$X\widetilde{N}_{\Gamma_{1}}\subset\mathrm{im}(\phi_{\Gamma}).$

Let $m\in\ker(\phi_{\Gamma})$ be any element. Let us write $m=Xm^{\prime}$ for
some $m^{\prime}\in\widetilde{M}_{\Gamma_{1}}$ and take a lift
$\widehat{m}^{\prime}\in M$ of $m^{\prime}$ under the canonical projection
$\widetilde{M}\rightarrow\widetilde{M}_{\Gamma_{1}}$. Then

$\phi(X\widehat{m}^{\prime})\in X\widetilde{N}\cap
z\widetilde{N}=zX\widetilde{N}.$

Write $\phi(X\widehat{m}^{\prime})=Xz\widehat{n}$, with
$\widehat{n}\in\widetilde{N}$. Let $B(\phi)(m)$ denote the image of
$\widehat{n}$ in
$\mathrm{coker}(\phi_{\Gamma})=\widetilde{N}_{\Gamma_{1}}/\mathrm{im}(\phi_{\Gamma})$.
It is easy to see that $B(\phi)(m)$ does not depend on the choice of
$\widehat{n}$, and we have a well defined map

$B(\phi)\,:\,\ker(\phi_{\Gamma})\longrightarrow\mathrm{coker}(\phi_{\Gamma})\,,$

which we view as a variant of the Bockstein morphism. We assume in addition
that

  * D2)

The map $B(\phi)$ is an isomorphism.

Then the exact sequence (4.13) induces a trivialization

(4.14)
$i_{\phi}\,:\,{\det}_{E}(\widetilde{M}_{\Gamma_{1}})\otimes{\det}^{-1}_{E}(\widetilde{N}_{\Gamma_{1}})\simeq{\det}_{E}(\ker(\phi_{\Gamma}))\otimes{\det}_{E}^{-1}(\mathrm{coker}(\phi_{\Gamma}))\xrightarrow{B(\phi)}E\,.$

######  Example 4.7.

In this paragraph, let us consider the special case when
$\widetilde{M}=\widetilde{N}=\widetilde{\mathscr{H}}(\Gamma_{1}).$ Any
morphism
$\phi\,:\,\widetilde{\mathscr{H}}(\Gamma_{1})\rightarrow\widetilde{\mathscr{H}}(\Gamma_{1})$
can be written in the form

$\phi(\widetilde{m})=(a(z)+Xb(z)){\widetilde{m}},\qquad\widetilde{m}\in\widetilde{\mathscr{H}}(\Gamma_{1}),$

for some $a(z),b(z)\in{\mathscr{H}}(\Gamma_{1}).$ We have
$\widetilde{M}_{\Gamma_{1}}=\widetilde{N}_{\Gamma_{1}}=\widetilde{E},$ and
$\phi_{\Gamma}\,:\,\widetilde{E}\rightarrow\widetilde{E}$ is the
multiplication by $Xb(0)$ map. Therefore, condition (D1)) holds if and only if
$b(0)\neq 0.$ The map
$B(\phi)\,:\,X\widetilde{E}\rightarrow\widetilde{E}/X\widetilde{E}$ sends $X$
to $a^{*}(0):=\left.z^{-1}a(z)\right|_{z=0},$ and therefore (D2)) holds if and
only if $a^{*}(0)\neq 0.$

We refer the reader to Lemma 4.9 where we present criteria to check the
properties (D1)) and (D2)). We employ these criteria in Theorem 4.13 in the
context of eigencurve, to check their validity (under suitable assumptions)
for the module
$\widetilde{M}=\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)$.
This is key to the Iwasawa decent formalism for thick Selmer complexes that we
develop in §4.2.5.

#####

For any $\widetilde{\mathscr{H}}(\Gamma_{1})$-bases $\\{m_{i}\\}_{i=1}^{h}$
and $\\{n_{i}\\}_{i=1}^{h}$ of $\widetilde{M}$ and $\widetilde{N}$, let us set

$\widetilde{\mathbf{m}}=\bigwedge_{i=1}^{h}m_{i}\in{\det}_{\widetilde{\mathscr{H}}(\Gamma_{1})}\widetilde{M},\qquad\widetilde{\mathbf{n}}=\bigwedge_{i=1}^{h}n_{i}\in{\det}_{\widetilde{\mathscr{H}}(\Gamma_{1})}\widetilde{N}.$

Consider the families $\\{m_{i},Xm_{i}\\}_{i=1}^{h}$ and
$\\{n_{i},Xn_{i}\\}_{i=1}^{h}$ as $\mathscr{H}(\Gamma_{1})$-bases of
$\widetilde{M}$ and $\widetilde{N}$ and let us set

$\mathbf{m}=\bigwedge_{i=1}^{h}(m_{i}\wedge
Xm_{i})\in{\det}_{\mathscr{H}(\Gamma_{1})}(\widetilde{M}),\qquad\mathbf{n}=\bigwedge_{i=1}^{h}(n_{i}\wedge
Xn_{i})\in{\det}_{\mathscr{H}(\Gamma_{1})}(\widetilde{N}).$

Let $\mathbf{m}_{\Gamma}\in\det_{E}(\widetilde{M}_{\Gamma_{1}})$ and
$\mathbf{n}_{\Gamma}\in\det_{E}(\widetilde{N}_{\Gamma_{1}})$ denote the images
of $\mathbf{m}$ and $\mathbf{n}$.

The following statement is the analogue of the descent formula (3.60) in this
setting:

######  Proposition 4.8.

Assume that the map $\phi$ satisfies conditions (D1)) and (D2)). Retaining the
notation of §4.2.3 and §4.2.3, let us write

$\det(\phi)(\widetilde{\mathbf{m}}\otimes{\widetilde{\mathbf{n}}}^{-1})=a(z)+b(z)X.$

Then $a(z)$ has a zero of order one at $0$ and

$i_{\phi}(\mathbf{m}_{\Gamma}\otimes\mathbf{n}^{-1}_{\Gamma})=a^{*}(0)\cdot
b(0),$

where $a^{*}(0)=\left.z^{-1}a(z)\right|_{z=0}$.

###### Proof.

Since $\mathscr{H}(\Gamma_{1})$ is a Bézout ring, it is easy to see that we
can choose an $\widetilde{\mathscr{H}}(\Gamma_{1})$-basis
$\\{m_{i}\\}_{i=1}^{h}$ of $M$ such that $X\overline{m}_{1}$ is an $E$-basis
of $\ker(\phi_{\Gamma})$ (where $\overline{m}$ denotes the image of $m\in M$
under the projection $M\to M_{\Gamma}$). Fix an arbitrary
$\widetilde{\mathscr{H}}(\Gamma_{1})$-basis $\\{n_{i}\\}_{i=1}^{h}$ of $N$.
Then

$\phi(m_{i})=\underset{j=1}{\overset{h}{\sum}}(a_{ij}(z)+Xb_{ij}(z))n_{j},$

where $a_{1j}(0)=0$ for all $1\leqslant j\leqslant h.$ Set
$a_{1j}^{*}(0):=\left.z^{-1}a_{1j}(z)\right|_{z=0}.$

Using the fact that $a_{1j}(0)=0$ for $1\leqslant j\leqslant h,$ it is easy to
see that

(4.15) $a^{*}(0)\cdot b(0)=\det(A)\cdot\det(B)$

where

$A=\left(\begin{matrix}a_{11}^{*}(0)&a_{12}^{*}(0)&\ldots&a_{1h}^{*}(0)\\\
a_{11}(0)&a_{12}(0)&\ldots&a_{1h}(0)\\\ \cdots&\cdots&\cdots&\ddots\\\
a_{h1}(0)&a_{h2}(0)&\cdots&a_{hh}(0)\end{matrix}\right),\qquad
B=\left(\begin{matrix}b_{11}(0)&b_{12}(0)&\ldots&b_{1h}(0)\\\
a_{11}(0)&a_{12}(0)&\ldots&a_{1h}(0)\\\ \cdots&\cdots&\cdots&\ddots\\\
a_{h1}(0)&a_{h2}(0)&\cdots&a_{hh}(0)\end{matrix}\right).$

Then

$B(\phi)(Xm_{1})=\underset{j=1}{\overset{h}{\sum}}a_{ij}^{*}(0)n_{j}\mod{\mathrm{im}(\phi_{\Gamma})}\,,$

and $i_{\phi}(\mathbf{m}_{\Gamma}\otimes\mathbf{n}_{\Gamma}^{-1})$ is equal to
the determinant

$\det\left(\begin{array}[]{@{}cc|cc|cc|cc@{}}0&b_{11}(0)&0&b_{12}(0)&\cdots&\cdots&0&b_{1h}(0)\\\
a_{11}^{*}(0)&0&a_{12}^{*}(0)&0&\cdots&\cdots&a_{1h}^{*}(0)&0\\\ \hline\cr
a_{21}(0)&b_{21}(0)&a_{22}(0)&b_{22}(0)&\cdots&\cdots&a_{2h}(0)&b_{2h}(0)\\\
0&a_{21}(0)&0&a_{22}(0)&\cdots&\cdots&0&a_{2h}(0)\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\ddots&&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&&\ddots&\vdots&\vdots\\\ \hline\cr
a_{21}(0)&b_{21}(0)&a_{22}(0)&b_{22}(0)&\cdots&\cdots&a_{hh}(0)&b_{hh}(0)\\\
0&a_{h1}(0)&0&a_{h2}(0)&\cdots&\cdots&0&a_{hh}(0)\end{array}\right)$

Using elementary row and column operations, it is easy to see that this
determinant is equal to a determinant of the form

$-\det\left(\begin{array}[]{@{}c|c@{}}*&A\\\ \hline\cr
B&0\end{array}\right)=\det(A)\det(B)\,.$

Comparing this computation with (4.15), we obtain the lemma for our particular
choice of $\\{m_{i}\\}_{i=1}^{h}$.

The general case can be reduced to this particular case by a change of basis.
∎

#####

Let $\widetilde{M}$ be a $\widetilde{\mathscr{H}}(\Gamma_{1})$-module
satisfying the condition of Proposition 4.3. Recall that $\widetilde{M}$ has a
free resolution

$0\longrightarrow\widetilde{P}_{1}\xrightarrow{\,\,\phi\,\,}\widetilde{P}_{0}\longrightarrow\widetilde{M}\longrightarrow
0.$

Then
${\det}_{\widetilde{\mathscr{H}}(\Gamma_{1})}(\widetilde{M})\simeq{\det}_{\widetilde{\mathscr{H}}(\Gamma_{1})}(\widetilde{P}_{0})\otimes{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma_{1})}(\widetilde{P}_{1})$.

######  Lemma 4.9.

Assume in addition that $\dim_{E}({\widetilde{M}}^{\,\Gamma_{1}})=1$. Then:

i) $\widetilde{M}^{\Gamma_{1}}\subset X\widetilde{M}$.

ii) The map

$\displaystyle B\,:\,{\widetilde{M}}^{\,\Gamma_{1}}$
$\displaystyle\longrightarrow\widetilde{M}_{\Gamma_{1}},$ $\displaystyle m$
$\displaystyle\longmapsto y\mod{z\widetilde{M}},\qquad\textrm{where
$B(m)=Xy$}$

is well-defined (namely, it does not depend on the choice of $y$).

iii) The map $\phi\,:\,\widetilde{P}_{1}\rightarrow\widetilde{P}_{0}$
satisfies the condition (D1)) and we have a commutative diagram

$\textstyle{{\widetilde{M}}^{\,\Gamma_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B}$$\scriptstyle{\sim}$$\textstyle{\ker(\phi_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B(\phi)}$$\textstyle{{\widetilde{M}}_{\Gamma_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{coker}(\phi_{\Gamma}).}$

In particular, $\phi$ satisfies the condition (D2)) if and only if $B$ is an
isomorphism.

###### Proof.

The proof is straightforward and is omitted. ∎

#####

Let $\widetilde{C}^{\bullet}$ be a complex of
$\widetilde{\mathscr{H}}(\Gamma_{1})$-modules concentrated in a single even
degree $i$ and such that $H^{i}(\widetilde{C}^{\bullet})$ satisfies the
conditions of Proposition 4.3. Let

$0\longrightarrow\widetilde{P}_{1}\xrightarrow{\,\,\phi\,\,}\widetilde{P}_{0}\longrightarrow
H^{i}(\widetilde{C}^{\bullet})\longrightarrow 0$

be a free resolution of $H^{i}(\widetilde{C}^{\bullet}).$ There exists a
unique, up to a homotopy, lift of the resolution above to a quasi-isomorphism

$\textstyle{\widetilde{P}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\widetilde{P}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{C}^{i-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{C}^{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{C}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{C}^{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

Set
$\widetilde{C}^{\bullet}_{0}=\widetilde{C}^{\bullet}\otimes_{\widetilde{\mathscr{H}}(\Gamma_{1})}^{\mathbb{L}}E.$
The diagram above provides us with a trivialization

(4.16)
${\eta}_{0}\,:\,{\det}_{E}^{-1}(\widetilde{C}^{\bullet}_{0})\simeq{\det}_{E}^{-1}((\widetilde{P}_{1})_{\Gamma_{1}}\xrightarrow{\phi_{\Gamma}}(\widetilde{P}_{0})_{\Gamma_{1}})\xrightarrow{i_{\phi}}E,$

where $i_{\phi}$ is the map (4.14). If $\widetilde{P}_{\bullet}^{\prime}$ is
another free resolution of $H^{i}(\widetilde{C}^{\bullet}),$ then there exists
a unique, up to a homotopy, quasi-isomorphism
$\widetilde{P}_{\bullet}^{\prime}\simeq\widetilde{P}_{\bullet}.$ This implies
that ${\eta}_{0}$ does not depend on the choice of $\widetilde{P}_{\bullet}.$
The map ${\eta}_{0}$ should be seen as an analogue of the trivialization map
(3.57) in classical Iwasawa theory.

######  Remark 4.10.

Let us consider the special case when $h=2$ and consider the map
$\phi\,:\,\widetilde{M}\rightarrow\widetilde{N}$ between two free
$\widetilde{\mathscr{H}}(\Gamma_{1})$-modules of rank $2$ given by the
diagonal matrix of the form

$\left(\begin{matrix}a_{1}(z)+Xb_{1}(z)&0\\\
0&a_{2}(z)+Xb_{2}(z)\end{matrix}\right)$

with $a_{1}(0)=a_{2}(0)=0$ and $b_{1}(0),b_{2}(0)\neq 0.$ Then
$\ker(\phi_{\Gamma})=X\widetilde{M},$ and in particular,
$\dim_{E}(\ker(\phi_{\Gamma}))=\dim_{E}(\mathrm{coker}(\phi_{\Gamma}))=2.$ In
this situation, we still can define the isomorphisms
$B(\phi)\,:\,\ker(\phi_{\Gamma})\rightarrow\mathrm{coker}(\phi_{\Gamma})$ and
$i_{\phi}\,:\,{\det}_{E}(\widetilde{M}_{\Gamma_{1}})\otimes{\det}^{-1}_{E}(\widetilde{N}_{\Gamma_{1}})\rightarrow
E$ via the same formulae. An easy computation gives

$i_{\phi}(m_{\Gamma}\otimes
n_{\Gamma}^{-1})=a_{1}^{*}(0)a_{2}^{*}(0)b_{1}(0)b_{2}(0).$

On the other hand,
$\det(\phi)(\widetilde{m}\otimes\widetilde{n}^{-1})=a_{1}(z)a_{2}(z)+X(a_{1}(z)b_{2}(z)+b_{1}(z)a_{2}(z)).$
This shows that Proposition 4.8 can not be directly generalized to the case
when $\dim_{E}(\ker(\phi_{\Gamma}))>1.$

####  4.2.4. The Bockstein map for thick Selmer complexes

In the present subsection, we study the Bockstein morphism $B(\phi)$ in the
context of thick Selmer complexes. The main results of §4.2.4, which are
recorded as Theorem 4.13 (where we explicitly describe the Bockstein morphism
for thick Selmer complexes and its key properties), will be used in §4.2.5 to
prove various leading term formulae (cf. Theorems 4.14 and 4.15) employing our
Iwasawa descent formalism for thick Selmer complexes.

The reader familiar with [Nek06] will realize that the overall structure of
this portion of our work is akin to §11.6 and §11.7 in op. cit.: Nekovář
studies Bockstein morphisms and their basic properties in [Nek06, §11.6] to
establish a descent formalism, and applies this with his Selmer complexes in
[Nek06, §11.7] to prove what he calls “formulas of the Birch and Swinnerton-
Dyer type”. Note however that his formalism does not apply in the setting we
have placed ourselves.

#####

We start with several auxiliary lemmas.

######  Lemma 4.11.

Let $1\leqslant j\leqslant k-1$ be an integer. Assume that one of the
following conditions holds:

$j\neq\frac{k}{2}$.

$j=\frac{k}{2}$ and $H^{1}_{0}(V_{f^{*}}(k/2))=\\{0\\}$.

