption that $\nu_{\pi}(a)$ is
sufficiently large. ∎

######  Theorem 2.2.5.

Let $H$ be a self-dual quadratic $\mathcal{O}_{F}$-lattice of finite rank $k$.
Let $M$ be an integral quadratic $\mathcal{O}_{F}$-lattice of finite rank $r$.
Let $N\in\mathcal{O}_{F}$ be an element of valuation $n$, then

$\textup{Den}(H,M\obot\langle
N\rangle)=\sum\limits_{i=0}^{[n/2]}q^{(2-k+r)i}\cdot\textup{Pden}(H,\langle\pi^{-2i}N\rangle)\cdot\textup{Den}(H(N,i),M).$

###### Proof.

This is proved in [Zhu23, §7]. ∎

######  Remark 2.2.6.

Let’s assume that $H=H_{k}^{\varepsilon}$ where $k\geq 3$ is an integer, then
[Zhu23, Example 7.1.2] shows that for any positive integer $0\leq
i\leq[\frac{n}{2}]$ where $n=\nu_{\pi}(N)$,

$H_{k}^{\varepsilon}(N,i)\simeq\langle-N\pi^{-2i}\rangle\obot
H_{k-2}^{\varepsilon}.$

######  Corollary 2.2.7.

Let $k$ be a positive integer and $\varepsilon\in\\{\pm 1\\}$, let $M$ be a
quadratic lattice of rank $r$ over $\mathcal{O}_{F}$, then for any
$a\in\mathcal{O}_{F}$, we have the following identity,

$\lim\limits_{m\rightarrow\infty}\textup{Den}(H_{k}^{\varepsilon},M\obot\langle
a\pi^{m}\rangle)=\textup{Den}(H_{k-2}^{\varepsilon},M)\cdot\frac{\textup{Pden}(H_{k}^{\varepsilon},\langle\pi\rangle)}{1-q^{2-k+r}}.$

###### Proof.

By Theorem 2.2.5 and Remark 2.2.6, we know that for any positive integer $m$,

$\textup{Den}(H_{k}^{\varepsilon},M\obot\langle
a\pi^{m}\rangle)-q^{2-k+r}\textup{Den}(H_{k}^{\varepsilon},M\obot\langle
a\pi^{m-2}\rangle)=\textup{Pden}(H_{k}^{\varepsilon},\langle
a\pi^{m}\rangle)\cdot\textup{Den}(\langle-a\pi^{m}\rangle\obot
H_{k-2}^{\varepsilon},M).$

Notice that $\textup{Pden}(H_{k}^{\varepsilon},\langle a\pi^{m}\rangle)$ is
independent of the positive integer $m$ by Example 11. Taking
$m\rightarrow\infty$, Lemma 2.2.4 implies the following,

$(1-q^{2-k+r})\cdot\lim\limits_{m\rightarrow\infty}\textup{Den}(H_{k}^{\varepsilon},M\obot\langle
a\pi^{m}\rangle)=\textup{Den}(H_{k-2}^{\varepsilon},M)\cdot\textup{Pden}(H_{k}^{\varepsilon},\langle\pi\rangle).$

∎

######  Definition 2.2.8.

Let $n\geq 0$, for $\varepsilon\in\\{\pm 1\\}$, we define the normalizing
factors to be

$\displaystyle\textup{Nor}^{\varepsilon}(X,n)$
$\displaystyle=(1-\frac{1+(-1)^{n+1}}{2}\cdot\varepsilon
q^{-(n+1)/2}X)\prod\limits_{1\leq i<(n+1)/2}(1-q^{-2i}X^{2}).$

######  Lemma 2.2.9.

Let $L$ be a quadratic lattice of rank $n$. Let $N\in\mathcal{O}_{F}$. When
$p$ is odd, there exist polynomials $\textup{Den}^{\varepsilon}(X,L)$,
$\textup{Den}^{\flat\varepsilon}(X,L)$ and
$\textup{Den}^{\varepsilon}_{\Delta(N)}(X,L)\in\mathbb{Z}[X]$, such that for
positive integers $k$,

$\displaystyle\textup{Den}^{\varepsilon}(X,L)\,\big{|}_{X=q^{-k}}=\frac{\textup{Den}(H_{2k+n+1}^{\varepsilon},L)}{\textup{Nor}^{\varepsilon}(q^{-k},n)},\,\,\,\textup{Den}^{\flat\varepsilon}(X,L)\,\big{|}_{X=q^{-k}}=\frac{\textup{Den}(H_{2k+n}^{\varepsilon},L)}{\textup{Nor}^{\varepsilon}(q^{-k},n-1)}\cdot(1-\varepsilon\chi(L)q^{-k})$
$\displaystyle\textup{Den}^{\varepsilon}_{\Delta(N)}(X,L)\,\big{|}_{X=q^{-k}}=\left\\{\begin{aligned}
&\frac{\textup{Den}(\delta_{F}(N)\obot
H_{2k+n-2}^{\varepsilon},L)}{\textup{Nor}^{\varepsilon}(q^{-k},n-1)},&\textup{When
$\pi\,|\,N$};\\\ &\frac{\textup{Den}(\delta_{F}(N)\obot
H_{2k+n-2}^{\varepsilon},L)}{\textup{Nor}^{\varepsilon\chi_{F}(N)}(q^{-k},n)},&\textup{When
$\pi\nmid N$}.\end{aligned}\right.$

We define the derived local density of $L$ and the derived local density of
$L$ with level $N$ to be

$\partial\textup{Den}^{\varepsilon}(L)=-\frac{\textup{d}}{\textup{d}X}\bigg{|}_{X=1}\textup{Den}^{\varepsilon}(X,L),\,\,\,\partial\textup{Den}^{\varepsilon}_{\Delta(N)}(L)=-\frac{\textup{d}}{\textup{d}X}\bigg{|}_{X=1}\textup{Den}^{\varepsilon}_{\Delta(N)}(X,L).$

When $p=2$ and $n$ is odd, the polynomial $\textup{Den}^{+}(X,L)$ with the
same evaluation formulas at $X=q^{-k}$ exist. When $p=2$ and $n$ is even, the
polynomial $\textup{Den}^{+}_{\Delta(N)}(X,L)$ with the same evaluation
formula at $X=q^{-k}$ exists. The derived local density
$\partial\textup{Den}^{+}(L)$ of $L$ is defined similarly.

###### Proof.

This is a combination of [LZ22, Lemma 3.3.2, Definition 3.4.1, Definition
3.5.2] ∎

We have the following functional equation for
$\textup{Den}^{\varepsilon}(X,L)$.

######  Theorem 2.2.10.

When the residue characteristic of $\mathcal{O}_{F}$ is not 2. Let $L$ be a
quadratic lattice of rank $n$. Then

$\displaystyle\textup{Den}^{\varepsilon}(X,L)$
$\displaystyle=w^{\varepsilon}(L)\cdot
X^{\textup{val}(L)}\cdot\textup{Den}^{\varepsilon}(\frac{1}{X},L),$
$\displaystyle\textup{Den}^{\flat\varepsilon}(X,L)$
$\displaystyle=(q^{1/2}X)^{2[\frac{\textup{val}(L)}{2}]}\cdot\textup{Den}^{\flat\varepsilon}(\frac{1}{qX},L).$

where $\textup{val}(L)$ is the $\pi$-adic valuation of the moment matrix of
$L$ under an $\mathcal{O}_{F}$-basis, and the sign of the functional equation
is equal to

$w^{\varepsilon}(L)\coloneqq(\textup{det}\,L,-(-1)^{\tbinom{n+1}{2}}u)_{F}\cdot\epsilon(L)\in\\{\pm
1\\},$

where $u\in\mathcal{O}_{F}^{\times}$ such that $u$ is a square if
$\varepsilon=1$, is not a square if $\varepsilon=-1$.

###### Proof.

Both of these two formulas are proved by Ikeda [Ike17, Theorem 4.1 (1),(2)]
when $n$ is odd and $\varepsilon=+1$, the same proof works in general. ∎

######  Remark 2.2.11.

When the field $F$ is dyadic, similar functional equation has been proved by
Ikeda and Katsurada [IK22, Proposition 2.1]. In this article, we will need the
functional equation in the following case: Let $F=\mathbb{Q}_{2}$ and
$L=\langle N,t\rangle$ for some $N,t\in\mathbb{Z}$. When $-tN$ is not a
square, let $-d$ be the fundamental discriminant of the field extension
$\mathbb{Q}(-tN)/\mathbb{Q}$. When $-tN$ is a square, let $d=-1$, then there
exists a positive number $c$ such that

$4Nt=c^{2}d.$

We have the following functional equation:

(12)
$\textup{Den}^{\flat+}(X,L)=(2^{1/2}X)^{2\nu_{2}(c)}\cdot\textup{Den}^{\flat+}(\frac{1}{2X},L).$

###  3\. Eisenstein series and Whittaker function

####  3.1. Incoherent Eisenstein series

For an positive integer $r$, let $\mathsf{W}_{r}$ be the standard symplectic
space over $\mathbb{Q}$ of dimension 2$r$. Let
$P=MN\subset\textup{Sp}(\mathsf{W}_{r})$ be the standard Siegel parabolic
subgroup, which take the following form under the standard basis of
$\mathsf{W}_{r}$,

$\displaystyle M(\mathbb{Q})$
$\displaystyle=\left\\{m(a)=\begin{pmatrix}a&0\\\
0&{}^{t}a^{-1}\end{pmatrix}:a\in\textup{GL}_{r}(\mathbb{Q})\right\\},$
$\displaystyle N(\mathbb{Q})$
$\displaystyle=\left\\{n(b)=\begin{pmatrix}1_{2}&b\\\
0&1_{2}\end{pmatrix}:b\in\textup{Sym}_{r}(\mathbb{Q})\right\\}.$

Let $\mathbb{A}$ be the ad$\grave{\textup{e}}$le ring over $\mathbb{Q}$. There
is an isomorphism
$\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\textup{Sp}(\mathsf{W}_{r})({\mathbb{A}})\times\mathbb{C}^{1}$
with the multiplication on the latter is given by the global Rao cycle,
therefore we can write an element of $\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})$
as $(g,t)$ where $g\in\textup{Sp}(\mathsf{W}_{r})({\mathbb{A}})$ and
$t\in\mathbb{C}^{1}$.

Let $P(\mathbb{A})=M(\mathbb{A})N(\mathbb{A})$ be the standard Siegel
parabolic subgroup of $\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})$ where

$\displaystyle M(\mathbb{A})$
$\displaystyle=\left\\{(m(a),t):a\in\textup{GL}_{r}(\mathbb{A}),t\in\mathbb{C}^{1}\right\\},$
$\displaystyle N(\mathbb{A})$
$\displaystyle=\left\\{n(b):b\in\textup{Sym}_{r}(\mathbb{A})\right\\}.$

Recall the following incoherent collection of rank $3$ quadratic spaces
$\mathbb{V}=\\{\mathbb{V}_{v}\\}$ over $\mathbb{A}$ we defined in (2),

$\mathbb{V}_{v}=\delta_{v}(N)\otimes\mathbb{Q}_{v}\,\,\textup{if $v<\infty$,
and $\mathbb{V}_{\infty}$ is positive definite.}$

then we can verify immediately that
$\prod\limits_{v}\epsilon(\mathbb{V}_{v})=-1$.

Let
$\chi:\mathbb{A}^{\times}/\mathbb{Q}^{\times}\rightarrow\mathbb{C}^{\times}$
be the quadratic character given by
$\chi(x)=\prod\limits_{v\leq\infty}\chi_{v}(x_{v})=\prod\limits_{v\leq\infty}(x_{v},-N)_{v}$
for all $x=(x_{v})\in\mathbb{A}^{\times}$. Fix the standard additive character
$\psi:\mathbb{A}/\mathbb{Q}\rightarrow\mathbb{C}^{\times}$ such that
$\psi_{\infty}(x)=e^{2\pi ix}$. We may view $\chi$ as a character on
$M(\mathbb{A})$ by

$\chi(m(a),t)=\chi(\textup{det}(a))\cdot\gamma(\textup{det}(a),\psi)^{-1}\cdot
t.$

and extend it to $P(\mathbb{A})$ trivially on $N(\mathbb{A})$. Here
$\gamma(\textup{det}(a),\psi)$ is the Weil index (see the work of Kudla
[Kud97, p. 548]). We define the degenerate principal series to be the
unnormalized smooth induction

$I(s,\chi)\coloneqq\textup{Ind}_{P(\mathbb{A})}^{\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})}(\chi\cdot|\cdot|_{\mathbb{Q}}^{s+(r+1)/2}),\,\,\,s\in\mathbb{C}.$

For a standard section $\Phi(-,s)\in I(s,\chi)$ (i.e., its restriction to the
standard maximal compact subgroup of $\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})$
is independent of $s$), we define the associated Siegel Eisenstein series

$E(g,s,\Phi)=\sum\limits_{\gamma\in
P(\mathbb{Q})\backslash\textup{Sp}(\mathbb{Q})}\Phi(\gamma g,s),$

which converges for $\textup{Re}(s)\gg 0$ and admits meromorphic continuation
to $s\in\mathbb{C}$.

Recall that $\mathscr{S}(\mathbb{V}^{r})$ is the space of Schwartz functions
on $\mathbb{V}^{r}$. The fixed choice of $\chi$ and $\psi$ gives a Weil
representation $\omega=\omega_{\chi,\psi}$ of
$\textup{Mp}(\mathsf{W}_{r,\mathbb{A}})\times\textup{O}(\mathbb{V})$ on
$\mathscr{S}(\mathbb{V}^{r})$. For
$\boldsymbol{\varphi}\in\mathscr{S}(\mathbb{V}^{r})$, define a function

$\Phi_{\boldsymbol{\varphi}}(g)\coloneqq\omega(g)\boldsymbol{\varphi}(0),\,\,\,g\in\textup{Mp}(\mathsf{W}_{r,\mathbb{A}}).$

Then $\Phi_{\boldsymbol{\varphi}}(g)\in I(0,\chi)$. Let
$\Phi_{\boldsymbol{\varphi}}(-,s)\in I(s,\chi)$ be the associated standard
section, known as the standard Siegel–Weil section associated to
$\boldsymbol{\varphi}$. For
$\boldsymbol{\varphi}\in\mathscr{S}(\mathbb{V}^{r})$, we write
$E(g,s,\boldsymbol{\varphi})\coloneqq E(g,s,\Phi_{\boldsymbol{\varphi}})$.

In this paper, we will mainly work on the case that $r=1$ or $2$.

####  3.2. Whittaker functions and local densities

Let $v$ be a finite place of $\mathbb{Q}$. Define the local degenerate
principal series to be the unnormalized smooth induction

$I_{v}(s,\chi_{v})\coloneqq\textup{Ind}_{P(\mathbb{Q}_{v})}^{\textup{Mp}(\mathsf{W}_{r,v})}(\chi_{v}\cdot|\cdot|_{v}^{s+(r+1)/2}),\,\,\,s\in\mathbb{C}.$

For any function $\Phi_{v}\in I_{v}(s,\chi_{v})$, we define the local
Whittaker function by the following integral,

(13)
$W_{T,v}(g_{v},s,\Phi_{v})=\int\limits_{N(\mathbb{Q}_{v})}\Phi_{v}(w_{r}^{-1}n(b)g_{v},s)\psi_{v}(-\textup{tr}(\frac{1}{2}Tb))\textup{d}n(b),\,\,\,w_{r}=\begin{pmatrix}0&1_{r}\\\
-1_{r}&0\end{pmatrix}.$

The function $W_{T,v}(g_{v},s,\Phi_{v})$ has analytic continuation to
$s\in\mathbb{C}$.

The fixed choice of $\chi_{v}$ and $\psi_{v}$ gives a local Weil
representation $\omega_{v}=\omega_{\chi_{v},\psi_{v}}$ of
$\textup{Mp}(\mathsf{W}_{r,v})\times\textup{O}(\mathbb{V}_{v})$ on the
Schwartz function space $\mathscr{S}(\mathbb{V}_{v}^{r})$. We define the local
Whittaker function associated to $\boldsymbol{\varphi}_{v}$ and
$T\in\textup{Sym}_{r}(\mathbb{Q}_{v})$ to be

(14) $W_{T,v}(g_{v},s,\boldsymbol{\varphi}_{v})\coloneqq
W_{T,v}(g_{v},s,\Phi_{\boldsymbol{\varphi}_{v}}),$

where
$\Phi_{\boldsymbol{\varphi}_{v}}(g_{v})\coloneqq\omega_{v}(g_{v})\boldsymbol{\varphi}_{v}(0)\in
I_{v}(0,\chi_{v})$ and $\Phi_{\boldsymbol{\varphi}_{v}}(-,s)$ is the
associated standard section.

The relationship between Whittaker functions and local densities is encoded in
the following proposition.

