# Arithmetic level raising on triple product of Shimura curves and Gross-
Schoen Diagonal cycles I: Ramified case

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

In this article we study the Gross-Schoen diagonal cycle on a triple product
of Shimura curves at a place of bad reduction. We relate the image of the
diagonal cycle under the Abel-Jacobi map to certain period integral that
governs the central critical value of the Garrett-Rankin type triple product
$L$-function via level raising congruences. As an application we prove certain
rank $0$ case of the Bloch-Kato conjecture for the symmetric cube motive of a
weight $2$ modular form.

###### 2000 Mathematics Subject Classification:

Primary 11G18, 11R34, 14G35

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 The symmetric cube motive
    3. 1.3 Bipartite Euler system and other applications
    4. 1.4 Notations and conventions
  2. 2 Review of weight spectral sequence
    1. 2.1 Nearby cycles on semi-stable schemes
    2. 2.2 Monodromy filtration and spectral sequence
    3. 2.3 Examples in dimension $1$ and $3$
    4. 2.4 Potential map and Galois cohomology
  3. 3 Arithmetic level raising for Shimura curves
    1. 3.1 Shimura curves and Shimura sets
    2. 3.2 The $p$-adic upper half plane
    3. 3.3 Cerednick-Drinfeld uniformization
    4. 3.4 Ramified level raising on Shimura curves
  4. 4 Arithmetic level raising for triple product of Shimura curves
    1. 4.1 Semistable model of triple product of Shimura curves
    2. 4.2 Cohomology of blow-ups
    3. 4.3 Level raising on triple product of Shimura curves
    4. 4.4 Proof of the arithmetic level raising
    5. 4.5 Diagonal cycle classes and the first reciprocity law
  5. 5 The Bloch-Kato conjecture for the triple product motive
    1. 5.1 Selmer groups of triple product motive
    2. 5.2 The symmetric cube motive
    3. 5.3 Proof of Theorem 5.8

##  1\. Introduction

Let $\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ be a triple of normalized
newforms in $S^{\mathrm{new}}_{2}(\Gamma_{0}(N))^{3}$ of level $N\geq 5$. We
assume that $N=N^{+}N^{-}$ is a factorization of $N$ such that
$(N^{+},N^{-})=1$ and $N^{-}$ is square-free with odd number of prime factors.
We fix a prime $l\geq 5$ which will be used as the residual characteristic of
the coefficient rings throughout this article. For each $i=1,2,3$, let
$\mathrm{V}_{i}=\mathrm{V}_{f_{i},\lambda_{i}}$ be the $\lambda_{i}$-adic
Galois representation attached $f_{i}$ given by Eichler-Shimura construction
where $\lambda_{i}$ is a place over $l$ in the Hecke field $\mathbf{Q}(f_{i})$
of $f_{i}$. Then we can attach a _Garret-Rankin type triple product
$L$-function_

$L(f_{1}\otimes f_{2}\otimes f_{3},s)$

to the triple product Galois representation
$\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}$. The parity of the
order of vanishing of $L(f_{1}\otimes f_{2}\otimes f_{3},s)$ at the central
critical point $s=2$ is controlled by the the _global root number_
$\epsilon\in\\{1,-1\\}$ which can be factored into $\epsilon=\prod_{v\mid
N\infty}\epsilon_{v}$ of local root numbers $\epsilon_{v}\in\\{\pm 1\\}$. In
our setting, $\epsilon_{\infty}=-1$ since the weights $(2,2,2)$ of
$\underline{\mathbf{f}}$ are _balanced_. In this article, we will assume the
following assumption.

($\epsilon=1$) The product of local root numbers $\prod_{v\mid
N}\epsilon_{v}=-1$ and therefore $\epsilon=1$.

One can also attach a more arithmetic object to the triple tensor product
Galois representation
$\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}$ called the Bloch-
Kato Selmer group. We remind its definition below. Denote by
$\mathrm{B_{cris}}$ the crystalline period ring with respect to
$\mathbf{Q}_{l}$. The _triple product Bloch-Kato Selmer group_

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))$

is the subset of classes
$s\in\mathrm{H}^{1}(\mathbf{Q},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))$
such that
${\rm{loc}}_{l}(s)\in\mathrm{H}^{1}_{f}(\mathbf{Q}_{l},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))$
where $\mathrm{loc}_{l}$ is the localization map to the local Galois
cohomology group at $l$ and
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{l},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))$
is given by the Bloch-Kato local condition

$\ker[\mathrm{H}^{1}(\mathbf{Q}_{l},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q}_{l},(\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3})\otimes\mathrm{B}_{\mathrm{cris}}(-1))].$

The _Bloch-Kato conjecture_ predicts a relationship between the order of the
vanishing of the triple product $L$-function $L(f_{1}\otimes f_{2}\otimes
f_{3},s)$ at $s=2$ and the rank of
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))$.
The present article will be concerned with the _rank 0 case_ of the Bloch-Kato
conjecture for $\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}$.
More precisely, we are concerned with the following conjecture.

######  Conjecture 1.

Suppose that the central critical value $L(f_{1}\otimes f_{2}\otimes f_{3},2)$
is non-zero, then we have

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}(-1))=0.$

One of the most successful approaches to prove such conjectures is via the
_Euler-Kolyvagin system_ arguments. In practice, one need to impose various
assumptions on the triple $\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ to
apply the Euler system argument. We will formulate a more precise conjecture
with a few additional assumptions in the last section of this article, see
Conjecture 5.4. This article makes the first step towards the construction of
such an Euler-Kolyvagin system for the triple product representation. More
precisely, the main result of this article concerns an explicit reciprocity
formula á la Bertonilli-Darmon for the representation
$\mathrm{V}_{1}\otimes\mathrm{V}_{2}\otimes\mathrm{V}_{3}$. Such formula,
loosely speaking, relates cycles classes on Shimura varieties to period
integrals that govern the central critical values of automorphic $L$-functions
via level raising congruences among automorphic forms. We refer the readers to
the recent article [LTXZZ] for the most general results along this line. In
this article, we relate the _Gross-Schoen diagonal cycle class_ on the triple
product of Shimura curves at a place of bad reduction to a period integral
that represents the algebraic part of the triple product $L$-function. Note
that our results are not subsumed by [LTXZZ]. In the companion article [Wang],
we will study the Gross-Schoen diagonal cycle class on the triple product of
Shimura curves at a place of good reduction and so-called second reciprocity
law is proved there. In the case when $\underline{\mathbf{f}}=(f,f,f)$ for a
single modular form $f$ of weight two, our constructions here and in [Wang]
produce the desired Euler-Kolyvagin system for the symmetric cube
representation of $f$ and we can give some evidences to the Bloch-Kato
conjecture for the symmetric cube representation of $f$ in the rank $0$ and
rank $1$ case.

###  1.1. Main results

In order to state our results precisely, we introduce more notations. For
$i=1,2,3$, let $\mathbf{Q}(f_{i})$ be the Hecke field of $f_{i}$. For
simplicity, we assume that
$E=\mathbf{Q}(f_{1})=\mathbf{Q}({f_{2}})=\mathbf{Q}({f_{3}})$ in this
introduction. Let $\lambda$ be a place of $E$ above $l$ and $E_{\lambda}$ be
the completion of $E$ at $\lambda$. Let
$\mathcal{O}=\mathcal{O}_{E_{\lambda}}$ be the ring of integers of
$E_{\lambda}$. We denote by $\varpi$ a uniformizer of $\mathcal{O}$ and set
$\mathcal{O}_{n}=\mathcal{O}/\varpi^{n}$ for any $n\geq 1$. Let
$\phi_{i}:\mathbb{T}\rightarrow\mathcal{O}$ be the natural morphism form the
$l$-adic Hecke algebra to $\mathcal{O}$ corresponding to the Hecke eigensystem
of $f_{i}$ and let $\phi_{i,n}:\mathbb{T}\rightarrow\mathcal{O}_{n}$ be the
reduction of $\phi_{i}$ modulo $\varpi^{n}$. Here the $l$-adic Hecke algebra
$\mathbb{T}$ is the $l$-adic completion of the Hecke algebra that acts
faithfully on the subspace of $S_{2}(\Gamma_{0}(N))$ that is new at primes
dividing $N^{-}$. In particular, $\phi_{i}$ sends the Hecke operator $T_{p}$
to the $p$-th Fourier coefficient $a_{p}(f)$ of $f$ for $p\nmid N$. We denote
by $I_{i,n}$ the kernel of $\phi_{i,n}$ and by $\mathfrak{m}_{i}$ the maximal
ideal in $\mathbb{T}$ containing $I_{i,n}$. Let
$\mathfrak{m}_{\underline{\mathbf{f}}}=(\mathfrak{m}_{1},\mathfrak{m}_{2},\mathfrak{m}_{3})$.
We will always assume that the maximal ideals $\mathfrak{m}_{i}$ are
_residually irreducible_ in the sense explained below. Let

$\rho_{i}:G_{\mathbf{Q}}\rightarrow\mathrm{GL}(\mathrm{V}_{i})$

be the Galois representation attached to $f_{i}$. Then we denote by
$\bar{\rho}_{i}$ the residual Galois representation of $\rho_{i}$. We say
$\mathfrak{m}_{i}$ is residually irreducible if $\bar{\rho}_{i}$ is absolutely
irreducible. We introduce the notion of _n-admissible primes_ for
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ in Definition 5.10. A prime $p$
is $n$-admissible for the triple $\underline{\mathbf{f}}$ if

  1. (1)

$p\nmid Nl$;

  2. (2)

$l\nmid p^{2}-1$;

  3. (3)

$\varpi^{n}\mid p+1-\epsilon_{p,i}a_{p}(f_{i})$ with $i=1,2,3$ and
$\epsilon_{i}\in\\{\pm 1\\}$;

  4. (4)

$\epsilon_{p,1}\epsilon_{p,2}\epsilon_{p,3}=1$.

This is an extension of the notion of $n$-admissible prime in [BD3] to the
triple product setting which can be loosely interpreted as those primes $p$
for which one can find a triple
$\underline{\mathbf{f}}^{[p]}=(f^{[p]}_{1},f^{[p]}_{2},f^{[p]}_{3})$ of weight
$2$ newforms of level $pN$ that are congruent to
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ modulo $\varpi^{n}$. We highlight
the additional condition $(4)$. This condition is imposed to make a sign
change for the triple product $L$-function attached to
$\underline{\mathbf{f}}^{[p]}=(f^{[p]}_{1},f^{[p]}_{2},f^{[p]}_{3})$ at $p$.
Indeed the local sign at $p$ is given by
$-\epsilon_{p,1}\epsilon_{p,2}\epsilon_{p,3}$ by [GK, 1.3] which is $-1$ by
our assumption. This sign change makes it reasonable to consider a cycle class
attached to the triple $\underline{\mathbf{f}}^{[p]}$ in light of the Bloch-
Kato conjecture of odd rank. For the triple $\underline{\mathbf{f}}^{[p]}$ and
$i=1,2,3$, we have morphisms
$\phi^{[p]}_{i}:\mathbb{T}^{[p]}\rightarrow\mathcal{O}$ and
$\phi^{[p]}_{i,n}:\mathbb{T}^{[p]}\rightarrow\mathcal{O}_{n}$ defined
similarly as before with $\mathbb{T}^{[p]}$ the $l$-adic Hecke algebra
corresponding to the subspace of $S_{2}(\Gamma_{0}(pN))$ that is new at primes
dividing $pN^{-}$ . Let $I^{[p]}_{i,n}$ be the kernel of $\phi^{[p]}_{i,n}$
and let $\mathfrak{m}^{[p]}_{i}$ be the maximal ideal containing
$I^{[p]}_{i,n}$ in $\mathbb{T}^{[p]}$. Let
$\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}=(\mathfrak{m}^{[p]}_{1},\mathfrak{m}^{[p]}_{2},\mathfrak{m}^{[p]}_{3})$.
We fix a such $n$-admissible prime $p$ for $\underline{\mathbf{f}}$. For the
cycle class, it is natural to consider the diagonal cycle on the triple
product of Shimura curves. The diagonal cycles on the triple product of curves
are generally referred to as the _Gross-Schoen diagonal cycles_. These cycles
are introduced and studied in [GS]. Their connections to tripe product
$L$-functions are given in [GK], [YZZ]. To define the Shimura curves in our
case, we need to introduce the following quaternion algebras. Let $B$ be the
definite quaternion algebra over $\mathbf{Q}$ of discriminant $N^{-}$ and
$B^{\prime}$ be the indefinite quaternion algebra over $\mathbf{Q}$ of
discriminant $pN^{-}$. Then one can associate a Shimura set
$X^{B}=X^{B}_{N^{+},N^{-}}$ to $B$ and a Shimura curve
$X=X^{B^{\prime}}_{N^{+},pN^{-}}$ over $\mathbf{Q}$ to $B^{\prime}$. We refer
the reader to §2.1 for the constructions. In particular we have an integral
model of $\mathfrak{X}$ of $X$ over $\mathbf{Z}_{(p)}$. Note that since $p$ is
ramified in $B^{\prime}$, the completion of $\mathfrak{X}$ at its special
fiber admits _Cerednick-Drinfeld uniformization_. We consider the diagonal
morphism

$\theta:\mathfrak{X}\rightarrow\mathfrak{X}^{3}$

of $\mathfrak{X}$ into the triple fiber product $\mathfrak{X}^{3}$. We obtain
thus a cycle class
$\theta_{*}[\mathfrak{X}\otimes\mathbf{Q}]\in\mathrm{CH}^{2}(\mathfrak{X}^{3}\otimes\mathbf{Q})$
in the Chow group of $\mathfrak{X}^{3}\otimes\mathbf{Q}$. Since the triple
$\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}$ is residually irreducible, the
Künneth formula implies that

$\mathrm{H}^{3}(\mathfrak{X}^{3}\otimes{{\mathbf{Q}}^{\mathrm{ac}}},\mathcal{O}(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}=(\otimes^{3}_{i=1}\mathrm{H}^{1}(X_{\mathbf{Q}^{\mathrm{ac}}},\mathcal{O}(1))_{\mathfrak{m}^{[p]}_{i}})(-1)$

and
$\mathrm{H}^{*}(\mathfrak{X}^{3}\otimes{{\mathbf{Q}}^{\mathrm{ac}}},\mathcal{O}(2))_{\mathfrak{m}_{\underline{\mathbf{f}}}^{[p]}}$
vanishes identically for $*\neq 3$. Thus the cycle class map and the
Hochschild-Serre spectral sequence induces the following Abel-Jacobi map

$\mathrm{AJ}_{\underline{\mathbf{f}}}:\mathrm{CH}^{2}(\mathfrak{X}^{3}\otimes\mathbf{Q})\rightarrow\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{3}(\mathfrak{X}^{3}\otimes{{\mathbf{Q}}^{\mathrm{ac}}},\mathcal{O}(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}).$

We denote by $\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})$ the Galois module
over $\mathcal{O}_{n}$ given by

$\otimes^{3}_{i=1}\mathrm{H}^{1}(X_{{\mathbf{Q}}^{\mathrm{ac}}},\mathcal{O}(1)){/I^{[p]}_{i,n}}$

The Abel-Jacobi map composed with the canonical map

$\mathrm{H}^{3}(X^{3}\otimes{{\mathbf{Q}}^{\mathrm{ac}}},\mathcal{O}(2))_{\mathfrak{m}_{\underline{\mathbf{f}}}^{[p]}}\rightarrow\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1)$

gives rise to the following mod $\varpi^{n}$-version of the Abel-Jacobi map

$\mathrm{AJ_{\underline{\mathbf{f}},n}}:\mathrm{CH}^{2}(\mathfrak{X}^{3}\otimes\mathbf{Q})\rightarrow\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1)).$

We thus obtain a global cohomology class
$\Theta^{[p]}_{n}\in\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
by applying the map $\mathrm{AJ_{\underline{\mathbf{f}},n}}$ to the cycle
$\theta_{*}[\mathfrak{X}\otimes\mathbf{Q}]\in\mathrm{CH}^{2}(\mathfrak{X}^{3}\otimes\mathbf{Q})$.