Then,

${}_{\mathrm{Iw}}^{2}(V_{k}(j))^{\Gamma}=\\{0\\}={}_{\mathrm{Iw}}^{2}(V_{k}(j))_{\Gamma}\,.$

###### Proof.

The commutative diagram with exact rows

$\textstyle{H^{2}_{\mathrm{Iw}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\mathrm{Iw}}(V_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\mathrm{Iw}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

induces the exact sequence

${}_{\mathrm{Iw}}^{2}(V_{f}(j))\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{k}(j))\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{f}(j))\,.$

Let $A$ (resp. $B\subset{}_{\mathrm{Iw}}^{2}(V_{f}(j))$) denote the coimage
(resp. the image) of the first (resp. the second) map. Consider the exact
sequence

$0\longrightarrow
A\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{k}(j))\longrightarrow B\longrightarrow
0\,.$

By the snake lemma, we have an exact sequence

$0\longrightarrow
A^{\Gamma}\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{k}(j))^{\Gamma}\longrightarrow
B^{\Gamma}\longrightarrow
A_{\Gamma}\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{k}(j))_{\Gamma}\longrightarrow
B_{\Gamma}\longrightarrow 0\,.$

Recall that that if $M$ is a finitely generated torsion $\Lambda_{E}$-module,
then $M$ is of finite dimension over $E$, and

(4.17) $M^{\Gamma}=\\{0\\}\,\iff\,M_{\Gamma}=\\{0\\},$

cf. (3.58). Since ${}_{\mathrm{Iw}}^{2}(V_{f}(k/2))_{\Gamma}\simeq
H^{1}_{0}(V_{f^{*}}(k/2))^{*}$, we have
${}^{2}_{\mathrm{Iw}}(V_{f}(j))^{\Gamma}=\\{0\\}={}^{2}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}$
in both cases (a) and (b). This implies that the modules $A^{\Gamma}$,
$B_{\Gamma}$, and therefore $A_{\Gamma}$ and $B^{\Gamma}$, vanish. Thus

${}_{\mathrm{Iw}}^{2}(V_{k}(j))^{\Gamma}=\\{0\\}={}_{\mathrm{Iw}}^{2}(V_{k}(j))_{\Gamma},$

as required. ∎

######  Lemma 4.12.

Let $1\leqslant j\leqslant k-1$ be an integer. Assume that one of the
following conditions holds:

$j\neq\frac{k}{2}$.

$j=\frac{k}{2}$ and $H^{1}_{0}(V_{f}(k/2))=\\{0\\}=H^{1}_{0}(V_{f^{*}}(k/2))$.

Then the rows of the following commutative diagram are exact and the vertical
morphisms are injective:

(4.18)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{k}}$$\scriptstyle{\textup{res}_{p}}$$\textstyle{H^{1}_{\\{p\\}}(V_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{res}_{p}}$$\textstyle{H^{1}_{\\{p\\}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{k}}$$\textstyle{H^{1}(\mathbb{Q}_{p},V_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

In particular, for every such $j$ we have

$\textup{res}_{p}\circ\iota_{k}\left(H^{1}_{\\{p\\}}(V_{f}(j))\right)=\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{k}(j))\right)\cap\iota_{k}\left(H^{1}(\mathbb{Q}_{p},V_{f}(j))\right)=\iota_{k}\circ\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{f}(j))\right)\,.$

###### Proof.

Since $H^{0}(\mathbb{Q}_{p},V_{f}(j))=\\{0\\}=H^{0}(V_{f}(j))$, the rows of
the diagram are exact except possibly in the right-most surjection on the top.

We have an exact sequence

$0\longrightarrow H^{1}_{\mathrm{Iw}}(V_{f}(j))\longrightarrow
H^{1}_{\mathrm{Iw}}(V_{k}(j))\longrightarrow
H^{1}_{\mathrm{Iw}}(V_{f}(j))\longrightarrow H^{2}_{\mathrm{Iw}}(V_{f}(j))$

of $\Lambda_{E}$-modules, where $H^{1}_{\mathrm{Iw}}(V_{f}(j))$ is free of
rank one and $H^{2}_{\mathrm{Iw}}(V_{f}(j))$ is a $\Lambda_{E}$-torsion module
(cf. [Kat04], Theorem 13.4). Therefore, $H^{1}_{\mathrm{Iw}}(V_{k}(j))$ is
free of rank $2$ over $\Lambda_{E}$. It follows from Lemma 3.42 and Lemma 4.11
that

$H^{1}_{\\{p\\}}(V_{f}(j))=H^{1}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}\quad\hbox{
and }\quad
H^{1}_{\\{p\\}}(V_{k}(j))=H^{1}_{\mathrm{Iw}}(V_{k}(j))_{\Gamma}\,.$

Therefore, $\dim_{E}H^{1}_{\\{p\\}}(V_{f}(j))=1$ and
$\dim_{E}H^{1}_{\\{p\\}}(V_{k}(j))=2.$ This shows that the map
$H^{1}_{\\{p\\}}(V_{k}(j))\rightarrow H^{1}_{\\{p\\}}(V_{f}(j))$ is
surjective, and we have proved that the rows of the diagram are exact.

It follows from the vanishing $H^{1}_{0}(V_{f}(j))=\\{0\\}$ that the left-most
and the right-most vertical maps are injective. Therefore, the vertical map in
the middle is injective as well. The final assertion can be proved by a
diagram chase.

∎

#####

We now prove the main result of this section. Let $g$ denote the map

(4.19)
$\left(H^{1}_{\mathrm{Iw}}(V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\right)\oplus\left(D_{k}[j]\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\right)\xrightarrow{\textup{res}_{p}\,+\,\mathrm{Exp}_{\mathbf{D}_{k}(\chi^{j})}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\,,$

where we use the notation from Section 2.1.2. If
$\mathscr{L}_{\mathrm{Iw}}^{\mathrm{cr}}(V_{f})\neq 0,$ the map $g$ is
injective, and we regard (4.19) as a resolution of $\mathrm{coker}(g)$. Since
$H^{0}(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),V_{k}^{*}(1-j))=\\{0\\},$ we also
have $H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))=\\{0\\},$ and the sequence

$0\longrightarrow H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\longrightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\longrightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\longrightarrow 0$

is exact. Since $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))$ is free over
$\Lambda_{E}$, this implies that
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))$ is
$\widetilde{\Lambda}_{E}$-free, and therefore
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)$
is $\widetilde{\mathscr{H}}(\Gamma)$-free.

Let us denote by
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{(\mathds{1})}(V_{k}(j),\alpha)$ the isotypic
component corresponding to the trivial character of
$\Delta=\mathrm{Gal}(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$. Note that the
$\Gamma_{1}$-invariants (resp. $\Gamma_{1}$-coinvariants) of this component
coincide with the $\Gamma$-invariants (resp. $\Gamma$-coinvariants) of
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)$.

######  Theorem 4.13.

Let $1\leqslant j\leqslant k-1$ be an integer such that either
$j\neq\frac{k}{2}$, or else $j=\frac{k}{2}$ and $L(f^{*},\frac{k}{2})\neq 0$.

We have

$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\simeq\mathrm{coker}(g)^{\Gamma}\,,\qquad\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}\simeq\mathrm{coker}(g)_{\Gamma}\,.$

Moreover:

Assume that the condition (GP)) holds. There exists a commutative diagram

(4.20)
$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{B}$$\textstyle{XD_{k}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\delta}}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\dfrac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))},}$

where the vertical map on the right $\widetilde{\delta}$ is given by

$\widetilde{\delta}(Xd)=-(j-1)!\left(\frac{1-p^{j-1}\alpha^{-1}}{1-p^{-j}\alpha}\right)\left(\exp^{*}_{V^{(\alpha)}_{f}(j)}\right)^{-1}({d}),\qquad
d\in D_{k}[j].$

The following conditions are equivalent.

  * 1)

$\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\neq 0$ .

  * 2)

The condition (GP)) holds,
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}$ is
a one dimensional $E$-vector space and the map

$B\,:\,\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}$

defined as in Lemma 4.9 is an isomorphism.

###### Proof.

In all cases we consider,
$H^{1}_{\mathrm{f}}(V_{f}(\frac{k}{2})=H^{1}_{\mathrm{f}}(V_{f^{*}}(\frac{k}{2}))=0.$
Therefore we can apply Lemmas 4.11 and 4.12. By the definition of the Selmer
complex, we have an exact sequence

$0\longrightarrow\mathrm{coker}(g)\longrightarrow\mathbf{R}^{2}{\boldsymbol{\Gamma}}_{\mathrm{Iw}}(V_{k}(j),\alpha)\longrightarrow{}_{\mathrm{Iw}}^{2}(V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\longrightarrow
0\,.$

Applying the snake lemma to the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{{}_{\mathrm{Iw}}^{2}(V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}^{2}(V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

obtained from this exact sequence and Lemma 4.11, we deduce that

(4.21)
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\simeq\mathrm{coker}(g)^{\Gamma}\quad\hbox{
and
}\quad\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}\simeq\mathrm{coker}(g)_{\Gamma}.$

This proves the very first assertion in our theorem.

Since

$\left(H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\right)^{\Gamma}=\\{0\\}\,,$

it follows that

$\textrm{coker}(g)^{\Gamma}=\ker(g_{\Gamma}):=\ker\left(H^{1}_{\mathrm{Iw}}(V_{k}(j))_{\Gamma}\oplus
D_{k}[j]\xrightarrow{\textup{res}_{p}\,+\,\mathrm{Exp}_{\mathbf{D}_{k}(\chi^{j}),0}}H^{1}(\mathbb{Q}_{p},V_{k}(j))\right).$

Moreover, by Proposition 2.4 and Proposition 2.6, we have

$\ker\left(\mathrm{Exp}_{\mathbf{D}_{k}(\chi^{j}),0}\right)=XD_{k}[j]\subset
D_{k}[j]\quad,\qquad\textrm{im}\left(\mathrm{Exp}_{\mathbf{D}_{k}(j),0}\right)=H^{1}(\mathbb{Q}_{p},XV^{(\beta)}_{k}(j)).$

Note that $XD_{k}[j]\simeq D^{(\alpha)}[j]$ and $(XV_{k})^{(\beta)}(j)\simeq
V^{(\beta)}_{f}(j).$ Since

$\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{f}(j))\right)\cap
H^{1}(\mathbb{Q}_{p},V^{(\beta)}_{f}(j))=H^{1}_{\beta}(V_{f}(j))=\\{0\\}$

and $H^{1}_{\mathrm{Iw}}(V_{k}(j))_{\Gamma}\hookrightarrow
H^{1}_{\\{p\\}}(V_{k}(j))$, it follows from Lemma 4.12 that

$\textup{res}_{p}\left(H^{1}_{\mathrm{Iw}}(V_{k}(j))_{\Gamma}\right)\cap\mathrm{Exp}_{\mathbf{D}_{k}(j),0}(D_{k}[j])=\\{0\\}\,.$

This yields a canonical isomorphism

$\textrm{coker}(g)^{\Gamma}\simeq\ker\left(\mathrm{Exp}_{\mathbf{D}_{k}(\chi^{j}),0}\right)=XD_{k}[j]\simeq
D^{(\alpha)}[j]\,.$

On the other hand, it follows from Lemma 4.12 that

$\dim_{E}\left(H^{1}_{\\{p\\}}(V_{k}(j))\right)=2\quad\hbox{ and
}\quad\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{k}(j))\right)\cap\iota_{k}\left(H^{1}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))\right)=\\{0\\}\,.$

Therefore,

$\textrm{coker}(g)_{\Gamma}\simeq\textrm{coker}(g_{\Gamma}):=\frac{H^{1}(\mathbb{Q}_{p},V_{k}(j))}{\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{k}(j))\right)+\iota_{k}\left(H^{1}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j)\right)}\xrightarrow{\sim}\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{f}(j))\right)}\,,$

where the last isomorphism is induced by the projection $V_{k}(j)\rightarrow
V_{f}(j)$. In particular,

$\dim_{E}\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}=\dim_{E}\left(\textrm{coker}(g)_{\Gamma}\right)=1.$

Let
$0\rightarrow\widetilde{P}_{1}\xrightarrow{\phi}\widetilde{P}_{0}\rightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\to
0$ be a free resolution. Since
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)$
is free, there exists a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(H^{1}_{\mathrm{Iw}}(V_{k}(j))\oplus
D_{k}[j]\right)\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{k}(j))\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\widetilde{P}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with exact rows and columns. Applying the snake lemma, we obtain the following
cartesian squares, where all maps are isomorphisms:

$\textstyle{\mathrm{coker}(g)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(g_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker(\phi_{\Gamma})}$
$\textstyle{\mathrm{coker}(g)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(\phi_{\Gamma}).}$

Therefore, we have a commutative diagram

(4.22)
$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{B}$$\textstyle{\ker(\phi_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{B(\phi)}$$\textstyle{\ker(g_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B(g)}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{coker}({\phi}_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{coker}(g_{\Gamma})\,.}$

By the definitions of the isomorphisms above, we have a commutative diagram

(4.23)
---
$\textstyle{{\rm
ker}(g_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{B(g)}$$\textstyle{XD_{k}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\delta}}$$\textstyle{\textrm{coker}(g_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}\left(H^{1}_{\\{p\\}}(V_{f}(j))\right)}}$

where the vertical map on the right is given by

$\widetilde{\delta}(Xd)=\mathrm{pr}_{0}(y),\quad\textrm{where $d\in D_{k}[j]$
and $\mathrm{Exp}_{V^{(\alpha)}_{x_{0}}(j),j}(d)=(\gamma-1)y.$}$

The identity (3.47) allows us to write

$y=\ell(V_{f}(j))\cdot\mathrm{Exp}_{\alpha,j}(x),$

where $\ell(V_{f}(j))=\underset{\begin{subarray}{c}1-k+j\leqslant i\leqslant
j-1\\\ i\neq 0\end{subarray}}{\prod}\ell_{i}.$ Therefore, using (1.23),

$\displaystyle\widetilde{\delta}(Xd)=\left(\underset{\begin{subarray}{c}1\leqslant
i\leqslant k-1\\\ i\neq
k-j\end{subarray}}{\prod}(i+j-k)\right)\mathrm{pr}_{0}\circ\mathrm{Exp}_{\alpha,j}(x)=-(j-1)!\left(\frac{1-p^{j-1}\alpha^{-1}}{1-p^{-j}\alpha}\right)\left(\exp^{*}_{V^{(\alpha)}_{f}(j)}\right)^{-1}(d).$

Now the proof of (ii) follows from (4.22) and (4.23).

The implication

$\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\neq 0\quad\implies{\quad\eqref{item_C4}}$

is explained in Remark 0.4.3(ii). We now use (3.14) to see that the map
$\widetilde{\delta}$ is an isomorphism if and only if
$\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\neq 0$. This concludes the proof of (ii).
∎

####  4.2.5. Iwasawa descent

The main results in this subsection are Theorems 4.14 and 4.15, where we prove
various leading term formulae, employing Iwasawa descent for thick Selmer
complexes which we established in the earlier portions (§4.2.3 and §4.2.4) of
the present chapter.

