# Algebraic Factorization of Adjoint Representations of Coleman Families: Non-
critical Case

Fırtına Küçük

(Date: September 2023)

###### Abstract.

The main objective of this article is to establish the $p$-adic Artin
formalism for the algebraic $p$-adic $L$-functions attached to the adjoint
representations of Coleman families of modular forms. In particular, we prove
a factorization formula involving the determinants of appropriate Selmer
complexes, using tools from rigid geometry, homological algebra, Euler
systems, and $p$-adic Hodge theory. This work extends an earlier result of
Palvannan to the $p$-non-ordinary setting.

###### Contents

  1. 1 Introduction
  2. 2 Affinoid Spaces and the Weight Space
  3. 3 $\left(\varphi,\Gamma\right)$-Modules and $p$-adic Hodge Theory
  4. 4 Eigencurve, Coleman Families and Triangulations
  5. 5 Selmer Complexes
  6. 6 Rankin-Selberg Products and Beilinson-Flach Elements
  7. 7 Algebraic Factorization

##  1\. Introduction

###  1.1. Introduction and Background

The main objective of this article is to prove the factorization of the
algebraic $p$-adic $L$-function attached to the representation:

$\displaystyle{\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})(\psi\chi_{\rm cyc}^{j})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{End}_{A}(\mathbf{V}_{\mathcal{F}})(\psi\chi_{\rm
cyc}^{j}),$

where $\mathcal{F}$ is a Coleman family over an affinoid algebra $A$ passing
through a non-$\theta$-critical $p$-stabilization of a newform $f\in
S_{k+2}(\Gamma_{1}(N),\varepsilon)$, $\mathbf{V}_{\mathcal{F}}$ is the big
Galois representation attached to $\mathcal{F}$, $\psi$ is a non-trivial
Dirichlet character and $\chi_{\rm cyc}$ is the $p$-adic cyclotomic character.
Define the representation ${{\rm Ad}^{0}}(\mathbf{V}_{\mathcal{F}})$ as:

$\displaystyle{{\rm Ad}^{0}}(\mathbf{V}_{\mathcal{F}})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\ker({\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})\xrightarrow{{\rm
Tr}}A).$

We then have the following decomposition of the representations:

${\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})(\psi\chi_{\rm cyc}^{j})\cong{{\rm
Ad}^{0}}(\mathbf{V}_{\mathcal{F}})(\psi\chi_{\rm cyc}^{j})\oplus
A(\psi\chi_{\rm cyc}^{j}).$

Let $V_{f}$ denote the Deligne’s (cohomological) representation attached to
$f$ over a $p$-adic field $E$, and let
${\mathrm{Ad}}(V_{f})\mathrel{\mathop{\mathchar 58\relax}}={\rm
End}_{E}(V_{f})$ and ${\mathrm{Ad}}^{0}(V_{f})\mathrel{\mathop{\mathchar
58\relax}}=\ker({\mathrm{Ad}}(V_{f})\xrightarrow{{\rm Tr}}E)$. Artin formalism
asserts that due to the definition of motivic $L$-functions, the $L$-function
attached to ${\mathrm{Ad}}(f)$ factors as:

$L({\mathrm{Ad}}(V_{f})(\psi),s)=L({{\rm Ad}^{0}}(V_{f})(\psi),s)L(\psi,s).$

Establishing the $p$-adic Artin formalism is, however, far from obvious, as
${\mathrm{Ad}}(V_{f})$ does not have any critical $L$-values (in the sense of
Deligne). The first result in this direction (following the spirit of Gross’
article [Gro80]) is due to Dasgupta in [Das16], where the modular form of
interest is $p$-ordinary:

######  Theorem (Dasgupta).

Let $f$ be a $p$-ordinary cuspidal eigenform and $\psi$ be a finite order
Dirichlet character with conductor coprime to $p$. Then we have the
factorization of $p$-adic $L$-functions

$\displaystyle L_{p}(f\otimes
f\otimes\psi,\sigma)=L_{p}({\mathrm{Sym}}^{2}(f)\otimes\psi,\sigma)L_{p}(\varepsilon\psi,\sigma/(z\cdot\kappa)),$

where $\kappa(z)=z^{k}$.

In [Pal17], Palvannan proved that the analogous factorization holds for the
associated algebraic $p$-adic $L$-functions (i.e. corresponding Greenberg
Selmer groups and their divisors). In [AL21], Arlandini and Loeffler
generalized the result of Dasgupta in [Das16] to the case where $f$ is
$p$-non-ordinary and $\psi$ is trivial (see Theorem 6.1).