######  Proposition 3.2.1.

Suppose $v\neq\infty$. Let $L$ be an integral quadratic
$\mathbb{Z}_{v}$-lattice of rank $1$ or $2$. Suppose that the quadratic form
of $L$ is represented by a number $t\in\mathbb{Q}_{v}$ (when the rank of $L$
is 1) or a matrix $T\in\textup{Sym}_{r}(\mathbb{Q}_{v})$ after a choice of
$\mathbb{Z}_{v}$-basis of $L$, we have the following identity,

(15)
$W_{t,v}(1,k+\frac{1}{2},1_{\delta_{v}(N)})=(-1,-N)_{v}|2N|_{v}^{1/2}\cdot\gamma(\mathbb{V}_{v})\cdot\textup{Den}(\delta_{v}(N)\obot
H_{2k}^{+},L),\,\,\,\textup{when the rank of $L$ is 1}.$ (16)
$W_{T,v}(1,k,1_{\delta_{v}(N)^{2}})=|2|_{v}^{3/2}\cdot|N|_{v}\cdot\gamma(\mathbb{V}_{v})^{2}\cdot\textup{Den}(\delta_{v}(N)\obot
H_{2k}^{+},L),\,\,\,\textup{when the rank of $L$ is 2}.$

where the constant
$\gamma(\mathbb{V}_{v})=\gamma(\textup{det}(\mathbb{V}_{v}),\psi_{v})^{-1}\cdot\epsilon(\mathbb{V}_{v})\cdot\gamma(\psi_{v})^{-3}$,
$\gamma(\textup{det}(\mathbb{V}_{v}),\psi_{v})$ and $\gamma(\psi_{v})$ are
Weil indexes (cf. Appendix of [Rao93]).

###### Proof.

Both of the two formulas are proved in [KRY06, Lemma 5.7.1]. ∎

For the place $v=\infty$, there is also a well-developed theory about the
Whittaker function at $\infty$. Let $W_{T,\infty}(g,s,\Delta(N)^{2})$ (resp.
$W_{t,\infty}(g,s,\Delta(N))$) be the Whittaker functions for the Gaussian
function on the space $\mathbb{V}_{\infty}^{2}$ (resp. $\mathbb{V}_{\infty}$),
both of them can be computed explicitly, for example, the works of Kudla,
Rapoport and Yang [KRY04, Lemma 8.6], [KRY06, Theorem 5.2.7, Proposition
5.7.7].

######  Definition 3.2.2.

For a symmetric matrix $T\in\textup{Sym}_{r}(\mathbb{Q})$, and a factorizable
section $\Phi=\otimes_{v}\Phi_{v}\in I(s,\chi)$ where $\Phi_{\infty}$ is the
standard section associated to the Gaussian function on the space
$\mathbb{V}_{\infty}^{r}$, for a complex number $s\in\mathbb{C}$ such that
$\textup{Re}(s)\gg 0$, we define the global Whittaker function to be

$W_{T}(g,s,\boldsymbol{\varphi})=W_{T,\infty}(g_{\infty},s,\Delta(N)^{r})\cdot\prod\limits_{v<\infty}W_{T,v}(g_{v},s,\boldsymbol{\varphi}_{v}),\,\,g=(g_{v})_{v}\in\textsf{Mp}(\mathsf{W}_{r,\mathbb{A}}).$

the above product can be meromorphically continued to the whole complex plane.

####  3.3. Fourier coefficients

We have a Fourier expansion of the Siegel Eisenstein series defined above,

$E(g,s,\boldsymbol{\varphi})=\sum\limits_{T\in\textup{Sym}_{r}(\mathbb{Q})}E_{T}(g,s,\boldsymbol{\varphi}),$

where

$E_{T}(g,s,\boldsymbol{\varphi})=\int_{\textup{Sym}_{r}(\mathbb{Q})\backslash\textup{Sym}_{r}(\mathbb{A})}E(n(b)g,s,\boldsymbol{\varphi})\psi(-\textup{tr}(Tb))\textup{d}n(b),$

the Haar measure $\textup{d}n(b)$ is normalized to be self-dual with respect
to $\psi$.

######  Lemma 3.3.1.

Let $T\in\textup{Sym}_{r}(\mathbb{Q})$ be a nonsingular matrix, let
$\boldsymbol{\varphi}\in\mathscr{S}(\mathbb{V}^{r})$ be a factorizable
Schwartz function, we have

$E_{T}(g,s,\boldsymbol{\varphi})=W_{T}(g,s,\boldsymbol{\varphi}).$

###### Proof.

This lemma follows from unravelling the integral defining the function
$E_{T}$, see [LZ22, §11.2]. ∎

####  3.4. Classical incoherent Eisenstein series

The hermitian symmetric domain for $\textup{Sp}(\mathsf{W}_{r})$ is the Siegel
upper half space

$\mathbb{H}_{r}=\\{\mathsf{z}=\mathsf{x}+i\mathsf{y}\,\,|\,\,\mathsf{x}\in\textup{Sym}_{r}(\mathbb{R}),\mathsf{y}\in\textup{Sym}_{r}(\mathbb{R})_{>0}\\}.$

When $r=1$, let $\tau=x+iy\in\mathbb{H}^{+}$ with $x,y\in\mathbb{R}$ and $y$
is positive. Define the classical incoherent Eisenstein series to be

(17) $E(\tau,s,\boldsymbol{\varphi})=y^{-3/4}\cdot
E(g_{\tau},s,\boldsymbol{\varphi}),\,\,g_{\tau}=[n(x)m(y^{1/2}),1]\in\textup{Mp}(\mathsf{W}_{r,\mathbb{A}}).$

In this paper, we will focus on the case that
$\boldsymbol{\varphi}=1_{\Delta(N)\otimes\hat{\mathbb{Z}}}\otimes\boldsymbol{\varphi}_{\infty}\in\mathscr{S}(\mathbb{V})$,
where $1_{\Delta(N)\otimes\hat{\mathbb{Z}}}$ is the characteristic function of
the rank $3$ $\hat{\mathbb{Z}}$-lattice $\Delta(N)\otimes\hat{\mathbb{Z}}$ and
$\boldsymbol{\varphi}_{\infty}$ is the Gaussian function
$\boldsymbol{\varphi}_{\infty}(x)=e^{-\pi q_{\infty}(x)}$. For the fixed
choice of the Schwartz function $\boldsymbol{\varphi}$ above and any
$t\in\mathbb{Q}$, we write

(18) $E(\tau,s,\Delta(N))\coloneqq
E(\tau,s,1_{\Delta(N)\otimes\hat{\mathbb{Z}}}\otimes\boldsymbol{\varphi}_{\infty}),\,\,\,E_{t}(\tau,s,\Delta(N))\coloneqq
E_{t}(\tau,s,1_{\Delta(N)\otimes\hat{\mathbb{Z}}}\otimes\boldsymbol{\varphi}_{\infty}).$

We will also need the normalized genus $1$ Eisenstein series
$\mathcal{E}(\tau,s,\Delta(N))$ defined as follows,

(19)
$\mathcal{E}(\tau,s,\Delta(N))=C_{N}(s)E(\tau,s-\frac{1}{2},\Delta(N)),\,\,\textup{where}\,\,C_{N}(s)=-\frac{s}{4\pi}\Lambda(2s)\left(\prod\limits_{p|N}(1-p^{-2s})\right)N^{(3s+1)/2}.$

Similarly, for any $t\in\mathbb{Q}$,
$\mathcal{E}_{t}(\tau,s,\Delta(N))=C_{N}(s)E_{t}(\tau,s-\frac{1}{2},\Delta(N))$.

When $r=2$, let $\mathsf{z}=\mathsf{x}+i\mathsf{y}\in\mathbb{H}_{2}$ with
$\mathsf{x},\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})$ and
$\mathsf{y}={{}^{t}a}\cdot a$ is positive definite. Define the classical
incoherent Eisenstein series to be

(20)
$E(\mathsf{z},s,\boldsymbol{\varphi})=\chi_{\infty}(m(a))^{-1}|\textup{det}(m(a))|^{-3/2}\cdot
E(g_{\mathsf{z}},s,\boldsymbol{\varphi}),\,\,g_{\mathsf{z}}=[n(x)m(a),1]\in\textup{Mp}(\mathsf{W}_{r,\mathbb{A}}).$

We write the central derivatives as,

(21) $\partial\textup{Eis}(\mathsf{z},\boldsymbol{\varphi})\coloneqq
E^{\prime}(\mathsf{z},0,\boldsymbol{\varphi}),\,\,\partial\textup{Eis}_{T}(\mathsf{z},\boldsymbol{\varphi})\coloneqq
E_{T}^{\prime}(\mathsf{z},0,\boldsymbol{\varphi}).$

Then we have a Fourier expansion

$\partial\textup{Eis}(\mathsf{z},\boldsymbol{\varphi})=\sum\limits_{T\in\textup{Sym}_{2}(\mathbb{Q})}\partial\textup{Eis}_{T}(\mathsf{z},\boldsymbol{\varphi}).$

In this paper, we will focus on the case that
$\boldsymbol{\varphi}=1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty}\in\mathscr{S}(\mathbb{V}^{2})$,
where
$\boldsymbol{\varphi}=1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}$
is the characteristic function of the rank $3$ $\hat{\mathbb{Z}}$-lattice
$\Delta(N)\otimes\hat{\mathbb{Z}}$ and $\boldsymbol{\varphi}_{\infty}$ is the
Gaussian function
$\boldsymbol{\varphi}_{\infty}(x)=e^{-\pi\,\textup{tr}\,T(x)}$. For the fixed
choice of the Schwartz function $\boldsymbol{\varphi}$ above and any
$T\in\textup{Sym}_{2}(\mathbb{Q})$, we write

(22) $\displaystyle E(\mathsf{z},s,\Delta(N)^{2})\coloneqq
E(\mathsf{z},s,1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty}),\,\,\,E_{T}(\mathsf{z},s,\Delta(N)^{2})\coloneqq
E_{T}(\mathsf{z},s,1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty}),$
(23)
$\displaystyle\partial\textup{Eis}(\mathsf{z},\Delta(N)^{2})\coloneqq\partial\textup{Eis}(\mathsf{z},1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty}),\,\,\,\partial\textup{Eis}_{T}(\mathsf{z},\Delta(N)^{2})\coloneqq\partial\textup{Eis}_{T}(\mathsf{z},1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty}).$

Similarly, for any $T\in\textup{Sym}_{2}(\mathbb{Q})$,
$W_{T}(g,s,\Delta(N)^{2})\coloneqq
W_{T}(g,s,1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}}\otimes\boldsymbol{\varphi}_{\infty})$.

######  Lemma 3.4.1.

Let $T=\textup{diag}\\{0,t\\}$ with $t\neq 0$, let
$\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a positive definite symmetric
matrix, then

$E_{T}(i\mathsf{y},s,\Delta(N)^{2})=y_{1}^{s/2}y_{2}^{-3/4}W_{t}(g_{iy_{2}},s+\frac{1}{2},\Delta(N))+(y_{1}y_{2})^{-3/4}W_{T}(g_{i\mathsf{y}},s,\Delta(N)^{2}).$

###### Proof.

This lemma follows from [SSY22, Lemma 4.11, formula (38)]. ∎

###  4\. Singular Fourier coefficients of Eisenstein series

In this section, we study the Fourier coefficient
$E_{T}(\mathsf{z},s,\Delta(N)^{2})$ of the Eisenstein series defined in
$\S$3.4 when the matrix $T$ is singular.

We first fix some notations. For any prime number $p$, we define the local
zeta function $\zeta_{p}(s)=(1-p^{-s})^{-1}$, we also define
$\zeta_{\infty}(s)=\pi^{-s/2}\Gamma(\frac{s}{2})$. Let
$\Lambda(s)=\prod\limits_{v}\zeta_{v}(s)$ be the completed zeta function where
$v$ ranges over all the places of $\mathbb{Q}$, then $\Lambda(s)$ can be
meromorphically continued to the whole complex plane and satisfies the
functional equation $\Lambda(s)=\Lambda(1-s)$.

####  4.1. The rank of $T$ is $0$

In this case, the matrix $T=\mathbf{0}_{2}$.

######  Lemma 4.1.1.

Let $n=\nu_{p}(N)\geq 0$ be the $p$-adic valuation of $N$, let $A_{p}(s)$ be
the following function for $s\in\mathbb{C}$,

$A_{p}(s)=|N|_{p}\cdot\begin{cases}\frac{1}{1+p^{-1-s}},&\textup{if $n=0$;}\\\
\frac{1}{1+p^{-1-s}}\cdot\frac{1-p^{(1-s)(n+1)}}{1-p^{1-s}}-p^{-2s}\frac{1}{1+p^{-1-s}}\cdot\frac{1-p^{(1-s)(n-1)}}{1-p^{1-s}},&\textup{if
$n\geq 1$.}\end{cases}$

The following identity holds,

$W_{\mathbf{0}_{2},p}(1,k,1_{\delta_{p}(N)^{2}})=|2|_{p}^{3/2}(N,-1)_{p}\cdot\frac{\zeta_{p}(2k-1)}{\zeta_{p}(2k+2)}\cdot
A_{p}(k).$

###### Proof.

For any rank $2$ nondegenerate quadratic lattice $M$ and integer $k\geq 0$,

(24)
$W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})=|2|^{3/2}|N|_{p}(N,-1)_{p}\cdot\textup{Nor}^{+}(X,1)\cdot\textup{Den}_{\Delta(N)}^{+}(X,M)\big{|}_{X=p^{-k}}.$

Suppose the Gross-Keating invariant of the quadratic lattice $M$ is
$\textup{GK}(M)=(a,b)$ for integers $a\leq b$ and $a,b\gg n$.
$\bullet$ If $n\equiv a\,\,(\textup{mod}\,2)$, by the formula in [Wed07,
$\S$2.11], we have

$\displaystyle\textup{Den}^{+}(X,M\obot\langle N\rangle)$
$\displaystyle=\frac{(1-(pX)^{n+1})(1-(pX^{2})^{(n+a)/2-i})}{(1-pX)(1-pX^{2})}+p^{(n+a)/2}X^{a}\frac{(1-X^{n+1})(1-(\tilde{\xi}X)^{b-a+1})}{(1-X)(1-(\tilde{\xi}X))}$
$\displaystyle-\frac{p^{(n+a)/2-1}X^{b+2}}{1-p^{-1}X^{2}}\left(\frac{1-X^{n+1}}{1-X}-\frac{(p^{-1}X^{2})^{(n+a)/2}-p^{(n-a)/2+1}X^{a-1}}{1-pX^{-1}}\right).$

where $\tilde{\xi}\in\\{\pm 1\\}$ depends on the lattice $M$.
$\bullet$ If $n\nequiv a\,\,(\textup{mod}\,2)$, by the formula in [Wed07,
$\S$2.11], we have

$\displaystyle\textup{Den}^{+}(X,M\obot\langle N\rangle)$
$\displaystyle=\frac{1-(pX)^{n+1}}{(1-pX)(1-pX^{2})}-\frac{p^{(n+a+1)/2}(X^{n+a+1}-X^{a})}{(1-X^{-1})(1-pX^{2})}$
$\displaystyle-\frac{p^{(n+a-1)/2}X^{b+1}}{1-p^{-1}X^{2}}\left(\frac{1-X^{n+1}}{1-X}-\frac{(p^{-1}X^{2})^{(n+a+1)/2}-p^{(n-a+1)/2}X^{a}}{1-pX^{-1}}\right).$

In both cases, when $0<X\leq p^{-1}$, we have the following formula

$\lim_{a,b\rightarrow\infty}\textup{Den}^{+}(X,M\obot\langle
N\rangle)=\frac{1-(pX)^{n+1}}{(1-pX)(1-pX^{2})}.$

therefore when $0<X\leq p^{-1}$,

$\displaystyle\lim_{a,b\rightarrow\infty}\textup{Den}^{+}_{\Delta(N)}(X,M)$
$\displaystyle=\lim_{a,b\rightarrow\infty}\left(\textup{Den}^{+}(X,M\obot\langle
N\rangle)-X^{2}\textup{Den}^{+}(X,M\obot\langle p^{-2}N\rangle)\right)$
$\displaystyle=\frac{1-X^{2}+(p^{n-1}-p^{n+1})X^{n+1}}{(1-pX)(1-pX^{2})}.$

Note that $\textup{Nor}^{+}(X,1)=1-p^{-1}X$, the lemma follows by combining
(24) and the following formula,

$W_{\mathbf{0}_{2},p}(1,k,1_{\delta_{p}(N)^{2}})=|2|^{3/2}|N|_{p}(N,-1)_{p}\cdot(1-p^{-k-1})\cdot\lim_{a,b\rightarrow\infty}\textup{Den}^{+}_{\Delta(N)}(p^{-k},M)$

∎

In the following, let $\Lambda(s)=\Gamma(s)\pi^{-s/2}\zeta(s)$ be the
normalized Riemann zeta function.