#####

Let us set

$\mathbf{R\Gamma}(V_{k}(j),\alpha):=\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\otimes^{\mathbf{L}}_{\widetilde{\mathscr{H}}(\Gamma)}\widetilde{E}.$

Note that

$\mathbf{R\Gamma}(V_{k}(j),\alpha)\simeq\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\otimes^{\mathbf{L}}_{{\mathscr{H}}(\Gamma)}E$

as complexes of $E$-vector spaces. For an integer $n$ that verifies (L1)) and
(L2)), let us put

$\mathbf{R\Gamma}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right):=\mathbf{R\Gamma}_{\mathrm{Iw}}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right).$

More explicitly,

$\mathbf{R\Gamma}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)=\begin{cases}\mathbf{R\Gamma}_{\mathrm{f}}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j)\right)&\textrm{if
$\ell\neq p$}\\\ N_{k}[j][-1]&\textrm{if $\ell=p$}.\end{cases}$

We will consider the fundamental line

$\displaystyle\Delta_{\widetilde{E}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right):=$
$\displaystyle\,\,{\det}_{\mathcal{O}_{E}^{(n)}}\left(\mathbf{R\Gamma}_{S}\left(T_{k}^{(n)}(j)\right)\oplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\right)\right)\otimes_{\mathcal{O}_{E}^{(n)}}{\det}_{\mathcal{O}_{E}^{(n)}}\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\right).$

over $\mathcal{O}^{(n)}_{E}$. Note that

$\Delta_{\widetilde{E}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\simeq\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\otimes_{\Lambda^{(n)}}\mathcal{O}_{E}^{(n)}\,.$

We will also consider a fundamental line over $\mathcal{O}_{E}$, which we
define on setting

(4.24) $\Delta_{E}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right):=\\\
{\det}_{\mathcal{O}_{E}}\left(\mathbf{R\Gamma}_{S}\left(T_{k}^{(n)}(j)\right)\bigoplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\right)\right)\otimes_{\mathcal{O}_{E}}{\det}_{\mathcal{O}_{E}}\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\right).$

We remark that

$\Delta_{E}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\simeq{\det}_{\mathcal{O}_{E}}\Delta_{\widetilde{E}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right).$

#####

By the functoriality of
$\otimes^{\mathbf{L}}_{\Lambda^{(n)}}\widetilde{\mathcal{O}}_{E},$ there
exists a commutative diagram

$\textstyle{\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{(\alpha)}_{\mathrm{Iw},V_{k}(j)}}$$\scriptstyle{\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\widetilde{\mathcal{O}}_{E}}$$\textstyle{{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma)}\mathbf{R\Gamma}(V_{k}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes^{\mathbf{L}}_{\mathrm{\widetilde{\mathscr{H}}(\Gamma)}}\widetilde{E}}$$\textstyle{\Delta_{\widetilde{E}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Theta}_{V_{k}(j)}^{(\alpha)}}$$\textstyle{{\det}^{-1}_{\widetilde{E}}\mathbf{R\Gamma}(V_{k}(j),\alpha).}$

Applying the functor ${\det}_{\mathcal{O}_{E}}(-)$ on the bottom row of this
diagram, we obtain a map

$\Theta_{V_{k}(j)}^{(\alpha)}\,:\,\Delta_{E}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\longrightarrow{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{k}(j),\alpha)\,.$

#####

Let

$i_{V_{k}(j)}\,:\,{\det}_{E}^{-1}\mathbf{R\Gamma}(V_{k}(j),\alpha)\rightarrow
E,$

denote the trivialization map (4.16) for the complex
$\widetilde{C}^{\bullet}=\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha).$

######  Theorem 4.14.

Let $1\leqslant j\leqslant k-1$ be an integer such that either
$j\neq\frac{k}{2}$, or else $j=\frac{k}{2}$ and $L(f^{*},\frac{k}{2})\neq 0$.
Assume that $\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\neq 0$. Then there exists a
canonical isomorphism

$\mu\,\,:\,\,{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{k}(j),\alpha)\xrightarrow{\,\sim\,}{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{f}(j),\alpha)\otimes{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{f}(j),\beta)$

with the following properties:

  * i)

The following diagram commutes:

(4.25) |  |
---|---|---
$\textstyle{{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{k}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{i_{V_{k}(j)}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{f}(j),\alpha)\otimes{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{V_{f}(j)}^{(\alpha)}\,\otimes\,i_{V_{f}(j)}^{(\beta)}}$$\textstyle{E,}$

where $i_{V_{f}(j)}^{(\alpha)}$ and $i_{V_{f}(j)}^{(\beta)}$ denote the
trivialization maps provided by Lemma 3.38 as in Section 3.5.2, and
$\ell^{*}(V_{f}(j)):=\underset{\begin{subarray}{c}i=1-k\\\
i\neq-j\end{subarray}}{\overset{-1}{\prod}}(i+j)$ .

  * ii)

The map $\mu$ sends

$\Theta_{V_{k}(j)}^{(\alpha)}\left(\Delta_{E}(T^{(n)}_{k}(j),N_{k}^{(n)}[j])\right)$

onto

$C_{\mathrm{K}}\cdot{\frac{b(\mathds{1},j)}{a(\mathds{1},j)}}\cdot\Theta_{V_{x_{0}}(j)}^{(\alpha)}\left(\Delta_{E}(T_{x_{0}}(j),\alpha)\right)\otimes_{\mathcal{O}_{E}}\Theta_{V_{x_{0}}(j)}^{(\beta)}\left(\Delta_{E}(T_{x_{0}}(j),\beta)\right)$

where $C_{\mathrm{K}}$ is the constant defined in (2.44).

  * iii)

We have

$i_{V_{k}(j)}\circ\Theta_{V_{k}(j)}^{(\alpha)}\left(\Delta_{E}(T^{(n)}_{k}(j),N_{k}^{(n)}[j])\right)=\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0).$

###### Proof.

It follows from the condition (GP)) (which holds since we assume
$\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\neq 0$, cf. Remark 0.4.3(ii)) that we
have an exact sequence

(4.26) $0\longrightarrow XD_{k}[j]\longrightarrow
D_{k}[j]\xrightarrow{\kappa_{j}}D^{(\beta)}[j]\longrightarrow 0\,,\qquad
XD_{k}\simeq D^{(\alpha)},$

where $\kappa_{j}$ is the map defined in Proposition 2.4. Let
$\mathbf{R\Gamma}(XV_{k}(j),D_{k}[j])$ denote the Selmer complex associated to
the diagram

$\textstyle{\mathbf{R\Gamma}_{S}(XV_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\oplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},XV_{k}(j))}$$\textstyle{\underset{\ell\in
S}{\oplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},XV_{k}(j),D_{k}[j])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the local condition at $p$ is given by

$\mathrm{Exp}_{\mathbf{D}_{k}(\chi^{j}),j,0}\,:\,D_{k}[j]\longrightarrow
H^{1}(\mathbb{Q}_{p},XV_{k}(j)),$

and the local conditions at $p\neq\ell\in S$ by the unramified local
conditions. Note that we may naturally identify $XV_{k}\subset V_{k}$ with the
representation $V_{f}$. The exact sequence (4.26) and Proposition 2.6 show
that we have a commutative diagram

(4.27) |  |
---|---|---
$\textstyle{XD_{k}[j][-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},XV_{k}(j),D_{k}[j])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\frac{b(\mathds{1},j)}{a(\mathds{1},j)}\cdot\kappa_{j}}$$\scriptstyle{\mathrm{Exp}_{V_{f}^{(\alpha)}(j),0}}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\mathrm{Exp}}_{\beta,j,0}}$$\textstyle{H^{1}(\mathbb{Q}_{p},XV_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},V_{f}(j))\,.}$

Here $\widetilde{\mathrm{Exp}}_{\beta,j,0}$ denotes the projection of the map
$\widetilde{\mathrm{Exp}}_{\beta,j}$ defined in (2.7), and
$D^{(\alpha)}[j][-1]$ means that the module $D^{(\alpha)}[j]$ is placed in
degree one. This diagram gives rise to an exact triangle

(4.28)
$XD_{k}[j][-1]\longrightarrow\mathbf{R\Gamma}(XV_{k}(j),D_{k}[j])\longrightarrow\mathbf{R\Gamma}(V_{f}(j),\beta)\,.$

Let $\mathbf{R\Gamma}_{0}(V_{k}/XV_{k}(j),\alpha)$ denote the Selmer complex
associated to the diagram

$\textstyle{\mathbf{R\Gamma}_{S}(V_{k}/XV_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},V_{k}/XV_{k}(j))}$$\textstyle{\underset{\ell\in
S\setminus\\{p\\}}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{f}}(\mathbb{Q}_{\ell},V_{k}/XV_{k}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with strict local conditions at $p$. Note that we canonically have
$V_{k}/XV_{k}\simeq V_{f}$. The exact sequence of local conditions

$0\longrightarrow\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},XV_{k}(j),D_{k}[j]))\longrightarrow\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},V_{k}(j),D_{k}[j])\longrightarrow\underset{\ell\in
S\setminus\\{p\\}}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{f}}(\mathbb{Q}_{\ell},V_{k}/XV_{k}(j))\longrightarrow
0$

provides us with an exact triangle

$\mathbf{R\Gamma}(XV_{k}(j),D_{k}[j])\longrightarrow\mathbf{R\Gamma}(V_{k}(j),\alpha)\longrightarrow\mathbf{R\Gamma}_{0}(V_{k}/XV_{k}(j),\alpha)\longrightarrow\mathbf{R\Gamma}(XV_{k}(j),D_{k}[j])[1]\,.$

Combined with (4.28), we obtain canonical isomorphisms

where we use the isomorphism $XD_{k}\simeq D^{(\alpha)}$ to write

${\det}_{E}\mathbf{R\Gamma}_{0}(V_{f}(j),\alpha)\otimes{\det}_{E}XD_{k}[j]\simeq{\det}_{E}\mathbf{R\Gamma}_{0}(V_{f}(j),\alpha)\otimes{\det}_{E}D^{(\alpha)}[j]\simeq{\det}_{E}\mathbf{R\Gamma}(V_{f}(j),\alpha)\,.$

We define $\mu$ as the composition of the maps in (4.2.5). Let
$0\rightarrow\widetilde{P}_{1}\xrightarrow{\phi}\widetilde{P}_{0}\rightarrow
0$ be a free resolution of
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha).$ By Theorem
4.13 and Proposition 3.41, we have the following commutative diagram

$\textstyle{\ker(f_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B(\phi)}$$\scriptstyle{\sim}$$\textstyle{XD_{k}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\delta}}$$\scriptstyle{\sim}$$\textstyle{D^{(\alpha)}[j]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\sim}$$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Bock}_{2}}$$\textstyle{\textrm{coker}(\phi_{\Gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\quad.}$

Since the Bockstein map $\mathrm{Bock}_{2}$ trivializes
${\det}_{E}\mathbf{R\Gamma}(V_{f}(j),\alpha)$ and $B(\phi)$ trivializes
${\det}_{E}\mathbf{R\Gamma}(V_{k}(j),\alpha),$ it is immediate from the
construction of $\mu$ that the diagram (4.25) commutes. Using the diagram
(4.27), it is now easy to check that the map $\mu$ sends

$\Theta_{V_{k}(j)}^{(\alpha)}(\Delta_{E}(T^{(n)}_{k}(j)),N_{k}^{(n)}[j])$

onto

$a\cdot\Theta_{V_{f}(j)}^{(\alpha)}\left(\Delta_{E}(T_{f}(j),\alpha)\right)\otimes_{\mathcal{O}_{E}}\Theta_{V_{f}(j)}^{(\beta)}\left(\Delta_{E}(T_{f}(j),\beta)\right)\,,$

where $a\in E$ is such that
$aN_{\beta}={\frac{b(\mathds{1},j)}{a(\mathds{1},j)}}\kappa_{0}(N_{\alpha}),$
which is precisely Part (ii) in the statement of our theorem.

Finally, (iii) follows from (i), (ii) and Lemma 3.38, which tells us that

$\displaystyle
i_{V_{x_{0}}(j)}^{(\alpha)}\circ\Theta_{V_{f}(j)}^{(\alpha)}\left(\Delta_{E}(T_{f}(j),\alpha)\right)=\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(j),N_{\alpha}[j],\mathds{1},0),$
$\displaystyle
i_{V_{x_{0}}(j)}^{(\beta)}\circ\Theta_{V_{f}(j)}^{(\beta)}\left(\Delta_{E}(T_{f}(j),\beta)\right)=\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0).$

This completes the proof of our theorem. ∎

#####

The following theorem is evidence for the infinitesimal thickening
$\widetilde{\mathbf{MC}}(f)$ of the $\theta$-critical Main Conjecture, as it
shows (among other things) that it is concurrent with the conclusion of
Theorem 2.23 and slope-zero Main Conjecture $\mathbf{MC}(f,\beta)$.

######  Theorem 4.15.

Let $1\leqslant j\leqslant k-1$ be an integer such that either
$j\neq\frac{k}{2}$, or else $j=\frac{k}{2}$ and $L(f^{*},\frac{k}{2})\neq 0.$
Assume that

$\mathscr{L}^{\mathrm{cr}}(V_{f}(k-j))\neq 0$ ,

the infinitesimal thickening $\widetilde{\mathbf{MC}}(f)$ of the
$\theta$-critical Main Conjecture is valid.

Then,

(4.33)
$L^{[1]}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*};\chi^{j})^{\pm}\sim_{p}C_{\mathrm{K}}\cdot{\mathcal{E}_{N}(f^{*};\chi^{j})\cdot\frac{b(\mathds{1},k-j)}{a(\mathds{1},k-j)}}\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k-j),N_{\beta}[k-j],\mathds{1},0)^{\pm}\,.$

###### Proof.

The Infinitesimal Main Conjecture reads

$\widetilde{\mathcal{E}}_{N}^{\iota}\cdot\mathbf{L}_{\mathrm{Iw},\alpha}\left(T_{k}^{(n)}(k),N_{k}[k]\right)=\Lambda^{(n)}\widetilde{L}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})^{\iota},\qquad\textrm{for
$n\gg 0.$}$

Twisting both sides, we deduce that

$\left(\mathrm{Tw}_{j}\circ\widetilde{\mathcal{E}}_{N}^{\iota}\right)\cdot\mathbf{L}_{\mathrm{Iw},\alpha}\left(T_{k}^{(n)}(k-j),N_{k}[k-j]\right)=\Lambda^{(n)}\left(\mathrm{Tw}_{-j}\circ\widetilde{L}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})\right)^{\iota},\qquad\textrm{for
$n\gg 0.$}$

We can rewrite the final equality in the form

$\left(\mathrm{Tw}_{j}\circ\widetilde{\mathcal{E}}_{N}^{\iota}\right)\cdot\mathbf{L}_{\mathrm{Iw},\alpha}\left(T_{k}^{(n)}(k-j),N_{k}[k-j]\right)=\left(a+\frac{X}{\varpi_{E}^{n}}b\right)\left(\mathrm{Tw}_{-j}\circ\widetilde{L}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})\right)^{\iota},$

where $a+\frac{X}{\varpi_{E}^{n}}b\in\Lambda^{(n)}$ is a unit; this is
equivalent to saying that $a\in\Lambda^{\times}$ and $b\in\Lambda.$

By Theorem 4.13, $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(k-j),\alpha)^{\Gamma}$
is one-dimensional, and we can apply our descent formalism with the lattice

$\Theta_{\mathrm{Iw},V_{k}({k-}j)}^{(\alpha)}(\Delta_{\mathrm{Iw}}(T_{k}^{(n)}(k-j),N_{k}[k-j])\subset{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma)}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(k-j),\alpha).$

Recall that

$\widetilde{L}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})=L_{\mathrm{K},\alpha^{*}}^{[0]}(f^{*},\xi^{*})+XL_{\mathrm{K},\alpha^{*}}^{[1]}(f^{*},\xi^{*})\,.$

The function $L_{\mathrm{K},\alpha^{*}}^{[0]}(f^{*},\xi^{*})$ has a simple
zero at all characters $\chi^{j}$ (where $j$ is as in the theorem), and we
denote by $L_{\mathrm{K},\alpha^{*}}^{[0],*}(f^{*},\xi^{*},\chi^{j})$ its
special value at $\chi^{j}.$ Proposition 4.8 implies that the map
$i_{V_{k}({k-}j)}$ sends

${\mathcal{E}_{N}(f^{*};\chi^{j})^{2}\cdot}\Theta^{(\alpha)}_{V_{k}({k-}j)}\left(\Delta_{E}(T_{k}^{(n)}(k-j),N_{k}^{(n)}[k-j])\right)$

onto

(4.35) $a(0)\cdot
L_{\mathrm{K},\alpha^{*}}^{[0],*}(f^{*},\xi^{*};\chi^{j})\left(a\cdot
L_{\mathrm{K},\alpha^{*}}^{[1]}(f^{*},\xi^{*};\chi^{j})+{\frac{b}{\varpi_{E}^{n}}}\cdot
L_{\mathrm{K},\alpha^{*}}^{[0]}(f^{*},\xi^{*},\chi^{j})\right)\\\
={a(0)^{2}\cdot L_{\mathrm{K},\alpha^{*}}^{[0],*}(f^{*},\xi^{*};\chi^{j})\cdot
L_{\mathrm{K},\alpha^{*}}^{[1]}(f^{*},\xi^{*};\chi^{j}).}$

On the other hand, Theorem 4.14 yields

$i_{V_{k}(k-j)}\circ\Theta_{V_{k}({k-}j)}^{(\alpha)}\left(\Delta_{E}(T^{(n)}_{k}({k-}j),N_{k}^{(n)}[{k-}j])\right)=\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(j),N_{\alpha}[k-j],\mathds{1},0)\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k-j),N_{\beta}[k-j],\mathds{1},0).$

Therefore,

$\displaystyle L_{\mathrm{K},\alpha^{*}}^{[0],*}(f^{*},\xi^{*};\chi^{j})$
$\displaystyle\,\cdot L_{\mathrm{K},\alpha^{*}}^{[1]}(f^{*},\xi^{*};\chi^{j})$
$\displaystyle\sim_{p}C_{\mathrm{K}}\cdot\mathcal{E}_{N}(f^{*};\chi^{j})^{2}\cdot\frac{b(\mathds{1},k-j)}{a(\mathds{1},k-j)}\cdot\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(k-j),N_{\alpha}[k-j],\mathds{1},0)\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0).$

Note that Conjecture $\mathbf{MC}(f,\alpha)$ (which follows from
$\widetilde{\mathbf{MC}}(f)$ by Proposition 4.6) implies that

$L_{\mathrm{K},\alpha^{*}}^{[0],*}(f^{*},\xi^{*};\chi^{j})\sim_{p}\mathcal{E}_{N}(f^{*};\chi^{j})\cdot\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{x_{0}}(k-j),N_{\alpha}[k-j],\mathds{1},0)\,.$

Hence,

$L_{\mathrm{K},\alpha^{*}}^{[1]}(f^{*},\xi^{*};\chi^{j})\sim_{p}C_{\mathrm{K}}\cdot\mathcal{E}_{N}({f^{*};\chi^{j}})\cdot{\frac{b(\mathds{1},k-j)}{a(\mathds{1},k-j)}}\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k-j),N_{\beta}[k-j],\mathds{1},0)\,,$

and the proof of our theorem is complete. ∎

######  Corollary 4.16.

The formula (4.33) can be rewritten in the form

$L^{[1]}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*};\chi^{j})^{\pm}\sim_{p}C_{\mathrm{K}}\cdot\mathcal{E}_{N}(f^{*};\chi^{j})\cdot\frac{e_{p,\alpha^{*}}(f^{*},\mathds{1},j)}{e_{p,\beta^{*}}(f^{*},\mathds{1},j)}\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k-j),N_{\beta}[k-j],\mathds{1},0)^{\pm}\,.$

###### Proof.

This follows from the equalities

$\frac{b(\mathds{1},k-j)}{a(\mathds{1},k-j)}=\frac{e_{p,\alpha}(f,\mathds{1},k-j)}{e_{p,\beta}(f\mathds{1},k-j)}=\frac{e_{p,\alpha^{*}}(f^{*},\mathds{1},j)}{e_{p,\beta^{*}}(f^{*}\mathds{1},j)}\,.$

∎

######  Remark 4.17.

It follows from Theorem 2.23 (see especially Equations (2.45) and (2.48)) that

$L^{[1]}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*};\chi^{j})^{\pm}=C_{K}\cdot\mathcal{E}_{N}(f^{*};\chi^{j})\cdot\frac{e_{p,\alpha^{*}}(f^{*},\mathds{1},j)}{e_{p,\beta^{*}}(f^{*}\mathds{1},j)}\cdot
L_{\mathrm{K},\beta^{*}}(f^{*},\xi^{*};\chi^{j})^{\pm}.$

Combining this formula with Corollary 4.16, we deduce that

$L_{\mathrm{K},\beta^{*}}(f^{*},\xi^{*};\chi^{j})^{\pm}\sim_{p}\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k-j),N_{\beta}[k-j],\mathds{1},0)^{\pm}\,.$

This also follows from the slope-zero Main Conjecture 3.32 via Iwasawa
theoretic descent. This shows that the conjecture $\widetilde{\mathbf{MC}}(f)$
is compatible with the slope-zero Main Conjecture $\mathbf{MC}(f,\beta)$.