Our main goal in this article is to establish the $p$-adic Artin formalism for
the algebraic counterparts of these $p$-adic $L$-functions, where $f$ is
$p$-non-ordinary and $\psi$ is non-trivial. In order to accomplish this, we
will appeal to Selmer complexes, which were developed by Nekovář in [Nek06]
and generalized by Pottharst (in [Pot13] and [Pot12]) and Benois (see
[Ben14],[Ben21]) to the case where the local conditions at $p$ come from
$(\varphi,\Gamma)$-modules, and the coefficients of the representations arise
from affinoid algebras. In an upcoming work, following the spirit of this
article, we will be interested in the case where $f$ is $\theta$-critical,
which will require a further investigation.

###  1.2. Statement of the main result

Dasgupta proved his result by interpolating $f$ in a Hida family and stated
the result by specializing the $p$-adic $L$-function attached to Rankin-
Selberg product of Hida families. On the contrary, we deduce our theorem
regarding the product of Coleman families by first proving the analogous
theorem for a single specialization, then generalize it to families using
control theorems for Selmer complexes. Therefore we will first state our main
result for a single specialization. Let us present the basic notation and
summarize our results below:

To simplify the notation, let

$\displaystyle V_{\mathrm{Ad}}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Ad}}(V_{f})(\psi\chi_{\rm cyc}^{j}),$ $\displaystyle
V_{{\rm Ad}^{0}}$ $\displaystyle\mathrel{\mathop{\mathchar 58\relax}}={{\rm
Ad}^{0}}(V_{f})(\psi\chi_{\rm cyc}^{j}),$ $\displaystyle
V_{1}\mathrel{\mathop{\mathchar 58\relax}}$ $\displaystyle=E(\psi\chi_{\rm
cyc}^{j}).$

Let $D_{\mathrm{Ad}}$, $D_{{\rm Ad}^{0}}$, and $D_{1}$ be certain
$(\varphi,\Gamma)$-modules (see Section 3, Section 5.4 and Section 7.2 for
details) attached to $V_{\mathrm{Ad}}$, $V_{{\rm Ad}^{0}}$, and $V_{1}$,
respectively. These $(\varphi,\Gamma)$-modules are the choice of local
conditions at $p$. For the data

$(V,D_{V})\in\\{(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}}),(V_{{{\rm
Ad}^{0}}},D_{{{\rm Ad}^{0}}}),(V_{1},D_{1})\\},$

let $\mathbf{R\Gamma}_{\rm Iw}(V,D_{V})$ denote the Pottharst style Selmer
complex attached to $(V,D_{V})$. We will denote the cohomologies of these
complexes by $H^{i}(V,D_{V})$. Let $D_{V}^{\perp}$ denote the orthogonal
complement of $D_{V}$ with respect to Grothendieck duality. Finally let
$\mathcal{H}=\mathcal{H}_{E}$ denote the large Iwasawa algebra introduced by
Perrin-Riou.

######  Theorem (Theorem 7.5).

Under the hypotheses i-viii of Section 7.1 we have:

  1. (1)

For the data

$(V,D_{V})\in\\{(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}}),(V_{{{\rm
Ad}^{0}}},D_{{{\rm Ad}^{0}}}),(V_{1},D_{1})\\},$

the Selmer complexes $\mathbf{R\Gamma}_{\rm Iw}(V,D_{V})$ and
$\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp})$ lie in the category of
perfect $\mathcal{H}$-modules with perfect amplitude at $[1,2]$.

  2. (2)

The cohomologies $H^{i}_{\rm Iw}(V,D_{V})$ and $H^{i}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp})$ vanish except $i\neq 2$, and we have the
following short exact sequences:

$\displaystyle 0\rightarrow H^{2}_{\rm Iw}(V_{{{\rm Ad}^{0}}},D_{{{\rm
Ad}^{0}}})$ $\displaystyle\rightarrow H^{2}_{\rm
Iw}(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}})\rightarrow H^{2}_{\rm
Iw}(V_{1},D_{1})\rightarrow 0,$ $\displaystyle 0\rightarrow H^{2}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})\rightarrow$ $\displaystyle H^{2}_{\rm
Iw}(V_{{\mathrm{Ad}}}^{*}(1),D_{{\mathrm{Ad}}}^{\perp})\rightarrow H^{2}_{\rm
Iw}(V_{{{\rm Ad}^{0}}}^{*}(1),D_{{{\rm Ad}^{0}}}^{\perp})\rightarrow 0.$

  3. (3)

For the pairs $(V,D_{V})$ as above, $H^{2}_{\rm Iw}(V,D_{V})$ and $H^{2}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp})$ are torsion, and we obtain the following
factorizations:

$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(V_{{{\rm Ad}^{0}}},D_{{{\rm
Ad}^{0}}}))\widehat{\otimes}_{\mathcal{H}}\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{1},D_{1})),$ $\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{{\mathrm{Ad}}}^{*}(1),D_{{\mathrm{Ad}}}^{\perp}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(V_{{{\rm
Ad}^{0}}}^{*}(1),D_{{{\rm
Ad}^{0}}}^{\perp}))\widehat{\otimes}_{\mathcal{H}}\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})).$