######  Proposition 4.1.2.

Let $\mathsf{z}=\mathsf{x}+i\mathsf{y}\in\mathbb{H}_{2}$, we have

$\partial\textup{Eis}_{\mathbf{0}_{2}}(\mathsf{z},\Delta(N)^{2})=\log\textup{det}(\mathsf{y})+2-4\frac{\Lambda^{\prime}(-1)}{\Lambda(-1)}-\sum\limits_{p|N}\frac{-np^{n+1}+2p^{n}+np^{n-1}-2}{p^{n-1}(p^{2}-1)}\cdot\log
p$

###### Proof.

By similar arguments in [SSY22, $\S$4.4], we have

$(\textup{det}\mathsf{y})^{3/4}\cdot
E_{\mathbf{0}_{2}}(\mathsf{z},s,\Delta(N)^{2})=W_{\mathbf{0}_{2}}(g_{\mathsf{z}},s,\Delta(N)^{2})+\sum\limits_{\gamma\in\Gamma_{\infty}\backslash\textup{SL}_{2}(\mathbb{Z})}B(m(\gamma)g,s)+\Phi_{1_{\Delta(N)^{2}}}(g,s)$

where the derivative of the middle term
$\sum\limits_{\gamma\in\Gamma_{\infty}\backslash\textup{SL}_{2}(\mathbb{Z})}B(m(\gamma)g,s)$
at $s=0$ is 0, and

$W_{\mathbf{0}_{2}}(g_{\mathsf{z}},s,\Delta(N)^{2})=\textup{det}(\mathsf{y})^{-s/2+3/4}\frac{(s-1)\Lambda(2s-1)}{(s+1)\Lambda(2s+2)}\cdot\prod\limits_{p|N}A_{p}(s),\,\,\,\Phi_{1_{\Delta(N)^{2}}}(g,s)=\textup{det}(\mathsf{y})^{s/2+3/4}.$

The proposition follows from combining the above formulas and Lemma 4.1.1. ∎

####  4.2. The rank of $T$ is 1.

Let $N$ be a positive integer, let $t$ be a nonzero integer, we fix a $2\times
2$ matrix $T=\textup{diag}\\{0,t\\}$. Now we are going to define a quadratic
character $\chi_{t}:\mathbb{A}^{\times}\rightarrow\mathbb{C}^{\times}$: When
$-tN$ is not a square in $\mathbb{Q}$, the character $\chi_{t}$ is the
quadratic Dirichlet character attached to the quadratic field extension
$\mathbb{Q}(-tN)/\mathbb{Q}$; when $-tN$ is a square, let $\chi_{t}$ be the
trivial character. When $-tN$ is not a square, let $-d$ be the fundamental
discriminant of the field extension $\mathbb{Q}(-tN)/\mathbb{Q}$. When $-tN$
is a square, let $d=-1$, then there exists a positive number $c$ such that

$4Nt=c^{2}d.$

For any prime number $p$, we define the local zeta function associated to
$\chi_{t}$ to be $L_{p}(s,\chi_{t})=(1-\chi_{t}(p)p^{-s})^{-1}$ and
$L_{\infty}(s,\chi_{t})$ to be the following,

$L_{\infty}(s,\chi_{t})=|d|^{s/2}\cdot\begin{cases}\pi^{-(s+1)/2}\Gamma(\frac{s+1}{2}),&\textup{if
$t>0$};\\\ \pi^{-s/2}\Gamma(\frac{s}{2}),&\textup{if $t<0$}.\end{cases}$

Let $\Lambda(s,\chi_{t})\coloneqq\prod\limits_{v}L_{v}(s,\chi_{t})$ be the
completed $L$-function of the quadratic character $\chi_{t}$, then
$\Lambda(s,\chi_{t})$ can be meromorphically continued to the whole complex
plane and satisfies the functional equation
$\Lambda(s,\chi_{t})=\Lambda(1-s,\chi_{t})$.

Define the following function for positive integers $k$,

(25) $g_{p}(k)=\frac{\textup{Den}(H_{2k+4}^{+},\langle
t,Np^{-2}\rangle)}{\textup{Den}(H_{2k+4}^{+},\langle t,N\rangle)},$

Notice that if the $p$-adic valuation of $N$ is $0$ or $1$, the function
$g_{p}(k)=0$.

######  Lemma 4.2.1.

The function $g_{p}(k)$ is a rational function in $p^{-k}$, hence it can be
meromorphically defined over $\mathbb{C}$. The function $g_{p}$ also has the
following functional equation,

$g_{p}(k)=p^{2k+1}g(-k-1).$

###### Proof.

We only consider the case that the $p$-adic valuation of $N$ is greater or
equal to $2$. By definition made in Lemma 2.2.9, when $k$ is a positive
integer,

$\displaystyle g_{p}(k)=\frac{\textup{Den}(H_{2k+4}^{+},\langle
t,p^{-2}N\rangle)}{\textup{Den}(H_{2k+4}^{+},\langle
t,N\rangle)}=\frac{\textup{Den}^{\flat+}(p^{-k-1},\langle
t,p^{-2}N\rangle)}{\textup{Den}^{\flat+}(p^{-k-1},\langle
t,N\rangle)}=\frac{\textup{Den}^{\flat+}(X,\langle
t,p^{-2}N\rangle)}{\textup{Den}^{\flat+}(X,\langle
t,N\rangle)}\bigg{|}_{X=p^{-k-1}}.$

Therefore $g_{p}(k)$ is a rational function in $p^{-k}$ by Lemma 2.2.9, hence
it can be meromorphically defined over $\mathbb{C}$. The functional equation
is a consequence of Theorem 2.2.10. ∎

######  Lemma 4.2.2.

Let $p$ be a prime number, for any positive integers $k$,

(26)
$\displaystyle\frac{W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})}{\textup{Den}^{\flat+}(p^{-k},\langle
t,N\rangle)}$
$\displaystyle=|2|_{p}|N|_{p}^{1/2}\gamma(\mathbb{V}_{p})^{2}\cdot\frac{1-p^{-2k}g_{p}(k-1)}{1-\chi_{t}(p)p^{-k}}\cdot\begin{cases}1-p^{-2k-2},&\textup{if
$p\nmid N$;}\\\ 1-p^{-k-1},&\textup{if $p\,|\,N$.}\end{cases},$ (27)
$\displaystyle\frac{W_{t,p}(1,k+\frac{1}{2},1_{\delta_{p}(N))}}{\textup{Den}^{\flat+}(p^{-k-1},\langle
t,N\rangle)}$
$\displaystyle=|2N|_{p}^{1/2}\gamma(\mathbb{V}_{p})\cdot(-1,N)_{p}\cdot\frac{1-p^{-2k-1}g_{p}(k)}{1-\chi_{t}(p)p^{-k-1}}\cdot\begin{cases}1-p^{-2k-2},&\textup{if
$p\nmid N$;}\\\ 1-p^{-k-1},&\textup{if $p\,|\,N$.}\end{cases}.$

###### Proof.

By Proposition 3.2.1 and Theorem 2.2.5, we know that

$\displaystyle
W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})=|2|_{p}^{3/2}|N|_{p}\gamma(\mathbb{V}_{p})^{2}\cdot\lim\limits_{m\rightarrow\infty}\textup{Den}(\delta_{p}(N)\obot
H_{2k}^{+},\langle t,p^{m}\rangle)$
$\displaystyle=|2|_{p}^{3/2}|N|_{p}\gamma(\mathbb{V}_{p})^{2}\cdot\lim\limits_{m\rightarrow\infty}\frac{\textup{Den}(H_{2k+4}^{+},\langle
N,t,p^{m}\rangle)-p^{-2k}\textup{Den}(H_{2k+4}^{+},\langle
Np^{-2},t,p^{m}\rangle)}{\textup{Pden}(H_{2k+4}^{+},\langle N\rangle)}.$

By Lemma 2.2.7, we have

$\displaystyle W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})=$
$\displaystyle|2|_{p}^{3/2}|N|_{p}\gamma(\mathbb{V}_{p})^{2}\cdot\frac{\textup{Pden}(H_{2k+4}^{+},\langle
p\rangle)}{\textup{Pden}(H_{2k+4}^{+},\langle N\rangle)}$
$\displaystyle\cdot\frac{\textup{Den}(H_{2k+2}^{+},\langle
N,t\rangle)-p^{-2k}\textup{Den}(H_{2k+2}^{+},\langle
Np^{-2},t\rangle)}{1-p^{-2k}}$ $\displaystyle=$
$\displaystyle|2|_{p}^{3/2}|N|_{p}\gamma(\mathbb{V}_{p})^{2}\cdot\textup{Den}(H_{2k+2}^{+},\langle
N,t\rangle)\cdot\frac{\textup{Pden}(H_{2k+4}^{+},\langle
p\rangle)}{\textup{Pden}(H_{2k+4}^{+},\langle
N\rangle)}\cdot(1-p^{-2k}g_{p}(k)).$

Then the formula (26) follows from the definition of local density polynomial
in Lemma 2.2.9.

For the formula (27), we have the following identity by Proposition 3.2.1 and
Theorem 2.2.5,

$\displaystyle W_{t,p}(1,k+\frac{1}{2},1_{\delta_{p}(N)})$
$\displaystyle=|2N|_{p}^{1/2}\gamma(\mathbb{V}_{p})\cdot(-1,N)_{p}\cdot\frac{\textup{Den}(H_{2k+4}^{+},\langle
N,t\rangle)-p^{-2k-1}\textup{Den}(H_{2k+4}^{+},\langle
Np^{-2},t\rangle)}{\textup{Pden}(H_{2k+4}^{+},\langle N\rangle)}$
$\displaystyle=|2N|_{p}^{1/2}\gamma(\mathbb{V}_{p})\cdot(-1,N)_{p}\cdot\textup{Den}(H_{2k+4}^{+},\langle
N,t\rangle)\cdot\frac{1-p^{-2k-1}g_{p}(k)}{\textup{Pden}(H_{2k+4}^{+},\langle
N\rangle)}.$

Then the formula (27) also follows from the definition of local density
polynomial in Lemma 2.2.9. ∎

For any prime number $p$, we consider the following meromorphic function for
$s\in\mathbb{C}$,

(28)
$\beta_{p}(s)=\frac{1+p^{s-1}}{1+p^{-s-1}}\cdot\frac{1-p^{-2s}g_{p}(s-1)}{1-g_{p}(s-1)}.$

######  Corollary 4.2.3.

Let $p$ be a prime number, the following identity holds for any integer $k$,

$\displaystyle\frac{W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})\cdot\frac{\zeta_{p}(2k)}{L_{p}(k,\chi_{t})}}{W_{t,p}(1,\frac{1}{2}-k,1_{\delta_{p}(N)})\cdot\frac{\zeta_{p}(2-2k)}{L_{p}(1-k,\chi_{t})}}=$
$\displaystyle|2|_{p}|N|_{p}^{1/2}(-1,N)_{p}\gamma(\mathbb{V}_{p})\cdot
p^{(1-2k)\nu_{p}(c)}\cdot\frac{\zeta_{p}(2k)}{\zeta_{p}(2k+2)}$
$\displaystyle\times\begin{cases}1,&\textup{if $p\nmid N$;}\\\
\beta_{p}(k),&\textup{if $p\,|\,N$}.\end{cases}$

###### Proof.

By Theorem 2.2.10 and the formula (12) in Remark 12, there is a functional
equation,

$\textup{Den}^{\flat+}(p^{-k},\langle
N,t\rangle)=p^{(1-2k)\nu_{p}(c)}\cdot\textup{Den}^{\flat+}(p^{k-1},\langle
N,t\rangle).$

The corollary follows from the functional equation above and Lemma 4.2.2. ∎

######  Proposition 4.2.4.

Let $T$ be a $2\times 2$ matrix of rank 1 which can be diagonalized to
$\textup{diag}\\{0,t\\}$ with $t\neq 0$. Let
$\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a positive definite symmetric
matrix, then for any complex number $k$, we have

$\displaystyle E_{T}(i\mathsf{y},k,\Delta(N)^{2})$
$\displaystyle=y_{1}^{k/2}E_{t}(iy_{2},\frac{1}{2}+k,\Delta(N))$
$\displaystyle+y_{1}^{-k/2}\cdot\frac{k-1}{k+1}\cdot
N^{-k}\cdot\frac{\Lambda(2-2k)}{\Lambda(2+2k)}\cdot\prod\limits_{p|N}\beta_{p}(k)\cdot
E_{t}(iy_{2},\frac{1}{2}-k,\Delta(N)).$

###### Proof.

When $-tN$ is not a square, let $-d$ be the fundamental discriminant of the
field extension $\mathbb{Q}(-tN)/\mathbb{Q}$. When $-tN$ is a square, let
$d=-1$, then there exists a positive number $c$ such that

$4Nt=c^{2}d.$

For any complex number $k$, the following identity is proved in [SSY22, Lemma
4.14] (see also [KRY06, Proposition 5.7.7]),

(29) $\displaystyle
W_{T,\infty}(g_{i\mathsf{y}},k,\Delta(N)^{2})\cdot\frac{\zeta_{\infty}(2k)}{L_{\infty}(k,\chi_{t})}=$
$\displaystyle-2iy_{1}^{-k/2+3/4}\cdot\frac{1-k}{1+k}\cdot\frac{\zeta_{\infty}(2k)}{\zeta_{\infty}(2k+2)}\cdot
c^{2k-1}N^{1/2-k}$ $\displaystyle\times
W_{t,\infty}(g_{iy_{2}},\frac{1}{2}-k,\Delta(N))\cdot\frac{\zeta_{\infty}(2-2k)}{L_{\infty}(1-k,\chi_{t})}.$

Recall that by Definition 3.2.2,
$W_{T}(g_{i\mathsf{y}},k,\Delta(N)^{2})=W_{T,\infty}(g_{i\mathsf{y}},k,\Delta(N)^{2})\cdot\prod\limits_{p}W_{T,p}(1,k,1_{\delta_{p}(N)^{2}})$
and
$W_{t}(g_{iy_{2}},k,\Delta(N))=W_{t,\infty}(g_{iy_{2}},k,\Delta(N))\cdot\prod\limits_{p}W_{t,p}(1,k,1_{\delta_{p}(N)})$.