##  Chapter 5 Thick Selmer groups and derivatives of critical $p$-adic
$L$-functions

Throughout §5, we shall work in the setting of §2.4.2. In particular, we
assume throughout that the hypothesis (GP)) holds true.

###  5.1. Recap and motivation

Before we describe our results in the present chapter, we summarize our
results in the earlier portions and explain how our results in §5 complement
those.

Recall that in §2.4.4, we have introduced a two-variable $p$-adic $L$-function
$L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi)\in\mathscr{H}_{\mathcal{X}}(\Gamma)$.
We proved that we can write it in the form

(5.1)
$L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi)=\ell(V_{f}(j))\,L_{\mathrm{K},\alpha}^{[0],\mathrm{imp},\pm}(f,\xi)+X\cdot
L_{\mathrm{K},\alpha}^{[1],\pm}(f,\xi)+\ldots\,,$

where
$\ell(V_{f}(j))=\left(\underset{i=1-k}{\overset{-1}{\prod}}\ell^{\iota}_{i}\right)$
(cf. Theorem 2.23(i)).

The interpolation properties of $L_{\mathrm{K},\alpha}^{[1]}(f,\xi)$ in terms
of the critical values of the Hecke $L$-function of $f$ (cf. Theorem 2.23(ii))
suggest that the _thick $p$-adic $L$-function_
$\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi):=L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi)\pmod{X^{2}}$
also carries arithmetic information, that is not readily visible in
$L_{\mathrm{K},\alpha}^{[0],\pm}(f,\xi):=L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi)\pmod{X}$.

Based on this insight, we developed a theory of thick Selmer complexes in
Chapter 4 and introduced what we call the thick fundamental line. We
formulated the Main Conjecture $\widetilde{\mathbf{MC}}(f)$ in terms of these
to reflect the arithmetic properties of the thick $p$-adic $L$-function
$L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi)\pmod{X^{2}}$.

####  5.1.1. Special values

Let $j$ be an integer and $r$ a natural number. In what follows, we consider
the function ${L}_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi,\chi^{s})$, which
is analytic on $j+p\mathbb{Z}_{p}$, and we denote by

$\left.\frac{\partial^{r}}{\partial
s^{r}}{L}_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi,\chi^{s})\right|_{s=j}$

its derivative at $s=j$.

##### The non-central critical $j$

Let us now fix an integer $j\neq\frac{k}{2}$ with $1\leqslant j\leqslant k-1$.
In view of (5.1), we have the following linearization of
$L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi,\chi^{s})$ about $X=0$, $s=j$:

(5.2) $\displaystyle\begin{aligned}
L_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi,\chi^{s})\equiv\,(s-j)\cdot\ell^{*}(V_{f}(j))\cdot
L_{\mathrm{K},\alpha}^{[0],\mathrm{imp},\pm}(f,\xi,\chi^{j})+X\cdot&L_{\mathrm{K},\alpha}^{[1],\pm}(f,\xi,\chi^{j})\\\
&\mod(X^{2},X(s-j),(s-j)^{2})\,.\end{aligned}$

Here, $\ell^{*}(V_{f}(j)):=\underset{{\begin{subarray}{c}i=1-k\\\
i\neq-j\end{subarray}}}{\overset{-1}{\prod}}(i+j)$.

Theorem 2.23 and Proposition 2.27 allows us to compute the linearization
$\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi,\chi^{s})$ of the thick
$p$-adic $L$-function about $s=j$ in terms of the Hecke $L$-value $L(f,j)$,
assuming that the $\mathcal{L}$-invariant $\mathscr{L}^{\rm cr}(V_{f}(j))$ is
nonzero.

##### Non-critical $j$

We may assume without loss of generality that $j\geq k$. Granted the validity
of the $\theta$-critical main conjecture $\mathbf{MC}(f,\alpha)$ (which is
equivalent, assuming that $\mathscr{L}^{\rm cr}_{\rm Iw}$ is nonzero, to the
slope-zero main conjecture $\mathbf{MC}(f,\beta)$, cf. Proposition 3.34), we
have

$L_{\mathrm{K},\alpha}^{[0],\pm}(f,\xi,\chi^{j})\neq 0\,.$

This value can be explained (up to a $p$-adic unit) in terms of fundamental
arithmetic invariants; see Theorem 3.45.

In summary, assuming that $\mathscr{L}^{\rm cr}_{\rm Iw}(V_{f})\neq 0$ and the
slope-zero main conjecture $\mathbf{MC}(f,\beta)$ holds true, the leading term
of the thick $p$-adic $L$-function
$\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi,\chi^{s})$ about $s=j$ equals
$L_{\mathrm{K},\alpha}^{[0],\pm}(f,\xi,\chi^{j})$, whose arithmetic meaning is
explained by Theorem 3.45(i).

##### Special value at the central critical point

Let us assume that $k$ is an even integer. In view of our discussion in
Sections 5.1.1–5.1.1, the case that remains to understand the arithmetic
meaning of $\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi)$ concerns the
central critical point $j=\frac{k}{2}$.

Suppose first that $L(f,\frac{k}{2})\neq 0$. Then the treatment of
$\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi,\chi^{s})$ about
$s=\frac{k}{2}$ is identical to the treatment of other critical values (cf.
§5.1.1 and Theorem 4.15). We will therefore assume until the end of §5.1.1
that $L(f,\frac{k}{2})=0$. Let us put
$r:={\mathrm{ord}}_{s=\frac{k}{2}}L(f,s)$ .

Let us assume that the conditions (BK)), (ht)) and (h-PR)) as well as the
slope-zero main conjecture $\mathbf{MC}(f,\beta)$ hold true. It follows from
(5.1) combined with Theorem 2.23, Proposition 2.27, Proposition 3.25(iii), and
the proof of Proposition 3.23 that

(5.3) $\displaystyle\begin{aligned}
\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(\mathcal{X},\xi,\chi^{s})\equiv&\left(s-\frac{k}{2}\right)^{r}\cdot\dfrac{\ell^{*}(V_{f}(j))}{(r-1)!}\cdot\frac{\partial^{r-1}}{\partial
s^{r-1}}\left.L_{\mathrm{K},\alpha}^{[0],\mathrm{imp},\pm}(f,\xi,\chi^{s})\right|_{s=\frac{k}{2}}\\\
&\,\,+X\cdot\left(s-\frac{k}{2}\right)\cdot\frac{d}{ds}\left.L_{\mathrm{K},\alpha}^{[1],\pm}(f,\xi,\chi^{s})\right|_{s=\frac{k}{2}}\qquad\mod\left(X\left(s-\frac{k}{2}\right)^{2},\left(s-\frac{k}{2}\right)^{r+1}\right).\end{aligned}$

Moreover, since the $\theta$-critical main conjecture $\mathbf{MC}(f,\alpha)$
holds true in this scenario, Theorem 3.47(ii) provides us with an
interpretation of the partial derivative

$\frac{\partial^{r-1}}{\partial
s^{r-1}}\left.L_{\mathrm{K},\alpha}^{[0],\mathrm{imp},\pm}(f,\xi,\chi^{s})\right|_{s=\frac{k}{2}}$

in terms of the fundamental arithmetic invariants (among which is the $p$-adic
regulator $R_{\alpha}(T_{f}(k/2))$).

As the next task at hand, we need to compute the derivative

$\frac{d}{ds}L_{\mathrm{K},\alpha}^{[1],\pm}(f,\xi,\chi^{s})_{|_{s=\frac{k}{2}}}$

in terms of $p$-adic heights. A strategy paralleling the discussion in the
previous paragraph would require

  * •

the truth of the thick Main Conjecture $\widetilde{\mathbf{MC}}(f)$,

  * •

a descent formalism for the thick fundamental line, extending Theorem 4.15(ii)
to cover the case when $j=\frac{k}{2}$ and $r>0$.

This doesn’t seem tractable; see Remark 5.16.

We will instead use a different method to compute
$\frac{d}{ds}\widetilde{L}_{\mathrm{K},\alpha}^{\pm}(f,\xi,\chi^{s})_{|_{s=\frac{k}{2}}}$
when $r=1$ in terms of $p$-adic heights on the thick Selmer complexes (cf.
Theorem 5.29). In this regard, our results in the present chapter are
supplementary to those we have established in Chapters 3 and 4.

######  Remark 5.1.

Proposition 3.19 and Corollary 3.20 shows that when $r=1$, (5.3) is valid
under a weaker set of assumptions:

  * •

The hypothesis (PR)) holds.

  * •

The height pairing $\langle\,,\,\rangle_{\beta}$ is nonzero.

  * •

Either (SZ1)) or (SZ2)) is valid, to guarantee that the slope-zero main
conjecture $\mathbf{MC}(f,\beta)$ holds true; cf. [SU14, Wan15], [Ski16,
Theorem 2.5.2], [Kat04, Theorem 15.2].

As we have explained in the proof of Corollary 3.21, all these three
conditions hold true when $f=f_{A}$ is the normalized newform attached to an
elliptic curve $A/\mathbb{Q}$.

####  5.1.2. This chapter

Our main objective in §5 is twofold.

Our first goal is to illustrate the relation of thick Selmer groups and the
Bloch–Kato Selmer groups in an explicit manner; see in particular Corollary
5.14.

Our second goal is to compute, when $r=1$, the height pairings we have
introduced in §1.3.1 on thick Selmer groups; cf. Theorem 5.29.

###  5.2. Selmer complexes revisited

####  5.2.1. Notation

Throughout §5.2, we work in the setting of Section 2.1. We study the
arithmetic invariants associated to the following objects:

  * •

A free $\widetilde{E}$-module $\widetilde{V}$ of rank $2$ equipped with a
continuous action of $G_{\mathbb{Q},S}.$

  * •

We set $V=\ker(\widetilde{V}\xrightarrow{X}\widetilde{V})$ and assume that the
restriction of $V$ on the decomposition group at $p$ splits as

$V=V^{(\alpha)}\oplus V^{(\beta)},$

where $V^{(\alpha)}$ and $V^{(\beta)}$ are one-dimensional crystalline
representations with Hodge–Tate weights $1-k$ and $0$ respectively, with
$\alpha$ and $\beta$ the $\varphi$-eigenvalues on
${\mathbf{D}}_{\mathrm{cris}}(V^{(\alpha)})$ and
${\mathbf{D}}_{\mathrm{cris}}(V^{(\beta)}).$ We assume that $\alpha\neq
p^{1-k}$. Set
$\mathbf{D}^{(\alpha)}:=\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V^{(\alpha)})$
and

(5.4) $\mathbf{D}:=t^{k-1}\mathbf{D}^{(\alpha)}.$

  * •

A non-saturated crystalline $(\varphi,\Gamma)$-submodule
$\widetilde{\mathbf{D}}\subset\mathbf{D}_{\mathrm{rig},\widetilde{E}}^{\dagger}(\widetilde{V})$
of rank one over the relative Robba ring $\mathcal{R}_{\widetilde{E}}$ such
that

$\ker(\widetilde{\mathbf{D}}\xrightarrow{X}\widetilde{\mathbf{D}})\simeq\mathbf{D}.$

In this section, we assume that $k$ is even. We are interested in the central
twist of the representations $V$ and $\widetilde{V}$ and set

$\displaystyle{}^{\rm c}V:=V(\chi^{\frac{k}{2}}),$ $\displaystyle{}^{\rm
c}\widetilde{V}:=V(\chi^{\frac{k}{2}})$ $\displaystyle{}^{\rm
c}\mathbf{D}:=\mathbf{D}(\chi^{\frac{k}{2}}),$ $\displaystyle{}^{\rm
c}\widetilde{\mathbf{D}}:=\widetilde{\mathbf{D}}(\chi^{\frac{k}{2}}).$

The $G_{\mathbb{Q}_{p}}$-representation ${}^{\rm c}V$ is split in our setting:

(5.5) $\left.{}^{\rm c}V\right|_{G_{\mathbb{Q}_{p}}}={}^{\rm
c}V^{(\alpha)}\oplus{}^{\rm c}V^{(\beta)}\,.$

The associated $(\varphi,\Gamma)$-modules can be explicitly described:

$\mathbf{D}^{\dagger}_{\mathrm{rig},E}({}^{\rm
c}V^{(\alpha)})=\mathscr{R}_{E}(\chi^{1-\frac{k}{2}}\nu_{\alpha})\qquad,\qquad\mathbf{D}^{\dagger}_{\mathrm{rig},E}({}^{\rm
c}V^{(\beta)})=\mathscr{R}_{E}(\chi^{\frac{k}{2}}\nu_{\beta})\,,$

where $\nu_{\alpha}$ and $\nu_{\beta}$ are unramified characters such that
$\nu_{\alpha}(\mathrm{Fr}_{p})=\alpha/p^{k-1}$ and
$\nu_{\beta}(\mathrm{Fr}_{p})=\beta.$

####  5.2.2. Punctual Selmer complexes

#####

Our main objective in §5.2.2 is to examine the Selmer complexes
$\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm c}\mathbf{D}^{(\alpha)})$ and
$\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm c}\mathbf{D})$ associated to the data
$({}^{\rm c}V,{}^{\rm c}\mathbf{D}^{(\alpha)})$ and $({}^{\rm c}V,{}^{\rm
c}\mathbf{D})$ as in Section 1.2.2. In particular, we illustrate the
degenerate properties of $\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm c}\mathbf{D})$.
This will serve as an added justification for our definition and extensive
discussion in Section 5.2.4 of the thick Selmer complexes. We first examine
the Selmer complex $\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D}^{(\alpha)}).$

######  Proposition 5.2.

There exists a canonical isomorphism

(5.6) $\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D}^{(\alpha)})\simeq\mathbf{R\Gamma}^{\mathrm{imp}}({}^{\rm
c}V,\alpha)$

in the derived category. Here $\mathbf{R\Gamma}^{\mathrm{imp}}({}^{\rm
c}V,\alpha)$ is the Selmer complex of Perrin-Riou given as in §3.5.2, where we
take $V$ in place of $V_{f}$.

