# Flach system on Quaternionic Hilbert–Blumenthal surfaces and distinguished
periods

Haining Wang  address: Shanghai Center for Mathematical Sciences, Fudan
University, No,2005 Songhu Road, Shanghai,200438, China.
<EMAIL_ADDRESS>
###### Abstract.

We study arithmetic properties of certain quaternionic periods of Hilbert
modular forms arising from base change of elliptic modular forms. These
periods which we call the distinguished periods are closely related to the
notion of distinguished representation that appear in work of
Harder–Langlands–Rapoport, Lai, Flicker–Hakim on the Tate conjectures for the
Hilbert–Blumenthal surfaces and their quaternionic analogues. In particular,
we prove an integrality result on the ratio of the distinguished period and
the quaternionic Petersson norm associated to the modular form. Our method is
based on an Euler system argument initiated by Flach by producing elements in
the motivic cohomologies of the quaternionic Hilbert–Blumenthal surfaces with
control of their ramification behaviours.

###### 2000 Mathematics Subject Classification:

Primary 11G18, 11R34, 14G35

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Strategy of the proof
    3. 1.3 Notations and conventions
  2. 2 Quaternionic Hilbert–Blumenthal surfaces
    1. 2.1 A quaternionic Shimura surface
    2. 2.2 Integral model for the quaternionnic Shimura surface
    3. 2.3 Supersingular locus of the quaternionic Shimura surface
  3. 3 Quaternionic Hirzebruch–Zagier divisor
    1. 3.1 Shimura curves and Drinfeld uniformization
    2. 3.2 Quaternionic Hirzebruch–Zagier divisor
    3. 3.3 Tate cycles on the quaternionic Shimura surface
  4. 4 Flach classes and reciprocity formula
    1. 4.1 Flach class of the quaternionic Shimura surface
    2. 4.2 Local bevaviours of Flach classes
  5. 5 Distinguished representation and base change
    1. 5.1 Dinstinguished representation
    2. 5.2 Asai representation and base change
  6. 6 Bounding the adjoint Selmer groups
    1. 6.1 Generalities on Selmer groups
    2. 6.2 Statement of the main result
    3. 6.3 The Flach system argument
    4. 6.4 Comparison of quaternionic periods

##  1\. Introduction

In this article, we study arithmetic properties of certain integral periods of
quaternionic Hilbert modular forms arising from base change. These periods are
closely related to the notion of distinguished representation which play a
prominent role in the work of Harder–Langlands–Rapoport [HLR], Lai [Lai],
Flicker–Hakim [FH] on the Tate conjectures for the Hilbert–Blumenthal surfaces
and their quaternionic analogues. We will therefore call these periods the
distinguished periods. While the representation theoretic properties of these
distinguished periods have been amply studied in the literature and have also
been generalized and studied for higher rank groups using the relative trace
formula, the arithmetic properties seem to be less well understood. We will
provide some results in this direction below. More precisely, we will compare
the distinguished period with another well-known period, namely the Petersson
norm of the associated definite quaternionic modular form. We will show that
the $\lambda$-adic valuation of the distinguished period will be greater than
or equal to the $\lambda$-adic valuation of the Petersson norm for a prime
$\lambda$ of the Hecke field over a fixed rational prime $l$. It is well-known
that the Petersson norm of a definite quaternionic modular form can be related
to congruence module of the quaternionic modular form and hence our result
shows that the distinguished period also captures this information. The
distinguished period should also contain the information of the base change
congruence module studied by Hida [Hid] and Urban–Tilouine [UT], but we will
address this question in another occasion.

This result should be viewed as an analogue of the integrality result of ratio
of Petersson norms of modular and indefinite quaternionic modular forms proved
by Prasana in [Pra] where he also shows that the positive valuation part of
this ratio should be related to the local Tamagawa factors which measures
quantitatively the level lowering congruences of the modular form. Our result
should also be reminiscent of the result of Ribet–Takahashi [RT] comparing the
degree of the modular parametrization of elliptic curve by a classical modular
curve and the degree of the parametrization of the elliptic curve by an
indefinite Shimura curve. The results of Prasana have been extended to a
program to understand of ratio of Petersson norms of general quaternionic
Hilbert modular forms, [IP1], [IP2]. It would be interesting to see if our
method could be useful in this program.

The method to study these distinguished periods is based on the so-called
Flach system of Jacquet–Langlands type introduced in the companion article
[Wang]. We will realize the $\lambda$-adic valuation of the distinguished
period as a natural bound for the length of certain subspace of the Bloch–Kato
Selmer group of the middle degree cohomology group of a suitable quaternionic
Hilbert–Blumenthal surface. This subspace turns out to be isomorphic to the
Bloch–Kato Selmer group of the adjoint Galois representation attached to the
definite quaternionic modular form whose base-change defines the distinguished
period. The length of the latter Selmer group can be determined using the
Taylor–Wiles method and is equal to the $\lambda$-adic valuation of the
Petersson norm of the definite quaternionic modular form and hence is less
than equal to the $\lambda$-adic valuation of the distinguished period. This
shows the desired integrality result of the ratio of periods as mentioned
above. We remark that Flach system of Jacquet–Langlands type is a simplified
geometric Euler system in the sense of [Weston3] that should reflect
Jacquet–Langlands correspondence in the cohomologies of Shimura varieties. In
[Wang], this is reflected using arithmetic level raising results on the
product of two Shimura curves. In this article, this is realized using the
Tate conjecture for the special fiber of the quaternionic Hilbert–Blumenthal
surface proved in [TX], see also [Lan]. The version of the Tate conjectures
for special fibers of Shimura varieties needed to construct similar Flach
systems as in this article have been proved for a large class of Shimura
varieties in [TX] and [XZ], therefore we expect the strategy in this article
and [Wang] should be useful to study the integrality questions of ratio of
periods coming from definite Shimura sets for other reductive groups. In a
very similar setting, we will construct a Flach system of Jacquet–Langlands
type on Picard modular surfaces and study the period of automorphic forms on
$\mathrm{U}(3)$ coming from theta lifts of automorphic forms of
$\mathrm{U}(2)$. We will also consider distinguished period arising from base
change to a real quadratic extension of a unitary group $\mathrm{U}(n)$, our
method seems to also work in this case and we will pursue this in another
occasion.

It should also be interesting to note that usually automorphic periods have
good connections to special values of $L$-functions and Selmer groups can be
studied in terms of these special values through Euler system arguments
relating these automorphic periods and hence the special values to the size of
the Selmer group via the so-called reciprocity formula. Here we turn things
around and use known bounds for Selmer groups to study integral periods.

###  1.1. Main results

We will introduce some notations before we state our main results. Let $l\geq
5$ be a fixed prime. Let $f$ be a normalized newform in
$\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ whose associated automorphic
representation of $\mathrm{GL}_{2}(\mathbf{A})$ is denoted by $\pi$. We will
assume that $\mathrm{N}$ admits a decomposition
$\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$ with
$(\mathrm{N}^{+},\mathrm{N}^{-})$ such that $\mathrm{N}^{-}$ is square-free
and consists of odd number of prime factors. Let $\overline{\mathrm{B}}$ be
the definite quaternion algebra of discriminant $\mathrm{N}^{-}$. Then $f$
admits a normalized Jacquet–Langlands transfer $f^{\dagger}$ to an automorphic
form for the group
$\mathrm{G}(\overline{\mathrm{B}})=\overline{\mathrm{B}}^{\times}$ whose
associated automorphic representation will be denoted by
$\pi^{\circ}=\pi^{\overline{\mathrm{B}}}$. We will be concerned with the base
change $\phi^{\dagger}$ of $f^{\dagger}$ by a real quadratic field
$\mathrm{F}$. Let $\mathrm{E}=\mathbf{Q}(f)$ be a Hecke field of $f$ and
$\iota_{l}:\overline{\mathbf{Q}}\hookrightarrow\mathbf{C}_{l}$ be a fixed
embedding which defines a place $\lambda$ in $\mathrm{E}$. We will denote by
$\mathrm{E}_{\lambda}$ the completion of $\mathrm{E}$ at $\lambda$ and by
$\mathcal{O}_{\lambda}$ its valuation ring. We will also denote by $\lambda$
the maximal ideal of $\mathcal{O}_{\lambda}$ and will fix a uniformizer
$\varpi$ for the ideal $\lambda$. Let $k_{\lambda}$ be the residue field of
$\mathcal{O}_{\lambda}$.

Let $\overline{\mathrm{Q}}=\overline{\mathrm{B}}\otimes\mathrm{F}$ be the
definite quaternion algebra obtained by base change to $\mathrm{F}$ and let
$\mathrm{G}(\overline{\mathrm{Q}})$ be the algebraic group defined by the Weil
restriction of $\overline{\mathrm{Q}}^{\times}$ from $\mathrm{F}$ to
$\mathbf{Q}$. Then $\phi^{\dagger}$ is contained in the automorphic
representation $\pi^{\overline{\mathrm{Q}}}$ of
$\mathrm{G}(\overline{\mathrm{Q}})$ obtained by base change of $\pi^{\circ}$
to $\mathrm{F}$. The automorphic forms $\phi^{\dagger}$ can be realized as a
smooth function valued in $\mathcal{O}_{\lambda}$ on certain Shimura set
$\mathrm{Z}(\overline{\mathrm{Q}})$ associated to the group
$\mathrm{G}(\overline{\mathrm{Q}})$ with suitable level structure. The Shimura
set $\mathrm{Z}(\overline{\mathrm{B}})$ defined similarly from
$\mathrm{G}(\overline{\mathrm{B}})$ can be embedded in
$\mathrm{Z}(\overline{\mathrm{Q}})$ and this can be seen as an analogue of the
Hirzebruch–Zagier morphism for the Shimura sets. We define the distinguished
period by

$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}\phi^{\dagger}(z).$

Recall a cupsidal automorphic representation $\Pi$ of
$\mathrm{GL}_{2}(\mathbf{A}_{\mathrm{F}})$ is called distinguished if there is
a function $\phi$ in $\Pi$ such that

$\mathcal{P}^{\mathrm{GL}_{2}}_{\mathrm{dis}}(\phi)=\int_{\mathrm{GL}_{2}(\mathbf{Q})\backslash\mathrm{GL}_{2}(\mathbf{A})}\phi(g)dg$

is non-vanishing. It is known that $\Pi$ is distinguished only if
$\Pi_{\infty}$ is in the discrete series and $\Pi$ comes from a cupsidal
automorphic representation $\pi$ of $\mathrm{GL}_{2}(\mathbf{A})$ via base
change. The notion of distinguished representation has played an prominent
role in the proof of the Tate conjecture of Hilbert–Blumenthal surfaces by
[HLR] and the quaternionic Hilbert–Blumenthal surfaces by [Lai] and [FH].
Moreover it is known that being distinguished is preserved under
Jacquet–Langlands correspondence, see Proposition 5.1 for the precise
statement.

The above discussion should have explained that the distinguished period is of
great arithmetic and representation theoretic interest and also why we stick
to the situation when the period is defined by an automorphic form coming from
base change. This set-up also leads us to consider a closely related period of
$f$ namely the Petersson norm of $f^{\dagger}$, this is defined by

$\mathcal{P}(f^{\dagger})=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}f^{\dagger}(z)^{2}.$

and is referred to as the quaternionic period of $f$ in [Wang]. Our main
result will be concerned with the $\lambda$-integrality of the ratio
$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})/\mathcal{P}(f^{\dagger})$ of these
two periods. Before we state it, we introduce several assumptions that are
imposed on the Galois representation attached to $f$ or equivalently to
$f^{\dagger}$. Let
$\rho_{\pi^{\circ}}:\mathrm{G}_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\mathrm{E}_{\lambda})=\mathrm{GL}(\mathrm{V}_{\pi^{\circ}})$
be the Galois representation associated to $f^{\dagger}$ by the
Eichler–Shimura construction and Jacquet–Langlands correspondence. Recall also
that $\pi^{\circ}$ is the automorphic representation corresponding to
$f^{\dagger}$. We denote by $\overline{\rho}_{\pi^{\circ}}$ the residual
representation of $\rho_{\pi^{\circ}}$. Let $\Sigma^{+}$ be the set of primes
dividing $\mathrm{N}^{+}$. Let $\Sigma^{-}_{\mathrm{ram}}$ be the set of $r$
dividing $\mathrm{N}^{-}$ such that $l\mid r^{2}-1$ and
$\Sigma^{-}_{\mathrm{mix}}$ be the set of $r$ dividing $\mathrm{N}^{-}$ such
that $l\nmid r^{2}-1$.

######  Assumption 1.1.

We make the following assumptions on $\bar{\rho}_{\pi^{\circ}}$:

  1. (1)

$\bar{\rho}_{\pi^{\circ}}|_{\mathrm{G}_{\mathbf{Q}(\zeta_{l})}}$ is absolutely
irreducible;

  2. (2)

The image of $\bar{\rho}_{\pi^{\circ}}$ contains
$\mathrm{GL}_{2}(\mathbf{F}_{l})$;

  3. (3)

$\overline{\rho}_{\pi^{\circ}}$ is minimal at primes in $\Sigma^{+}$ in the
sense that all the liftings of
$\overline{\rho}_{\pi^{\circ}}|_{\mathrm{G}_{\mathbf{Q}_{r}}}$ are minimally
ramified for $r\in\Sigma^{+}$;

  4. (4)

$\overline{\rho}_{\pi^{\circ}}$ is ramified at primes in
$\Sigma^{-}_{\mathrm{ram}}$.

Let $\mathrm{T}_{\pi^{\circ}}$ be a suitable Galois stable lattice in
$\rho_{\pi^{\circ}}$ which we will make precise in the main body of this
article and $\mathrm{T}_{\pi^{\circ},n}$ be the reduction of
$\mathrm{T}_{\pi^{\circ}}$ modulo $\lambda^{n}$. We will let
$\mathrm{M}_{n}=\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ},n})$ be the symmetric
square of $\mathrm{T}_{\pi^{\circ},n}$.

######  Theorem 1.2.

Let $f\in\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ be a newform of weight $2$
with $\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$ such that $\mathrm{N}^{-}$ is
squarefree and has odd number of prime factors. Let $f^{\dagger}$ be the
automorphic form on $\mathrm{Z}(\overline{\mathrm{B}})$ corresponding to $f$
under the Jacquet–Langlands correspondence and let $\phi^{\dagger}$ be the
base change of $f^{\dagger}$ considered as an automorphic form on
$\mathrm{Z}(\overline{\mathrm{Q}})$.

  1. (1)

We assume that the residual Galois representation
$\overline{\rho}_{\pi^{\circ}}$ satisfies Assumption 1.1;

  2. (2)

We further assume that $\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})=0$ for every
$n\geq 1$ where
$\Delta_{n}=\mathrm{Gal}(\mathbf{Q}(\mathrm{M}_{n})/\mathbf{Q})$ for the
splitting field $\mathbf{Q}(\mathrm{M}_{n})$ of the Galois module
$\mathrm{M}_{n}$.

Then we have the following inequality

$\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))\geq\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger})).$

It should also be interesting to study the integrality question of the ratio
between the distinguished period $\mathcal{P}_{\mathrm{dis}}(f_{\mathrm{F}})$
for the base-change $f_{\mathrm{F}}$ of $f$ and the classical Petersson norm
$\langle f,f\rangle$ of $f$. The $p$-adic reciprocity formuals proved in
[LLZ], [LSZ] should be more suitable for this purpose. Our main result is
clearly the compact version for this more classical ratio under the
Jacquet–Langlands correspondence. We also suspect that when the $\lambda$-adic
valuation of this ratio is positive, then the positive part may be related to
the base-change congruence ideal as defined by Hida [Hid]. We hope to return
to this question in the near future.

###  1.2. Strategy of the proof

We now briefly outline the proof for the main integrality result in Theorem
1.2. Let $\mathrm{V}_{\pi^{\circ}}$ be the representation space of
$\rho_{\pi^{\circ}}$ and recall that we have fixed a lattice
$\mathrm{T}_{\pi^{\circ}}$ of $\mathrm{V}_{\pi^{\circ}}$. We define the
divisible Galois module $\mathcal{M}_{\pi^{\circ}}$ by

$0\rightarrow\mathrm{T}_{\pi^{\circ}}\rightarrow\mathrm{V}_{\pi^{\circ}}\rightarrow\mathcal{M}_{\pi^{\circ}}\rightarrow
0.$

The key step towards proving the integrality of the ratio of the distinguished
period and the quaternionic period is to prove the following theorem bounding
the Bloch–Kato Selmer group of $\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$ in
terms of $\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$.

######  Theorem 1.3.

Let $\nu=\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$
and $\eta=\varpi^{\nu}$.

  1. (1)

We assume that the residual Galois representation
$\overline{\rho}_{\pi^{\circ}}$ is absolutely irreducible.

  2. (2)

We further assume that

$\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})=0$

for every $n\geq 1$.

