(3.41) ${\rm rank}\,H^{1}_{\star}(V_{f^{*}}/T_{f^{*}}({k}/{2}))^{\vee}={\rm
rank}\,H^{1}_{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))^{\vee}+1\,.$

which in turn shows, using [MR04, Lemma 4.1.1], that

(3.42)
$\dim\,H^{1}_{\star}(V_{f^{*}}({k}/{2}))=\dim\,H^{1}_{0}(V_{f^{*}}({k}/{2}))+1\,.$

On passing to Pontryagin duals in (3.40) and using local Tate duality, we
obtain the exact sequence

(3.43) $0\longrightarrow
H^{1}_{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))\longrightarrow
H^{1}_{\star}(V_{f^{*}}/T_{f^{*}}({k}/{2}))\longrightarrow
H^{1}_{\star}(\mathbb{Q}_{p},V_{f^{*}}/T_{f^{*}}({k}/{2}))\longrightarrow 0\,$

and in turn the commutative diagram

(3.44)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{0}(V_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\star}(V_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\star}(\mathbb{Q}_{p},V_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\star}(V_{f^{*}}/T_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\star}(\mathbb{Q}_{p},V_{f^{*}}/T_{f^{*}}({k}/{2}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the surjection on the first row at the very right is a consequence of
(3.42) together with the fact that $\dim
H^{1}_{\star}(\mathbb{Q}_{p},V_{f^{*}}({k}/{2}))=1$, and the vertical
surjection on the very right follows from the definitions.

Using the snake lemma with the diagram (3.44), we deduce that

(3.45)
$\\#{}_{\star}(T_{f^{*}}(k/2))\,=\,\\#{}_{0}(T_{f^{*}}(k/2))\cdot\left[H^{1}_{\star}(\mathbb{Q}_{p},T_{f^{*}}({k}/{2}))_{\rm
tf}:\frac{H^{1}_{\star}(T_{f^{*}}({k}/{2}))_{\rm
tf}}{H^{1}_{0}(T_{f^{*}}({k}/{2}))_{\rm tf}}\right]^{-1}\,.$

######  Remark 3.26.

In view of our analysis in Propositions 3.19 and 3.23, we expect that

  * •

$H^{1}_{\beta}(V_{f}(\frac{k}{2}))=H^{1}_{\\{p\\}}(V_{f}(\frac{k}{2}))$ if and
only if $L(f^{*},\frac{k}{2})=0$ ,

  * •

$H^{1}_{\alpha}(V_{f}(\frac{k}{2}))\neq H^{1}_{\\{p\\}}(V_{f}(\frac{k}{2}))$
in all cases.

###  3.3. Modules of algebraic $p$-adic $L$-functions

####  3.3.1. Selmer complexes for modular forms

In this subsection, we define Perrin-Riou’s modules of algebraic $p$-adic
$L$-functions associated to the Hecke eigenvalues $\alpha$ and $\beta$, using
the formalism of Selmer complexes as in [Ben14]. The reader will notice that
the general constructions could be greatly simplified and made more explicit
for the slope-$0$ eigenvalue $\beta$. However, for our further purposes, it is
important to present a coherent picture allowing the unified treatment of both
cases, working with abstract objects with finer functorial properties.

#####

Let us fix an $\mathcal{O}_{E}$-lattice $T_{f}$ of $V_{f}$ and for
$\star\in\\{\alpha,\beta\\}$, let us set
$D^{(\star)}[j]:={\mathbf{D}}_{\mathrm{cris}}(V_{f}^{(\star)}(j))$ and denote
by $N_{\star}[j]\subset D^{(\star)}[j]$ the $\mathcal{O}_{E}$-lattice
generated by $\eta_{f}^{\star}[j].$ We consider the following local conditions
$\left(\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\star)\right)_{\ell\in
S}$ on the level of complexes:

  * •

For all $\ell\in S\setminus\\{p\\}$, we shall take the unramified local
condition (cf. (1.33)):

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\star):=\left[(V_{f}(j)\otimes_{\mathbb{Z}_{p}}\Lambda)^{I_{\ell}}\xrightarrow{\mathrm{Fr}_{\ell}-1}(V_{f}(j)\otimes_{\mathbb{Z}_{p}}\Lambda)^{I_{\ell}}\right]\,.$

Recall that

$\mathbf{R}^{i}\mathbf{\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\star)=\begin{cases}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j))&\text{if
$i\neq 1,$}\\\ 0&\text{if $i\neq 1.$}\end{cases}$

  * •

The local condition at $p$ is given by the complex

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j),\star):=D^{(\star)}[j]\otimes_{\mathcal{O}_{E}}\Lambda[-1]$

concentrated in degree $1$, together with the morphism (1.22)

$\mathbf{Exp}_{\star,j}\,:\,D^{(\star)}[j]\otimes\Lambda\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\,.$

Here, we have used the quasi-isomorphism

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\simeq\left[\mathbf{D}^{\dagger}(V_{f}(j))\xrightarrow{\psi-1}\mathbf{D}^{\dagger}(V_{f}(j))\right]$

to define the exponential map on the level of complexes.

We denote by $\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\alpha)$
and $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)$ the Selmer complexes
associated to the diagrams

(3.46)
$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j)))}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\alpha)\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j)))}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\beta)\,\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

#####

We remark that, from the point of view of Perrin-Riou’s theory [PR95], the
complex $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)$ is the algebaric
counterpart of the $p$-adic $L$-function
$L_{\mathrm{S},\beta^{*}}(f^{*},\xi^{*}).$ In Section 3.4, we will see that
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\alpha)$ is naturally
an algebraic counterpart of the cyclotomic improvement
$L_{\mathrm{S},\alpha^{*}}^{[0],\mathrm{imp}}(f^{*},\xi^{*})$ of the $p$-adic
$L$-function $L_{\mathrm{S},\alpha^{*}}^{[0]}(f^{*},\xi^{*})$. One may also
introduce “a non-improved version” of this complex on replacing
$\mathbf{Exp}_{\alpha,j}$ by the map

$\mathbf{Exp}_{V_{f}^{(\alpha)}(j),j}\,:\,D^{(\alpha)}{[j]}\otimes\Lambda\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)$

and considering the Selmer complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ of
$\mathscr{H}(\Gamma)$-modules associated to the diagram

$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}}}(V_{f}(j))\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\bigl{(}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j)))\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)\bigr{)}}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\bigl{(}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},V_{f}(j),\alpha)\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)\bigr{)}\,\,,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in which the component at $p$ of the vertical map is
$\mathbf{Exp}_{V_{f}^{(\alpha)}(j),j}.$ Recall that the maps
$\mathbf{Exp}_{V_{f}^{(\alpha)}(j),j}$ and $\mathbf{Exp}_{\alpha,j}$ are
related by the formula

(3.47)
$\mathbf{Exp}_{V_{f}^{(\alpha)}(j),j}=\ell(V_{f}(j))\cdot\mathbf{Exp}_{\alpha,j}\,,\qquad\qquad\ell(V_{f}(j))=\underset{i=1-k+j}{\overset{j-1}{\prod}}\ell_{i}.$

#####

For $\star\in\\{\alpha,\beta\\}$, set
$\Omega_{p}(T_{f}^{(\star)},N_{\star})=|b|,$ where $b\in E^{*}$ is such that
${\mathbf{D}}_{\mathrm{cris}}(T_{f}^{(\star)})={b}N_{\star}$ (cf. Section
1.1.7). If we choose $T_{f}$ such that
$\Omega_{p}(T_{f}^{(\star)},N_{\star})\in\mathcal{O}_{E}^{*}$, then thanks to
(1.22), we have an integral version of the diagrams (3.46):

$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}}}(T_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j)))}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\alpha)\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}}}(T_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j)))}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\beta)\,\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Whenever $\Omega_{p}(T_{f}^{(\star)},N_{\star})\in\mathcal{O}_{E}^{*}$, we let
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)$ and
$\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f}(j),\beta)$ denote the Selmer complexes
associated to these diagrams.

#####

The basic properties of these complexes are summarized in the following:

######  Proposition 3.27.

The $\Lambda$-module

${}^{2}_{\mathrm{Iw}}(T_{f}(j)):=\ker\left(H^{2}_{\mathrm{Iw},S}(T_{f}(j))\longrightarrow\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\right)$

is finitely generated and $\Lambda$-torsion.

Assume that the condition (S)) holds true for some integer $j=j_{0}$, and let
$\star\in\\{\alpha,\beta\\}$. Then the following statements are valid:

For all $j\in\mathbb{Z},$ the map

$f_{\star}\,:\,H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}^{(\star)}(j))\xrightarrow{\textup{res}_{p}\,+\,\iota_{\star}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))$

is injective. Here, $\iota_{\star}$ is the morphism induced from the natural
injection $T_{f}^{(\star)}\to T_{f}$.

$\mathbf{R}^{i}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(j),\alpha)=\\{0\\}=\mathbf{R}^{i}\mathbf{\Gamma}_{{\mathrm{Iw}}}(T_{f}(j),\beta)$
if $i\neq 2,3.$

The $\Lambda$-modules
$\mathbf{R}^{2}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(j),\alpha)$
and $\mathbf{R}^{2}\mathbf{\Gamma}_{{\mathrm{Iw}}}(T_{f}(j),\beta)$ sit in the
exact sequences

$\displaystyle
0\longrightarrow\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))}{H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)}\longrightarrow\mathbf{R}^{2}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(j),\alpha)\longrightarrow{}^{2}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow
0\,,$ $\displaystyle
0\longrightarrow\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))}{H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus\mathrm{Exp}_{\beta,j}(N_{\beta}[j]\otimes\Lambda)}\longrightarrow\mathbf{R}^{2}\mathbf{\Gamma}_{{\mathrm{Iw}}}(T_{f}(j),\beta)\longrightarrow{}^{2}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow
0\,.$

$\mathbf{R}^{3}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(j),\alpha)\simeq
H^{0}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}/T_{f^{*}}(k-j))^{\wedge}\simeq\mathbf{R}^{3}\mathbf{\Gamma}_{{\mathrm{Iw}}}(T_{f}(j),\beta)\,.$

The analogous properties hold for the complexes
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\alpha)$ and
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta).$ In addition,

$\mathbf{R}^{3}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(V_{f}(j),\alpha)=\\{0\\}=\mathbf{R}^{3}\mathbf{\Gamma}_{{\mathrm{Iw}}}(V_{f}(j),\beta)\,.$

###### Proof.

This statement is proved in [Kat04, Theorem 12.4].

For each $\star\in\\{\alpha,\beta\\}$, we have

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

inside $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))$. Indeed, this claim
holds unconditionally if $\star=\beta$, and when $\star=\alpha$, it follows
from condition (S)) (cf. Section 3.2.3). By [Kat04, Theorem 12.4],
$H^{1}_{\mathrm{Iw}}(T_{f}(j))$ is a torsion-free $\Lambda$-module. Since
$H^{1}(\mathbb{Q}_{p},T_{f}^{(\star)}(j))$ is a free $\Lambda$-module of rank
one, we deduce that

$\textup{res}_{p}\left(H^{1}_{\mathrm{Iw}}(T_{f}(j))\right)\cap
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}^{(\star)}(j))=\\{0\\}.$

This is a restatement of (ii).

These portions follow easily from earlier remarks; cf. [Ben14, Theorem 4] for
similar arguments. For the reader’s convenience, we reproduce these arguments
of op. cit. below.

It follows from the definition of a Selmer complex that we have an exact
triangle

$\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))[-1]\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f}(j))\oplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\alpha)\right).$

The associated long exact sequence of cohomology reads

(3.48) $\begin{aligned} \ldots\longrightarrow\underset{\ell\in
S}{\oplus}H^{i-1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\longrightarrow\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\longrightarrow
H^{i}_{\mathrm{Iw}}(T_{f}(j))\oplus\underset{\ell\in
S}{\bigoplus}\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\alpha)\\\
\longrightarrow\underset{\ell\in
S}{\oplus}H^{i}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\longrightarrow\ldots\end{aligned}$

Thanks to this exact sequence, it is clear that
$\mathbf{R}^{0}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)=0$.
Moreover,

$\mathbf{R}^{1}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)=\ker\left(H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus(N_{\alpha}[j]\otimes\Lambda)\rightarrow\underset{\ell\in
S}{\oplus}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\right)\subset\ker(f_{\alpha})=0$

by Part (ii) of our proposition.

Note that for all $\ell\neq p$, we have

$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))=H^{1}(\mathbb{Q}_{\ell}(\zeta_{p^{\infty}})/\mathbb{Q}_{\ell},(\Lambda\otimes_{\mathbb{Z}_{p}}T_{f}(j)^{I_{\ell}})^{\iota})=:\mathbf{R}^{1}\boldsymbol{\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\alpha),$

where $I_{\ell}$ denotes the inertia group at $\ell$ (cf. [PR92], Section
2.2.4). Using the sequence (3.48) with $i=2$, we obtain the exact sequence

$H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus(N_{\alpha}[j]\otimes\Lambda)\longrightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\longrightarrow
H^{2}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\,.$

We therefore conclude that the induced sequence

$0\longrightarrow\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))}{H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)}\longrightarrow\mathbf{R}^{2}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(j),\alpha)\longrightarrow{}^{2}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow
0$

is exact. Finally, using (3.48) with $i=3$, we have

$H^{2}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow\underset{\ell\in
S}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\longrightarrow\mathbf{R}^{3}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\longrightarrow
0\,.$

On comparing this exact sequence with the Poitou–Tate exact sequence, we
deduce that

$\mathbf{R}^{3}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\simeq
H^{0}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}/T_{f^{*}}(k-j))^{\wedge}.$

The proof of these facts for other complexes is completely analogous. We also
note that

$\mathbf{R}^{3}\boldsymbol{\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\star)=H^{0}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}(k-j))^{*}=0$

because $V_{f^{*}}^{(\alpha^{*})}\simeq E(\nu_{\alpha^{*}}\chi^{1-k})$ and
$V_{f^{*}}^{(\beta^{*})}\simeq E(\nu_{\beta^{*}})$ with non-trivial unramified
characters $\nu_{\alpha^{*}}$ and $\nu_{\beta^{*}}.$ This completes the proof
of our proposition. ∎

####  3.3.2. Projective resolutions and determinants

#####

In this subsection, we recall some basic facts about projective resolutions
over the rings $\mathcal{O}_{E}[[\Gamma_{1}]],$
$\mathcal{O}_{E}[[\Gamma_{1}]][1/p]$, and $\mathscr{H}(\Gamma_{1}),$ which
will be used in the rest of the book.

Let $R$ be an integral domain. Throughout, projective $R$-module always means
a finitely generated projective $R$-module. Recall that an $R$-module $M$ is
perfect if there exists a finite projective resolution $P_{\bullet}\rightarrow
M\rightarrow 0$ of $M$. If $P_{\bullet}=(P_{i})_{i\geqslant 0}$ is finite, its
length is defined as the largest $i$ such that $P_{i}\neq 0$.

Every finitely generated module over a principal ideal domain is perfect and
admits a projective resolution of length $1$. If $R$ is a regular local ring
of dimension $d,$ then, by a classical theorem of Serre (cf. [Mat89], §19)
every finitely generated $R$-module admits a projective resolution of length
$d$. In particular, every finitely generated module over
$\mathcal{O}_{E}[[\Gamma_{1}]]$ admits a projective resolution of length $2$.

#####

An integral domain is a Bézout ring if every finitely generated ideal of $R$
is principal. It follows immediately From this definition that a noetherian
Bézout ring is principal. An archetypal example of a non-principal Bézout ring
is the ring of analytic functions on the open unit disc over a complete non-
archimedean field (cf. [Laz62]). In particular, $\mathscr{H}(\Gamma_{1})$ is a
Bézout ring.

Some results about finitely generated modules over a principal ideal domain
extend, with some modifications, to modules over a Bézout ring. For example,
every finitely generated torsion-free module over a Bézout ring $R$ is free
(cf. [Kap49]). Recall that an $R$-module is finitely presented if it is
isomorphic to a quotient of the form $F/N$, where $F$ and $N$ are finitely
generated and $F$ is free. As a result, we deduce that every finitely
presented module over a Bézout ring admits a free resolution of length $1$. We
need a very slight extension of this fact.

######  Lemma 3.28.

Every finitely presented $\mathscr{H}(\Gamma)$-module admits a projective
resolution of length $1$.

###### Proof.

Recall the canonical decomposition (1.1):

$\mathscr{H}(\Gamma)=\underset{\eta\in
X(\Delta)}{\oplus}\mathscr{H}(\Gamma_{1})_{\eta}\,.$

Note that
$\mathscr{H}(\Gamma_{1})_{\eta}=\mathscr{H}(\Gamma_{1})\delta_{\eta}$ are
isomorphic to $\mathscr{H}(\Gamma_{1})$ as rings. Correspondingly, any
$\mathscr{H}(\Gamma)$-module $M$ decomposes into a direct sum

$M=\underset{\eta\in X(\Delta)}{\oplus}M^{(\eta)},$

where each $M^{(\eta)}$ is an $\mathscr{H}(\Gamma)_{\eta}$-module. Since an
$\mathscr{H}(\Gamma)$-module $P$ is projective if and only if $P^{(\eta)}$ is
$\mathscr{H}(\Gamma)_{\eta}$-projective for all $\eta\in X(\Delta),$ the proof
of our lemma follows from the case of Bézout rings. ∎

######  Remark 3.29.

The same argument proves the well-known fact that any finitely generated
$\Lambda$-module admits a projective resolution of length $2$.