###### Proof.

We have the following commutative diagram of local conditions at $p$:

$\textstyle{{\mathbf{D}}_{\mathrm{cris}}({}^{\rm
c}V^{(\alpha)})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{Exp}_{\alpha,k/2,0}}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{(\alpha)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}V).}$

where the left vertical map $\mathbf{Exp}_{\alpha,k/2,0}$ is induced by the
map $\mathbf{Exp}_{\alpha,k/2}\,:\,\mathfrak{D}({}^{\rm
c}\mathbf{D}^{(\alpha)})[-1]\rightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}({}^{\rm
c}\mathbf{D}^{(\alpha)}).$ In degree one, this map coincides with the
exponential map

$\mathrm{Exp}_{\alpha,k/2,0}\,:\,{\mathbf{D}}_{\mathrm{cris}}({}^{\rm
c}\mathbf{D}^{(\alpha)})\longrightarrow H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{(\alpha)})\,.$

Equation (1.23) reads:

$\mathrm{Exp}_{\alpha,k/2,0}(x)=\frac{(-1)^{k/2}}{(k/2-1)!}(\exp^{*}_{{}^{\rm
c}\mathbf{D}^{(\alpha)}})^{-1}\left((1-\alpha^{-1}p^{k/2-1})(1-\alpha
p^{-k/2})^{-1}(x)\right).$

Since $v_{p}(\alpha)=k-1$ and $\exp^{*}_{{}^{\rm
c}\mathbf{D}^{(\alpha)}}\,:\,H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{(\alpha)})\rightarrow{\mathbf{D}}_{\mathrm{cris}}({}^{\rm
c}\mathbf{D}^{(\alpha)})$ is an isomorphism, the map
$\mathrm{Exp}_{\alpha,k/2,0}$ is an isomorphism. Moreover
$H^{i}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}^{(\alpha)})=0$ for $i\neq 1.$ Thus
$\mathbf{Exp}_{\alpha,k/2,0}$ is a quasi-isomorphism, and therefore the
corresponding Selmer complexes are quasi-isomorphic. ∎

We turn our attention to the cohomology of
$\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}).$

######  Lemma 5.3.

We have $H^{i}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})=0$ (for $i\neq 1$) and
$\dim_{E}H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})=1.$ Moreover, the natural
map $H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})\rightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}^{(\alpha)})$ induced by the
inclusion ${}^{\rm c}\mathbf{D}\hookrightarrow{}^{\rm
c}\mathbf{D}^{(\alpha)}$, is the zero map.

###### Proof.

The computation of $H^{i}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})$ follows easily
from the Poincaré duality and the Euler characteristic formula for the
cohomology of $(\varphi,\Gamma)$-modules.

Let us consider the following commutative diagram:

$\textstyle{\mathscr{D}_{\textrm{cris}}({}^{\rm
c}\mathbf{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\scriptstyle{\sim}$$\textstyle{{\mathbf{D}}_{\mathrm{cris}}({}^{\rm
c}V^{(\alpha)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}V^{(\alpha)})}$

The Hodge–Tate weights of ${}^{\rm c}\mathbf{D}$ are positive, and therefore
the left vertical map is an isomorphism. The right vertical map is the zero
map, since the Hodge–Tate weight of the $1$-dimensional
$G_{\mathbb{Q}_{p}}$-representation ${}^{\rm c}V_{f}^{(\alpha)}$ is negative.
Therefore the bottom row is the zero map. ∎

######  Remark 5.4.

Let us consider the tautological exact sequence

$0\longrightarrow{}^{\rm c}\mathbf{D}\longrightarrow{}^{\rm
c}\mathbf{D}^{(\alpha)}\longrightarrow\,^{c}\mathbb{D}\longrightarrow 0\,,$

where ${}^{c}\mathbb{D}:={}^{\rm c}\mathbf{D}^{(\alpha)}/{}^{\rm
c}\mathbf{D}.$ It follows from Lemma 5.3 that the map
$\partial_{0}\,:\,H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D})\rightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})$ is an isomorphism. It can be
described explicitly as follows. Let $x\in
H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D})$ and let $d\in{}^{\rm
c}\mathbf{D}^{(\alpha)}$ denote any lift of $x$. Set
$y=((\varphi-1)d,(\gamma-1)d).$ Then $y\in
C^{1}_{\varphi,\gamma}({}^{c}\mathbf{D})$ and $\partial_{0}(x)=y.$

######  Proposition 5.5.

The Selmer group $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D})$ splits canonically into the direct sum

(5.15) $\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{0}({}^{\rm
c}V)\oplus H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}),$
$\displaystyle[(z,(z_{\ell}^{+}),(\lambda_{\ell}))]$
$\displaystyle\longmapsto\quad\quad\left(\,[z]\,\,,\,\,[z_{p}^{+}]\,\right).$

###### Proof.

This is proved in a manner identical to [Ben20, Proposition 7.1.2]. The exact
sequence (1.31) combined with the fact that $H^{0}(\mathbb{Q}_{p},{}^{\rm
c}V)=\\{0\\}$ shows that

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D})=\ker\left(H^{1}(\mathbb{Q},{}^{\rm
c}V)\oplus\left(\underset{\ell\in S\setminus\\{p\\}}{\bigoplus}H^{1}_{\rm
f}(\mathbb{Q}_{\ell},{}^{\rm c}V)\oplus H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D})\right)\xrightarrow{f_{S}-g_{S}}\underset{\ell\in
S}{\bigoplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V)\right).$

Since $H^{1}_{\rm f}(\mathbb{Q}_{\ell},{}^{\rm
c}V)\xrightarrow{f_{\ell}}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V)$ is the natural
inclusion for $\ell\in S\setminus\\{p\\}$ and $H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D})\xrightarrow{g_{p}}H^{1}(\mathbb{Q}_{p},{}^{\rm c}V)$ is the
zero-map by Lemma 5.3, we have a natural identification

$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V,{}^{\rm
c}\mathbf{D})\,\,\,$ $\displaystyle=\,\ker\left(H^{1}(\mathbb{Q},{}^{\rm
c}V)\longrightarrow\underset{\ell\in
S\setminus\\{p\\}}{\bigoplus}\frac{H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V)}{H^{1}_{\rm f}(\mathbb{Q}_{\ell},{}^{\rm c}V)}\oplus
H^{1}(\mathbb{Q}_{p},{}^{\rm c}V)\right)\oplus H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D})$ $\displaystyle\simeq H^{1}_{0}({}^{\rm c}V)\oplus
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})\,,$
$\displaystyle[(z,(z_{\ell}^{+}),(\lambda_{\ell}))]$
$\displaystyle\longmapsto\quad\quad\left(\,[z]\,\,,\,\,[z_{p}^{+}]\,\right)$

where the second equality follows from the definition of the fine Selmer group
$H^{1}_{0}({}^{\rm c}V)$, cf. §1.2.3. This completes the proof of our
proposition. ∎

######  Remark 5.6.

Let $f$ be an eigenform with the critical stabilisation $f^{\alpha},$ and let
$V:=V_{f}$, $\mathbf{D}:=t^{k-1}\mathbf{D}^{(\alpha)}.$ Proposition 5.5 tells
us that $R^{1}\boldsymbol{\Gamma}({}^{\rm c}V,{}^{\rm c}\mathbf{D})$ extends
the fine Selmer group $H^{1}_{0}({}^{\rm c}V)$, rather than the Bloch–Kato
Selmer group, by the local cohomology group $H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D})$. This is an indication as to why the Selmer complex
$\mathbf{R\Gamma}({}^{\rm c}V,{}^{\rm c}\mathbf{D})$ may not capture some of
the arithmetic invariants of $f$ (that the Bloch–Kato Selmer group encodes).

#####

We continue to work in the setting of §5.2.2. Assume that we have a Galois
equivariant non-degenerate bilinear pairing

(5.16) $V^{\prime}\otimes V\longrightarrow E(1-k).$

It follows from the non-degeneracy of this pairing that the
$G_{\mathbb{Q}_{p}}$-representation $V^{\prime}$ also splits into the direct
sum $V^{\prime}=V^{\prime(\alpha)}\oplus V^{\prime(\beta)},$ where
$V^{\prime(\alpha)}$ (respectively $V^{\prime(\beta)}$) is the orthogonal
complement of $V^{(\beta)}$ (respectively $V^{(\alpha)}$). Set
$\mathbf{D}^{\prime(\alpha)}:=\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V^{\prime(\alpha)})$
and $\mathbf{D}^{\prime}:=t^{k-1}\mathbf{D}^{\prime(\alpha)}.$ The pairing
(5.16) induces

${}^{\rm c}V^{\prime}\otimes{}^{\rm c}V\longrightarrow E(1)\,,$

which gives rise to the $p$-adic height pairing

$\left<\,,\,\right>_{{}^{\rm c}\mathbf{D}^{\prime},{}^{\rm
c}\mathbf{D}}=\left<\,,\,\right>^{\textup{cyc}}_{{}^{\rm
c}\mathbf{D}^{\prime},{}^{\rm c}\mathbf{D}}\,:\,\mathbf{R\Gamma}({}^{\rm
c}V^{\prime},{}^{\rm c}\mathbf{D}^{\prime})\otimes\mathbf{R\Gamma}({}^{\rm
c}V,{}^{\rm c}\mathbf{D})\longrightarrow E[-2]\,.$

######  Proposition 5.7.

We have a commutative diagram

(5.20) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
56.3456pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime},{}^{\rm
c}\mathbf{D}^{\prime})\otimes\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V,{}^{\rm
c}\mathbf{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
65.12234pt\raise 6.84795pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.07426pt\hbox{$\scriptstyle{\left<\,\,,\,\,\right>_{{}^{\rm
c}\mathbf{D}^{\prime},{}^{\rm c}\mathbf{D}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.3456pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.3456pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 110.3456pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}_{0}({}^{\rm c}V^{\prime})\otimes
H^{1}_{0}({}^{\rm
c}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
61.20761pt\raise-46.70514pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.2063pt\hbox{$\scriptstyle{\left<\,\,,\,\,\right>_{\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
110.3456pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.3456pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
110.3456pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where the left vertical map is induced by the decomposition (5.15), and the
$p$-adic height pairing $\left<\,\,,\,\,\right>_{\alpha}$ is induced by the
composition

$H_{0}^{1}({}^{\rm c}V)\longrightarrow{}_{\mathrm{Iw}}({}^{\rm
c}V)^{\Gamma}\longrightarrow{}_{\mathrm{Iw}}({}^{\rm
c}V)_{\Gamma}\longrightarrow H_{0}^{1}({}^{\rm c}V^{\prime})^{*}\,,$

(cf. §3.5.3).

###### Proof.

Employing Part (iii) of Proposition 1.9 with $\mathbf{D}_{1}^{\prime}=0$ and
taking $\mathbf{D}_{2}^{\prime}=\mathbf{D}^{\prime}$ to be the
$(\varphi,\Gamma)$-module associated to any splitting module, we infer that
the restriction of any $p$-adic height pairing (associated to any splitting
module) to $H^{1}_{0}$ is independent of the choice of the said splitting
module, and therefore coincides with $\left<\,\,,\,\,\right>_{\alpha}.$

Since the $p$-adic height pairing factors through the Bockstein morphisms by
its definition (cf. Equation (1.38)), the proof follows from the commutative
diagram

$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V^{\prime},{}^{\rm
c}\mathbf{D}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime},{}^{\rm
c}\mathbf{D}^{\prime})}$$\textstyle{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the horizontal maps are the Bockstein maps, together with the fact that
$H^{2}(\mathbb{Q}_{p},{{}^{\rm c}\mathbf{D}^{\prime}})=\\{0\\}$, which we have
checked in Lemma 5.3. ∎

######  Corollary 5.8.

In the setting of Theorem 5.7, assume that $H^{1}_{0}({}^{\rm c}V)=\\{0\\}$.
Then the height pairing

$\left<\,\,,\,\,\right>_{{}^{\rm c}\mathbf{D}^{\prime},{}^{\rm
c}\mathbf{D}}\,\,:\,\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime},{}^{\rm
c}\mathbf{D}^{\prime})\otimes\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V,{}^{\rm c}\mathbf{D})\longrightarrow E$

on the extended Selmer groups vanishes.

######  Remark 5.9.

Corollary 5.8 is further evidence as to why the Selmer complex attached to a
$\theta$-critical modular form $f_{\alpha}$ does not capture all the
arithmetic invariants associated with $f$. Together with Remark 5.6, they
serve as a justification for the study of what we call thick Selmer complexes
in §5.2.4 below.

####  5.2.3. Thick Selmer groups

#####

In this section, we study various Selmer groups naturally attached to the
representation ${}^{\rm c}\widetilde{V}=\widetilde{V}(\chi^{k/2}).$ The
relevant Selmer complex will be introduced in the next section.

######  Definition 5.10.

The thick Greenberg Selmer group is defined by setting

$\mathrm{Sel}_{\alpha}({}^{\rm c}\widetilde{V}):=\ker\left({H^{1}_{S}({}^{\rm
c}\widetilde{V})}\longrightarrow\underset{\ell\neq
p}{\bigoplus}\frac{H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}\widetilde{V})}{H^{1}_{\rm f}(\mathbb{Q}_{\ell},{}^{\rm
c}\widetilde{V})}\bigoplus\frac{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})}{H^{1}_{\rm f}(\mathbb{Q}_{p},X\cdot{}^{\rm
c}\widetilde{V})}\right).$

The sequence (4.1) induces the exact sequence

(5.25) $0\longrightarrow{H^{1}_{S}({}^{\rm
c}V)}\xrightarrow{[X]}{H^{1}_{S}({}^{\rm
c}\widetilde{V}})\longrightarrow{H^{1}_{S}({}^{\rm c}V)}$

where the injection on the left is because ${H^{0}_{S}({}^{\rm c}V)}=\\{0\\}$.

######  Proposition 5.11.

Assume that $H^{0}(\mathbb{Q}_{\ell},{}^{\rm c}V)=0$ for all $\ell\in
S\setminus\\{p\\}.$ Then the sequence (5.25) gives rise to an exact sequence

$0\longrightarrow{H^{1}_{\rm f}({}^{\rm
c}V)}\xrightarrow{[X]}\mathrm{Sel}_{\alpha}({}^{\rm
c}\widetilde{V})\rightarrow H^{1}_{0}({}^{\rm c}V)\,.$

###### Proof.

Let $\ell\neq p$ be a prime in $S$. Since
$\dim_{E}H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm
c}V)=\dim_{E}H^{0}(\mathbb{Q}_{\ell},{}^{\rm c}V),$ we have that

$H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm c}V)=0,\qquad\forall\ell\neq p.$

It then follows from the short exact sequence

$H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm c}V)\longrightarrow
H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm c}\widetilde{V})\longrightarrow
H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm c}V)$

that $H^{1}_{\mathrm{f}}(\mathbb{Q}_{\ell},{}^{\rm c}\widetilde{V})=0$. For
$\ell=p,$ the tautological exact sequence

$0\longrightarrow X{}^{\rm c}V^{(\alpha)}\longrightarrow{}^{\rm
c}\widetilde{V}/X{}^{\rm c}V^{(\beta)}\longrightarrow{}^{\rm
c}V\longrightarrow 0$

induces the exact sequence

$0\longrightarrow H^{1}(\mathbb{Q}_{p},X{}^{\rm c}V^{(\alpha)})\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V}/X{}^{\rm
c}V^{(\beta)})\longrightarrow H^{1}(\mathbb{Q}_{p},{}^{\rm c}V),$

where $H^{1}(\mathbb{Q}_{p},X{}^{\rm c}V^{(\alpha)})\simeq
H^{1}(\mathbb{Q}_{p},X{}^{\rm c}V)/H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},X{}^{\rm
c}V).$ Let us set

$\displaystyle U_{\mathrm{f}}:=\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V)\oplus
H^{1}(\mathbb{Q}_{p},X{}^{\rm c}V^{(\alpha)}),$
$\displaystyle\widetilde{U}:=\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}\widetilde{V})\oplus H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V}/X{}^{\rm
c}V^{(\beta)}),$ $\displaystyle U_{0}:=\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V)\oplus
H^{1}(\mathbb{Q}_{p},{}^{\rm c}V).$

We then have a commutative diagram with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H^{1}_{S}({}^{\rm
c}V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H^{1}_{S}({}^{\rm
c}\widetilde{V})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H_{S}^{1}({}^{\rm
c}V)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{\mathrm{f}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{0}\,.}$

The exact sequence of the kernels of vertical maps reads

$0\longrightarrow{H^{1}_{\rm f}({}^{\rm
c}V)}\xrightarrow{[X]}\mathrm{Sel}_{\alpha}({}^{\rm
c}\widetilde{V})\xrightarrow{\,\pi\,}H^{1}_{0}({}^{\rm c}V)\,,$

and the proposition is proved.

∎

####  5.2.4. Thick Selmer complexes

In this Section, we are interested in Selmer complexes attached to the
representation ${}^{\rm c}\widetilde{V}:=\,\widetilde{V}(\chi^{k/2}).$ The
complex $\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})$ (defined following the formalism of §1.2.2) will be
called the _thick Selmer complex_ and its cohomology groups the _thick
(extended) Selmer groups_.

Our primary objective in §5.2.4 is to compute the thick Selmer group
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})$ explicitly and show that it is indeed the extension
of the true (Bloch–Kato) Selmer group $H^{1}_{\rm f}({}^{\rm c}V)$ in an
important case of interest (e.g. when the Hecke $L$-function of the dual form
$f^{*}$ has at most a simple zero at $\frac{k}{2}$). This will be the first
indication that the thick Selmer complexes are the correct algebraic objects
that capture arithmetic properties of the refinement $(f,\alpha)$ of $f$ (in
the sense of Bellaïche), including their extremal exceptional zeros as well as
those encoded in the Bloch–Kato Selmer group (which conjecturally explain the
leading terms of complex $L$-functions).