Then we have

$\mathrm{length}_{\mathcal{O}_{\lambda}}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\leq\nu.$

The proof of this theorem relies on the construction of the so-called Flach
system of Jacquet–Langlands type on certain Shimura surface which we now
describe in more details. Let $p$ be a prime which is _inert_ in $\mathrm{F}$.
We can obtain another quaternion algebra $\mathrm{B}$ by switching the local
invariants of $\overline{\mathrm{B}}$ at $p$ and $\infty$. Then
$\mathrm{Q}=\mathrm{B}\otimes\mathrm{F}$ is a totally indefinite quaternion
algebra over $\mathrm{F}$ obtained from $\overline{\mathrm{Q}}$ by switching
the invariants at the two archimedean places of $\mathrm{F}$. Note that
$\pi^{\overline{\mathrm{Q}}}$ admits a Jacquet–Langlands transfer
$\pi^{\mathrm{Q}}$ as a representation of $\mathrm{G}(\mathrm{Q})$ which is
the algebraic group given by the Weil restriction of $\mathrm{Q}^{\times}$
from $\mathrm{F}$ to $\mathbf{Q}$. Let
$\rho_{\pi^{\mathrm{Q}}}:\mathrm{G}_{\mathrm{F}}\rightarrow\mathrm{GL}_{2}(\mathrm{E}_{\lambda})=\mathrm{GL}(\mathrm{V}_{\pi^{\mathrm{Q}}})$
be the Galois representation of $\mathrm{G}_{\mathrm{F}}$ attached to
$\pi^{\mathrm{Q}}$. Then it is well-known that the Asai representation
$\mathrm{As}(\mathrm{\rho_{\pi^{\mathrm{Q}}}})=\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})$
can be realized on the middle degree cohomology of certain Shimura surface
$\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$ which we will refer to as the
quaternionic Hilbert–Blumenthal surface. Roughly speaking these Shimura
surfaces defines a moduli problem $\mathrm{X}(\mathrm{Q})$ which classifies
abelian fourfold with quaternionic multiplication by $\mathrm{Q}$ and
additional structures. The integral cohomology of
$\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$ with coefficient in
$\mathcal{O}_{\lambda}$ defines a natural lattice
$\mathrm{T}_{\pi^{\mathrm{Q}}}$ in $\mathrm{V}_{\pi^{\mathrm{Q}}}$ and we
define the divisible Galois module $\mathcal{M}_{\pi^{\mathrm{Q}}}$ using the
exact sequence

$0\rightarrow\mathrm{T}_{\pi^{\mathrm{Q}}}\rightarrow\mathrm{V}_{\pi^{\mathrm{Q}}}\rightarrow\mathcal{M}_{\pi^{\mathrm{Q}}}\rightarrow
0.$

In our set-up, the Asai representation
$\mathrm{As}(\mathcal{M}_{\pi^{\mathrm{Q}}})(-1)$ splits into

$\mathrm{As}(\mathcal{M}_{\pi^{\mathrm{Q}}})(-1)=\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})\oplus\mathrm{E}_{\lambda}/\mathcal{O}_{\lambda}(\omega_{\mathrm{F}/\mathbf{Q}})$

where $\omega_{\mathrm{F}/\mathbf{Q}}$ is the quadratic character associated
to $\mathrm{F}$. Since $\mathrm{As}(\mathrm{T}_{\pi^{\mathrm{Q}}})$ is
realized in the $\pi^{\mathrm{Q}}$-isotopic part
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2))$
of the middle degree cohomology of $\mathrm{X}(\mathrm{Q})$ and
$\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})(1)$ is the Kummer dual of
$\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$, we are naturally led to
construct elements in the Galois cohomology group
$\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2)))$.
For this purpose, we consider the motivic cohomology group
$\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathbf{Z}(2)))$
for the surface $\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$ whose elements
consist of pairs $(\mathrm{Z},f)$ where $\mathrm{Z}$ is a curve on
$\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$ and $f$ is a rational function on
$\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$ with trivial Weil divisors. Let
$\mathrm{X}(\mathrm{B})\otimes\mathbf{Q}$ be the Shimura curve associated to
the indefinite quaternion algebra $\mathrm{B}$, then we have the quaternionic
Hirzebruch–Zagier morphism

$\theta:\mathrm{X}(\mathrm{B})\otimes\mathbf{Q}\rightarrow\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$

which can even be defined integrally. Then we define the Flach element by the
pair

$\Theta^{[p]}=(\theta_{\ast}\mathrm{X}(\mathrm{B})\otimes\mathbf{Q},p).$

The construction of this element is inspired by that of [Flach1] which is used
to bound the symmetric square Selmer group for an elliptic curve. However, for
our purpose, we do not need to include the Siegel modular unit in our
definition (which is also not available for us as our cycle class is given by
a compact Shimura curve). On the other hand, the original construction of the
Flach element has applications to Iwasawa theory. In a closely related
setting, this is carried out in [LLZ]. Note that they consider the case when
the Hilbert modular form is not arising from base change and hence our result
is of complimentary nature to their work. There is an Abel–Jacobi map

$\mathrm{AJ}_{\pi^{\mathrm{Q}}}:\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathbf{Z}(2))\rightarrow\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2)))$

defined using the Chern character map from the $K$-theory to étale cohomology.
The class $\kappa^{[p]}$ defined by
$\mathrm{AJ}_{\pi^{\mathrm{Q}}}(\Theta^{[p]})$ is called the Flach class in
our setting. The local ramification behaviour of the class $\kappa^{[p]}$ will
be analyzed. In particular at $p$, we are concerned with singular quotient

$\partial_{p}(\kappa^{[p]})\in\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{p},\mathcal{O}_{\lambda}(2)))$

of $\kappa^{[p]}$. Note that
$\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{p},\mathcal{O}_{\lambda}(2))$
is isomorphic to
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))^{\mathrm{G}_{\mathbf{F}_{p}}}$.
If we choose $p$ carefully, then
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))^{\mathrm{G}_{\mathbf{F}_{p}}}$
is isomorphic to
$\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})][\pi^{\overline{\mathrm{Q}}}]$
the $\pi^{\overline{\mathrm{Q}}}$-isotypic component of the space of the
$\mathcal{O}_{\lambda}$-valued functions on
$\mathrm{Z}(\overline{\mathrm{Q}})$ by the Tate conjecture for
$\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p}$ proved in [TX]. In
this case, we can view $\partial_{p}(\kappa^{[p]})$ as an element of
$\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})][\pi^{\overline{\mathrm{Q}}}]$.
There is a natural pairing $(\cdot,\cdot)$ on this space
$\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})]$. Then we can prove
the following key reciprocity formula

$(\partial_{p}(\kappa^{[p]}),\phi^{\dagger})=\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}).$

Then an Euler system argument allows us to show that
$\eta=\varpi^{\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))}$
annihilates the Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
under some technical assumptions. Using this annihilation result, it can be
shown that
$\mathrm{leng}_{\mathcal{O}_{\lambda}}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
admits an upper-bound given by
$\nu=\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$.

On the other hand, the Selmer group of the adjoint representation
$\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$ is closely related to the
deformation theory of the residual representation
$\overline{\rho}_{\pi^{\circ}}$. In fact, as a by-product of the so-called
$\mathrm{R}=\mathrm{T}$ theorem proved using the Taylor–Wiles method, it can
be shown that
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
has a subspace
$\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
whose length is given by the $\lambda$-valuation of certain congruence number
$\eta(\mathrm{N}^{+},\mathrm{N}^{-})$ that detects congruences between the
modular form $f$ and other modular forms in
$\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ which are new at primes dividing
$\mathrm{N}^{-}$. Finally, it is well–known that the $\lambda$-valuation of
this congruence number is exactly that of the Petersson norm of $f^{\dagger}$
under our assumptions. This finally proves that
$\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))\geq\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger}))$.

###  1.3. Notations and conventions

We will use common notations and conventions in algebraic number theory and
algebraic geometry. The cohomology theories appear in this article will be
understood as the étale cohomologies. For a field $\mathrm{K}$, we denote by
$\mathrm{K}^{\mathrm{ac}}$ the separable closure of $\mathrm{K}$ and denote by
$\mathrm{G}_{\mathrm{K}}:=\mathrm{Gal}(\mathrm{K}^{\mathrm{ac}}/\mathrm{K})$
the Galois group of $\mathrm{K}$. For a finite place $v$ of $\mathrm{K}$,
$\mathrm{Frob}_{v}$ is the arithmetic Frobenius at $v$.

We let $\mathbf{A}_{\mathrm{F}}$ be the ring of adèles over a number field
$\mathrm{F}$ and $\mathbf{A}^{\infty}_{\mathrm{F}}$ be the subring of finite
adèles. For a finite place $v$ of $\mathrm{F}$,
$\mathbf{A}^{\infty,v}_{\mathrm{F}}$ is the prime-to-$v$ part of
$\mathbf{A}^{\infty}_{\mathrm{F}}$. When $\mathrm{F}=\mathbf{Q}$, then we will
omit the subscript $\mathrm{F}$ from these notations.

When $\mathrm{K}$ is a local field, we denote by $\mathcal{O}_{\mathrm{K}}$
its valuation ring and by $k$ its residue field. We let
$\mathrm{I}_{\mathrm{K}}$ be the inertia subgroup of
$\mathrm{G}_{\mathrm{K}}$. For a $\mathrm{G}_{\mathrm{K}}$-module
$\mathrm{M}$,

  1. (1)

the finite part $\mathrm{H}^{1}_{\mathrm{fin}}(\mathrm{K},\mathrm{M})$ of
$\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$ is defined to be
$\mathrm{H}^{1}(k,\mathrm{M}^{\mathrm{I}_{\mathrm{K}}})$;

  2. (2)

the singular part $\mathrm{H}^{1}_{\mathrm{sin}}(\mathrm{K},\mathrm{M})$ of
$\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$ is defined to be the quotient of
$\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$ by the image of
$\mathrm{H}^{1}_{\mathrm{fin}}(\mathrm{K},\mathrm{M})$ in
$\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$ via inflation;

  3. (3)

let $x$ be an element of $\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$, we call the
image of $x$ in $\mathrm{H}^{1}_{\mathrm{sin}}(\mathrm{K},\mathrm{M})$ the
singular residue of $x$ written as $\partial_{p}(x)$.

Let $\mathrm{K}$ be a number field and let $\mathrm{K}_{v}$ be the completion
of $\mathrm{K}$ at a place $v$. Suppose
$x\in\mathrm{H}^{1}(\mathrm{K},\mathrm{M})$, then we will write
$\mathrm{loc}_{v}(x)$ the image of $x$ under the restriction map

$\mathrm{loc}_{v}:\mathrm{H}^{1}(\mathrm{K},\mathrm{M})\rightarrow\mathrm{H}^{1}(\mathrm{K}_{v},\mathrm{M}).$

We will denote by $\partial_{v}(x)$ the image of $x$ in
$\mathrm{H}^{1}_{\mathrm{sin}}(\mathrm{K}_{v},\mathrm{M})$ under the composite
map of

$\mathrm{loc}_{v}:\mathrm{H}^{1}(\mathrm{K},\mathrm{M})\rightarrow\mathrm{H}^{1}(\mathrm{K}_{v},\mathrm{M})$

and the natural map
$\mathrm{H}^{1}(\mathrm{K}_{v},\mathrm{M})\rightarrow\mathrm{H}^{1}_{\mathrm{sin}}(\mathrm{K}_{v},\mathrm{M})$.

### Acknowledgements

We would like to thank Ming-Lun Hsieh for introducing the notion of
distinguished representation to the author and pointing out useful references.
We would like to thank Matteo Tamiozzo for useful email communications.

##  2\. Quaternionic Hilbert–Blumenthal surfaces

###  2.1. A quaternionic Shimura surface

Let $\mathrm{F}$ be a real quadratic field of discriminant
$\mathrm{D}_{\mathrm{F}}$. Let $\mathrm{Q}=\mathrm{B}\otimes\mathrm{F}$ for an
indefinite quaternion algebra $\mathrm{B}$ with discriminant $\mathrm{N}^{-}p$
which is square free consisting of even number of prime factors. Suppose that
$p$ is a prime inert in $\mathrm{F}$. Given this $\mathrm{Q}$, let
$\mathrm{G}(\mathrm{Q})$ be the algebraic group over $\mathbf{Q}$ defined by
the Weil restriction of $\mathrm{Q}^{\times}$ from $\mathrm{F}$ to
$\mathbf{Q}$. Fix an open compact subgroup $\mathrm{K}$ of
$\mathrm{G}(\mathrm{Q})(\mathbf{A}^{\infty})$ we can associate to it a Shimura
variety $\mathrm{Sh}_{\mathrm{K}}(\mathrm{Q})$ over $\mathbf{Q}$ whose complex
points are given by

$\mathrm{Sh}_{\mathrm{K}}(\mathrm{Q})(\mathbf{C})=\mathrm{G}(\mathrm{Q})(\mathbf{Q})\backslash\mathcal{D}\times\mathrm{G}(\mathrm{Q})(\mathbf{A}^{\infty})/\mathrm{K}$

where we denote by $\mathcal{D}=\mathcal{H}^{\pm 2}$ product of two copies of
the upper and lower half planes.

Let $\widetilde{\mathrm{G}}(\mathrm{Q})$ be the algebraic group over
$\mathbf{Q}$ whose $\mathrm{R}$-points are given by

$\widetilde{\mathrm{G}}(\mathrm{Q})(\mathrm{R})=\\{x\in(\mathrm{Q}\otimes\mathrm{R})^{\times}:\mathrm{N}^{\circ}(x)\in\mathrm{R}^{\times}\\}$

for any $\mathbf{Q}$-algebra $\mathrm{R}$. Fix an open compact subgroup
$\widetilde{\mathrm{K}}$ of
$\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{A}^{\infty})$, we can associate to
it a Shimura variety ${\mathrm{Sh}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})$
which represents the following moduli problem over $\mathbf{Q}$. It associates
to a scheme $\mathrm{S}\in\mathrm{Sch}/\mathbf{Q}$ the isomorphism classes of
triples $(\mathrm{A},\iota,\overline{\eta})$ up to isogeny where

  1. (1)

$\mathrm{A}$ is an abelian scheme over $\mathrm{S}$ of relative dimension $4$;

  2. (2)

$\iota:\mathrm{B}\otimes\mathrm{F}\rightarrow\operatorname{End}^{0}(\mathrm{A})$
is a homomorphism such that

$\iota(b\otimes a)^{\ast}=\iota(b^{\ast}\otimes a)$

where the first $(\cdot)^{\ast}$ means the Rosati involution of
$\operatorname{End}^{0}(\mathrm{A})=\operatorname{End}_{\mathrm{S}}(\mathrm{A})\otimes\mathbf{Q}$
while the second $(\cdot)^{\ast}$ means the main involution on $\mathrm{B}$;

  3. (3)

$\overline{\eta}$ is a $\mathrm{K}$-equivalence class of
$\mathrm{B}\otimes\mathrm{F}$-equivariant isomorphisms

$\eta:\widehat{\mathrm{V}}(\mathrm{A})\xrightarrow{\sim}\mathrm{B}\otimes\mathrm{F}(\mathbf{A}^{\infty})$

where
$\widehat{\mathrm{V}}(\mathrm{A})=\prod_{v}\mathrm{T}_{v}(\mathrm{A})\otimes\mathbf{Q}$
and which preserves Weil-pairing on the left-hand-side and the reduced trace
pairing on the right-hand-side up to a scalar in $\mathbf{A}^{\infty\times}$.

We will refer to [Kot] for the precise definitions of the notations used here.
We require that the following Kottwitz conditions are satisfied

$\mathrm{det}(\iota(b\otimes
a);\mathrm{Lie}(\mathrm{A}))=\mathrm{N}^{\circ}(b)\mathrm{N}^{2}_{\mathrm{F}/\mathbf{Q}}(a)$

for $b\otimes a\in\mathrm{B}\otimes\mathrm{F}$. It is known that the
polarization datum that are usually included in the above moduli problem can
be omitted, see [Zin, Lemma 3.8], [LT, Remark 2.5].

######  Remark 2.1.

It is clear that the above moduli problem is the same as the
$\mathrm{GSpin}$-type Shimura variety defined by Kulda–Raopoport in [KR1, §1]
via Morita equivalence.

For $\widetilde{\mathrm{K}}$ sufficiently small, this moduli problem is
representable by a quasi-projective smooth scheme over $\mathbf{Q}$. Note that
when $\mathrm{B}\otimes\mathrm{F}=\mathrm{M}_{2}(\mathrm{F})$, then we recover
the classical Hilbert–Blumenthal surface. As before, the $\mathbf{C}$-points
of $\mathrm{Sh}_{\widetilde{\mathrm{K}}}(\mathrm{Q})(\mathbf{C})$ can be
described by the double coset space

$\mathrm{Sh}_{\widetilde{\mathrm{K}}}(\mathrm{Q})(\mathbf{C})=\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{Q})\backslash\mathcal{D}\times\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{A}^{\infty})/\widetilde{\mathrm{K}}.$

When $\widetilde{\mathrm{K}}$ is the restriction of $\mathrm{K}$, there is a
canonical map
$\mathrm{Sh}_{\widetilde{\mathrm{K}}}(\mathrm{Q})\rightarrow\mathrm{Sh}_{\mathrm{K}}(\mathrm{Q})$
extending the natural map

$\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{Q})\backslash\mathcal{D}\times\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{A}^{\infty})/\widetilde{\mathrm{K}}\rightarrow\mathrm{G}(\mathrm{Q})(\mathbf{Q})\backslash\mathcal{D}\times\mathrm{G}(\mathrm{Q})(\mathbf{A}^{\infty})/\mathrm{K}$

on $\mathbf{C}$-points. The PEL-type Shimura surface
$\mathrm{Sh}_{\widetilde{\mathrm{K}}}({\mathrm{Q}})$ will facilitate the study
of the geometry of $\mathrm{Sh}_{\mathrm{K}}({\mathrm{Q}})$ whose cohomology
carries the automorphic representation of interest. We will refer to
$\mathrm{Sh}_{\mathrm{K}}({\mathrm{Q}})$ as the quaternionic
Hilbert–Blumenthal surface associated to $\mathrm{Q}$ or simply as the
_quaternionic Shimura surface_. In the case when
$\mathrm{Q}=\mathrm{M}_{2}(\mathrm{F})$,
$\mathrm{Sh}_{\mathrm{K}}({\mathrm{Q}})$ agrees with the classical
Hilbert–Blumenthal surface.

###  2.2. Integral model for the quaternionnic Shimura surface

Let $\mathcal{O}_{\mathrm{B}}\subset\mathrm{B}$ be a maximal order stable
under the involution $\ast$ and we consider the order
$\mathcal{O}_{\mathrm{Q}}=\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$
in $\mathrm{B}\otimes\mathrm{F}$. Let $p$ be a prime which is inert in
$\mathrm{F}$ and divides the discriminant of $\mathrm{B}$,
$\widetilde{\mathrm{K}}^{p}\subset\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{A}^{\infty,p})$
be a compact open subgroup, we can define a moduli problem over
$\mathbf{Z}_{(p)}$ which associates to each
$\mathrm{S}\in\mathrm{Sch}/\mathbf{Z}_{(p)}$ the set of isomorphism classes of
triples $(\mathrm{A},\iota,\overline{\eta}^{p})$ where

  1. (1)

$\mathrm{A}$ is an abelian scheme over $\mathrm{S}$ of relative dimension $4$
up to prime to $p$-isogeny;

  2. (2)

$\iota:\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}\rightarrow\operatorname{End}(\mathrm{A})\otimes\mathbf{Z}_{(p)}$
is a homomorphism such that

$\iota(b\otimes a)^{\ast}=\iota(b^{\ast}\otimes a)$

where the first $(\cdot)^{\ast}$ means the Rosati involution of
$\operatorname{End}_{\mathrm{S}}(\mathrm{A})\otimes\mathbf{Z}_{(p)}$ while the
second $(\cdot)^{\ast}$ means the main involution on $\mathrm{B}$.