#####

In this subsection, we recall for the convenience of the reader the
trivialization of the determinant of a torsion module over an integral domain,
and refer to [KM76] for the proofs and further details.

Let $R$ be a domain and let $Q(R)$ denote its field of fractions. Let
$C^{\bullet}$ be a perfect complex of $R$-modules. Assume that
$C^{\bullet}\otimes_{R}Q(R)$ is acyclic. Suppose that $P^{\bullet}$ is a
complex of projective modules quasi-isomorphic to the perfect complex
$C^{\bullet}.$ Then $P^{\bullet}\otimes_{R}Q(R)$ is acyclic, and therefore has
a canonical trivialization

${\det}_{Q(R)}^{-1}\left(P^{\bullet}\otimes_{R}Q(R)\right)\xrightarrow{\,\,\sim\,\,}Q(R).$

(We work with ${\det}^{-1}$ rather than with $\det$. This is because
${\det}^{-1}$ is a natural generalization of the characteristic polynomial of
an Iwasawa module.) Composing this isomorphism with the canonical map

${\det}^{-1}_{R}(C^{\bullet})\simeq{\det}^{-1}_{R}(P^{\bullet})\xhookrightarrow{\quad}{\det}^{-1}_{Q(R)}(P^{\bullet}\otimes
Q(R))$

we obtain a trivialization

(3.49)
$\vartheta_{C^{\bullet}}\,:\,{\det}^{-1}_{R}(C^{\bullet})\xhookrightarrow{\quad}Q(R).$

Note that if $P^{\prime\bullet}$ is another complex of projective modules
quasi-isomorphic to $C^{\bullet},$ we then have a diagram

$\textstyle{{\det}^{-1}_{Q(R)}\bigl{(}P^{\bullet}\otimes_{R}Q(R)\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}\simeq}$$\textstyle{{\det}^{-1}_{R}(C^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q(R)\,.}$$\textstyle{{\det}^{-1}_{Q(R)}\bigl{(}P^{\prime\bullet}\otimes_{R}Q(R)\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The triangle on the left of this diagram commutes tautologically, and the
commutativity of the triangle on the right follows easily from the definition
of the trivialization map [KM76, pp. 32–34]. Therefore,
$\vartheta=\vartheta_{C^{\bullet}}$ does not depend on the choice of
$P^{\bullet}$.

#####

We need the following partial generalization of the previous construction to
the case of an arbitrary ring $R$ (is not necessarily an integral domain). Let
$M$ be a torsion $R$-module that admits a projective resolution
$P_{\bullet}=(P_{1}\xrightarrow{\phi}P_{0})$ of length one by finitely
generated free modules of rank $r$. The map
$\det{(\phi)}\,:\,\wedge^{r}P_{1}\rightarrow\wedge^{r}P_{0}$ induces a map

${\det}_{R}^{-1}\left(P_{1}\xrightarrow{\phi}P_{0}\right)={\det}_{R}(P_{1})\otimes{\det}_{R}^{-1}({P}_{0})\xrightarrow{\det(\phi)}{R},$

which we also denote by $\det(\phi).$ The arguments of [KM76] apply verbatim
and show that the composition

(3.50)
$\vartheta_{M}\,:\,{\det}^{-1}_{R}(M)\simeq{\det}^{-1}_{R}(P_{1}\xrightarrow{\phi}P_{0})\xhookrightarrow{\quad}R$

does not depend on the choice of $P_{\bullet}$ . If $R$ is an integral domain,
$\vartheta_{M}$ coincides with the map (3.49) where $M$ is viewed as a complex
concentrated in degree $0.$

####  3.3.3. Modules of algebraic $p$-adic $L$-functions

#####

Let us assume that the hypothesis (S)) holds true for some choice of
$j=j_{0}$. Then by Proposition 3.27, the cohomology of the complexes
$\mathbf{R\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ and
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)$ are $\Lambda_{E}$-torsion,
and we have injective maps

$i_{\mathrm{Iw},V_{f}(j)}^{(\alpha),\mathrm{imp}}\,:\,{\det}_{\Lambda_{E}}^{-1}\mathbf{R\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha)\hookrightarrow\Lambda_{E}\,,\qquad
i_{\mathrm{Iw},V_{f}(j)}^{(\beta)}\,:\,{\det}_{\Lambda_{E}}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\hookrightarrow\Lambda_{E}$

for every integer $j$. The Iwasawa theoretic version of the fundamental line
of $T_{f}(j)$ associated to local conditions
$\left(\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\star)\right)_{\ell\in
S}$ is given as follows, cf. [PR95, Ben14]:

$\Delta_{\mathrm{Iw}}(T_{f}(j),N_{\star}[j]):={\det}^{-1}_{\Lambda}\left(\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f}(j))\bigoplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\star))\right)\right)\otimes{\det}_{\Lambda}\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\right).$

It follows from definitions that we have canonical morphisms

$\displaystyle\Theta_{\mathrm{Iw},V_{f}(j)}^{(\alpha),\mathrm{imp}}\,:\,\Delta_{\mathrm{Iw}}(T_{f}(j),N_{\alpha}[j])\hookrightarrow{\det}_{\Lambda_{E}}^{-1}\mathbf{R\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha),$
$\displaystyle\Theta_{\mathrm{Iw},V_{f}(j)}^{(\beta)}\,:\,\Delta_{\mathrm{Iw}}(T_{f}(j),N_{\beta}[j])\hookrightarrow{\det}_{\Lambda_{E}}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta).$

######  Definition 3.30.

Assume that the hypothesis (S)) holds true for some $j=j_{0}$. Then for any
integer $j$, the free $\Lambda_{\mathcal{O}_{E}}$-modules

$\displaystyle\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j]):=i^{(\alpha),\mathrm{imp}}_{{\mathrm{Iw},V_{f}(j)}}\circ\Theta_{\mathrm{Iw},V_{f}(j)}^{(\alpha),\mathrm{imp}}\bigl{(}\Delta_{\mathrm{Iw}}(T_{f}(j),N_{\alpha}[j])\bigr{)},$
$\displaystyle\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j]):=i_{\mathrm{Iw},V_{f}(j)}^{(\beta)}\circ\Theta_{\mathrm{Iw},V_{f}(j)}^{(\alpha)}\bigl{(}\Delta_{\mathrm{Iw}}T_{f}(j),N_{\beta}[j])\bigr{)}$

of rank one are called the modules of (algebraic) $p$-adic $L$-functions.

We remark that the Iwasawa theoretic properties of
$\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j])$ associated to the
slope-zero eigenvalue $\beta$ are covered by Perrin-Riou’s general theory
developed in [PR95]. In contrast, the module
$\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j])$
associated to the critical eigenvalue $\alpha$ does not fit in the framework
of op. cit. and will be studied in the remainder of this section.

We record the first properties of these modules in the following:

######  Proposition 3.31.

Assume that $\Omega_{p}(T_{f}^{(\star)},N_{\star})\in\mathcal{O}_{E}^{*}$ for
$\star\in\\{\alpha,\beta\\}.$ The following statements hold true for all
$m\in\mathbb{Z}$, under the validity of (S)) for some integer $j=j_{0}$:

One has

$\displaystyle\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(m),N_{\alpha}[m])=\mathrm{char}_{\Lambda}\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(m))}{H^{1}_{\mathrm{Iw}}(T_{f}(m))+\mathrm{Exp}_{\alpha,m}(N_{\alpha}[j]\otimes\Lambda)}\right)\cdot\mathrm{char}_{\Lambda}({}^{2}_{\mathrm{Iw}}(T_{f}(m)))\,,$
$\displaystyle\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(m),N_{\beta}[m])=\mathrm{char}_{\Lambda}\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(m))}{H^{1}_{\mathrm{Iw}}(T_{f}(m))+\mathrm{Exp}_{\beta,m}(N_{\beta}[j]\otimes\Lambda)}\right)\cdot\mathrm{char}_{\Lambda}({}^{2}_{\mathrm{Iw}}(T_{f}(m)))\,.$

$\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(m),N_{\alpha}[m])=\mathscr{L}_{\mathrm{Iw}}^{\rm
cr}(V_{f}(m))\cdot\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(m),N_{\beta}[m]).$

###### Proof.

This follows immediately from Proposition 3.27. Note that
$H^{0}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}/T_{f^{*}}(k-m))$ is finite and
hence

$\mathrm{char}_{\Lambda}\left(\mathbf{R}^{3}\mathbf{\Gamma}^{\mathrm{imp}}_{{\mathrm{Iw}}}(T_{f}(m),\alpha)\right)=\Lambda=\mathrm{char}_{\Lambda}\left(\mathbf{R}^{3}\mathbf{\Gamma}_{{\mathrm{Iw}}}(T_{f}(m),\beta)\right).$

It follows from the definition of the Iwasawa theoretic critical
$\mathcal{L}$-invariant and the elementary properties of determinants that

$\mathrm{char}_{\Lambda}\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(m))}{H^{1}_{\mathrm{Iw}}(T_{f}(m))+\mathrm{Exp}_{\alpha,j}(N_{\alpha}[m]\otimes\Lambda)}\right)=\mathscr{L}_{\mathrm{Iw}}^{\mathrm{cr}}(V_{f}(m))\cdot\mathrm{char}_{\Lambda}\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(m))}{H^{1}_{\mathrm{Iw}}(T_{f}(m))+\mathrm{Exp}_{\beta,m}(N_{\beta}[m]\otimes\Lambda)}\right).$

This combined with Part (i) proves the asserted equality. ∎

###  3.4. Main Conjecture for $\theta$-critical forms

#### 3.4.1.

In §3.4, we will always assume that $T_{f}$ is chosen so that

$\Omega_{p}(T_{f}^{(\star)},N_{\star})\in\mathcal{O}_{E}^{*},\qquad\star\in\\{\alpha,\beta\\}\,.$

The slope-zero ($p$-ordinary) Iwasawa Main Conjecture in the formulation of
Perrin-Riou reads:

######  Conjecture 3.32 ($p$-ordinary Iwasawa Main Conjecture).

  * $\mathbf{MC}(f,\beta)$

$\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(k),N_{\beta}[k])=(L_{\mathrm{S},\beta^{*}}(f^{*},\xi^{*})^{\iota})\,.$

Proposition 3.31 allows us to translate $\mathbf{MC}(f,\beta)$ to the
classical version of the slope-zero Iwasawa main conjecture. Furthermore,
under the validity of (SZ1)) or (SZ1)), the conjecture $\mathbf{MC}(f,\beta)$
holds true; cf. [SU14, Wan15], [Ski16, Theorem 2.5.2], [Kat04, Theorem 15.2].

Note that $\mathbf{MC}(f,\beta)$ is invariant under cyclotomic twists, namely,
it is equivalent to the assertion that

$i_{\mathrm{Iw},T_{f}(k-j)}^{(\beta)}\circ\Theta_{\mathrm{Iw},V_{f}(k-j)}^{(\beta)}\left(\Delta_{\mathrm{Iw}}(T_{f}(k-j),\beta)\right)=(\mathrm{Tw}_{j}\circ
L_{\mathrm{S},\beta^{*}}(f^{*},\xi^{*})^{\iota}),\qquad\forall
j\in\mathbb{Z}\,.$

(We recall that $\mathrm{Tw}_{j}\circ
F(\gamma-1)^{\iota}:=F(\chi^{j}(\gamma)\gamma-1)^{\iota}$.)

We next formulate the critical Iwasawa Main Conjecture in the style of Perrin-
Riou:

######  Conjecture 3.33 (Critical Iwasawa Main Conjecture).

We have $\mathscr{L}^{\mathrm{cr}}_{\mathrm{Iw}}(V_{f})\neq 0$ and

  * $\mathbf{MC}(f,\alpha)$

$\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(k),N_{\alpha}[k])^{\pm}=\left(\lambda(f^{*})^{\pm}L^{\mathrm{imp},\pm}_{\mathrm{S},\alpha^{*}}(f^{*},\xi^{*})^{\iota}\right)\,,$

where $\lambda(f^{*})^{\pm}$ is the constant defined in Corollary 2.25.

We may of course rewrite $\mathbf{MC}(f,\alpha)$ in the following equivalent
form:
$\mathbf{MC}(f,\alpha)$
$\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(k),N_{\alpha}[k])=\left(L^{\mathrm{imp}}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})^{\iota}\right),$
where $L^{\mathrm{imp}}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})$ is defined in
Theorem 2.23. It is in its original formulation where the $p$-adic
$L$-function $L^{[0],\mathrm{imp}}_{\mathrm{S},\alpha^{*}}(f^{*},\xi^{*})$,
which is constructed in a purely analytic manner, makes an appearance.

Moreover, Proposition 3.31, allows to rewrite $\mathbf{MC}(f,\alpha)$ in the
following explicit form:

${\rm
char}_{\Lambda}\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(k))}{H^{1}_{\mathrm{Iw}}(T_{f}(k))+\mathrm{Exp}_{\alpha,k}(N_{\alpha}[k]\otimes\Lambda)}\right)^{\pm}\cdot\mathrm{char}_{\Lambda}({}^{2}_{\mathrm{Iw}}(T_{f}(k)))^{\pm}={\left(\lambda(f^{*})^{\pm}L^{[0],\mathrm{imp},\pm}_{\mathrm{S},\alpha^{*}}(f^{*},\xi^{*})\right).}$

We may link the main conjectures $\mathbf{MC}(f,\alpha)$ and
$\mathbf{MC}(f,\beta)$ with one another:

######  Proposition 3.34.

Assume that $\mathscr{L}_{\mathrm{Iw}}^{\rm cr}(V_{f})\neq 0$. Then the
conjectures $\mathbf{MC}(f,\alpha)$ and $\mathbf{MC}(f,\beta)$ are equivalent.

###### Proof.

This follows from Propositions 2.27 and 3.31. ∎

We remind the reader that the non-vanishing $\mathscr{L}_{\mathrm{Iw}}^{\rm
cr}(V_{f})\neq 0$ of the Iwasawa theoretic critical $\mathcal{L}$-invariant is
equivalent to the requirement that the conditions ($p$Bei)) and (S)) holds
true for some integer $j=j_{0}$. See §3.2.2 where we have an extensive
discussion on the validity of (S)) in the scenario when the eigenform $f$ has
CM.

######  Remark 3.35.

Proposition 3.34 tells us that the Iwasawa theory of the critical
stabilization of $f^{*}$ is equivalent to that of its ordinary stabilization.
In Chapter 4, we shall formulate a refined version of $\mathbf{MC}(f,\alpha)$
that relates the infinitesimal thickening
$\widetilde{L}_{\mathrm{S},\alpha^{*}}(f^{*},\xi^{*})$ of the critical
$p$-adic $L$-function to the Iwasawa theoretic invariants of the infinitesimal
deformation of $V_{f}$ along the eigencurve.

#### 3.4.2.

We may formulate the critical Main Conjecture in an equivalent form which
involves the non-improved $p$-adic $L$-function
$L_{\mathrm{S},\alpha^{*}}^{{[0]}}(f^{*},\xi^{*})$ and the non-improved Selmer
complex $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(k),\alpha).$ Namely, it follows
from the identity (3.47) (where also the factor $\ell(V_{f}(j))$ is defined)
that we have a commutative diagram

(3.51)
$\textstyle{\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j),\alpha)\otimes_{\Lambda}\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Exp}_{V_{f}(j),j}}$$\scriptstyle{{\,}\cdot\,\ell(V_{f}(j))}$$\textstyle{\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\textstyle{\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j),\alpha)\otimes_{\Lambda}\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Exp}_{\alpha,j}\otimes\mathrm{id}}$$\textstyle{\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\,.}$

for the local conditions at $p$.

The diagram (3.51) induces a canonical morphism

(3.52)
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(T_{f}(j),\alpha)\otimes_{\Lambda}\mathscr{H}(\Gamma)\,.$

######  Proposition 3.36.

Assume that $\mathscr{L}_{\mathrm{Iw}}(V_{f})\neq 0$. Then the following
statements hold true:

i) We have

(3.53)
$\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)=0\quad\textrm{
if $i\neq 2.$}$

ii) The $\mathscr{H}(\Gamma)$-module
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ is a
torsion module. Moreover, it is perfect and admits a projective resolution of
length $1$.