Let $\mathbf{V}_{\mathrm{Ad}}$ denote
${\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})(\psi\chi_{\rm cyc}^{j})$, we define
$\mathbf{V}_{{\rm Ad}^{0}}$ and $\mathbf{V}_{1}$ similarly. For $\mathbf{V}$
one of those representations, let $\mathbf{D}_{\mathbf{V}}$ denote the local
condition at $p$ attached to $\mathbf{V}$, and let
$\mathbf{D}_{\mathbf{V}}^{\perp}$ denote the corresponding dual local
condition. From the theorem above we deduce the following:

######  Theorem (Theorem 7.8).

Under our hypotheses in Section 7.1, we have:

  1. (1)

For the data

$(\mathbf{V},\mathbf{D}_{\mathbf{V}})\in\\{(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}}),(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm Ad}^{0}}}),(\mathbf{V}_{1},\mathbf{D}_{1})\\},$

the Selmer complexes $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})$ and $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})$ lie in the category of
perfect complexes of $\mathcal{H}_{A}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{H}\widehat{\otimes}_{E}A$-modules with perfect amplitude
at $[1,2]$.

  2. (2)

The cohomologies $H^{i}_{\rm Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})$ and
$H^{i}_{\rm Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})$ vanish
except $i\neq 2$, and we have the short exact sequences:

$\displaystyle 0\rightarrow H^{2}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm Ad}^{0}}})$ $\displaystyle\rightarrow H^{2}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}})\rightarrow
H^{2}_{\rm Iw}(\mathbf{V}_{1},\mathbf{D}_{1})\rightarrow 0,$ $\displaystyle
0\rightarrow H^{2}_{\rm
Iw}(\mathbf{V}_{1}^{*}(1),\mathbf{D}_{1}^{\perp})\rightarrow$ $\displaystyle
H^{2}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp})\rightarrow
H^{2}_{\rm Iw}(\mathbf{V}_{{{\rm Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm
Ad}^{0}}}^{\perp})\rightarrow 0.$

  3. (3)

We have the following factorizations:

$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm
Ad}^{0}}}))\widehat{\otimes}_{\mathcal{H}_{A}}\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{1},\mathbf{D}_{1})),$ $\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm
Ad}^{0}}}^{\perp}))\widehat{\otimes}_{\mathcal{H}_{A}}\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{1}^{*}(1),\mathbf{D}_{1}^{\perp})).$

For the perfectness of the complexes above, we prove and use the following
theorem, in which we generalize the arguments of [BCS23, §4.5] to our setting:

######  Theorem (Theorem 5.21).

Assume that the following conditions hold:

  1. (i)

$H^{3}_{\rm Iw}(\mathbf{V},\mathbf{D})=0$.

  2. (ii)

For each prime ideal $\mathfrak{p}$ of
$\Lambda_{A}[1/p]\mathrel{\mathop{\mathchar
58\relax}}=\Lambda[1/p]\widehat{\otimes}A$, we have
$(\mathbf{V}\otimes_{A}(\Lambda_{A}[1/p]^{\iota})/\mathfrak{p})^{G_{{\mathbf{Q}},S}}=0$,

  3. (iii)

For each $\ell\in S_{f}$, $\ell\neq p$, $H^{1}(I_{\ell},\mathbf{V})$ is
projective as $A$-module.

Then $\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D})\in\mathscr{D}_{\rm
parf}^{[1,2]}(\mathcal{H}_{A})$.

We also establish a partial result on comparison of algebraic $p$-adic
$L$-functions (i.e. determinants of appropriate Selmer complexes) attached to
$\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}(\psi\chi_{\rm
cyc}^{j})$ and $\mathbf{V}_{\mathrm{Ad}}$, see Theorem 7.9 for the exact
statement.

We will end this section by pointing out some key differences between the
result of Palvannan and our theorems. Firstly, Palvannan assumes that $f$ is
$p$-ordinary, in this case there is a
$\mathrm{Gal}(\overline{{\mathbf{Q}}}_{p}/{{\mathbf{Q}}_{p}})$-stable subspace
of $V_{f}$ (and $\mathbf{V}_{\mathbf{f}}$, where $\mathbf{f}$ is the Hida
family passing through $f$) which satisfies the Panchishkin condition.
Palvannan forms the local conditions attached to the relevant representations
by using this subspace. In the $p$-non ordinary case, which we assume, $V_{f}$
and $\mathbf{V}_{\mathcal{F}}$ are irreducible as
$\mathrm{Gal}(\overline{{\mathbf{Q}}}_{p}/{{\mathbf{Q}}_{p}})$
representations, hence there is no such subspace. This is why we need to work
with Pottharst style Selmer complexes, which allows us to choose our local
conditions as $(\varphi,\Gamma)$-modules that comes from the “triangulation”
of $\mathbf{V}_{\mathcal{F}}$ (see Theorem 4.7 for details). We should note
that the theory of $(\varphi,\Gamma)$-modules are very closely related to
$p$-adic Hodge theory. We also use the advantage of appealing to the theory of
Selmer complexes, since it is very convenient to use the language of derived
categories.