On the other hand, let
$\underline{\mathbf{f}}^{B}=(f^{B}_{1},f^{B}_{2},f^{B}_{3})\in
S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 3}$ be the _Jacquet-Langlands transfer_
of $\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ in the space of _quaternionic
modular forms_ $S^{B}_{2}(N^{+},\mathcal{O})^{3}$ as in [BD3, Definition 1.1].
We consider the following period integral

$I(f^{B}_{1},f^{B}_{2},f^{B}_{3})=\sum_{z\in
X^{B}}f^{B}_{1}(z)f^{B}_{2}(z)f^{B}_{3}(z).$

By the main result of [KH] which resolves a conjecture of Jacquet, this period
integral is non-vanishing if $L(f_{1}\otimes f_{2}\otimes f_{3},2)$ is non-
vanishing. Our first goal of this article is to provide an explicit relation
between the cohomology class $\Theta^{[p]}_{n}$ and the period integral
$I(f^{B}_{1},f^{B}_{2},f^{B}_{3})$. To compare these two objects, we pass the
class $\Theta^{[p]}_{n}$ living in the global Galois cohomology group
$\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
to its _singular quotient_
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
at $p$ whose definition is recalled in (2.9). The singular quotient
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
itself has a geometric description and this description is referred to as the
_ramified arithmetic level raising_ in the recent literature [LT], [LTXZZ].
Here the word ramified refers to the fact that the relevant Shimura variety
has bad reduction at the prime $p$. Our first main result is such an
arithmetic level raising theorem for the triple product of Shimura curves.

######  Theorem 2 (Ramified arithemtic level raising ).

Let $p$ be an $n$-admissible prime for the triple
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$. For each $i=1,2,3$, assume that

  1. (1)

the maximal ideal $\mathfrak{m}_{i}$ is residually irreducible;

  2. (2)

each $S^{B}_{2}(N^{+},\mathcal{O})_{\mathfrak{m}_{i}}$ is a free rank $1$
module over $\mathbb{T}_{\mathfrak{m}_{i}}$.

Then the singular quotient
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
is free of rank $3$ over $\mathcal{O}_{n}$ and we have an isomorphism

(1.1)
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))\cong\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O})/I_{i,n}.$

This theorem is proved in Theorem 4.7 and Corollary 4.11. We remark that the
hypothesis $(2)$ on the forms $f_{i}$ is called _$l$ -isolated_ in [BD3]. One
can achieve hypothesis $(2)$ by imposing conditions on the residual Galois
representation $\bar{\rho}_{i}$ of $\rho_{i}$ and then applying the Taylor-
Wiles argument refined by Diamond in [Dia]. For example, one can impose the
following conditions in [CH1, Hypothesis $\mathrm{CR}^{+}$].

######  Assumption 3 ($\mathrm{CR}^{+}$).

For $i=1,2,3$, the residual Galois representation $\bar{\rho}_{i}$ satisfies
the following assumptions

  1. (1)

$\bar{\rho}_{i}$ is absolutely irreducible when restricted to
$G_{\mathbf{Q}(\sqrt{p^{*}})}$ where $p^{*}=(-1)^{\frac{p-1}{2}}p$;

  2. (2)

If $q\mid N^{-}$ and $q\equiv\pm 1\mod l$, then $\bar{\rho}_{i}$ is ramified;

  3. (3)

If $q\mid\mid N^{+}$ and $q\equiv 1\mod l$, then $\bar{\rho}_{i}$ is ramified;

  4. (4)

The Artin conductor $N_{\bar{\rho}_{i}}$ is prime to $N/N_{\bar{\rho}_{i}}$.

We refer the reader to [CH1, Proposition 6.8] for an exposition of the Taylor-
Wiles argument mentioned above to achieve hypothesis $(2)$ in Theorem 2. The
proof of Theorem 2 is geometric in nature by analyzing the semistable model of
$X^{3}$ at an $n$-admissible prime $p$. To calculate the singular quotient, we
use the machineries developed in [Liu2], in particular the so-called
_potential map_ which we give a down-to-earth introduction in the first part
of the article. The right-hand side of the isomorphism (1.1) can be made even
more geometrically in terms of the component groups of the Shimura curves at
$p$. Let $\mathcal{J}$ be the Néron model of the Jacobian
$\mathrm{Jac}(X_{\mathbf{Q}_{p^{2}}})$ of $X_{\mathbf{Q}_{p^{2}}}$ and let
$\Phi$ be the group of connected components of the special fiber of
$\mathcal{J}$, then the isomorphism in (1.1) comes more canonically from the
following isomorphism

(1.2)
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))\cong\oplus^{3}_{j=1}(\otimes^{3}_{i=1}\Phi_{\mathcal{O}}/I^{[p]}_{i,n}).$

Comparing to the pioneering works on arithmetic level raising [BD3], [Liu2],
[LTXZZ], one of the main difference in our setting is that the singular
quotient is of rank $3$ in our settings as opposed to being of rank $1$ in all
the previously mentioned works. This interesting phenomenon suggests some
richer structures hidden in our setting which we do not completely understand
yet. Nevertheless, the above theorem still provides the natural setting for
relating the cohomology class $\Theta^{[p]}_{n}$ to the period integral
$I(f^{B}_{1},f^{B}_{2},f^{B}_{3})=\sum_{z\in
X^{B}}f^{B}_{1}(z)f^{B}_{2}(z)f^{B}_{3}(z).$ In order to state this relation,
we introduce a natural pairing

$\displaystyle(\hphantom{a},\hphantom{a}):\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O})[I_{i,n}]\times\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O})/(I_{i,n})\rightarrow\mathcal{O}_{n}$

in (4.7). Let $\partial_{p}\Theta^{[p]}_{n}$ be the image of
$\Theta^{[p]}_{n}$ in
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$.
For $j=1,2,3$, we denote by

$\partial^{(j)}_{p}\Theta^{[p]}_{n}\in\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O}){/I_{i,n}}$

its projection to the $j$-th copy of

$\oplus^{3}_{j=1}(\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O}){/I_{i,n}}).$

Then our second main result is the following explicit reciprocity formula
which we will call the _first reciprocity law_.

######  Theorem 4.

Let $p$ be an $n$-admissible prime for the triple $\underline{\mathbf{f}}$. We
assume the assumptions in Theorem 2 are satisfied. Let
$\phi_{1}\otimes\phi_{2}\otimes\phi_{3}\in\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O})[I_{i,n}]$.
Then we have the following formula,

(1.3)
$(\partial^{(j)}_{p}\Theta^{[p]}_{n},\phi_{1}\otimes\phi_{2}\otimes\phi_{3})=(p+1)^{3}\sum_{z\in
X^{B}}\phi_{1}(z)\phi_{2}(z)\phi_{3}(z)$

for $j=1,2,3$.

This formula is the analogue of the _first reciprocity law_ in [BD3, Theorem
4.1], the _congruence formulae_ in [Liu1, Theorem 4.11] and [Liu2, Theorem
4.5]. The right-hand side of (1.3) obviously represents the period integral
$I(f^{B}_{1},f^{B}_{2},f^{B}_{3})$ and this formula suggests that the class
$\Theta^{[p]}_{n}$ should be useful to bound the Selmer group. However as we
explained before, the singular quotient is of rank $3$ and the class
$\Theta^{[p]}_{n}$ itself is not enough to bound the Selmer group. Combining
the present work with the companion article [Wang] suggests the following
intuitive picture: the Selmer group of the level raised triple
$\underline{\mathbf{f}}^{[p]}$ skips the possibility of being rank $1$ and
goes directly to rank $3$ from the rank $0$ Selmer group for the original
triple $\underline{\mathbf{f}}$. Therefore it should be reasonable to expect
three global classes that fill up the singular quotient. We invite the reader
to compare this with the paper of Darmon-Rotger [DR2]. They considered the so-
called _unbalanced case_ where the weights of the triple of modular forms are
$(2,1,1)$. The singular quotient at $p$ there is a direct sum of four one-
dimensional spaces. They managed to find $4$ global classes starting from the
primitive diagonal cycle class similar to our $\Theta^{[p]}_{n}$ in this
article using techniques from $p$-adic deformations (Hida families) which fill
up the singular quotient and such that each class only appears in exactly one
of the four one-dimensional spaces. Using these classes, they can bound
certain equivariant part of the Mordell-Weil group of an elliptic curve but
not the Selmer group. This suggests that we should be able modify the class
$\Theta^{[p]}_{n}$ and construct three classes which each sits in exactly one
of the three direct summands of

$\oplus^{3}_{j=1}(\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O}){/I_{i,n}}).$

Moreover these three classes should satisfy a similar reciprocity law as the
one we obtained for $\Theta^{[p]}_{n}$ in Theorem 4. Using these conjectural
classes, we are indeed able to bound the Selmer group for the triple product
motive. Therefore we regard the present work as the first step towards the
rank $0$ case of the Bloch-Kato conjecture for the triple product motive of
modular forms.

###  1.2. The symmetric cube motive

It is natural to consider the degenerate case when $f_{1}=f_{2}=f_{3}=f$ for a
single modular form $f\in S^{\mathrm{new}}_{2}(\Gamma_{0}(N))$. We denote by
$\mathrm{V}_{f}$ the Galois representation attached to $f$. In this case the
triple tensor product representation factors as

$\mathrm{V}_{f}^{\otimes
3}(-1)=\mathrm{Sym}^{3}\mathrm{V}_{f}(-1)\oplus\mathrm{V}_{f}\oplus\mathrm{V}_{f}.$

Correspondingly, we have a factorization of the $L$-function

$L(f\otimes f\otimes f,s)=L(\mathrm{Sym}^{3}f,s)L(f,s-1)^{2}.$

We introduce in this case the notion of _ $(n,1)$-admissible primes_ for $f$.
A prime $p$ is an $(n,1)$-admissible prime for $f$ if

  1. (1)

$p\nmid Nl$;

  2. (2)

$l\nmid p^{2}-1$;

  3. (3)

$\varpi^{n}\mid p+1-a_{p}(f)$.

Comparing to the definition of an $n$-admissible prime for a triple
$(f_{1},f_{2},f_{3})$, we require here that $\epsilon_{p,i}=1$ for all
$i=1,2,3$. We let $\underline{\mathbf{f}}=(f,f,f)$. Let
$\mathrm{M}^{\diamond}_{n}(\underline{\mathbf{f}})(-1)$ be the symmetric cube
component of $\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1)$. Then we find
that the singular quotient
$\mathrm{H}^{1}_{\mathrm{sing}}(\mathbf{Q}_{p},\mathrm{M}^{\diamond}_{n}(\underline{\mathbf{f}})(-1))$
at $p$ is in fact of rank $1$. Therefore the Euler system argument can be
applied here. However, the factorization of the $L$-function and the Galois
representation suggest that the results concerning the rank $0$ case of the
Bloch-Kato conjecture here will be conditional on the non-vanishing of
$L(f,1)$. Therefore we have the following result.

######  Theorem 5.

Suppose that the modular form $f$ satisfies the following assumptions:

  1. (1)

the maximal ideals $\mathfrak{m}_{f}$ are all residually irreducible;

  2. (2)

the $\mathbb{T}_{\mathfrak{m}_{f}}$-module
$S^{B}_{2}(N^{+},\mathcal{O})_{\mathfrak{m}_{f}}$ is free of rank $1$;

  3. (3)

the residual Galois representation $\bar{\rho}_{f}$ is surjective;

  4. (4)

the value $L(f,1)$ is non-vanishing.

Assume that value $L(\mathrm{Sym}^{3}f,2)$ is non-zero. Then the Bloch-Kato
Selmer group for $\mathrm{Sym}^{3}\mathrm{V}_{f}(-1)$

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Sym}^{3}\mathrm{V}_{f}(-1))=0.$

This theorem is proved in the last section of this article and is based on a
familiar Euler system argument, see [Gro], [BD3], [Liu1] and [Liu2] for a few
examples. A similar result in the rank $1$ case for the Bloch-Kato Selmer
group of $\mathrm{Sym}^{3}\mathrm{V}_{f}(-1)$ will be proved in [Wang].

###  1.3. Bipartite Euler system and other applications

We close this introduction by discussing some applications and questions the
present work points to. In the companion work [Wang], we prove the _second
reciprocity law_ in our setting which should have applications to the rank $1$
case of the Bloch-Kato conjecture for the triple product motive of modular
forms. It also implies that the diagonal cycle classes $\Theta^{[p]}_{n}$ and
the period integrals $I(f^{B}_{1},f^{B}_{2},f^{B}_{3})$ form an analogue of
the Bipartite Euler system in the sense of Howard [How]. Note that the
definition there requires the only Frobenius eigenvalues on the local Galois
representation at an $n$-admissible prime $p$ are $p$ and $1$ and therefore
the singular quotient is of rank $1$. This is obviously different from our
setting. Therefore what we have for the triple product motive is only an
analogue of his Bipartite Euler system. On the other hand, the classes
$\Theta^{\diamond[p]}_{n}$ we produced for the symmetric cube motive of
modular forms do form a Bipartite Euler system in his sense.

Secondly, by replacing the $l$-adic étale cohomology and $l$-adic Abel-Jacobi
map in the present work with crystalline cohomology and the $p$-adic Abel-
Jacobi map one should be able to prove certain type of $p$-adic Gross-Zagier
formulas for certain triple product $p$-adic $L$-functions. See [Hsi] for the
relevant construction of $p$-adic $L$-functions and see [BD2] for the type of
$p$-adic Gross-Zagier formula for Heegner points on Shimura curves.

###  1.4. Notations and conventions

We will use common notations and conventions in algebraic number theory and
algebraic geometry. The cohomologies of schemes appear in this article will be
understood as computed over the étale sites.

For a field $K$, we denote by $K^{\mathrm{ac}}$ a separable closure of $K$ and
put $G_{K}:=\mathrm{Gal}(K^{ac}/K)$ the Galois group of $K$. We let
$\mathbf{A}$ be the ring of adèles over $\mathbf{Q}$ and $\mathbf{A}^{\infty}$
be the subring of finite adèles. For a prime $p$, $\mathbf{A}^{\infty,(p)}$ is
the prime-to-$p$ part of $\mathbf{A}^{\infty}$.

Let $K$ be a local field with ring of integers $\mathcal{O}_{K}$ and residue
field $k$. We let $I_{K}$ be the inertia subgroup of $G_{K}$. Suppose
$\mathrm{M}$ is a $G_{K}$-module. Then the finite part
$\mathrm{H}^{1}_{\mathrm{fin}}(K,\mathrm{M})$ of
$\mathrm{H}^{1}(K,\mathrm{M})$ is defined to be $H^{1}(k,\mathrm{M}^{I_{K}})$
and the singular part $\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$ of
$\mathrm{H}^{1}(K,\mathrm{M})$ is defined to be the quotient of
$\mathrm{H}^{1}(K,\mathrm{M})$ by the image of
$\mathrm{H}^{1}_{\mathrm{fin}}(K,\mathrm{M})$.

Two quaternion algebras will be used in this article: the definite quaternion
algebra $B$ with discriminant $N^{-}$ and the indefinite quaternion algebra
$B^{\prime}$ with discriminant $pN^{-}$.

### Acknowledgements

We would like to thank Henri Darmon and Liang Xiao for valuable discussions
about this work. We would like to thank Minglun Hsieh for introducing the work
of Bertonili-Darmon to the author in graduate school and for his interest to
the present work. We also acknowledge the deep debt this article owes to the
pioneering works of Bertonili-Darmon and Yifeng Liu. We are truly grateful to
Pengfei Guan for his generous support in the unsettling period of time during
which this article was written.