  3. (3)

$\overline{\eta}^{p}$ is a $\mathrm{K}^{p}$-equivalence class of
$\mathcal{O}_{\mathrm{B}}^{?}\otimes\mathcal{O}_{\mathrm{F}}$-equivariant
isomorphisms

$\eta^{p}:\widehat{\mathrm{V}}^{p}(\mathrm{A})\xrightarrow{\sim}\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}(\mathbf{A}^{\infty,p})$

where $\widehat{\mathrm{V}}^{p}(\mathrm{A})=\prod_{v\neq
p}\mathrm{T}_{v}(\mathrm{A})\otimes\mathbf{Q}$ and $\eta^{p}$ preserves Weil-
pairing on the left-hand-side and the reduced trace pairing on the right-hand-
side up to a scalar in $\mathbf{A}^{\infty,p\times}$;

We will refer to [Kot] for the precise definitions of the notations used here.
We require that the following Kottwitz conditions are satisfied

$\mathrm{det}(\iota(b\otimes
a);\mathrm{Lie}(\mathrm{A}))=\mathrm{N}^{\circ}(b)\mathrm{N}^{2}_{\mathrm{F}/\mathbf{Q}}(a)$

for $b\otimes a\in\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$
which is understood as an identity of polynomial functions with coefficient in
$\mathcal{O}_{\mathrm{S}}$ as in [Kot]. It follows from [Kot, §5] that the
this moduli problem is representable by a smooth quasi-projective scheme
$\widetilde{\mathrm{X}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})$ over
$\mathbf{Z}_{(p)}$. Let
$\widetilde{\mathrm{K}}=\widetilde{\mathrm{K}}_{p}\tilde{\mathrm{K}}^{p}$
where $\widetilde{\mathrm{K}}_{p}$ is the intersection of
$(\mathcal{O}_{\mathrm{Q}}\otimes\mathbf{Z}_{p})^{\times}$ with
$\widetilde{\mathrm{G}}(\mathrm{Q})(\mathbf{Q}_{p})$. Then the generic fiber
$\widetilde{\mathrm{X}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})\otimes\mathbf{Q}$
is given by ${\mathrm{Sh}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})$ defined in
the previous subsection. The canonical map
${\mathrm{Sh}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})\rightarrow\mathrm{Sh}_{\mathrm{K}}(\mathrm{Q})$
extends to the integral model $\mathrm{X}_{\mathrm{K}}(\mathrm{Q})$ of
$\mathrm{Sh}_{\mathrm{K}}(\mathrm{Q})$ and gives rise to a finite map

(2.1)
$\widetilde{\mathrm{X}}_{\widetilde{\mathrm{K}}}(\mathrm{Q})\longrightarrow{\mathrm{X}}_{\mathrm{K}}(\mathrm{Q})$

between schemes over $\mathbf{Z}_{(p)}$.

We will now fix a definite choice of the level structure for later purpose. We
will do this by introducing a slight variant of the above moduli problem. Let
$\mathrm{N}^{+}$ be an integer that coprime to $\mathrm{N}^{-}p$ and we put
$\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$. Let $d\geq 4$ be another integer.
Then we consider the moduli problem over
$\mathbf{Z}[1/\mathrm{N}\mathrm{D}_{\mathrm{F}}]$ that assigns for each
$\mathrm{S}\in\mathrm{Sch}/\mathbf{Z}[1/\mathrm{N}\mathrm{D}_{\mathrm{F}}]$
the set of isomorphism classes of $4$-tuples
$(\mathrm{A},\iota,\mathrm{C}_{\mathrm{N}^{+}},\alpha_{d})$ where

  1. (1)

$\mathrm{A}$ is an abelian scheme over $\mathrm{S}$ of relative dimension $4$;

  2. (2)

$\iota:\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}\rightarrow\operatorname{End}(\mathrm{A})$
is a homomorphism such that

$\iota(b\otimes a)^{\ast}=\iota(b^{\ast}\otimes a)$

where the first $(\cdot)^{\ast}$ means the Rosati involution of
$\operatorname{End}_{\mathrm{S}}(\mathrm{A})$ while the second
$(\cdot)^{\ast}$ means the main involution on $\mathrm{B}$;

  3. (3)

$\mathrm{C}_{\mathrm{N}^{+}}$ is an
$\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$-stable flat subgroup
of $\mathrm{A}[\mathfrak{n}^{+}]$ for
$\mathfrak{n}^{+}=\mathrm{N}^{+}\mathcal{O}_{\mathrm{F}}$ such that at every
geometric point, it is isomorphic to
$(\mathcal{O}_{\mathrm{F}}/\mathfrak{n}^{+})^{2}$;

  4. (4)

$\alpha_{d}:(\mathcal{O}_{\mathrm{F}}/d)_{\mathrm{S}}^{2}\hookrightarrow\mathrm{A}[\mathfrak{d}]$
is an $\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$-equivariant
injection of group schemes over $\mathrm{S}$ for the ideal
$\mathfrak{d}=d\mathcal{O}_{\mathrm{F}}$.

This moduli problem is representable by a scheme
$\widetilde{\mathrm{X}}_{\mathrm{N}^{+},d}(\mathrm{Q})$. The corresponding
integral model for the Shimura variety associated to the group
$\mathrm{G}(\mathrm{Q})$ will be denoted by
$\mathrm{X}_{\mathrm{N}^{+},d}(\mathrm{Q})$ and there is a finite morphism

$\widetilde{\mathrm{X}}_{\mathrm{N}^{+},d}(\mathrm{Q})\rightarrow\mathrm{X}_{\mathrm{N}^{+},d}(\mathrm{Q})$

as in (2.1). Since we will not need the precise form of the level structure,
we will refer the reader to [LT, Example 2.12] for the precise description of
the open compact subgroup $\mathrm{K}_{\mathrm{N}^{+},d}$ defining the level
structure. When there is no danger of confusion, we will simply write
$\mathrm{X}(\mathrm{Q})$ for the scheme
$\mathrm{X}_{\mathrm{N}^{+},d}(\mathrm{Q})$.

###  2.3. Supersingular locus of the quaternionic Shimura surface

Let $\mathbf{F}$ be a fixed algebraic closure of $\mathbf{F}_{p}$. We set
$\mathrm{D}=\mathrm{B}\otimes\mathbf{Q}_{p}$ to be the quaternionic division
algebra over $\mathbf{Q}_{p}$. Let $\mathcal{L}$ be an isocrystal of height
$4$ over $\mathbf{F}$ with an action
$\iota:\mathrm{D}\otimes\mathbf{Q}_{p^{2}}\rightarrow\operatorname{End}(\mathcal{L})$
of $\mathrm{D}\otimes\mathbf{Q}_{p^{2}}$ on $\mathcal{L}$. Since we always
have $\mathrm{D}\otimes\mathbf{Q}_{p^{2}}=\mathrm{M}_{2}(\mathbf{Q}_{p^{2}})$,
we have $\mathcal{L}=\mathcal{N}^{2}$ and $\mathcal{N}$ is equipped with an
action of $\mathbf{Q}_{p^{2}}$. We will use covariant Dieudonné theory
throughout this article. A Dieudonné lattice $\mathrm{M}$ is a lattice in
$\mathcal{N}$ with the property that
$p\mathrm{M}\subset\mathrm{F}\mathrm{M}\subset\mathrm{M}$. A Dieudonné lattice
is superspecial if $\mathrm{F}^{2}\mathrm{M}=p\mathrm{M}$ and in this case
$\mathrm{F}=\mathrm{V}$. We are concerned with a Dieudonné lattice
$\mathrm{M}$ with an additional endomorphism
$\iota:\mathbf{Z}_{p^{2}}\rightarrow\operatorname{End}(\mathrm{M})$. We can
decompose $\mathrm{M}$ as $\mathrm{M}_{0}\oplus\mathrm{M}_{1}$ according to
the action of $\mathbf{Z}_{p^{2}}$.

Let $(\mathrm{Nilp})$ be the category of $\mathrm{W}_{0}$-schemes over which
$p$ is locally nilpotent. We consider the set valued functor $\mathcal{M}$
that sends $\mathrm{S}\in(\mathrm{Nilp})$ to the isomorphism classes of the
collection
$(\mathrm{X},\iota_{\mathrm{X}},\lambda_{\mathrm{X}},\rho_{\mathrm{X}})$
where:

  1. (1)

$\mathrm{X}$ is a $p$-divisible group of dimension $2$ and height $4$ over
$\mathrm{S}$;

  2. (2)

$\iota_{\mathrm{X}}:\mathbf{Z}_{p^{2}}\rightarrow\operatorname{End}(\mathrm{X})$
is an action of $\mathbf{Z}_{p^{2}}$ on $\mathrm{X}$ defined over
$\mathrm{S}$;

  3. (3)

$\rho_{\mathrm{X}}:\mathrm{X}\times_{\mathrm{S}}\overline{\mathrm{S}}\rightarrow\mathbb{X}\times_{\mathbf{F}}\overline{\mathrm{S}}$
is an $\mathbf{Z}_{p^{2}}$-linear quasi-isogeny over $\overline{\mathrm{S}}$
which is the special fiber of $\mathrm{S}$ at $p$.

We require that $\iota_{\mathrm{X}}$ satisfies the Kottwitz condition

$\mathrm{det}(\iota_{\mathrm{X}}(a);\mathrm{Lie}(\mathrm{X}))=\mathrm{N}_{\mathbf{Q}_{p^{2}}/\mathbf{Q}_{p}}(a)$

for $a\in\mathbf{Z}_{p^{2}}$. For
$\rho_{\mathrm{X}}:\mathrm{X}\times_{\mathrm{S}}\overline{\mathrm{S}}\rightarrow\mathbb{X}\times_{\mathbf{F}}\overline{\mathrm{S}}$,
we require that
$\rho_{\mathrm{X}}^{-1}\circ\lambda_{\mathbb{X}}\circ\rho_{\mathrm{X}}=c(\rho_{\mathrm{X}})\lambda_{\mathrm{X}}$
for a $\mathbf{Q}_{p}$-multiple $c(\rho_{\mathrm{X}})$. This moduli problem is
representable by a formal scheme $\mathcal{M}$, locally formally of finite
type over $\mathrm{Spf}(\mathrm{W}_{0})$. We will be mainly concerned with the
underlying reduced closed subscheme $\mathcal{M}_{\mathbf{F}}$ of
$\mathcal{M}$. Let
$x=(\mathrm{X},\iota_{\mathrm{X}},\lambda_{\mathrm{X}},\rho_{\mathrm{X}})$ be
a point in $\mathcal{M}(\mathbf{F})$ and let $\mathrm{M}$ be the Dieudonné
lattice of $\mathrm{X}$. The action of $\mathbf{Z}_{p^{2}}$-action on
$\mathrm{X}$ gives rise to a grading of the Dieudonné module
$\mathrm{M}=\mathrm{M}_{0}\oplus\mathrm{M}_{1}$. For this lattice, we always
have

$\displaystyle
p\mathrm{M}_{0}\subset^{1}\mathrm{V}\mathrm{M}_{1}\subset^{1}\mathrm{M}_{0}$
$\displaystyle
p\mathrm{M}_{1}\subset^{1}\mathrm{V}\mathrm{M}_{0}\subset^{1}\mathrm{M}_{1}.$

We say $i\in\\{0,1\\}$ is a critical index for $\mathrm{M}$ with respect to
the $\mathbf{Z}_{p^{2}}$-action if
$\mathrm{V}^{2}\mathrm{M}_{i}=p\mathrm{M}_{i}$ and in this case we say $i$ is
a critical index of the point $x$. It is clear that $\mathrm{M}$ is
superspecial if $0$ and $1$ are both critical indices of $\mathrm{M}$ and in
this case we say $x$ is a superspecial point.

######  Lemma 2.2.

We have the following statements:

  1. (1)

for any Dieudonné lattice $\mathrm{M}$ associated to a point in
$\mathcal{M}(\mathbf{F})$, at least one $i\in\\{0,1\\}$ is critical for
$\mathrm{M}$;

  2. (2)

we have a partition of the scheme
$\mathcal{M}_{\mathbf{F}}=\mathcal{M}^{\circ}_{\mathbf{F}}\cup\mathcal{M}^{\bullet}_{\mathbf{F}}$
where $\mathcal{M}^{\circ}_{\mathbf{F}}$ consists those points of
$\mathcal{M}_{\mathbf{F}}$ such that $i=0$ is a critical index of the
associated Dieudonné lattice and $\mathcal{M}^{\bullet}_{\mathbf{F}}$ consists
those points of $\mathcal{M}_{\mathbf{F}}$ such that $i=1$ is a critical index
of the associated Dieudonné lattice;

  3. (3)

the irreducible components of $\mathcal{M}^{\circ}_{\mathbf{F}}$ and
$\mathcal{M}^{\bullet}_{\mathbf{F}}$ are projective lines. These two family of
projective lines intersect at the superspecial points of
$\mathcal{M}_{\mathbf{F}}$.

###### Proof.

The first part $(1)$ follows from [KR1, Lemma 4.2]. The second part $(2)$
follows from $(1)$. The third part $(3)$ is [KR1, Proposition 4.4] but we will
recall briefly the construction of those projective lines: suppose
$\mathrm{M}$ is associated to a point in $\mathcal{M}^{\circ}_{\mathbf{F}}$,
then $\Lambda_{0}=\mathrm{M}^{p\mathrm{V}^{-2}}_{0}$ is a lattice over
$\mathbf{Z}_{p^{2}}$, then we associate the projective line given by
$\mathbf{P}(\Lambda_{0}/p\Lambda_{0})$. Suppose $\mathrm{M}$ is associated to
a point in $\mathcal{M}^{\bullet}_{\mathbf{F}}$, then
$\Lambda_{1}=\mathrm{M}^{p\mathrm{V}^{-2}}_{1}$ is a lattice over
$\mathbf{Z}_{p^{2}}$, then we associate the projective line given by
$\mathbf{P}(\Lambda_{1}/p\Lambda_{1})$. It is clear these projective lines are
defined over $\mathbf{F}_{p^{2}}$. ∎

######  Remark 2.3.

We will refer to those projective lines in $\mathcal{M}^{\circ}$ as projective
lines of $\circ$-type and those projective lines in $\mathcal{M}^{\bullet}$ as
projective lines of $\bullet$-type.

Let $\overline{\mathrm{Q}}$ be the quaternion algebra obtained from
$\mathrm{Q}$ by switching the invariants at the archimedean places of
$\mathrm{F}$, then we define the algebraic group
$\mathrm{G}(\overline{\mathrm{Q}})$ over $\mathbf{Q}$ given by the Weil
restriction of $\overline{\mathrm{Q}}^{\times}$ from $\mathrm{F}$ to
$\mathbf{Q}$. Let $\mathrm{K}$ be an open compact subgroup of
$\mathrm{G}(\mathrm{Q})(\mathbf{A}^{\infty})$, then $\mathrm{K}$ can be viewed
as an open compact subgroup of
$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A}^{\infty})$. Let
$\overline{\mathrm{X}}(\mathrm{Q})$ be the special fiber of the quaternionic
surface $\mathrm{X}(\mathrm{Q})$ over $\mathbf{F}$. The Rapoport–Zink
uniformization theorem [RZ, Theorem 6.1], [She, Theorem 1.2] furnishes the
following description of the supersingular locus
$\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$ of
$\overline{\mathrm{X}}(\mathrm{Q})$.

######  Proposition 2.4.

The supersingular locus $\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$ of
$\overline{\mathrm{X}}(\mathrm{Q})$ is pure of dimension $1$.

  1. (1)

We have an isomorphism from the double quotient

$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{Q})\backslash\mathcal{M}_{\mathbf{F}}\times\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A}^{\infty,p})/\mathrm{K}^{p}$

and $\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$ which descends to an
isomorphism over $\mathbf{F}_{p^{2}}$;

  2. (2)

The irreducible components of
$\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$ are projective lines which
are parametrized by two copies of

$\mathrm{Z}(\overline{\mathrm{Q}})=\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{Q})\backslash\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A}^{\infty})/\mathrm{K}^{p}\mathrm{K}_{p}$

where $\mathrm{K}_{p}$ is
$(\mathcal{O}_{\overline{\mathrm{Q}}}\otimes\mathbf{Z}_{p})^{\times}$. We
denote these two copies by $\mathrm{Z}^{\circ}(\overline{\mathrm{Q}})$ and
$\mathrm{Z}^{\bullet}(\overline{\mathrm{Q}})$ respectively, they parametrize
those projective lines of $\circ$-type and those projective lines of
$\bullet$-type respectively;

  3. (3)

The superspecial locus $\overline{\mathrm{X}}^{\mathrm{ssp}}(\mathrm{Q})$ is a
discrete set of points parametrized by

$\mathrm{Z}_{\mathrm{Iw}}(\overline{\mathrm{Q}})=\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{Q})\backslash\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A}^{\infty})/\mathrm{K}^{p}\mathrm{Iw}_{p}$

where $\mathrm{Iw}_{p}$ is the Iwahori subgroup of $\mathrm{K}_{p}$.

###### Proof.

The first part $(1)$ follows from the Rapoport–Zink uniformization theorem and
the fact that the Weil descent datum for $\mathcal{M}$ is effective over
$\mathbf{Z}_{p^{2}}$. The second part $(2)$ and the third part $(3)$ follow
from the first part and descriptions in Lemma 2.2 immediately. ∎

######  Remark 2.5.

By the previous proposition we can write
$\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})=\overline{\mathrm{X}}^{\circ}(\mathrm{Q})\cup\overline{\mathrm{X}}^{\bullet}(\mathrm{Q})$
where

  1. (1)

$\overline{\mathrm{X}}^{\circ}(\mathrm{Q})$ is a $\mathbf{P}^{1}$-bundle over
$\mathrm{Z}^{\circ}(\mathrm{Q})$ which we will call the $\circ$-component for
$\overline{\mathrm{X}}(\mathrm{Q})$;

  2. (2)

$\overline{\mathrm{X}}^{\bullet}(\mathrm{Q})$ is a $\mathbf{P}^{1}$-bundle
over $\mathrm{Z}^{\bullet}(\mathrm{Q})$ which we will call the
$\bullet$-component for $\overline{\mathrm{X}}(\mathrm{Q})$.