Another technical difference is that in the non-ordinary case, the $p$-adic
$L$-functions are no longer integral, this is why we use $\mathcal{H}$, the
algebra of locally analytic distributions on $\Gamma_{0}$, for cyclotomic
deformations. Note that $\mathcal{H}$ is a non-Noetherian ring. Fortunately
there is a rich theory of coadmissible modules over $\mathcal{H}$ thanks to
Schneider and Teitelbaum [ST03], which is applied to the theory of Selmer
complexes by Pottharst and Benois. We also require some tools from rigid
geometry, as our coefficient ring is an affinoid algebra and $\mathcal{H}$ is
the global sections of the structure sheaf on rigid open unit ball.

Finally we utilize Beilinson-Flach Euler system in Section 6 to establish the
analogue of the AD-TOR assumption of [Pal17] in our setting. In order to do
this, we adapt the results of Loeffler and Zerbes in [LZ16] to our setting,
combining with the ideas from [LZ19] and [BLV21].

###  1.3. Notation and Conventions

Let $p$ always be an odd prime. For a field $F$ (with characteristic $0$) we
will denote $\mathrm{Gal}(\overline{F}/F)$ by $G_{F}$, and when
$F={{\mathbf{Q}}_{p}}$ we will write simply $G_{p}=G_{{{\mathbf{Q}}_{p}}}$. If
$F$ is a finite extension of ${\mathbf{Q}}$ or ${{\mathbf{Q}}_{p}}$, we will
denote the ring of integers of $F$ by $\mathcal{O}_{F}$.

Let ${\mathbf{Q}}^{\rm cyc}=\bigcup_{n\geq 0}{\mathbf{Q}}(\zeta_{p^{n}})$ be
the compositum of the $p$-power cyclotomic extensions of ${\mathbf{Q}}$, and
let $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}({\mathbf{Q}}^{\rm
cyc}/{\mathbf{Q}})\cong{\mathbf{Z}}_{p}^{\times}$. Let
$\Delta\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}({\mathbf{Q}}(\zeta_{p})/{\mathbf{Q}})$ be the torsion
part of $\Gamma$, and for all $n\geq 0$, let
${\mathbf{Q}}_{n}\mathrel{\mathop{\mathchar
58\relax}}={\mathbf{Q}}(\zeta_{p^{n+1}})^{\Delta}$,
${\mathbf{Q}}_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\bigcup_{n\geq
0}{\mathbf{Q}}_{n}=({\mathbf{Q}}^{\rm cyc})^{\Delta}$ be the cyclotomic
${{\mathbf{Z}}_{p}}$ extension of ${\mathbf{Q}}$, and
$\Gamma_{n}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}({\mathbf{Q}}_{\infty}/{\mathbf{Q}}_{n})$. Then we have
$\Gamma\cong\Gamma_{0}\times\Delta$. Under this isomorphism, we also have the
decomposition of characters $\chi_{\rm cyc}=\omega\cdot\langle\chi_{\rm
cyc}\rangle$, where $\omega$ is the Teichmüller character. Note that the
$p$-adic cyclotomic character $\chi_{\rm cyc}$ gives a canonical isomorphism
$\chi_{\rm cyc}\mathrel{\mathop{\mathchar
58\relax}}\Gamma\xrightarrow{\sim}{\mathbf{Z}}_{p}^{\times}$, and we have
$\Gamma_{0}\cong(1+p{{\mathbf{Z}}_{p}})\cong{{\mathbf{Z}}_{p}}$ and
$\Delta\cong({\mathbf{Z}}/p{\mathbf{Z}})^{\times}$.

For a finite extension $K$ of ${{\mathbf{Q}}_{p}}$ we can also define $K^{\rm
cyc}=K\cdot{\mathbf{Q}}^{\rm cyc}=\bigcup_{n\geq 0}K(\zeta_{p^{n}})$,
$\Gamma_{K}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Gal}(K^{\rm cyc}/K)$,
$\Delta_{K}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}(K(\zeta_{p})/K)$, $K_{n}\mathrel{\mathop{\mathchar
58\relax}}=K(\zeta_{p^{n}})^{\Delta}$, $K_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=(K^{\rm cyc})^{\Delta}$, and
$\Gamma_{K}^{0}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}(K_{\infty}/K)$. For our purposes we will assume that
$K$ is unramified over ${{\mathbf{Q}}_{p}}$.