Notice that both of the functions $W_{T}(g_{i\mathsf{y}},k,\Delta(N)^{2})$ and
$W_{t}(g_{iy_{2}},k,\Delta(N))$ can be meromorphically continued to whole
complex plane. By combining the above identity (29), Corollary 4.2.3, and the
functional equations $\Lambda(s)=\Lambda(1-s)$,
$\Lambda(s,\chi_{t})=\Lambda(1-s,\chi_{t})$, we get

$W_{T}(g_{i\mathsf{y}},k,\Delta(N)^{2})=y_{1}^{-k/2+3/4}\cdot\frac{k-1}{k+1}\cdot
N^{-k}\cdot\frac{\Lambda(2-2k)}{\Lambda(2+2k)}\cdot\prod\limits_{p|N}\beta_{p}(k)\cdot
W_{t}(g_{iy_{2}},\frac{1}{2}-k,\Delta(N)).$

Then Lemma 3.4.1, Lemma 3.3.1 and formulas (17), (20) imply that

$\displaystyle E_{T}(i\mathsf{y},k,\Delta(N)^{2})$
$\displaystyle=y_{1}^{k/2}E_{t}(iy_{2},\frac{1}{2}+k,\Delta(N))$
$\displaystyle+y_{1}^{-k/2}\cdot\frac{k-1}{k+1}\cdot
N^{-k}\cdot\frac{\Lambda(2-2k)}{\Lambda(2+2k)}\cdot\prod\limits_{p|N}\beta_{p}(k)\cdot
E_{t}(iy_{2},\frac{1}{2}-k,\Delta(N))$

∎

######  Corollary 4.2.5.

Let $T$ be a $2\times 2$ matrix of rank 1 which can be diagonalized to
$\textup{diag}\\{0,t\\}$ with $t\neq 0$. Let
$\mathsf{y}=\textup{diag}\\{y_{1},y_{2}\\}$ be a positive definite symmetric
matrix, we have

$\displaystyle E_{T}^{\prime}(i\mathsf{y},0,\Delta(N)^{2})=$ $\displaystyle
2E_{t}^{\prime}(iy_{2},\frac{1}{2},\Delta(N))$ $\displaystyle+\left(\log
y_{1}+2+\log
N+\frac{4\Lambda^{\prime}(2)}{\Lambda(2)}-\sum\limits_{p|N}\beta_{p}^{\prime}(0)\right)E_{t}(iy_{2},\frac{1}{2},\Delta(N)).$

###### Proof.

This follows from Proposition 4.2.4. ∎

##  Part II The Geometric Side

###  5\. The geometry of the modular curve $\mathcal{X}_{0}(N)$

####  5.1. Integral model

Let $N$ be a positive integer, the classical modular curve
$\mathcal{Y}_{0}(N)_{\mathbb{C}}$ over $\mathbb{C}$ is defined as the
following smooth $1$-dimensional complex curve,

$\mathcal{Y}_{0}(N)_{\mathbb{C}}\coloneqq\textup{GL}_{2}(\mathbb{Q})\backslash\mathbb{H}^{\pm}\times\textup{GL}_{2}(\mathbb{A}_{f})/\Gamma_{0}(N)(\hat{\mathbb{Z}})\simeq\Gamma_{0}(N)\backslash\mathbb{H}^{+},$

where $\mathbb{H}^{\pm}=\mathbb{C}\backslash\mathbb{R}$ and
$\mathbb{H}^{+}=\\{z=x+iy\in\mathbb{C}:x\in\mathbb{R},y\in\mathbb{R}_{>0}\\}$
is the upper half plane. The group $\Gamma_{0}(N)(\hat{\mathbb{Z}})$ is the
following open compact subgroup of $\textup{GL}_{2}(\mathbb{A}_{f})$,

$\Gamma_{0}(N)(\hat{\mathbb{Z}})=\left\\{x=\begin{pmatrix}a&b\\\
Nc&d\end{pmatrix}\in\textup{GL}_{2}(\hat{\mathbb{Z}})\,\,:\,\,a,b,c,d\in\hat{\mathbb{Z}}\right\\},$

and
$\Gamma_{0}(N)=\Gamma_{0}(N)(\hat{\mathbb{Z}})\bigcap\textup{SL}_{2}(\mathbb{Z})$.
The modular curve $\mathcal{Y}_{0}(N)_{\mathbb{C}}$ parametrizes cyclic
$N$-isogenies of elliptic curves over the complex number $\mathbb{C}$ by the
following map:

$\Gamma_{0}(N)\tau\in\Gamma_{0}(N)\backslash\mathbb{H}^{+}\cdot\tau\longmapsto(E_{\tau}\stackrel{{\scriptstyle\times
N}}{{\rightarrow}}E_{N\tau}),$

where for a given $\tau\in\mathbb{H}^{+}$, the elliptic curve $E_{\tau}$ is
given by the complex torus $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$. The
smooth curve $\mathcal{Y}_{0}(N)_{\mathbb{C}}$ is not proper, its
compactification
$\mathcal{X}_{0}(N)_{\mathbb{C}}\coloneqq\mathcal{Y}_{0}(N)_{\mathbb{C}}\cup\\{\textup{cusps}\\}$
is a smooth projective curve over $\mathbb{C}$.

Katz and Mazur [KM85] constructs an integral model of $\mathcal{Y}_{0}(N)$
over $\mathbb{Z}$ by extending the concept of cyclic isogeny to arbitrary base
scheme, we refer the readers to [KM85, §3(3.4)] and the arthor’s previous work
[Zhu23, §4] for details. The moduli interpretation of the stack
$\mathcal{Y}_{0}(N)$ is given as follows: for a scheme $S$, the objects of the
groupoid $\mathcal{Y}_{0}(N)(S)$ are cyclic $N$-isogenies
$(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E^{\prime})$ where $E$ and
$E^{\prime}$ are elliptic curves over $S$, a morphism between two cyclic
isogenies
$(E_{1}\stackrel{{\scriptstyle\pi_{1}}}{{\rightarrow}}E_{1}^{\prime})$ and
$(E_{2}\stackrel{{\scriptstyle\pi_{2}}}{{\rightarrow}}E_{2}^{\prime})$ is a
pair of isomorphisms of elliptic curves
$a:E_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{2}$ and
$a^{\prime}:E_{1}^{\prime}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}E_{2}^{\prime}$
such that $a^{\prime}\circ\pi_{1}=\pi_{2}\circ a$.

Cesnavicius [Ces17] defined a compactification of $\mathcal{Y}_{0}(N)$ as the
moduli stack of cyclic $N$-isogenies between generalized elliptic curves, we
denote this compactification by $\mathcal{X}_{0}(N)$, its complex fiber is
isomorphic to $\mathcal{X}_{0}(N)_{\mathbb{C}}$.

There are two natural morphisms from $\mathcal{Y}_{0}(N)$ to
$\mathcal{Y}_{0}(1)$ given as follows

$\displaystyle p_{1},p_{2}:\mathcal{Y}_{0}(N)$
$\displaystyle\longrightarrow\mathcal{Y}_{0}(1)$
$\displaystyle(E_{1}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E_{2})$
$\displaystyle\stackrel{{\scriptstyle p_{1}}}{{\longmapsto}}E_{1};$
$\displaystyle(E_{1}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E_{2})$
$\displaystyle\stackrel{{\scriptstyle p_{2}}}{{\longmapsto}}E_{2}.$

Both of the morphism
$p_{1},p_{2}:\mathcal{Y}_{0}(N)\rightarrow\mathcal{Y}_{0}(1)$ extends to
morphisms from $\mathcal{X}_{0}(N)$ to $\mathcal{X}_{0}(1)$, we still use the
symbols $p_{1}$ and $p_{2}$ to denote them.

######  Theorem 5.1.1.

The stack $\mathcal{X}_{0}(N)$ is a regular proper 2-dimensional Deligne-
Mumford stack, both of the morphisms $p_{1}$ and $p_{2}$ are finite flat of
degree $\psi(N)\coloneqq N\cdot\prod\limits_{p|N}(1+p^{-1})$ over
$\mathcal{X}_{0}(1)$. Moreover, the stack $\mathcal{X}_{0}(N)$ is smooth over
$\mathbb{Z}[\frac{1}{N}]$

###### Proof.

This is proved in [Ces17, Theorem 5.13], the degree of $p_{1}$ and $p_{2}$ are
computed in [KM85, (13.4.9)]. ∎

######  Lemma 5.1.2.

Let $N=M_{1}\cdot M_{2}$ where $\textup{g.c.d.}(M_{1},M_{2})=1$, there is a
natural isomorphism

$\mathcal{X}_{0}(N)\simeq\mathcal{X}_{0}(M_{1})\times_{p_{1},\mathcal{X}_{0}(1),p_{1}}\mathcal{X}_{0}(M_{2}).$

###### Proof.

This follows from [KM85, Corollary 1.10.15]. ∎

####  5.2. The Atkin-Lehner involution on $\mathcal{X}_{0}(N)$

######  Lemma 5.2.1.

Let $S$ be a scheme, and $\pi:E\rightarrow E^{\prime}$ be a cyclic $N$-isogeny
between elliptic curves over $S$, then $\pi^{\vee}:E^{\prime}\rightarrow E$ is
also a cyclic $N$-isogeny.

###### Proof.

We only need to show that the order $N$ quotient group scheme
$E[N]/\textup{ker}(\pi)$ is cyclic, and this is proved in [KM85, Corollary
5.5.4(3)]. ∎

With Lemma 5.2.1, we can define the Atkin-Lehner involution $W_{N}$ on the
stack $\mathcal{Y}_{0}(N)$ by the following,

$\displaystyle W_{N}:\mathcal{Y}_{0}(N)$
$\displaystyle\longrightarrow\mathcal{Y}_{0}(N)$
$\displaystyle(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E^{\prime})$
$\displaystyle\longmapsto(E^{\prime}\stackrel{{\scriptstyle\pi^{\vee}}}{{\rightarrow}}E)$

######  Lemma 5.2.2.

The Atkin-Lehner involution $W_{N}$ extends to the compactified stack
$\mathcal{X}_{0}(N)$.

###### Proof.

By [Ces17, Proposition 4.2.7 (c)], the cyclicity condition of a $N$-isogeny is
closed, hence $\mathcal{X}_{0}(N)$ is a closed substack of
$\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)$. Let
$\iota:\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)\rightarrow\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)$
be the involution which switches the two copies of $\mathcal{X}_{0}(1)$, let
$\mathcal{X}_{0}(N)^{\prime}=\mathcal{X}_{0}(N)_{\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1),\iota}\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)$
be the image of $\mathcal{X}_{0}(N)$ under the involution $W_{N}$, it is also
a closed substack of $\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)$, but
$\mathcal{X}_{0}(N)$ and $\mathcal{X}_{0}(N)^{\prime}$ have common dense open
dense substack $\mathcal{Y}_{0}(N)$, hence
$\mathcal{X}_{0}(N)=\mathcal{X}_{0}(N)^{\prime}$, therefore the Atkin-Lehner
involution $W_{N}$ extends to the compactified stack $\mathcal{X}_{0}(N)$. ∎

####  5.3. Cusps of the modular curve $\mathcal{X}_{0}(N)$

Let $\mathcal{H}\coloneqq\mathcal{X}_{0}(1)\times\mathcal{X}_{0}(1)$ be the
product of the smooth Deligne-Mumford stack $\mathcal{X}_{0}(1)$. Let
$P_{\infty}(1):\textup{Spec}\,\mathbb{Z}\rightarrow\mathcal{X}_{0}(1)$ be the
cuspidal divisor of the stack $\mathcal{X}_{0}(1)$ which corresponds to the
standard $1$-gon over $\mathbb{Z}$ (we refer to [Con07, §2.1] for detailed
definitions of standard $1$-gon). We can view $P_{\infty}(1)$ as an element in
$\textup{CH}^{1}(\mathcal{X}_{0}(1))$, then we define the following element in
$\textup{CH}^{1}(\mathcal{H})$:

$\mathsf{Cusp}(\mathcal{H})\coloneqq
P_{\infty}(1)\times\mathcal{X}_{0}(1)+\mathcal{X}_{0}(1)\times P_{\infty}(1).$

The two curves $P_{\infty}(1)\times\mathcal{X}_{0}(1)$ and
$\mathcal{X}_{0}(1)\times P_{\infty}(1)$ intersects transversally at the point
$(P_{\infty}(1),P_{\infty}(1))$.

The natural morphism
$\mathcal{X}_{0}(N)\stackrel{{\scriptstyle(p_{1},p_{2})}}{{\longrightarrow}}\mathcal{H}$
is a closed immersion. We define the cuspidal divisor of $\mathcal{X}_{0}(N)$
to be the pullback of $\mathsf{Cusp}(\mathcal{H})$, i.e.,

$\mathsf{Cusp}(\mathcal{X}_{0}(N))=(p_{1},p_{2})^{\ast}\mathsf{Cusp}(\mathcal{H})\in\textup{CH}^{1}(\mathcal{X}_{0}(N)).$

Let $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(N))$ be the formal completion of
$\mathcal{X}_{0}(N)$ along the cuspidal divisor
$\textup{Cusp}(\mathcal{X}_{0}(N))$. It is well-known that
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))\simeq\textup{Spf}\,\mathbb{Z}[[q]]$.
The two morphisms
$p_{1},p_{2}:\mathcal{X}_{0}(N)\rightarrow\mathcal{X}_{0}(1)$ induce two
morphisms from $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(N))$ to
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))$, we still use $p_{1}$ and
$p_{2}$ to denote them, both of them are finite flat of degree of $\psi(N)$.

######  Proposition 5.3.1.

Let $N=p^{n}$ for some non-negative integer $n$, then the formal scheme
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))$ is a disjoint union as
follows:

$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))=\coprod\limits_{\begin{subarray}{c}-n\leq
a\leq n\\\ a\equiv n\,\textup{mod}\,2\end{subarray}}\mathsf{C}^{a}(p^{n}),$

where every $\mathsf{C}^{a}(p^{n})$ is finite flat over
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))$ via the morphisms $p_{1}$ and
$p_{2}$, such that $W_{N}(\mathsf{C}^{a}(p^{n}))=\mathsf{C}^{-a}(p^{n})$. Let
$\textup{deg}_{1}(\mathsf{C}^{a}(p^{n}))$ (resp.
$\textup{deg}_{2}(\mathsf{C}^{a}(p^{n}))$) be the finite flat degree of
$\mathsf{C}^{a}(p^{n})$ over $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))$ via
the morphism $p_{1}$ (resp. $p_{2}$), then

$\textup{deg}_{1}(\mathsf{C}^{a}(p^{n}))=\begin{cases}\varphi(p^{(n-a)/2}),&\textup{if
$0\leq a\leq n$};\\\ p^{-a}\varphi(p^{(n+a)/2}),&\textup{if $-n\leq a<0$}.\\\
\end{cases}$
$\textup{deg}_{2}(\mathsf{C}^{a}(p^{n}))=\begin{cases}p^{a}\varphi(p^{(n-a)/2}),&\textup{if
$0\leq a\leq n$};\\\ \varphi(p^{(n+a)/2}),&\textup{if $-n\leq a<0$}.\\\
\end{cases}$

More explicitly, if we view $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))$
as a $\mathbb{Z}[[q]]$-formal scheme via the morphism $p_{1}$, then

(30)
$\mathsf{C}^{a}(p^{n})\simeq\begin{cases}\textup{Spf}\,\mathbb{Z}[\zeta_{p^{(n-a)/2}}][[q]],&\textup{if
$0\leq a\leq n$};\\\
\textup{Spf}\,\mathbb{Z}[\zeta_{p^{(n+a)/2}}][[q]][z]/(z^{p^{-a}}-\zeta_{p^{(n+a)/2}}q),&\textup{if
$-n\leq a<0$},\\\ \end{cases}$

where for any $k\geq 1$, $\zeta_{p^{k}}$ is a primitive $p^{k}$-th root of
unity.

###### Proof.

It is proved by Edixhoven in [Edi90, §1.2.2] that if we view
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))$ as a
$\mathbb{Z}[[q]]$-formal scheme via the morphism $p_{1}$, then

(31)
$\textup{Spf}\,\mathbb{Z}[[q]]\coprod\textup{Spf}\,\mathbb{Z}[[q^{p^{-n}}]]\coprod\bigsqcup\limits_{\begin{subarray}{c}c+d=n\\\
c\geq
d>0\end{subarray}}\textup{Spf}\,\mathbb{Z}[\zeta_{p^{d}}][[q]]\coprod\bigsqcup\limits_{\begin{subarray}{c}c+d=n\\\
d>c>0\end{subarray}}\textup{Spf}\,\mathbb{Z}[\zeta_{p^{c}}][[q]][z]/(z^{p^{d-c}}-\zeta_{p^{c}}q),$

therefore $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))$ is a disjoint
union of $n+1$ formal schemes, let $\mathsf{C}^{a}(p^{n})$ be one of the
formal schemes according to formula (30), then
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))=\coprod\limits_{\begin{subarray}{c}-n\leq
a\leq n\\\ a\equiv n\,\textup{mod}\,2\end{subarray}}\mathsf{C}^{a}(p^{n})$ by
(31). The finite flat degree of $\mathsf{C}^{a}(p^{n})$ over
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))$ via the morphisms $p_{1}$ and
$p_{2}$ can be computed explicitly by (30). ∎

######  Remark 5.3.2.

The isomorphism

$\displaystyle\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))\simeq$
$\displaystyle\textup{Spf}\,\mathbb{Z}[[q]]\coprod\textup{Spf}\,\mathbb{Z}[[q^{p^{-n}}]]\coprod\bigsqcup\limits_{\begin{subarray}{c}c+d=n\\\
c\geq
d>0\end{subarray}}\textup{Spf}\,\mathbb{Z}[\zeta_{p^{d}}][[q]]\coprod\bigsqcup\limits_{\begin{subarray}{c}c+d=n\\\
d>c>0\end{subarray}}\textup{Spf}\,\mathbb{Z}[\zeta_{p^{c}}][[q]][z]/(z^{p^{d-c}}-\zeta_{p^{c}}q),$

and Lemma 5.1.2 imply that the formal scheme
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(N))$ has a unique closed and open
formal subscheme which is isomorphic to $\textup{Spf}\,\mathbb{Z}[[q]]$ as
$\mathbb{Z}[[q]]$-scheme via the morphism $p_{1}$, we denote the underlying
scheme by $P_{\infty}(N)$, it corresponds to a morphism
$P_{\infty}(N):\textup{Spec}\,\mathbb{Z}\rightarrow\mathcal{X}_{0}(N)$. When
$N=1$, this definition agrees with our previous definition of $P_{\infty}(1)$.

Let $P_{0}(N)=W_{N}\circ
P_{\infty}(N):\textup{Spec}\,\mathbb{Z}\rightarrow\mathcal{X}_{0}(N)$ be the
composition of the automorphism $W_{N}$ of $\mathcal{X}_{0}(N)$ and the
$\mathbb{Z}$-point $P_{\infty}(N)$. When $N=1$, we have
$P_{\infty}(1)=P_{0}(1)$.