######  Remark 2.6.

Note that the description from the previous Proposition 2.4 can also be
obtained using the so-called isogeny trick instead of using Rapoport–Zink
uniformization theorem, see the main result of [TX] which treats much more
general quaternionic Shimura varieties.

##  3\. Quaternionic Hirzebruch–Zagier divisor

###  3.1. Shimura curves and Drinfeld uniformization

Let $\mathrm{B}$ be the indefinite quaternion algebra over $\mathbf{Q}$ with
discriminant $p\mathrm{N}^{-}$ considered in the last section. We denote by
$\mathrm{G}(\mathrm{B})$ the algebraic group over $\mathbf{Q}$ given by
$\mathrm{B}^{\times}$. We fix an open compact subgroup $\mathrm{K}$ of
$\mathrm{G}(\mathrm{B})(\mathbf{A}^{\infty})$ such that
$\mathrm{K}^{p}\mathrm{K}_{p}$ where $\mathrm{K}^{p}$ is sufficiently small
and $\mathrm{K}_{p}=(\mathcal{O}_{\mathrm{B}}\otimes\mathbf{Z}_{p})^{\times}$.
We consider the moduli problem over $\mathbf{Z}_{(p)}$ which assigns each
$\mathrm{S}\in\mathrm{Sch}/\mathbf{Z}_{(p)}$ the set of triples
$(\mathrm{A},\iota,\overline{\eta})$ where

  1. (1)

$\mathrm{A}$ is an abelian scheme over $\mathrm{S}$ of relative dimension $2$,
up to prime to $p$-isogeny;

  2. (2)

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

  3. (3)

$\overline{\eta}^{p}$ is a $\mathrm{K}^{p}$-equivalence class of
$\mathcal{O}_{\mathrm{B}}$-equivariant isomorphisms

$\eta^{p}:\widehat{\mathrm{V}}^{p}(\mathrm{A})\xrightarrow{\sim}\mathcal{O}_{\mathrm{B}}\otimes\mathbf{A}^{\infty,p}$

where $\widehat{\mathrm{V}}^{p}(\mathrm{A})=\prod_{v\neq
p}\mathrm{T}_{v}(\mathrm{A})\otimes\mathbf{Q}$ and $\eta^{p}$ preserves Weil-
pairing on the left-hand-side and the reduced trace pairing on the right-hand-
side up to a scalar in $(\mathbf{A}^{\infty,p})^{\times}$.

It is well known this moduli problem is representable by a projective scheme
over $\mathbf{Z}_{(p)}$ of relative dimension $1$ denoted by
$\mathrm{X}_{\mathrm{K}}(\mathrm{B})$. Its generic fiber
$\mathrm{Sh}_{\mathrm{K}}(\mathrm{B})=\mathrm{X}_{\mathrm{K}}(\mathrm{B})_{\mathbf{Q}}$
is the canonical model of the Shimura variety determined by $\mathrm{B}$ and
$\mathrm{K}$ over $\mathbf{Q}$. The set of $\mathbf{C}$-points of
$\mathrm{Sh}_{\mathrm{K}}(\mathrm{B})$ is given by the following double coset
space

$\mathrm{Sh}_{\mathrm{K}}(\mathrm{B})(\mathbf{C})=\mathrm{G}(\mathrm{B})(\mathbf{Q})\backslash\mathcal{H}^{\pm}\times\mathrm{G}(\mathrm{B})(\mathbf{A}^{\infty})/\mathrm{K}.$

We will now fix a definite choice of the level structure as we did for the
quaternionic Hilbert–Blumenthal surface. We introduce a moduli problem over
$\mathbf{Z}[1/\mathrm{N}d]$ that assigns each
$\mathrm{S}\in\mathrm{Sch}/\mathbf{Z}[1/\mathrm{N}d]$ the set of isomorphism
classes of $4$-tuples
$(\mathrm{A},\iota,\mathrm{C}_{\mathrm{N}^{+}},\alpha_{d})$ where

  1. (1)

$\mathrm{A}$ is an abelian scheme over $\mathrm{S}$ of relative dimension $2$;

  2. (2)

$\iota:\mathcal{O}_{\mathrm{B}}\rightarrow\operatorname{End}_{\mathrm{S}}(\mathrm{A})$
is a homomorphism which is special in the sense of [BC, 131–132];

  3. (3)

$\mathrm{C}_{\mathrm{N}^{+}}$ is an $\mathcal{O}_{\mathrm{B}}$-stable finite
flat subgroup of $\mathrm{A}[\mathrm{N}^{+}]$ such that at every geometric
point, it is cyclic of order $(\mathrm{N}^{+})^{2}$;

  4. (4)

$\alpha_{d}:(\mathbf{Z}/d\mathbf{Z})^{2}_{\mathrm{S}}\hookrightarrow\mathrm{A}[d]$
is an $\mathcal{O}_{\mathrm{B}}$-equivariant injection of group schemes over
$\mathrm{S}$.

This moduli problem is representable by a scheme
${\mathrm{X}}_{\mathrm{N}^{+},d}(\mathrm{B})$ once $d\geq 4$. When there is no
danger of confusion, we will denote this scheme simply by
$\mathrm{X}(\mathrm{B})$. We will refer the readers to [Wang, §2.1] for the
precise definition of the open compact subgroup corresponding to the level
structure in the above moduli problem.

Let $\mathbf{F}$ be an algebraic closure of $\mathbf{F}_{p}$ and
$\mathrm{W}_{0}=\mathrm{W}(\mathbf{F})$ be the ring of Witt vectors of
$\mathbf{F}$ whose fraction field is denoted by $\mathrm{K}_{0}$. Let
$\mathrm{D}$ be the quaternion division algebra over $\mathbf{Q}_{p}$ and
$\mathcal{O}_{\mathrm{D}}$ be the maximal order of $\mathrm{D}$. We will
present $\mathcal{O}_{\mathrm{D}}$ as
$\mathcal{O}_{\mathrm{D}}=\mathbf{Z}_{p^{2}}[\Pi]/(\Pi^{2}-p)$ with $\Pi
a=\sigma(a)\Pi$ for $a\in\mathbf{Z}_{p^{2}}$. We first recall the notion of a
_special formal $\mathcal{O}_{D}$-module_. Let $\mathrm{S}$ be a
$\mathrm{W}_{0}$-scheme. A special formal $\mathcal{O}_{\mathrm{D}}$-module
over $\mathrm{S}$ is a formal $p$-divisible group $\mathrm{X}$ of dimension
$2$ and height $4$ with an $\mathcal{O}_{\mathrm{D}}$-action
$\iota:\mathcal{O}_{\mathrm{D}}\rightarrow\operatorname{End}_{\mathrm{S}}(\mathrm{X})$
such that $\mathrm{Lie}(\mathrm{X})$ is a locally free
$\mathbf{Z}_{p^{2}}\otimes\mathcal{O}_{\mathrm{S}}$-module of rank $1$. We fix
a special $\mathcal{O}_{\mathrm{D}}$-module $\mathbb{X}$ over $\mathbf{F}$
whose Dieudonné module is denoted by $\mathbb{M}$.

Let $\mathcal{M}_{\mathrm{Dr}}$ be the set valued functor that sends
$S\in(\mathrm{Nilp})$ to the set of isomorphism classes of pairs
$(\mathrm{X},\rho_{\mathrm{X}})$ where

  1. (1)

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

  2. (2)

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

The functor $\mathcal{M}_{\mathrm{Dr}}$ is represented by a formal scheme over
$\mathrm{W}_{0}$ which we also denote by $\mathcal{M}_{\mathrm{Dr}}$. The
formal scheme $\mathcal{M}_{\mathrm{Dr}}$ decomposes into a disjoint union

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

according to the height $i$ of the quasi-isogeny $\rho_{\mathrm{X}}$. Each
formal scheme $\mathcal{M}_{\mathrm{Dr},i}$ is isomorphic to the _$p$ -adic
upper half plane_ $\mathcal{H}_{p}$. The group
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$ acts naturally on the formal scheme
$\mathcal{M}_{\mathrm{Dr}}$ and each $\mathcal{M}_{\mathrm{Dr},i}$ affords an
action of the group

$\mathrm{GL}^{0}_{2}(\mathbf{Q}_{p}):=\\{g\in\mathrm{GL}_{2}(\mathbf{Q}_{p}):\mathrm{ord}_{p}(\mathrm{det}(g))=0\\}.$

Let $(\mathrm{X},\rho)\in\mathcal{M}_{\mathrm{Dr}}(\mathbf{F})$ and
$\mathrm{M}$ be the Dieudonné lattice of $\mathrm{X}$. The action of
$\mathbf{Z}_{p^{2}}$ on $\mathrm{X}$ induces a grading
$\mathrm{M}=\mathrm{M}_{0}\oplus\mathrm{M}_{1}$ that satisfies

(3.1) $\displaystyle
p\mathrm{M}_{0}\subset^{1}\mathrm{V}\mathrm{M}_{1}\subset^{1}\mathrm{M}_{0}\hphantom{aa}p\mathrm{M}_{1}\subset^{1}\mathrm{V}\mathrm{M}_{0}\subset^{1}\mathrm{M}_{1};$
$\displaystyle
p\mathrm{M}_{0}\subset^{1}\Pi\mathrm{M}_{1}\subset^{1}\mathrm{M}_{0}\hphantom{aa}p\mathrm{M}_{1}\subset^{1}\Pi\mathrm{M}_{0}\subset^{1}\mathrm{M}_{1}.$

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

(3.2)
$\displaystyle\Pi:\mathrm{M}_{0}/\mathrm{V}\mathrm{M}_{1}\rightarrow\mathrm{M}_{1}/\mathrm{V}\mathrm{M}_{0},$
$\displaystyle\Pi:\mathrm{M}_{1}/\mathrm{V}\mathrm{M}_{0}\rightarrow\mathrm{M}_{0}/\mathrm{V}\mathrm{M}_{1}.$

Since both $\mathrm{M}_{0}/\mathrm{V}\mathrm{M}_{1}$ and
$\mathrm{M}_{1}/\mathrm{V}\mathrm{M}_{0}$ are one dimensional and the
composite of the two maps is obviously zero, we can conclude that there is an
$i\in\\{0,1\\}$ such that $\Pi\mathrm{M}_{i}\subset\mathrm{V}\mathrm{M}_{i}$.
Since both $\Pi\mathrm{M}_{i}$ and $\mathrm{V}\mathrm{M}_{i}$ are of colength
$1$ in $\mathrm{M}_{i+1}$, we conclude that they are in fact equal to each
other. We say that $i$ is a _critical index_ for $\mathrm{M}$ with respect to
the $\mathrm{D}$-action if $\mathrm{V}\mathrm{M}_{i}=\Pi\mathrm{M}_{i}$ and a
critical index always exists for $\mathrm{M}$. Let $\tau=\Pi^{-1}\mathrm{V}$
and it acts as an automorphism on $\mathrm{M}_{i}$ if $i$ is a critical index.
If $0$ is a critical index, then we set $\Lambda_{0}=\mathrm{M}^{\tau=1}_{0}$
and this is a $\mathbf{Z}_{p}$-lattice of rank $2$ and we associate to it the
projective line $\mathbf{P}(\Lambda_{0}/p\Lambda_{0})$. Then
$\mathrm{V}\mathrm{M}_{1}/p\mathrm{M}_{0}\subset^{1}\mathrm{M}_{0}/p\mathrm{M}_{0}=\Lambda_{0}/p\Lambda_{0}\otimes\mathbf{F}$
gives a point on $\mathbf{P}(\Lambda_{0}/p\Lambda_{0})(\mathbf{F})$. If $1$ is
a critical index, then we similarly put
$\Lambda_{1}=\Pi\mathrm{M}^{\tau=1}_{1}$ and we again associate to it the
projective line $\mathbf{P}(\Lambda_{1}/p\Lambda_{1})$. Similarly
$\mathrm{V}\mathrm{M}_{0}/p\mathrm{M}_{1}\subset^{1}\mathrm{M}_{1}/p\mathrm{M}_{1}=\Lambda_{1}/p\Lambda_{1}\otimes\mathbf{F}$
gives a point on $\mathbf{P}(\Lambda_{1}/p\Lambda_{1})(\mathbf{F})$. This
discussion leads to the following lemma.

######  Lemma 3.1.

The space $\mathcal{M}_{\mathrm{Dr},\mathbf{F}}$ admits the following
descriptions.

  1. (1)

For any Dieudonné lattice $\mathrm{M}$ associated to a point in
$\mathcal{M}_{\mathrm{Dr}}(\mathbf{F})$, at least one $i\in\\{0,1\\}$ is
critical;

  2. (2)

We have a partition of the scheme
$\mathcal{M}_{\mathrm{Dr},\mathbf{F}}=\mathcal{M}^{\circ}_{\mathrm{Dr},\mathbf{F}}\cup\mathcal{M}^{\bullet}_{\mathrm{Dr},\mathbf{F}}$
where $\mathcal{M}^{\circ}_{\mathrm{Dr},\mathbf{F}}$ consists those points of
$\mathcal{M}_{\mathrm{Dr},\mathbf{F}}$ such that $i=0$ is a critical index and
$\mathcal{M}^{\bullet}_{\mathrm{Dr},\mathbf{F}}$ consists those points of
$\mathcal{M}_{\mathrm{Dr},\mathbf{F}}$ such that $i=1$ is a critical index;

  3. (3)

The irreducible components of $\mathcal{M}^{\circ}_{\mathrm{Dr},\mathbf{F}}$
and $\mathcal{M}^{\bullet}_{\mathrm{Dr},\mathbf{F}}$ are projective lines.
These two family of projective lines intersect at the superspecial points of
$\mathcal{M}_{\mathrm{Dr},\mathbf{F}}$.

The Cerednick–Drinfeld uniformization theorem provides the following
proposition describing the special fiber
$\overline{\mathrm{X}}(\mathrm{B})_{\mathbf{F}}$ of $\mathrm{X}(\mathrm{B})$.
Let $\overline{\mathrm{B}}$ be the definite quaternion algebra obtained from
$\mathrm{B}$ by switching the invariant at $p$ and $\infty$. Then we define
$\mathrm{G}(\overline{\mathrm{B}})$ to be algebraic group defined by
$\overline{\mathrm{B}}^{\times}$. Let $\mathrm{K}$ be an open compact subgroup
of $\mathrm{G}(\mathrm{B})(\mathbf{A}^{\infty})$, then $\mathrm{K}^{p}$ can be
viewed naturally as an open compact of
$\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A}^{\infty,p})$.

######  Proposition 3.2.

We have the following descriptions of the scheme
$\overline{\mathrm{X}}(\mathrm{B})$.

  1. (1)

We have an isomorphism from the double quotient

$\mathrm{G}(\overline{\mathrm{B}})(\mathbf{Q})\backslash\mathcal{M}_{\mathrm{Dr},\mathbf{F}}\times\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A}^{\infty,p})/\mathrm{K}^{p}$

to $\overline{\mathrm{X}}(\mathrm{B})_{\mathbf{F}}$ which descends to an
isomorphism over $\mathbf{F}_{p^{2}}$;

  2. (2)

The irreducible components of $\overline{\mathrm{X}}(\mathrm{B})$ are
projective lines which are parametrized by two copies of the Shimura set

$\mathrm{Z}(\overline{\mathrm{B}})=\mathrm{G}(\overline{\mathrm{B}})(\mathbf{Q})\backslash\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A}^{\infty,p})/\mathrm{K}^{p}\overline{\mathrm{K}}_{p}$

where $\overline{\mathrm{K}}_{p}$ is the group
$(\mathcal{O}_{\overline{\mathrm{B}}}\otimes\mathbf{Z}_{p})^{\times}$. We
denote these two copies by $\mathrm{Z}^{\circ}(\overline{\mathrm{B}})$ and
$\mathrm{Z}^{\bullet}(\overline{\mathrm{B}})$, they parametrize those
projective lines corresponding to critical index $i=0$ resp. critical index
$i=1$;

  3. (3)

The superspecial locus $\overline{\mathrm{X}}(\mathrm{B})$ is a discrete set
of points parametrized by

$\mathrm{Z}_{\mathrm{Iw}}(\overline{\mathrm{B}})=\mathrm{G}(\overline{\mathrm{B}})(\mathbf{Q})\backslash\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A}^{\infty,p})/\mathrm{K}^{p}\mathrm{Iw}_{p}$

where $\mathrm{Iw}_{p}$ is the Iwahori subgroup of
$\overline{\mathrm{K}}_{p}$.

###### Proof.

This is a well-known result following from the Cerednick–Drinfeld
uniformization theorem for the Shimura curve $\mathrm{X}(\mathrm{B})$. For
example, see the proof of [Wang, Proposition 3.2] for more details. ∎

######  Remark 3.3.

By the previous proposition we can write
$\overline{\mathrm{X}}(\mathrm{B})=\overline{\mathrm{X}}^{\circ}(\mathrm{B})\cup\overline{\mathrm{X}}^{\bullet}(\mathrm{B})$
where

  1. (1)

$\overline{\mathrm{X}}^{\circ}(\mathrm{B})$ is a $\mathbf{P}^{1}$-bundle over
$\mathrm{Z}^{\circ}(\overline{\mathrm{B}})$ which we will call the
$\circ$-component;

  2. (2)

$\overline{\mathrm{X}}^{\bullet}(\mathrm{B})$ is a $\mathbf{P}^{1}$-bundle
over $\mathrm{Z}^{\bullet}(\overline{\mathrm{B}})$ which we will call the
$\bullet$-component.