Note that $\Gamma\cong\Gamma_{{{\mathbf{Q}}_{p}}}$,
$\Delta\cong\Delta_{{{\mathbf{Q}}_{p}}}$,
$\Gamma_{0}\cong\Gamma_{{{\mathbf{Q}}_{p}}}^{0}$.

We define the Iwasawa algebra $\Lambda_{E}(\Gamma)$ as

$\Lambda_{E}(\Gamma)=\mathcal{O}_{E}[[\Gamma]]\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{H}\mathcal{O}_{E}[\Gamma/H].$

We also set the large Iwasawa algebra, introduced by Perrin-Riou in [PR94] as

$\mathcal{H}_{E}(T)\mathrel{\mathop{\mathchar 58\relax}}=\\{f\in
E[[T]]\mathrel{\mathop{\mathchar 58\relax}}f\textnormal{ converges in the open
unit ball}\\}.$

If $V$ is a $p$-adic representation with coefficients in a finite extension of
${{\mathbf{Q}}_{p}}$ which is Hodge-Tate, then we have the following Hodge-
Tate decomposition:

$V\otimes{{\mathbf{C}}_{p}}\cong\bigoplus_{i\in{\mathbf{Z}}}{{\mathbf{C}}_{p}}(i)^{h_{i}},$

where $h_{i}=0$ for all but finitely many integers $i$. The indices $i$ for
which $h_{i}\neq 0$ will be called the Hodge-Tate weights of $V$. In that
case, the jumps appearing in the de Rham filtration of $\mathbf{D}_{\rm
dR}(V)$ will be the negatives of the Hodge-Tate weights of $V$. In particular,
the Hodge-Tate weight of the $p$-adic cyclotomic character will be $+1$.

Let $M$ be a pure motive, which is geometric. The $L$-function attached to $M$
is defined as:

$L(M,s)\mathrel{\mathop{\mathchar 58\relax}}=\prod_{\ell}\det(1-{\rm
Frob}^{-1}_{\ell}X|M_{p}^{I_{\ell}})_{X=\ell^{-s}}^{-1},$

where $\ell$ runs through all integer primes with $p\neq\ell$. Here the
geometric Frobenius at $\ell$ is denoted by ${\rm Frob}^{-1}_{\ell}$. For
instance, the $L$-function attached to a newform $f\in
S_{k+2}(\Gamma_{0}(N))$, which is defined as

$L(f,s)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{n=1}^{\infty}\frac{a_{n}(f)}{n^{s}}$

will differ from the motivic $L$-function $L(M(f),s)$ up to a finitely many
Euler factors, where $M(f)$ denotes the motive attached to $f$ by Scholl in
[Sch90]. The $p$-adic realization $V_{f}\mathrel{\mathop{\mathchar
58\relax}}=M_{p}(f)$ of $M(f)$ is Deligne’s cohomological representation
attached to $f$, and with our conventions on the Hodge-Tate weights, $V_{f}$
has the Hodge-Tate weights $\\{-1-k,0\\}$.

To fix the possible confusion with our conventions when we can consider the
twists by Dirichlet characters, we will assume that if $\psi$ denotes a
Dirichlet character, then $\psi(p)=\psi({\rm Frob}^{-1}_{p})$, as done in
[AL21]. We also note that we will write $\psi+j$ to mean the character
$\psi\chi_{\rm cyc}^{j}$.

###  1.4. Acknowledgements

This article is derived from my PhD thesis [Kü23], which was supervised by
Assoc. Prof. Kâzım Büyükboduk and funded by University College Dublin PhD
Advance Scheme. Therefore, I would first like to thank my supervisor for
suggesting this problem and for his incredible support during my Ph.D and the
preparation of my thesis and this article. I would also like to thank to my
colleagues Dr. Erman Işık and Dr. Daniele Casazza, and also to the external
examiner for my viva voce, Prof. Matteo Longo, for their valuable suggestions.

##  2\. Affinoid Spaces and the Weight Space

In this section, we will review definitions and some properties of affinoid
algebras, their spectra and the weight space. The details on affinoid algebras
and affinoid spaces can be found in [Bos14], and the details on the weight
space can be found in [Bel21] and [BB20].

###  2.1. Affinoid Algebras

######  Definition 2.1 (Tate algebra).

Let $E$ be a finite extension of ${{\mathbf{Q}}_{p}}$. The $E$-algebra

$E\langle x_{1},...,x_{n}\rangle\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\sum_{\alpha\in{\mathbf{N}}^{n}}c_{\alpha}x^{\alpha}\in
E[[x_{1},...,x_{n}]]\mathrel{\mathop{\mathchar
58\relax}}\mathinner{\\!\left\lvert c_{\alpha}\right\rvert}\rightarrow
0\textrm{\ as }\alpha\rightarrow\infty\right\\}$

is called the Tate algebra.