###### Proof.

Assume that $\mathscr{L}_{\mathrm{Iw}}(V_{f})\neq 0$. Mimicking the proof of
Proposition 3.27, we deduce that
$\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)=0$ if $i\neq
2$ and that $\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$
is $\mathscr{H}(\Gamma)$-torsion. Let us consider the exact sequence

$0\longrightarrow\displaystyle\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))\otimes\mathscr{H}}{H^{1}_{\mathrm{Iw}}(V_{f}(j))\otimes\mathscr{H}+\mathrm{Exp}_{V_{f}(j),j}(N_{\alpha}[j]\otimes\mathscr{H})}\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\longrightarrow{}^{2}_{\mathrm{Iw}}(V_{f}(j))\otimes\mathscr{H}\longrightarrow
0,$

where we write $\otimes\mathscr{H}$ in place of
$\otimes_{\Lambda_{E}}\mathscr{H}(\Gamma)$ to save space.

For each $\eta\in X(\Delta)$, the $\eta$-isotypic components of
$H^{1}_{\mathrm{Iw}}(V_{f}(j))$,
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))$ and
${}^{2}_{\mathrm{Iw}}(V_{f}(j))$ are finitely generated modules over the
principal ideal domain $\mathcal{O}_{E}[[\Gamma_{1}]][1/p].$ Since
$N_{\alpha}[j]\otimes\mathscr{H}(\Gamma)$ is free of rank one, the
$\eta$-components of the left and right terms of the exact sequence above are
finitely presented and therefore admit projective resolutions of length 1. The
behaviour of the projective dimension in short exact sequences (cf. [Sta23,
Lemma 10.109.9(ii)]) tells us that the projective dimension of
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)^{(\eta)}$ is
bounded by $1$ for each $\eta$. The proposition now follows from Lemma 3.28. ∎

From Proposition 3.36, we obtain a canonical trivialization

$i_{\mathrm{Iw},V_{f}(j)}^{(\alpha)}\,:\,{\det}_{\mathscr{H}(\Gamma)}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\longrightarrow\mathscr{H}(\Gamma)$

and a canonical map

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

######  Lemma 3.37.

Set

$\mathbf{L}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j]):=i_{\mathrm{Iw},V_{f}(j)}^{(\alpha)}\circ\Theta_{\mathrm{Iw},V_{f}(j)}^{(\alpha)}\bigl{(}\Delta_{\mathrm{Iw}}(T_{f}(j),N_{\alpha}[j])\bigr{)}.$

Then

$\mathbf{L}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j])=\ell(V_{f}(j))\cdot\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(j),N_{\alpha}[j]).$

###### Proof.

Consider the following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))\otimes\mathscr{H}}{H^{1}_{\mathrm{Iw}}(T_{f}(j))\otimes\mathscr{H}+\mathrm{Exp}_{V_{f}(j),j}(N_{\alpha}[j]\otimes\mathscr{H})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{2}_{\mathrm{Iw}}(T_{f}(j))\otimes\mathscr{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\left(\frac{H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))}{H^{1}_{\mathrm{Iw}}(T_{f}(j))+\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)}\right)\otimes\mathscr{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha)\otimes\mathscr{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{2}_{\mathrm{Iw}}(T_{f}(j))\otimes\mathscr{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The second row of this diagram is the exact sequence computing the second
cohomology of $\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\alpha)$
(cf. Proposition 3.27) tensored with $\mathscr{H}(\Gamma)$. Here, we have
written $\otimes\mathscr{H}$ in place of
$\otimes_{\Lambda}\mathscr{H}(\Gamma)$ to save space. The first row is the
analogous exact sequence for
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha),$ and the vertical maps are
induced by the morphism (3.52). Since

$\mathrm{Exp}_{V_{f}(j),j}(N_{\alpha}[j]\otimes_{E}\mathscr{H}(\Gamma))=\ell(V_{f}(j))\cdot\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes_{E}\mathscr{H}(\Gamma)),$

the proof of the lemma follows by taking the determinants in the diagram
above. ∎

Therefore, the conjecture $\mathbf{MC}(f,\alpha)$ can be written in the
following equivalent form:

(3.54)
$\mathbf{L}_{\mathrm{Iw},\alpha}(T_{f}(k),N_{\alpha}[k])^{\pm}=\left(\lambda(f^{*})^{\pm}L_{\mathrm{S},\alpha^{*}}^{[0],\pm}(f^{*},\xi^{*})^{\iota}\right)=\left(L_{\mathrm{K},\alpha^{*}}^{\pm}(f^{*},\xi^{*})^{\iota}\right),$

where $L_{\mathrm{K},\alpha^{*}}^{\pm}(f^{*},\xi^{*})$ is defined in Theorem
2.23.

###  3.5. Iwasawa theoretic descent

In this final section of the present chapter, we develop Iwasawa descent for
the modules of algebraic $p$-adic $L$-functions (cf. Definition 3.30), which
we utilize to prove the leading term formulae for these. Our main results in
this regard are Theorem 3.45, which applies to both non-critical and non-
central critical values, and Theorem 3.47, which focuses on central critical
values. We would like to emphasize also Theorem 3.44, which highlights a
significant difference between the modules of $p$-adic $L$-functions
associated with the $\theta$-critical $p$-stabilization and that attached to
the slope-zero $p$-stabilization. This disparity becomes apparent, for
example, in their orders of vanishing at the central critical point (cf.
Theorem 3.44(iv)). Considering the Iwasawa main conjectures
$\mathbf{MC}(f,\alpha)$ and $\mathbf{MC}(f,\beta)$ in conjunction with
Proposition 2.27, this difference can be accounted by a pole of our Iwasawa
theoretic $\mathcal{L}$-invariant at the central critical point.

####  3.5.1. General formalism

#####

We recall the formalism of Iwasawa descent. To simplify notation, let us set
$z=\gamma_{1}-1$. Let $\mathscr{K}(\Gamma_{1})$ denote the field of fractions
of $\mathscr{H}(\Gamma_{1})$.

Let $C^{\bullet}$ be a perfect complex of $\mathscr{H}(\Gamma_{1})$-modules.
Assume that
$C^{\bullet}\otimes_{\mathscr{H}(\Gamma_{1})}\mathscr{K}(\Gamma_{1})$ is
acyclic and let us write

$\vartheta_{\mathrm{Iw}}\,\,:\,\,{\det}^{-1}_{\mathscr{H}(\Gamma_{1})}C^{\bullet}\longrightarrow\mathscr{K}(\Gamma_{1})$

for the associated trivialization map (3.49). We then have
$\vartheta_{\mathrm{Iw}}({\det}^{-1}_{\mathscr{H}(\Gamma_{1})}C^{\bullet})=f\mathscr{H}(\Gamma_{1}),$
for some $f\in\mathscr{K}(\Gamma_{1})$. Let $r$ be the unique integer such
that $z^{-r}f\in\mathscr{H}_{0}(\Gamma_{1})^{\times}$ is a unit in the
localization $\mathscr{H}_{0}(\Gamma_{1})$ of $\mathscr{H}(\Gamma_{1})$ with
respect to the principal ideal $z\mathscr{H}(\Gamma_{1})$.

Let us define
$C^{\bullet}_{0}:=C^{\bullet}\otimes^{\mathbf{L}}_{\mathscr{H}(\Gamma_{1})}E$,
where the tensor product is with respect to the augmentation map
$\mathscr{H}(\Gamma_{1})\to E$. We have a natural distinguished triangle

$C^{\bullet}\xrightarrow{[z]}C^{\bullet}\longrightarrow
C^{\bullet}_{0}\longrightarrow C^{\bullet}[1]\,.$

In degree $n$, this triangle gives rise to a short exact sequence

(3.55) $0\longrightarrow H^{n}(C^{\bullet})_{\Gamma_{1}}\longrightarrow
H^{n}(C^{\bullet}_{0})\longrightarrow
H^{n+1}(C^{\bullet})^{\Gamma_{1}}\longrightarrow 0.$

We say that $C^{\bullet}$ is semi-simple (at the augmentation ideal) if the
composition of the natural map

(3.56) $\mathrm{Bock}_{n}\,:\,H^{n}(C^{\bullet})^{\Gamma_{1}}\longrightarrow
H^{n}(C^{\bullet})\longrightarrow H^{n}(C^{\bullet})_{\Gamma_{1}}$

is an isomorphism for all $n$. If $C^{\bullet}$ is semi-simple, then there
exists a canonical trivialisation of ${\det}_{E}C^{\bullet}_{0}$ given by

(3.57) $\displaystyle\begin{aligned}
\vartheta_{0}\,\,:\,\,{\det}_{E}^{{-1}}C^{\bullet}_{0}&=\underset{n\in\mathbb{Z}}{\otimes}{\det}_{E}^{(-1)^{{n-1}}}H^{n}(C_{0}^{\bullet})\\\
&\xrightarrow[\eqref{eqn_2022_05_06_1540}]{\sim}\underset{n\in\mathbb{Z}}{\otimes}\left({\det}^{(-1)^{{n-1}}}_{E}H^{n}(C^{\bullet})_{\Gamma_{1}}\otimes{\det}^{(-1)^{{n-1}}}_{E}H^{n+1}(C^{\bullet})^{\Gamma_{1}}\right)\\\
&\xrightarrow{\,\,\,\sim\,\,\,}\underset{n\in\mathbb{Z}}{\otimes}\left({\det}^{(-1)^{{n-1}}}_{E}H^{n}(C^{\bullet})_{\Gamma_{1}}\otimes{\det}^{(-1)^{{n}}}_{E}H^{n}(C^{\bullet})^{\Gamma_{1}}\right)\xrightarrow[\eqref{eqn:
semisimplicity projection}]{\sim}E\,.\end{aligned}$

Note that the complex $C^{\bullet}_{0}$ is not necessarily acyclic and that
$\vartheta_{0}$ differs from the trivialization (3.49) in general.

######  Lemma 3.38.

Let $C^{\bullet}$ be a perfect complex of $\mathscr{H}(\Gamma_{1})$-modules.
Assume that
$C^{\bullet}\otimes_{\mathscr{H}(\Gamma_{1})}\mathscr{K}(\Gamma_{1})$ is
acyclic as well as that $C^{\bullet}$ is semi-simple. Then,

$r=\displaystyle\underset{n\in\mathbb{Z}}{\sum}(-1)^{n}\dim_{E}H^{n}(C^{\bullet})^{\Gamma_{1}}\,,$

and there exists a commutative diagram

$\textstyle{{\det}^{-1}_{\mathscr{H}(\Gamma_{1})}C^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z^{-r}\vartheta_{\mathrm{Iw}}}$$\scriptstyle{\otimes_{\mathscr{H}(\Gamma_{1})}^{\mathbf{L}}E}$$\textstyle{\mathscr{H}_{0}(\Gamma_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}^{-1}_{E}{C^{\bullet}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta_{0}}$$\textstyle{E}$

in which the right vertical arrow is the augmentation map. The same holds for
complexes of $\Lambda_{E}$-modules.

###### Proof.

See [BG03, Lemma 8.1]. This is a particular case of Nekovář’s descent theory
[Nek06]. Note that Burns and Greither consider complexes over $\Lambda_{E}$
but since $\mathscr{H}(\Gamma_{1})$ is a Bézout ring, all their arguments work
in our case and are omitted here. ∎

#####

The following classical result is a particular case of Lemma 3.38. Let $M$ is
a finitely generated torsion $\mathcal{O}_{E}[[\Gamma_{1}]]$-module. Then
$M_{E}:=M\otimes_{\mathcal{O}_{E}}E$ is of finite dimension over $E$, and the
tautological exact sequence

$0\longrightarrow M_{E}^{\Gamma_{1}}\longrightarrow
M_{E}\xrightarrow{{[z]}}M_{E}\longrightarrow(M_{E})_{\Gamma_{1}}\longrightarrow
0$

shows that $\dim_{E}(M_{E}^{\Gamma_{1}})=\dim_{E}(M_{E})_{\Gamma_{1}}$. In
particular,

(3.58) $M_{E}^{\Gamma_{1}}=\\{0\\}\,\iff\,(M_{E})_{\Gamma_{1}}=\\{0\\}\,.$

Let $g=g({z})\in\Lambda{(\Gamma_{1})}$ be a generator of the characteristic
ideal of $M$, and let $r(M)$ denote the order of vanishing of the formal power
series $g$ at $z=0$. Assume that $M_{E}$ is semi-simple, i.e. that the natural
map

$M_{E}^{\Gamma_{1}}\longrightarrow(M_{E})_{\Gamma_{1}}$

is an isomorphism. Then,

(3.59) $r(M)=\dim_{E}M_{E}^{\Gamma_{1}}\,,$

and

(3.60) $\lim_{s\to
0}\frac{g(\chi(\gamma_{1})^{s}-1)}{s^{r(M)}}\sim_{p}(\log\chi(\gamma_{1}))^{r(M)}\left[M_{\Gamma_{1}}:M^{\Gamma_{1}}\right]\,,$

where “$\sim_{p}$” means equality up to multiplication by a $p$-adic unit.

We will repeatedly use these general observations in what follows. The
following lemma was used in the proof of Proposition 3.19.

######  Lemma 3.39.

Let $M$ be a finitely generated torsion $\Lambda(\Gamma_{1})$-module. Assume
that $M_{E}^{\Gamma_{1}}$ and $(M_{E})_{\Gamma_{1}}$ are one-dimensional
$E$-vector spaces. Then the natural map
$M_{E}^{\Gamma_{1}}\rightarrow(M_{E})_{\Gamma_{1}}$ is an isomorphism if and
only if $r(M)=1.$

###### Proof.

Let $A$ denote the $z$-primary component of $M_{E}.$ Since $\Lambda_{E}$ is
principal, $M_{E}^{\Gamma_{1}}=A^{\Gamma_{1}}$ and
$(M_{E})_{\Gamma_{1}}=A_{\Gamma_{1}}.$ Since $A$ is a direct sum of cyclic
modules of the form $E[z]/(z^{m}),$ from our assumptions if follows
immediately that $A\simeq E[z]/(z)$ if and only if the map
$M_{E}^{\Gamma_{1}}\rightarrow(M_{E})_{\Gamma_{1}}$ is an isomorphism. ∎

####  3.5.2. The non-central case

#####

In this section, we apply Lemma 3.38 to the trivializations

$i_{\mathrm{Iw},V_{f}(j)}^{(\alpha),\mathrm{imp}}\,:\,{\det}^{-1}_{\Lambda_{E}}\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(j),\alpha)\longrightarrow\Lambda_{E}\,,\qquad\qquad
i_{\mathrm{Iw},V_{f}(j)}^{(\beta)}\,:\,{\det}^{-1}_{\Lambda_{E}}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\longrightarrow\Lambda_{E},$

Let us set

$\mathbf{R\Gamma}^{\mathrm{imp}}(V_{f}(j),\alpha):=\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\otimes_{\Lambda_{E}}^{\mathbf{L}}E\,,\qquad\qquad\mathbf{R\Gamma}(V_{f}(j),\beta):=\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\otimes_{\Lambda_{E}}^{\mathbf{L}}E\,.$

#####

Let us denote by
$\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j],\mathds{1})$
(resp. $\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1})$) the
isotypic component of the $\Lambda_{\mathcal{O}_{E}}$-module
$\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j])$
(resp. $\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j])$) that
corresponds to the trivial character $\mathds{1}\in X(\Delta)$.

Let
$f_{\alpha}^{\mathrm{imp}}({z})\in\mathrm{Frac}(\mathcal{O}_{E}[[\Gamma_{1}]])$
(resp. $f_{\beta}({z})\in\mathrm{Frac}(\mathcal{O}_{E}[[\Gamma_{1}]])$) denote
a generator of
$\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j],\mathds{1})$
(resp. $\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\alpha}[j],\mathds{1})$)
and let $r_{\alpha}(j)$ (resp. $r_{\beta}(j)$) denote the order of vanishing
of $f_{\alpha}^{\mathrm{imp}}({z})$ (resp. of $f_{\beta}(z)$) at $z=0$. We set

$\displaystyle\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp},*}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)=\lim_{s\to
0}s^{-r_{\alpha}(j)}f_{\alpha}^{\mathrm{imp}}(\chi(\gamma_{1})^{s}-1),$
$\displaystyle\mathbf{L}_{\mathrm{Iw},\beta}^{*}(T_{f}(j),N_{\beta}[j],\mathds{1},0)=\lim_{s\to
0}s^{-r_{\beta}(j)}f_{\beta}(\chi(\gamma_{1})^{s}-1).$

These quantities depend on the choices of $f_{\alpha}^{\mathrm{imp}},$
$f_{\beta},$ and $\gamma_{1}$ only up to multiplication by a unit in
$\mathcal{O}_{E}$.