###  3.2. Quaternionic Hirzebruch–Zagier divisor

Let $(\mathrm{A},\iota,\mathrm{C}_{\mathrm{N}^{+}},\alpha_{d})$ be an element
of $\mathrm{X}(\mathrm{B})(\mathrm{S})$ for a scheme $\mathrm{S}$ over
$\mathbf{Z}[1/\mathrm{N}d\mathrm{D}_{\mathrm{F}}]$, we define

  1. (1)

$\widetilde{\mathrm{A}}=\mathrm{A}\otimes\mathcal{O}_{\mathrm{F}}$ is the
abelian scheme given by the Serre’s tensor construction from $\mathrm{A}$;

  2. (2)

$\widetilde{\iota}:\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}\hookrightarrow\operatorname{End}(\mathrm{A})\otimes\mathcal{O}_{\mathrm{F}}\hookrightarrow\operatorname{End}(\widetilde{\mathrm{A}})$
be the morphism given by $\iota\otimes\mathcal{O}_{\mathrm{F}}$;

  3. (3)

$\widetilde{\mathrm{C}}_{\mathrm{N}^{+}}\subset\widetilde{\mathrm{A}}[\mathfrak{n}^{+}]$
be the $\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$-stable finite
flat subgroup of $\widetilde{\mathrm{A}}$ given by

$\widetilde{\mathrm{C}}_{\mathrm{N}^{+}}=\mathrm{C}_{\mathrm{N}^{+}}\otimes\mathcal{O}_{\mathrm{F}};$

  4. (4)

$\widetilde{\alpha}_{d}:(\mathcal{O}_{\mathrm{F}}/d)^{2}\hookrightarrow\mathrm{A}[\mathfrak{d}]$
is the $\mathcal{O}_{\mathrm{B}}\otimes\mathcal{O}_{\mathrm{F}}$-equivariant
injection

$\widetilde{\alpha}_{d}:(\mathbf{Z}/d)^{2}\otimes\mathcal{O}_{\mathrm{F}}\hookrightarrow\mathrm{A}\otimes\mathcal{O}_{\mathrm{F}}[d]=\widetilde{\mathrm{A}}[\mathfrak{d}].$

given by $\alpha_{d}\otimes\mathcal{O}_{\mathrm{F}}$.

This defines an element of
$(\widetilde{\mathrm{A}},\widetilde{\iota},\widetilde{\mathrm{C}}_{\mathrm{N}^{+}},\widetilde{\alpha}_{d})\in\widetilde{\mathrm{X}}(\mathrm{Q})(\mathrm{S})$
over $\mathbf{Z}[1/\mathrm{N}d\mathrm{D}_{\mathrm{F}}]$ which gives a map from
$\mathrm{X}(\mathrm{B})\otimes\mathbf{Z}[1/\mathrm{N}d\mathrm{D}_{\mathrm{F}}]$
to $\widetilde{\mathrm{X}}(\mathrm{Q})$. We therefore obtain the quaternionic
Hirzebruch–Zagier morphism over
$\mathbf{Z}[1/\mathrm{N}d\mathrm{D}_{\mathrm{F}}]$

$\theta:\mathrm{X}(\mathrm{B})\otimes\mathbf{Z}[1/\mathrm{N}d\mathrm{D}_{\mathrm{F}}]\rightarrow\mathrm{X}(\mathrm{Q})$

by composing the previously defined map with the canonical map
$\widetilde{\mathrm{X}}(\mathrm{Q})\rightarrow\mathrm{X}(\mathrm{Q})$.

Let $(\mathrm{X},\iota,\rho_{\mathrm{X}})$ be an element of
$\mathcal{M}_{\mathrm{Dr}}(\mathrm{S})$ with $\mathrm{S}$ in
$(\mathrm{Nilp})$, then we can restrict
$\iota:\mathcal{O}_{\mathrm{D}}\rightarrow\operatorname{End}_{\mathrm{S}}(\mathrm{X})$
to ${\mathbf{Z}_{p^{2}}}$ and
$(\mathrm{X},\iota_{|_{\mathbf{Z}_{p^{2}}}},\rho_{\mathrm{X}})$ gives rise to
an element of $\mathcal{M}(\mathrm{S})$ and hence we can define a morphism
$\mathcal{M}_{\mathrm{Dr}}\rightarrow\mathcal{M}$. Let
$\mathrm{M}=\mathrm{M}_{0}\oplus\mathrm{M}_{1}$ be the Dieudonné lattice of
$\mathrm{X}$. Suppose that $i\in\\{0,1\\}$ is a critical index for
$\mathrm{M}$ with respect to the action of $\mathcal{O}_{\mathrm{D}}$, then
$\mathrm{V}\mathrm{M}_{i}=\Pi\mathrm{M}_{i}$. It follows that
$\mathrm{V}^{2}\mathrm{M}_{i}=p\mathrm{M}_{i}$ therefore $i$ is also a
critical index for $\mathrm{M}$ with respect to the restricted action of
$\mathbf{Z}_{p^{2}}$. Therefore the morphism
$\mathcal{M}_{\mathrm{Dr}}\rightarrow\mathcal{M}$ respects the partition with
respect to the two notion of critical indices and thus to the $\circ$ and
$\bullet$ components, that is we have morphisms
$\mathcal{M}^{\circ}_{\mathrm{Dr}}\rightarrow\mathcal{M}^{\circ}$ and
$\mathcal{M}^{\bullet}_{\mathrm{Dr}}\rightarrow\mathcal{M}^{\bullet}$.

######  Lemma 3.4.

We have the following statements on the reduction of the Hirzebruch–Zagier
morphism.

  1. (1)

The image of the induced morphism on the special fiber

$\overline{\theta}:\overline{\mathrm{X}}(\mathrm{B})\rightarrow\overline{\mathrm{X}}(\mathrm{Q}).$

is contained in the supersingular locus
$\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$ of
$\overline{\mathrm{X}}(\mathrm{Q})$.

  2. (2)

The map $\overline{\theta}$ respects the partition according to the critical
indices and induces a map

$\overline{\theta}^{?}:\overline{\mathrm{X}}^{?}(\mathrm{B})\rightarrow\overline{\mathrm{X}}^{?}(\mathrm{Q})$

which restricts to the canonical embedding of

$\vartheta^{?}:\mathrm{Z}^{?}(\overline{\mathrm{B}})\rightarrow\mathrm{Z}^{?}(\overline{\mathrm{Q}})$

for $?\in\\{\circ,\bullet\\}$.

###### Proof.

The first part $(1)$ follows from Proposition 2.4 $(1)$ and 3.2 $(1)$. The
second part $(2)$ follows from the previous discussions on the compatibilities
of the two notions of critical indices. Note a similar result is obtained in
[Lan]. ∎

###  3.3. Tate cycles on the quaternionic Shimura surface

Suppose that $\pi^{\mathrm{Q}}$ is a cuspidal automorphic representation of
$\mathrm{G}(\mathrm{Q})$ of weight $(2,2)$ and trivial central character. We
will always assume that $\pi^{\mathrm{Q}}$ is _non-dihedral_ and is defined
over $\mathrm{E}_{\lambda}$. Suppose that $\pi^{\mathrm{Q}}$ appears in the
middle degree cohomology

$\mathrm{H}^{2}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathrm{E}_{\lambda})=\bigoplus\limits_{\Pi^{\mathrm{Q}}}\mathrm{H}^{2}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathrm{E}_{\lambda})[\Pi^{\mathrm{Q}}]\otimes(\Pi^{\mathrm{Q}\infty})^{\mathrm{K}_{\mathrm{N}^{+},d}}$

of $\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}$. We can associate to
$\pi^{\mathrm{Q}}$ a two dimensional $\lambda$-adic representation
$(\rho_{\pi^{\mathrm{Q}},\lambda},\mathrm{V}_{\pi^{\mathrm{Q}}})$ of
$\mathrm{G}_{\mathrm{F}}$ using the construction of Blasius–Rogawski [BR] and
Taylor [Tay].

The Asai representation associated to the representation
$\mathrm{V}_{\pi^{\mathrm{Q}}}$ will be denoted by

$(\mathrm{As}(\rho_{\pi^{\mathrm{Q}},\lambda}),\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})),$

this is a representation of $\mathrm{G}_{\mathbf{Q}}$ isomorphic to the tensor
induction
$\otimes\mathrm{Ind}^{\mathrm{G}_{\mathbf{Q}}}_{\mathrm{G}_{\mathrm{F}}}\mathrm{V}_{\pi^{\mathrm{Q}}}$
of $\mathrm{V}_{\pi^{\mathrm{Q}}}$ from $\mathrm{G}_{\mathrm{F}}$ to
$\mathrm{G}_{\mathrm{Q}}$. It will be important for us to note that
$(\mathrm{As}(\rho_{\pi^{\mathrm{Q}},\lambda})(-1),\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1))$
is realized on the cohomology
$\mathrm{H}^{2}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathrm{E}_{\lambda}(1))$
given by the same construction as in [BL].

######  Lemma 3.5.

The $\lambda$-adic Galois representation

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathrm{E}_{\lambda}(1))=\mathrm{H}^{2}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathrm{E}_{\lambda}(1))[\pi^{\mathrm{Q}}]$

is isomorphic to $m(\pi^{\mathrm{Q}},d)$ copies of
$\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1)$ for some
$m(\pi^{\mathrm{Q}},d)\geq 1$ .

###### Proof.

The same proof for [Liu1, Lemma 3.9] works here for our quaternionic
Hilbert–Blumenthal surface as well noting that
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathrm{E}_{\lambda}(1))$
is a semi-simple Galois module by [Nek2]. ∎

Let
$\mathbb{T}_{\mathrm{F}}=\mathbb{T}^{\mathfrak{d}\mathfrak{n}}_{\mathrm{F}}$
be the Hecke algebra for the group $\mathrm{G}(\mathrm{Q})$ consisting of
prime-to-$\mathfrak{d}\mathfrak{n}$ Hecke operators where $\mathfrak{n}$ is
the ideal $\mathrm{N}\mathcal{O}_{\mathrm{F}}$ and $\mathfrak{d}$ is the ideal
$d\mathcal{O}_{\mathrm{F}}$. The associated Hecke eigensystem of
$\pi^{\mathrm{Q}}$ are given by the traces of the Frobenius elements of the
representation $\overline{\rho}_{\pi^{\mathrm{Q}}}$. Then the traces of the
Frobenius elements of the residual representation
$\overline{\rho}_{\pi^{\mathrm{Q}}}$ of ${\rho}_{\pi^{\mathrm{Q}}}$ defines a
morphism

$\psi_{\pi^{\mathrm{Q}}}:\mathbb{T}_{\mathrm{F}}\rightarrow
k_{\lambda}\phantom{aaa}\mathrm{T}_{v}\mapsto\mathrm{Tr}\overline{\rho}_{\pi^{\mathrm{Q}}}(\mathrm{Fr}_{v}).$

We define

(3.3)
$\mathfrak{m}_{\mathrm{F}}=\mathrm{ker}(\psi_{\pi^{\mathrm{Q}}}:\mathbb{T}_{\mathrm{F}}\rightarrow
k_{\lambda})$

to be the maximal ideal given by the kernel of this morphism and call it the
maximal ideal associated to $\overline{\rho}_{\pi^{\mathrm{Q}}}$.

Now we will consider the cohomology of
$\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}$ with integral
coefficients. To simplify the notations, we will set

$\mathrm{H}^{i}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(j))=\mathrm{H}^{i}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{{\lambda}}(j))_{\mathfrak{m}_{\mathrm{F}}}$

for integers $i,j\geq 0$ and call it the $\pi^{\mathrm{Q}}$-isotypic component
of
$\mathrm{H}^{i}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{{\lambda}}(j))$.
In particular,
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(1))$
defines an integral lattice in
$\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1)$ which we will denote by
$\mathrm{As}(\mathrm{T}_{\pi^{\mathrm{Q}}})(-1)$ which in turn determines an
integral lattice $\mathrm{T}_{\pi^{\mathrm{Q}}}$ of
$\mathrm{V}_{\pi^{\mathrm{Q}}}$.

######  Definition 3.6.

We say the number $d$ that appears in the level structure for
$\mathrm{X}(\mathrm{Q})$ is clean for $\mathrm{T}_{\pi^{\mathrm{Q}}}$ if

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(1))=m(\pi^{\mathrm{Q}},d)\mathrm{As}(\mathrm{T}_{\pi^{\mathrm{Q}}})(-1)$

with $m(\pi^{\mathrm{Q}},d)$ defined in Lemma 3.5.

This terminology is first introduced in [Liu2] from a slightly different
situation. Intuitively, this condition means that the $d$-new forms occurring
on
$\mathrm{H}^{2}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(1))$
will not be congruent to the $d$-old forms. From here on, we will always
choose $d$ to be clean for $\mathrm{T}_{\pi^{\mathrm{Q}}}$. Note that by the
proper base change theorem, we can identify

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))=\mathrm{H}^{2}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{{\lambda}}(1))_{\mathfrak{m}_{\mathrm{F}}}$

with
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{p},\mathcal{O}_{\lambda}(1))$
and thus with $m(\pi^{\mathrm{Q}},d)$-copies of
$\mathrm{As}(\mathrm{T}_{\pi^{\mathrm{Q}}})(-1)|_{\mathrm{G}_{\mathbf{Q}_{p}}}$.

Let $\pi^{\overline{\mathrm{Q}}}$ be the Jacquet-Langlands transfer of
$\pi^{\mathrm{Q}}$ to $\mathrm{G}(\overline{\mathrm{Q}})$. We define similarly
the $\pi^{\overline{\mathrm{Q}}}$-isotypic component of the cohomology
$\mathrm{H}^{0}(\mathrm{Z}^{?}(\overline{\mathrm{Q}}),\mathcal{O}_{\lambda})=\mathcal{O}_{\lambda}[\mathrm{Z}^{?}(\overline{\mathrm{Q}})]$
of the Shimura set $\mathrm{Z}(\overline{\mathrm{Q}})$ by

$\mathcal{Z}^{?}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})=\mathcal{O}_{\lambda}[\mathrm{Z}^{?}(\overline{\mathrm{Q}})]_{\mathfrak{m}_{\mathrm{F}}}$

for $?\in\\{\circ,\bullet\\}$. If we would like to identify these spaces for
$?\in\\{\circ,\bullet\\}$, then we will omit the superscript
$?\in\\{\circ,\bullet\\}$ from the notations. The description of the
supersingular locus of $\overline{\mathrm{X}}(\mathrm{Q})$ in Lemma 2.4 and
the cycle class map provides the following map

$\mathcal{C}:\mathcal{O}_{\lambda}[\mathrm{Z}^{\circ}(\overline{\mathrm{Q}})]\oplus\mathcal{O}_{\lambda}[\mathrm{Z}^{\bullet}(\overline{\mathrm{Q}})]\rightarrow\mathrm{H}^{2}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))^{\mathrm{Fr}_{p^{2}}}$

which can be considered as a geometric realization of the Jacquet–Langlands
correspondence between the group $\mathrm{G}(\mathrm{Q})$ and
$\mathrm{G}(\overline{\mathrm{Q}})$.

######  Definition 3.7.

We say the prime $p$ is _admissible_ for $\mathrm{T}_{\pi^{\mathrm{Q}}}$ if
its associated Galois representation satisfies the following condition

  1. (1)

$p$ is an inert prime in $\mathrm{F}$;

  2. (2)

$a_{p^{2}}(\pi^{\mathrm{Q}})\not\in\\{2p,-2p,-p^{2}-1,p^{2}+1\\}$ where
$a_{p^{2}}(\pi^{\mathrm{Q}})=\mathrm{tr}(\mathrm{Fr}_{p^{2}},\mathrm{T}_{\pi^{\mathrm{Q}}})$.

We say the prime $p$ is $n$-_admissible_ for $\mathrm{T}_{\pi^{\mathrm{Q}}}$
if it is admissible and

$a_{p^{2}}(\pi^{\mathrm{Q}})\not\in\\{2p,-2p,-p^{2}-1,p^{2}+1\\}\mod\lambda^{n}.$

We now record the following key fact on the Tate conjecture for the special
fiber of the quaternionic Shimura surface $\mathrm{X}(\mathrm{Q})$ proved in
[TX] which directly inspires our construction of the Flach system on
$\mathrm{X}(\mathrm{Q})$.

######  Proposition 3.8.

Suppose $p$ is an admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$. Then
the restriction of the map $\mathcal{C}$ induces an isomorphism

$\mathcal{C}_{\pi^{\mathrm{Q}}}:\mathcal{Z}^{\circ}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\oplus\mathcal{Z}^{\bullet}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\xrightarrow{\sim}\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))^{\mathrm{Fr}_{p^{2}}}.$

Moreover, the action of the Frobenius element of
$\mathrm{Gal}(\mathbf{F}_{p^{2}}/\mathbf{F}_{p})$ on the right hand switches
the two factors on the left hand side.

###### Proof.

This follows from the main result of [TX], see in particular section $1.1$ of
[TX] for this particular case. Note as explained in [Liu1], we can upgrade the
result of [TX] to integral coefficient using the assumption that
$\mathrm{tr}(\mathrm{Fr}_{p^{2}},\mathrm{T}_{\pi^{\mathrm{Q}}})\mod\lambda\not\in\\{2p,-2p\\}$.
∎

##  4\. Flach classes and reciprocity formula

###  4.1. Flach class of the quaternionic Shimura surface

Let $\mathrm{X}$ be a proper smooth variety of finite type over a field
$\mathrm{K}$. For an integer $d$, consider the complex

(4.1)
$\bigoplus_{x\in\mathrm{X}^{d-1}}\mathrm{K}_{2}k(x)\rightarrow\bigoplus_{x\in\mathrm{X}^{d}}k(x)^{\times}\xrightarrow{d_{1}}\bigoplus_{x\in\mathrm{X}^{d+1}}\mathbf{Z}$

where $\mathrm{X}^{i}$ denotes the set of points of codimension $i$ on the
variety $\mathrm{X}$, $\mathrm{K}_{2}k(x)$ is Milnor K-group of the field
$k(x)$, the first map is the so-called tame symbol map and the second is given
by the divisor map. The _motivic cohomology group_ of $\mathrm{X}$

$\mathrm{H}^{2d+1}_{\mathcal{M}}(\mathrm{X},\mathbf{Z}(d+1))$

is defined to be the cohomology of this complex: elements of
$\mathrm{H}^{2d+1}_{\mathcal{M}}(\mathrm{X},\mathbf{Z}(d+1))$ are represented
by the formal sums $\sum_{i}(\mathrm{Z}_{i},f_{i})$ of pairs consisting of a
codimension $d$ cycle $\mathrm{Z}_{i}$ on $\mathrm{X}$ and a non-zero rational
function $f_{i}$ on $\mathrm{Z}_{i}$ such that
$\sum_{i}\mathrm{div}_{\mathrm{Z}_{i}}(f_{i})=0$ as a Weil divisor on
$\mathrm{Z}_{i}$. This group is also known as the higher Chow group
$\mathrm{CH}^{d}(\mathrm{X},1)$ of $\mathrm{X}$ in the literature. There is a
Chern character map

(4.2)
$\mathrm{ch}:\mathrm{H}^{2d+1}_{\mathcal{M}}(\mathrm{X},\mathbf{Z}(d+1))\rightarrow\mathrm{H}^{2d+1}(\mathrm{X},\mathbf{Z}_{l}(d+1))$

given by the coniveau spectral sequences in K-theory and in étale cohomology.