##  2\. Review of weight spectral sequence

###  2.1. Nearby cycles on semi-stable schemes

Let $K$ be a henselian discrete valuation field with residue field $k$ of
characteristic $p$ and valuation ring $\mathcal{O}_{K}$. We fix a uniformizer
$\pi$ of $\mathcal{O}_{K}$. We set $S=\mathrm{Spec}(\mathcal{O}_{K})$,
$s=\mathrm{Spec}(k)$ and $\eta=\mathrm{Spec}(K)$. Let $K^{\mathrm{ac}}$ be a
separable closure of $K$ and $K_{\mathrm{ur}}$ the maximal unramified
extension of $K$ in $K^{\mathrm{ac}}$. We denote by $k^{\mathrm{ac}}$ the
residue field of $K_{\mathrm{ur}}$. Let
$I_{K}=\mathrm{Gal}(K^{\mathrm{ac}}/K_{\mathrm{ur}})\subset
G_{K}=\mathrm{Gal}(K^{\mathrm{ac}}/K)$ be the inertia group. Let $l$ be a
prime different from $p$. We set $t_{l}:I_{K}\rightarrow\Lambda(1)$ to be the
canonical surjection given by

$\sigma\mapsto(\sigma(\pi^{1/l^{m}})/\pi^{1/l^{m}})_{m}$

for every $\sigma\in I_{K}$.

Let $\mathfrak{X}$ be a _strict semi-stable scheme_ over $S$ purely of
relative dimension $n$ which we also assume to be proper. This means that
$\mathfrak{X}$ is locally of finite presentation and Zariski locally étale
over

$\mathrm{Spec}(\mathcal{O}_{K}[X_{1},\cdots,X_{n}]/(X_{1}\cdots X_{r}-\pi))$

for some integer $1\leq r\leq n$. We let $X_{k}$ be the special fiber of
$\mathfrak{X}$ and $X_{k^{\mathrm{ac}}}$ be its base-change to
$k^{\mathrm{ac}}$. Let $X=\mathfrak{X}_{\eta}$ be the generic fiber of
$\mathfrak{X}$ and $X_{K_{\mathrm{ur}}}$ be its base-change to
$K_{\mathrm{ur}}$. We have the following natural maps

$\displaystyle i:X_{k}\rightarrow\mathfrak{X},$ $\displaystyle
j:X\rightarrow\mathfrak{X},$
$\displaystyle\bar{i}:X_{k^{\mathrm{ac}}}\rightarrow\mathfrak{X}_{\mathcal{O}_{K_{\mathrm{ur}}}},$
$\displaystyle\bar{j}:X_{K_{\mathrm{ur}}}\rightarrow\mathfrak{X}_{\mathcal{O}_{K_{\mathrm{ur}}}}.$

Throughout this paper, we fix a prime $l\geq 5$. Let $\Lambda$ be
$\mathbf{Z}/l^{v},\mathbf{Z}_{l}$ or a finite extension of $\mathbf{Z}_{l}$.
We define the _Nearby cycle sheaf_ by

$R^{q}\Psi(\Lambda)=\bar{i}^{*}R^{q}\bar{j}_{*}\Lambda$

and the _Nearby cycle complex_ by

$R\Psi(\Lambda)=\bar{i}^{*}R\bar{j}_{*}\Lambda.$

We regard the latter as an object in the derived category
$D^{+}(X_{k^{\mathrm{ac}}},\Lambda[I_{K}])$ of sheaves of $\Lambda$-modules
with continuous $I_{K}$-actions. By proper base change, we always have

$\mathrm{H}^{*}(X_{k^{\mathrm{ac}}},R\Psi(\Lambda))=\mathrm{H}^{*}(X_{K^{\mathrm{ac}}},\Lambda).$

Let $D_{1},\cdots,D_{m}$ be the set of irreducible components of $X_{k}$. For
each index set $I\subset\\{1,\cdots,m\\}$ of cardinality $p$, we set
$X_{I,k}=\cap_{i\in I}D_{i}$. This is a smooth scheme of dimension $n-p$. For
$1\leq p\leq m-1$, let

(2.1)
$X^{(p)}_{k}=\bigsqcup_{I\subset\\{1,\cdots,m\\},\mathrm{Card}(I)=p+1}X_{I,k}$

and

(2.2) $a_{p}:X^{(p)}_{k}\rightarrow X_{k}$

be the projection, we have $a_{p*}\Lambda=\wedge^{p+1}a_{0*}\Lambda$. Consider
the Kummer exact sequence in the case $\Lambda=\mathbf{Z}/l^{v}$

(2.3)
$0\rightarrow\Lambda(1)\rightarrow\mathcal{O}^{*}_{X}\rightarrow\mathcal{O}^{*}_{X}\rightarrow
0.$

Let $\partial(\pi)\in i^{*}R^{1}j_{*}\Lambda(1)$ be the image of $\pi$ under
the coboundary map by applying $i^{*}Rj_{*}$ to the above exact sequence. We
let $\theta:\Lambda_{X_{k}}\rightarrow i^{*}R^{1}j_{*}\Lambda(1)$ be the map
sending $1$ to $\partial(\pi)$ and $\delta:\Lambda_{X_{k}}\rightarrow
a_{0*}\Lambda$ be the canonical map. Then we have the following results
regarding to the resolution of the Nearby cycle sheaf.

######  Proposition 2.1.

We have the following.

  1. (1)

There is an isomorphism of exact sequences

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  2. (2)

For $p\geq 0$, we have an exact sequence

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  3. (3)

For $p\geq 0$, we have a quasi-isomorphism of complexes

$R^{p}\Psi(\Lambda)[-p]\xrightarrow{\sim}[a_{p*}\Lambda(-p)\xrightarrow{\delta\wedge}\cdots\xrightarrow{\delta\wedge}a_{n*}\Lambda(-p)\rightarrow
0]$

###### Proof.

The first two statements are taken from [Sai, Corollary 1.3] and the last one
is an immediate consequence of the first two. ∎

###  2.2. Monodromy filtration and spectral sequence

Suppose $A$ is an object in an abelian category and $N$ is a nilpotent
endomorphism on $A$. We define two filtrations on $A$.

  * •

We define the kernel filtration $F_{\bullet}$ by putting
$F_{p}A=\ker(N^{p+1}:A\rightarrow A)$ for $p\geq 0$.

  * •

We define the image filtration $G^{\bullet}$ by putting
$G^{q}A=\mathrm{im}(N^{q}:A\rightarrow A)$ for $q>0$.

Using these two filtrations,

  * •

we define the convolution filtration $M_{\bullet}A$ by
$M_{r}A=\oplus_{p-q=r}F_{p}A\cap G^{q}A$.

To calculate the graded piece of the convolution filtration, we define the
induced $G^{\bullet}$-filtration on the graded piece of $F_{\bullet}$ by
$G^{q}Gr^{F}_{p}A=\mathrm{im}(G^{q}A\cap F_{p}A\rightarrow Gr^{F}_{p}A)$. Then
we see that the $r$-th graded piece of the convolution filtration
$M_{\bullet}A$ is given by

$Gr^{M}_{r}A\cong\bigoplus_{p-q=r}Gr^{q}_{G}Gr^{F}_{p}A.$

The convolution filtration in this case is known as the _Monodromy filtration_
and it is characterized by

  1. (1)

$M_{n}A=0$ and $M_{-n-1}A=0$.

  2. (2)

$N:A\rightarrow A$ sends $M_{r}A$ into $M_{r-2}A$ for $r\in\mathbf{Z}$.

  3. (3)

$N^{r}:Gr^{M}_{r}A\rightarrow Gr^{M}_{-r}A$ is an isomoprhism.

Let $A=R\Psi(\Lambda)$. Let $T$ be an element in $I_{K}$ such that $t_{l}(T)$
is a generator of $\Lambda(1)$ then $T$ induces a nilpotent operator $T-1$ on
$R\Psi(\Lambda)$. Let $N=(T-1)\otimes\breve{T}$ where
$\breve{T}\in\Lambda(-1)$ be the dual of $t_{l}(T)$. Then with respect to $N$,
we have the following characterization of the Monodromy filtration on
$R\Psi(\Lambda)$

  1. (1)

$M_{n}R\Psi(\Lambda)=0$ and $M_{-n-1}R\Psi(\Lambda)=0$.

  2. (2)

$N:R\Psi(\Lambda)(1)\rightarrow R\Psi(\Lambda)$ sends $M_{r}R\Psi(\Lambda)(1)$
into $M_{r-2}R\Psi(\Lambda)$ for $r\in\mathbf{Z}$.

  3. (3)

$N^{r}:Gr^{M}_{r}R\Psi(\Lambda)(r)\rightarrow Gr^{M}_{-r}R\Psi(\Lambda)$ is an
isomoprhism.

We can use Proposition 2.1 to calculate the Monodromy filtration on
$R\Psi(\Lambda)$. In addition to Proposition 2.1, we need the following
results in [Sai, Lemma 2.5, Corollary 2.6].

  1. (1)

The kernel filtration $F_{p}R\Psi(\Lambda)$ is given by the canonical
truncated filtration $\tau_{\leq p}R\Psi(\Lambda)$ and therefore

(2.4) $Gr^{F}_{p}R\Psi(\Lambda)\cong
R^{p}\Psi(\Lambda)[-p]\cong[a_{p*}\Lambda(-p)\xrightarrow{\delta\wedge}\cdots\xrightarrow{\delta\wedge}a_{n*}\Lambda(-p)\rightarrow
0].$

  2. (2)

The image filtration $G^{q}Gr^{F}_{p}R\Psi(\Lambda)$ on
$Gr^{F}_{p}R\Psi(\Lambda)$ is given by the truncation in (2.4) to $p+q$
position

$G^{q}Gr^{F}_{p}R\Psi(\Lambda)=[a_{p+q*}\Lambda(-p)\xrightarrow{\delta\wedge}\cdots\xrightarrow{\delta\wedge}a_{n*}\Lambda(-p)\rightarrow
0].$

  3. (3)

Combining the above two results, we have

$Gr^{q}_{G}Gr^{F}_{p}R\Psi(\Lambda)=a_{p+q*}\Lambda[-p-q](-p)$

and therefore we arrive at the following equation

$Gr^{M}_{r}R\Psi(\Lambda)=\bigoplus_{p-q=r}a_{p+q*}\Lambda[-p-q](-p).$

The Monodromy filtration induces the _weight spectral sequence_

(2.5)
$\mathrm{E}^{p,q}_{1}=\mathrm{H}^{p+q}(X_{k^{\mathrm{ac}}},Gr^{M}_{-p}R\Psi(\Lambda))\Rightarrow\mathrm{H}^{p+q}(X_{k^{\mathrm{ac}}},R\Psi(\Lambda))=\mathrm{H}^{p+q}(X_{K^{\mathrm{ac}}},\Lambda).$

The $\mathrm{E}_{1}$-term of this spectral sequence can be made explicit by

$\displaystyle\mathrm{H}^{p+q}(X_{k^{\mathrm{ac}}},Gr^{M}_{-p}R\Psi(\Lambda))$
$\displaystyle=\bigoplus_{i-j=-p,i\geq 0,j\geq
0}\mathrm{H}^{p+q-(i+j)}(X^{(i+j)}_{k^{\mathrm{ac}}},\Lambda(-i))$
$\displaystyle=\bigoplus_{i\geq\mathrm{max}(0,-p)}\mathrm{H}^{q-2i}(X^{(p+2i)}_{k^{\mathrm{ac}}},\Lambda(-i)).$

This spectral sequence is first found by Rapoport-Zink in [RZ] and thus is
also known as the Rapoport-Zink spectral sequence.

###  2.3. Examples in dimension $1$ and $3$

We will make the weight spectral sequence explicit in dimension $1$ and
dimension $3$ which are the only cases that will be used in the computations
later. The following convention will used throughout this article: we will
write $\mathrm{H}^{*}(a_{p*}\Lambda)$ instead of
$\mathrm{H}^{*}(X_{k^{\mathrm{ac}}},a_{p*}\Lambda)=\mathrm{H}^{*}(X^{(p)}_{k^{\mathrm{ac}}},\Lambda)$.

####  2.3.1. One dimensional case

: Let $\mathfrak{X}$ be a relative curve over
$\mathrm{Spec}(\mathcal{O}_{K})$. Then we immediately calculate that

(2.6) $\displaystyle Gr^{M}_{-1}R\Psi(\Lambda)=a_{1*}\Lambda[-1],$
$\displaystyle Gr^{M}_{0}R\Psi(\Lambda)=a_{0*}\Lambda,$ $\displaystyle
Gr^{M}_{1}R\Psi(\Lambda)=a_{1*}\Lambda[-1](-1).$

The $\mathrm{E}_{1}$-page of the weight spectral sequence is given by

${2}$${\mathrm{H}^{0}(a_{1*}\Lambda)(-1)}$${\mathrm{H}^{2}(a_{0*}\Lambda)}$${1}$${\mathrm{H}^{1}(a_{0*}\Lambda)}$${0}$${\mathrm{H}^{0}(a_{0*}\Lambda)}$${\mathrm{H}^{0}(a_{1*}\Lambda)}$
${-1}$${0}$${1}$

and it clearly degenerates at the $\mathrm{E}_{2}$-page. We therefore have the
Monodromy filtration

$0\subset^{\mathrm{E}^{1,0}_{2}}M_{1}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)\subset^{\mathrm{E}^{0,1}_{2}}M_{0}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)\subset^{\mathrm{E}^{-1,2}_{2}}M_{-1}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)$

with graded pieces given by

$\displaystyle
Gr^{M}_{-1}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)=\ker[\mathrm{H}^{0}(a_{1*}\Lambda(-1))\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda)]$
$\displaystyle
Gr^{M}_{0}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)=\mathrm{H}^{1}(a_{0*}\Lambda);$
$\displaystyle
Gr^{M}_{1}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)=\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)];$

where $\tau$ is the _Gysin morphism_ and $\rho$ is the _restriction morphism_.
Note that we have the following commutative diagram

(2.7)
${\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda(1))}$${\ker[\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda)(1)]}$${\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)}$${\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)].}$$\scriptstyle{N}$$\scriptstyle{N}$

In this case, we recover the _Picard-Lefschetz formula_ if we identify
$\mathrm{H}^{0}(a_{1*}\Lambda)$ with the _vanishing cycles_
$\oplus_{x}R\Phi(\Lambda)_{x}$ on $X_{k^{\mathrm{ac}}}$ where $x$ runs through
the set of singular points $X^{(1)}_{k}$. We refer the readers to [Ill,
Example 2.4.6] for the details about this case.