There is an explicit description of the cusps of the complex modular curve
$\mathcal{X}_{0}(N)_{\mathbb{C}}$ (cf. [DS05, §3.8]),

$P_{\frac{aQ}{N}},\,\,\textup{where}\,\,Q|N\,\,\textup{and}\,\,a\in(\mathbb{Z}/(Q,N/Q))^{\times}.$

Especially, when $N=p^{n}$ for some non-negative integer $n$, the cusps of the
complex modular curve $\mathcal{X}_{0}(p^{n})_{\mathbb{C}}$ are

$P_{\frac{a}{p^{k}}},\,\,\textup{where}\,\,0\leq k\leq
n\,\,\textup{and}\,\,a\in(\mathbb{Z}/p^{\textup{min}\\{k,n-k\\}})^{\times}.$

######  Lemma 5.3.3.

Let $P$ be a cusp of the complex modular curve
$\mathcal{X}_{0}(p^{n})_{\mathbb{C}}$, let $\textup{ra}_{1}(P)$ (resp.
$\textup{ra}_{2}(P)$) be the ramification degree of the morphism $p_{1}$
(resp. $p_{2}$) at $P$. Suppose $P=P_{\frac{a}{p^{k}}}$ for some integer
$0\leq k\leq n$ and $a\in(\mathbb{Z}/p^{\textup{min}\\{k,n-k\\}})^{\times}$,
then

(32) $\textup{ra}_{1}(P_{a/p^{k}})=\begin{cases}1,&\textup{if $\frac{n}{2}\leq
k\leq n$};\\\ p^{n-2k},&\textup{if $0\leq k<\frac{n}{2}$}.\\\ \end{cases}$
(33) $\textup{ra}_{2}(P_{a/p^{k}})=\begin{cases}p^{2k-n},&\textup{if
$\frac{n}{2}\leq k\leq n$};\\\ 1,&\textup{if $0\leq k<\frac{n}{2}$}.\\\
\end{cases}$

###### Proof.

For any cusp point $P$, let $\textup{Stab}(P)\in\Gamma(p^{n})$ be the
stabilizer of the cusp. For the cusp point $P_{\infty}(p^{n})$, we have

$\textup{Stab}(P_{\infty}(p^{n}))=\left\\{\begin{pmatrix}1&x\\\
0&1\end{pmatrix}:x\in\mathbb{Z}.\right\\}.$

Let $A\in\mathbb{Z}$ be a lift of $a$ to $\mathbb{Z}$, then $A$ is prime to
$p$, hence there exists integers $b,d$ such that $Ab-dp^{k}=1$. Let

$\gamma=\begin{pmatrix}A&d\\\ p^{k}&b\end{pmatrix},$

then
$\textup{Stab}(P_{\frac{a}{p^{k}}})=\gamma\textup{Stab}(P_{\infty}(p^{n}))\gamma^{-1}\cap\Gamma_{0}(p^{n})$.
By simple calculations,

$\gamma\textup{Stab}(P_{\infty}(p^{n}))\gamma^{-1}\cap\Gamma_{0}(p^{n})=\left\\{\begin{pmatrix}1-p^{k}Ax&xA^{2}\\\
-p^{2k}x&1+p^{k}Ax\end{pmatrix}:x\in
p^{n-2k}\mathbb{Z}\cap\mathbb{Z}.\right\\},$

the formula (32) follows from the calculations above. Notice that
$p_{2}=p_{1}\circ W_{N}$, hence the formula (33) follows from (32). ∎

######  Corollary 5.3.4.

Let $P_{\frac{a}{p^{k}}}$ be a cusp of the complex modular curve
$\mathcal{X}_{0}(p^{n})_{\mathbb{C}}$, where $0\leq k\leq n$ and
$a\in(\mathbb{Z}/p^{\textup{min}\\{k,n-k\\}})^{\times}$, then
$P_{\frac{a}{p^{k}}}$ belongs to the component $\mathsf{C}^{2k-n}(p^{n})$ of
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(p^{n}))$, i.e.,
$P_{\frac{a}{p^{k}}}\in\mathsf{C}^{2k-n}(p^{n})(\mathbb{C})$.

###### Proof.

This follows from formula (30) in Proposition 5.3.1, and formulas (32) and
(33) in Lemma 33. ∎

####  5.4. Reduction mod $p$ of $\mathcal{X}_{0}(N)$

Let $p$ be a fixed prime, the reduction mod $p$ of the stack
$\mathcal{X}_{0}(N)$ has been studied extensively in [KM85, Chapter 13]. For
an $\mathbb{F}$-scheme $S$, an integer $m\geq 0$ and an elliptic curve over
$S$, let $E^{(p^{m})}$ be the $m$-th Frobenius twist of $E$, we use
$F^{m}:E\rightarrow E^{(p^{m})}$ to denote the $m$-th iterated Frobenius
morphism which has degree $p^{m}$, and $V^{m}:E^{(p^{m})}\rightarrow E$ be the
$m$-th iterated Verschiebung morphism which is also the dual morphism of
$F^{m}$.

######  Lemma 5.4.1.

For any two integer $m\geq 0$, the $m$-th iterated Frobenius morphism $F^{m}$
and the $m$-th iterated Verschiebung morphism $V^{m}$ are cyclic of degree
$p^{m}$.

###### Proof.

This is proved in [KM85, Thereom 13.3.5 (2)]. ∎

Let $n\geq 0$ be an integer, for any integer $a$ such that $0\leq a\leq n$ and
$n\equiv a\,\,(\textup{mod}\,2)$, let $t=\frac{n-a}{2}$, we define two stacks
$\mathcal{Y}^{a}$ and $\mathcal{Y}^{-a}$ in the following paragraphs.

Let $\mathcal{Y}^{a}$ be the following stack: for an $\mathbb{F}$-scheme $S$,
the groupoid $\mathcal{Y}^{a}(S)$ consists of cyclic $p^{n}$-isogenies
$(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E^{\prime})$ such that there
exists an isomorphism $\varepsilon:E^{(p^{a})}\simeq E^{\prime}$ making the
following diagram commutes,

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{n-t}}$$\scriptstyle{\pi}$$\textstyle{E^{\prime}}$$\textstyle{E^{(p^{n-t})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{(p^{t})}}$$\scriptstyle{\sim}$$\textstyle{E^{\prime(p^{t})}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{t}}$

Let $\mathcal{Y}^{-a}$ be the following stack: for an $\mathbb{F}$-scheme $S$,
the groupoid $\mathcal{Y}^{-a}(S)$ consists of cyclic $p^{n}$-isogenies
$(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E^{\prime})$ such that there
exists an isomorphism $\varepsilon:E\simeq E^{\prime(p^{a})}$ making the
following diagram commutes,

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{t}}$$\scriptstyle{\pi}$$\textstyle{E^{\prime}}$$\textstyle{E^{(p^{t})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{(p^{t})}}$$\scriptstyle{\sim}$$\textstyle{E^{\prime(p^{n-t})}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n-t}}$
(34) $\mathcal{Y}^{a}\rightarrow\mathcal{Y}_{0}(1)\times\mathcal{Y}_{0}(1).$

######  Theorem 5.4.2.

Let $n\geq 0$ be an integer, let
$\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}\coloneqq\mathcal{Y}_{0}(p^{n})\times_{\mathbb{Z}}\mathbb{F}$
be the reduction mod $p$ of the stack $\mathcal{Y}_{0}(p^{n})$. For any
integer $a$ such that $-n\leq a\leq n$ and $n\equiv a\,\,(\textup{mod}\,2)$,
the stack $\mathcal{Y}^{a}$ has the following properties:
(a) The stack $\mathcal{Y}^{a}$ is a 1-dimensional Deligne-Mumford stack.
(b) The natural morphism
$\mathcal{Y}^{a}\rightarrow\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}$ is a closed
immersion, the composite morphism
$\mathcal{Y}^{a}\rightarrow\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}\rightarrow\mathcal{Y}_{0}(1)_{\mathbb{F}}$
is an isomorphism if $a\geq 0$; is finite flat of degree $p^{-a}$ if $a\leq
0$.
(c) The following equality holds: $W_{N}(\mathcal{Y}^{a})=\mathcal{Y}^{-a}$.
(d) The stack $\mathcal{Y}^{a}$ contains every supersingular point of
$\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}$. Let
$P=(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}E^{\prime})$ be a
supersingular $\mathbb{F}$-point of $\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}$. Let
$\mathcal{O}_{P}$ be the completed local ring of the stack
$\mathcal{Y}_{0}(1)_{\mathbb{F}}\times\mathcal{Y}_{0}(1)_{\mathbb{F}}$ at $P$,
let $\mathcal{O}_{a,P}$ be the completed local ring of $\mathcal{Y}^{a}$ at
the point corresponding to the pair $(E,E^{\prime})$, then there exists an
isomorphism $\mathcal{O}_{P}\simeq\mathbb{F}[[t_{1},t_{2}]]$ such that the
closed immersion
$\mathcal{Y}^{a}\rightarrow\mathcal{Y}_{0}(1)_{\mathbb{F}}\times\mathcal{Y}_{0}(1)_{\mathbb{F}}$
induces the following isomorphism,

$\mathcal{O}_{a,P}\simeq\begin{cases}\mathbb{F}[[t_{1},t_{2}]]/(t_{1}-t_{2}^{p^{a}}),&\textup{if
$a\geq 0$;}\\\ \mathbb{F}[[t_{1},t_{2}]]/(t_{1}^{p^{-a}}-t_{2}),&\textup{if
$a\leq 0$.}\\\ \end{cases}$

(e) Over the open substack
$\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}^{\textup{ord}}=\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}-\\{\textit{supersingular
points}\\}$ of $\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}$, the morphism
$\bigsqcup\limits_{\begin{subarray}{c}-n\leq a\leq n\\\ n\equiv
a\,(\textup{mod}\,2)\end{subarray}}\mathcal{Y}^{a}\rightarrow\mathcal{Y}_{0}(p^{n})_{\mathbb{F}}^{\textup{red}}$
is an isomorphism.

###### Proof.

These are the main results of [KM85, §13]. ∎

######  Remark 5.4.3.

Let $\mathcal{X}^{a}$ be the scheme-theoretic closure of $\mathcal{Y}^{a}$
inside
$\mathcal{X}_{0}(p^{n})_{\mathbb{F}}\coloneqq\mathcal{X}_{0}(p^{n})\times_{\mathbb{Z}}\mathbb{F}$,
then Theorem 5.4.2 is still true if we replace all the symbol $\mathcal{Y}$ by
the symbol $\mathcal{X}$.

######  Theorem 5.4.4.

Let $n=\nu_{p}(N)$ be the $p$-adic valuation of $N$, let $N_{p}=p^{-n}N$. For
any integer $a$ such that $-n\leq a\leq n$ and $n\equiv
a\,\,(\textup{mod}\,2)$, let
$\mathcal{X}_{p}^{a}(N)=\mathcal{X}^{a}\times_{p_{1},\mathcal{X}_{0}(1),p_{1}}\mathcal{X}_{0}(N_{p})$
be a closed substack of $\mathcal{X}_{0}(N)$, then we have the following
identity in $\textup{CH}^{1}(\mathcal{X}_{0}(N))$:

$\textup{div}(p)=\sum\limits_{\begin{subarray}{c}-n\leq a\leq n\\\ n\equiv
a\,(\textup{mod}\,2)\end{subarray}}\varphi(p^{(n-|a|)/2})\cdot\mathcal{X}^{a}_{p}(N).$

###### Proof.

The multiplicities of the irreducible components $\mathcal{X}_{p}^{a}(N)$ are
byproducts of [KM85, Theorem 13.3.5, Theorem 13.4.6]. ∎

We now consider the reduction mod $p$ of cusps. Let $n=\nu_{p}(N)$ be the
$p$-adic valuation of $N$. For any integer $a$ such that $-n\leq a\leq n$ and
$n\equiv a\,\,(\textup{mod}\,2)$, let $N_{p}=p^{-n}N$. Let
$\mathsf{C}^{a}(p^{n},N_{p})\coloneqq\mathsf{C}^{a}(p^{n})\times_{p_{1},\mathcal{X}_{0}(1),p_{1}}\mathcal{X}_{0}(N_{p})$,
and $\mathsf{C}^{a}_{p}(p^{n},N_{p})$ be its reduction mod $p$. Let
$\mathcal{C}_{p}^{n}(p^{n},N_{p})$ (resp. $\mathcal{C}_{p}^{-n}(p^{n},N_{p})$)
be the curve $\mathcal{X}_{p}^{n}(N)$ (resp. $\mathcal{X}_{p}^{-n}(N)$) over
$\mathbb{F}$, and $\mathcal{C}_{p}^{a}(p^{n},N_{p})$ be the non-reduced curve
over $\mathbb{F}$ corresponding to
$p^{(n-|a|)/2-1}(p-1)\mathcal{X}^{a}_{p}(N)$ when $-n<a<n$.

######  Proposition 5.4.5.

Let $n=\nu_{p}(N)$ be the $p$-adic valuation of $N$, let $N_{p}=p^{-n}N$. For
any integer $a$ such that $-n\leq a\leq n$ and $n\equiv
a\,\,(\textup{mod}\,2)$, the formal scheme $\mathsf{C}^{a}_{p}(p^{n},N_{p})$
is the formal completion of the curve $\mathcal{C}_{p}^{a}(p^{n},N_{p})$ along
its cuspidal locus.

###### Proof.

Let $\deg_{1}(\mathcal{C}_{p}^{a}(p^{n},N_{p}))$ (resp.
$\deg_{2}(\mathcal{C}_{p}^{a}(p^{n},N_{p}))$) be the finite flat degree of the
curve $\mathcal{C}_{p}^{a}(p^{n},N_{p})$ over $\mathcal{X}_{p}(1)$ via the
morphism $p_{1}$ (resp. $p_{2}$). By the definition of the curve
$\mathcal{C}_{p}^{a}(p^{n},N_{p})$, we have

$\textup{deg}_{1}(\mathcal{C}_{p}^{a}(p^{n},N_{p}))=\psi(N_{p})\cdot\begin{cases}\varphi(p^{(n-a)/2}),&\textup{if
$0\leq a\leq n$};\\\ p^{-a}\varphi(p^{(n+a)/2}),&\textup{if $-n\leq a<0$}.\\\
\end{cases}$
$\textup{deg}_{2}(\mathcal{C}_{p}^{a}(p^{n},N_{p}))=\psi(N_{p})\cdot\begin{cases}p^{a}\varphi(p^{(n-a)/2}),&\textup{if
$0\leq a\leq n$};\\\ \varphi(p^{(n+a)/2}),&\textup{if $-n\leq a<0$}.\\\
\end{cases}$

Notice that the finite flat degree of the formal scheme
$\mathsf{C}^{a}(p^{n},N_{p})$ over
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(1))$ via the morphism $p_{1}$ (resp.
$p_{2}$) is calculated in Proposition 5.3.1 and coincides with the formulas
above. Since the formal completion of $\mathcal{X}_{0}(N)_{\mathbb{F}_{p}}$
along its cuspidal locus is a disjoint union of $n+1$ formal schemes
$\mathsf{C}^{a}_{p}(p^{n},N_{p})$, the proposition follows. ∎

As a short summary, we take the example that $N=p^{n}$ and draw the following
table.