Next suppose that $\mathcal{O}$ is a complete local ring with fraction field
$K$ and residue field $k$ of characteristic $p$ different from $l$. Let
$\mathfrak{X}$ be a proper regular scheme over $\mathcal{O}$, let $\mathrm{X}$
be its generic fiber and $\overline{\mathrm{X}}$ be its special fiber. The
motivic cohomology
$\mathrm{H}^{2d}_{\mathcal{M}}(\overline{\mathrm{X}},\mathbf{Z}(d))$ in this
case agree with the usual Chow group $\mathrm{CH}^{d}(\overline{\mathrm{X}})$.
There is a map

(4.3)
$\mathrm{div}:\mathrm{H}^{2d+1}_{\mathcal{M}}(\mathrm{X},\mathbf{Z}(d+1))\rightarrow\mathrm{H}^{2d}_{\mathcal{M}}(\overline{\mathrm{X}},\mathbf{Z}(d))$

defined by sending a pair $(\mathrm{Z},f)$ to the divisor of $f$ on the
closure $\mathcal{Z}$ of $\mathrm{Z}$ in $\mathfrak{X}$. Note that this
divisor is entirely supported on the special fiber $\overline{\mathcal{Z}}$ of
$\mathcal{Z}$.

From the definition, an element of the motivic cohomology of
$\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes{\mathbf{Q}},\mathbf{Z}(2))$
can be represented by a curve on the surface
$\mathrm{X}(\mathrm{Q})\otimes{\mathbf{Q}}$ and a rational function on this
curve with trivial Weil divisors. We will consider the element represented by
the pair

$\Theta^{[p]}=(\theta_{\ast}\mathrm{X}(\mathrm{B})\otimes{\mathbf{Q}},p)$

where $\theta_{\ast}\mathrm{X}(\mathrm{B})\otimes\mathbf{Q}$ is the image of
$\mathrm{X}(\mathrm{B})\otimes\mathbf{Q}$ under the Hirzebruch–Zagier morphism
$\theta$ and we will refer to this element as the Flach element in the motivic
cohomology of the quaternionic Hilbert–Blumenthal surface
$\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}$.

######  Proposition 4.1.

Suppose that the residual Galois representation
$\overline{\rho}_{\pi^{\mathrm{Q}}}$ attached to $\pi^{\mathrm{Q}}$ has non-
solvable image. Then the $\pi^{\mathrm{Q}}$-isotypic component

$\mathrm{H}^{i}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathcal{O}_{\lambda})=\mathrm{H}^{i}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathcal{O}_{{\lambda}})_{\mathfrak{m}_{\mathrm{F}}}$

of
$\mathrm{H}^{i}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathcal{O}_{\lambda})$
vanishes unless $i=2$.

###### Proof.

This is the main theorem of [CT], see [CT, Theorem 7.5.2]. ∎

######  Lemma 4.2.

Assume that the residual Galois representation
$\overline{\rho}_{\pi^{\mathrm{Q}}}$ attached to $\pi^{\mathrm{Q}}$ has non-
solvable image. Then we have an isomorphism

$\mathrm{H}^{3}(\mathrm{X}(\mathrm{Q}),\mathcal{O}_{\lambda}(2))_{\mathfrak{m}_{\mathrm{F}}}=\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2))).$

###### Proof.

This follows from the Hoschchild–Serre spectral sequence applied to the
cohomology
$\mathrm{H}^{3}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathcal{O}_{{\lambda}}(2))_{\mathfrak{m}_{\mathrm{F}}}$
and the fact that
$\mathrm{H}^{i}(\mathrm{X}(\mathrm{Q})\otimes{\overline{\mathbf{Q}}},\mathcal{O}_{{\lambda}})_{\mathfrak{m}_{\mathrm{F}}}\neq
0$ only for $i=2$. ∎

The Chern character map in (4.2) induces a map

$\mathrm{ch}_{\mathcal{O}_{\lambda}}:\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathbf{Z}(2))\rightarrow\mathrm{H}^{3}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathcal{O}_{\lambda}(2))_{\mathfrak{m}_{\mathrm{F}}}.$

Suppose the Galois representation $\overline{\rho}_{\pi^{\mathrm{Q}}}$
associated to $\mathfrak{m}_{\mathrm{F}}$ has non-solvable image, then it
induces the following Abel–Jacobi map

$\mathrm{AJ}_{\pi^{\mathrm{Q}}}:\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q},\mathbf{Z}(2))\rightarrow\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2)))$

in light of the previous lemma. The image of the Flach element $\Theta^{[p]}$
under this map will be denoted by

(4.4) $\kappa^{[p]}=\mathrm{AJ}_{\pi^{\mathrm{Q}}}(\Theta^{[p]})$

and will be referred to as the Flach class and it will play a pivotal role for
our Euler system argument later.

###  4.2. Local bevaviours of Flach classes

To use the Euler system argument, we will have to analyze the local behaviours
of the Flach class $\kappa^{[p]}$ at all primes $r$ of $\mathbf{Q}$. Let
$\mathrm{N}(\pi^{\mathrm{Q}})$ be the product of primes at which the Asai
representation
$((\mathrm{As}(\rho_{\pi^{\mathrm{Q}},\lambda})(-1),\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1)))$
is ramified.

######  Lemma 4.3.

For any prime $r\nmid\mathrm{N}(\pi^{\mathrm{Q}})$, the Flach class is
unramified:

$\mathrm{loc}_{r}(\kappa^{[p]})\in\mathrm{H}^{1}_{\mathrm{fin}}(\mathbf{Q}_{r},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{r},\mathcal{O}_{\lambda}(2))).$

###### Proof.

Since
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{r},\mathcal{O}_{\lambda}(2))$
is unramified at $r$. It follows that

(4.5)
$\displaystyle\mathrm{H}^{1}_{\mathrm{sin}}(\mathbf{Q}_{r},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{r},\mathcal{O}_{\lambda}(2)))$
$\displaystyle=(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{r},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{r}}}$
$\displaystyle=(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{r},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{r}}}.$

On the other hand, we have a commutative diagram by [Weston1, Theorem 3.1.1]

${\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}_{r},\mathbf{Z}(2))}$${\mathrm{H}^{1}(\mathbf{Q}_{r},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{r},\mathcal{O}_{\lambda}(2)))}$${\mathrm{H}^{2}_{\mathcal{M}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{r},\mathbf{Z}(1))}$${(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{r},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{r}}}}$$\scriptstyle{\mathrm{AJ}_{\pi^{\mathrm{Q}}}}$$\scriptstyle{\mathrm{div}_{r}}$$\scriptstyle{\partial_{r}}$$\scriptstyle{\mathrm{cl}}$

where

  1. (1)

the top horizontal map is the Abel–Jacobi map $\mathrm{AJ}_{\pi^{\mathrm{Q}}}$
for $\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}_{r}$;

  2. (2)

the bottom horizontal map is given by the usual cycle class map of the special
fiber $\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{r}$;

  3. (3)

the right vertical map $\partial_{r}$ is the singular quotient map at $r$
under the identification of (4.5).

By definition, it is clear that $\mathrm{div}_{r}(\Theta^{[p]})$ vanishes in
$\mathrm{H}^{2}_{\mathcal{M}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{r},\mathbf{Z}(1))$,
indeed $\mathrm{div}_{r}(\Theta^{[p]})$ is only supported at the special fiber
of $\theta_{\ast}\mathrm{X}(\mathrm{B})$ at $p$. Thus,
$\partial_{r}(\kappa^{[p]})$ vanishes and hence
$\mathrm{loc}_{r}(\kappa^{[p]})$ lies in the finite part. ∎

Next we turn to the more interesting case of analyzing
$\mathrm{loc}_{p}(\Theta^{[p]})$. First we give a slight refinement of
Proposition 3.8.

######  Lemma 4.4.

Suppose $p$ is an admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$, then
we have an isomorphism

$\mathcal{Z}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\xrightarrow{\sim}(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p}}}.$

###### Proof.

By Proposition 3.8,
$(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p}}}$
can be identified as the invariant space of
$(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p^{2}}}}$
under the action of $\mathrm{Gal}(\mathbf{F}_{p^{2}}/\mathbf{F}_{p})$. Hence
it can be identified as the diagonal image of
$\mathcal{Z}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})$ in
$\mathcal{Z}^{\circ}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\oplus\mathcal{Z}^{\bullet}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})$
as the non-trivial element of
$\mathrm{Gal}(\mathbf{F}_{p^{2}}/\mathbf{F}_{p})$ interchanges the
$\circ$-component and the $\bullet$-component of the supersingular locus of
$\overline{\mathrm{X}}^{\mathrm{ss}}(\mathrm{Q})$. The lemma is proved. ∎

Now we state the _reciprocity formula_ for the class $\kappa^{[p]}$. First we
define a bilinear pairing

$(\cdot,\cdot):\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})]\times\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})]\rightarrow\mathcal{O}_{\lambda}$

on the space $\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})]$ by

$(\zeta,\phi)=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{Q}})}\zeta\phi(z).$

######  Proposition 4.5.

Let $p$ be an admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$ and $\phi$
be any element in
$\mathcal{Z}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})$. Then we
have the following identity

$(\partial_{p}(\kappa^{[p]}),\phi)=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}\phi(z)$

and here the sum is taken over elements in the image of
$\mathrm{Z}(\overline{\mathrm{B}})$ in $\mathrm{Z}(\overline{\mathrm{Q}})$ via
the canonical embedding $\nu$.

###### Proof.

By definition, it is clear that
$\mathrm{div}_{p^{2}}(\Theta^{[p]})=[\mathrm{X}^{\circ}(\mathrm{B})]+[\mathrm{X}^{\bullet}(\mathrm{B})]$
as an element in the motivic cohomology

$\mathrm{H}^{2}_{\mathcal{M}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{p^{2}},\mathbf{Z}(1))$

which can be identified with the usual Chow group
$\mathrm{CH}^{1}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{p^{2}})$.
We will still consider the commutative diagram used in the previous case

(4.6)
${\mathrm{H}^{3}_{\mathcal{M}}(\mathrm{X}(\mathrm{Q})\otimes\mathbf{Q}_{p^{2}},\mathbf{Z}(2))}$${\mathrm{H}^{1}(\mathbf{Q}_{p^{2}},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}}_{p},\mathcal{O}_{\lambda}(2)))}$${\mathrm{H}^{2}_{\mathcal{M}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\mathbf{F}_{p^{2}},\mathbf{Z}(1))}$${(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p^{2}}}}}$$\scriptstyle{\mathrm{AJ}_{\pi^{\mathrm{Q}}}}$$\scriptstyle{\mathrm{div}_{p^{2}}}$$\scriptstyle{\partial_{p^{2}}}$$\scriptstyle{\mathrm{cl}}$

and we are concerned with the singular residue
$\partial_{p^{2}}(\kappa^{[p]})$. By the commutativity of this diagram,
$\partial_{p^{2}}(\kappa^{[p]})$ agrees with the cycle class of
$[\mathrm{X}^{\circ}(\mathrm{B})]+[\mathrm{X}^{\bullet}(\mathrm{B})]$ in
$(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p^{2}}}}$.

It follows that under the identification

(4.7)
$\mathcal{Z}^{\circ}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\oplus\mathcal{Z}^{\bullet}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\xrightarrow{\sim}\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1))^{\mathrm{Fr}_{p^{2}}},$

$\partial_{p^{2}}(\kappa^{[p]})$ is given by
$(\mathbf{1}_{\mathrm{Z}^{\circ}(\overline{\mathrm{B}})},\mathbf{1}_{\mathrm{Z}^{\bullet}(\overline{\mathrm{B}})})$
where $\mathbf{1}_{\mathrm{Z}^{?}(\overline{\mathrm{B}})}$ is the
characteristic function of $\mathrm{Z}^{?}(\overline{\mathrm{B}})$ for
$?\in\\{\circ,\bullet\\}$. This follows from the fact that the isomorphism in
(4.7) is given by the cycle class map and statements of Lemma 3.4. Thus under
the identification

$\mathcal{Z}_{\pi^{\overline{\mathrm{Q}}}}(\overline{\mathrm{Q}})\xrightarrow{\sim}(\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\overline{\mathrm{X}}(\mathrm{Q})\otimes\overline{\mathbf{F}}_{p},\mathcal{O}_{\lambda}(1)))^{\mathrm{G}_{\mathbf{F}_{p}}},$

the singular residue $\partial_{p}(\kappa^{[p]})$ is simply given by
$\mathbf{1}_{\mathrm{Z}(\overline{\mathrm{B}})}$ and hence we have

$\displaystyle(\partial_{p}(\kappa^{[p]}),\phi)$
$\displaystyle=(\mathbf{1}_{\mathrm{Z}(\overline{\mathrm{B}})},\phi)=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}\phi(z).$

∎

##  5\. Distinguished representation and base change

###  5.1. Dinstinguished representation

We recall the definition of a distinguished representation and its
quaternionic variant. Let $\Pi$ be a cuspidal automorphic representation of
$\mathrm{GL}_{2}(\mathbf{A}_{\mathrm{F}})$ with trivial central character,
then $\Pi$ is called _distinguished_ if there exists an automorphic form
$\phi$ in the space of $\Pi$ such that the period

$\mathcal{P}^{\mathrm{GL}_{2}}_{\mathrm{dis}}(\phi)=\int_{\mathrm{GL}_{2}(\mathbf{Q})\backslash\mathrm{GL}_{2}(\mathbf{A}_{\mathrm{F}})}\phi(g)dg$

is non-vanishing. It is known that $\pi$ is distinguished if it satisfies the
following properties:

  1. (1)

$\Pi_{\infty}$ is in the discrete series;

  2. (2)

$\Pi$ is in the image of base-change of an automorphic representation $\pi$ of
$\mathrm{GL}_{2}(\mathbf{A})$.

We will consider the quaternionic variant of the above period for the
quaternion algebra
$\overline{\mathrm{Q}}=\overline{\mathrm{B}}\otimes\mathrm{F}$. Suppose that
$\pi^{\overline{\mathrm{Q}}}$ is a cuspidal automorphic representation of
$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A})$. Then we say
$\pi^{\overline{\mathrm{Q}}}$ is distinguished if there exists an automorphic
form $\phi$ in the space of $\pi^{\overline{\mathrm{Q}}}$ such that the period

$\mathcal{P}_{\mathrm{dis}}(\phi)=\int_{\mathrm{G}(\overline{\mathrm{B}})(\mathbf{Q})\backslash\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A})}\phi(g)dg$

is non-vanishing. For a fixed $\phi$, we will call
$\mathcal{P}_{\mathrm{dis}}(\phi)$ the distinguished period of $\phi$.

These distinguished representations appear naturally in the proof of Tate
conjectures for Hilbert–Blumenthal surfaces and its quaternionic variant
[HLR], [Lai], [FH]. In fact, it can be shown that the Tate classes defined on
abelian extensions of $\mathbf{Q}$ are all supported on the isotypic component
of the distinguished representations. We will not use this fact in the sequel.

The following proposition shows that the notion of being distinguished is
compatible with respect to the Jacquet–Langlands correspondence.

######  Proposition 5.1.

Suppose that $\Pi$ is a cuspidal automorphic representation of
$\mathrm{GL}_{2}(\mathbf{A}_{\mathrm{F}})$ which corresponds to a cuspidal
automorphic representation $\pi^{\overline{\mathrm{Q}}}$ of
$\mathrm{G}(\overline{\mathrm{Q}})$ under the Jacquet–Langlands
correspondence. Then $\pi^{\overline{\mathrm{Q}}}$ is distinguished with
respect to $\mathrm{G}(\overline{\mathrm{B}})$ if and only if $\pi$ is
distinguished with respect to $\mathrm{GL}_{2}(\mathbf{A})$ and for each prime
$r$ dividing the discriminant $\mathrm{D}({\overline{\mathrm{B}}})$ of
$\overline{\mathrm{B}}$ which is inert in $\mathrm{F}$, the local
representation $\pi^{\overline{\mathrm{Q}}}_{r}$ is not a principal series
rerpesentation $\mathrm{I}(\mu_{1},\mu_{2})$ with $\mu_{i}$ trivial on
$\mathbf{Q}^{\times}_{r}$ for $i=1,2$.

###### Proof.

This is a special case of the main theorem of Flicker–Hakim, see [FH, Theorem
0.3]. ∎

###  5.2. Asai representation and base change

In light of the above proposition and our reciprocity formula in Proposition
4.5, we will from here on restrict our attention to the situation given by the
following datum:

  1. (1)

$f$ is a weight $2$ newform in $\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ which
defines a cuspidal automorphic representation $\pi$ of
$\mathrm{GL}_{2}(\mathbf{A})$;

  2. (2)

$\pi$ admits a Jacquet–Langlands transfer to a cuspidal automorphic
representation $\pi^{\circ}=\pi^{\overline{\mathrm{B}}}$ of
$\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A})$;

  3. (3)

$\pi^{\overline{\mathrm{Q}}}$ is the cuspidal automorphic representation of
$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A})$ given by the base change of
$\pi^{\overline{\mathrm{B}}}$ to
$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A})$;

  4. (4)

$\pi^{\mathrm{Q}}$ is the cuspidal automorphic representation of
$\mathrm{G}({\mathrm{Q}})(\mathbf{A})$ obtained from
$\pi^{\overline{\mathrm{Q}}}$ via the Jacquet–Langlands transfer from
$\pi^{\overline{\mathrm{Q}}}$ to $\mathrm{G}({\mathrm{Q}})(\mathbf{A})$.

Let $(\rho_{\pi^{\circ}},\mathrm{V}_{\pi^{\circ}})$ be the Galois
representation attached to $\pi^{\circ}$ and
$(\rho_{\pi^{\mathrm{Q}}},\mathrm{V}_{\pi^{\mathrm{Q}}})$ be the Galois
representation attached to $\pi^{\mathrm{Q}}$. By construction, the Galois
representation attached to $\pi^{\overline{\mathrm{Q}}}$ is the same as
$(\rho_{\pi^{\mathrm{Q}}},\mathrm{V}_{\pi^{\mathrm{Q}}})$. In the situation
above, the Asai representation
$\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1)$ can be related to the
representation $\mathrm{V}_{\pi^{\circ}}$ as in the following lemma.