#####

For $\star\in\\{\alpha,\beta\\}$, let us set

(3.61)
$\Delta_{E}(T_{f}(j),\star)=\Delta_{\mathrm{Iw}}(T_{f}(j),\star)\otimes_{\Lambda}\mathcal{O}_{E}\,.$

Then, assuming that the Selmer complexes
$\mathbf{R\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ and
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)$ are semi-simple, Lemma 3.38
provides us with a pair of commutative diagrams

(3.62)
$\textstyle{{\det}_{\mathscr{H}(\Gamma)}^{-1}\mathbf{R\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z^{-r_{\alpha}(j)}i_{\mathrm{Iw},V_{f}(j)}^{(\alpha)}}$$\scriptstyle{\otimes_{\mathscr{H}(\Gamma)}^{\mathbf{L}}E}$$\textstyle{\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}_{E}^{-1}\mathbf{R\Gamma}^{\mathrm{imp}}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{V_{f}(j)}^{(\alpha),\mathrm{imp}}}$$\textstyle{E}$
$\textstyle{{\det}_{\mathscr{H}(\Gamma)}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z^{-r_{\beta}(j)}i_{\mathrm{Iw},V_{f}(j)}^{(\beta)}}$$\scriptstyle{\otimes_{\mathscr{H}(\Gamma)}^{\mathbf{L}}E}$$\textstyle{\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}_{E}^{-1}\mathbf{R\Gamma}(V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{V_{f}(j)}^{(\beta)}}$$\textstyle{E\,\,.}$

This would in turn furnish us with the following interpretation of special
values of algebraic $p$-adic $L$-functions:

(3.63)
$\displaystyle\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp},*}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)=i_{V_{f}(j)}^{(\alpha),\mathrm{imp}}\left(\Delta_{E}(T_{f}(j),\alpha)\right),$
$\displaystyle\mathbf{L}_{\mathrm{Iw},\beta}^{*}(T_{f}(j),N_{\beta}[j],\mathds{1},0)=i_{V_{f}(j)}^{(\beta)}\left(\Delta_{E}(T_{f}(j),\beta)\right).$

We now employ this approach in variety of scenarios.

Proposition 3.27 provides us with the following exact sequences, which will
play a key role in the remainder of this chapter:

(3.64) $\displaystyle
0\longrightarrow\mathrm{coker}(g_{\alpha})\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(T_{f}(j),\alpha)\longrightarrow{}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow
0\,,$ (3.65) $\displaystyle
0\longrightarrow\mathrm{coker}(g_{\beta})\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(T_{f}(j),\beta)\longrightarrow{}_{\mathrm{Iw}}(T_{f}(j))\longrightarrow
0\,,$

where

(3.66) ${g}_{\star}\,:\,H^{1}_{\mathrm{Iw}}(T_{f}(j))\oplus
N_{\star}[j]\otimes\Lambda\xrightarrow{\textup{res}_{p}\,+\,\mathrm{Exp}_{\star,j}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j)),\qquad\qquad\star\in\\{\alpha,\beta\\}\,.$

######  Proposition 3.40.

Suppose that $j\neq k/2$ is a strictly positive integer.

Assume that we are in one of the following situations:

  * a)

$1\leqslant j\leqslant k-1$ .

  * b)

$j\geqslant k$ and the conjecture ($p$Bei)) holds.

Then the following statements hold true:

  * 1)

$\mathbf{R\Gamma}(V_{f}(j),\beta)$ is acyclic and $r_{\beta}(j)=0.$

  * 2)

The cohomology groups of
$\mathbf{R\Gamma}(T_{f}(j),\beta):=\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f}(j),\beta)\otimes_{\Lambda}\mathcal{O}_{E}$
have finite cardinality, and we have

$\left|\mathbf{L}_{\mathrm{Iw},\beta}^{*}(T_{f}(j),N_{\beta}[j],\mathds{1},0)\right|^{-1}=\frac{\\#\mathbf{R}^{2}\boldsymbol{\Gamma}(T_{f}(j),\beta)}{\\#\mathbf{R}^{1}\boldsymbol{\Gamma}(T_{f}(j),\beta)}.$

Assume that we are in one of the following scenarios:

  * a)

$1\leqslant j\leqslant k-1$ and the condition (S)) holds .

  * b)

$j\geqslant k$ and the the both conditions (S)) and ($p$Bei)) hold.

Then the statements 1-2) above also hold for the complexes
$\mathbf{R\Gamma}^{\mathrm{imp}}(V_{f}(j),\alpha),$
$\mathbf{R\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)$ and the module
$\mathbf{L}^{\mathrm{imp}}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j]).$ In
particular, $r_{\alpha}(j)=0$ and

$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp},*}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}=\frac{\\#\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)}{\\#\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)}\,.$

###### Proof.

We prove (i), treating the scenarios (a) and (b) separately. Our proof uses
two statements well-known to the experts, Lemma 3.42 and Lemma 3.43, whose
proofs are deferred until §3.5.2.

Consider the short exact sequence (3.55), which reads

$0\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}^{i}(V_{f}(j),\beta)_{\Gamma}\longrightarrow\mathbf{R}^{i}\boldsymbol{\Gamma}(V_{f}(j),\beta)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}^{i+1}(V_{f}(j),\beta)^{\Gamma}\longrightarrow
0\,.$

in our case. Under our running assumptions,
$\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta){=\\{0\\}}$ if
$i\neq 2$. Therefore, we have natural identifications

$\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(j),\beta)=\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)^{\Gamma}\,,\qquad\qquad\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(j),\beta)=\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)_{\Gamma}.$

Let us apply the snake lemma to the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

that we obtain from the exact sequence (3.65) tensored with $\mathbb{Q}_{p}$,
where $\gamma$ is a topological generator and the vertical arrows are the
multiplication-by-$(\gamma-1)$ maps. Noting that, under our assumptions, we
have
${}_{\mathrm{Iw}}(V_{f}(j))^{\Gamma}={\\{0\\}}={}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}$
by Lemma 3.43 below, we deduce that

(3.67)
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)^{\Gamma}=\mathrm{coker}(g_{\beta})^{\Gamma}=\ker\left(H^{1}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}\oplus
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))_{\Gamma}\longrightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))_{\Gamma}\right),$
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)_{\Gamma}=\mathrm{coker}(g_{\beta})_{\Gamma}=\mathrm{coker}\left(H^{1}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}\oplus
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))_{\Gamma}\longrightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))_{\Gamma}\right).$

Using Lemma 3.42, we infer that

$H^{1}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}=H^{1}_{\\{p\\}}(V_{f}(j))\,\,,\qquad
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))_{\Gamma}=H^{1}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))\,.$

Hence, (3.67) can be recast as

(3.68)
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)^{\Gamma}=\mathrm{coker}(g_{\beta})^{\Gamma}=\ker\left(H^{1}_{\\{p\\}}(V_{f}(j))\oplus
H^{1}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))\longrightarrow
H^{1}(\mathbb{Q}_{p},V_{f}(j))\right),$
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)_{\Gamma}=\mathrm{coker}(g_{\beta})_{\Gamma}=\mathrm{coker}\left(H^{1}_{\\{p\\}}(V_{f}(j))\oplus
H^{1}(\mathbb{Q}_{p},V_{f}^{(\beta)}(j))\longrightarrow
H^{1}(\mathbb{Q}_{p},V_{f}(j))\right).$

Since we have $H^{1}_{\beta}(V_{f}(j))=0$ thanks to our running hypotheses, we
deduce that

$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)^{\Gamma}=\\{0\\}=\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\beta)_{\Gamma}\,.$

This completes the proof that the complex $\mathbf{R\Gamma}(V_{f}(j),\beta)$
is acyclic.

It follows from the acyclicity of $\mathbf{R\Gamma}(V_{f}(j),\beta)$ that
$r_{\beta}(j)=0$ and the map $i_{V_{f}(j)}^{(\beta)}$ coincides with the
composition of the isomorphisms

${\det}_{E}^{-1}\mathbf{R\Gamma}(V_{f}(j),\beta)\xrightarrow{\sim}\bigotimes_{i\in\mathbb{Z}}{\det}_{E}^{(-1)^{i-1}}\mathbf{R}^{i}\boldsymbol{\Gamma}(V_{f}(j),\beta)\xrightarrow{\sim}E.$

Since
$\mathbf{R\Gamma}^{i}(T_{f}(j),{\beta})\otimes_{\mathcal{O}_{E}}E=\mathbf{R\Gamma}^{i}(V_{f}(j),{\beta})=\\{0\\}$
for all $i\in\mathbb{Z}$, it follows that the cohomology groups of the complex
$\mathbf{R\Gamma}(T_{f}(j),{\beta})$ have finite cardinality. The part (i) of
the proposition follows from the commutativity of the diagram (3.62) and the
fact that
$\Delta_{\mathrm{Iw}}(T_{f}(j),\beta)\simeq{\det}_{\Lambda}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f}(j),\beta)$.

The proof of part (ii) is entirely analogous, and we omit it here. ∎

#####

Let us apply Lemma 3.38 with the complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$. This computation will be
used in Chapter 4.

Let us put

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

######  Proposition 3.41.

Let $1\leqslant j\leqslant k-1$ be an integer such that $j\neq k/2$. Assume
that condition (S)) holds. Then the following statements hold true:

We have

$\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\simeq
D^{(\alpha)}\,,\qquad\qquad\mathbf{R}^{2}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\simeq\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j)}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))}\,,$

and $\mathbf{R}^{i}\boldsymbol{\Gamma}(V_{f}(j),\alpha)=0$ for $i\neq 1,2.$

The complex $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ is semi-simple.
The Bockstein map together with the isomorphisms in Part (i) give rise to a
commutative diagram

(3.69)
$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Bock_{2}}}$$\textstyle{D^{(\alpha)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j)}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))}\,,}$

where the map $\delta$ is given by

$\delta(d)=\ell^{*}(V_{f}(j))\cdot\left(\mathrm{pr}_{0}\circ\mathrm{Exp}_{\alpha,j}(d)\right)\,,\qquad\ell^{*}(V_{f}(j))=\underset{i=1-k}{\overset{-1}{\prod}}(i+j)$

or, in more explicit form,

$\delta(d)=-(j-1)!\left(\frac{1-p^{j-1}\alpha^{-1}}{1-p^{-j}\alpha}\right)\left(\exp^{*}_{V^{(\alpha)}_{x_{0}}(j)}\right)^{-1}(d)\,.$

Both squares in the following diagram commute:

$\textstyle{\Delta_{E}(T_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}^{-1}_{E}\mathbf{R\Gamma}^{\mathrm{imp}}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{V_{f}(j)}^{(\alpha),\mathrm{imp}}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell^{*}(V_{f}(j))}$$\textstyle{\Delta_{E}(T_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\det}^{-1}_{E}\mathbf{R\Gamma}(V_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{V_{f}(j)}^{(\alpha)}}$$\textstyle{E.}$

###### Proof.

The proof is purely formal. We remark that the map

$\mathrm{Exp}_{V(j),j,0}\,:\,D^{(\alpha)}\longrightarrow
H^{1}(\mathbb{Q}_{p},V_{f}(j))$

is the zero-map and therefore, the kernel and the cokernel of the map

$H^{1}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}\oplus
D^{(\alpha)}\xrightarrow{\textup{res}_{p}+\mathrm{Exp}_{V_{f}(j)j,0}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},V_{f}(j))_{\Gamma}$

are isomorphic to $D^{(\alpha)}$ and
$\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(j))}{\textup{res}_{p}(H^{1}_{\\{p\\}}(V_{f}(j)))}$,
respectively. The proof of (i) can be completed mimicking the arguments used
in the proof of Proposition 3.40.

It follows from the definition of the Bockstein map that

$\delta(d)=\mathrm{pr}_{0}(y)\mod H^{1}_{\\{p\\}}(V_{f}(j)),\quad\textrm{where
$\mathrm{Exp}_{V_{f}(j),j}(y)=d.$}$

Taking (3.47) into account together with the formulae (1.23), we conclude the
proof of Part (ii).

This assertion directly follows from definitions. ∎

#####

We prove give a proof of the two lemmas used in the proof of Proposition 3.40.
Both statements are well-known (cf. [PR95, Proposition 3.4.2(i)]), but we
record them here for the reader’s convenience.

######  Lemma 3.42.

For any $p$-adic representation $W$ of $G_{\mathbb{Q},S}$ with
$H^{0}(\mathbb{Q}_{p},W^{*}(1))=0$, we have a short exact sequence

$0\longrightarrow H^{1}_{\mathrm{Iw}}(W)_{\Gamma}\longrightarrow
H^{1}_{\\{p\\}}(W)\longrightarrow{}^{2}_{\mathrm{Iw}}(W)^{\Gamma}\longrightarrow
0\,.$

###### Proof.

Consider the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}^{2}(W)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{Iw}}(W)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\mathrm{Iw}}(W)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},W)_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{1}(\mathbb{Q}_{\ell},W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S\setminus\\{p\\}}{\oplus}H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},W)^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,.}$

The rows and the middle column of this diagram are exact. We explain that the
right-most column is also exact. We have

$H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)^{\Gamma}\simeq\left(H^{0}(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),W^{*}(1))_{\Gamma}\right)^{*}$

by local duality. It follows from (3.58) that

$\dim_{E}H^{0}(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),W^{*}(1))_{\Gamma}=\dim_{E}H^{0}(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),W^{*}(1))^{\Gamma}=\dim_{E}H^{0}(\mathbb{Q}_{p},W^{*}(1))=0,$

where the final vanishing is our running assumption. Hence
$H^{2}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)^{\Gamma}=0$. This shows that the right-
most column in the diagram above is indeed exact, and in turn, allows us to
define the dotted arrow. Moreover, we have

$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},W)_{\Gamma}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\rm
f}(\mathbb{Q}_{\ell},W)$

for all $\ell\neq p$. The lemma follows from this fact via a diagram chase. ∎

######  Lemma 3.43.

There exists a canonical isomorphism

${}^{2}_{\mathrm{Iw}}(V_{f}(j))_{\Gamma}\simeq
H^{1}_{\mathrm{0}}(V_{f^{*}}(k-j))$

Assume that $H^{1}_{\mathrm{0}}(V_{f^{*}}(k-j))=0.$ Then,

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

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

###### Proof.

Part i) is proved in [PR95, Section 3.3.4] (see also [Ben14], Section 6.1.3).
Part ii) follows from this fact and Lemma 3.42. ∎

####  3.5.3. The central critical case

In this section, we assume that $k$ is even and consider the case
$j=\frac{k}{2}$. Proposition 3.41 applies also with $j=\frac{k}{2}$ when
$r_{\rm an}(\frac{k}{2})=0$. We therefore assume $r_{\rm an}(\frac{k}{2})\geq
1$.

#####

Recall the $p$-adic heights

$\left<\,,\,\right>_{\star}:H^{1}_{\star}(V_{f^{*}}({k}/{2}))\otimes
H^{1}_{\star}(V_{f}({k}/{2}))\longrightarrow
E,\qquad\qquad\star\in\\{\alpha,\beta\\},$

associated to splitting submodules
$D_{\star}[k/2]\subset{\mathbf{D}}_{\mathrm{cris}}(V_{f}(k/2))$, cf. [Ben20].
Recall that
$H^{1}_{\beta}(V_{f^{*}}({k}/{2}))=H^{1}_{\mathrm{f}}(V_{f^{*}}({k}/2))$ and
$\left<\,,\,\right>_{\beta}$ coincides with the classical $p$-adic height
pairing introduced in [Nek93].

Let us define the associated regulator

$R_{\star}(T_{f}({k}/{2})):=\left|\det\left<e_{i}^{*},e_{j}\right>_{\star}\right|^{-1}\,,$

(as above, the subscript “$\star$” stands for $\alpha$ or $\beta$) for any
choice of bases $\\{e_{i}^{*}\\}$ of
$H^{1}_{\star}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}}$ and $\\{e_{i}\\}$ of
$H^{1}_{\star}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}}$. It is clear that
$R_{\star}(T_{f}({k}/{2}))$ depends on the choice of these bases only up to
multiplication by a $p$-adic unit.

In what follows, whenever $M$ and $N$ are free submodules of
$H^{1}_{\star}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}}$ and
$H^{1}_{\star}(T_{f}({k}/{2}))_{\mathrm{tf}}$ of the same rank and with bases
$\\{m_{i}\\}$ and $\\{n_{j}\\}$, respectively, we write
$\left<M,N\right>_{\star}$ as a shorthand for
$\det\left<m_{i},n_{j}\right>_{\star}$.

######  Theorem 3.44.

Assume that the conditions (BK)) and (h-PR)) hold true. Then:

$H^{1}_{\alpha}(V_{f}(k/2))=H^{1}_{0}(V_{f}(k/2))$ and the pairing
$\left<\,,\,\right>_{\alpha}$ coincides with the restriction of
$\left<\,,\,\right>_{\beta}$ on $H^{1}_{0}(V_{f}(k/2)).$

There exist canonical isomorphisms

$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\simeq
H^{1}_{0}(V_{f}(k/2))\,,$
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\simeq
H^{1}_{0}(V_{f^{*}}(k/2))^{*}$
$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(k/2),\beta)\simeq
H^{1}_{\mathrm{f}}(V_{f}(k/2))\,,$
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}(V_{f}(k/2),\beta)\simeq
H^{1}_{\mathrm{f}}(V_{f^{*}}(k/2))^{*}.$

The Bockstein maps are given by the diagrams

$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Bock}_{2}}$$\textstyle{H^{1}_{0}(V_{f}(k/2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left<\,,\,\right>_{\alpha}}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{0}(V_{f^{*}}(k/2))^{*}}$
$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(k/2),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Bock}_{2}}$$\textstyle{H^{1}_{\mathrm{f}}(V_{f}(k/2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left<\,,\,\right>_{\beta}}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}(V_{f}(k/2),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\mathrm{f}}(V_{f^{*}}(k/2))^{*}}$

where the right vertical maps in both diagrams are induced by the $p$-adic
height pairing.