We continue to work in the setting of §5.2.2. In particular, we assume that
the conditions (C1))–(C3)) and (GP)) hold true.

######  Proposition 5.12.

The following assertions are valid:

$H^{i}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{\mathbf{D}})=0$ for $i\neq 1$, and
the exponential map induces an isomorphism

$\exp\,:\,{\mathscr{D}}_{\mathrm{cris}}({}^{\rm
c}\widetilde{\mathbf{D}})\xrightarrow{\sim}H^{i}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{\mathbf{D}}).$

We have a quasi-isomorphism

$\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})\simeq\mathbf{R\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\,,$

where the complex $\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ is as in
§4.2.5, where we replace $V_{k}$ with $\widetilde{V}.$

###### Proof.

On taking the cohomology of the exact sequence

$0\longrightarrow{}^{\rm c}\mathbf{D}\xrightarrow{[X]}{}^{\rm
c}\widetilde{\mathbf{D}}\longrightarrow{}^{\rm c}\mathbf{D}\longrightarrow 0$

and taking into account that $H^{i}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D})=0$
for $i\neq 1,$ we obtain that $H^{i}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{\mathbf{D}})=0$ for $i\neq 1.$ Moreover, ${}^{\rm
c}\widetilde{\mathbf{D}}$ is crystalline of weight $k/2\geqslant 1$ and
$\alpha\neq p^{k/2-1}.$ Therefore, the exponential map is an isomorphism, as
claimed.

The proof is entirely analogous to the proof of Proposition 5.2. We have the
following commutative diagram of local conditions at $p$:

$\textstyle{{\mathscr{D}}_{\mathrm{cris}}({}^{\rm
c}\widetilde{\mathbf{D}})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{Exp}_{{}^{\rm
c}\widetilde{\mathbf{D}},0}}$$\scriptstyle{\widetilde{\mathrm{E}}_{\alpha,k/2}^{(\mathds{1})}}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{\mathbf{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R\Gamma}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\,.}$

Here, the horizontal maps correspond to the local conditions at $p$ for the
complexes $\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ and
$\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},{}^{\rm c}\widetilde{\mathbf{D}})$,
respectively. Recall that the $(\varphi,\Gamma)$-module ${}^{\rm
c}\widetilde{\mathbf{D}}$ is not saturated in
$\mathbf{D}^{\dagger}_{\mathrm{rig},E}({}^{\rm c}\widetilde{V})$ and has
Hodge–Tate weights $(k/2,k/2).$ Therefore, the vertical map on the left is an
isomorphism in degree one. Moreover
$\mathbf{R}\boldsymbol{\Gamma}^{i}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{\mathbf{D}})=0$ for $i\neq 1.$ Thence, the vertical map on the
left is a quasi-isomorphism. This concludes the proof of this portion.

∎

We will compute $\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ (in turn
also $\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})$), assuming that $H^{1}_{0}({}^{\rm c}V)=0.$ In this
respect, the present section can be thought of as a supplement to §4. Set
$\widetilde{D}={\mathscr{D}}_{\mathrm{cris}}(\widetilde{\mathbf{D}})$ and let
us put ${}^{c}\widetilde{D}:=\widetilde{D}[k/2]$.

######  Proposition 5.13.

Assume that the hypothesis (GP)) holds true and that $H^{1}_{0}({}^{\rm
c}V)=0$. Then:

We have the following natural short exact sequences:

(5.34) $0\longrightarrow
X{}^{c}\widetilde{D}\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\longrightarrow\mathrm{Sel}_{\alpha}({}^{\rm
c}\widetilde{V})\longrightarrow 0\,.$

If in addition the condition () ‣ 3.23) holds for $V$, i.e. if the map

$H^{1}_{\rm f}(V)\longrightarrow H^{1}(\mathbb{Q}_{p},V)$

is nonzero, then there exists an exact sequence

(5.35) $0\longrightarrow H^{1}_{0}({}^{\rm
c}\widetilde{V})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\longrightarrow{}^{c}\widetilde{D}\longrightarrow 0\,.$

###### Proof.

Since $\mathbf{R\Gamma}({}^{\rm
c}\widetilde{V},\alpha):=\mathbf{R\Gamma}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V},\alpha)\otimes^{\mathbf{L}}_{\mathscr{H}(\Gamma)}E,$ we have

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)=\ker\left({H^{1}_{S}({}^{\rm
c}\widetilde{V})}\oplus\left(\underset{\ell\in
S\setminus\\{p\\}}{\bigoplus}H^{1}_{\rm f}(\mathbb{Q}_{\ell},{}^{\rm
c}\widetilde{V})\oplus{}^{c}\widetilde{D}\right)\xrightarrow{f_{S}-g_{S}}\underset{\ell\in
S}{\bigoplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}\widetilde{V})\right),$

where $g_{p}=\widetilde{\mathrm{E}}_{\alpha,k/2}^{(\mathds{1})}$ (cf. the
proof of Proposition 5.12). Thanks to this description of
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ and since
$g_{\ell}$ is injective for $\ell\neq p$, the module
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ fits in an
exact sequence

(5.36) $\displaystyle\begin{aligned}
0\longrightarrow\ker&\left({}^{c}\widetilde{D}\xrightarrow{g_{p}}H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\right)\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\\\
&\qquad\qquad\qquad\longrightarrow\ker\left({H^{1}_{S}({}^{\rm
c}\widetilde{V})}\xrightarrow{\textup{res}_{S}}\frac{\underset{\ell\in
S}{\bigoplus}H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}\widetilde{V})}{{\rm
im}(g_{S})}\right)\longrightarrow 0\,.\end{aligned}$

We note that

(5.37) ${\rm im}(g_{\ell})=\begin{cases}H^{1}_{{\rm
f}}(\mathbb{Q}_{\ell},{}^{\rm c}\widetilde{V})&\hbox{ if }\ell\neq p\,,\\\
H^{1}_{{\rm f}}(\mathbb{Q}_{p},X\cdot{}^{\rm c}\widetilde{V})&\hbox{ if
}\ell=p\,,\end{cases}$

where this follows from definitions if $\ell\neq p$ (cf. §1.2.1), whereas in
the case $\ell=p$, we refer to Corollary 2.8. Therefore, the rightmost term in
(5.36) equals $\mathrm{Sel}_{\alpha}({}^{\rm c}\widetilde{V})$.

Using (2.6) and Proposition 2.6 we observe that

$\ker\left({}^{c}\widetilde{D}\xrightarrow{\,g_{p}\,}H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\right)=X{}^{c}\widetilde{D}={\mathbf{D}}_{\mathrm{cris}}({}^{c}V^{(\alpha)})\,.$

This combined with the discussion in the previous paragraph completes the
proof of the exactness of the first sequence.

It follows from definitions that $H^{1}_{0}({}^{\rm
c}\widetilde{V})\subset\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha).$ Set $H^{1}_{/0}({}^{\rm
c}\widetilde{V}):=H^{1}_{S}({}^{\rm c}\widetilde{V})/H^{1}_{0}({}^{\rm
c}\widetilde{V}).$ We then have an exact sequence

(5.38) $0\longrightarrow H^{1}_{0}({}^{\rm
c}\widetilde{V})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\longrightarrow\ker\left(H^{1}_{/0}({}^{\rm
c}\widetilde{V})\oplus{}^{c}\widetilde{D}\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V})\right)\longrightarrow 0\,.$

The localisation map $H^{1}_{/0}({}^{\rm c}\widetilde{V})\rightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V})$ is injective, and it follows
from the exact sequence (5.25) and the condition () ‣ 3.23) that

$\mathrm{im}\left(H^{1}_{/0}({}^{\rm c}\widetilde{V})\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V})\right)=H^{1}_{\rm
f}(\mathbb{Q}_{p},X{}^{\rm c}\widetilde{V})\,.$

On the other hand, it follows from (GP)) that

$\mathrm{im}\left({}^{c}\widetilde{D}\xrightarrow{\,g_{p}\,}H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\right)=H^{1}_{\rm f}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{V}).$

Therefore, the map

$\ker\left(H^{1}_{/0}({}^{\rm
c}\widetilde{V})\oplus{}^{c}\widetilde{D}\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\right)\longrightarrow{}^{c}\widetilde{D},\qquad(x,y)\mapsto
y$

is an isomorphism. Using this isomorphism, we can rewrite the exact sequence
(5.38) as (5.35). This concludes the proof of our proposition. ∎

#####

Using the isomorphisms
$X{}^{c}\widetilde{D}\simeq{\mathscr{D}}_{\mathrm{cris}}(X{}^{\rm
c}\widetilde{\mathbf{D}})\overset{\exp}{\simeq}H^{1}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{\mathbf{D}})$ and $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\simeq\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},{}^{\rm c}\widetilde{\mathbf{D}})$, we can rewrite the exact
sequence (5.34) in the following form:

$0\longrightarrow H^{1}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{\mathbf{D}})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})\longrightarrow\mathrm{Sel}_{\alpha}({}^{\rm
c}\widetilde{V})\longrightarrow 0\,.$

We record the following consequence of this fact:

######  Corollary 5.14.

Assume that $H^{1}_{0}(V)=0$ and the condition (GP)) holds. Then there exist
an exact sequence

$0\longrightarrow H^{1}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{\mathbf{D}})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})\xrightarrow{\,\Delta\,}H^{1}_{\rm f}({}^{\rm
c}V)\longrightarrow 0\,.$

If, in addition, the condition () ‣ 3.23) holds, then we have an isomorphism

$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})\xrightarrow{\,\sim\,}{}^{c}\widetilde{D},$
$\displaystyle[x^{\mathrm{sel}}]:=[(x,(x^{+}_{\ell})_{\ell\in
S},(\lambda_{\ell})_{\ell\in S})]\longmapsto\log_{{}^{\rm
c}\widetilde{\mathbf{D}}}(x^{+}_{p})\,,$

and the following diagram commutes:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{\mathbf{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},{}^{\rm
c}\widetilde{\mathbf{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{H^{1}_{\rm
f}({}^{\rm
c}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X{}^{c}\widetilde{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{c}\widetilde{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{c}D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,.}$

The following proposition computes the Bockstein map on the complex
$\mathbf{R\Gamma}({}^{\rm c}\widetilde{V},\alpha).$

######  Proposition 5.15.

Suppose that $H^{1}_{0}({}^{\rm c}V)=0$ and $H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V)=0$ for all $\ell\in S\setminus\\{p\\}.$ Assume that the hypotheses () ‣
3.23) and (GP)) hold true. Then:

There exist natural isomorphisms

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\simeq{}^{c}\widetilde{D}\,\,,\qquad\qquad\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\simeq\displaystyle\frac{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})}{\textup{res}_{p}\left(H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{V})\right)}\,.$

In particular, $\mathbf{R}^{i}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)$ is a free $\widetilde{E}$-module of rank one
$(i=1,2)$.

We have a commutative diagram

$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Bock_{2}}}$$\textstyle{{}^{c}\widetilde{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\frac{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})}{\textup{res}_{p}\left(H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{V})\right)}\,,}$

where the right vertical map $\vartheta$ is given by the following rule: Let
$d\in{}^{c}\widetilde{D}$ and let
$\widehat{d}\in{}^{c}\widetilde{D}\otimes\Lambda$ be any lift of $d.$ Then
there exist $x\in H^{1}_{\mathrm{Iw}}(X{}^{\rm
c}\widetilde{V})\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)$ and $y\in
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)$ such that

$\mathrm{Exp}_{{}^{\rm
c}\widetilde{V}}(\widehat{d})=\textup{res}_{p}(x)+(\gamma-1)y\,.$

Then $\vartheta(d):=\mathrm{pr}_{0}(y)\mod H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{V})$. In particular, $\mathrm{im}(\vartheta)\subset
H^{1}(\mathbb{Q}_{p},X{}^{\rm c}\widetilde{V})$ .

###### Proof.

Since $H^{1}_{0}({}^{\rm c}V)=0,$ it follows from Lemma 4.11 that
${}^{2}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})^{\Gamma}=\\{0\\}={}^{2}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})_{\Gamma}$, and

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)=\mathrm{coker}(g)^{\Gamma}\,,\qquad\qquad\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)=\mathrm{coker}(g)_{\Gamma}\,,$

where $g$ is the map (4.19) with $j=\frac{k}{2}$:

$\left(H^{1}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\right)\oplus\left({}^{c}\widetilde{D}\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\right)\xrightarrow{\textup{res}_{p}\,+\,\mathrm{Exp}_{{}^{\rm
c}\widetilde{\mathbf{D}}}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\otimes_{\widetilde{\Lambda}_{E}}\widetilde{\mathscr{H}}(\Gamma)\,.$

Therefore, as in the previous chapter, we infer that
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ and
$\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ can be
naturally identified with the kernel and the cokernel, respectively, of the
map

$g_{\Gamma}\,:\,H^{1}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})_{\Gamma}\oplus{}^{c}\widetilde{D}\xrightarrow{\textup{res}_{p}\,+\,\mathrm{Exp}_{{}^{\rm
c}\widetilde{\mathbf{D}},0}}H^{1}(\mathbb{Q}_{p},{}^{\rm c}\widetilde{V})\,.$

Recall that in Corollary 5.14 (see also the proof of Proposition 5.13), we
already proved that

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)\simeq\ker(g_{\Gamma})\simeq{}^{c}\widetilde{D}.$

This can be deduced also using the arguments below, which utilize to compute
$\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha).$

Recall that $H^{1}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{\mathbf{D}})_{\Gamma}\xrightarrow{\sim}H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{\mathbf{D}})=H^{1}_{\mathrm{f}}({}^{\rm
c}\widetilde{\mathbf{D}})$. In what follows, we shall use the indicated
isomorphism to identify these spaces. It follows from Lemma 4.12 that
$H^{1}_{\mathrm{Iw}}({}^{\rm c}\widetilde{V})_{\Gamma}=H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{V})$ sits in the exact sequence

$0\longrightarrow H^{1}_{\\{p\\}}(X{}^{\rm c}\widetilde{V})\longrightarrow
H^{1}_{\mathrm{Iw}}({}^{\rm c}\widetilde{V})_{\Gamma}\longrightarrow
H^{1}_{\\{p\\}}({}^{\rm c}V)\longrightarrow 0\,,$

which can be extended to the following commutative diagram with exact rows

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(X{}^{\rm
c}\widetilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{res}_{p}}$$\textstyle{H^{1}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{res}_{p}}$$\textstyle{H^{1}_{\\{p\\}}({}^{\rm
c}V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{res}_{p}}$$\textstyle{0}$$\textstyle{H^{1}_{\\{p\\}}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(\mathbb{Q}_{p},{}^{\rm
c}\widetilde{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(\mathbb{Q}_{p},{}^{\rm
c}V)}$

where the vertical arrows are injections. Since the restriction map
$H^{1}_{\\{p\\}}(X{}^{\rm
c}\widetilde{V})\xrightarrow{\textup{res}_{p}}H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{V})$ is an isomorphism, and

$\mathrm{im}\left(\mathrm{Exp}_{{}^{\rm
c}\widetilde{\mathbf{D}},0}\right)=H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{V}),$

we have

(5.51) $\mathrm{im}\left(\mathrm{Exp}_{{}^{\rm
c}\widetilde{\mathbf{D}},0}\right)=H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},X{}^{\rm
c}\widetilde{V})=\textup{res}_{p}(H^{1}_{\\{p\\}}(X{}^{\rm c}\widetilde{V})).$

As a result, we have ${\rm
im}(g_{\Gamma})=\textup{res}_{p}(H^{1}_{\mathrm{Iw}}({}^{\rm
c}\widetilde{V})_{\Gamma})=\textup{res}_{p}(H^{1}_{\\{p\\}}({}^{\rm
c}\widetilde{V}))\,,$ and the second isomorphism in (i) follows.

The proof that $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}\widetilde{V},\alpha)$ is a free $\widetilde{E}$-module of rank one follows
from the first isomorphism in (i), whereas the fact that
$\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm c}\widetilde{V},\alpha)$ is a free
$\widetilde{E}$-module of rank one from the second isomorphism in (i) combined
with Lemma 4.12.

The formula for $\vartheta$ in (ii) follows directly from the definition of
the Bockstein map and the proofs of the remaining claims in this portion are
clear. ∎

######  Remark 5.16.

In this remark, we return back to the setting of Chapter 4 and assume that
$\widetilde{V}=V_{k}$ is the infinitesimal thickening of the $p$-adic Galois
representation attached to a critical modular form of weight $k$ along the
eigencurve.