We will also denote $E\langle x_{1},...,x_{n}\rangle$ by $T_{n}$, if $E$ is
inferred from the context. Tate algebras are also characterized by the
following property given in [Bos14, §2.2 Lemma 1]:

######  Proposition 2.2.

A formal power series with $n$ variables

$f=\sum_{\alpha\in{\mathbf{N}}^{n}}c_{\alpha}x^{\alpha}\in
E[[x_{1},...,x_{n}]]$

lies in $T_{n}$ iff $f$ converges on the closed unit ball

$B^{n}(\overline{E})\mathrel{\mathop{\mathchar
58\relax}}=\\{x=(x_{1},...,x_{n})\in\overline{E}^{n}\mathrel{\mathop{\mathchar
58\relax}}\mathinner{\\!\left\lvert x_{i}\right\rvert}\leq 1\\}.$

Some properties of the Tate algebras are as follows: For each $n$, $T_{n}$ is
UFD, Noetherian, Jacobson, and regular of Krull dimension $n$. For each
maximal ideal $\mathfrak{m}$ of $T_{n}$, the residue field
$T_{n}/\mathfrak{m}$ is a finite extension of $E$. The Tate algebra $T_{n}$ is
also complete with respect to the Gauss norm:

$\mathinner{\\!\left\lVert f\right\rVert}\mathrel{\mathop{\mathchar
58\relax}}={\rm max}_{\alpha}\mathinner{\\!\left\lvert
c_{\alpha}\right\rvert},$

where

$f(x)=\sum_{\alpha}c_{\alpha}x^{\alpha}\in T_{n}.$

In particular, $T_{1}$ is an Euclidean domain, hence it is a PID. This last
property will be useful when we discuss nice affinoid neighborhoods in
Definition 4.1.

######  Definition 2.3 (Affinoid algebra).

An $E$-algebra $A$ is called an affinoid algebra over $E$ if there exists an
$E$-algebra epimorphism $e\mathrel{\mathop{\mathchar
58\relax}}T_{n}\twoheadrightarrow A$ from a Tate algebra $T_{n}$ to $A$.

In other words, affinoid algebras are quotients of Tate algebras by some
ideals. Affinoid algebras are Noetherian and Jacobson. There is a norm on
affinoid algebras induced by the Gauss norm on Tate algebras, and all
$E$-algebra morphisms between affinoid algebras are continuous. Affinoid
algebras also satisfy Noether normalization theorem, i.e. for some $d$ there
exists a finite monomorphism $T_{d}\rightarrow A$, and in particular the Krull
dimension is $d$.

There exists a category of affinoid $E$-algebras with morphisms given by
$E$-algebra maps which admits pushouts, where we give the universal property
of pushouts in the following (see [Bos14, §3.1 Proposition 2, Appendix A
Proposition 6]):

######  Proposition 2.4 (Completed tensor product).

Let $R$, $A_{1}$, $A_{2}$ be affinoid $E$-algebras such that $A_{1}$ and
$A_{2}$ are $R$-algebras. Then there exists a completed tensor product
$A_{1}\widehat{\otimes}_{R}A_{2}$ of $A_{1}$ and $A_{2}$ over $R$ satisfying
the following universal property: There are $R$-algebra homomorphisms
$i_{1}\mathrel{\mathop{\mathchar 58\relax}}A_{1}\rightarrow
A_{1}\widehat{\otimes}_{R}A_{2}$ and $i_{2}\mathrel{\mathop{\mathchar
58\relax}}A_{2}\rightarrow A_{1}\widehat{\otimes}_{R}A_{2}$, and for any
$R$-algebra $A$ lying in this category and $R$-algebra homomorphisms
$f_{1}\mathrel{\mathop{\mathchar 58\relax}}A_{1}\rightarrow A$ and
$f_{2}\mathrel{\mathop{\mathchar 58\relax}}A_{2}\rightarrow A$, there exists a
unique $R$-algebra homomorphism $f\mathrel{\mathop{\mathchar
58\relax}}A_{1}\widehat{\otimes}_{R}A_{2}\rightarrow A$ such that the
following diagram commutes:

${A}$${A_{1}}$${A_{1}\widehat{\otimes}_{R}A_{2}}$${A_{2}.}$$\scriptstyle{f_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{\exists!f}$$\scriptstyle{i_{2}}$$\scriptstyle{f_{2}}$

The completed tensor product $A_{1}\widehat{\otimes}_{R}A_{2}$ can be
constructed so that the maps $i_{1}\mathrel{\mathop{\mathchar
58\relax}}A_{1}\rightarrow A_{1}\widehat{\otimes}_{R}A_{2}$ and
$i_{2}\mathrel{\mathop{\mathchar 58\relax}}A_{2}\rightarrow
A_{1}\widehat{\otimes}_{R}A_{2}$ can be given as $a_{1}\mapsto a_{1}\otimes 1$
and $a_{2}\mapsto 1\otimes a_{2}$, respectively.