In particular, the complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}({k/2}),\alpha)$ (resp.
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(k/2),\beta)$) is semi-simple if and only
if the $p$-adic height pairing $\left<\,,\,\right>_{\alpha}$ (resp.
$\left<\,,\,\right>_{\beta}$) is non-degenerate.

We have

$r_{\alpha}(k/2)=\dim_{E}H^{1}_{0}(V_{f}(k/2))\,,\qquad\qquad
r_{\beta}(k/2)=\dim_{E}H^{1}_{\rm f}(V_{f}(k/2))\,.$

If in addition (ht)) holds, we have $r_{\beta}(k/2)=r_{\alpha}(k/2)+1$.

###### Proof.

We refer the reader to [PR95, Proposition 3.4.2(ii) and Section 3.3.4] and
[Col00] for the proofs of the results (1))–(3)) recorded below.

  * 1)

$H^{1}_{0}(V_{f^{*}}(k/2))$ and
$H^{1}_{u}(V_{f}(k/2)):=H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\cap
H^{1}_{\mathrm{f}}(V_{f}(k/2))$ are orthogonal to one another under the
pairing $\left<\,,\,\right>_{\beta}$. Therefore,

(3.70)
$\left<H^{1}_{\mathrm{f}}(T_{f^{*}}(k/2))_{\mathrm{tf}},H^{1}_{\mathrm{f}}(T_{f}(k/2))_{\mathrm{tf}}\right>_{\beta}=\left<H^{1}_{0}(T_{f^{*}}(k/2))_{\mathrm{tf}},\frac{H^{1}_{\mathrm{f}}(T_{f}(k/2))_{\mathrm{tf}}}{H^{1}_{u}(T_{f}(k/2))_{\mathrm{tf}}}\right>_{\beta}\left<\frac{H^{1}_{\mathrm{f}}(T_{f^{*}}(k/2))_{\mathrm{tf}}}{H^{1}_{0}(T_{f^{*}}(k/2))_{\mathrm{tf}}},H^{1}_{u}(T_{f}(k/2))_{\mathrm{tf}}\right>_{\beta}.$

  * 2)

The Iwasawa-theoretic version of the Poitou–Tate exact sequence provides us
with a map

$H^{1}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}(k/2))^{*}\longrightarrow{}_{\mathrm{Iw}}(V_{f}(k/2))\,.$

Since $H^{0}(V_{f^{*}}(\frac{k}{2}))=0,$ we have
$H^{1}(V_{f^{*}}(\frac{k}{2}))\simeq
H^{1}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}(\frac{k}{2}))^{\Gamma}.$ This
gives rise to a commutative diagram

(3.71)
$\textstyle{H^{1}(\mathbb{Q}(\zeta_{p^{\infty}}),V_{f^{*}}(k/2))^{*}_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}(V_{f^{*}}(k/2))^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{0}(V_{f^{*}}(k/2))^{*},}$

where the vertical morphisms are surjective and the horizontal ones are
isomorphisms.

  * 3)

Consider the composition of morphisms

$\mu\,:\,H^{1}_{\mathrm{f}}(V_{f}(k/2))\longrightarrow{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\longrightarrow{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\longrightarrow
H^{1}_{0}(V_{f^{*}}(k/2))^{*},$

where the first map is provided by Lemma 3.42, the middle map is the natural
projection and the last map is provided by diagram (3.71). Then $\mu$
coincides with the pairing induced by
$\log^{-1}\chi(\gamma)\left<\,\cdot\,,\,\cdot\,\right>_{\beta}$.

By Proposition 3.23, we have

$H^{1}_{\alpha}(V_{f}(k/2))=H^{1}_{0}(V_{f}(k/2))\,,$

and by Proposition 3.25,

$H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\cap
H^{1}_{0}(V_{f}(k/2))=\\{0\\}\,.$

We note that the proofs of these portions of Proposition 3.23 and Proposition
3.25 do not rely on the hypothesis (ht)). Therefore, Lemma 3.42 gives rise to
an isomorphism

$H^{1}_{0}(V_{f}(k/2))\simeq{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\,.$

Consider the map (3.66) for $j=\frac{k}{2}.$ Since

$\textup{res}_{p}\left(H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\right)\cap
H^{1}_{\alpha}(\mathbb{Q}_{p},V_{f}(k/2))=\\{0\\},$

we infer that $\ker({g}_{\alpha})_{\Gamma}\otimes\mathbb{Q}_{p}$ and
$\mathrm{coker}({g}_{\alpha})_{\Gamma}\otimes\mathbb{Q}_{p}$ vanish. Tensoring
the exact sequence (3.64) with $\mathbb{Q}_{p}$ and applying the snake lemma,
we obtain an exact sequence

(3.72)
$0\longrightarrow\mathrm{coker}({g}_{\alpha})^{\Gamma}\otimes\mathbb{Q}_{p}\longrightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\longrightarrow{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\\\
\longrightarrow\mathrm{coker}({g}_{\alpha})_{\Gamma}\otimes\mathbb{Q}_{p}\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\longrightarrow{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\longrightarrow
0\,.$

Hence,

$\displaystyle\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\simeq{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\simeq
H^{1}_{0}(V_{f}(k/2)),$
$\displaystyle\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\simeq{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\simeq
H^{1}_{0}(V_{f^{*}}(k/2))^{*}.$

A diagram chase shows that the diagram

$\textstyle{\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Bock}_{2}}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(V_{f}(k/2),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}}$

commutes. This fact, combined with (1))–(3)), concludes the proof of the
portion of the theorem concerning the complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}^{\mathrm{imp}}(V_{f}(k/2),\alpha).$

We now turn our attention to the case of
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(k/2),\beta).$ Recall that we have

(3.73)
$H^{1}_{\beta}(V_{f}(k/2))=H^{1}_{\mathrm{f}}(V_{f}(k/2))=H^{1}_{\\{p\\}}(V_{f}(k/2))$

by Proposition 3.25; we remark once again that the proof of this portion of
Proposition 3.25 does not rely on the hypothesis (ht)). Since we have
$H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\subset H^{1}_{\\{p\\}}(V_{f}(k/2))$
by Lemma 3.42, it follows from (3.73) that

(3.74) $H^{1}_{u}(V_{f}(k/2))=H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}.$

Moreover, we have a canonical decomposition

(3.75)
$H^{1}_{\\{p\\}}(V_{f}(k/2))=H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\oplus
H^{1}_{0}(V_{f}(k/2))\,.$

We therefore have canonical isomorphisms

$\mathrm{coker}(g_{\beta})^{\Gamma}\otimes\mathbb{Q}_{p}\simeq
H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\,,\qquad\qquad\mathrm{coker}(g_{\beta})_{\Gamma}\otimes\mathbb{Q}_{p}\simeq\frac{H^{1}(\mathbb{Q}_{p},V_{f}(k/2))}{H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},V_{f}(k/2))}.$

The Selmer complex $\mathbf{R\Gamma}(V_{f}(k/2),\beta)$ is associated to the
diagram

$\textstyle{\mathbf{R\Gamma}_{S}(V_{f}(k/2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\oplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},V_{f}(k/2))}$$\textstyle{\underset{\ell\in
S}{\oplus}\mathbf{R\Gamma}(\mathbb{Q}_{\ell},V_{f}(k/2),\beta),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where
$\mathbf{R\Gamma}(\mathbb{Q}_{\ell},V_{f}(k/2),\beta):=\mathbf{R\Gamma}_{\mathrm{f}}(\mathbb{Q}_{\ell},V_{f}(k/2))$
for $\ell\neq p$ and the local condition at $p$ is given by

$\mathbf{R\Gamma}(\mathbb{Q}_{p},V_{f}(k/2),\beta):=D^{(\beta)}[k/2]$

together with the map

$D^{(\beta)}[k/2][-1]\xrightarrow{\mathbf{Exp}_{\beta,k/2,0}}\mathbf{R\Gamma}(\mathbb{Q}_{p},V_{f}(k/2))\,.$

The map $\mathbf{Exp}_{\beta,k/2,0}$ induces

$\mathrm{Exp}_{\beta,k/2,0}\,:\,D^{(\beta)}[k/2]\longrightarrow
H^{1}(\mathbb{Q}_{p},V_{f}(k/2))\,,$

and the image of the map $\mathrm{Exp}_{\beta,k/2,0}$ is the Bloch–Kato
subgroup $H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},V_{f}(k/2))$. Therefore,

$\mathbf{R}^{1}\boldsymbol{\Gamma}(V_{f}(k/2),\beta)\simeq
H^{1}_{\mathrm{f}}(V_{f}(k/2))\,.$

It is not difficult to see that we have a commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\beta})^{\Gamma}\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{R}^{1}(V_{f}(k/2),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\\{p\\}}(V_{f}(k/2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{0}(V_{f}(k/2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\,,}$

where the bottom row is induced by the decomposition (3.75) and all vertical
maps are isomorphisms. Since the bottom row is exact, then so is the upper
row. Therefore, we have the following commutative diagram with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\beta})^{\Gamma}\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{1}(V_{f}(k/2),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Bock_{2}}}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))^{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\beta})_{\Gamma}\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}(V_{f}(k/2),\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that the right-most vertical map in this diagram admits an interpretation
in terms of the $p$-adic height pairing, cf. (3)). The left-most vertical map
fits in the diagram

$\textstyle{\mathrm{coker}(g_{\beta})^{\Gamma}\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Upsilon}$$\textstyle{\mathrm{coker}(g_{\beta})_{\Gamma}\otimes\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\frac{H^{1}(\mathbb{Q}_{p},V_{f}(k/2))}{H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},V_{f}(k/2))},}$

where $\Upsilon$ is the unique map that ensures that this diagram commutes. We
have the following interpretation of the right vertical map in this diagram in
terms of $p$-adic height pairing, cf. [PR95]: The diagram

$\textstyle{\displaystyle\frac{H^{1}_{\mathrm{f}}(V_{f^{*}}(k/2))}{H^{1}_{0}(V_{f^{*}}(k/2))}\times
H^{1}_{\mathrm{Iw}}(V_{f}(k/2))_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left<\,,\,\right>_{\beta}}$$\scriptstyle{(\textup{res}_{p},{\Upsilon})}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},V_{f^{*}}(k/2))\times\displaystyle\frac{H^{1}(\mathbb{Q}_{p},V_{f}(k/2))}{H^{1}_{\mathrm{f}}(\mathbb{Q}_{p},V_{f}(k/2))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$local
duality$\textstyle{E}$

commutes.

Putting these facts together, we see that $\mathrm{Bock}_{2}$ is an
isomorphism if and only if the pairing $\left<\,,\,\right>_{\beta}$ is non-
degenerate. This completes the proof of the portion of the theorem concerning
the complex $\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(k/2),\beta)$. ∎

####  3.5.4. Birch and Swinnerton-Dyer-style formulae

In §3.5.4, we prove $p$-adic Birch and Swinnerton-Dyer style formulae for our
algebraic $p$-adic $L$-functions. These results will not be used in the
remainder of this paper, but we include them here for the sake of
completeness. We remark that they were proved at a great level of generality
in [PR95] in the non-$\theta$-critical case (see also [Ben14]), and the proofs
in the $\theta$-critical scenario (which we present below) are very similar
granted the input from the earlier portions of §3.5.

#####

As before, let us set

$e_{p,\alpha}(f,\mathds{1},j)=\left(1-\alpha^{-1}p^{j-1}\right)\cdot\left(1-\beta
p^{-j}\right)\,,\qquad
e_{p,\beta}(f,\mathds{1},j)=\left(1-\beta^{-1}p^{j-1}\right)\cdot\left(1-\alpha
p^{-j}\right).$

We denote by

$\mathrm{Tam}^{0}(T_{f}(j)):=\underset{\ell\in
S}{\prod}\mathrm{Tam}_{\ell}(T_{f}(j))$

the product of local Tamagawa numbers of $V_{f}(j)$ and by

$\mathrm{Tam}_{\\{p\\}}^{0}(T_{f}(j)):=\underset{\ell\in
S\setminus\\{p\\}}{\prod}\mathrm{Tam}_{\ell}(T_{f}(j))$

the product of local Tamagawa numbers away from $p$. Recall that we have set
$\Omega_{p}(T_{f}^{(\star)},N_{\star}):=|b|,$ where $b\in E^{*}$ is such that
${\mathbf{D}}_{\mathrm{cris}}(T_{f}^{(\star)})=bN_{\star}$. Throughout §3.5.4,
we will assume that
$\Omega_{p}(T_{f}^{(\star)},N_{\star})\in\mathcal{O}_{E}^{*}$ for
$\star\in\\{\alpha,\beta\\}.$

######  Theorem 3.45.

Suppose $j\geqslant k$. Assume that the conditions ($p$Bei)) and (S)) hold.
Then $r_{\alpha}(j)=0$ and

$\left|\mathbf{L}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}=\left|(j-k)!\cdot
e_{p,\alpha}(f,\mathds{1},j)\right|^{-1}\cdot\frac{\\#{}_{\rm
f}(T_{f}(j))\cdot
R_{\alpha}(T_{f}(j))\cdot\mathrm{Tam}^{0}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}.$

Suppose $1\leqslant j\leqslant k-1$ and $j\neq\frac{k}{2}$. Assume that the
condition(S)) holds. Then $r_{\alpha}(j)=0$ and

$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}=\frac{\\#{}_{\alpha}(T_{f}(j))\cdot\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\cdot
w(T_{f}^{(\beta)}(j))\cdot w(T^{(\beta)}_{f^{*}}(k-j)),$

where
$w(T_{f}^{(\beta)}(j))=\\#H^{0}(\mathbb{Q}_{p},V_{f}^{(\beta)}/T_{f}^{(\beta)}(j))$
and
$w(T^{(\beta)}_{f^{*}}(k-j))=\\#H^{0}(\mathbb{Q}_{p},V^{(\beta)}_{f^{*}}/T^{(\beta)}_{f^{*}}(k-j)).$

iii) Suppose $j\geqslant k.$ Assume that the condition($p$Bei)) holds and
$H^{1}_{\beta}(V_{f}(j))=0.$ Then $r_{\beta}(j)=0$ and

$\left|\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0)\right|^{-1}=\left|(j-1)!\cdot
e_{p,\beta}(f,\mathds{1},j)\right|^{-1}\cdot\frac{\\#{}_{\rm f}(T_{f}(j))\cdot
R_{\beta}(T_{f}(j))\cdot\mathrm{Tam}^{0}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}.$

iv) Suppose $1\leqslant j\leqslant k-1$ and $j\neq\frac{k}{2}$. Then
$r_{\beta}(j)=0$ and

$\left|\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0)\right|^{-1}=\left|(j-1)!\cdot
e_{p,\beta}(f,\mathds{1},j)\right|^{-1}\cdot\frac{\\#{}_{\rm
f}(T_{f}(j))\cdot\mathrm{Tam}^{0}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}.$

######  Remark 3.46.

The formulae for
$\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)$
and $\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0)$ in
Theorem 3.45 are concurrent with Propositions 3.14 and 3.16, in view of
$\mathbf{MC}(f,\alpha)$, $\mathbf{MC}(f,\beta)$ and Proposition 2.27. Note
that, we can express
$\left|\mathbf{L}_{\mathrm{Iw},\alpha}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}$
also in the form

$\left|(j-k)!\cdot
e_{p,\alpha}(f,\mathds{1},j)\cdot\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\right|^{-1}\cdot\frac{\\#{}_{\rm
f}(T_{f}(j))\cdot
R_{\beta}(T_{f}(j))\cdot\mathrm{Tam}^{0}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}$

in the situation of Theorem 3.45(i), and

$\left|\mathscr{L}^{\mathrm{cr}}(V_{f}(j))\right|^{-1}\frac{\\#{}_{\mathrm{f}}(T_{f}(j))\cdot\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\cdot
w(T_{f}^{(\alpha)}(j))\cdot w(T^{(\alpha)}_{f^{*}}(k-j))$

in the situation of Theorem 3.45(ii). These expressions are akin to those that
appear in the $p$-adic Birch and Swinnerton-Dyer conjecture in the non-
critical scenario, cf. [MTT86, Ben14].