The discussion in §5.2 applies verbatim when the pair
$({}^{c}\widetilde{V},{}^{c}\widetilde{\mathbf{D}})$ is replaced with
$(\widetilde{V}(\chi^{j}),\widetilde{\mathbf{D}}(\chi^{j}))$ for integers
$1\leq j\leq k-1$, without demanding that $k$ is even. It is required that $k$
is even only when one is interested (as we are in the remainder of this
chapter) in the interpretation of the central critical values in terms of
$p$-adic height pairings.

Proposition 5.15(ii) tells us that the thick Selmer group
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{k},{}^{\rm c}\mathbf{D}_{k})$
behaves in a manner reminiscent of the behaviour of the extended Selmer groups
in the presence of exceptional zeros. The exceptional zero that
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{k},{}^{\rm c}\mathbf{D}_{k})$
accounts for is that of thick $p$-adic $L$-function
$\widetilde{L}_{\mathrm{K},\eta}^{\pm}(f,\xi)$, cf. (5.1).

The kernel of $\vartheta$ contains $X{}^{c}D_{k}$. We infer from the
commutative diagram in Proposition 5.15(ii) that the complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}({}^{\rm c}V_{k},\alpha)$ is not semi-simple.
It seems that the modification of the Bockstein map used in the earlier
sections of Chapter 4 can not be extended to this case.

It is not clear whether or not the map $\vartheta$ is nonzero. The requirement
that $\vartheta\neq 0$ appears to be a refinement of (GP)).

Based on the absence of a duality theory over $\mathcal{O}_{\mathcal{X}}$
combined with the previous remarks, it is not clear to us whether or not the
derivative of the secondary $p$-adic $L$-function
$L^{[1],\pm}_{K,\alpha^{*}}(f^{*},\xi^{*})$ at the central critical point can
be computed in terms of $p$-adic heights. See, however, Theorem 5.29(ii) below
where we compute this quantity in terms of the second-order derivative of an
$\mathcal{O}_{\mathcal{X}}$-adic regulator (at $X=0$). Nonetheless, the
arithmetic meaning of $L^{[1],\pm}_{K,\alpha^{*}}(f^{*},\xi^{*})$ at the
central critical point, even granted the infinitesimal thickening
$\widetilde{\mathbf{MC}}(f)$ of the $\theta$-critical Main Conjecture, remains
mysterious.

###  5.3. $p$-adic regulators and leading term formulae

Our goal in this section is to compute the leading terms of the critical
$p$-adic $L$-functions given as in §2, in terms of the regulators associated
to the $p$-adic heights we have introduced in §1.3. As in the earlier sections
in this chapter, we shall solely concentrate on the central critical values.

####  5.3.1. The Selmer complex $\mathbf{R\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})$

##### Central critical twists

As before, we let $V_{\mathcal{X}}$ and $V^{\prime}_{\mathcal{X}}$ denote the
free $\mathcal{O}_{\mathcal{X}}$-modules of rank $2$ as in §2.4.2, which are
endowed with continuous actions of $G_{\mathbb{Q},S}$. Recall that the Galois
representations $V_{\mathcal{X}}$ and $V^{\prime}_{\mathcal{X}}$ interpolate
Deligne’s representations over the affinoid neighborhood $\mathcal{X}$
contained in the eigencurve, about the point $x_{0}$ that corresponds to the
$\theta$-critical eigenform $f_{\alpha}\in
S_{k}(\Gamma_{1}(N)\cap\Gamma_{0}(p),\varepsilon_{f})$.

######  Definition 5.17.

The universal $\mathscr{H}_{\mathcal{W}}$-valued cyclotomic character
$\bbchi:\Gamma\to\mathcal{O}_{\mathcal{W}}^{*}$ is given by

$\bbchi(\gamma):=\chi(\gamma)^{k}\exp\left(\log_{p}(1+Y)\frac{\log_{p}(\left<\chi(\gamma)\right>)}{\log_{p}(1+p)}\right)\,,\qquad\gamma\in\Gamma\,.$

We continue to assume that $k$ is even and we consider the _central critical
twist_ ${}^{\rm c}V_{\mathcal{X}}:=V_{\mathcal{X}}(\bbchi^{\frac{1}{2}})$ of
$V_{\mathcal{X}}$. Put ${}^{\rm
c}\mathbf{D}_{\mathcal{X}}:=\mathbf{D}_{\mathcal{X}}(\bbchi^{\frac{1}{2}})$.
The pairing $(\,,\,)$ given as in (2.4.2) gives rise to an
$\mathcal{O}_{\mathcal{W}}$-linear pairing

(5.52) ${}^{\rm c}V_{\mathcal{X}}^{\prime}\otimes{}^{\rm
c}V_{\mathcal{X}}\longrightarrow\mathcal{O}_{\mathcal{W}}(\chi).$

For each $x\in\mathcal{X}^{\mathrm{cl}}(E)\setminus\\{x_{0}\\},$ the
$\varphi$-module ${\mathbf{D}}_{\mathrm{cris}}(V_{x}^{\prime})$ decomposes
into the direct sum

${\mathbf{D}}_{\mathrm{cris}}(V_{x}^{\prime})={\mathbf{D}}_{\mathrm{cris}}(V_{x}^{\prime})^{\varphi=\alpha^{\prime}(x)}\oplus{\mathbf{D}}_{\mathrm{cris}}(V_{x}^{\prime})^{\varphi=\beta^{\prime}(x)},\qquad\textrm{where
$\alpha^{\prime}(x)=\alpha(x)\varepsilon_{f}^{-1}(p),$ and
$\beta^{\prime}(x)=\beta(x)\varepsilon_{f}^{-1}(p),$ }$

and we denote by $\mathbf{D}_{\mathcal{X}}^{\prime}$ the unique free
$\mathscr{R}_{\mathcal{X}}$-submodule of rank one of
$\mathbf{D}^{\dagger}_{\mathrm{rig},\mathcal{X}}(V_{\mathcal{X}}^{\prime})$
such that
$\mathbf{D}^{\prime}_{x}={\mathbf{D}}_{\mathrm{cris}}(V_{x}^{\prime})^{\varphi=\alpha^{\prime}(x)}$
for all $x\in\mathcal{X}^{\mathrm{cl}}(E)\setminus\\{x_{0}\\}.$ Set ${}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime}:={}^{\rm
c}\mathbf{D}_{\mathcal{X}}(\bbchi^{\frac{1}{2}}).$ The modules ${}^{\rm
c}\mathbf{D}^{\prime}_{\mathcal{X}}$ and ${}^{\rm c}\mathbf{D}_{\mathcal{X}}$
are orthogonal to each other under the canonical pairing induced from (5.52).

#####

In this section, we study the Selmer complexes $\mathbf{R\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})$ and
$\mathbf{R\Gamma}({}^{\rm c}V^{\prime}_{\mathcal{X}},{}^{\rm
c}\mathbf{D}^{\prime}_{\mathcal{X}})$ (see §1.2.2 for general definitions).

######  Lemma 5.18.

Let $\ell\in S\setminus\\{p\\}$ be a prime. Then the following statements hold
true:

$H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V_{x})=H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V^{\prime}_{x})=\\{0\\}$ at any classical point $x\in\mathcal{X}(E),$

$H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V_{\mathcal{X}})=H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V_{\mathcal{X}}^{\prime})=\\{0\\}$ whenever $\mathcal{X}$ is sufficiently
small.

For sufficiently small $\mathcal{X}$, the $\mathcal{O}_{\mathcal{X}}$-modules
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})$ and $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime}_{\mathcal{X}},{}^{\rm c}\mathbf{D}^{\prime}_{\mathcal{X}})$ are
free.

###### Proof.

It follows from the local-global compatibility of Langlands correspondence
that

$H^{0}(\mathbb{Q}_{\ell},{}^{\rm
c}V_{x})=\\{0\\}=H^{2}(\mathbb{Q}_{\ell},{}^{\rm c}V_{x})$

for all $x\in\mathcal{X}^{\rm cl}(E)$ and $\ell\in S\setminus\\{p\\}$. By the
local Euler characteristic formula, we then conclude that
$H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V_{x})=\\{0\\}$ for all
$x\in\mathcal{X}^{\rm cl}(E)$. We deduce from the self-duality of ${}^{\rm
c}V_{x}$ that $H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V_{x}^{\prime})=\\{0\\}$.

It follows from Part i) that $H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V^{\prime}_{\mathcal{X}})/\mathfrak{m}_{x}H^{1}(\mathbb{Q}_{\ell},{}^{\rm
c}V_{\mathcal{X}}^{\prime})=\\{0\\}$ for every $x\in\mathcal{X}^{\rm cl}(E)$,
thence also that $H^{1}(\mathbb{Q}_{\ell},{}^{\rm c}V_{\mathcal{X}}^{\prime})$
is a torsion $\mathcal{O}_{\mathcal{X}}$-module with no support at any
classical point in $\mathcal{X}(E)$ (including $x=x_{0}$). On shrinking
$\mathcal{X}$ as necessary, we can ensure that it has no support anywhere on
$\mathcal{X}$.

Since $\mathcal{O}_{\mathcal{X}}$ is a PID, it suffices to prove that both
modules are torsion-free.

Since the $\mathcal{O}_{\mathcal{X}}$-module
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})$ is finitely generated, we can ensure (on
shrinking $\mathcal{X}$) that the torsion submodule of
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})$ is contained in

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})[X^{\infty}]:=\bigcup_{n}\ker\left(\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\xrightarrow{[X^{n}]}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})\right)\,.$

It therefore suffices to check that

${\rm im}\left(\mathbf{R}^{0}\boldsymbol{\Gamma}({}^{\rm c}V_{x_{0}},{}^{\rm
c}\mathbf{D}_{x_{0}})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\right)=\ker\left(\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\xrightarrow{[X]}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})\right)\,.$

This is clear, since $\mathbf{R}^{0}\boldsymbol{\Gamma}({}^{\rm
c}V_{x_{0}},{}^{\rm c}\mathbf{D}_{x_{0}})=\\{0\\}$.

The proof of the freeness of $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime}_{\mathcal{X}},{}^{\rm c}\mathbf{D}^{\prime}_{\mathcal{X}})$ is
identical and is omitted here. ∎

#####

Let us set
$\,{}^{c}\mathbb{D}_{\mathcal{X}}:=\mathbf{D}^{\dagger}_{\mathrm{rig},\mathcal{X}}({}^{\rm
c}V_{\mathcal{X}})/{}^{\rm c}\mathbf{D}_{\mathcal{X}},$ so that we have the
following tautological exact sequence:

$0\longrightarrow{}^{\rm
c}\mathbf{D}_{\mathcal{X}}\longrightarrow\mathbf{D}^{\dagger}_{\mathrm{rig},\mathcal{X}}({}^{\rm
c}V_{\mathcal{X}})\longrightarrow\,^{c}\mathbb{D}_{\mathcal{X}}\longrightarrow
0\,.$

We also set
$\,{}^{c}\mathbb{D}_{\mathcal{X}}^{\prime}:=\mathbf{D}^{\dagger}_{\mathrm{rig},\mathcal{X}}({}^{\rm
c}V^{\prime}_{\mathcal{X}})/{}^{\rm c}\mathbf{D}^{\prime}_{\mathcal{X}}$ .

######  Lemma 5.19.

The following assertions are valid for sufficiently small $\mathcal{X}$:

$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})=\\{0\\}.$

Assume that the condition (GP)) holds. Then the support of the
$\mathcal{O}_{\mathcal{X}}$-module
$H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})_{\rm tor}$ is
$\\{x_{0}\\}$, and $H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})_{\rm
tor}\simeq\mathcal{O}_{\mathcal{X}}/X\mathcal{O}_{\mathcal{X}}$.

The assertions i-ii) hold if we replace $\,{}^{c}\mathbb{D}_{\mathcal{X}}$
with $\,{}^{c}\mathbb{D}^{\prime}_{\mathcal{X}}$.

###### Proof.

Let us consider the short exact sequence

(5.53)
$0\longrightarrow\,^{c}\mathbb{D}_{\mathcal{X}}\xrightarrow{X-X(x)}\,^{c}\mathbb{D}_{\mathcal{X}}\xrightarrow{\mathrm{sp}_{x}}\,^{c}\mathbb{D}_{x}\longrightarrow
0\,.$

The exactness of the sequence

$0\longrightarrow
H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\xrightarrow{X-X(x)}H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\xrightarrow{\mathrm{sp}_{x}}H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x})$

for each $x\in\mathcal{X}^{\rm cl}(E)$ and the vanishing
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x})=0$ whenever $x\neq x_{0}$ shows
(on shrinking $\mathcal{X}$ as necessary) that
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})$ can only have support
at $x_{0}$; in particular it is a torsion $\mathcal{O}_{\mathcal{X}}$-module.

On the other hand, the exact sequence

$0\longrightarrow H^{0}(\mathbb{Q}_{p},{{}^{\rm
c}\mathbf{D}}_{\mathcal{X}})\longrightarrow H^{0}(\mathbb{Q}_{p},{}^{\rm
c}V_{\mathcal{X}})\longrightarrow
H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\longrightarrow
H^{1}(\mathbb{Q}_{p},{{}^{\rm c}\mathbf{D}}_{\mathcal{X}})$

together with the fact that $H^{0}(\mathbb{Q}_{p},{}^{\rm
c}V_{\mathcal{X}})=0$ show that
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})$ injects into
$H^{1}(\mathbb{Q}_{p},{{}^{\rm c}\mathbf{D}}_{\mathcal{X}})_{\rm tor}$. Since
we have

$0=H^{0}(\mathbb{Q}_{p},{{}^{\rm
c}\mathbf{D}}_{x})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\mathbb{Q}_{p},{{}^{\rm
c}\mathbf{D}}_{\mathcal{X}})[\mathfrak{m}_{x}]$

for each $x\in\mathcal{X}^{\rm cl}(E)$ (including $x=x_{0}$), we can ensure
that $H^{1}(\mathbb{Q}_{p},{{}^{\rm c}\mathbf{D}}_{\mathcal{X}})_{\rm tor}=0$
(shrinking $\mathcal{X}$, if necessary). This concludes the proof that
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})=0$.

We note that the vanishing of
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})$ does not imply the
vanishing of $H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x_{0}}).$ More precisely,
using the long exact sequence in $G_{\mathbb{Q}_{p}}$-cohomology that the
short exact sequence (5.53) gives rise to, we obtain an isomorphism

$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x})\simeq
H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})[\mathfrak{m}_{x}]\,$

for any $x$. The $\mathcal{O}_{\mathcal{X}}$-module
$H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})_{\rm tor}$ is finitely
generated. Since $H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x})=0$ if
$x\in\mathcal{X}^{\mathrm{cl}}(E)\setminus\\{x_{0}\\}$ and
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{x_{0}})\simeq
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}_{x_{0}})$ is of dimension one over
$E$, we can assume, on shrinking $\mathcal{X}$ as necessary, that the support
of $H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})_{\rm tor}$ is
$\\{x_{0}\\}$.

Since $H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}_{x_{0}})\simeq\mathcal{O}_{\mathcal{X}}/X\mathcal{O}_{\mathcal{X}},$
we only need to prove that
$H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})[X^{2}]=H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})[X]$.
By the exactness of the sequence

$0\longrightarrow H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{k})\longrightarrow
H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\xrightarrow{[X^{2}]}H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})$

we reduce to proving that
$\dim_{E}H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{k})=1.$ To that end, let us
consider the exact sequence

$0\longrightarrow{}^{\rm
c}\mathbf{D}_{k}\longrightarrow\mathbf{D}^{\dagger}_{\mathrm{rig}}({}^{\rm
c}V_{k})\longrightarrow{\,{}^{c}\mathbb{D}}_{k}\longrightarrow 0\,,$

which induces an exact sequence

$H^{0}(\mathbb{Q}_{p},{\,{}^{c}\mathbb{D}}_{k})\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}\mathbf{D}_{k})\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}V_{k})\,.$

Let $V_{k}^{(\alpha)}$ be the maximal crystalline sub-representation of
$V_{k}$ defined in §2.1.1 and denoted there by $\widetilde{V}^{(\alpha)}.$
Then the quotient ${}^{\rm c}V_{k}\Big{/}{}^{c}V_{k}^{(\alpha)}\simeq{}^{\rm
c}V_{x_{0}}^{(\beta)}$ has Hodge-Tate weight $-\frac{k}{2}$, and therefore the
map

$H^{1}(\mathbb{Q}_{p},{}^{c}V_{k}^{(\alpha)})\longrightarrow
H^{1}(\mathbb{Q}_{p},{}^{\rm c}V_{k})$

is injective. We have a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}_{\mathrm{cris}}({}^{\rm
c}\mathbf{D}_{x_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}_{\mathrm{cris}}({}^{\rm
c}\mathbf{D}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{{\mathbf{D}}_{\mathrm{cris}}({}^{c}V_{k}^{(\alpha)})\big{/}\operatorname{Fil}^{0}{\mathbf{D}}_{\mathrm{cris}}({}^{c}V_{k}^{(\alpha)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\mathbb{Q}_{p},{\,{}^{c}\mathbb{D}}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},{}^{c}V_{k}^{(\alpha)}),}$

where the first row is exact and the image of the right-most map on this row
is nonzero thanks to the condition (GP)). This shows that
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{k})$ is of dimension one over $E$, as
required. ∎

To simplify the notation, we denote by $H^{1}_{{}^{\rm c}\mathbf{D}}({}^{\rm
c}V_{\mathcal{X}})$ the Greenberg-style Selmer group attached to the
$(\varphi,\Gamma)$-module ${}^{\rm c}\mathbf{D}_{\mathcal{X}}$, namely

$H^{1}_{{}^{\rm c}\mathbf{D}}({}^{\rm
c}V_{\mathcal{X}}):=\ker\left(H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}})\xrightarrow{\bigoplus_{\ell\in
S}{r_{\ell}}}H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\oplus\bigoplus_{p\neq\ell\in
S}H^{1}(\mathbb{Q}_{\ell}^{\textup{ur}},{}^{\rm c}V_{\mathcal{X}})\right)\,.$

(cf. §1.2.3). In view of Lemma 5.18, we have

$H^{1}_{{}^{\rm c}\mathbf{D}}({}^{\rm
c}V_{\mathcal{X}}):=\ker\left(H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}})\xrightarrow{\,r_{p}\,}H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\right)\,.$

######  Proposition 5.20.

There exists a canonical isomorphism

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\simeq H^{1}_{{}^{\rm c}\mathbf{D}}({}^{\rm
c}V_{\mathcal{X}}).$

###### Proof.

We have the following exact sequence thanks the definition of Selmer
complexes:

(5.58) $0\longrightarrow
H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})\longrightarrow
H^{1}_{{}^{\rm c}\mathbf{D}}({}^{\rm c}V_{\mathcal{X}})\longrightarrow 0.$

The proof of the proposition follows since
$H^{0}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}})=0$ by Lemma 5.19. ∎

####  5.3.2. The rank-one case

In this section, we maintain the previous notations and conventions and assume
in addition that

(5.59) $r_{\mathrm{an}}(k/2):=\mathrm{ord}_{s=\frac{k}{2}}L(f,s)=1\,.$

as well as the hypothesis (PR)) hold true for the eigenform $f$.