###  2.2. Affinoid and Rigid Spaces

######  Definition 2.5 (Affinoid space).

$X$ is called an affinoid space over $E$ if $X=\mathrm{Spm}(A)$ for some
affinoid algebra $A$ over $E$, where $\mathrm{Spm}(A)$ is the maximal spectra
of $A$ (i.e. the set of maximal ideals of $A$). $X$ is equipped with the
following data:

  * •

$X$ has the Zariski topology, where the closed subsets of $X$ are precisely
the following:

$V(\mathfrak{a})\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathrm{Spm}(A)\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{a}\subseteq\mathfrak{m}_{x}\\},$

where $\mathfrak{a}$ is an ideal of $A$, and $\mathfrak{m}_{x}$ is the
corresponding maximal ideal in $A$ to the point $x$,

  * •

$A$ is the ring of “functions” on $X$ viewed as follows: if $f\in A$ and
$x\in\mathrm{Spm}(A)$, then $f(x)=f{\rm\ mod\ }\mathfrak{m}_{x}\in
A/\mathfrak{m}_{x}$.

Note that in this case, for an ideal $\mathfrak{a}$ of $A$, the closed subset
$V(\mathfrak{a})$ of $\mathrm{Spm}(A)$ is precisely

$V(\mathfrak{a})\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathrm{Spm}(A)\mathrel{\mathop{\mathchar
58\relax}}f(x)=0{\rm\ for\ all\ }f\in\mathfrak{a}\\}.$

The morphisms of affinoid spaces are defined so that the category of affinoid
spaces is the opposite category of the category of affioid algebras. For
instance, the pullbacks (fibered products) are defined as
$\mathrm{Spm}(A_{1})\times_{\mathrm{Spm}(R)}\mathrm{Spm}(A_{2})\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Spm}(A_{1}\widehat{\otimes}_{R}A_{2})$ and the natural maps
$p_{1}\mathrel{\mathop{\mathchar 58\relax}}X_{1}\times_{X}X_{2}\rightarrow
X_{1}$ and $p_{2}\mathrel{\mathop{\mathchar
58\relax}}X_{1}\times_{X}X_{2}\rightarrow X_{2}$ (which correspond to $i_{1}$
and $i_{2}$ in the opposite category) are the usual projections
$(x_{1},x_{2})\mapsto x_{1}$ and $(x_{1},x_{2})\mapsto x_{2}$ respectively.

Zariski topology on $\mathrm{Spm}(A)$ enjoys many similar properties to the
topology on the prime spectrum ${\rm Spec}(A)$. For instance, the sets of the
form

$D_{f}\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathrm{Spm}(A)\mathrel{\mathop{\mathchar 58\relax}}f(x)\neq
0\\}\cong\mathrm{Spm}(A_{f})$

for $f\in A$ forms a basis for this topology. There are more interesting
topologies, such as canonical topology, weak and strong Grothendieck
topologies, which are finer than the Zariski topology. The details can be
found on Chapter 3 and 4 of [Bos14].

######  Definition 2.6 (Affinoid subdomain).

A subset $U\subseteq X=\mathrm{Spm}(A)$ is called an affinoid subdomain of $X$
if there exists a morphism of affinoid spaces $\iota\mathrel{\mathop{\mathchar
58\relax}}X^{\prime}\rightarrow X$ such that $\iota(X^{\prime})\subseteq U$
and this map is universal with this property, i.e. for any morphism
$\varphi\mathrel{\mathop{\mathchar 58\relax}}Y=\mathrm{Spm}(B)\rightarrow X$
with $\varphi(Y)\subseteq U$, there exists a unique morphism
$\varphi^{\prime}\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X^{\prime}$
such that $\varphi=\iota\varphi^{\prime}$.

It turns out that in this case, $\iota$ is a bijection from $X^{\prime}$ onto
$U$, and $X^{\prime}$ is unique up to unique isomorphism, hence we can regard
$U=X^{\prime}=\mathrm{Spm}(A_{U})$. Also for any maximal ideal
$\mathfrak{m}_{x}\in U$, we have
$A_{\iota(\mathfrak{m}_{x})}\cong{A_{U}}_{\mathfrak{m}_{x}}$ and
$A/{\iota(\mathfrak{m}_{x})}^{n}\cong{A_{U}}/{\mathfrak{m}_{x}}^{n}$ for any
$n\in{\mathbf{N}}$ (see [Bos14, §3.3 Lemma 10]).