###### Proof.

Applying the snake lemma to the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(T_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{{}_{\mathrm{Iw}}(T_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\gamma-1]}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{coker}(g_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}_{\mathrm{Iw}}(T_{f}(j),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{\mathrm{Iw}}(T_{f}(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with exact rows which we obtain from the exact sequence (3.64), we have an
exact sequence

(3.76)
$0\rightarrow\mathrm{coker}(g_{\alpha})^{\Gamma}\rightarrow\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)\rightarrow{}_{\mathrm{Iw}}(T_{f}(j))^{\Gamma}\\\
\rightarrow\mathrm{coker}(g_{\alpha})_{\Gamma}\rightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)\rightarrow{}_{\mathrm{Iw}}(T_{f}(j))_{\Gamma}\rightarrow
0.$

By Proposition 3.40, all terms of this sequence are finite groups under our
running assumptions. Moreover, $r_{\alpha}(j)=0$ and

(3.77)
$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}=\frac{\\#\mathbf{R}^{2}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)}{\\#\mathbf{R}^{1}\boldsymbol{\Gamma}^{\mathrm{imp}}(T_{f}(j),\alpha)}=\frac{\\#\mathrm{coker}({g}_{\alpha})_{\Gamma}}{\\#\mathrm{coker}({g}_{\alpha})^{\Gamma}}\cdot\frac{\\#{}_{\mathrm{Iw}}(T_{f}(j))_{\Gamma}}{\\#{}_{\mathrm{Iw}}(T_{f}(j))^{\Gamma}}.$

The arguments of [PR95, Section 3.3.4] (see also [Col00], computations
following Proposition 3.9, pp. 34-35) show that

(3.78)
$\frac{\\#{}_{\mathrm{Iw}}(T_{f}(j))_{\Gamma}}{\\#{}_{\mathrm{Iw}}(T_{f}(j))^{\Gamma}}=\frac{\left[H^{1}(\mathbb{Q}_{p},T_{f}(j)):H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))_{\Gamma}\right]}{\left[H^{1}_{\\{p\\}}(T_{f}(j)):H^{1}_{\mathrm{Iw}}(T_{f}(j))_{\Gamma}\right]}\cdot\frac{{\\#{}_{0}(T_{f^{*}}(k-j))}\cdot\mathrm{Tam}_{\\{p\\}}^{0}(T_{f}(j))}{\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\,.$

On the other hand, the snake lemma gives

(3.79)
$\frac{\\#\mathrm{coker}({g}_{\alpha})_{\Gamma}}{\\#\mathrm{coker}({g}_{\alpha})^{\Gamma}}=\left[H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}(j))_{\Gamma}:H^{1}_{\mathrm{Iw}}(T_{f}(j))_{\Gamma}+\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}\right].$

Since $\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}$ is
$\mathbb{Z}_{p}$-free, we have

(3.80) $\displaystyle\begin{aligned}
&\left[H^{1}(\mathbb{Q}_{p},T_{f}(j)):H^{1}_{\\{p\\}}(T_{f}(j))+\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}\right]=\\\
&\hskip
56.9055pt\left[H^{1}(\mathbb{Q}_{p},T_{f}(j))_{\mathrm{tf}}:H^{1}_{\\{p\\}}(T_{f}(j))_{\mathrm{tf}}+H^{1}_{\star}(\mathbb{Q}_{p},T_{f}(j))_{\mathrm{tf}}\right]\\\
&\hskip
71.13188pt\times\frac{\\#H^{0}(\mathbb{Q}_{p},V_{f}/T_{f}(j))\left[H^{1}_{\star}(\mathbb{Q}_{p},T_{f}(j))_{\mathrm{tf}}:\mathrm{Exp}_{\star,j}(N_{\star}[j]\otimes\Lambda)_{\Gamma}\right]}{\\#H^{0}(V_{f}/T_{f}(j))}.\end{aligned}$

Combining formulae (3.77), (3.78) (3.79) and (3.80) with Lemma 3.13(ii), we
deduce that

(3.81) $\displaystyle\begin{aligned}
&\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{\mathrm{imp}}(T_{f}(j),N_{\alpha}[j],\mathds{1},0)\right|^{-1}=\\\
&\hskip
14.22636pt\frac{\\#{}_{\alpha}(T_{f}(j))\cdot\\#H^{0}(\mathbb{Q}_{p},V_{f}/T_{f}(j))\cdot\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\cdot\left[H^{1}(\mathbb{Q}_{p},T_{f}^{(\alpha)}(j))_{\mathrm{tf}}:\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}\right].\end{aligned}$

Assume first that $j\geqslant k$. It follows from the definition of the
Tamagawa numbers and (1.23) that

(3.82) $\displaystyle\begin{aligned}
\left[H^{1}(\mathbb{Q}_{p},T^{(\alpha)}_{f}(j))_{\mathrm{tf}}:\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}\right]&=|(j-k)!|^{-1}\left|1-\alpha^{-1}p^{j-1}\right|^{-1}\frac{\mathrm{Tam}_{p}(T_{f}^{(\alpha)}(j))}{\\#H^{0}(\mathbb{Q}_{p},V_{f}^{(\alpha)}/T_{f}^{(\alpha)}(j))}\,.\end{aligned}$

Putting together (3.81) and (3.82) with Proposition 3.17, we infer that

$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(j),\mathds{1},0)\right|^{-1}=\left|(j-k)!\cdot
e_{p,\alpha}(f,\cdot\mathds{1},j))\right|^{-1}\frac{\\#{}_{\rm
f}(T_{f}(j))\cdot
R_{\alpha}(T_{f}(j))\cdot\mathrm{Tam}^{0}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\,.$

This concludes the proof of Theorem 3.45(i).

Let us now assume that $1\leqslant j\leqslant k-1$ and $j\neq\frac{k}{2}.$
Since $\mathrm{Exp}_{\alpha,j}$ establishes an isomorphism between
$\mathfrak{D}(T_{f}^{(\alpha)}(j))$ and
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}^{(\alpha)}(j)),$ we have

(3.83)
$\left[H^{1}(\mathbb{Q}_{p},T^{(\alpha)}_{f}(j)):\mathrm{Exp}_{\alpha,j}(N_{\alpha}[j]\otimes\Lambda)_{\Gamma}\right]=\\#H^{0}(\mathbb{Q}_{p},V^{(\beta)}_{f^{*}}/T^{(\beta)}_{f^{*}}(k-j)).$

Therefore,

(3.84)
$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}(j),\mathds{1},0)\right|^{-1}=\\\
\frac{\\#{}_{\alpha}(T_{f}(j))\cdot\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}(j))}{\\#H^{0}(V_{f}/T_{f}(j))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}(k-j))}\cdot\\#H^{0}(\mathbb{Q}_{p},V_{f}^{(\beta)}/T_{f}^{(\beta)}(j))\cdot\\#H^{0}(\mathbb{Q}_{p},V^{(\beta)}_{f^{*}}/T^{(\beta)}_{f^{*}}(k-j)).$

This completes the proof of Theorem 3.45(ii).

The assertions in (iii) and (iv) concerning
$\mathbf{L}_{\mathrm{Iw},\beta}(T_{f}(j),N_{\beta}[j],\mathds{1},0)$ can be
proved by the same argument (and, in any case, these statements are covered by
Perrin-Riou’s general descent formalism). ∎

#####

In §3.5.4, we assume that $k$ is even and consider the scenario when
$j=\frac{k}{2}$ is the central critical point for the Hecke $L$-function of
$f$.

######  Theorem 3.47.

Assume that $k\geqslant 2$ is even and that the conditions (BK)), (ht)) and
(h-PR)) hold true. Then:

We have $r_{\beta}(\frac{k}{2})=\dim_{E}H^{1}_{\rm f}(V_{f^{*}}(\frac{k}{2}))$
and

$\displaystyle\left|\mathbf{L}_{\mathrm{Iw},\beta}^{*}(T_{f}({k}/{2}),N_{\alpha}[k/2],\mathds{1},0)\right|^{-1}$
$\displaystyle\qquad\qquad\qquad\qquad=|({k}/{2}-1)!\cdot
e_{p,\beta}(f,\mathds{1},k/2))|^{-1}\cdot\frac{\\#{}_{\rm
f}(T_{f^{*}}({k}/{2}))\cdot
R_{\beta}(T_{f}({k}/{2}))\cdot\mathrm{Tam}^{0}(T_{f}({k}/{2}))}{\\#H^{0}(V_{f}/T_{f}({k}/{2}))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))}.$

We have $r_{\alpha}(\frac{k}{2})=\dim_{E}H^{1}_{0}(V_{f^{*}}(\frac{k}{2})).$
The $p$-adic height pairing

$\left<\,,\,\right>_{\alpha}:\quad H^{1}_{0}(V_{f^{*}}(k/2))\otimes
H^{1}_{0}(V_{f}(k/2))\longrightarrow E$

is non-degenerate and coincides with the restriction of
$\left<\,,\,\right>_{\beta}$ to $H^{1}_{0}(V_{f}(k/2))$. Moreover,

(3.85)
$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}({k}/{2}),N_{\alpha}[k/2],\mathds{1},0)\right|^{-1}\\\
=\frac{\\#{}_{\alpha}(T_{f}({k}/{2}))\cdot
R_{\alpha}(T_{f}({k}/{2}))\cdot\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}({k}/{2}))}{\\#H^{0}(V_{f}/T_{f}({k}/{2}))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))}\cdot
w(T_{f}^{(\beta)}({k}/{2}))\cdot w(T^{(\beta)}_{f^{*}}({k}/{2})),$

where

$w(T_{f}^{(\beta)}({k}/{2})):=\\#H^{0}(\mathbb{Q}_{p},V_{f}^{(\beta)}/T_{f}^{(\beta)}({k}/{2}))\quad\hbox{
and }\quad
w(T^{(\beta)}_{f^{*}}({k}/{2})):=\\#H^{0}(\mathbb{Q}_{p},V^{(\beta)}_{f^{*}}/T^{(\beta)}_{f^{*}}({k}/{2})).$

###### Proof.

This portion is covered by the general descent formalism of Perrin-Riou (which
our argument below to prove (ii) parallels), and its proof will be omitted.

We already proved as part of Theorem 3.44 all the assertions except for the
leading term formula.

It follows from Lemma 3.38 and our explicit description of the Bockstein map
in Theorem 3.44(iii) that

(3.86)
$\left|\mathbf{L}_{\mathrm{Iw},\alpha}^{*}(T_{f}({k}/{2}),N_{\alpha}[k/2],\mathds{1},0)\right|^{-1}=\left[{}_{\mathrm{Iw}}(T_{f}({k}/{2}))_{\Gamma}:{}_{\mathrm{Iw}}(T_{f}({k}/{2}))^{\Gamma}\right]\cdot\left[\\#\mathrm{coker}(g_{\alpha})_{\Gamma}:\\#\mathrm{coker}(g_{\alpha})^{\Gamma}\right].$

Recall that we have
$H^{1}_{u}(T_{f}(k/2))=H^{1}_{\mathrm{Iw}}(T_{f}(k/2))_{\Gamma}$ under our
assumptions; cf. (3.74). We use the following computation from [PR95, Section
3.3.4] (see also [Col00], the discussion following Proposition 3.5):

(3.87) $\displaystyle\begin{aligned}
&\left[{}_{\mathrm{Iw}}(T_{f}({k}/{2}))_{\Gamma}:{}_{\mathrm{Iw}}(T_{f}({k}/{2}))^{\Gamma}\right]=\\\
&\hskip
19.91684pt(\log\chi(\gamma_{1}))^{-r_{\alpha}(k/2)}\cdot\frac{\\#{}_{0}(T_{f^{*}}({k}/{2}))\mathrm{Tam}^{0}_{\\{p\\}}(T_{f}({k}/{2}))\left[H^{1}(\mathbb{Q}_{p},T_{f}({k}/{2})):H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}({k}/{2}))_{\Gamma}\right]}{\\#H^{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))}\\\
&\hskip
227.62204pt\times\left|\left<H^{1}_{0}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}},\frac{H^{1}_{\\{p\\}}(T_{f}({k}/{2}))}{H^{1}(T_{f}(k/2))^{\rm
u}}\right>_{\alpha}\right|^{-1}.\end{aligned}$

Note that we have used here the fact that
$H^{1}_{\\{p\\}}(T_{f}(\frac{k}{2}))=H^{1}_{\rm f}(T_{f}(\frac{k}{2}))$ under
our running assumptions. Let us write

(3.88)
$\left<H^{1}_{0}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}},\frac{H^{1}_{\\{p\\}}(T_{f}({k}/{2}))}{H^{1}(T_{f}(k/2))^{\rm
u}}\right>_{\alpha}=\\\
\frac{\left<H^{1}_{0}(T_{f^{*}}({k}/{2}))_{\mathrm{tf}},H^{1}_{0}(T_{f}({k}/{2}))_{\mathrm{tf}}\right>_{\alpha}}{\left[H^{1}_{\\{p\\}}(T_{f}({k}/{2}))_{\mathrm{tf}}:H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tf}}+H^{1}_{0}(T_{f}(k/2))_{\mathrm{tf}}\right]}\cdot\frac{\\#H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tor}}}{\\#H^{0}(V_{f}/T_{f}(k/2))}\,.$

The formula (3.79) applies in our case, and mimicking the computations from
the proof of Theorem 3.45, we deduce that

(3.89) $\displaystyle\begin{aligned}
\frac{\\#\mathrm{coker}(g_{\alpha})_{\Gamma}}{\\#\mathrm{coker}(g_{\alpha})^{\Gamma}}\cdot&\left[H^{1}(\mathbb{Q}_{p},T_{f}({k}/{2})):H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{f}({k}/{2}))_{\Gamma}\right]=w(T_{f}^{(\beta)}(k/2))w(T_{f^{*}}^{(\beta)}(k/2))\\\
&\hskip
106.69783pt\times\frac{\left[H^{1}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}:H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tf}}+H^{1}_{\alpha}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}\right]}{\\#H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tor}}}\,.\end{aligned}$

It is easy to check that

(3.90) $\displaystyle\begin{aligned}
&\frac{\left[H^{1}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}:H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tf}}+H^{1}_{\alpha}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}\right]}{\left[H^{1}_{\\{p\\}}(T_{f}({k}/{2}))_{\mathrm{tf}}:H^{1}(T_{f}(k/2))^{\rm
u}_{\mathrm{tf}}+H^{1}_{0}(T_{f}(k/2))_{\mathrm{tf}}\right]}=\\\ &\hskip
113.81102pt\left[H^{1}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}:H^{1}_{\alpha}(\mathbb{Q}_{p},T_{f}(k/2))_{\mathrm{tf}}+\frac{H^{1}_{\\{p\\}}(T_{f}({k}/{2}))_{\mathrm{tf}}}{H^{1}_{0}(T_{f}(k/2))_{\mathrm{tf}}}\right]\,.\end{aligned}$

The formulae (3.87)–(3.90) together with Proposition 3.39 show that

(3.91)
$\left[{}_{\mathrm{Iw}}(T_{f}({k}/{2}))_{\Gamma}:{}_{\mathrm{Iw}}(T_{f}({k}/{2}))^{\Gamma}\right]\cdot\left[\\#\mathrm{coker}(g_{\alpha})_{\Gamma}:\\#\mathrm{coker}(g_{\alpha})^{\Gamma}\right]\\\
\qquad\quad\sim_{p}(\log\chi(\gamma_{1}))^{-r_{\alpha}(k/2)}\times\frac{{}_{\alpha}(T_{f^{*}}(k/2))\cdot
R_{\alpha}(T_{f}(k/2))\cdot\mathrm{Tam}_{\\{p\\}}^{0}(T_{f}(k/2))}{\\#H^{0}(V_{f}/T_{f}({k}/{2}))\cdot\\#H^{0}(V_{f^{*}}/T_{f^{*}}({k}/{2}))}\cdot
w(T_{f}^{(\beta)}({k}/{2}))\cdot w(T^{(\beta)}_{f^{*}}({k}/{2})).$

Part (ii) of our theorem follows from this computation and (3.86). ∎

##  Chapter 4 Main Conjectures for the infinitesimal deformation

In this chapter, we develop Iwasawa theory for the infinitesimal deformation
$V_{k}$ of Deligne’s representation $V_{f}$ along the eigencurve. This
involves the definition of the _thick_ fundamental line in §4.1 in this
context, which we then use to define the infinitesimal thickening of the
module of algebraic $p$-adic $L$-functions; cf. Equation (4.12).

The infinitesimal thickening $\widetilde{\mathbf{MC}}(f)$ of the main
conjecture asserts that the infinitesimal thickening of the module of
algebraic $p$-adic $L$-functions is generated by the thick $p$-adic
$L$-function
$\widetilde{L}_{\mathrm{S},\alpha^{*}}^{\pm}(f^{*},\xi^{*})^{\iota}$ up to a
canonical fudge factor.

To prove the leading term formulae for the infinitesimal thickening of the
module of algebraic $p$-adic $L$-functions (cf. Theorem 4.14), we rely on the
Iwasawa descent formalism for the infinitesimal thickening of the module of
algebraic $p$-adic $L$-functions, which we develop in §4.2.5. Our descent
formalism dwells on the general discussion in §4.2.3 and the integrality
properties we establish in the opening section of this chapter (cf. Lemma
4.1).

As a further motivation for the contents of the present chapter, we recall
from Chapter 3 the degenerate behaviour of the Selmer complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}(T_{f},\alpha)$ when $f_{\alpha}$ is
$\theta$-critical. The key insight in our study, as we shall illustrate in the
present chapter as well as the next, is that the Selmer complex attached to
the infinitesimal deformation $V_{k}$ (which we call the thick Selmer complex)
encodes strictly more information. This is illustrated by Theorems 4.14 and
4.15, which tell us that the thick Selmer complex “sees” also the special
values of the secondary $p$-adic $L$-function
$L^{[1]}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})^{\pm}$.