In what follows, thanks to our discussion in Remark 3.18, one may drop the
assumption (PR)) from the statements below whenever (CM)) holds true (and it
does hold true when $k=2$).

#####

We denote by ${}^{c}\mathbf{z}_{0}(\mathcal{X},\xi)\in H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}}^{\prime})$ the image of the big Beilinson–Kato element
$\mathbf{z}(\mathcal{X},\xi)\in
H^{1}_{S}(V_{\mathcal{X}}^{\prime}\,\widehat{\otimes}\,\Lambda^{\iota}(1))$
under the obvious projection, where the latter element is given as in §2.4.3.
Note that (5.59) implies (thanks to Kato’s reciprocity laws) that

(5.60) ${}^{c}\mathbf{z}_{0}(x_{0},\xi)\in H^{1}_{\rm f}({}^{\rm
c}V_{x_{0}}^{\prime})\,.$

Recall that the condition (PR)) implies (thanks to the Euler system machinery)
that

(5.61) $\dim H^{1}_{0}({}^{\rm c}V_{x_{0}}^{\prime})=0\,,\qquad\dim H^{1}_{\rm
f}({}^{\rm c}V_{x_{0}}^{\prime})=1\,$

(cf. Proposition 3.19). This in turn shows that the parity conjecture
asserting that

(5.62) $\mathrm{ord}_{s=\frac{k}{2}}L(f,s)\equiv\dim H^{1}_{\rm f}({}^{\rm
c}V_{x_{0}}^{\prime})\mod 2$

holds true.

######  Proposition 5.21.

Suppose that (5.59) and (PR)) hold. Then on sufficiently small $\mathcal{X}$
we have

$\dim_{E}H^{1}_{\rm f}({}^{\rm c}V_{x}^{\prime})=1$

for all $x\in\mathcal{X}^{\rm cl}(E)$. Moreover,
${}^{c}\mathbf{z}_{0}(x,\xi)\in H^{1}_{\rm f}({}^{\rm c}V_{x}^{\prime})$ and
the truth of the parity conjecture is constant on $\mathcal{X}^{\rm cl}(E)$.

###### Proof.

The condition (PR)) shows that
$\textup{res}_{p}({}^{c}\mathbf{z}_{0}(\mathcal{X},\xi))\in
H^{1}(\mathbb{Q}_{p},{}^{\rm c}V_{\mathcal{X}}^{\prime})$ is nonzero. Since
$H^{1}(\mathbb{Q}_{p},{}^{\rm c}V^{\prime}_{\mathcal{X}})$ is a free
$\mathcal{O}_{\mathcal{X}}$-module of rank $2$, it follows (on shrinking
$\mathcal{X}$ as necessary) that

$\textup{res}_{p}({}^{c}\mathbf{z}_{0}(x,\xi))=\textup{res}_{p}({}^{c}\mathbf{z}_{0}(\mathcal{X},\xi))(x)\neq
0$

for all $x\in\mathcal{X}^{\rm cl}(E)$.

The constancy of the sign of the functional equation in families (cf. [PX15],
§3.6) together with (5.59) shows that
$\mathrm{ord}_{s=\frac{w(x)}{2}}L(f_{x},s)$ is odd for all
$x\in\mathcal{X}^{\rm cl}(E)$. In particular, $L(f_{x},\frac{w(x)}{2})=0$.
This together with Kato’s reciprocity laws (and the conclusion of the
preceding paragraph) show that

$0\neq{}^{c}\mathbf{z}_{0}(x,\xi)\in H^{1}_{\rm f}({}^{\rm c}V_{x}^{\prime}).$

for all $x$ as above. Now a standard application of the Euler system machinery
(with the Beilinson–Kato Euler system for the eigenform $f_{x}$) shows that
$\dim H^{1}_{\rm f}({}^{\rm c}V_{x}^{\prime})=1$, which has the same parity as
$\mathrm{ord}_{s=\frac{w(x)}{2}}L(f_{x},s)$. ∎

Recall that we have an isomorphism

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\simeq H^{1}_{\mathbf{D}^{\prime}}({}^{\rm
c}V_{\mathcal{X}}^{\prime}).$

######  Theorem 5.22.

Suppose that $r_{\mathrm{an}}(k/2)=1$ and assume that the conditions (PR)) and
(GP)) hold. Then for sufficiently small $\mathcal{X}$, the following
assertions are valid:

Both $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})$ and $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime}_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})$ are
free of rank one as $\mathcal{O}_{\mathcal{X}}$-modules.

The $\mathcal{O}_{\mathcal{X}}$-module $H^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime})$, and therefore
also the isomorphic module $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})$, is
generated by the element $X\cdot{}^{c}\mathbf{z}_{0}(\mathcal{X},\xi).$

###### Proof.

Equation (5.61) together with (5.15) show that

$\dim\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{x_{0}}^{\prime},{}^{\rm
c}\mathbf{D}_{x_{0}}^{\prime})=1.$

The control theorem for Selmer complexes yields an injection

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})/\mathfrak{m}_{x_{0}}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\hookrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{x_{0}}^{\prime},{}^{\rm c}\mathbf{D}_{x_{0}}^{\prime}),$

which in turn shows that

${\rm
rank}_{\mathcal{O}_{\mathcal{X}}}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\leqslant 1.$

Proposition 5.21 tells us that, on shrinking $\mathcal{X}$ as necessary,
$H^{1}_{\rm f}({}^{\rm
c}V_{x}^{\prime})=\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime}_{x},{}^{\rm c}\mathbf{D}_{x}^{\prime})$ is 1-dimensional for
every $x\in\mathcal{X}^{\rm cl}(E)\setminus\\{x_{0}\\}$. Consider the exact
sequence

$0\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})/\mathfrak{m}_{x}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V^{\prime}_{x},{}^{\rm
c}\mathbf{D}_{x}^{\prime})\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})[\mathfrak{m}_{x}]\longrightarrow 0,$

induced from the control theorem. Since
$\mathbf{R}^{2}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})$ is a finitely generated
$\mathcal{O}_{\mathcal{X}}$-module, on further shrinking $\mathcal{X}$, one
can ensure that this exact sequence induces an isomorphism

$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})/\mathfrak{m}_{x}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})\simeq
H^{1}_{\rm f}({}^{\rm c}V_{x}^{\prime}),\qquad\forall x\in\mathcal{X}^{\rm
cl}(E).$

This shows that $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\neq\\{0\\}$. Recalling from Lemma 5.18
that $\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})$ is
free, and combining with the conclusion of the previous paragraph, we deduce
that

${\rm
rank}_{\mathcal{O}_{\mathcal{X}}}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})=1\,,$

as required.

The argument to prove that ${\rm
rank}_{\mathcal{O}_{\mathcal{X}}}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm c}\mathbf{D}_{\mathcal{X}})=1$ proceeds in a similar
way.

Let us put $P^{\prime}:=X\cdot{}^{c}\mathbf{z}_{0}(\mathcal{X},\xi).$ We shall
first prove that $P^{\prime}\in H^{1}_{\mathbf{D}^{\prime}}({}^{\rm
c}V_{\mathcal{X}}^{\prime}).$ Recall that

$H^{1}_{{}^{\rm c}\mathbf{D}^{\prime}}({}^{\rm
c}V_{\mathcal{X}}^{\prime})=\ker\left(H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}}^{\prime})\xrightarrow{\,r_{p}\,}H^{1}(\,^{c}\mathbb{D}_{\mathcal{X}}^{\prime})\right).$

We need to check that $P^{\prime}$ verifies the relevant local condition under
our running hypotheses. For each $x\in\mathcal{X}^{\rm cl}(E),$ we have a
commutative diagram

$\textstyle{H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}}^{\prime})/\mathfrak{m}_{x}H^{1}_{S}({}^{\rm
c}V_{\mathcal{X}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{p,x}}$$\textstyle{H^{1}(\,^{c}\mathbb{D}_{\mathcal{X}}^{\prime})/\mathfrak{m}_{x}H^{1}(\,^{c}\mathbb{D}_{\mathcal{X}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{S}({}^{\rm
c}V_{x}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\,^{c}\mathbb{D}^{\prime}_{x}),}$

where the vertical maps are injective. If $x\neq x_{0},$ then

$\ker\left(H^{1}_{S}({}^{\rm c}V_{x}^{\prime})\longrightarrow
H^{1}(\,^{c}\mathbb{D}^{\prime}_{x})\right))\simeq H^{1}_{\rm f}({}^{\rm
c}V^{\prime}_{x}),$

and it follows from Proposition 5.21 that $r_{p,x}(P^{\prime})=0.$ Moreover,
we also have $r_{p,x_{0}}(P^{\prime})=0.$ Since the support of
$H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}_{\mathcal{X}}^{\prime})_{\rm
tor}=\\{x_{0}\\},$ this implies that $P^{\prime}\in\ker(r_{p})=H^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime}).$

Let $Q^{\prime}$ be a generator of $H^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime}).$ Then
$P^{\prime}=a(X)Q^{\prime}$ for some $a(X)\in\mathcal{O}_{\mathcal{X}}.$ To
prove that $P^{\prime}$ generates $H^{1}_{\mathbf{D}^{\prime}}({}^{\rm
c}V_{\mathcal{X}}^{\prime})$ for sufficiently small $\mathcal{X}$, we only
need to check that $a(0)\neq 0.$ Suppose on the contrary that $a$ is divisible
by $X$ and put $a(X)=Xb(X)$. Then by definitions, we have

${}^{c}\mathbf{z}_{0}(\mathcal{X},\xi)-bP^{\prime}\in{H}^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime})[X]=\\{0\\}\,$

where the vanishing is because the $\mathcal{O}_{\mathcal{X}}$-module
$\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})={H}^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime})$ is torsion-free
(cf. Lemma 5.18). This in turn shows that
${}^{c}\mathbf{z}_{0}(\mathcal{X},\xi)\in{H}^{1}_{{}^{\rm
c}\mathbf{D}^{\prime}}({}^{\rm c}V_{\mathcal{X}}^{\prime})$ and hence

$\textup{res}_{p}({}^{c}\mathbf{z}_{0}(x_{0},\xi))\in\ker\left(H^{1}(\mathbb{Q}_{p},{}^{\rm
c}V_{x_{0}}^{\prime})\longrightarrow
H^{1}(\mathbb{Q}_{p},\,^{c}\mathbb{D}^{\prime}_{x_{0}})\right)={\rm
im}\left(H^{1}(\mathbb{Q}_{p},{}^{\rm
c}\mathbf{D}^{\prime}_{x_{0}})\longrightarrow H^{1}(\mathbb{Q}_{p},{}^{\rm
c}V_{x_{0}}^{\prime})\right)\stackrel{{\scriptstyle\rm Lemma~{}\ref{lemma:
coboundary map}}}{{=}}\\{0\\},$

contradicting the hypothesis (PR)).

∎

####  5.3.3. $\mathcal{O}_{\mathcal{X}}$-adic height pairing

#####

Let $Q\in\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})$ be any element. For each $x\in\mathcal{X}^{\rm
cl}(E)$ (resp. $w\in\mathcal{W}^{\rm cl}(E)$), we let
$Q_{x}\in\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm c}V_{x},{}^{\rm
c}\mathbf{D}_{x})$ (resp. $Q_{w}\in\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{w},{}^{\rm c}\mathbf{D}_{w})$) denote the image of $Q$ under the
appropriate morphism. We similarly define, for a given
$Q^{\prime}\in\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm c}\mathbf{D}_{\mathcal{X}}^{\prime})$,
$Q^{\prime}_{?}$ for $?=y,w$.

Following our notation in Section 2.2.1, let us set

$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})^{\circ}:=\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\otimes_{\mathcal{O}_{\mathcal{W}}}\mathcal{O}_{\mathcal{X}},$
$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})^{\circ}:=\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\otimes_{\mathcal{O}_{\mathcal{W}}}\mathcal{O}_{\mathcal{X}}.$

The $\mathcal{O}_{\mathcal{W}}$-bilinear pairing ${}^{\rm
c}V_{\mathcal{X}}^{\prime}\otimes_{\mathcal{O}_{\mathcal{W}}}{}^{\rm
c}V_{\mathcal{X}}\rightarrow\mathcal{O}_{\mathcal{W}}$ induces the cyclotomic
$p$-adic height pairing

$\langle\,,\,\rangle_{{}^{\rm c}\mathbf{D}^{\prime},{}^{\rm
c}\mathbf{D}}\,:\,\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\,\otimes_{\mathcal{O}_{\mathcal{W}}}\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\longrightarrow\mathcal{O}_{\mathcal{W}}.$

We extend this pairing by linearity to a $\mathcal{O}_{\mathcal{X}}$-bilinear
map

$\langle\,,\,\rangle_{\mathbf{D}}\,:\,\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})^{\circ}\,\otimes\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})^{\circ}\longrightarrow\mathcal{O}_{\mathcal{X}}\,,$
$\left\langle\sum_{i}a_{i}^{\prime}\otimes z_{i}^{\prime},\sum_{j}a_{j}\otimes
z_{j}\right\rangle_{\mathbf{D}}:=\sum_{i,j}a_{i}^{\prime}a_{j}\cdot\langle
z_{i}^{\prime},z_{j}\rangle_{{}^{\rm c}\mathbf{D}^{\prime},{}^{\rm
c}\mathbf{D}}.$

######  Definition 5.23.

We define the pairing

$\langle\,,\,\rangle_{\mathcal{X}}\,:\,\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}}^{\prime},{}^{\rm
c}\mathbf{D}_{\mathcal{X}}^{\prime})\,\otimes\mathbf{R}^{1}\boldsymbol{\Gamma}({}^{\rm
c}V_{\mathcal{X}},{}^{\rm
c}\mathbf{D}_{\mathcal{X}})\longrightarrow\mathcal{O}_{\mathcal{X}}$

on setting

$\left\langle Q^{\prime},Q\right\rangle_{\mathcal{X}}:=\frac{1}{2}\langle
Q^{\prime},X\otimes Q+1\otimes XQ\rangle_{\mathbf{D}}$

and call it the $\mathcal{O}_{\mathcal{X}}$-adic height pairing.

We remark that

$\left\langle Q^{\prime},Q\right\rangle_{\mathcal{X}}:=\frac{1}{2}\langle
X\otimes Q^{\prime}+1\otimes XQ^{\prime},Q\rangle_{\mathbf{D}}.$

thank to the property (Adj)). Moreover, for any
$x\in\mathcal{X}^{\mathrm{cl}}(E)\setminus\\{x_{0}\\}$ we have

$X\otimes Q^{\prime}+1\otimes XQ^{\prime}=1\otimes
2X(x)Q^{\prime}+(X-X(x))\otimes Q^{\prime}+1\otimes(X-X(x))Q^{\prime}.$

Thence, recalling from (2.18) the specialization morphism ${\rm sp}_{x}$, we
have

$\mathrm{sp}_{x}(X\otimes Q^{\prime}+1\otimes
XQ^{\prime})=2X(x)Q^{\prime}_{x}.$