Cusp points | $\frac{a}{p^{n}}$ | $\frac{a}{p^{n-1}}$ | $\cdots$ | $\frac{a}{p^{k}}$ | $\cdots$ | $\frac{a}{p}$ | $a$
---|---|---|---|---|---|---|---
Choice of $a$ | 1 | $(\mathbb{Z}/p\mathbb{Z})^{\times}$ | $\cdots$ | $(\mathbb{Z}/(p^{k},p^{n-k})\mathbb{Z})^{\times}$ | $\cdots$ | $(\mathbb{Z}/p\mathbb{Z})^{\times}$ | 1
Ramification via $p_{1}$ | 1 | 1 | $\cdots$ | $\textup{max}\\{1,p^{n-2k}\\}$ | $\cdots$ | $p^{n-2}$ | $p^{n}$
Ramification via $p_{2}$ | $p^{n}$ | $p^{n-2}$ | $\cdots$ | $\textup{max}\\{1,p^{2k-n}\\}$ | $\cdots$ | $1$ | $1$
Components | $\mathsf{C}^{n}(p^{n})$ | $\mathsf{C}^{n-2}(p^{n})$ | $\cdots$ | $\mathsf{C}^{2k-n}(p^{n})$ | $\cdots$ | $\mathsf{C}^{2-n}(p^{n})$ | $\mathsf{C}^{-n}(p^{n})$
Mod $p$ | $\mathcal{C}_{p}^{n}(p^{n})$ | $\mathcal{C}_{p}^{n-2}(p^{n})$ | $\cdots$ | $\mathcal{C}_{p}^{2k-n}(p^{n})$ | $\cdots$ | $\mathcal{C}_{p}^{2-n}(p^{n})$ | $\mathcal{C}_{p}^{-n}(p^{n})$

###  6\. Arithmetic intersection theory on $\mathcal{X}_{0}(N)$

####  6.1. Arithmetic Chow groups
$\widehat{\textup{CH}}^{\bullet}(\mathcal{X}_{0}(N))$

We apply the arithmetic intersection theory developed by Henri Gillet in
[Gil82] and [Gil09] to the regular proper flat Deligne-Mumford stack
$\mathcal{X}_{0}(N)$. We obtain the following arithmetic Chow ring of
$\mathcal{X}_{0}(N)$,

$\widehat{\textup{CH}}^{\bullet}(\mathcal{X}_{0}(N))=\bigoplus\limits_{n=0}^{2}\widehat{\textup{CH}}^{n}(\mathcal{X}_{0}(N)).$

Roughly speaking, a class in $\widehat{\textup{CH}}^{n}(\mathcal{X}_{0}(N))$
is represented by an arithmetic cycle $(\mathcal{Z},g_{\mathcal{Z}})$, where
$\mathcal{Z}$ is a codimension $n$ closed substack of $\mathcal{X}_{0}(N)$,
with $\mathbb{C}$-coefficients, and $g_{\mathcal{Z}}$ is a Green current for
$\mathcal{Z}(\mathbb{C})$, i.e., $g_{\mathcal{Z}}$ is a current on the proper
smooth complex curve $\mathcal{X}_{0}(N)_{\mathbb{C}}$ of degree $(n-1,n-1)$
for which there exists a smooth $\omega$ such that

$dd^{c}(g)+\delta_{\zeta}=[\omega].$

holds; here $[\omega]$ is the current defined by integration against the
smooth form $\omega$. The rational arithmetic cycles are those of the form
$\widehat{\textup{div}}(f)=(\textup{div}(f),\iota_{\ast}[-\textup{log}(|\tilde{f}|^{2})])$,
where $f\in\kappa(\mathcal{Z})^{\times}$ is a rational function on a
codimension $n-1$ integral substack
$\iota:\mathcal{Z}\hookrightarrow\mathcal{X}_{0}(N)$, together with classes of
the form $(0,\partial\eta+\overline{\partial}\eta^{\prime})$. By definition,
the arithmetic Chow group $\textup{CH}^{n}(\mathcal{X}_{0}(N))$ is the
quotient of the space of arithmetic cycles by the $\mathbb{C}$-subspace
spanned by those rational cycles.

Let $\mathcal{Z}$ be an irreducible codimension $2$ cycle on
$\mathcal{X}_{0}(N)$, then $\mathcal{Z}$ is a Deligne-Mumford stack over
$\mathbb{F}_{p}$ for some prime number $p$ and the groupoid
$\mathcal{Z}(\overline{\mathbb{F}}_{p})$ would be a singleton with a finite
automorphism group $\textup{Aut}(\mathcal{Z})$, the rational function field
$\kappa(\mathcal{Z})$ of $\mathcal{Z}$ is a finite extension of
$\mathbb{F}_{p}$. Clearly $\delta_{\mathcal{Z}}=0$ because
$\mathcal{Z}(\mathbb{C})=\varnothing$.

Let $(\mathcal{Z},g)=(\sum\limits_{i}n_{i}[\mathcal{Z}_{i}],g)$ be an
arithmetic cycle of codimension $2$ where each $\mathcal{Z}_{i}$ is an
irreducible codimension $2$ cycle on $\mathcal{X}_{0}(N)$. We define the
degree map as follows,

(35)
$\displaystyle\widehat{\textup{deg}}:\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$
$\displaystyle\longrightarrow\mathbb{C},$ $\displaystyle[(\mathcal{Z},g)]$
$\displaystyle\longmapsto\sum\limits_{i}n_{i}\cdot\frac{\textup{log}\,|\kappa(\mathcal{Z}_{i})|}{|\textup{Aut}(\mathcal{Z}_{i})|}+\frac{1}{2}\int\limits_{\mathcal{X}_{0}(N)(\mathbb{C})}g.$

here the integration $\int\limits_{\mathcal{X}_{0}(N)_{\mathbb{C}}}g$ is the
integration of the constant function $1$ on $\mathcal{X}_{0}(N)_{\mathbb{C}}$
against the $(1,1)$-current $g$. It is a finite number since the stack
$\mathcal{X}_{0}(N)$ is proper. This number is independent of the choice of
representing element $(\mathcal{Z},g)$ as a consequence of the product formula
(cf. [Gil82, 3.4.3]).

####  6.2. Extended arithmetic Chow group
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N),\mathcal{S})$

######  Definition 6.2.1.

Let $\mathcal{S}$ be the set of cusps on the complex curve
$\mathcal{X}_{0}(N)_{\mathbb{C}}$. For $P\in\mathcal{S}$ and $\varepsilon>0$,
denote by $B_{\varepsilon}(P)\subset\mathcal{X}_{0}(N)_{\mathbb{C}}$ the open
disk of radius $\varepsilon$ centered at $P$ and
$\mathcal{X}_{0}(N)_{\varepsilon}=\mathcal{X}_{0}(N)_{\mathbb{C}}\backslash\bigcup\limits_{P\in\mathcal{S}}B_{\varepsilon}(P)$.
Let $t$ be a local parameter at a point $P\in\mathcal{S}$, for a line bundle
$\mathcal{L}$ on $\mathcal{X}_{0}(N)$, a singular metric $h_{\mathcal{L}}$ on
the induced complex line bundle $\mathcal{L}_{\infty}$ on
$\mathcal{X}_{0}(N)_{\mathbb{C}}$ is called hermitian, logarithmically
singular (with respect to $\mathcal{S}$), if the following two conditions
hold:
(a) $h_{\mathcal{L}}$ is a smooth, hermitian metric on $\mathcal{L}_{\infty}$
restricted to $\mathcal{Y}_{0}(N)_{\mathbb{C}}$;
(b) for each $P\in\mathcal{S}$ and any section $l$ of $\mathcal{L}$, there
exists a real number $\alpha_{\widehat{\mathcal{L}},l,P}$ and a positive,
continuous function $\varphi_{\widehat{\mathcal{L}},l,P}$ defined on
$B_{\varepsilon}(P)$ and smooth away from the origin such that the equality

$h_{\mathcal{L}}\left(l(t)\right)=-\textup{log}(|t|^{2})^{\alpha_{\widehat{\mathcal{L}},l,P}}\cdot|t|^{\textup{ord}_{P}(l)}\cdot\varphi_{\widehat{\mathcal{L}},l,P}(t)$

holds for all $t\in B_{\varepsilon}(P)\backslash\\{P\\}$; furthermore, there
exist positive constants $\beta_{\widehat{\mathcal{L}},l,P}$ and
$\rho_{\widehat{\mathcal{L}},l,P}$ such that the inequalities

$\bigg{|}\frac{\partial\varphi_{\widehat{\mathcal{L}},l,P}(t)}{\partial
t}\bigg{|}\leq\frac{\beta_{\widehat{\mathcal{L}},l,P}}{|t|^{1-\rho_{\widehat{\mathcal{L}},l,P}}},\,\,\,\,\bigg{|}\frac{\partial\varphi_{\widehat{\mathcal{L}},l,P}(t)}{\partial\overline{t}}\bigg{|}\leq\frac{\beta_{\widehat{\mathcal{L}},l,P}}{|t|^{1-\rho_{\widehat{\mathcal{L}},l,P}}},\,\,\,\,\bigg{|}\frac{\partial^{2}\varphi_{\widehat{\mathcal{L}},l,P}(t)}{\partial
t\partial\overline{t}}\bigg{|}\leq\frac{\beta_{\widehat{\mathcal{L}},l,P}}{|t|^{2-\rho_{\widehat{\mathcal{L}},l,P}}}.$

######  Example 6.2.2.

Let $\pi^{\textup{univ}}:E^{\textup{univ}}\rightarrow E^{\prime\textup{univ}}$
be the universal cyclic $N$-isogeny between generalized elliptic curves
$E^{\textup{univ}}\stackrel{{\scriptstyle
p^{\textup{univ}}}}{{\rightarrow}}\mathcal{X}_{0}(N)$ and
$E^{\prime\textup{univ}}\stackrel{{\scriptstyle
p^{\prime\textup{univ}}}}{{\rightarrow}}\mathcal{X}_{0}(N)$ over the modular
curve $\mathcal{X}_{0}(N)$. Let $\omega_{N}\coloneqq
p^{\textup{univ}}_{\ast}\left(\Omega^{1}_{E^{\textup{univ}}/\mathcal{X}_{0}(N)}\right)$.
This bundle can be metrized on the complex curve
$\mathcal{Y}_{0}(N)_{\mathbb{C}}$ in the following way: let $f$ be a section
of $\omega_{N}$, for any $\tau=u+iv\in\mathcal{H}^{+}$, the metric $\|\cdot\|$
at $\tau$ is determined by the formula

$\|f\|_{\tau}^{2}=2\sqrt{\pi}e^{-\frac{\gamma}{2}}v\cdot|f(\tau)|^{2},$

where $\gamma=-\Gamma^{\prime}(1)$ is the Euler-Mascheroni constant, this
metric is hermitian and logarithmically singular with respect to the set
$\mathcal{S}$ by the work of Du and Yang [DY19, Theorem 5.1]. In the rest of
this paper, we will use $\widehat{\omega}_{N}=(\omega_{N},\|\cdot\|)$ to
denote this hermitian, logarithmically singular line bundle.

Let $\widehat{\textup{Pic}}(\mathcal{X}_{0}(N),\mathcal{S})$ be the group of
isomorphism classes of hermitian, logarithmically singular line bundle on
$\mathcal{X}_{0}(N)$ with respect to the set $\mathcal{S}$, we also denote it
by $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N),\mathcal{S})$ and call it the
extended arithmetic Chow group of $\mathcal{X}_{0}(N)$ with respect to the set
$\mathcal{S}$.

######  Definition 6.2.3.

Let $\widehat{\mathcal{L}}=(\mathcal{L},h_{\mathcal{L}})$ and
$\widehat{\mathcal{M}}=(\mathcal{M},h_{\mathcal{M}})$ be two hermitian,
logarithmically singular line bundles on $\mathcal{X}_{0}(N)$ with respect to
the set $\mathcal{S}$, let $l,m$ be non-trivial, global sections, whose
induced divisors on $Xc_{\mathbb{C}}$ have no points in common. Then the
generalized arithmetic intersection number
$\widehat{\mathcal{L}}\cdot\widehat{\mathcal{M}}$ is defined by

$\widehat{\mathcal{L}}\cdot\widehat{\mathcal{M}}\coloneqq(l,m)_{\textup{fin}}+\langle
l,m\rangle_{\infty};$

here $(l,m)_{\textup{fin}}$ is defined by Serre’s Tor-formula, which
specializes to

$(l,m)_{\textup{fin}}=\sum\limits_{x\in\mathcal{X}_{0}(N)}\log\\#\mathcal{O}_{\mathcal{X}_{0}(N),x}/(l_{x},m_{x}),$

where $l_{x},m_{x}$ are the local equations of $l,m$ respectively at the point
$x\in\mathcal{X}_{0}(N)$ and

(36) $\displaystyle\langle l,m\rangle_{\infty}=$
$\displaystyle-\log(h_{\mathcal{M}}(m))(\textup{div}(l)-\sum\limits_{P\in\mathcal{S}}\textup{ord}_{P}(l)\cdot
P)+\sum\limits_{P\in\mathcal{S}}\textup{ord}_{P}(l)\left(\alpha_{\widehat{\mathcal{M}},m,P}-\log(\varphi_{\widehat{\mathcal{M}},m,P}(0))\right)$
$\displaystyle-\lim\limits_{\varepsilon\rightarrow
0}\left(\sum\limits_{P\in\mathcal{S}}\textup{ord}_{P}(l)\cdot\alpha_{\widehat{\mathcal{M}},m,P}\cdot\log(-\log\varepsilon^{2})+\int\limits_{\mathcal{X}_{0}(N)_{\varepsilon}}\log
h_{\mathcal{L}}(l)\cdot c_{1}(\widehat{\mathcal{M}})\right).$

######  Proposition 6.2.4.

The formula (36) induces a bilinear, symmetric pairing

$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N),\mathcal{S})\times\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N),\mathcal{S})\longrightarrow\mathbb{C}.$

###### Proof.

This is proved by Kühn in [Küh01]. ∎

######  Example 6.2.5.

The pairing $\widehat{\omega}_{N}\cdot\widehat{\omega}_{N}$ has been computed
by Kühn [Küh01] and Bost, adjusting for the normalization of the metric in
[DY19, Lemma 7.3], the result is

$\widehat{\omega}_{N}\cdot\widehat{\omega}_{N}=\langle\widehat{\omega}_{N},\widehat{\omega}_{N}\rangle=\frac{\psi(N)}{24}\left(\frac{1}{2}-\frac{\Lambda^{\prime}(-1)}{\Lambda(-1)}\right).$

####  6.3. Intersection numbers of irreducible components of
$\mathcal{X}_{0}(N)_{\mathbb{F}_{p}}$

Let $\widehat{\mathcal{X}}^{a}_{p}(N)=(\mathcal{X}^{a}_{p}(N),0)$ be the
corresponding class in the arithmetic Chow group
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$. Recall that there is an
intersection pairing
$\langle\cdot,\cdot\rangle:\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))\times\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))\rightarrow\mathbb{C}$.

######  Lemma 6.3.1.

Let $n=\nu_{p}(N)$ be the $p$-adic valuation of $N$, let $N_{p}=p^{-n}N$, for
integers $a,b$ such that $-n\leq a\neq b\leq n$ and $a,b\equiv
n\,\,(\textup{mod}\,2)$, we have

(37)
$\begin{split}\langle\widehat{\mathcal{X}}^{a}_{p}(N),\widehat{\mathcal{X}}^{b}_{p}(N)\rangle=\log(p)\cdot\begin{cases}\frac{\psi(N_{p})(p-1)}{24}p^{\textup{min}\\{|a|,|b|\\}},&\textup{if
$ab\geq 0$;}\\\ \frac{\psi(N_{p})(p-1)}{24},&\textup{if $ab\leq 0$.}\\\
\end{cases}\end{split}$

For integer $a$ such that $-n\leq a\leq n$ and $a\equiv
n\,\,(\textup{mod}\,2)$, we have

(38)
$\begin{split}\langle\widehat{\mathcal{X}}^{a}_{p}(N),\widehat{\mathcal{X}}^{a}_{p}(N)\rangle=\log(p)\cdot\begin{cases}-\frac{\psi(N_{p})p^{|a|}}{12},&\textup{if
$|a|\neq n$;}\\\ -\frac{\psi(N_{p})(p-1)p^{n-1}}{24},&\textup{if $|a|=n$.}\\\
\end{cases}\end{split}$

###### Proof.

Let first assume $a,b\geq 0$ and $a\neq b$, then by the definition of the
pairing $\langle\cdot,\cdot\rangle$ and (d), (e) of Theorem 5.4.2,

$\displaystyle\langle\widehat{\mathcal{X}}^{a}_{p}(N),\widehat{\mathcal{X}}^{b}_{p}(N)\rangle$
$\displaystyle=\sum\limits_{P\in\mathcal{X}^{\textup{ss}}_{0}(N)(\mathbb{F})}\frac{1}{\\#\textup{Aut}(P)}\cdot\textup{Length}\left(\mathbb{F}[[t_{1},t_{2}]]/(t_{1}-t_{2}^{p^{a}},t_{1}-t_{2}^{p^{b}})\right)\cdot\log(p)$
$\displaystyle=\sum\limits_{P\in\mathcal{X}^{\textup{ss}}_{0}(N)(\mathbb{F})}\frac{1}{\\#\textup{Aut}(P)}\cdot
p^{\textup{min}\\{a,b\\}}\cdot\log(p).$

By (e) of Theorem 5.4.2, the stack $\mathcal{X}_{0}(p^{n})$ has the same
number of supersingular $\mathbb{F}$-points as the stack $\mathcal{X}_{0}(1)$,
while the later one has been computed explicitly by the following formula (for
example, see [KM85, Corollary 12.4.6]),

$\sum\limits_{P\in\mathcal{X}^{\textup{ss}}_{0}(1)(\mathbb{F})}\frac{1}{\\#\textup{Aut}(P)}=\frac{p-1}{24}.$

Moreover, by Lemma 5.1.2 we know that
$\mathcal{X}_{0}(N)_{\mathbb{F}}\simeq\mathcal{X}_{0}(p^{n})_{\mathbb{F}}\times_{\mathcal{X}_{0}(1)_{\mathbb{F}}}\mathcal{X}_{0}(N_{p})_{\mathbb{F}}$,
recall the fact that $\mathcal{X}_{0}(N_{p})_{\mathbb{F}}$ is finite flat of
degree $\psi(N_{p})$ over $\mathcal{X}_{0}(1)_{\mathbb{F}}$, hence

$\sum\limits_{P\in\mathcal{X}^{\textup{ss}}_{0}(N)(\mathbb{F})}\frac{1}{\\#\textup{Aut}(P)}=\frac{\psi(N_{p})(p-1)}{24},$

therefore
$\langle\widehat{\mathcal{X}}^{a}_{p}(N),\widehat{\mathcal{X}}^{b}_{p}(N)\rangle=\log(p)\cdot\frac{\psi(N_{p})(p-1)}{24}p^{\textup{min}\\{a,b\\}}$.
The other cases of formula (37) can be proved similarly.