####  2.3.2. Three dimensional case

Let $\mathfrak{X}$ be a relative threefold over
$\mathrm{Spec}(\mathcal{O}_{K})$. We can also easily list the graded pieces of
the Monodromy filtration on $R\Psi(\Lambda)$

$\displaystyle Gr^{M}_{-3}R\Psi(\Lambda)=a_{3*}\Lambda[-3],$ $\displaystyle
Gr^{M}_{-2}R\Psi(\Lambda)=a_{2*}\Lambda[-2],$ $\displaystyle
Gr^{M}_{-1}R\Psi(\Lambda)=a_{1*}\Lambda[-1]\oplus a_{3*}\Lambda[-3](-1),$
$\displaystyle Gr^{M}_{0}R\Psi(\Lambda)=a_{0*}\Lambda\oplus
a_{2*}\Lambda[-1](-1),$ $\displaystyle
Gr^{M}_{1}R\Psi(\Lambda)=a_{1*}\Lambda[-1](-1)\oplus a_{3*}\Lambda[-3](-2),$
$\displaystyle Gr^{M}_{2}R\Psi(\Lambda)=a_{2*}\Lambda[-2](-2),$ $\displaystyle
Gr^{M}_{3}R\Psi(\Lambda)=a_{3*}\Lambda[-3](-3).$

The $\mathrm{E}_{1}$-page of the weight spectral sequence is given below

(2.8)
${6}$${\mathrm{H}^{0}(a_{3*}\Lambda)(-3)}$${\mathrm{H}^{2}(a_{2*}\Lambda)(-2)}$${\mathrm{H}^{4}(a_{1*}\Lambda)(-1)}$${\mathrm{H}^{6}(a_{0*}\Lambda)}$${5}$${\mathrm{H}^{1}(a_{2*}\Lambda)(-2)}$${\mathrm{H}^{3}(a_{1*}\Lambda)(-1)}$${\mathrm{H}^{5}(a_{0*}\Lambda)}$${4}$${\mathrm{H}^{0}(a_{2*}\Lambda)(-2)}$${\mathrm{H}^{2}(a_{1*}\Lambda)(-1)\oplus\mathrm{H}^{0}(a_{3*}\Lambda)(-2)}$${\mathrm{H}^{4}(a_{0*}\Lambda)\oplus\mathrm{H}^{2}(a_{2*}\Lambda)(-1)}$${\mathrm{H}^{4}(a_{1*}\Lambda)}$${3}$${\mathrm{H}^{1}(a_{1*}\Lambda)(-1)}$${\mathrm{H}^{3}(a_{0*}\Lambda)\oplus\mathrm{H}^{1}(a_{2*}\Lambda)(-1)}$${\mathrm{H}^{3}(a_{1*}\Lambda)}$${2}$${\mathrm{H}^{0}(a_{1*}\Lambda)(-1)}$${\mathrm{H}^{2}(a_{0*}\Lambda)\oplus\mathrm{H}^{0}(a_{2*}\Lambda)(-1)}$${\mathrm{H}^{2}(a_{1*}\Lambda)\oplus\mathrm{H}^{0}(a_{3*}\Lambda)(-1)}$${\mathrm{H}^{2}(a_{2*}\Lambda)}$${1}$${\mathrm{H}^{1}(a_{0*}\Lambda)}$${\mathrm{H}^{1}(a_{1*}\Lambda)}$${\mathrm{H}^{1}(a_{2*}\Lambda)}$${0}$${\mathrm{H}^{0}(a_{0*}\Lambda)}$${\mathrm{H}^{0}(a_{1*}\Lambda)}$${\mathrm{H}^{0}(a_{2*}\Lambda)}$${\mathrm{H}^{3}(a_{3*}\Lambda)}$
${-3}$${-2}$${-1}$${0}$${1}$${2}$${3}$

and it does not necessarily degenerates at $\mathrm{E}_{2}$.

###  2.4. Potential map and Galois cohomology

Let $\mathrm{M}$ be a $G_{K}$-module over $\Lambda$, then we have the
following exact sequence of Galois cohomology groups

(2.9)
$0\rightarrow\mathrm{H}^{1}_{\mathrm{fin}}(K,\mathrm{M})\rightarrow\mathrm{H}^{1}(K,\mathrm{M})\xrightarrow{\partial_{p}}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})\rightarrow
0$

where

(2.10)
$\mathrm{H}^{1}_{\mathrm{fin}}(K,\mathrm{M})=\mathrm{H}^{1}(k,\mathrm{M}^{I_{K}})$

is called the _unramified_ or _finite_ part of the cohomology group
$\mathrm{H}^{1}(K,\mathrm{M})$ and
$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$ defined as the quotient of
$\mathrm{H}^{1}(K,\mathrm{M})$ by its finite part is called the _singular
quotient_ of $\mathrm{H}^{1}(K,\mathrm{M})$. The natural quotient map
$\mathrm{H}^{1}(K,\mathrm{M})\xrightarrow{\partial_{p}}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$
will be referred to as the _singular quotient map_. Let
$x\in\mathrm{H}^{1}(K,\mathrm{M})$, we call the element
$\partial_{p}(x)\in\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$ the _singular
residue_ of $x$. Let
$\mathrm{M}=\mathrm{H}^{n}(X_{K^{\mathrm{ac}}},\Lambda(r))$ be $r$-th twist of
the middle degree cohomology of $X_{K^{\mathrm{ac}}}$. We review in this
section how to calculate the singular quotient using the weight spectral
sequence recalled above. In [Liu2], the author postulated certain situations
where the singular quotient can be calculated by a formula similar to the
Picard-Lefschetz formula. What will be presented below is the same as his
construction of the _potential map_. We will not recall his general machinery
but only the case of a curve or a threefold. We need the following elementary
lemma.

######  Lemma 2.2.

Let $\mathrm{M}=\mathrm{H}^{n}(X_{K^{\mathrm{ac}}},\Lambda(r))$, then we have

(2.11)
$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})\cong(\frac{\mathrm{M}(-1)}{N\mathrm{M}})^{G_{k}}$

###### Proof.

This is well-known. The details can be found in [Liu2, Lemma 2.6] for example.
∎

####  2.4.1. One dimensional case

In this case, let $\mathrm{M}=\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda(1))$,
we can use the Picard-Lefschetz formula. Recall the diagram

${\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda(1))}$${\ker[\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))]}$${\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda)}$${\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)]}$$\scriptstyle{N}$$\scriptstyle{N}$

in (2.7). Then we have

(2.12) $\displaystyle\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$
$\displaystyle\cong(\frac{\mathrm{M}(-1)}{N\mathrm{M}})^{G_{k}}\cong(\frac{\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)]}{N\ker[\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))]})^{G_{k}}$

Composing $\tau$ and $\rho$, we have

(2.13)
$\displaystyle(\frac{\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)]}{N\ker[\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1)))]})^{G_{k}}\cong\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))]^{G_{k}}.$

####  2.4.2. Three dimensional case

In this case, in light of later applications we will only consider
$\mathrm{M}=\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))$. We put the
following assumptions on $\mathrm{M}$.

######  Assumption 2.3.

We assume the weight spectral sequence satisfy the following assumptions.

  1. (1)

The weight spectral sequence which converges to
$\mathrm{M}=\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))$ degenerates in its
$\mathrm{E}_{2}$-page. Thus it induces the following filtration

$\displaystyle
0\subset^{\mathrm{E}^{3,0}_{2}(2)}M_{3}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))\subset^{\mathrm{E}^{2,1}_{2}(2)}M_{2}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))$
$\displaystyle\subset^{\mathrm{E}^{1,2}_{2}(2)}M_{1}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))\subset^{\mathrm{E}^{0,3}_{2}(2)}M_{0}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))$
$\displaystyle\subset^{\mathrm{E}^{-1,4}_{2}(2)}M_{-1}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))\subset^{\mathrm{E}^{-2,5}_{2}(2)}M_{-2}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda(2))$
$\displaystyle\subset^{\mathrm{E}^{-3,6}_{2}(2)}M_{-3}\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))=\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2)).$

  2. (2)

The term $\mathrm{E}^{i,3-i}_{2}(2)$ in the above filtration which has a non-
trivial subquotient that is $G_{k}$-invariant is only
$\mathrm{E}^{-1,4}_{2}(2)$.

  3. (3)

The term $\mathrm{E}^{i,3-i}_{2}(2)(-1)=\mathrm{E}^{i,3-i}_{2}(1)$ in the
above filtration which has a non-trivial subquotient which is
$G_{k}$-invariant is only $\mathrm{E}^{1,2}_{2}(1)$.

Under the assumptions above, we have the following commutative diagram

${\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))^{G_{k}}}$${(\mathrm{E}^{-1,4}_{2}(2))^{G_{k}}}$${\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(1))^{G_{k}}}$${(\mathrm{E}^{1,2}_{2}(1))^{G_{k}}}$$\scriptstyle{N}$$\scriptstyle{N}$

and from we obtain

$\displaystyle\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})$
$\displaystyle\cong(\frac{\mathrm{M}(-1)}{N\mathrm{M}})^{G_{k}}\cong(\frac{\mathrm{E}^{1,2}_{2}(1)}{N\mathrm{E}^{-1,4}_{2}(2)})^{G_{k}}$

We have

$\mathrm{E}^{-1,4}_{2}(2)=\frac{\ker[\mathrm{H}^{2}(a_{1*}\Lambda(1))\oplus\mathrm{H}^{0}(a_{3*}\Lambda)\xrightarrow{(\tau+\rho,\tau)}\mathrm{H}^{4}(a_{0*}\Lambda(2))\oplus\mathrm{H}^{2}(a_{2*}\Lambda(1))]}{\mathrm{im}[\mathrm{H}^{0}(a_{2*}\Lambda)\xrightarrow{(\tau,\rho)}\mathrm{H}^{2}(a_{1*}\Lambda(1))\oplus\mathrm{H}^{0}(a_{3*}\Lambda)]}$

and

$\mathrm{E}^{1,2}_{2}(1)=\frac{\ker[\mathrm{H}^{2}(a_{1*}\Lambda(1))\oplus\mathrm{H}^{0}(a_{3*}\Lambda)\xrightarrow{(\rho,\tau)}\mathrm{H}^{2}(a_{2*}\Lambda(1))]}{\mathrm{im}[\mathrm{H}^{2}(a_{0*}\Lambda(1))\oplus\mathrm{H}^{0}(a_{2*}\Lambda)\xrightarrow{(\rho,\tau+\rho)}\mathrm{H}^{2}(a_{1*}\Lambda(1))\oplus\mathrm{H}^{0}(a_{3*}\Lambda)]}.$

Then we find that

(2.14)
$\displaystyle(\frac{\mathrm{E}^{1,2}_{2}(1)}{N\mathrm{E}^{-1,4}_{2}(2)})^{G_{k}}=(\frac{\mathrm{im}[\mathrm{H}^{2}(a_{1*}\Lambda(1))\xrightarrow{\tau}\mathrm{H}^{4}(a_{0*}\Lambda(2))]}{\tau\mathrm{im}[\mathrm{H}^{2}(a_{0*}\Lambda(1))\xrightarrow{\rho}\mathrm{H}^{2}(a_{1*}\Lambda(1))]})^{G_{k}}.$

We will denote by

$A^{2}(X_{k},\Lambda)^{0}=\mathrm{im}[\mathrm{H}^{2}(a_{1*}\Lambda(1))\xrightarrow{\tau}\mathrm{H}^{4}(a_{0*}\Lambda(2))]^{G_{k}}$

which appear on the numerator of (2.14) and by

$A_{2}(X_{k},\Lambda)^{0}=\mathrm{im}[\mathrm{H}^{2}(a_{0*}\Lambda(1))\xrightarrow{\rho}\mathrm{H}^{2}(a_{1*}\Lambda(1))]^{G_{k}}$

which appear on the denominator of (2.14). Let

$A^{2}(X_{k},\Lambda)^{0}\xrightarrow{\eta}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2)))$

be the natural map and we have the following exact sequence

(2.15)
$A_{2}(X_{k},\Lambda)^{0}\xrightarrow{\nabla}A^{2}(X_{k},\Lambda)^{0}\xrightarrow{\eta}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2)))\rightarrow
0.$

the natural map $\nabla$ induced by $\tau$ is called the potential map, see
[Liu2, Definition 2.5].

We will now explain how this map is related to the image of the Abel-Jacobi
map. Consider the cycle class map

$\mathrm{cl}:\mathrm{CH}^{2}(X_{K},\Lambda)\rightarrow\mathrm{H}^{4}(X_{K},\Lambda(2))$

where $\mathrm{CH}^{2}(X_{K},\Lambda)$ is the Chow group of $X_{K}$ with
coefficient in $\Lambda$. Let $\mathrm{CH}^{2}(X_{K},\Lambda)_{0}$ be the
kernel of the composite map

$\mathrm{CH}^{2}(X_{K},\Lambda)\xrightarrow{\mathrm{cl}}\mathrm{H}^{4}(X_{K},\Lambda(2))\rightarrow\mathrm{H}^{4}(X_{K^{\mathrm{ac}}},\Lambda(2)).$

Then the cycle class map induces the following _Abel-Jacobi map_

$\mathrm{AJ}_{\Lambda}:\mathrm{CH}^{2}(X_{K},\Lambda)_{0}\rightarrow\mathrm{H}^{1}(K,\mathrm{H}^{3}(X_{K^{\mathrm{ac}}},\Lambda(2))).$

Suppose that $z\in\mathrm{CH}^{2}(X_{K},\Lambda)_{0}$. Let $\mathcal{Z}$ be
the Zariski closure of $z$ in $\mathfrak{X}$ and $z^{*}$ be the cycle of
codimension $2$ in $\mathfrak{X}$ supported on $\mathcal{Z}$ and whose
restriction to $X_{K}$ is $z$. Let
$\tilde{z}\in\mathrm{H}^{4}(a_{0*}\Lambda(2))^{G_{k}}=\mathrm{H}^{4}(X^{(0)}_{k^{\mathrm{ac}}},\Lambda(2))^{G_{k}}$
be the image of $z^{*}$ under the following composite maps

$\displaystyle\mathrm{H}^{4}_{\mathcal{Z}}(\mathfrak{X},\Lambda(2))\rightarrow\mathrm{H}^{4}(\mathfrak{X},\Lambda(2))\rightarrow\mathrm{H}^{4}(X_{k},\Lambda(2))\rightarrow\mathrm{H}^{4}(X_{k^{\mathrm{ac}}},\Lambda(2))^{G_{k}}\rightarrow\mathrm{H}^{4}(X^{(0)}_{k^{\mathrm{ac}}},\Lambda(2))^{G_{k}}.$

We have the following results of Liu [Liu2].

######  Proposition 2.4.

The element $\tilde{z}$ belongs to $A^{2}(X_{k},\Lambda)^{0}$. Moreover we
have

$\partial_{p}\mathrm{AJ}_{\Lambda}(z)=\eta(\tilde{z})$

in
$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{3}(X_{k^{\mathrm{ac}}},\Lambda(2)))$.

###### Proof.

The first statement follows easily from [Liu2, Lemma 2.17] and the second is a
special case of [Liu2, Theorem 2.18]. ∎

##  3\. Arithmetic level raising for Shimura curves

###  3.1. Shimura curves and Shimura sets

Let $N$ be a positive integer with a factorization $N=N^{+}N^{-}$ with $N^{+}$
and $N^{-}$ coprime to each other. We assume that $N^{-}$ is square-free and
is a product of odd number of primes. Let $B$ be the definite quaternion
algebra over $\mathbf{Q}$ with discriminant $N^{-}$ and let $B^{\prime}$ be
the indefinite quaternion algebra over $\mathbf{Q}$ with discriminant
$pN^{-}$. Let $\mathcal{O}_{B^{\prime}}$ be a maximal order of $B^{\prime}$
and let $\mathcal{O}_{B^{\prime},N^{+}}$ be the Eichler order of level $N^{+}$
in $\mathcal{O}_{B^{\prime}}$. We let $G^{\prime}$ be the algebraic group over
$\mathbf{Q}$ given by $B^{\prime\times}$ and let $K^{\prime}_{N^{+}}$ be the
open compact of $G^{\prime}(\mathbf{A}^{\infty})$ defined by
$\hat{\mathcal{O}}^{\times}_{B^{\prime},N^{+}}$. We let $G$ be the algebraic
group over $\mathbf{Q}$ given by $B^{\times}$. Note that we have an
isomorphism
$G^{\prime}(\mathbf{A}^{\infty,(p)})\xrightarrow{\sim}G(\mathbf{A}^{\infty,(p)})$
and via this isomorphism we will view $K^{\prime p}$ as an open compact
subgroup of $G(\mathbf{A}^{\infty,(p)})$ for any open compact subgroup
$K^{\prime}$ of $G^{\prime}(\mathbf{A}^{\infty})$.