######  Lemma 5.2.

We have an isomorphism

$\mathrm{As}(\mathrm{V}_{\pi^{\mathrm{Q}}})(-1)\cong\mathrm{Sym}^{2}(\mathrm{V}_{\pi^{\circ}})(-1)\oplus\mathrm{E}_{\lambda}(\omega_{\mathrm{F}/\mathbf{Q}}).$

Note that $\mathrm{Sym}^{2}(\mathrm{V}_{\pi^{\circ}})(-1)$ is also isomorphic
to $\mathrm{Ad}^{0}(\mathrm{V}_{\pi^{\circ}})$. Similarly, we also have a such
decomposition integrally

$\mathrm{As}(\mathrm{T}_{\pi^{\mathrm{Q}}})(-1)\cong\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ}})(-1)\oplus\mathcal{O}_{\lambda}(\omega_{\mathrm{F}/\mathbf{Q}})$

for the lattice $\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ}})(-1)$ in
$\mathrm{Sym}^{2}(\mathrm{V}_{\pi^{\circ}})(-1)$.

##  6\. Bounding the adjoint Selmer groups

###  6.1. Generalities on Selmer groups

We will consider a general Galois representation
$\rho:\mathrm{G}_{\mathbf{Q}}\rightarrow\mathrm{GL}(\mathrm{V})$ over
$\mathrm{E}_{\lambda}$. Suppose that $\mathrm{T}$ is a Galois stable lattice
in $\mathrm{V}$ and set $\mathcal{M}=\mathrm{V}/\mathrm{T}$, this is called
the divisible Galois module associated to $\rho$. We recall some relevant
definitions and facts concerning the Bloch–Kato Selmer group for
$\mathcal{M}$. These Galois modules fit in the exact sequence

$0\rightarrow\mathrm{T}\xrightarrow{i}\mathrm{V}\xrightarrow{\mathrm{pr}}\mathcal{M}\rightarrow
0.$

Let $\mathrm{M}_{n}=\mathcal{M}[\lambda^{n}]$ and
$\mathrm{T}_{n}=\mathrm{T}/\lambda^{n}\mathrm{T}$. Let
$i_{n}:\mathrm{M}_{n}\hookrightarrow\mathcal{M}$ and
$\mathrm{pr}_{n}:\mathrm{T}\rightarrow\mathrm{T}_{n}$ be the natural inclusion
and reduction maps. The Galois module $\mathrm{M}_{n}$ will be referred to as
the finite Galois module.

We define the local Bloch–Kato conditions using the following recipe:

  1. (1)

for $v\neq l$, we define
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{v},\mathrm{V})=\mathrm{H}^{1}_{\mathrm{fin}}(\mathbf{Q}_{v},\mathrm{V})$;

  2. (2)

for $v=l$, we define
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{p},\mathrm{V})=\mathrm{ker}\\{\mathrm{H}^{1}(\mathbf{Q}_{l},\mathrm{V})\rightarrow\mathrm{H}^{1}(\mathbf{Q}_{l},\mathrm{V}\otimes\mathrm{B}_{\mathrm{cris}})\\}$;

  3. (3)

we define
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathcal{M})=\mathrm{pr}_{\ast}\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathrm{V})$
for each prime $r$;

  4. (4)

we define
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathrm{M}_{n})=i^{\ast}_{n}\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathcal{M})$
for each prime $r$.

We will be interested in the Bloch–Kato Selmer group of $\mathcal{M}$ defined
by

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M})=\mathrm{ker}\\{\mathrm{H}^{1}(\mathbf{Q},\mathcal{M})\rightarrow\prod_{r}\frac{\mathrm{H}^{1}(\mathbf{Q}_{r},\mathcal{M})}{\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathcal{M})}\\}.$

The Bloch–Kato Selmer group of $\mathrm{M}_{n}$ is defined by the same recipe
replacing $\mathcal{M}$ by $\mathrm{M}_{n}$. Moreover we have

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M})=\lim\limits_{\longrightarrow}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})$

and an exact sequence

$0\rightarrow\mathrm{pr}_{\ast}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{V})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M})\rightarrow\Sha(\mathbf{Q},\mathcal{M})\rightarrow
0$

defining the _Tate-Shafarevich group_ $\Sha(\mathbf{Q},\mathcal{M})$ for
$\mathcal{M}$. The proof of following lemma is an easy diagram chase and
therefore will be omitted.

######  Lemma 6.1.

Suppose we have an exact sequence of divisible or finite Galois modules

$0\rightarrow\mathcal{M}^{\prime}\rightarrow\mathcal{M}\rightarrow\mathcal{M}^{\prime\prime}\rightarrow
0.$

Then it induces an exact sequence

$\displaystyle
0\rightarrow\mathrm{H}^{0}(\mathbf{Q},\mathcal{M}^{\prime})\rightarrow\mathrm{H}^{0}(\mathbf{Q},\mathcal{M})\rightarrow\mathrm{H}^{0}(\mathbf{Q},\mathcal{M}^{\prime\prime})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M}^{\prime})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathcal{M}^{\prime\prime})\rightarrow
0.$

###  6.2. Statement of the main result

We now consider the datum given in the beginning of §5.2. Let
$f\in\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ be a newform of level
$\mathrm{N}$ whose associated automorphic representation of
$\mathrm{GL}_{2}(\mathbf{A})$ is given by $\pi$. Suppose that $\mathrm{N}$
admits a decomposition $\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$. Under the
assumption that $\mathrm{N}^{-}$ is square-free and consists of odd number of
prime divisors, $f$ admits a normalized Jacquet–Langlands transfer
$f^{\dagger}$ as an automorphic form realized in
$\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{B}})]$. Here normalized
means that there is an element $z\in\mathrm{Z}(\overline{\mathrm{B}})$ such
that $f^{\dagger}(z)$ is non-zero modulo $\lambda$. The base change of
$f^{\dagger}$ to $\mathrm{F}$ can be regarded as an automorphic form realized
in $\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{Q}})]$ which we will
denote by $\phi^{\dagger}$. We will choose $\phi^{\dagger}$ such that it is
normalized as well in the same sense for $f^{\dagger}$. In the discussion of
§5.2, the automorphic form $f^{\dagger}$ is contained in an automorphic
representation of $\mathrm{G}(\overline{\mathrm{B}})(\mathbf{A})$ which is
denoted by $\pi^{\circ}=\pi^{\overline{\mathrm{B}}}$ and $\phi^{\dagger}$ is
contained in the automorphic representation $\pi^{\overline{\mathrm{Q}}}$ of
$\mathrm{G}(\overline{\mathrm{Q}})(\mathbf{A})$ which is the base change of
$\pi^{\circ}$ to $\mathrm{F}$. Recall also that $\pi^{\overline{\mathrm{Q}}}$
admits a Jacquet–Langlands transfer $\pi^{\mathrm{Q}}$ to
$\mathrm{G}(\mathrm{Q})(\mathbf{A})$.

For the convenience of the reader, we recall some notations concerning Galois
modules and period integrals that we will use later:

  1. (1)

Let $(\rho_{\pi^{\circ}},\mathrm{V}_{\pi^{\circ}})$ be the Galois
representation associated to $\pi^{\circ}$ and $\mathrm{T}_{\pi^{\circ}}$ be
the Galois stable lattice that we have fixed in $\S 5.2$. Then we define the
divisible Galois module $\mathcal{M}_{\pi^{\circ}}$ by

$0\rightarrow\mathrm{T}_{\pi^{\circ}}\rightarrow\mathrm{V}_{\pi^{\circ}}\rightarrow\mathcal{M}_{\pi^{\circ}}\rightarrow
0.$

  2. (2)

Let $(\rho_{\pi^{\mathrm{Q}}},\mathrm{V}_{\pi^{\mathrm{Q}}})$ be the Galois
representation associated to $\pi^{\mathrm{Q}}$ and
$\mathrm{T}_{\pi^{\mathrm{Q}}}$ be the Galois stable lattice we have fixed in
$\S 5.2$. Then we define the divisible Galois module
$\mathcal{M}_{\pi^{\mathrm{Q}}}$ by

$0\rightarrow\mathrm{T}_{\pi^{\mathrm{Q}}}\rightarrow\mathrm{V}_{\pi^{\mathrm{Q}}}\rightarrow\mathcal{M}_{\pi^{\mathrm{Q}}}\rightarrow
0.$

  3. (3)

We have a natural decomposition

$\mathrm{As}(\mathcal{M}_{\pi^{\mathrm{Q}}})(-1)=\mathrm{Sym}^{2}(\mathcal{M}_{\pi^{\circ}})(-1)\oplus\mathrm{E}_{\lambda}/\mathcal{O}_{\lambda}(\omega_{\mathrm{F}/\mathbf{Q}}).$

  4. (4)

Note that $\mathrm{Sym}^{2}(\mathcal{M}_{\pi^{\circ}})(-1)$ is the same as
$\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$. We will set

$\mathrm{M}_{n}=\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}[\lambda^{n}])$

and

$\mathrm{N}_{n}=\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ},n})$

for $\mathrm{T}_{\pi^{\circ},n}=\mathrm{T}_{\pi^{\circ}}\mod\lambda^{n}$ and
$n\geq 1$.

  5. (5)

Recall we have an embedding
$\vartheta:\mathrm{Z}(\overline{\mathrm{B}})\rightarrow\mathrm{Z}(\overline{\mathrm{Q}})$
of Shimura sets as in Lemma 3.4 and thus we can consider the integral
distinguished period defined by

$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}\phi^{\dagger}(z)$

where the sum is taken over elements of the image of
$\vartheta:\mathrm{Z}(\overline{\mathrm{B}})\rightarrow\mathrm{Z}(\overline{\mathrm{Q}})$.

The following theorem will be the main theorem on bounding the Selmer group
using the distinguished period.

######  Theorem 6.2.

Let $f\in\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ be a newform of weight $2$
with $\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$ such that $\mathrm{N}^{-}$ is
squarefree and has odd number of prime factors. Let $f^{\dagger}$ be the
normalized automorphic form in
$\mathcal{O}_{\lambda}[\mathrm{Z}(\overline{\mathrm{B}})]$ corresponding to
$f$ under the Jacquet–Langlands correspondence and $\phi^{\dagger}$ be the
base-change of $f^{\dagger}$ which is also normalized. Let
$\nu=\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$ and
$\eta=\varpi^{\nu}$. Suppose the following conditions hold.

  1. (1)

The residual Galois representation $\overline{\rho}_{\pi^{\circ}}$ is
absolutely irreducible;

  2. (2)

The image of $\overline{\rho}_{\pi^{\circ}}$ contains
$\mathrm{GL}_{2}(\mathbf{F}_{p})$;

  3. (3)

We further assume that $\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})=0$ for every
$n\geq 1$ where
$\Delta_{n}=\mathrm{Gal}(\mathbf{Q}(\mathrm{M}_{n})/\mathbf{Q})$ for the
splitting field $\mathbf{Q}(\mathrm{M}_{n})$ of the Galois module
$\mathrm{M}_{n}$.

Then $\eta$ annihilates the Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$,
moreover we have

$\mathrm{length}_{\mathcal{O}_{\lambda}}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\leq\nu.$

###  6.3. The Flach system argument

To prove the main theorem, we will show that under the assumptions of the main
theorem, $\eta$ annihilates each finite Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})$ with $n\geq 1$. Here the
argument follows closely that of [Flach1] and [Weston1]. We define

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{n}(2))$

to be the reduction of
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2))$
modulo $\lambda^{n}$.

######  Lemma 6.3.

For each $n\geq 1$, the Galois module

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{n}(1))$

is isomorphic to $m(\pi^{\mathrm{Q}},d)$ copies of
$\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ},n})(-1)\oplus\mathcal{O}_{n}(\omega_{\mathrm{F}/\mathbf{Q}})$.
In particular, there is a projection from
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{n}(2))$
to $\mathrm{N}_{n}=\mathrm{Sym}^{2}(\mathrm{T}_{\pi^{\circ},n})$.

###### Proof.

Recall that we have an isomorphism

$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(1))\cong
m(\pi^{\mathrm{Q}},d)(\mathrm{Sym}^{2}\mathrm{T}_{\pi^{\circ}}(-1)\oplus\mathcal{O}_{\lambda}(\omega_{\mathrm{F}/\mathbf{Q}}))$

by the definition of $d$-cleaness of $\mathrm{T}_{\pi^{\mathrm{Q}}}$. The
lemma follows from this immediately from Proposition 4.1 which implies that
$\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(1))$
is torsion free. ∎

We consider the Flach class

$\kappa^{[p]}\in\mathrm{H}^{1}(\mathbf{Q},\mathrm{H}^{2}_{\pi^{\mathrm{Q}}}(\mathrm{X}(\mathrm{Q})\otimes\overline{\mathbf{Q}},\mathcal{O}_{\lambda}(2)))$

defined as in 4.4 and we will use this class to construct annihilator of the
Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$. By
the above lemma, we can project this class to
$\mathrm{H}^{1}(\mathbf{Q},\mathrm{N}_{n})$ and we will denote by
$\kappa^{[p]}_{n}$ the resulting class. We also recall that
$\eta=\varpi^{\nu}$ where
$\nu=\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$.

######  Lemma 6.4.

Let $p$ be an $n$-admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$.

  1. (1)

The singular quotient $\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})$
is free of rank one over $\mathcal{O}_{n}$.

  2. (2)

The quotient of $\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})$ by the
module generated by $\partial_{p}(\kappa^{[p]}_{n})$ is annihlated by $\eta$.

###### Proof.

By the definition of an $n$-admissible prime, it is easily calculated that
among the Frobenius eigenvalues at $p$ for $\mathrm{N}_{n}$, the number of
times the value $1$ can only appear once. Since we have

$\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})=\operatorname{Hom}(\mathbf{Z}_{l}(1),\mathrm{N}_{n})^{\mathrm{G}_{\mathbf{F}_{p}}}=(\mathrm{N}_{n}(-1))^{\mathrm{G}_{\mathbf{F}_{p}}},$

$\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})$ is free of rank one
over $\mathcal{O}_{n}$. It follows from the reciprocity law proved in
Proposition 4.5 that the quotient of
$\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})$ by the module generated
by $\partial_{p}(\kappa^{[p]}_{n})$ is annihilated by $\eta$. ∎

######  Lemma 6.5.

Let $p$ be an $n$-admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$ and
define

$\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})=\mathrm{ker}\\{\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathrm{H}^{1}(\mathbf{Q}_{p},\mathrm{M}_{n})\\}.$

Then we have

$\eta\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\subset\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n}).$

###### Proof.

We consider the element $\kappa^{[p]}_{n}$ in
$\mathrm{H}^{1}(\mathbf{Q},\mathrm{N}_{n})$. We have verified that
$\mathrm{loc}_{r}(\kappa^{[p]}_{n})$ lies in the finite part for all $r\nmid
p\mathrm{N}$. On the other hand, the same proofs for [Flach1, Lemma 2.8] and
[Flach1, Lemma 2.10] carry over here and shows that
$\mathrm{loc}_{r}(\kappa^{[p]}_{n})$ lie in
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathrm{M}_{n})$ for $r\mid\mathrm{N}$.
Under the local Tate duality

$\langle\cdot,\cdot\rangle_{r}:\mathrm{H}^{1}(\mathbf{Q}_{r},\mathrm{N}_{n})\times\mathrm{H}^{1}(\mathbf{Q}_{r},\mathrm{M}_{n})\rightarrow\mathcal{O}_{n},$

it is well–known that our local conditions
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathrm{N}_{n})$ are orthogonal to
$\mathrm{H}^{1}_{f}(\mathbf{Q}_{r},\mathrm{M}_{n})$ and at $r=p$ induces a
perfect pairing

$\langle\cdot,\cdot\rangle_{p}:\mathrm{H}^{1}_{\mathrm{sin}}(\mathbf{Q}_{p},\mathrm{N}_{n})\times\mathrm{H}^{1}_{\mathrm{fin}}(\mathbf{Q}_{p},\mathrm{M}_{n})\rightarrow\mathcal{O}_{n}.$

Let $s$ be any element in $\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})$,
then by global class field theory

$\sum_{r}\langle\mathrm{loc}_{r}(s),\mathrm{loc}_{r}(\kappa^{[p]})\rangle_{r}=0.$

This identity reduces to
$\langle\mathrm{loc}_{p}(s),\mathrm{loc}_{p}(\kappa^{[p]})\rangle_{p}=0$ by
the discussions above. It follows that
$\langle\mathrm{loc}_{p}(s),\partial_{p}(\kappa^{[p]})\rangle_{p}=0$. Since we
know that $\eta\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{n})$ is
contained in the line generated by $\partial_{p}\kappa^{[p]}$ by Lemma 6.4,
$\eta\mathrm{loc}_{p}(s)$ has to vanish by the perfectness of the above
pairing. Therefore
$\eta\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\subset\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})$
follows from this. ∎

######  Lemma 6.6.

Let $\mathrm{F}_{n}=\mathbf{Q}(\mathrm{M}_{n})$ be the splitting field for the
Galois module $\mathrm{M}_{n}$ and
$\Delta_{n}=\mathrm{Gal}(\mathrm{F}_{n}/\mathbf{Q})$. Then we have

$\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})\subset\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})$

where $\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})$ is considered as a subgroup
of $\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}_{n})$ via inflation.

###### Proof.

Let $s\in\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})$ and consider the
exact sequence

(6.1)
$0\rightarrow\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})\rightarrow\mathrm{H}^{1}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathrm{H}^{1}(\mathrm{F}_{n},\mathrm{M}_{n})^{\Delta_{n}}=\operatorname{Hom}_{\Delta_{n}}(\mathrm{G}_{\mathrm{F}_{n}},\mathrm{M}_{n})\rightarrow
0.$

Let $\psi:\mathrm{G}_{\mathrm{F}_{n}}\rightarrow\mathrm{M}_{n}$ be the image
of $s$ in
$\operatorname{Hom}_{\Delta_{n}}(\mathrm{G}_{\mathrm{F}_{n}},\mathrm{M}_{n})$.
Then we need to show that $\psi=0$. Let $\tilde{s}$ be the cocycle
representing $s$ and let $\mathrm{F}^{\prime}_{n}$ be the fixed field of the
kernel of $\tilde{s}$. Let $\Gamma$ be
$\mathrm{Gal}(\mathrm{F}^{\prime}_{n}/\mathrm{F}_{n})$, then it is clear that
$\psi$ factors through $\psi:\Gamma\rightarrow\mathrm{M}_{n}$. Let $\tau$ be
the Frobenius element at $p$ in $\Delta_{n}$ and fix a lift $\tau^{\prime}$ to
$\mathrm{Gal}(\mathrm{F}^{\prime}_{n}/\mathbf{Q})$. Let $g$ be any element in
$\Gamma$. By the Chebatorev density theorem we can find a place $v^{\prime}$
of $\mathrm{F}^{\prime}_{n}$ such that
$\mathrm{Fr}_{\mathrm{F}^{\prime}_{n}/\mathrm{F}_{n}}(v^{\prime})=\tau^{\prime}g$.
Let $v$ be the place under $v^{\prime}$ in $\mathrm{F}_{n}$ which necessarily
lies over $p$.