Recall that we have $V_{f}\simeq V_{x_{0}}$, where $x_{0}$ is the critical
point on the eigencurve which corresponds to $f_{\alpha}$. We assume
throughout this chapter that the ramification index of the weight map
$\mathcal{X}\rightarrow\mathcal{W}$ at $x_{0}$ is equal to $e=2$. Recall the
tautological exact sequence

(4.1) $0\longrightarrow V_{f}\longrightarrow V_{k}\longrightarrow
V_{f}\longrightarrow 0,$

where the copy of $V_{f}$ on the left arises from its identification with
$V[x_{0}]$.

We note that $V_{k}$ can be viewed as a Galois representation with
coefficients in the ring $\widetilde{E}:=E[X]/(X^{2})$. One may also forget
the Hecke-module structure and consider $V_{k}$ as a $4$-dimensional Galois
representation over $E$.

###  4.1. The fundamental line

####  4.1.1. Lattices

We recall various definitions from §2.4.5 for the convenience of the reader.
Let us put $\widetilde{E}:=E[X]/X^{2}$. For each $n\geqslant 0,$ recall the
unital $O_{E}$-subalgebra

$\mathcal{O}_{E}^{(n)}=O_{E}+\varpi_{E}^{-n}O_{E}X\,\subset E[X]/X^{2}$

of $\widetilde{E}$, where $\varpi_{E}$ is a uniformizer of $E$. Set

$\mathcal{O}_{E}^{(\infty)}:=\underset{n\geqslant
1}{\cup}\mathcal{O}_{E}^{(n)}.$

Let $T_{f}$ be an $\mathcal{O}_{E}$-lattice in $V_{f}$ which is stable under
the action of $G_{\mathbb{Q},S}.$

######  Lemma 4.1.

The following assertions hold true:

For any $n\gg 0$ there exists a free $\mathcal{O}_{E}^{(n)}$-submodule
$T_{k}^{(n)}\subset V_{k}$ of rank two such that

  * L1)

$T_{k}^{(n)}$ is stable under the action of $G_{\mathbb{Q},S}$ ,

  * L2)

$T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}=T_{f}$.

Suppose that the natural number $n$ verifies the conclusions of Part (i). Let
$m$ be another natural number such that $T_{k}^{(m)}$ is a free
$\mathcal{O}_{E}^{(m)}$-submodule of $V_{k}$ satisfying the properties
(L1))–(L2)) with $n$ replaced by $m$. Then there exists
$s\geqslant\max\\{m,n\\}$ such that

$T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}^{(s)}_{E}=T_{k}^{(m)}\otimes_{\mathcal{O}_{E}^{(m)}}\mathcal{O}^{(s)}_{E}.$

###### Proof.

Let $\\{v_{1},v_{2}\\}$ be a basis of $T_{f}.$ Fix arbitrary lifts
$\widetilde{v}_{1},\widetilde{v}_{2}\in V_{k}$ of $v_{1},v_{2}$ under the
surjection $V_{k}\twoheadrightarrow V_{f}$ and set
$\widetilde{T}=\mathcal{O}_{E}^{(0)}\widetilde{v}_{1}+\mathcal{O}_{E}^{(0)}\widetilde{v}_{2}$.
Note that we have

$g\cdot\widetilde{v}_{i}=a_{i,1}(g)\widetilde{v}_{1}+a_{i,2}(g)\widetilde{v}_{2}+X(b_{i,1}(g)\widetilde{v}_{1}+b_{i,2}(g)\widetilde{v}_{2})$

with $a_{i,j}(g)\in\mathcal{O}_{E}$ for any $g\in G_{\mathbb{Q},S}$, since
$\widetilde{v}_{i}\mapsto v_{i}$ under the $G_{\mathbb{Q},S}$-equivariant
morphism $V_{k}\twoheadrightarrow V_{f}$. This discussion shows that

$g\cdot\widetilde{v}_{i}\in\bigcup_{m\geq
0}(\widetilde{T}+\varpi_{E}^{-m}X\widetilde{T})\,.$

It follows from the compactness of the Galois group that there exists
$n\geqslant 0$ such that

$g\cdot\widetilde{v}_{i}\in\widetilde{T}+{\varpi_{E}^{-n}}X\widetilde{T},\qquad\forall
g\in G_{\mathbb{Q},S},\quad i=1,2.$

Set
$T_{k}^{(n)}=\mathcal{O}_{E}^{(n)}\widetilde{v}_{1}+\mathcal{O}_{E}^{(n)}\widetilde{v}_{2}$.
Thanks to our choice of $n$, the ${\mathcal{O}}_{E}^{(n)}$-module
$T_{k}^{(n)}$ is $G_{\mathbb{Q},S}$-stable. Furthermore, we have
$T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}O_{E}=T_{f}$ by construction and
our proof is complete.

It is clear that for each $m\geq n$, the $\ \mathcal{O}_{E}^{(m)}$-module
$T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(m)}$ verifies the
properties (L1))–(L2)). This proves the first part of the lemma.

It follows from the compactness that there exists $s\geqslant\max\\{m,n\\}$
such that

$T_{k}^{(n)}\subset
T_{k}^{(m)}\otimes_{\mathcal{O}_{E}^{(m)}}\mathcal{O}_{E}^{(s)}$ and
$T_{k}^{(m)}\subset
T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(s)}.$

Then

$T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(s)}\subset
T_{k}^{(m)}\otimes_{\mathcal{O}_{E}^{(m)}}\mathcal{O}_{E}^{(s)}\subset
T_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(s)},$

and the proposition is proved. ∎

Recall that
$\Lambda^{(n)}:=\mathcal{O}_{E}^{(n)}\otimes_{\mathcal{O}_{E}}\Lambda$ and let
us set
$\widetilde{\mathscr{H}}(\Gamma)=\widetilde{E}\otimes_{E}\mathscr{H}(\Gamma)$.
We have decompositions of these algebras into $\Delta$-isotypic components:

$\Lambda^{(n)}=\underset{i=1}{\overset{p-1}{\oplus}}\Lambda^{(n)}_{\omega^{i}},\qquad\widetilde{\mathscr{H}}(\Gamma)=\underset{i=1}{\overset{p-1}{\oplus}}\widetilde{\mathscr{H}}_{\omega^{i}}(\Gamma).$

Let us put

$\Lambda^{(\infty)}:=\mathcal{O}_{E}^{(\infty)}\otimes_{\mathcal{O}_{E}}\Lambda=\varinjlim_{n}\Lambda^{(n)}.$

####  4.1.2. Complexes

#####

Let us fix a natural number $n$ for which the properties (L1))–(L2)) are
verified. Recall that $\iota$ denotes the canonical involution on $\Lambda.$
We consider the global and local Iwasawa cohomology complexes

(4.2)
$\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{k}^{(n)}(j))\,:=\mathbf{R\Gamma}(G_{\mathbb{Q},S},T_{k}^{(n)}(j)\otimes_{\mathcal{O}_{E}}\Lambda^{\iota})\,,\qquad\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\,:=\,\mathbf{R\Gamma}(G_{\mathbb{Q}_{\ell}},T_{k}^{(n)}(j)\otimes_{\mathcal{O}_{E}}\Lambda^{\iota})$

as objects of the derived category $\mathscr{D}(\Lambda^{(n)})$ of
$\Lambda^{(n)}$-modules. One then has

$\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\Lambda\simeq\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{f}(j))\,,\qquad\mathbf{R\Gamma}_{{\mathrm{Iw}}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\Lambda\simeq\mathbf{R\Gamma}_{{\mathrm{Iw}}}(\mathbb{Q}_{\ell},T_{f}(j)).$

#####

Since the action of $G_{\mathbb{Q},S}$ (respectively $G_{\mathbb{Q}_{\ell}}$)
on $V_{k}$ factors through a quotient containing a pro-$p$-group of finite
index, the following properties follow from the general results of Flach
[Fla00] (see especially Propositions 4.1, 4.2, 5.2 and Remark 3.1 in op.
cit.):

  * $\bullet$

The complexes $\mathbf{R\Gamma}_{S}(T_{k}^{(n)}(j))$ and
$\mathbf{R\Gamma}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))$ ($\ell\in S$) are perfect
objects in the derived category $\mathscr{D}(\mathcal{O}_{E}^{(n)})$ of
$\mathcal{O}_{E}^{(n)}$-modules. The complexes
$\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{k}^{(n)}(j))$ and
$\mathbf{R\Gamma}_{{\mathrm{Iw}}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))$ are
perfect objects in the derived category $\mathscr{D}(\Lambda^{(n)})$ of
$\Lambda^{(n)}$-modules.

  * $\bullet$

For $m\geqslant n,$ we have

$\displaystyle\mathbf{R\Gamma}_{S}(T_{k}^{(n)}(j))\otimes_{\mathcal{O}_{E}^{(n)}}^{\mathbf{L}}\mathcal{O}_{E}^{(m)}\simeq\mathbf{R\Gamma}_{S}(T_{k}^{(n)}(j)\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(m)}),$
$\displaystyle\mathbf{R\Gamma}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\otimes_{\mathcal{O}_{E}^{(n)}}^{\mathbf{L}}\mathcal{O}_{E}^{(m)}\simeq\mathbf{R\Gamma}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j)\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(m)}),$
$\displaystyle\mathbf{R\Gamma}_{\mathrm{Iw},S}(T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\Lambda^{(m)}\simeq\mathbf{R\Gamma}_{\mathrm{Iw},S}(T_{k}^{(n)}(j)\otimes_{\Lambda^{(n)}}\Lambda^{(m)}),$
$\displaystyle\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\Lambda^{(m)}\simeq\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j)\otimes_{\Lambda^{(n)}}\Lambda^{(m)}).$

#####

Set $D_{k}:={\mathscr{D}}_{\mathrm{cris}}\left(\mathbf{D}_{k}\right).$ Recall
that we denote by $N_{\alpha}$ the $\mathcal{O}_{E}$-lattice of
$D^{(\alpha)}:={\mathbf{D}}_{\mathrm{cris}}(V_{f}^{(\alpha)})$ generated by
$\eta_{f}^{\alpha}$. Let us fix a free $\mathcal{O}_{E}^{(n)}$-submodule
$N_{k}^{(n)}\subset D_{k}$ of rank one such that

$N_{k}^{(n)}\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}=N_{\alpha}.$

For each non-archimedean place $\ell\neq p$, we consider the usual unramified
local condition at $\ell$:

$\mathbf{R\Gamma}_{\text{\rm
Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}):=\mathbf{R\Gamma}_{{\mathrm{Iw}},\rm
f}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j)):=\,\left[(T_{k}^{(n)}(j))^{I_{\ell}}\otimes\Lambda^{\iota}\xrightarrow{1-f_{\ell}}(T_{k}^{(n)}(j))^{I_{\ell}}\otimes\Lambda^{\iota}\right],$

where $I_{\ell}$ is the inertia subgroup at $\ell$ and $f_{\ell}$ is the
geometric Frobenius.

#####

At $\ell=p$, we shall consider the following Perrin-Riou–style local
conditions, see [Ben14] for a further discussion. Let
$\mathbf{D}^{\dagger}_{E}$ denote the functor which associates to a $p$-adic
$G_{\mathbb{Q}_{p}}$-representation its overconvergent
$(\varphi,\Gamma)$-module [CC98]. There exists a quasi-isomorphism

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j))\simeq\left[\mathbf{D}^{\dagger}_{E}(T_{k}^{(n)}(j))\xrightarrow{\psi-1}\mathbf{D}^{\dagger}_{E}(T_{k}^{(n)}(j))\right],$

in the derived category of $\Lambda^{(n)}$-modules, cf. [Ben14, Proposition
A5]. Since $\mathscr{H}(\Gamma)$ is flat over $\Lambda,$ we have

(4.3)
$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\widetilde{\mathscr{H}}(\Gamma)\simeq\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j))\otimes_{\Lambda}^{\mathbf{L}}\mathscr{H}(\Gamma)\\\
\simeq\left[\mathbf{D}^{\dagger}_{E}(T_{k}^{(n)}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\xrightarrow{\psi-1}\mathbf{D}^{\dagger}_{E}(T_{k}^{(n)}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\right].$

Moreover, as
$\mathbf{D}^{\dagger}_{E}(T_{k}^{(n)}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\simeq\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V_{k}(j))$
(cf. [Pot12]), we have the following isomorphism in the derived category
$\mathscr{D}(\mathscr{H}(\Gamma))$ of $\mathscr{H}(\Gamma)$-modules:

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}^{\mathbf{L}}\widetilde{\mathscr{H}}(\Gamma)\simeq\left[\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V_{k}(j))\xrightarrow{\psi-1}\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V_{k}(j))\right].$

Let us set

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}):=({N_{k}^{(n)}}[j]\otimes_{\mathcal{O}_{E}}\Lambda)[-1].$

We will denote the Perrin-Riou exponential map

$\text{\rm
Exp}_{\mathbf{D}_{k}(j),j}\,:\,\mathfrak{D}(\mathbf{D}_{k}[j])\longrightarrow\mathbf{D}^{\dagger}_{\mathrm{rig},E}(V_{k}(j))^{\psi=1}$

simply by $\text{\rm Exp}_{\mathbf{D}_{k}(j)}.$ The map $\text{\rm
Exp}_{\mathbf{D}_{k}(j)}$ induces a morphism

$\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]})\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma),$

which we shall take as the local condition at $p.$

#####

Consider the diagram

$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma)}$$\textstyle{\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]})\right)\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma)\,\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note that the objects of this diagram depend on $V_{k}$ and $D_{k}$ rather
than $T_{k}^{(n)}$ and $N_{k}^{(n)}$ (since $p$ is inverted), and we denote by
$\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)$ the Selmer
complex associated to this diagram. By definition, the complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)$ (to which we
henceforth refer as the _thick_ Selmer complex) sits in the following
distinguished triangle:

(4.4) $\displaystyle\begin{aligned}
\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)\longrightarrow&\left(\mathbf{R\Gamma}_{\mathrm{Iw},S}(T_{k}^{(n)}(j))\bigoplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}\right)\right)\right)\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma)\\\
&\qquad\qquad\qquad\qquad\qquad\longrightarrow\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{k}^{(n)}(j))\right)\otimes_{\Lambda^{(n)}}\widetilde{\mathscr{H}}(\Gamma)\xrightarrow{[+1]}\,.\end{aligned}$

Let us define the fundamental line
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),{N_{k}^{(n)}}[j]\right)$ on setting

$\displaystyle\Delta_{\mathrm{Iw}}$
$\displaystyle\left(T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}\right):=$
$\displaystyle{\det}^{-1}_{\Lambda^{(n)}}\left(\mathbf{R\Gamma}_{\mathrm{Iw},S}(T_{k}^{(n)}(j))\bigoplus\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}\right)\right)\right)\otimes{\det}_{\Lambda^{(n)}}\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}\left(\mathbb{Q}_{\ell},T_{k}^{(n)}(j)\right)\right).$

It follows from the exact triangle (4.4) that we have a natural injection

(4.5)
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),{N_{k}^{(n)}[j]}\right)\longrightarrow{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma)}\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)\,.$

#####

Suppose $m\geqslant n$ are natural numbers both verifying the conditions (L1))
and (L2)). We then have an isomorphism

(4.6)
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(m)}\simeq\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j)\otimes_{\mathcal{O}_{E}^{(n)}}\mathcal{O}_{E}^{(m)},N_{k}^{(m)}[j]\right),$

which is compatible with the morphisms (4.5) in the obvious sense. Set

(4.7)
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right):=\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)\otimes_{\mathcal{O}_{E}^{(n)}}{\mathcal{O}_{E}^{(\infty)}}.$

Since the rings $\mathcal{O}_{E}^{(n)}$ and $\mathcal{O}_{E}^{(\infty)}$ are
local, $\Delta_{\mathrm{Iw}}\left(T_{k}^{(n)}(j),N_{k}^{(n)}[j]\right)$ and
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right)$ are
free modules of rank one over $\mathcal{O}_{E}^{(n)}$ and
$\mathcal{O}_{E}^{(\infty)}$, respectively. It follows from (4.6) and Lemma
4.1(ii) that the module
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right)$ (to
which we henceforth refer to the _thick_ fundamental line) does not, up to
canonical isomorphism, depend on the choice of $n$ and the lift $T_{k}^{(n)}$
of $T_{f}$. By extension of scalars, the map (4.5) induces a map

(4.8)
$\Theta_{\mathrm{Iw},V_{k}(j)}^{(\alpha)}\,:\,\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right)\longrightarrow{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma)}\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)\,.$

#####

Recall that

$N_{\alpha}\otimes_{\mathcal{O}_{E}}E=D^{(\alpha)}.$

Consider the Selmer complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{f}(j),\alpha\right)$ of
$\mathscr{H}_{E}(\Gamma)$-modules associated to the diagram

$\textstyle{\mathbf{R\Gamma}_{{\mathrm{Iw}},S}(T_{f}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j))\otimes_{\Lambda}\mathscr{H}(\Gamma)}$$\textstyle{\left(\underset{\ell\in
S}{\bigoplus}\mathbf{R\Gamma}_{\mathrm{Iw}}(\mathbb{Q}_{\ell},T_{f}(j),\alpha)\right)\otimes_{\Lambda}\mathscr{H}(\Gamma)\,\,,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which was introduced in Section 3.3.1.