Let $X=X^{B^{\prime}}_{N^{+},pN^{-}}$ be the Shimura curve over $\mathbf{Q}$
with level $K^{\prime}=K^{\prime}_{N^{+}}$. The complex points of this curve
is given by the following double coset

$X(\mathbf{C})=G^{\prime}(\mathbf{Q})\backslash\mathcal{H}^{\pm}\times
G^{\prime}(\mathbf{A}^{\infty})/K^{\prime}.$

There is a natural model $\mathfrak{X}$ over $\mathbf{Z}_{(p)}$ of $X$. Recall
it represents the following moduli problem. Let $S$ be a test scheme over
$\mathbf{Z}_{(p)}$ and then $\mathfrak{X}(S)$ is the set of triples
$(A,\iota,\bar{\eta})$ where

  1. (1)

$A$ is an $S$-abelian scheme of relative dimension $2$;

  2. (2)

$\iota:\mathcal{O}_{B^{\prime}}\hookrightarrow\operatorname{End}_{S}(A)$ is an
embedding which is special in the sense of [BC, 131-132];

  3. (3)

$\bar{\eta}$ is an equivalence class of isomorphisms

$\eta:V^{p}(A)\xrightarrow{\sim}B^{\prime}(\mathbf{A}^{\infty,(p)})$

up to multiplication by $K^{p}$, where

$V^{p}(A)=\prod_{q\neq p}T_{q}(A)\otimes\mathbf{Q}$

is the prime to $p$ rational Tate module of $A$.

It is well known this moduli problem is representable by a projective scheme
over $\mathbb{Z}_{(p)}$ of dimension $1$. We will usually consider the base
change of $\mathfrak{X}$ to $\mathbf{Z}_{p^{2}}$ and we will denote it by the
same notation. Let $\mathbf{F}_{q}$ be a finite extension of $\mathbf{F}_{p}$,
we will denote by $X_{\mathbf{F}_{q}}$ the base change of the special fiber
$X_{\mathbf{F}_{p}}$ of $\mathfrak{X}$ to $\mathbf{F}_{q}$.

Let $K$ be the open compact subgroup of $G(\mathbf{A}^{\infty})$ given by the
Eichler order $\mathcal{O}_{B,N^{+}}$. We define the _Shimura set_ $X^{B}$ by
the following double coset

(3.1) $X^{B}=G(\mathbf{Q})\backslash G(\mathbf{A}^{\infty})/K.$

Let $K_{0}(p)$ be the open compact subgroup of $G(\mathbf{A}^{\infty})$ given
by the Eichler order $\mathcal{O}_{B,pN^{+}}$ of level $pN^{+}$ in $B$. We
define $X^{B}_{0}(p)$ by the double coset

(3.2) $X^{B}_{0}(p)=G(\mathbf{Q})\backslash G(\mathbf{A}^{\infty})/K_{0}(p).$

Let $\Lambda$ be a ring. We will use the following notations throughout this
article.

  * •

We denote by $\mathrm{H}^{0}(X^{B},\Lambda)$ the set of continuous function on
$X^{B}$ valued in $\Lambda$. We will also write this space as
$S^{B}_{2}(N^{+},\Lambda)$ and refer to it as the space of _quaternionic
modular forms_ of level $N^{+}$

  * •

In the same way, we define $\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)$. We will
also write this space as $S^{B}_{2}(pN^{+},\Lambda)$ and refer to it as the
space of _quaternionic modular forms_ of level $pN^{+}$

  * •

Let $W$ be a scheme we will let $\mathbf{P}^{d}(W)$ be the
$\mathbf{P}^{d}$-bundle over $W$. For example $\mathbf{P}^{1}(X^{B})$ is the
$\mathbf{P}^{1}$-bundle over $X^{B}$.

###  3.2. The $p$-adic upper half plane

Let $k^{\mathrm{ac}}$ be an algebraic closure of $\mathbf{F}_{p}$ and let
$W_{0}=W(k^{\mathrm{ac}})$ be ring of Witt vectors of $k^{\mathrm{ac}}$ with
fraction field $K_{0}$. Let $D$ be the quaternion algebra over
$\mathbf{Q}_{p}$ and let $\mathcal{O}_{D}$ be the maximal order of $D$. We
write

$\mathcal{O}_{D}=\mathbf{Z}_{p^{2}}[\Pi]/(\Pi^{2}-p)$

with $\Pi a=\sigma(a)\Pi$ with $a\in\mathbf{Z}_{p^{2}}$. We first recall the
notion of a _special formal $\mathcal{O}_{D}$-module_. Let $S$ be a
$W_{0}$-scheme. A special formal $\mathcal{O}_{D}$-module over $S$ is a formal
$p$-divisible group $X$ of dimension $2$ and height $4$ with an
$\mathcal{O}_{D}$-action

$\iota:\mathcal{O}_{D}\rightarrow\operatorname{End}_{S}(X)$

such that $\mathrm{Lie}(X)$ is a locally free
$\mathbf{Z}_{p^{2}}\otimes\mathcal{O}_{S}$-module of rank $1$. We fix a
special $\mathcal{O}_{D}$-module $\mathbb{X}$ over $k^{\mathrm{ac}}$ whose
Dieudonné module is denoted by $\mathbb{M}$. Consider the functor
$\mathcal{M}$ on the category Nilp of $W_{0}$-schemes $S$ such that $p$ is
locally nilpotent in $\mathcal{O}_{S}$ such that $\mathcal{M}(S)$ classifies
the isomorphism classes of pairs $(X,\rho_{X})$ where

  1. (1)

$X$ is special formal $\mathcal{O}_{D}$-module;

  2. (2)

$\rho_{X}:X\times_{S}\bar{S}\rightarrow\mathbb{X}\times_{\mathbf{F}}\bar{S}$
is a quasi-isogney.

The functor $\mathcal{M}$ is represented by a formal scheme over $W_{0}$ which
we also denote by $\mathcal{M}$. The formal scheme $\mathcal{M}$ breaks into a
disjoint union

$\mathcal{M}=\bigsqcup_{i\in\mathbf{Z}}\mathcal{M}_{i}$

according to the height $i$ of the quasi-isogeny $\rho_{X}$. Each formal
scheme $\mathcal{M}_{i}$ is isomorphic to the _$p$ -adic upper half plane_
$\mathcal{H}_{p}$ base-changed to $W_{0}$. The group
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$ acts naturally on the formal scheme
$\mathcal{M}$ and each $\mathcal{M}_{i}$ affords an action of the group
$\mathrm{GL}^{0}_{2}(\mathbf{Q}_{p}):=\\{g\in\mathrm{GL}_{2}(\mathbf{Q}_{p}):\mathrm{ord}_{p}(\mathrm{det}(g))=0\\}$.
We review the descriptions of the special fiber of the formal scheme
$\mathcal{M}_{0}$. Since this description is well-known, see [KR] for example,
we content on explaining this on the level of points. Let
$(X,\rho)\in\mathcal{M}(k^{\mathrm{ac}})$ and let $M$ be the covariant
Dieudonné of $X$ and the action of $\mathbf{Z}_{p^{2}}$ on $X$ induced a
grading

(3.3) $M=M_{0}\oplus M_{1}$

that satisifies

(3.4) $\displaystyle
pM_{0}\subset^{1}VM_{1}\subset^{1}M_{0},\hphantom{a}pM_{1}\subset^{1}VM_{0}\subset^{1}M_{1}$
$\displaystyle pM_{0}\subset^{1}\Pi
M_{1}\subset^{1}M_{0},\hphantom{a}pM_{1}\subset^{1}\Pi M_{0}\subset^{1}M_{1}$

Since the action of $\Pi$ and $V$ commute, we have the induced maps

(3.5) $\displaystyle\Pi:M_{0}/VM_{1}\rightarrow M_{1}/VM_{0},$
$\displaystyle\Pi:M_{1}/VM_{0}\rightarrow M_{0}/VM_{1}.$

Since both $M_{0}/VM_{1}$ and $M_{1}/VM_{0}$ are of dimension $1$ and the
composite of the two maps is obviously zero, we can conclude that there is an
$i\in\mathbf{Z}/2\mathbf{Z}$ such that $\Pi M_{i}\subset VM_{i}$. Since both
$\Pi M_{i}$ and $VM_{i}$ are of colength $1$ in $M_{i+1}$, we conclude that
they are in fact equal to each other. We say that $i$ is a _critical index_ if
$VM_{i}=\Pi M_{i}$ and a critical index always exists for $M$. We let
$\tau=\Pi^{-1}V$ and it acts as an automorphism on $M_{i}$ if $i$ is a
critical index.

If $0$ is a critical index, then we set $L_{0}=M^{\tau=1}_{0}$ and we call it
a _vertex lattice of type $0$_. This is a $\mathbf{Z}_{p}$-lattice of rank $2$
and we associate to it the projective line $\mathbf{P}(L_{0}/p)$. Then
$VM_{1}/pM_{0}\subset^{1}M_{0}/pM_{0}=L_{0}/pL_{0}\otimes k^{\mathrm{ac}}$
gives a point on $\mathbf{P}(L_{0}/pL_{0})(k^{\mathrm{ac}})$. If $1$ is a
critical index, then we similarly put $L_{1}=\Pi M^{\tau=1}_{1}$ and we call
it a _vertetx lattice of type $1$_. We again associate to it the projective
line $\mathbf{P}(L_{1}/pL_{1})$. Similarly
$VM_{0}/pM_{1}\subset^{1}M_{1}/pM_{1}=L_{1}/pL_{1}\otimes k^{\mathrm{ac}}$
gives a point on $\mathbf{P}(L_{1}/pL_{1})(k^{\mathrm{ac}})$. This
construction gives all the irreducible components of the special fiber of
$\mathcal{M}_{0}$. If both $0$ and $1$ are critical indices, then we will
identify the point on $\mathbf{P}(L_{1}/pL_{1})$ given by
$VM_{0}/pM_{1}\subset^{1}M_{1}/pM_{1}=L_{1}/pL_{1}\otimes k^{\mathrm{ac}}$ and
the point on $\mathbf{P}(L_{0}/pL_{0})$ given by
$VM_{1}/pM_{0}\subset^{1}M_{0}/pM_{0}=L_{0}/pL_{0}\otimes k^{\mathrm{ac}}$.
Notice in this case the vertex lattices $L_{0},L_{1}$ satisfies the following
inclusions

$pL_{0}\subset L_{1}\subset L_{0}.$

We summarize the above discussion in the following proposition.

######  Proposition 3.1.

We have the following statements.

  1. (1)

The irreducible components of the special fiber of $\mathcal{M}_{0}$ can be
partitioned into two types according to whether $0$ or $1$ is a critical
index. Each irreducible component corresponds to a vertex lattice and is
isomorphic to a projective line.

  2. (2)

Two irreducible components corresponding two vertex lattices of the same type
will not intersect. Let $L_{0}$ be a vertex lattice of type $0$ and $L_{1}$ be
a vertex lattice of type $1$, then the corresponding irreducible components
intersect if an only if $pL_{0}\subset L_{1}\subset L_{0}$.

  3. (3)

The irreducible components are parametrized by
$\mathrm{GL}_{2}(\mathbf{Q}_{p})/\mathrm{GL}_{2}(\mathbf{Z}_{p})\times\mathbf{Z}/2\mathbf{Z}$.
The intersection points of the irreducible components are parametrized by
$\mathrm{GL}_{2}(\mathbf{Q}_{p})/\mathrm{Iw}_{p}$ where $\mathrm{Iw}_{p}$ is
the Iwahori subgroup of $\mathrm{GL}_{2}(\mathbf{Q}_{p})$.

###### Proof.

The first two points follow from the previous discussions. For the third
point, note that the stabilizer of a vertex lattice in
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$ is obviously
$\mathrm{GL}_{2}(\mathbf{Z}_{p})$. The condition $pL_{0}\subset L_{1}\subset
L_{0}$ defines a standard alcove in the Bruhat-Tits building of
$\mathrm{PGL}_{2}(\mathbf{Q}_{p})$ and therefore the stabilizer of the pair
$(L_{0},L_{1})$ is the Iwahori subgroup $\mathrm{Iw}_{p}$. ∎

###  3.3. Cerednick-Drinfeld uniformization

From here on, we will set $k=\mathbf{F}_{p^{2}}$, $K=\mathbf{Q}_{p^{2}}$ and
$\mathcal{O}_{K}=\mathbf{Z}_{p^{2}}$. Recall we have the integral model
$\mathfrak{X}$ of the Shimura curve $X$ over $\mathcal{O}_{K}$ and we denote
by $\mathfrak{X}^{\wedge}$ its completion along the ideal defined by $p$. The
Cerednick-Drinfeld uniformization theorem asserts that the
${\mathfrak{X}}^{\wedge}$ can be uniformized by the formal scheme
$\mathcal{M}$:

(3.6)
${\mathfrak{X}}^{\wedge}\xrightarrow{\sim}G(\mathbf{Q})\backslash\mathcal{M}\times
G(\mathbf{A}^{\infty,(p)})/K^{p}.$

It follows from the descriptions in Proposition 3.1 and the above
uniformization theorem that the irreducible components of the special fiber
$X_{k}$ are projective lines. It also follows that $\mathfrak{X}$ has strict
semistable reduction. More precisely, we have the following proposition.

######  Proposition 3.2.

We have the following descriptions of the scheme $X_{k}$.

  1. (1)

The scheme $X_{k}$ is a union $\mathbf{P}^{1}$-bundles over Shimura sets

$X_{k}=\mathbf{P}^{1}(X^{B}_{+})\cup\mathbf{P}^{1}(X^{B}_{-}).$

where both $X^{B}_{+}$ and $X^{B}_{-}$ are isomorphic to the Shimura set
$X^{B}$ as in (3.1).

  2. (2)

The intersection points of the two $\mathbf{P}^{1}$-bundles
$\mathbf{P}^{1}(X^{B}_{+})$ and $\mathbf{P}^{1}(X^{B}_{-})$ are given by

$\mathbf{P}^{1}(X^{B}_{+})\cap\mathbf{P}^{1}(X^{B}_{-})=X^{B}_{0}(p).$

This also can be identified with the set of singular points on $X_{k}$.

###### Proof.

For $(1)$, recall that there are two types of irreducible components of the
special fiber of $\mathcal{M}_{0}$ corresponding to vertex lattices of type
$0$ and vertex lattices of type $1$. By the Cerednick–Drinfeld uniformization
(3.6), we only need to notice that the irreducible components of corresponding
to vertex lattices of type $0$ or vertex lattice of type $1$ are uniformized
by

$G(\mathbf{Q})\backslash\mathrm{GL}_{2}(\mathbf{Q}_{p})/\mathrm{GL}_{2}(\mathbf{Z}_{p})\times
G(\mathbf{A}^{\infty,(p)})/K^{p}.$

This is just another way of writing $X^{B}$. We will define $X^{B}_{+}$ to be
the copy of $X^{B}$ corresponding to the vertex lattice of type $0$ and define
$X^{B}_{-}$ to be the copy of $X^{B}$ corresponding to the vertex lattice of
type $1$. For $(2)$, the Cerednick–Drinfeld uniformization (3.6) shows that
the set of intersection points are given by

$G(\mathbf{Q})\backslash\mathrm{GL}_{2}(\mathbf{Q}_{p})/\mathrm{Iw}_{p}\times
G(\mathbf{A}^{\infty,(p)})/K^{p}.$

Again this is another way of writing $X^{B}_{0}(p)$. The rest of the claims
are clear. ∎

We denote by $\pi_{+}:X^{B}_{0}(p)\rightarrow X^{B}_{+}$ and
$\pi_{-}:X^{B}_{0}(p)\rightarrow X^{B}_{-}$ the natural specialization maps.
These two maps give the Hecke correspondence of $X^{B}$. This can be seen as
follows: the map $\pi_{+}$ is induced by sending the pair of vertex lattices
$(L_{1}\subset L_{0})$ to $L_{0}$ and the map $\pi_{-}$ is induced by sending
$(L_{1}\subset L_{0})$ to $L_{1}$. The two lattices are related by the matrix
$\begin{pmatrix}0&p\\\ 1&0\\\ \end{pmatrix}$ that is if one chooses a suitable
basis $(e_{1},e_{2})$ for $L_{0}$ then $(e_{2},pe_{1})$ will be a basis for
$L_{1}$.