Since $s_{p}:=\mathrm{res}_{p}(s)$ is trivial,
$\tilde{s}|_{\mathrm{Gal}(\mathrm{F}^{\prime}_{n,v^{\prime}}/\mathbf{Q}_{p})}$
is a coboundary. Thus we have

$\tilde{s}(\tau^{\prime}g)\in(\tau^{\prime}g-1)\mathrm{M}_{n}=(\tau-1)\mathrm{M}_{n}.$

Taking $g=1$ gives $\tilde{s}(\tau^{\prime})\in(\tau-1)\mathrm{M}_{n}$. On the
other hand, the cocycle relation gives

$\tilde{s}(\tau^{\prime}g)=\tilde{s}(\tau^{\prime})+\tau\tilde{s}(g).$

It follows then $\tau\tilde{s}(g)\in(\tau-1)\mathrm{M}_{n}$. Hence
$\tilde{s}(g)\in(\tau-1)\mathrm{M}_{n}$ for any $g\in\Gamma$ as
$(\tau-1)\tilde{s}(g)\in(\tau-1)\mathrm{M}_{n}$. Thus the image of $\psi$ lie
in $(\tau-1)\mathrm{M}_{n}$. Note that $\psi$ is $\Delta_{n}$-equivariant and
$\mathrm{M}_{n}$ is irreducible. Since
$\mathrm{M}_{n}\neq(\tau-1)\mathrm{M}_{n}$ by the same reasoning as in Lemma
6.4, the image of $\psi$ is zero. ∎

######  Lemma 6.7.

Under the same assumptions of Theorem 6.2, $\eta$ annihilates the Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$.
Moreover
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
can be identified with $\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{\nu})$.

###### Proof.

For each $n$, we pick an $n$-admissible prime $p$ for
$\mathrm{T}_{\pi^{\mathrm{Q}}}$. By the previous Lemma 6.5 and Lemma 6.6, we
have

$\eta\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\subset\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n}).$

By the second assumption in the statement of the Theorem, we know
$\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})$ is trivial. Thus
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})$ is indeed annihilated by
$\eta$ for each $n$. Hence
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$ is
also annihilated by $\eta$ as desired.

For the in particular part, we abbreviate
$\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})$ by $\mathrm{M}$. Consider the
exact sequence

$0\rightarrow\mathrm{M}_{n}\rightarrow\mathrm{M}\xrightarrow{\varpi^{n}}\varpi^{n}\mathrm{M}\rightarrow
0.$

It induces a long exact sequence

$0\rightarrow\mathrm{H}^{0}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathrm{H}^{0}(\mathbf{Q},\mathrm{M})\rightarrow\mathrm{H}^{0}(\mathbf{Q},\varpi^{n}\mathrm{M})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M})\rightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\varpi^{n}\mathrm{M}).$

The assumptions in the statement of the lemma implies that there is an
isomorphism

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\cong\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M})[\varpi^{n}].$

It follows then we have
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{\nu})\cong\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M})[\varpi^{\nu}]\cong\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$.
∎

Next we will use the Bockstein pairing to upgrade the annihilation result
above to a result concerning the length of the Selmer group. We define the
first $p$-Tate–Shafarevich group of $\mathrm{M}_{n}$ by

${}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})=\ker\\{\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathrm{H}^{1}(\mathbf{Q}_{p},\mathrm{M}_{n})\\}.$

It is clear that
${}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})\subset\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})$.

Suppose that ${}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{n})$ is trivial, then
one can define the _Bockstein pairing_

$\\{\cdot,\cdot\\}_{n}:\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{N}_{n})\otimes\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{n})\rightarrow\mathbf{Q}_{l}/\mathbf{Z}_{l}$

associated to the exact sequence

$0\rightarrow\mathrm{N}_{n}\xrightarrow{\varpi^{n}}\mathrm{N}_{2n}\rightarrow\mathrm{N}_{n}\rightarrow
0$

where is the first map is the multiplication by $\varpi^{n}$ and the second
map is the natural projection map. We will not recall the definition of this
pairing but refer the reader to [Weston1] for some discussions.

Let $p$ be a $\nu$-admissible prime for $\mathrm{T}_{\pi^{\mathrm{Q}}}$ now.
Consider the submodule $\langle\kappa^{p}_{\nu}\rangle$ of
$\mathrm{H}^{1}(\mathbf{Q},\mathrm{N}_{\nu})$ generated by $\kappa^{p}_{\nu}$,
we observe that $\langle\kappa^{p}_{\nu}\rangle$ lies in
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{N}_{\nu})$ by Lemma 6.4.

######  Lemma 6.8.

Under the assumptions of Theorem 6.2, the restriction of
$\\{\cdot,\cdot\\}_{\nu}$ to $\langle\kappa^{p}_{\nu}\rangle$ is right non-
degenerate and it induces an inclusion

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\hookrightarrow\operatorname{Hom}(\langle\kappa^{p}_{\nu}\rangle,\mathbf{Q}_{l}/\mathbf{Z}_{l}).$

###### Proof.

Let $y\in\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{\nu})$. First note that
${}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{\nu})$ is trivial by Lemma 6.6.
Therefore it makes sense to compute the Bockstein pairing
$\\{\kappa^{p}_{\nu},y\\}_{\nu}$. By definition, this is done by choosing a
lift $\kappa^{p}_{2\nu}\in\mathrm{H}^{1}(\mathbf{Q},\mathrm{N}_{2\nu})$ of
$\kappa^{p}_{\nu}$ and defining

$\\{\kappa^{p}_{\nu},y\\}_{\nu}=\langle\frac{1}{\eta}\partial_{p}(\kappa^{p}_{2\nu}),\mathrm{loc}_{p}(y)\rangle_{\nu}$

using the Tate local duality at $p$.

Suppose now $y$ is chosen that $\\{\kappa^{p}_{\nu},y\\}_{\nu}=0$. By Lemma
6.4, the class $\frac{1}{\eta}\partial_{p}(\kappa^{p}_{2\nu})$ generates
$\mathrm{H}^{1}_{\sin}(\mathbf{Q}_{p},\mathrm{N}_{\nu})$. By the perfectness
of the Tate local duality, we have $\mathrm{loc}_{p}(y)=0$ and it follows that
$y\in\mathrm{H}^{1}_{\\{p\\}}(\mathbf{Q},\mathrm{M}_{\nu})$ and thus $y=0$ by
Lemma 6.6. This shows that we have an inclusion

$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{M}_{\nu})\hookrightarrow\operatorname{Hom}(\langle\kappa^{p}_{\nu}\rangle,\mathbf{Q}_{l}/\mathbf{Z}_{l}).$

Thus we have
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\hookrightarrow\operatorname{Hom}(\langle\kappa^{p}_{\nu}\rangle,\mathbf{Q}_{l}/\mathbf{Z}_{l})$
by Lemma 6.7. ∎

#### _Proof of Theorem 6.2 _

Now Theorem 6.2 follows from combining Lemma 6.7 and Lemma 6.8 which gives the
desired length estimate

$\mathrm{length}_{\mathcal{O}_{\lambda}}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\leq\nu.$

$\square$

###  6.4. Comparison of quaternionic periods

Recall the set up at the beginning of $\S 6.2$,
$f\in\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ is a modular form which is new at
primes dividing $\mathrm{N}^{-}$ and admits a normalized Jacquet–Langlands
transfer $f^{\dagger}$ to an automorphic form on
$\mathrm{Z}(\overline{\mathrm{B}})$. The base change of $f^{\dagger}$ to
$\mathrm{F}$ can be regarded as an automorphic form on
$\mathrm{Z}(\overline{\mathrm{Q}})$ which we denote by $\phi^{\dagger}$.

The purpose of this final section is to compare the distinguished period

$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}\phi^{\dagger}(z)$

and the _quaternionic period_

$\mathcal{P}(f^{\dagger})=\sum\limits_{z\in\mathrm{Z}(\overline{\mathrm{B}})}f^{\dagger
2}(z)=\langle f^{\dagger},f^{\dagger}\rangle$

which is known as the Petersson norm of $f^{\dagger}$. We would like to
compare $\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))$
and $\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger}))$ under suitable
assumptions on the Galois representation
$\rho_{\pi^{\circ}}:\mathrm{G}_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\mathrm{E}_{\lambda})$
associated to $\pi^{\circ}$. More precisely, we will show that the
$\lambda$-adic valuation of the ratio

$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})/\mathcal{P}(f^{\dagger})$

is non-negative. Results of this flavour have been by Prasana in [Pra, Theorem
5.2] and by Urban–Tilouine in [UT, Proposition 4.22] with quite different
method.

Let $\Sigma^{+}$ be the set of primes dividing $\mathrm{N}^{+}$ and let
$\Sigma^{-}_{\mathrm{ram}}$ be the set of primes $r$ dividing $\mathrm{N}^{-}$
such that $l\mid r^{2}-1$.

######  Assumption 6.9.

We make the following assumptions on $\bar{\rho}_{\pi^{\circ}}$:

  1. (1)

$\bar{\rho}_{\pi^{\circ}}|_{\mathrm{G}_{\mathbf{Q}(\zeta_{l})}}$ is absolutely
irreducible;

  2. (2)

The image of $\bar{\rho}_{\pi^{\circ}}$ contains
$\mathrm{GL}_{2}(\mathbf{F}_{l})$;

  3. (3)

$\overline{\rho}_{\pi^{\circ}}$ is _minimal_ at primes in $\Sigma^{+}$ in the
sense that all the liftings of
$\overline{\rho}_{\pi^{\circ}}|_{\mathrm{G}_{\mathbf{Q}_{r}}}$ are minimally
ramified for $r\in\Sigma^{+}$;

  4. (4)

$\overline{\rho}_{\pi^{\circ}}$ is ramified at primes in
$\Sigma^{-}_{\mathrm{ram}}$.

Now we can state the main result on comparing the periods
$\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger})$ and $\mathcal{P}(f^{\dagger})$.

######  Theorem 6.10.

Let $f\in\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ be a newform of weight $2$
with $\mathrm{N}=\mathrm{N}^{+}\mathrm{N}^{-}$ such that $\mathrm{N}^{-}$ is
squarefree and has odd number of prime factors. Let $f^{\dagger}$ be the
normalized automorphic form on $\mathrm{Z}(\overline{\mathrm{B}})$
corresponding to $f$ under the Jacquet–Langlands correspondence and
$\phi^{\dagger}$ be the base change of $f^{\dagger}$ considered as an
automorphic form on $\mathrm{Z}(\overline{\mathrm{Q}})$ which is also
normalized.

  1. (1)

We assume that the residual Galois representation
$\overline{\rho}_{\pi^{\circ}}$ satisfies Assumption 6.9;

  2. (2)

We further assume that $\mathrm{H}^{1}(\Delta_{n},\mathrm{M}_{n})=0$ for every
$n\geq 1$ where
$\Delta_{n}=\mathrm{Gal}(\mathbf{Q}(\mathrm{M}_{n})/\mathbf{Q})$ for the
splitting field $\mathbf{Q}(\mathrm{M}_{n})$ of the Galois module
$\mathrm{M}_{n}$.

Then we have the following inequality

$\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))\geq\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger})).$

To prove this theorem, we will study the Bloch–Kato Selmer group
$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
from the perspective of deformation theory of the residual representation
$\overline{\rho}_{\pi^{\circ}}$. Although the Bloch–Kato Selmer group itself
has less connection with the deformation theory of the residual representation
$\overline{\rho}_{\pi^{\circ}}$, we can introduce a smaller Selmer group as
follows. We define the local condition
$\mathrm{H}^{1}_{\mathrm{new}}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
and $\mathrm{H}^{1}_{\mathrm{new}}(\mathbf{Q}_{v},\mathrm{M}_{n})$ as in [Lun,
Definition 3.6], see also [KO, (3.3), (3.4)]. Let $\Sigma^{-}_{\mathrm{mix}}$
be the set of prime divisors of $\mathrm{N}^{-}$such that $l\nmid r^{2}-1$.
Then we define the Selmer group
$\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
by

$\mathrm{ker}\\{\mathrm{H}^{1}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\rightarrow\prod_{v\not\in\Sigma^{-}_{\mathrm{mix}}}\frac{\mathrm{H}^{1}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))}{\mathrm{H}^{1}_{f}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))}\times\prod_{v\in\Sigma^{-}_{\mathrm{mix}}}\frac{\mathrm{H}^{1}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))}{\mathrm{H}^{1}_{\mathrm{new}}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))}\\}.$

Let $\mathbb{T}=\mathbb{T}^{\mathrm{N}}$ be the prime-to-$\mathrm{N}$ Hecke
algebra for the group $\mathrm{G}(\overline{\mathrm{B}})$. Let

$\phi_{\pi^{\circ}}:\mathbb{T}\rightarrow k_{\lambda}$

be the morphism provided by the Hecke eigensystem corresponding to the trace
of Frobenius of $\overline{\rho}_{\pi^{\circ}}$. Then we obtain a maximal
ideal $\mathfrak{m}=\ker(\phi_{\pi^{\circ}})$. On the other hand, consider the
global deformation problem given by

$\mathcal{S}_{\mathrm{mix}}:=(\overline{\rho}_{\pi^{\circ}},\chi_{l},\Sigma^{+}\cup\Sigma^{-}_{\mathrm{ram}}\cup\Sigma^{-}_{\mathrm{mix}}\cup\\{l\\},\\{\mathcal{D}_{v}\\}_{v\in\Sigma^{+}\cup\Sigma^{-}_{\mathrm{ram}}\cup\Sigma^{-}_{\mathrm{mix}}\cup\\{l\\}})$

that classifies the deformations of $\overline{\rho}_{\pi^{\circ}}$ over an
$\mathcal{O}$-algebra which satisfy the following local deformation
conditions:

  1. (1)

for $v=l$, $\mathcal{D}_{l}$ classifies deformations which are
Fontaine–Laffaille crystalline;

  2. (2)

for $v\in\Sigma^{-}_{\mathrm{ram}}\cup\Sigma^{+}$, $\mathcal{D}_{v}$ is the
local deformation problem that classifies deformations that are minimally
ramified;

  3. (3)

for $v\in\Sigma^{-}_{\mathrm{mix}}$,
$\mathcal{D}_{v}=\mathcal{D}^{\mathrm{new}}_{v}$ is the local deformation
problem that classifies deformations that are new in the sense of [Lun,
Definition 3.6].

This global deformation problem is represented by the deformation ring
$\mathrm{R}_{\mathrm{mix}}$. Moreover it follows from [Lun, Proposition 4.1],
see also [KO, Theorem 3.14] that there is an isomorphism

(6.2) $\mathrm{R}_{\mathrm{mix}}\cong\mathbb{T}_{\mathfrak{m}}.$

We remark that although we use [Lun] and [KO] as references. This type of
$\mathrm{R}=\mathrm{T}$ theorem is first studied in [Kha] reversing the steps
taken in the original Taylor–Wiles method. Consider the congruence number
$\eta_{f}(\mathrm{N}^{+},\mathrm{N}^{-})$ defined in [PW11, §2.2] that detects
congruences between $f$ and modular forms in
$\mathrm{S}_{2}(\Gamma_{0}(\mathrm{N}))$ which are new at primes dividing
$\mathrm{N}^{-}$, the $\lambda$-valuation
$\mathrm{ord}_{\lambda}(\eta_{f}(\mathrm{N}^{+},\mathrm{N}^{-}))$ agrees with
the length of
$\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$
as a consequence of the above $\mathrm{R}=\mathrm{T}$ theorem. It also follows
that
$\mathcal{Z}_{\pi^{\circ}}(\overline{\mathrm{B}})=\mathcal{O}_{\lambda}[\mathrm{Z}_{\mathrm{N}^{+}}(\overline{\mathrm{B}})]_{\mathfrak{m}}$
is free over $\mathbb{T}_{\mathfrak{m}}$ of rank $1$. Hence [DDT, 4.17]
implies that $\eta_{f}(\mathrm{N}^{+},\mathrm{N}^{-})$ can be chosen to be the
Petersson norm $\mathcal{P}(f^{\dagger})$ of $f^{\dagger}$. Thus we have

$\mathrm{length}_{\mathcal{O}_{\lambda}}\phantom{.}\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))=\mathrm{ord}_{\lambda}(\eta_{f}(\mathrm{N}^{+},\mathrm{N}^{-}))=\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger})).$

On the other hand, we have an inclusion

$\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\hookrightarrow\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))$

by the definitions of the local conditions defining the Selmer groups: indeed,
one can show that

$\mathrm{H}^{1}_{\mathrm{new}}(\mathbf{Q}_{v},\mathrm{Ad}^{0}(\mathcal{M}_{\rho}))$

is trivial by [KO, Proposition 4.4]. Hence we have

$\mathrm{length}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\geq\mathrm{length}\phantom{.}\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))=\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger}))$

######  Remark 6.11.

In fact, we have proved in the companion work [Wang] that

$\mathrm{length}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))=\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger}))$

using a similar method as in this article.

#### _Proof of Theorem 6.10 _

By the main theorem 6.2, we have the following inequality

$\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))\geq\mathrm{length}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}})).$

On the other hand, the above discussion implies that

$\mathrm{length}\phantom{.}\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))\geq\mathrm{length}\phantom{.}\mathrm{H}^{1}_{\mathcal{S}}(\mathbf{Q},\mathrm{Ad}^{0}(\mathcal{M}_{\pi^{\circ}}))=\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger})).$

We have arrived at the desired inequality
$\mathrm{ord}_{\lambda}(\mathcal{P}_{\mathrm{dis}}(\phi^{\dagger}))\geq\mathrm{ord}_{\lambda}(\mathcal{P}(f^{\dagger}))$.
$\square$

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