Recall that the principal arithmetic divisor associated to the constant
function $p$ is

$\widehat{\textup{div}}(p)=(\textup{div}(p),-\log(p^{2})).$

It is 0 in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$, hence for any
integer $a$ such that $-n\leq a\leq n$ and $a\equiv n\,\,(\textup{mod}\,2)$,

(39)
$\displaystyle\langle\widehat{\mathcal{X}}^{a}_{p}(N),\mathcal{X}^{n}_{p}(N)+\mathcal{X}^{-n}_{p}(N)+\sum\limits_{\begin{subarray}{c}-n<b<n\\\
n\equiv
b\,(\textup{mod}\,2)\end{subarray}}p^{(n-|b|)/2-1}(p-1)\mathcal{X}^{b}_{p}(N)\rangle$
$\displaystyle=\langle\widehat{\mathcal{X}}^{a}_{p}(N),(\textup{div}(p),0)\rangle$
$\displaystyle=\langle\widehat{\mathcal{X}}^{a}_{p}(N),(0,\log(p^{2})\rangle=0.$

therefore formula (38) can be proved by combining (37) and (39). ∎

Let $n=\nu_{p}(N)\geq 0$ be an integer and $N_{p}=p^{-n}N$, let
$\widehat{\mathcal{X}}_{p}(N)$ be the following element in
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$,

$\widehat{\mathcal{X}}_{p}(N)=\frac{n}{2}\widehat{\mathcal{X}}^{n}_{p}(N)-\frac{n}{2}\widehat{\mathcal{X}}^{-n}_{p}(N)+\sum\limits_{\begin{subarray}{c}-n<a<n\\\
a\equiv n\,\textup{mod}\,2\end{subarray}}\frac{a}{2}\cdot
p^{(n-|a|)/2-1}(p-1)\widehat{\mathcal{X}}^{a}_{p}(N).$

######  Corollary 6.3.2.

We have
$W_{N}^{\ast}(\widehat{\mathcal{X}}_{p}(N))=-\widehat{\mathcal{X}}_{p}(N)$.
Let $n=\nu_{p}(N)\geq 0$ be an integer, then

$\langle\widehat{\mathcal{X}}_{p}(N),\widehat{\mathcal{X}}_{p}(N)\rangle=\frac{\psi(N)}{24}\cdot\frac{-np^{n+1}+2p^{n}+np^{n-1}-2}{p^{n-1}(p^{2}-1)}\cdot\log(p).$

###### Proof.

The fact
$W_{N}^{\ast}(\widehat{\mathcal{X}}_{p}(N))=-\widehat{\mathcal{X}}_{p}(N)$
follows from the definition of the element $\widehat{\mathcal{X}}_{p}(N)$. The
intersection number can be computed by Lemma 38. ∎

####  6.4. An explicit section of the Hodge line bundle

In this section, we define and study the associated divisor class of an
explicit rational section of the line bundle $\widehat{\omega}_{N}^{\otimes
12\varphi(N)}$.

For any positive integer $N$, define the following function $a_{N}$ on the set
of positive integers,

$a_{N}(t)=\sum\limits_{r|t}\mu(\frac{t}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)},$

where $\mu(\cdot)$ is the Möbius function and $r$ ranges over all the positive
integers dividing $t$.

######  Lemma 6.4.1.

The function $a_{N}$ has the following properties,
(a). $\sum\limits_{t|N}a_{N}(t)=\varphi(N)$ and
$\sum\limits_{t|N}t^{-1}a_{N}(t)=0$.
(b). Let $p$ be a prime number such that $p|N$, for any positive integer $t$
such that $pt|N$,

$a_{N}(pt)=pa_{p^{-1}N}(t).$

###### Proof.

The part (a) has been proved in [DY19, Lemma 3.2]. We prove part (b) as
follows. Let $n=\nu_{p}(N)\geq 1$ be the $p$-adic valuation of the integer
$N$, and $N_{p}=p^{-n}N$. By definition,

$\displaystyle a_{N}(pt)$
$\displaystyle=\sum\limits_{r|pt}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}r|pt\\\ p\nmid
r\end{subarray}}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}+\sum\limits_{\begin{subarray}{c}r|pt\\\
p|r\end{subarray}}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}r|pt\\\ p\nmid
r\end{subarray}}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}+\sum\limits_{r^{\prime}|t}\mu(\frac{t}{r^{\prime}})\mu(\frac{p^{-1}N}{r^{\prime}})\frac{\varphi(N)}{\varphi(p^{-1}N/r^{\prime})}$

If $n\geq 2$, we have $\mu(\frac{N}{r})=0$ when $p\nmid r$, then

$\displaystyle a_{N}(pt)$
$\displaystyle=\sum\limits_{r^{\prime}|t}\mu(\frac{t}{r})\mu(\frac{p^{-1}N}{r^{\prime}})\frac{\varphi(N)}{\varphi(p^{-1}N/r^{\prime})}$
$\displaystyle=p\sum\limits_{r^{\prime}|t}\mu(\frac{t}{r^{\prime}})\mu(\frac{p^{-1}N}{r^{\prime}})\frac{\varphi(p^{-1}N)}{\varphi(p^{-1}N/r^{\prime})}=pa_{p^{-1}N}(t).$

If $n=1$, we have $t|p^{-1}N=N_{p}$, especially, $p\nmid t$,

$\displaystyle a_{N}(pt)$
$\displaystyle=\sum\limits_{r|pt}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}$
$\displaystyle=\sum\limits_{r|t}\mu(\frac{pt}{r})\mu(\frac{N}{r})\frac{\varphi(N)}{\varphi(N/r)}+\sum\limits_{r^{\prime}|t}\mu(\frac{t}{r^{\prime}})\mu(\frac{p^{-1}N}{r^{\prime}})\frac{\varphi(N)}{\varphi(p^{-1}N/r^{\prime})}$
$\displaystyle=\sum\limits_{r|t}\mu(\frac{t}{r})\mu(\frac{p^{-1}N}{r})\frac{\varphi(p^{-1}N)}{\varphi(p^{-1}N/r)}+\sum\limits_{r^{\prime}|t}\mu(\frac{t}{r^{\prime}})\mu(\frac{p^{-1}N}{r^{\prime}})\frac{(p-1)\varphi(p^{-1}N)}{\varphi(p^{-1}N/r^{\prime})}=pa_{p^{-1}N}(t).$

∎

Let $\Delta$ be the modular discriminant function on the upper half plane
$\mathbb{H}^{+}$, it is a cusp form of weight $12$ and level $1$ with
expansion at $\infty$ given as follows,

(40) $\Delta(z)=q\prod\limits_{n\geq 1}(1-q^{n})^{24},\,\,\textup{$q=e^{2\pi
iz}$ where $z\in\mathbb{H}^{+}$}.$

The modular form $\Delta(z)$ is a global section of the line bundle
$\widehat{\omega}_{1}^{\otimes 12}$ by the definition made in Katz’s work
[Katz73]. Similarly, for any positive integer $t$, the modular form
$\Delta(tz)$ is a section of the line bundle $\widehat{\omega}_{t}^{\otimes
12}$. Now we construct an explicit section $\Delta_{N}$ of the line bundle
$\widehat{\omega}_{N}^{\otimes 12\varphi(N)}$ following the lines in [DY19,
§1],

(41) $\Delta_{N}(z)=\prod\limits_{t|N}\Delta(tz)^{a_{N}(t)}.$

where $t$ ranges over all the positive integers dividing $N$. We also define
the following section:

$\Delta_{N}^{0}(z)\coloneqq\Delta_{N}\big{|}_{W_{N},12\varphi(N)}(z)=\Delta_{N}(W_{N}z)\cdot(Nz^{2})^{-6\varphi(N)}.$

It is also a rational section of the metrized line bundle
$\widehat{\omega}_{N}^{\otimes 12\varphi(N)}$.

######  Lemma 6.4.2.

Let $p$ be a prime number such that $p|N$, let $n=\nu_{p}(N)\geq 1$ and
$N_{p}=p^{-n}N$, we have

$\Delta_{N}(z)=\frac{\Delta_{N_{p}}(p^{n}z)^{p^{n}}}{\Delta_{N_{p}}(p^{n-1}z)^{p^{n-1}}}.$

###### Proof.

By the definition of $\Delta_{N}$, we have

$\Delta_{N}(z)=\prod\limits_{t|N_{p}}\Delta(tz)^{a_{N}(t)}\cdot\prod\limits_{p|t|N}\Delta(tz)^{a_{N}(t)}.$

If $n\geq 2$, we have $a_{N}(t)=0$ when $t|N_{p}$. Combining Lemma 6.4.1, we
have

(42)
$\Delta_{N}(z)=\prod\limits_{t^{\prime}|p^{-1}N}\Delta(pt^{\prime}z)^{a_{N}(pt^{\prime})}=\left(\prod\limits_{t^{\prime}|p^{-1}N}\Delta(pt^{\prime}z)^{a_{p^{-1}N}(t^{\prime})}\right)^{p}=\Delta_{p^{-1}N}(pz)^{p}.$

If $n=1$, we have $a_{N}(t)=-a_{N_{p}}(t)$ when $t|N_{p}$. Combining Lemma
6.4.1, we have

(43)
$\Delta_{N}(z)=\left(\Delta_{N_{p}}(z)\right)^{-1}\cdot\prod\limits_{t^{\prime}|N_{p}}\Delta(pt^{\prime}z)^{a_{N}(pt^{\prime})}=\frac{\Delta_{N_{p}}(pz)^{p}}{\Delta_{N_{p}}(z)}.$

Therefore the lemma follows by induction based on formulas (42) and (43). ∎

Recall the definition of the cusps $P_{\infty}(N)$ and $P_{0}(N)$ in Remark
5.3.2. We have the following theorem which describes the divisors associated
to the rational sections $\Delta_{N}$ and $\Delta_{N}^{0}$ explicitly.

######  Theorem 6.4.3.

For any positive integer $N$, and any prime number $p$, let $n=\nu_{p}(N)\geq
0$ be the $p$-adic valuation of the integer $N$, then as an element in
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$, we have

$\textup{div}(\Delta_{N})=\psi(N)\varphi(N)P_{\infty}(N)+\sum\limits_{p|N}f_{p}(N),$
$\textup{div}(\Delta_{N}^{0})=\psi(N)\varphi(N)P_{0}(N)+\sum\limits_{p|N}f_{p}^{0}(N),$

where for any $p|N$,

(44) $f_{p}(N)=12p^{n-1}\varphi(N_{p})\sum\limits_{\begin{subarray}{c}-n\leq
a<n\\\ a\equiv n\,\textup{mod
2}\end{subarray}}\left(\frac{1-p}{2}(n-a)-1\right)\varphi(p^{(n-|a|)/2})\cdot\mathcal{X}_{p}^{a}(N).$
(45)
$f_{p}^{0}(N)=-6n\varphi(N)\mathcal{X}_{p}^{-n}(N)+12p^{n-1}\varphi(N_{p})\sum\limits_{\begin{subarray}{c}-n<a\leq
n\\\ a\equiv n\,\textup{mod
2}\end{subarray}}\varphi(p^{(n-|a|)/2})\left(\frac{1-p}{2}n-1\right)\cdot\mathcal{X}_{p}^{a}(N).$

###### Proof.

The expansion of $\Delta_{N}$ at $\infty$ is computed in [DY19, Proposition
3.3], there exist some integer $C_{N}(n)$ such that

$\Delta_{N}(z)=q^{\psi(N)\varphi(N)}\prod\limits_{n\geq
1}(1-q^{n})^{24C_{N}(n)},$

hence the vanishing order of $\Delta_{N}$ at $\infty$ is $\psi(N)\varphi(N)$,
and there is no vertical component of $\textup{div}(\Delta_{N})$ at the cusp
$\textsf{C}^{n}(p^{n},N_{p})$. Let $k$ be a positive integer such that $0\leq
k<n$, we first study the expansion of the modular form
$\Delta_{N_{p},p^{n}}(z)\coloneqq\Delta_{N_{p}}(p^{n}z)$ at the component
$\mathsf{C}^{2k-n}(p^{n},N_{p})$ of
$\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(N))$, since the point
$P_{\frac{1}{p^{k}}}$ belongs to this cusp by Corollary 5.3.4, let

$\gamma=\begin{pmatrix}1&0\\\
p^{k}&1\end{pmatrix},\,\,\gamma_{k}=\begin{pmatrix}p^{n-k}&-1\\\
1&0\end{pmatrix},\,\,n_{k}=\begin{pmatrix}p^{k}&1\\\ 0&p^{n-k}\end{pmatrix}.$

Note that $p^{n}\cdot\gamma z=\gamma_{k}n_{k}z$ for any $z\in\mathbb{C}$.

$\displaystyle\Delta_{N_{p},p^{n}}\big{|}_{\gamma,12\varphi(N_{p})}(z)$
$\displaystyle=\Delta_{N_{p}}(p^{n}\gamma z)\cdot
j(\gamma,z)^{-12\varphi(N_{p})}$
$\displaystyle=\Delta_{N_{p}}(\gamma_{k}n_{k}z)\cdot
j(\gamma_{k},n_{k}z)^{-12\varphi(N_{p})}\cdot\left(\frac{j(\gamma_{k},n_{k}z)}{j(\gamma,z)}\right)^{12\varphi(N_{p})}$
$\displaystyle=p^{-12\varphi(N_{p})(n-k)}\cdot\Delta_{N_{p}}\big{|}_{\gamma_{k},12\varphi(N_{p})}(n_{k}z).$

Moreover, for any $z\in\mathbb{H}^{+}$,

$\displaystyle\Delta_{N_{p}}\big{|}_{\gamma_{k},12\varphi(N_{p})}(z)$
$\displaystyle=\Delta_{N_{p}}(\frac{p^{n-k}z-1}{z})\cdot
j(\gamma_{k},z)^{-12\varphi(N_{p})}$
$\displaystyle=\Delta_{N_{p}}(-\frac{1}{z})\cdot z^{-12\varphi(N_{p})}$
$\displaystyle=\prod\limits_{t|N_{p}}\Delta(-\frac{t}{z})^{a_{N_{p}}(t)}\cdot
z^{-12\varphi(N_{p})}$
$\displaystyle=\prod\limits_{t|N_{p}}\Delta(\frac{z}{t})^{a_{N_{p}}(t)}\cdot\prod\limits_{t|N_{p}}t^{-12a_{N_{p}}(t)}.$

Let $C_{N_{p}}=\prod\limits_{t|N_{p}}t^{-12a_{N_{p}}(t)}$, we know that
$\nu_{p}(C_{N_{p}})=0$, and

$\Delta_{N_{p},p^{n}}\big{|}_{\gamma,12\varphi(N_{p})}(z)=p^{-12\varphi(N_{p})(n-k)}\cdot
C_{N_{p}}\cdot\prod\limits_{t|N_{p}}\Delta\left(\frac{p^{k}z+1}{p^{n-k}t}\right)^{a_{N_{p}}(t)}.$

Similarly, for $\Delta_{N_{p},p^{n-1}}(z)=\Delta_{N_{p}}(p^{n-1}z)$, we have

$\Delta_{N_{p},p^{n}}\big{|}_{\gamma,12\varphi(N_{p})}(z)=p^{-12\varphi(N_{p})(n-k-1)}\cdot
C_{N_{p}}\cdot\prod\limits_{t|N_{p}}\Delta\left(\frac{p^{k}z+1}{p^{n-k-1}t}\right)^{a_{N_{p}}(t)}.$

By definition,

$\displaystyle\Delta_{N}\big{|}_{\gamma,12\varphi(N)}(z)$
$\displaystyle=\frac{\Delta_{N_{p},p^{n}}^{p^{n}}}{\Delta_{N_{p},p^{n-1}}^{p^{n-1}}}\bigg{|}_{\gamma,12\varphi(N)}(z)=\frac{\left(\Delta_{N_{p},p^{n}}\big{|}_{\gamma,12\varphi(N_{p})}(z)\right)^{p^{n}}}{\left(\Delta_{N_{p},p^{n-1}}\big{|}_{\gamma,12\varphi(N_{p})}(z)\right)^{p^{n-1}}}$
(46)
$\displaystyle=p^{-12\varphi(N_{p})\left(p^{n}(n-k)-p^{n-1}(n-k-1)\right)}\cdot
C_{N_{p}}^{p^{n}-p^{n-1}}\cdot\prod\limits_{t|N_{p}}\left(\frac{\Delta\left(\frac{p^{k}z+1}{p^{n-k}t}\right)^{p^{n}}}{\Delta\left(\frac{p^{k}z+1}{p^{n-k-1}t}\right)^{p^{n-1}}}\right)^{a_{N_{p}}(t)}.$