The nontrivial element in $\mathrm{Gal}(k/\mathbf{F}_{p})$ acts on $X_{k}$ and
it permutes the two $\mathbf{P}^{1}$-bundles $\mathbf{P}^{1}(X^{B}_{+})$ and
$\mathbf{P}^{1}(X^{B}_{-})$. On the set $X^{B}_{0}(p)$ it acts by the
classical _Atkin-Lehner involution_. For this, see [BD1, §1.7] for example.

###  3.4. Ramified level raising on Shimura curves

Recall $N=N^{+}N^{-}$ with $N^{+}$ and $N^{-}$ defined previously. Let
$f\in\mathrm{S}^{\mathrm{new}}_{2}(\Gamma_{0}(N))$ be a newform of weight $2$.
Let $E=\mathbf{Q}(f)$ be the Hecke field of $f$. Let $\lambda$ be a place of
$E$ above $l$ and $E_{\lambda}$ be the completion of $E$ at $\lambda$. Let
$\varpi$ be a uniformizer of $\mathcal{O}=\mathcal{O}_{E_{\lambda}}$ and we
write $\mathcal{O}_{n}=\mathcal{O}/\varpi^{n}$ for $n\geq 1$. We let
$\mathbb{T}=\mathbb{T}_{N^{+},N^{-}}$, respectively
$\mathbb{T}^{[p]}=\mathbb{T}_{N^{+},pN^{-}}$, be the $l$-adic Hecke algebra
corresponding to the cusp forms of level $N=N^{+}N^{-}$, respectively of level
$Np=N^{+}N^{-}p$, which is new at primes dividing $N^{-}$, respectively at
primes dividing $pN^{-}$. Since $f$ is an eigenform, we have a morphism
$\phi_{f}:\mathbb{T}\rightarrow\mathcal{O}$ corresponding to the system of
eigenvalues of $f$. More precisely, we have $\phi_{f}(T_{v})=a_{v}(f)$ for
$v\nmid N$ and $\phi_{f}(U_{v})=a_{v}(f)$ for $v\mid N$.

######  Definition 3.3.

Let $n\geq 1$ be an integer. We say that a prime $p$ is _$n$ -admissible_ for
$f$ if

  1. (1)

$p\nmid Nl$;

  2. (2)

$l\nmid p^{2}-1$;

  3. (3)

$\varpi^{n}\mid p+1-\epsilon_{p}(f)a_{p}(f)$ for some
$\epsilon_{p}(f)\in\\{-1,1\\}$.

Let $\phi_{f,n}:\mathbb{T}\rightarrow\mathcal{O}_{n}$ be reduction of the map
$\phi_{f}:\mathbb{T}\rightarrow\mathcal{O}$ modulo $\varpi^{n}$. We denote by
$I_{f,n}$ the kernel of this map and $\mathfrak{m}_{f}$ the unique maximal
ideal of $\mathbb{T}$ containing $I_{f,n}$. We will say $\mathfrak{m}_{f}$ is
residually irreducible if the residue Galois representation $\bar{\rho}_{f}$
attached to $f$ is irreducible. The following result is known as the _ramified
arithmetic level raising for Shimura curves_ and was first proved in [BD3,
Theorem 5.15, Corollary 5.18]. The proof of the theorem below is inspired by
two lectures given by Liang Xiao at the Morningside center [Xiao] on the
subject.

######  Theorem 3.4.

Let $p$ be an $n$-admissible prime for $f$. We assume that

  1. (i)

the maximal ideal $\mathfrak{m}_{f}$ is residually irreducible;

  2. (ii)

The module $S^{B}_{2}(N^{+},\mathcal{O})_{\mathfrak{m}_{f}}$ is free of rank
$1$ over $\mathbb{T}_{\mathfrak{m}_{f}}$.

Then we have the following.

  1. (1)

There exists a surjective homomorphism
$\phi^{[p]}_{f,n}:\mathbb{T}^{[p]}\rightarrow\mathcal{O}_{n}$ such that
$\phi^{[p]}_{f,n}$ agrees with $\phi_{f,n}$ at all Hecke operators away from
$p$ and sends $U_{p}$ to $\epsilon_{p}(f)$. We will denote by $I^{[p]}_{f,n}$
the kernel of $\phi^{[p]}_{f,n}$.

  2. (2)

We have an isomorphism of $\mathcal{O}_{n}$-modules of rank $1$

$S^{B}_{2}(N^{+},\mathcal{O}){/I_{f,n}}\xrightarrow{\cong}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\mathcal{O}(1)){/I^{[p]}_{f,n}}).$

###### Proof.

In this proof we will set
$\mathrm{M}=\mathrm{H}^{1}(X_{k^{\mathrm{ac}}},\mathcal{O}(1))$ and
$\mathrm{M}_{n}=\mathrm{H}^{1}(X_{k^{\mathrm{ac}}},\mathcal{O}(1)){/I_{f,n}}$.
Then we use the following formula as proved in (2.12) and (2.13)

(3.7)
$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M})\cong\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\mathcal{O})\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\mathcal{O})\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\mathcal{O}(1))]^{G_{k}}.$

Here

$\mathrm{H}^{0}(a_{0*}\mathcal{O})=\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{+}),\mathcal{O})\oplus\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{-}),\mathcal{O})$

and we can identify it with the space $S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}$. Similarly under the Poincare duality, we can also identify

$\mathrm{H}^{2}(a_{0*}\mathcal{O}(1))=\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{+}),\mathcal{O}(1))\oplus\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{-}),\mathcal{O}(1))$

with $S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}$. The space
$\mathrm{H}^{0}(a_{1*}\mathcal{O})$ can be identified with
$S^{B}_{2}(pN^{+},\mathcal{O})$. Under these identifications, the composition
appearing in (3.7)

$\mathrm{H}^{0}(a_{0*}\mathcal{O})\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\mathcal{O})\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\mathcal{O}(1))$

is given by the _intersection matrix_

$\begin{pmatrix}-(p+1)&T_{p}\\\ T_{p}&-(p+1)\\\ \end{pmatrix}$

which we will also denote by $\nabla$. Since $p$ is $n$-admissible for $f$,
the singular quotient of $\mathrm{M}_{n}$

$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M}_{n})\cong\mathrm{coker}[S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n}}\xrightarrow{\nabla}S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n})}]$

is of rank one over $\mathcal{O}_{n}$ and is isomorphic to
$S^{B}_{2}(N^{+},\mathcal{O})_{/I_{f,n}}$. Note here the isomorphism between

$\mathrm{coker}[S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n}}\xrightarrow{\nabla}S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n})}]$

and $S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}_{/I_{f,n}}$ is induced by the map

$S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}_{/I_{f,n}}\rightarrow
S^{B}_{2}(N^{+},\mathcal{O})_{/I_{f,n}},\hphantom{aa}(x,y)\mapsto\frac{1}{2}(x+\epsilon_{p}(f)y)$

This proves the second part of the theorem.

By [BD3, Theorem 5.8] and [CH2, §3.5], the natural $U_{p}$-action on

$\mathrm{H}^{2}(a_{0*}\mathcal{O}(1))\cong
S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}$

is given by $(x,y)\mapsto(-py,x+T_{p}y)$. We consider the automorphism

$\delta:S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}\rightarrow
S^{B}_{2}(N^{+},\mathcal{O})^{\oplus 2}$

given by $(x,y)\mapsto(x+T_{p}y,y)$. Then a quick calculation gives us that
$\nabla\circ\delta=U^{2}_{p}-1$. This means that the quotient

$\frac{S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}}{(I_{f,n},U^{2}_{p}-1)}\cong\mathrm{coker}[S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n}}\xrightarrow{\nabla}S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n})}]$

is of rank $1$. Since $p$ is $n$-admissible for $f$, we see immediately
$U_{p}+\epsilon_{f}$ is invertible on $S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}_{/I_{f,n}}$. Therefore we have

$\frac{S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}}{(I_{f,n},U^{2}_{p}-1)}\cong\frac{S^{B}_{2}(N^{+},\mathcal{O})^{\oplus
2}}{(I_{f,n},U_{p}-\epsilon_{p}(f))}$

and the latter quotient is of rank $1$ over $\mathcal{O}_{n}$. Then the action
of $\mathbb{T}^{[p]}$ on this rank $1$ quotient gives the desired morphism
$\phi^{[p]}_{f,n}:\mathbb{T}^{[p]}\rightarrow\mathcal{O}_{n}$.

∎

We will now compare the ingredients in the above proof with those in [BD3].
Let $\mathcal{G}$ be the dual reduction graph of $X_{k}$. We denote by
$\mathcal{V}(\mathcal{G})$ the set of vertices and $\mathcal{E}(\mathcal{G})$
the set of edges. Then we have the following identifications

$\mathrm{H}^{0}(a_{0*}\Lambda)=\Lambda[\mathcal{V}(\mathcal{G})]$

and

$\mathrm{H}^{0}(a_{1*}\Lambda)=\Lambda[\mathcal{E}(\mathcal{G})]$

for $\Lambda$ equals to $\mathbf{Z}_{/l^{n}}$, $\mathbf{Z}_{l}$ or a finite
extension of $\mathbf{Z}_{l}$. Moreover under these identifications, the
restriction map

$\mathrm{H}^{0}(a_{0*}\Lambda)\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\Lambda)$

can be identified with

$\Lambda[\mathcal{V}(\mathcal{G})]\xrightarrow{d^{*}=-s^{*}+t^{*}}\Lambda[\mathcal{E}(\mathcal{G})].$

And the map

$\mathrm{H}^{0}(a_{1*}\Lambda)\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))$

can be identified with

$\Lambda[\mathcal{E}(\mathcal{G})]\xrightarrow{d_{*}=-s_{*}+t_{*}}\Lambda[\mathcal{V}(\mathcal{G})].$

Here $s,t$ are the source and target maps

$\mathcal{E}(\mathcal{G})\xrightarrow{s,t}\mathcal{V}(\mathcal{G})$

defined in an obvious manner. Let $\mathcal{J}$ be the Néron model of the
Jacobian $\mathrm{Jac}(X_{K})$ of $X_{K}$ and let $\Phi$ be the group of
connected components of special fiber $\mathcal{J}_{k}$. Denote by
$\mathcal{X}$, resp. $\mathcal{X}^{\vee}$, the _character group_ , resp. the
_cocharacter group_ of $\mathcal{J}_{k}$. The following proposition describes
$\Phi$, $\mathcal{X}$ and $\mathcal{X}^{\vee}$ in terms of the dual reduction
graph $\mathcal{G}$.

######  Proposition 3.5.

We have the following statements

  1. (1)

There is a Hecke module isomorphism

$\mathbf{Z}[\mathcal{V}(\mathcal{G})]_{0}/d_{*}d^{*}\cong\Phi$

where $\mathbf{Z}[\mathcal{V}(\mathcal{G})]_{0}$ be the image of
$\mathbf{Z}[\mathcal{E}(\mathcal{G})]\xrightarrow{d_{*}}\mathbf{Z}[\mathcal{V}(\mathcal{G})]$.

  2. (2)

The kernel of the map

$\mathbf{Z}[\mathcal{V}(\mathcal{G})]\xrightarrow{d^{*}}\mathbf{Z}[\mathcal{E}(\mathcal{G})]$

is Eisenstein.

  3. (3)

The cokernel of the map

$\mathbf{Z}[\mathcal{E}(\mathcal{G})]\xrightarrow{d_{*}}\mathbf{Z}[\mathcal{V}(\mathcal{G})]$

is Eisenstein.

  4. (4)

There is an isomorphism of Hecke modules

$\mathcal{X}=\ker[\mathbf{Z}[\mathcal{E}(\mathcal{G})]\xrightarrow{d_{*}}\mathbf{Z}[\mathcal{V}(\mathcal{G})]].$

  5. (5)

There is an isomorphism of Hecke modules

$\mathcal{X}^{\vee}=\mathrm{coker}[\mathbf{Z}[\mathcal{V}(\mathcal{G})]\xrightarrow{d^{*}}\mathbf{Z}[\mathcal{E}(\mathcal{G})]].$

###### Proof.

The first statement follows from [BLR1, §9.6, Theorem 1]. The second statement
is the well known “Ihara’s lemma” for definite quaternion algebras in view of
what we have discussed before. See [Rib2, Theorem 3.15] for the version we
need. ∎

######  Corollary 3.6 ([BD3, Corollary 5.18]).

Under the assumptions in Theorem 3.4, we have a more canonical isomorphism

$\Phi_{\mathcal{O}}/I^{[p]}_{f,n}\cong\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\mathcal{O}(1))/I^{[p]}_{f,n}).$

###### Proof.

Recall the following formula coming out of the weight spectral sequence for
Shimura curves

$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\mathcal{O}(1))/I^{[p]}_{f,n})\cong\mathrm{coker}[\mathrm{H}^{0}(a_{0*}\mathcal{O})/I^{[p]}_{f,n}\xrightarrow{\rho}\mathrm{H}^{0}(a_{1*}\mathcal{O})/I^{[p]}_{f,n}\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\mathcal{O}(1))/I^{[p]}_{f,n}]^{G_{k}}.$

From the discussion before and Proposition 3.5 $(1)$, we see that the right-
hand side of the above equation can be identified with
$\Phi_{\mathcal{O}}/I^{[p]}_{f,n}$. ∎

######  Lemma 3.7.

There is an isomorphism under the assumption in Theorem 3.4

$\mathcal{X}^{\vee}_{\mathcal{O}}/I^{[p]}_{f,n}\cong\Phi_{\mathcal{O}}/I^{[p]}_{f,n}\cong
S^{B}_{2}(N^{+},\mathcal{O})/I_{f,n}.$

###### Proof.

We have an exact sequence

$0\rightarrow\mathcal{X}_{\mathcal{O}}\rightarrow\mathcal{X}^{\vee}_{\mathcal{O}}\rightarrow\Phi_{\mathcal{O}}\rightarrow
0$

by [BD3, 5.11] which induces

$\mathcal{X}_{\mathcal{O}}/I^{[p]}_{f,n}\rightarrow\mathcal{X}^{\vee}_{\mathcal{O}}/I^{[p]}_{f,n}\rightarrow\Phi_{\mathcal{O}}/I^{[p]}_{f,n}\rightarrow
0.$

Let $J$ be the Jacobian of the curve $X$. By the proof of [BD3, Theorem 5.17],
the Galois module $\mathrm{Ta}_{l}(J)/I^{[p]}_{f,n}$ is of rank $2$ over
$\mathcal{O}_{n}$ and hence $\mathcal{X}^{\vee}_{\mathcal{O}}/I^{[p]}_{f,n}$
is of rank $1$ over $\mathcal{O}_{n}$ by the $p$-adic uniformization of $J$.
The result follows as $\Phi_{\mathcal{O}}/I^{[p]}_{f,n}\cong
S^{B}_{2}(N^{+},\mathcal{O})/I_{f,n}$ is also rank $1$ under our assumption by
the previous discussions. ∎

##  4\. Arithmetic level raising for triple product of Shimura curves

###  4.1. Semistable model of triple product of Shimura curves

Recall in the last section we have defined the Shimura curve
$X=X^{B^{\prime}}_{N^{+},pN^{-}}$ with discriminant $pN^{-}$ over
$\mathbf{Q}$. Let $X^{3}$ be the threefold fiber product of $X$ over
$\mathbf{Q}$. Recall that we have the integral model $\mathfrak{X}$ of $X$
defined over $\mathcal{O}_{K}$. Let $\mathfrak{X}^{3}$ be the threefold fiber
product of $\mathfrak{X}$ over $\mathcal{O}_{K}$. First, we analyze the
reduction of $\mathfrak{X}^{3}$. We denote by $X^{3}_{k}$ the special fiber of
$\mathfrak{X}^{3}$. By Proposition 3.2, we know each $X_{k}$ can be described
as the union $\mathbf{P}^{1}(X^{B}_{+})\cup\mathbf{P}^{1}(X^{B}_{-})$ and
therefore the special fiber $X^{3}_{k}$ can be described by the cube given
below.