The tautological exact sequence (4.1) induces a distinguished triangle

(4.9)
$\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{f}(j),\alpha\right)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{f}(j),\alpha\right)\longrightarrow\mathbf{R\Gamma}_{\mathrm{Iw}}\left(V_{f}(j),\alpha\right)[1]\,.$

####  4.1.3. Trivialization of the thick fundamental line

#####

Our goal in §4.1.3 is to establish an auxiliary statement (Proposition 4.3
below) that we will use to trivialize the thick fundamental line
$\Delta_{\mathrm{Iw}}(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j])$.

Let $R$ be a Bézout ring, which is simultaneously an $E$-algebra. Set
$\widetilde{R}:=R\otimes_{E}\widetilde{E}$. Then
$\widetilde{R}/X\widetilde{R}=R.$ If $\widetilde{M}$ is a
$\widetilde{R}$-module, we can and we will consider
$M:=\widetilde{M}/X\widetilde{M}$ as an $R$-module. Note that
$X^{2}\widetilde{M}=0.$ We shall consider $\widetilde{R}$-modules satisfying
the following additional condition:

  * (M)

$\ker({\widetilde{M}}\xrightarrow{X}{\widetilde{M}})=X{\widetilde{M}}\,.$

Note that each free $\widetilde{R}$-module verifies (M).

######  Lemma 4.2.

The following two conditions are equivalent:

$\widetilde{M}$ satisfies (M) .

The sequence

${0\longrightarrow M\longrightarrow\widetilde{M}\longrightarrow
M\longrightarrow 0,}$

where the map $M:=\widetilde{M}/X\widetilde{M}\to\widetilde{M}$ is induced by
multiplication by $X$, is exact.

###### Proof.

This is clear and the proof is omitted. ∎

######  Proposition 4.3.

Suppose that $\widetilde{M}$ is a finitely generated $\widetilde{R}$-module
satisfying (M). If $M$ is finitely presented over $R$, then $\widetilde{M}$
admits a free resolution of length $1$.

###### Proof.

Since $R$ is a Bézout ring and $M$ is finitely presented, it has a free
resolution

$0\longrightarrow P_{1}\longrightarrow P_{0}\longrightarrow M\longrightarrow
0$

of length 1 (cf. §3.3.2). We will construct a diagram of the form

(4.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{P}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}_{1}}$$\scriptstyle{r_{1}}$$\textstyle{\widetilde{P}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}_{0}}$$\scriptstyle{r_{0}}$$\textstyle{\widetilde{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

where $\widetilde{P}_{i}$ ($i=0,1$) are free $\widetilde{R}$-modules. Fix a
basis $\\{e_{0,i}\\}_{i}$ of $P_{0}.$ Let
$\widetilde{P}_{0}=\underset{i}{\oplus}\widetilde{R}\,\widetilde{e}_{0,i}$ be
an arbitrary free $\widetilde{R}$-module of rank ${\rm rank}_{R}\,P_{i}$,
generated by some set $\\{\widetilde{e}_{0,i}\\}_{i}.$ Let us denote by
${r_{0}\,:\,\widetilde{P}_{0}\rightarrow P_{0}}$ the map given by

$\sum_{i}\widetilde{a}_{i}\widetilde{e}_{0,i}\longmapsto\sum_{i}a_{i}e_{0,i},$

where $a_{i}$ stands for the image of $\widetilde{a}_{i}$ under the projection
$\widetilde{R}\twoheadrightarrow R$.

Let $\widetilde{m}_{i}\in\widetilde{M}$ be any lift of
$m_{i}:=f_{0}(e_{0,i})\in M$. We define $\widetilde{f}_{0}$ as the unique
$\widetilde{R}$-linear map such that
$\widetilde{f}_{0}(\widetilde{e}_{0,i})=\widetilde{m}_{i}$ for all $i.$ It is
easy to see by Nakayama’s lemma that $\widetilde{f}_{0}$ is surjective and the
square on the right in diagram (4.10) commutes.

Similarly, fix a basis $\\{e_{1,j}\\}_{j}$ of $P_{1}$ and set
$\widetilde{P}_{1}=\underset{j}{\oplus}\widetilde{A}\widetilde{e}_{1,j}.$ We
define the map $r_{1}\,:\,\widetilde{P}_{1}\rightarrow P_{1}$ as before, by
setting $r_{1}(\widetilde{e}_{1,j})=e_{1,j}.$ Take any lifts
$\widetilde{n}_{j}\in\widetilde{P}_{0}$ of the elements
$n_{j}:=f_{1}(e_{1,j})$ and define the $\widetilde{R}$-linear map
$g\,:\,\widetilde{P}_{1}\rightarrow P_{0}$ setting
$g(\widetilde{e}_{1,j})=\widetilde{n}_{j}.$ If we take $\widetilde{f}_{1}=g,$
then the square on the left in diagram (4.10) commutes, but the upper row is
not necessarily exact, and we therefore need to modify $g$.

For any $j$, it follows from the constructions above that
$\widetilde{f}_{0}\circ g(\widetilde{e}_{1,j})\in X\widetilde{M}.$ Hence
$\widetilde{f}_{0}\circ g(\widetilde{e}_{1,j})=Xx_{j}$ for some
$x_{j}\in\widetilde{M}.$ Take $y_{j}\in\widetilde{P}_{0}$ such that
$\widetilde{f}_{0}(y_{j})=x_{j}.$ Define the map $\widetilde{f}_{1}$ setting

$\widetilde{f}_{1}(\widetilde{e}_{1,j})=g(\widetilde{e}_{1,j})-Xy_{j},\qquad\forall
j\,.$

Then $\widetilde{f}_{0}\circ\widetilde{f}_{1}=0.$ In addition, since
$\widetilde{f}_{1}\equiv g\pmod{X},$ the diagram (4.10) is commutative.

We only need to check that the upper row of diagram (4.10) is exact. To that
end, let us first assume that $x\in\ker(\widetilde{f}_{1})$. Write
$x=\sum_{j}\widetilde{a}_{j}\widetilde{e}_{1,j}.$ Then
$r_{1}(x)\in\ker(f_{1}),$ and therefore $\sum_{j}a_{j}e_{1,j}=0,$ where
$a_{j}\in\mathcal{O}_{\mathcal{W}}$ denotes the image of $\widetilde{a}_{j}$.
We therefore infer that $a_{j}=0$ and hence $\widetilde{a}_{j}\in
X\widetilde{R}$ for all $j.$ Writing $\widetilde{a}_{j}=X\widetilde{b}_{j}$,
we see that $x=Xy$ for
$y=\sum_{j}\widetilde{b}_{j}\widetilde{e}_{1,j}\in\widetilde{P}_{1},$ and
$X\widetilde{f}_{1}(y)=0$ in $\widetilde{P}_{0}.$ Since $\widetilde{P}_{0}$
satisfies the condition (M), this implies that $\widetilde{f}_{1}(y)\in
X\widetilde{P_{0}},$ and therefore once again that $\widetilde{b}_{j}\in
X\widetilde{R}.$ This shows that $\widetilde{a}_{j}=0$ for all $j$, and hence
$x=0$.

We next check the exactness of the upper row at $\widetilde{P}_{0}$. Assume
that $x\in\widetilde{P}_{0}$ is such that $\widetilde{f}_{0}(x)=0$. It follows
from the commutativity of (4.10) and the exactness of the bottom row that
there exists $y\in\widetilde{P}_{1}$ such that $\widetilde{f}_{1}(y)\equiv
x\pmod{X\widetilde{P}_{0}}$. We may therefore assume without loss of
generality that $x\in X\widetilde{P}_{0}.$ Write $x=Xx^{\prime}$ with
$x^{\prime}\in\widetilde{P}_{0}.$ Then $X\widetilde{f}_{0}(x^{\prime})=0$ in
$\widetilde{M}.$ Since $\widetilde{M}$ satisfies (M), this implies that
$\widetilde{f}_{0}(x^{\prime})\in X\widetilde{M}.$ Hence $f_{0}\circ
r_{0}(x^{\prime})=0$ and it follows from the commutativity of (4.10) that
there exists $z\in\widetilde{P}_{1}$ such that $\widetilde{f}_{1}(z)\equiv
x^{\prime}\pmod{X}.$ Therefore $\widetilde{f}_{1}(Xz)=Xx^{\prime}=x,$ and the
exactness is checked. ∎

#####

Suppose that $\mathscr{L}_{\mathrm{Iw}}^{\rm cr}(V_{f})\neq 0$. By (3.53),
$\mathbf{R}^{i}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)=0$ for all
$i\neq 2$, and it follows from the global Euler–Poincaré characteristic
formula that
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)$ is torsion.
The distinguished triangle (4.9) then reduces to an exact sequence

$0\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\xrightarrow{[X]}\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)\longrightarrow\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)\longrightarrow
0.$

We infer using Lemma 4.2 that
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)$ verifies
the condition (M). By Proposition 3.36, for each $\eta\in X(\Delta),$ the
$\eta$-isotypic component
$\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}(V_{f}(j),\alpha)^{(\eta)}$ is
a finitely presented torsion module over $\mathscr{H}(\Gamma)_{\eta}$.
Therefore, for every integer $j$, we have a canonical trivialization (3.50)

(4.11)
$i_{\mathrm{Iw},V_{k}(j)}^{(\alpha)}\,:\,{\det}_{\widetilde{\mathscr{H}}(\Gamma)}^{-1}\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right)\xhookrightarrow{\quad}\widetilde{\mathscr{H}}(\Gamma)\,.$

These maps are compatible with one another under twisting.

Paralleling the discussion in §3.3, the map (4.5) allows us to consider
$\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right)$ as
a ${\Lambda}^{(\infty)}$-submodule of
${\det}_{\widetilde{\mathscr{H}}(\Gamma)}^{-1}\mathbf{R}^{2}\boldsymbol{\Gamma}_{\mathrm{Iw}}\left(V_{k}(j),\alpha\right).$

######  Definition 4.4.

We define the free $\Lambda^{(\infty)}$-module of rank one

(4.12)
${\mathbf{L}}_{\mathrm{Iw},\alpha}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right):=i_{\mathrm{Iw},V_{k}(j)}^{(\alpha)}\left(\Delta_{\mathrm{Iw}}\left(T_{k}^{(\infty)}(j),N_{k}^{(\infty)}[j]\right)\right)$

and call it the infinitesimal thickening of the module of algebraic $p$-adic
$L$-functions associated with $T_{k}^{(\infty)}(j)$.

As we have remarked at the start of Chapter 3, this terminology is inspired by
that of Perrin-Riou [PR95, §2]. The nomenclature can be justified by the
Iwasawa main conjecture $\widetilde{\mathbf{MC}}(f)$ below, which predicts
that this module is indeed generated by the infinitesimal thickening of the
critical $p$-adic $L$-function. Its subtle relation with the modules of
“punctual” $p$-adic $L$-functions (that are given by Definition 3.30) is the
subject of Theorem 4.14.

###  4.2. Main Conjectures for the infinitesimal thickening of critical
$p$-adic $L$-functions

#### 4.2.1.

We are now ready to formulate the infinitesimal thickening of the Iwasawa Main
Conjecture for $\theta$-critical forms $\mathbf{MC}(f,\alpha)$. Recall that we
denote by $\widetilde{\mathcal{E}}_{N}$ the image of
$\mathcal{E}_{N}=\underset{\ell\mid
N}{\prod}(1-a_{\ell}(x)\sigma_{\ell}^{-1})$ in $\Lambda^{(\infty)}.$

######  Conjecture 4.5.

We have $\mathscr{L}_{\mathrm{Iw}}^{\rm cr}(V_{f})\neq 0$ and

  * $\widetilde{\mathbf{MC}}(f)$

${\mathbf{L}}_{\mathrm{Iw},\alpha}\left(T_{k}^{(\infty)}(k),N_{k}^{(\infty)}[k]\right)^{\pm}=\left(\lambda^{\pm}(f^{*})\cdot\widetilde{L}_{\mathrm{S},\alpha^{*}}^{\pm}(f^{*},\xi^{*})^{\iota}\right)\,,$

where
$\left(\lambda^{\pm}(f^{*})\cdot\widetilde{L}_{\mathrm{S},\alpha^{*}}^{\pm}(f^{*},\xi^{*})^{\iota}\right)$
denotes the $\Lambda^{(\infty),\pm}$-module generated by
$\lambda^{\pm}(f^{*})\cdot\widetilde{L}_{\mathrm{S},\alpha^{*}}^{\pm}(f^{*},\xi^{*})^{\iota}.$

Using Proposition 2.32, this conjecture can be formulated in terms of Kato’s
$p$-adic $L$-function:

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

#### 4.2.2.

In what follows, we shall explore various consequences of Conjecture 4.5.

######  Proposition 4.6.

Conjecture $\widetilde{\mathbf{MC}}(f)$ implies $\mathbf{MC}(f,\alpha)$.

###### Proof.

This follows from constructions of Section 3.4.2 and the following commutative
diagram, where the vertical morphisms are induced by
$\otimes^{\mathbf{L}}_{\widetilde{\mathscr{H}}(\Gamma)}\mathscr{H}(\Gamma)$:

$\textstyle{\Delta_{\mathrm{Iw}}(T_{k}^{(\infty)}(k),N_{k}^{(\infty)}[k])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{\mathrm{Iw},V_{k}(k)}^{(\alpha)}}$$\textstyle{{\det}_{\widetilde{\mathscr{H}}(\Gamma)}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(k),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathrm{Iw},V_{k}(k)}^{(\alpha)}}$$\textstyle{\widetilde{\mathscr{H}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Delta_{\mathrm{Iw}}(T_{f}(k),N_{\alpha}[k])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{\mathrm{Iw},V_{f}(k)}^{(\alpha)}}$$\textstyle{{\det}_{\mathscr{H}(\Gamma)}^{-1}\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{f}(k),\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\mathrm{Iw},V_{f}(k)}^{(\alpha)}}$$\textstyle{\mathscr{H}(\Gamma).}$

∎

####  4.2.3. Descent formalism over $\widetilde{\mathscr{H}}(\Gamma)$

#####

Our main aim in the remainder of this chapter is to prove that Conjecture
$\widetilde{\mathbf{MC}}(f)$ is consistent with the slope-zero Main Conjecture
$\mathbf{MC}(f,\beta)$ and the interpolation properties of the secondary
$p$-adic $L$-function $L^{[1]}_{\mathrm{K},\alpha^{*}}(f^{*},\xi^{*})$. We
remark that the thick Selmer complex
$\mathbf{R\Gamma}_{\mathrm{Iw}}(V_{k}(j),\alpha)$ is not semi-simple, namely
the natural map

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

is not an isomorphism.

Our goal in §4.2.3 is to prove an analog of the descent formula (3.60) for
$\widetilde{\mathscr{H}}(\Gamma)$-modules.

#####

To simplify notation, let us put $z:=\gamma_{1}-1.$ Let $\widetilde{M}$ and
$\widetilde{N}$ be free $\widetilde{\mathscr{H}}(\Gamma_{1})$-modules of
finite rank $h$ and let

$\phi\,:\,\widetilde{M}\longrightarrow\widetilde{N}$

be an injective morphism. Then $\phi$ induces a map

$\det(\phi)\,:\,{\det}_{\widetilde{\mathscr{H}}(\Gamma_{1})}(\widetilde{M})\otimes{\det}^{-1}_{\widetilde{\mathscr{H}}(\Gamma_{1})}(\widetilde{N})\longrightarrow\widetilde{\mathscr{H}}(\Gamma_{1}).$

Let
$\widetilde{M}_{\Gamma_{1}}:=\widetilde{M}\otimes_{\widetilde{\mathscr{H}}(\Gamma_{1})}\widetilde{E}$
and
$\widetilde{N}_{\Gamma_{1}}:=\widetilde{N}\otimes_{\widetilde{\mathscr{H}}(\Gamma_{1})}\widetilde{E}.$
We have an induced morphism of free $\widetilde{E}$-modules

$\phi_{\Gamma}\,:\,\widetilde{M}_{\Gamma_{1}}\longrightarrow\widetilde{N}_{\Gamma_{1}}.$

Consider the tautological exact sequence

(4.13)
$0\longrightarrow\ker(\phi_{\Gamma})\longrightarrow\widetilde{M}_{\Gamma_{1}}\xrightarrow{\phi_{\Gamma}}\widetilde{N}_{\Gamma_{1}}\longrightarrow\mathrm{coker}(\phi_{\Gamma})\longrightarrow
0.$

of $\widetilde{E}$-modules. We assume that the following condition holds true:

  * D1)

$\dim_{E}\left(\ker(\phi_{\Gamma})\right)=1.$

It follows from this condition that $\ker(\phi_{\Gamma})\subset
X\widetilde{M}_{\Gamma_{1}}$ and