By the expansion of $\Delta$ at $\infty$ given in the formula (40), we know
that the last term in formula (46) doesn’t vanish. Recall that the cusp
$P_{\frac{1}{p^{k}}}$ lies in the component $\mathsf{C}^{2k-n}(p^{n},N_{p})$
of $\widehat{\mathsf{Cusp}}(\mathcal{X}_{0}(N))$ and mod $p$ reduction
$\mathsf{C}^{2k-n}_{p}(p^{n},N_{p})$ of the formal scheme
$\mathsf{C}^{2k-n}(p^{n},N_{p})$ is the completion of curve
$\mathcal{C}_{p}^{2k-n}(p^{n},N_{p})=p^{(n-|2k-n|)/2-1}(p-1)\mathcal{X}_{p}^{2k-n}(N)$
along its cuspidal locus, hence the multiplicity of
$\mathcal{C}_{p}^{2k-n}(p^{n},N_{p})$ in $\textup{div}(\Delta_{N})$ is
$-12\varphi(N_{p})\left(p^{n}(n-k)-p^{n-1}(n-k-1)\right)$. ∎

######  Corollary 6.4.4.

For any positive integer $N$, we have the following identity in
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$,

$\widehat{\omega}_{N}-W_{N}^{\ast}\widehat{\omega}_{N}=\sum\limits_{p|N}\widehat{\mathcal{X}}_{p}(N).$

###### Proof.

We know that

$\widehat{\omega}_{N}=(\textup{div}(\Delta_{N}),-\log\|\Delta_{N}\|^{2})=(\textup{div}(\Delta_{N}^{0}),-\log\|\Delta_{N}^{0}\|^{2}),$

therefore

$\displaystyle\widehat{\omega}_{N}-W_{N}^{\ast}\widehat{\omega}_{N}$
$\displaystyle=(\textup{div}(\Delta_{N}^{0}),-\log\|\Delta_{N}^{0}\|^{2})-W_{N}^{\ast}(\textup{div}(\Delta_{N}),-\log\|\Delta_{N}\|^{2})$
$\displaystyle=(\sum\limits_{p|N}\left(f_{p}^{0}(N)-W_{N}^{\ast}f_{p}(N)\right),0)=\sum\limits_{p|N}\widehat{\mathcal{X}}_{p}(N).$

∎

###  7\. Arithmetic special cycles

####  7.1. The codimension 1 case: Arithmetic special divisors in
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$

In the section, we define an element in the codimension 1 arithmetic Chow
group $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$ of $\mathcal{X}_{0}(N)$
for any pair $(t,y)$, where $t$ is an integer and $y$ is a positive real
number.

#####  7.1.1. Special divisors

We first define an element in the Chow group
$\textup{CH}^{1}(\mathcal{X}_{0}(N))$. When $t\leq 0$, we define
$\mathcal{Z}(t,\Delta(N))$ to be $0$ in the Chow group
$\textup{CH}^{1}(\mathcal{X}_{0}(N))$. Let’s now consider the case that $t>0$.

######  Definition 7.1.1.

For every positive integer $t$. The stack $\mathcal{Z}(t,\Delta(N))$ is
defined as follows, its fibered category over a scheme $S$ consists of the
following objects,

$((E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}E^{\prime}),j),$

where $(E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}E^{\prime})$ is an
object in $\mathcal{Y}_{0}(N)(S)$, $j\in\textup{Hom}_{S}(E,E^{\prime})$ is an
isogeny such that,

$j\circ\pi^{\vee}+\pi\circ j^{\vee}=0\,\,\,\textup{and}\,\,\,j^{\vee}\circ
j=t.$

It is a well-known fact that the stack $\mathcal{Z}(t,\Delta(N))$ is a
generalized Cartier divisor on the stack $\mathcal{X}_{0}(N)$, a detailed
proof can be found in [Zhu23, Proposition 4.3.7] (see also [SSY22, Lemma
2.3]), therefore it can be viewed as an element in the Chow group
$\textup{CH}^{1}(\mathcal{X}_{0}(N))$.

Up to now, we have already constructed elements $\mathcal{Z}(t,\Delta(N))$ in
the Chow group $\textup{CH}^{1}(\mathcal{X}_{0}(N))$ for any integer $t$, but
these elements need to be modified due to the fact that $\mathcal{Y}_{0}(N)$
is not compact. For any real number $s$, define the function $\beta_{s}$ to be

(47)
$\beta_{s}(r)=\int_{1}^{\infty}e^{-rt}t^{-s}\textup{d}t,\,\,\textup{where}\,\,r>0.$

Let $y>0$ be a positive real number, $t$ be an integer, define

(48)
$g(t,y,\Delta(N))=\begin{cases}\frac{\sqrt{N}}{2\pi\sqrt{y}}\beta_{3/2}(-4\pi
ty),&\textup{if $-Nt>0$ is a square;}\\\
\frac{\sqrt{N}}{2\pi\sqrt{y}},&\textup{if $t=0$;}\\\
0,&\textup{otherwise.}\end{cases}$

Recall that $\mathsf{Cusp}(\mathcal{X}_{0}(N))$ is the cuspidal divisor of the
stack $\mathcal{X}_{0}(N)$, we define the modified special divisor to be

(49)
$\mathcal{Z}^{\ast}(t,y,\Delta(N))=\mathcal{Z}(t,\Delta(N))+g(t,y,\Delta(N))\cdot\mathsf{Cusp}(\mathcal{X}_{0}(N)).$

It is an element in $\textup{CH}^{1}(\mathcal{X}_{0}(N))$.

#####  7.1.2. Green functions

Now we are going to construct the Green functions for these cycles, following
the work of Kudla [Kud97]. Let
$\mathbb{D}=\\{z\in\Delta(N)\otimes_{\mathbb{Z}}\mathbb{C}:(z,z)=0,\,(z,\overline{z})<0\\}\,/\,\mathbb{C}^{\ast}\subset\mathbb{P}(\Delta(N)\otimes\mathbb{C})$,
for any $x\in\Delta(N)\otimes\mathbb{R}$ and $[z]\in\mathbb{D}$, let
$\textup{R}(x,[z])=-|(x,z)|^{2}\cdot(z,\overline{z})^{-1}$. For any element in
$\tau=x+iy\in\mathbb{H}^{+}$, we let

$h(\tau)\coloneqq\frac{1}{\sqrt{N}y}\begin{pmatrix}Nx&N(x^{2}+y^{2})\\\
-1&-x\end{pmatrix}.$

We also define an element $c(\tau)$ of $\mathbb{D}$ associated to the element
$\tau$ as follows,

$c(\tau)=\left[\begin{pmatrix}-N\tau&-N\tau^{2}\\\
1&\tau\end{pmatrix}\right].$

Now we construct Green forms for the special cycles defined in $\S$7.1.1. For
any pair $(t,y)$ where $t$ is an integer and $y$ is a positive real number, we
define

$\mathfrak{g}(t,y,\Delta(N))([z])=\sum\limits_{\begin{subarray}{c}x\in\Delta(N),x\neq
0\\\ (x,x)=t\end{subarray}}\beta_{1}(2\pi\textup{R}(y^{1/2}x,[z])).$

It is invariant under the action of $\Gamma_{0}(N)$, hence the function
$\mathfrak{g}(d,y,\Delta(N))([z])$ is smooth on the open subset
$\mathcal{Y}_{0}(N)_{\mathbb{C}}-\mathcal{Z}(t)_{\mathbb{C}}$.

For the pair $(t,y)$, we also consider the following differential on the upper
half plane

(50)
$\omega(t,y,\Delta(N))=\sum\limits_{\begin{subarray}{c}x\in\Delta(N),x\neq
0\\\
(x,x)=t\end{subarray}}\left(y(x,h(\tau))^{2}-\frac{1}{2\pi}\right)e^{-2\pi\textup{R}(x,c(\tau))}\cdot\frac{\textup{d}\tau\wedge\textup{d}\overline{\tau}}{-2i\textup{Im}(\tau)^{2}}\in\Omega^{1,1}(\mathbb{H}^{+}).$

It is invariant under the action of $\Gamma_{0}(N)$ and can be extended to the
cusps, hence the differential $(1,1)$-form $\omega(t,y,\Delta(N))$ can be
descended to a differential $(1,1)$-form on the complex modular curve
$\mathcal{X}_{0}(N)_{\mathbb{C}}$.

Following [DY19], we modify the bundle $\widehat{\omega}_{N}$ in the following
way,

(51)
$\widehat{\omega}\coloneqq-\widehat{\omega}_{N}-W_{N}^{\ast}\widehat{\omega}_{N}=-2\widehat{\omega}_{N}+\sum\limits_{p|N}\widehat{\mathcal{X}}_{p}(N),$

notice that the last identity follows from Corollary 6.4.4.

######  Theorem 7.1.2.

For any pair $(t,y)$ such that $t$ is an integer and $y$ is a positive number,
(a). The function $\mathfrak{g}(t,y,\Delta(N))([z])$ is a Green function for
the modified divisor $\mathcal{Z}^{\ast}(t,y,\Delta(N))$, and the element
$(\mathcal{Z}^{\ast}(t,y,\Delta(N)),\mathfrak{g}(t,y,\Delta(N))([z]))$ belongs
to the extended arithmetic Chow group
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N),\mathcal{S})$, more precisely,

$\textup{d}\textup{d}^{c}\mathfrak{g}(t,y,\Delta(N))+\delta_{\mathcal{Z}^{\ast}(t,y,\Delta(N))_{\mathbb{C}}}=[\omega(t,y,\Delta(N))].$

$\bullet$ When $-Nt>0$ is a square, the function
$\mathfrak{g}(t,y,\Delta(N))([z])$ has log singularity at the cusps;
$\bullet$ When $t=0$, the function $\mathfrak{g}(0,y,\Delta(N))([z])$ has log-
log singularity at the cusps;
$\bullet$ Otherwise the function $\mathfrak{g}(t,y,\Delta(N))([z])$ decreases
exponentially at the cusps.
(b). The element
$\widehat{\omega}+(\mathcal{Z}^{\ast}(0,y,\Delta(N)),\mathfrak{g}(0,y,\Delta(N)))$
lies in the usual arithmetic Chow group
$\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$.

###### Proof.

The part (a) is proved in [DY19, Theorem 5.1], the part (b) is proved in
[DY19, Theorem 6.6]. ∎

######  Definition 7.1.3.

For any integer $t\in\mathbb{Z}$ and real number $y>0$, we define the
following element in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$,

$\widehat{\mathcal{Z}}(t,y,\Delta(N))=\begin{cases}(\mathcal{Z}^{\ast}(t,y,\Delta(N)),\mathfrak{g}(t,y,\Delta(N))),&\textup{if
$t\neq 0$;}\\\
\widehat{\omega}+(\mathcal{Z}^{\ast}(0,y,\Delta(N)),\mathfrak{g}(0,y,\Delta(N)))-(0,\log
y),&\textup{if $t=0$.}\end{cases}$

######  Lemma 7.1.4.

For any integer $t\in\mathbb{Z}$ and real number $y>0$, we have the following
identity in $\widehat{\textup{CH}}^{1}(\mathcal{X}_{0}(N))$,

$W_{N}^{\ast}\widehat{\mathcal{Z}}(t,y,\Delta(N))=\widehat{\mathcal{Z}}(t,y,\Delta(N)).$

###### Proof.

When $t=0$, this follows from the definition of $\widehat{\omega}$ in (51).
When $t\neq 0$, this is proved in [DY19, Proposition 6.8]. ∎

####  7.2. The codimension 2 case: Arithmetic special cycles in
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$

In the section, we define an element in the codimension 2 arithmetic Chow
group $\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$ of $\mathcal{X}_{0}(N)$
for any pair $(T,\mathsf{y})$, where $T\in\textup{Sym}_{2}(\mathbb{Q})$ is a
half-integral $2\times 2$ symmetric matrix and
$\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})$ is a positive definite $2\times 2$
matrix.

#####  7.2.1. The rank of $T$ is 2

In this case, the matrix $T$ is nonsingular. For any positive definite
$2\times 2$ matrix $\mathsf{y}$ and $T$-adimissible Schwartz function
$\boldsymbol{\varphi}\in\mathscr{S}(\mathbb{V}_{f}^{2})$, an element
$\widehat{\mathcal{Z}}(T,\mathsf{y},\boldsymbol{\varphi})\in\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$
has been made in [Zhu23, §4.3 (16)]. We define
$\widehat{\mathcal{Z}}(T,\mathsf{y})=\widehat{\mathcal{Z}}(T,\mathsf{y},1_{\left(\Delta(N)\otimes\hat{\mathbb{Z}}\right)^{2}})$.

#####  7.2.2. The rank of $T$ is 1

######  Lemma 7.2.1.

Let $T\in\textup{Sym}_{2}(\mathbb{Q})$ be a half-integral $2\times 2$
symmetric matrix such that $\textup{rank}(T)=1$, then there exists
$t\in\mathbb{Z}$ and an element $\gamma\in\textup{GL}_{2}(\mathbb{Z})$ such
that

$T={{}^{t}\gamma}\cdot\begin{pmatrix}0&0\\\ 0&t\end{pmatrix}\cdot\gamma.$

Moreover, the integer $t$ is uniquely determined by $T$, and is invariant upon
replacing $T$ by ${{}^{t}\sigma}T\sigma$ with
$\sigma\in\textup{GL}_{2}(\mathbb{Z})$.

###### Proof.

This is proved in [SSY22, Lemma 2.10]. ∎

######  Definition 7.2.2.

Let $T\in\textup{Sym}_{2}(\mathbb{Q})$ be a half-integral $2\times 2$
symmetric matrix such that $\textup{rank}(T)=1$, let
$\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})$ is a positive definite $2\times 2$
matrix. Let $t$ be the integer associated to $T$ by Lemma 7.2.1. Let
$y_{2}=t^{-1}\textup{tr}(T\mathsf{y})$ and
$y_{1}=\frac{\textup{det}\,\mathsf{y}}{y_{2}}$, both of $y_{1}$ and $y_{2}$
are positive numbers. Define

$\widehat{\mathcal{Z}}(T,\mathsf{y})=\widehat{\mathcal{Z}}(t,y_{2},\Delta(N))\cdot\widehat{\omega}-\left(0,\log
y_{1}\cdot\delta_{\mathcal{Z}^{\ast}(t,y_{2},\Delta(N))_{\mathbb{C}}}\right)$

where $\mathcal{Z}^{\ast}(t,y_{2},\Delta(N))_{\mathbb{C}}$ is the complex
points of the modified special cycle defined in (49).

######  Remark 7.2.3.

When the rank of $T$ is 1. We notice two invariance properties, which both
follows from the definitions:

(52)
$\widehat{\mathcal{Z}}({{}^{t}\gamma}T\gamma,\mathsf{y})=\widehat{\mathcal{Z}}(T,\gamma\mathsf{y}{{}^{t}\gamma}),\,\,\textup{for
any}\,\,\gamma\in\textup{GL}_{2}(\mathbb{Z}).$ (53)
$\widehat{\mathcal{Z}}\left(\begin{pmatrix}0&0\\\
0&t\end{pmatrix},\theta\mathsf{y}{{}^{t}\theta}\right)=\widehat{\mathcal{Z}}\left(\begin{pmatrix}0&0\\\
0&t\end{pmatrix},\mathsf{y}\right),\,\,\textup{for
any}\,\,\theta=\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}\,\,\textup{where
$x\in\mathbb{R}$}.$

#####  7.2.3. The rank of $T$ is 0

In this case, the matrix $T=\boldsymbol{0}_{2}$.

######  Definition 7.2.4.

Let $\mathsf{y}\in\textup{Sym}_{2}(\mathbb{R})$ be a positive definite matrix,
we define the following element in
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$,

$\widehat{\mathcal{Z}}(\mathbf{0}_{2},\mathsf{y})=\widehat{\omega}\cdot\widehat{\omega}+(0,\log\textup{det}\mathsf{y}\cdot[\Omega]).$

Now for any element $\mathsf{z}=\mathsf{x}+i\mathsf{y}\in\mathbb{H}_{2}$, we
consider the following generating series with coefficients in
$\widehat{\textup{CH}}^{2}(\mathcal{X}_{0}(N))$,

$\widehat{\phi}_{2}(\mathsf{z})=\sum\limits_{T\in\textup{Sym}_{2}(\mathbb{Q})}\widehat{\mathcal{Z}}(T,\mathsf{y})\cdot
q^{T}$