${{X}^{045}_{[+-+]}}$${{X}^{024}_{[+++]}}$${{X}^{015}_{[--+]}}$${{X}^{012}_{[-++]}}$${{X}^{345}_{[+--]}}$${{X}^{234}_{[++-]}}$${{X}^{135}_{[---]}}$${{X}^{123}_{[-+-]}}$

We will explain the meaning of the simplices in this cube.

  * •

The $0$-simplices are the vertices of the cube. They correspond to
$3$-dimensional strata in $X^{3}_{k}$. Consider the vertex labeled by
$X^{123}_{[-+-]}$ for example. The superscript $123$ has no real meaning and
is simply used for ordering the vertices. This ordering is inherited from
Liu’s paper [Liu2] where the labels have real meanings in terms of achimedean
places of a cubic field. The subscript $[-+-]$ means that $X^{123}_{[-+-]}$ is
of the form

$\mathbf{P}^{1}(X^{B}_{-})\times\mathbf{P}^{1}(X^{B}_{+})\times\mathbf{P}^{1}(X^{B}_{-}).$

  * •

The $1$-simplices are the edges of the cube. They correspond to
$2$-dimensional strata in $X^{3}_{k}$. For example, we will label the edge
between ${X}^{135}_{[---]}$ and ${X}^{123}_{[-+-]}$ by $X^{1235}_{[-\pm-]}$.
This means we will take the union on the superscript and take the intersection
on the subscript. Then $X^{1235}_{[-\pm-]}$ is of the form

$\mathbf{P}^{1}(X^{B}_{-})\times X^{B}_{0}(p)\times\mathbf{P}^{1}(X^{B}_{-}).$

  * •

The $2$-simplices are the faces of the cube. They correspond to
$1$-dimensional strata in $X^{3}_{k}$. We use similar convention as in the
last point. For example

$X^{01235}_{[-\pm\pm]}=\mathbf{P}^{1}(X^{B}_{-})\times X^{B}_{0}(p)\times
X^{B}_{0}(p).$

  * •

Finally, the $3$-simplex is the zero dimensional components given by

$X^{012345}_{[\pm\pm\pm]}=X^{B}_{0}(p)\times X^{B}_{0}(p)\times X^{B}_{0}(p).$

We will usually drop the subscript from the notations and put it back there
when we need to know the exact form of the strata. By an easy computation on
the local rings, we see that $\mathfrak{X}^{3}$ is not semi-stable. Following
the procedure in [GS, Example 6.15], we can obtain a semistable model denoted
by $\mathcal{Y}$ of $\mathfrak{X}^{3}$ over $\mathcal{O}_{K}$. More precisely,
to obtain $\mathcal{Y}$, we blow-up $\mathfrak{X}^{3}$ along the closed
subscheme $X^{135}$, then we blow-up the strict transform of $X^{024}$. We
denote by $\pi:\mathcal{Y}\rightarrow\mathfrak{X}^{3}$ the natural morphism
between these two schemes given by the aforementioned process. The generic
fiber $\mathcal{Y}_{K}$ agree with the generic fiber ${X}^{3}_{K}$ of
$\mathfrak{X}^{3}$. The special fiber of $\mathcal{Y}$ will be denoted by
$Y_{k}$ and it can be described by the following cube. The densely dotted line
on the cube correspond to the new intersections between three dimensional
strata caused by the blow-ups and they give new two dimensional strata.

(4.1)
${{Y}^{045}_{[+-+]}\vphantom{f^{*}}}$${{Y}^{024}_{[+++]}\hphantom{EEE}}$${{Y}^{015}_{[--+]}\vphantom{f^{*}}}$${{Y}^{012}_{[-++]}\hphantom{Ee}}$${{Y}^{345}_{[+--]}\vphantom{U}}$${{Y}^{234}_{[++-]}\vphantom{E}}$${{Y}^{135}_{[---]}\vphantom{M}}$${{Y}^{123}_{[-+-]}\vphantom{N}}$

We will explain the meaning of some of the simplices of this cube.

  * •

The $0$-simplices are the vertices of the cube. They correspond to
$3$-dimensional strata in ${Y}_{k}$. These are the strict transform of the
corresponding strata in $X^{3}_{k}$. For example, $Y^{012}$ is the strict
transform of $X^{012}$ under $\pi$.

  * •

The $1$-simplices are the edges on the cube. They correspond to
$2$-dimensional strata in ${Y}_{k}$. Notice that there are three types of
edges:

    1. (1)

those correspond to the original edges in the cube for $X^{3}_{k}$, for
example $Y^{0125}=Y^{012}\cap Y^{015}$;

    2. (2)

those correspond to the faces in the cube for $X^{3}_{k}$, for example
$Y^{01235}=Y^{012}\cap Y^{135}$;

    3. (3)

the one correspond to the “main diagonal” of the cube $Y^{012345}=Y^{024}\cap
Y^{135}$.

######  Proposition 4.1.

We have the following descriptions of the $2$-dimensional and $3$-dimensional
strata appear in the above list.

  1. (1)

The three dimensional strata

$Y^{i(i+1)(i+2)}$

is the blow-up of $X^{i(i+1)(i+2)}$ along the one dimensional strata
$X^{i(i+1)(i+2)(i+3)(i+5)}$ for $i\in\\{0,1,2,3,4,5\\}$.

  2. (2)

The three dimensional strata

$Y^{024}$

is the blow-up of $X^{024}$ along the zero dimensional strata $X^{012345}$
followed by the blow-up of the strict transform of $X^{01234}\cup
X^{01245}\cup X^{02345}$.

  3. (3)

The three dimensional strata

$Y^{135}$

is the blow-up of $X^{135}$ along the zero dimensional strata $X^{012345}$
followed by the blow-up of the strict transform of $X^{01235}\cup
X^{01345}\cup X^{12345}$.

  4. (4)

The two dimensional strata

$Y^{i(i+1)(i+2)(i+3)}$

maps isomorphically to $X^{i(i+1)(i+2)(i+3)}$ for $i\in\\{0,1,2,3,4,5\\}$.

  5. (5)

The two dimensional strata

$Y^{i(i+1)(i+2)(i+4)}$

is the blow-up of $X^{i(i+1)(i+2)(i+4)}$ along $X^{012345}$ for
$i\in\\{0,1,2,3,4,5\\}$.

  6. (6)

The two dimensional strata

$Y^{i(i+1)(i+2)(i+3)(i+5)}$

is a $\mathbf{P}^{1}$-bundle over $X^{i(i+1)(i+2)(i+3)(i+5)}$ for
$i\in\\{0,1,2,3,4,5\\}$. In fact, $Y^{i(i+1)(i+2)(i+3)(i+5)}$ is the
exceptional divisor of the blow-up $\pi:Y^{i(i+1)(i+2)}\rightarrow
X^{i(i+1)(i+2)}$.

  7. (7)

The two dimensional strata

$Y^{012345}$

is a $\mathbf{P}^{2}$-bundle over $X^{012345}$.

Moreover all the strata of dimension $2$ or dimension $3$ are given in the
above list.

###### Proof.

The proof of these statements are exactly the same as these given in [Liu2,
Proposition B.39]. Although we are working with different Shimura varities,
the underlying local models are the same. ∎

###  4.2. Cohomology of blow-ups

We insert here a quick discussion of how to compute cohomology of varieties
obtained by blow-ups, following [Ito, §3]. Let $F$ be an algebraically closed
field of any characteristic and let $X$ be a projective smooth irreducible
variety over $F$ of dimension $n$. Let $Y_{1},\cdots,Y_{r}\subset X$ be
mutually disjoint smooth closed irreducible subvarieties of codimension $d\geq
2$. Let $i:Y=\sqcup^{r}_{t=1}Y_{t}\hookrightarrow X$ be the closed immersion.
Let $\pi:\tilde{X}\rightarrow X$ be the blow-up of $X$ along $Y$ and
$\tilde{Y}$ be the strict transform of $Y$. These maps will fit in the
following cartesian diagram

(4.2)
${\tilde{Y}}$${\tilde{X}}$${Y}$${X.}$$\scriptstyle{\tilde{i}}$$\scriptstyle{\pi_{\mid
Y}}$$\scriptstyle{\pi}$$\scriptstyle{i}$

Since $Y$ is a disjoint union of mutually disjoint smooth irreducible closed
subvarieties of codimension $d$, $\tilde{Y}$ is a $\mathbf{P}^{d-1}$-bundle
over $Y$. Let
$\xi=c_{1}(\mathcal{O}_{\tilde{Y}}(1))\in\mathrm{H}^{2}(\tilde{Y},\Lambda(1))$
be the first Chern class of $\mathcal{O}_{\tilde{Y}}(1)$ of the
$\mathbf{P}^{d-1}$-bundle $\tilde{Y}$.

######  Proposition 4.2.

We have the following fomrula

$\displaystyle\mathrm{H}^{k}(\tilde{X},\Lambda(n))$
$\displaystyle\cong\mathrm{H}^{k-2}(Y,\Lambda(n-1))\oplus\cdots\oplus\mathrm{H}^{k-2-2(d-2)}(Y,\Lambda(n-(d-2)))\xi^{d-2}\oplus\mathrm{H}^{k}(X,\Lambda(n)).$

###### Proof.

This is a quite well-known results in [SGA5] and one can find a nice
exposition of this result in [Ito, (3.3)]. ∎

###  4.3. Level raising on triple product of Shimura curves

As in §2.1, $N$ is a positive integer which admits a factorization
$N=N^{+}N^{-}$ with $(N^{+},N^{-})=1$ such that $N^{-}$ is square free and has
odd number of prime factors. Let

$\displaystyle f_{1}=\sum_{n\geq 1}a_{n}(f_{1})q^{n},$ $\displaystyle
f_{2}=\sum_{n\geq 1}a_{n}(f_{2})q^{n},$ $\displaystyle f_{3}=\sum_{n\geq
1}a_{n}(f_{3})q^{n}$

be a triple of weight $2$ normalized newforms in
$S^{\mathrm{new}}_{2}(\Gamma_{0}(N))$. We write
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$. For each $i=1,2,3$, let
$E_{i}=\mathbf{Q}(f_{i})$ be the Hecke field of $f_{i}$. Let $\mathcal{O}_{i}$
be a finite extension of $\Lambda$ containing the ring of integers of $E_{i}$
with uniformizer $\varpi_{i}$. We will write $\mathbf{F}_{i}$ as the residue
field of $\mathcal{O}_{i}$. Let $p$ be a prime away from $N$. Recall that
$\mathbb{T}=\mathbb{T}_{N^{+},N^{-}}$, respectively
$\mathbb{T}^{[p]}=\mathbb{T}_{N^{+},N^{-}p}$, is the $l$-adic Hecke algebra
corresponding to the subspace of the cusp forms of level $N=N^{+}N^{-}$,
respectively of level $Np=N^{+}N^{-}p$, which are new at primes dividing
$N^{-}$, respectively at $pN^{-}$. Since $(f_{1},f_{2},f_{3})$ is a triple of
eigenforms, we have morphisms $\phi_{i}:\mathbb{T}\rightarrow\mathcal{O}_{i}$
corresponding to $f_{i}$ and
$\phi_{i,n}:\mathbb{T}\rightarrow\mathcal{O}_{i,n}$ corresponding to the
reduction of $\phi_{i}$ modulo $\varpi^{n}_{i}$ for $i=1,2,3$.

######  Definition 4.3.

Let $n\geq 1$ be an integer. We say that a prime $p$ is _$n$ -admissible_ for
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$ if

  1. (1)

$p\nmid Nl$;

  2. (2)

$l\nmid p^{2}-1$;

  3. (3)

$\varpi_{i}^{n}\mid p+1-\epsilon_{p,i}a_{p}(f_{i})$ with $\epsilon_{p,i}=\pm
1$ for $i=1,2,3$;

  4. (4)

$\epsilon_{p,1}\epsilon_{p,2}\epsilon_{p,3}=1$.

Let $p$ be an _$n$ -admissible_ prime for $\underline{\mathbf{f}}$. We know by
Theorem 3.4 that there are morphisms
$\phi^{[p]}_{i,n}:\mathbb{T}^{[p]}\rightarrow\mathcal{O}_{i,n}$ that agree
with $\phi_{i,n}:\mathbb{T}\rightarrow\mathcal{O}_{i,n}$ at all Hecke
operators except those at $p$ and such that
$\phi^{[p]}_{i,n}(U_{p})=\epsilon_{p,i}$ for $i=1,2,3$. We denote by
$I_{{i},n}$, respectively $I^{[p]}_{{i},n}$, the kernel of the map
$\phi_{i,n}$, respectively the kernel of the map $\phi^{[p]}_{i,n}$. We also
let $\mathfrak{m}_{i}$ be the maximal ideal of $\mathbb{T}$ containing
$I_{{i},n}$ and let $\mathfrak{m}^{[p]}_{i}$ be the maximal ideal of
$\mathbb{T}^{[p]}$ containing $I^{[p]}_{{i},n}$. We will always assume that
the maximal ideal $\mathfrak{m}_{i}$ is residually irreducible. Let
$\mathfrak{m}_{\underline{\mathbf{f}}}=(\mathfrak{m}_{1},\mathfrak{m}_{2},\mathfrak{m}_{3})$
and
$\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}=(\mathfrak{m}^{[p]}_{1},\mathfrak{m}^{[p]}_{2},\mathfrak{m}^{[p]}_{3})$.

There is an action of
$\mathbb{T}^{[p]}\times\mathbb{T}^{[p]}\times\mathbb{T}^{[p]}$ on the $l$-adic
cohomology of $\mathfrak{X}^{3}$ which extends to that of $\mathcal{Y}$. Let
$\Lambda=\mathbf{Z}_{l}$ from now on. By the Künneth formula it makes sense to
localize the cohomology

$\mathrm{H}^{3}(\mathcal{Y}\otimes{K^{\mathrm{ac}}},\Lambda(2))=\mathrm{H}^{3}(\mathfrak{X}^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))$

at the triple $\mathfrak{m}_{\underline{\mathbf{f}}^{[p]}}$.

(4.3)
$\mathrm{H}^{3}(\mathfrak{X}^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))_{\mathfrak{m}_{\underline{\mathbf{f}}^{[p]}}}=\otimes^{3}_{i=1}\mathrm{H}^{1}(X_{K^{\mathrm{ac}}},\Lambda(1))_{\mathfrak{m}^{[p]}_{i}}(-1).$

This follows from the fact that the $\mathrm{H}^{0}$ and $\mathrm{H}^{2}$ of
the Shimura curve $X_{K^{\mathrm{ac}}}$ are Eisenstein as Hecke modules and
the Künneth formula. The same reasoning implies that
$\mathrm{H}^{*}(\mathfrak{X}^{3}\otimes{K^{\mathrm{ac}}},\Lambda(r))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}$
is non-zero only when $*=3$ for any integer $r$. To analyze
$\mathrm{H}^{3}(\mathfrak{X}^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))$, we will
localize its weight spectral sequence whose first page is given by (2.8) at
the triple $\mathfrak{m}_{\underline{\mathbf{f}}^{[p]}}$ and write the
resulting spectral sequence by $\mathbb{E}(2)$. The “untwisted” spectral
sequence will be denoted by $\mathbb{E}$.

######  Lemma 4.4.

The $\mathrm{E}_{1}$-page of the spectral sequence $\mathbb{E}(2)$ is given
below

${6}$${\mathrm{H}^{0}(a_{3*}\Lambda(-1))_{*}}$${\mathrm{H}^{2}(a_{2*}\Lambda)_{*}}$${\mathrm{H}^{4}(a_{1*}\Lambda(1))_{*}}$${\mathrm{H}^{6}(a_{0*}\Lambda(2))_{*}}$${4}$${\mathrm{H}^{0}(a_{2*}\Lambda)_{*}}$${\mathrm{H}^{2}(a_{1*}\Lambda(1))_{*}\oplus\mathrm{H}^{0}(a_{3*}\Lambda)_{*}}$${\mathrm{H}^{4}(a_{0*}\Lambda(2))_{*}\oplus\mathrm{H}^{2}(a_{2*}\Lambda(1))_{*}}$${\mathrm{H}^{4}(a_{1*}\Lambda(2))_{*}}$${2}$${\mathrm{H}^{0}(a_{1*}\Lambda(1))_{*}}$${\mathrm{H}^{2}(a_{0*}\Lambda(2))_{*}\oplus\mathrm{H}^{0}(a_{2*}\Lambda(1))_{*}}$${\mathrm{H}^{2}(a_{1*}\Lambda(2))_{*}\oplus\mathrm{H}^{0}(a_{3*}\Lambda(1))_{*}}$${\mathrm{H}^{2}(a_{2*}\Lambda(2))_{*}}$${0}$${\mathrm{H}^{0}(a_{0*}\Lambda(2))_{*}}$${\mathrm{H}^{0}(a_{1*}\Lambda(2))_{*}}$${\mathrm{H}^{2}(a_{2*}\Lambda(2))}$${\mathrm{H}^{0}(a_{3*}\Lambda(2))_{*}}$
${-3}$${-2}$${-1}$${0}$${1}$${2}$${3}$

where the subscript ${*}$ means localization at
$\mathfrak{m}_{\underline{\mathbf{f}}^{[p]}}$. In particular it follows that
$\mathbb{E}(2)$ will degenerate on its $\mathrm{E}_{2}$-page.

###### Proof.

This follows from Proposition 4.2 and the explicit descriptions of
$X^{3}_{{k}}$ in (4.1) which allow us to remove all the odd degree cohomology
terms. ∎

######  Lemma 4.5.

The spectral sequence $\mathbb{E}(2)$ satisfies Assumption 2.3.

###### Proof.

We need to analyze the cohomology term $\mathrm{H}^{2}(a_{1*}\Lambda(1))$.
First we have

(4.4) $\displaystyle\mathrm{H}^{2}(a_{1*}\Lambda(1))=$
$\displaystyle\oplus^{5}_{i=0}\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+3)}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle\oplus\oplus^{5}_{i=0}\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+4)}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle\oplus\oplus^{5}_{i=0}\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+3)(i+5)}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle\oplus\mathrm{H}^{2}(Y^{012345}_{k^{\mathrm{ac}}},\Lambda(1))$

by Proposition 4.1. We explicate the terms in the direct sum on the righthand
side of (4.4).

  * •

For $\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+3)}_{k^{\mathrm{ac}}},\Lambda(1))$, we
only make it explicit for $i=0$. By Proposition 4.1 $(4)$, we have

$\displaystyle\mathrm{H}^{2}(Y^{0123}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle=\mathrm{H}^{2}(X^{0123}_{[-+\pm],k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle=\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{-}),\Lambda(1))\otimes\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda)\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)$
$\displaystyle\oplus\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{-}),\Lambda)\otimes\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda(1))\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda).$

  * •

For $\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+4)}_{k^{\mathrm{ac}}},\Lambda(1))$, we
only make it explicit for $i=0$. By Proposition 4.1 $(5)$, we have

$\displaystyle\mathrm{H}^{2}(Y^{0124}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle=\mathrm{H}^{2}(X^{0124}_{[\pm++],k^{\mathrm{ac}}},\Lambda(1))\oplus\mathrm{H}^{0}(X^{012345}_{k^{\mathrm{ac}}},\Lambda)$
$\displaystyle=\mathrm{H}^{0}(X^{B}_{0}(p)),\Lambda)\otimes\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda(1))\otimes\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda)$
$\displaystyle\oplus\mathrm{H}^{0}(X^{B}_{0}(p)),\Lambda)\otimes\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda)\otimes\mathrm{H}^{2}(\mathbf{P}^{1}(X^{B}_{+}),\Lambda(1))$
$\displaystyle\oplus\mathrm{H}^{0}(X^{012345},\Lambda).$

  * •

For $\mathrm{H}^{2}(Y^{i(i+1)(i+2)(i+3)(i+5)}_{k^{\mathrm{ac}}},\Lambda(1))$,
we again only make it explicit for $i=0$. By Proposition 4.1 $(6)$, we have

$\displaystyle\mathrm{H}^{2}(Y^{01235}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle=\mathrm{H}^{2}(\mathbf{P}^{1}(X^{01235}_{[-\pm\pm],k^{\mathrm{ac}}}),\Lambda(1))$
$\displaystyle=\mathrm{H}^{0}(X^{01235}_{[-\pm\pm],k^{\mathrm{ac}}}),\Lambda)$
$\displaystyle=\mathrm{H}^{0}(\mathbf{P}^{1}(X^{B}_{-}),\Lambda)\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda).$

  * •

By Proposition 4.1 $(7)$, we have

$\displaystyle\mathrm{H}^{2}(Y^{012345}_{k^{\mathrm{ac}}},\Lambda(1))$
$\displaystyle=\mathrm{H}^{0}(X^{012345}_{k^{\mathrm{ac}}},\Lambda)$
$\displaystyle=\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)\otimes\mathrm{H}^{0}(X^{B}_{0}(p),\Lambda)$

It follows from the above calculation that $G_{k}$ acts trivially on
$\mathrm{H}^{2}(a_{1*}\Lambda(1))$. Using this we can immediately verify that
Assumption 2.3 is satisfied for the spectral sequence $\mathbb{E}$. ∎

The above lemma allows us to apply the exact sequence in (2.15) to calculate

$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{3}(X^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))_{\mathfrak{m}_{\underline{\mathbf{f}}^{[p]}}}).$

For $i=1,2,3$, we will set

$\mathrm{T}^{[p]}_{i}=\mathrm{H}^{1}(X_{\mathbf{Q}^{\mathrm{ac}}},\mathcal{O}_{i}(1))_{\mathfrak{m}^{[p]}_{i}}$

and let $\mathrm{T}^{[p]}_{i,n}=\mathrm{T}^{[p]}_{i}/I^{[p]}_{i,n}$ be the
natural quotient. Let

$\mathrm{M}^{[p]}(\underline{\mathbf{f}})=\mathrm{T}^{[p]}_{1}\otimes\mathrm{T}^{[p]}_{2}\otimes\mathrm{T}^{[p]}_{3}$

which is a $G_{\mathbf{Q}}$-module over
$\mathcal{O}=\mathcal{O}_{1}\otimes\mathcal{O}_{2}\otimes\mathcal{O}_{3}$ and
let

$\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})=\mathrm{T}^{[p]}_{1,n}\otimes\mathrm{T}^{[p]}_{2,n}\otimes\mathrm{T}^{[p]}_{3,n}$

which is a $G_{\mathbf{Q}}$-module over
$\mathcal{O}_{n}=\mathcal{O}_{1,n}\otimes\mathcal{O}_{2,n}\otimes\mathcal{O}_{3,n}$.

######  Lemma 4.6.

Let $p$ be an $n$-admissible prime for $\underline{\mathbf{f}}$. The module
$\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})$ is unramified viewed as an
$\mathcal{O}_{n}[G_{K}]$-module and is isomorphic to

$\mathcal{O}_{n}\oplus\mathcal{O}^{\oplus 3}_{n}(1)\oplus\mathcal{O}^{\oplus
3}_{n}(2)\oplus\mathcal{O}_{n}(3).$

The singular quotient
$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
is free of rank $3$ over $\mathcal{O}_{n}$

###### Proof.

The first part follows from the fact that $p$ is an $n$-admissible prime for
$\underline{\mathbf{f}}$. In particular, $(3)$ in the definition of
$n$-admissible prime for $\underline{\mathbf{f}}$ implies that
$\mathrm{T}^{[p]}_{i,n}\cong\mathcal{O}_{i,n}\oplus\mathcal{O}_{i,n}(1)$ as an
$\mathcal{O}_{K}$-module. See also Lemma 5.6. For the second part, we have

$\displaystyle\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))$
$\displaystyle\cong\mathrm{H}^{1}(I_{K},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))^{G_{k}}$
$\displaystyle\cong\mathrm{Hom}(I_{K},\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))^{G_{k}}$
$\displaystyle\cong\mathrm{Hom}(\Lambda(1),\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))^{G_{k}}.$

Then the result follows from the first part. ∎

The following theorem is the _ramified arithmetic level raising for the triple
product of Shimura curves_ and is one of the main results in this article.
Recall that we denote by $\Phi$ the group of connected components of the
special fiber of the Néron model of the Jacobian of the Shimura curve $X_{K}$.

######  Theorem 4.7.

Let $p$ be an $n$-admissible prime for the triple
$\underline{\mathbf{f}}=(f_{1},f_{2},f_{3})$. For $i=1,2,3$, assume that

  1. (1)

the maximal ideal $\mathfrak{m}_{i}$ is residually irreducible;

  2. (2)

each $S^{B}_{2}(N^{+},\mathcal{O}_{i})_{\mathfrak{m}_{i}}$ is free of rank $1$
over $\mathbb{T}_{\mathfrak{m}_{i}}$.

Then we have an isomorphism

$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))\cong\oplus^{3}_{j=1}(\otimes^{3}_{i=1}S^{B}_{2}(N^{+},\mathcal{O}_{i}){/I_{i,n}}).$

More canonically, we have the isomorphism

$\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{M}^{[p]}_{n}(\underline{\mathbf{f}})(-1))\cong\oplus^{3}_{j=1}(\otimes^{3}_{i=1}\Phi_{\mathcal{O}_{i}}/I^{[p]}_{i,n}).$

###  4.4. Proof of the arithmetic level raising

To prove the preceding theorem, we need a different presentation of the
potential map. Recall that we have the following exact sequence

$A_{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\nabla}A^{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\eta}\mathrm{H}^{1}_{\mathrm{sing}}(K,\mathrm{H}^{3}(X^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}})\rightarrow
0$

where

$\displaystyle
A^{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}=\mathrm{im}[\mathrm{H}^{2}(a_{1*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\tau}\mathrm{H}^{4}(a_{0*}\Lambda(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}]^{G_{k}}$
$\displaystyle
A_{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}=\mathrm{im}[\mathrm{H}^{2}(a_{0*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\rho}\mathrm{H}^{2}(a_{1*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}]^{G_{k}}.$

######  Lemma 4.8.

We have the following statements.

  1. (1)

$A_{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\cong\mathrm{coker}[\mathrm{H}^{0}(a_{1*}\Lambda)_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}]^{G_{k}}$.

  2. (2)

$A^{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\cong\ker[\mathrm{H}^{4}(a_{0*}\Lambda(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\rho}\mathrm{H}^{4}(a_{1*}\Lambda(2))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}]^{G_{k}}$.

  3. (3)

The map $\nabla$ is induced by the composite $\tau\circ\rho$ under the above
isomorphisms.

###### Proof.

By the previous reasoning,
$\mathrm{H}^{2}(X^{3}\otimes{K^{\mathrm{ac}}},\Lambda(1))$ is zero as it is
Eisenstein and therefore $\mathbb{E}^{0,2}_{2}(1)$ is zero. It follows then
that

$\ker[\mathrm{H}^{2}(a_{0*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\rho}\mathrm{H}^{2}(a_{1*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}]\cong\mathrm{im}[\mathrm{H}^{0}(a_{1*}\Lambda)_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\xrightarrow{\tau}\mathrm{H}^{2}(a_{0*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}].$

Then $(1)$ is clear. The proof of $(2)$ is completely the same using the fact
that $\mathrm{H}^{4}(X^{3}\otimes{K^{\mathrm{ac}}},\Lambda(2))$ and thus
$\mathbb{E}^{0,4}_{2}(2)$ vanish this time. The rest of the claims follow by
constructions. ∎

######  Lemma 4.9.

We have the following isomorphisms

  1. (1)

$A_{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\cong\oplus^{5}_{i=0}\mathrm{H}^{0}(X^{i(i+1)(i+2)(i+3)(i+5)}_{k},\Lambda)_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\oplus\mathrm{H}^{0}(X^{012345}_{k},\Lambda)_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}$;

  2. (2)

$A^{2}(Y_{k},\Lambda)^{0}_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\cong\oplus^{5}_{i=0}\mathrm{H}^{2}(X^{i(i+1)(i+2)(i+3)(i+5)}_{k},\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\oplus\mathrm{H}^{0}(X^{012345}_{k},\Lambda)_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}$.

###### Proof.

The second part follows from the first part by duality and we only prove the
first part. We have by Proposition 4.1

$\mathrm{H}^{2}(a_{0*}\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}=\oplus^{5}_{i=0}\mathrm{H}^{2}(Y^{i(i+1)(i+2)}_{k^{\mathrm{ac}}},\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\oplus\mathrm{H}^{2}(Y^{024}_{k^{\mathrm{ac}}},\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}\oplus\mathrm{H}^{2}(Y^{135}_{k^{\mathrm{ac}}},\Lambda(1))_{\mathfrak{m}^{[p]}_{\underline{\mathbf{f}}}}.$

By Proposition 4.1 $(1)$ and Proposition 4.2, we have

$\displaystyle\mathrm{H}^{2}(Y^{i(i+1)(i+2)}_{k^{\mathrm{ac}}},\Lambda(1))=\mathrm{H}^{2}(X^{i(i+1)(i+2)}_{k^{\mathrm{ac}}},\Lambda(1))\oplus\mathrm{H}^{0}(X^{i(i+1)(i+2)(i+3)(i+5)}_{k^{\mathrm{ac}}},\Lambda);$
$\displaystyle\mathrm{H}^{2}(Y^{024}_{k^{\mathrm{ac}}},\Lambda(1))=\mathrm{H}^{2}(X^{024}_{k^{\mathrm{ac}}},\Lambda(1))\oplus\mathrm{H}^{0}(X^{01234}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{01245}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{02345}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{012345}_{k^{\mathrm{ac}}},\Lambda);$
$\displaystyle\mathrm{H}^{2}(Y^{135}_{k^{\mathrm{ac}}},\Lambda(1))=\mathrm{H}^{2}(X^{135}_{k^{\mathrm{ac}}},\Lambda(1))\oplus\mathrm{H}^{0}(X^{01235}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{01345}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{12345}_{k^{\mathrm{ac}}},\Lambda)\oplus\mathrm{H}^{0}(X^{012345}_{k^{\mathrm{ac}}},\Lambda).$

On the other hand, we have

$\displaystyle\mathrm{H}^{0}(a_{1*}\Lambda)=$
$\displaystyle\oplus^{5}_{i=0}\mathrm{H}^{0}(X^{i(i+1)(i+2)(i+3)}_{k^{\mathrm{ac}}},\Lambda)$
$\displaystyle\oplus\oplus^{5}_{i=0}\mathrm{H}^{0}(X^{i(i+1)(i+2)(i+4)}_{k^{\mathrm{ac}}},\Lambda)$