An important property of affinoid subdomains for our discussions is as follows
(see [Bos14, §4.1 Corollary 5]):

######  Proposition 2.7.

If $U\subseteq X=\mathrm{Spm}(A)$ is an affinoid subdomain of $X$, then the
restriction map $A\rightarrow A_{U}$ is flat.

######  Definition 2.8 (Admissible open subset).

A subset $U\subseteq X$ is called admissible open if there exists a covering
of $U$ by affinoid subdomains $\\{U_{i}\\}_{i}$ such that for each morphism of
affinoid spaces $\varphi\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$
with $\varphi(Y)\subseteq U$, there is a finite refinement of the covering
$\\{\varphi^{-1}(U_{i})\\}_{i}$ of $Y$.

For example, all Zariski open subsets of $X$ are admissible, in particular
they are covered by finitely many affinoid subdomains of $X$. Another
important example is the open unit ball. For $n\in{\mathbf{N}}$, let

$W_{n}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Spm}\left(E\langle
X/p^{1/n}\rangle\right)$

be the closed ball with radius $p^{-1/n}$. Then the open unit ball is
admissibly covered by these closed balls $W_{n}$.

##  3\. $\left(\varphi,\Gamma\right)$-Modules and $p$-adic Hodge Theory

###  3.1. Robba Rings and $(\varphi,\Gamma)$-modules

######  Definition 3.1.

Let $r\geq 0$, $A(r,1)\mathrel{\mathop{\mathchar
58\relax}}=\\{z\in{{\mathbf{C}}_{p}}\mathrel{\mathop{\mathchar
58\relax}}p^{-\frac{1}{r}}\leq\mathinner{\\!\left\lvert
z\right\rvert}_{p}<1\\}$, and $\pi$ denotes a formal variable. The Robba ring
$\mathcal{R}_{K}^{(r)}$ is defined as the set of formal series
$f(\pi)=\sum_{n\in{\mathbf{Z}}}a_{n}\pi^{n}$ with $a_{n}\in K$ such that
$f(\pi)$ converges in the annulus $A(r,1)$. If $A$ is an affinoid algebra over
${{\mathbf{Q}}_{p}}$, then we set
$\mathcal{R}_{K,A}^{(r)}\mathrel{\mathop{\mathchar
58\relax}}=A\widehat{\otimes}_{{{\mathbf{Q}}_{p}}}\mathcal{R}_{K}^{(r)}$ and
$\mathcal{R}_{K,A}\mathrel{\mathop{\mathchar
58\relax}}=\cup_{r>0}\mathcal{R}_{K,A}^{(r)}$. If $K$ is clear from the
context, we will denote those Robba rings simply by $\mathcal{R}_{A}^{(r)}$
and $\mathcal{R}_{A}$, respectively.

Note that for $s\leq r$, we have $A(r,1)\subseteq A(s,1)$ which implies that
$\mathcal{R}_{K}^{(s)}\subseteq\mathcal{R}_{K}^{(r)}$.

######  Definition 3.2.

Let $E$ be a finite extension of ${{\mathbf{Q}}_{p}}$ and
$f\in\mathcal{R}_{E}$. The continuous $E$-linear map
$\varphi\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{R}_{E}\to\mathcal{R}_{E}$ is defined as

$\varphi(f(\pi))\mathrel{\mathop{\mathchar 58\relax}}=f((1+\pi)^{p}-1).$

Similarly, for any $\gamma\in\Gamma$ we define the action of $\Gamma$ on
$\mathcal{R}_{E}$ by

$\gamma(f(\pi))\mathrel{\mathop{\mathchar 58\relax}}=f((1+\pi)^{\chi_{\rm
cyc}(\gamma)}-1).$

A $(\varphi,\Gamma)$-module over a Robba ring $\mathcal{R}_{A}$ is a finitely
generated locally free $\mathcal{R}_{A}$-module equipped with a
$\varphi$-semilinear map and a semilinear continuous action of $\Gamma$
satisfying some certain conditions (see [Ben21], [KPX14] for the details). The
$(\varphi,\Gamma)$-modules over $\mathcal{R}_{A}$ form a Tannakian category
denoted by $\mathbf{M}_{\mathcal{R}_{A}}^{\varphi,\Gamma}$.

Let $\mathbf{V}$ be a $p$-adic representation of $G_{p}$ with coefficients in
an affinoid algebra $A$. To each such representation $\mathbf{V}$ one can
attach a $(\varphi,\Gamma)$-module $\mathbf{D}_{{\rm rig},A}^{\dagger}(V)$ in
a functorial way. If $A=E$ is a finite extension of ${{\mathbf{Q}}_{p}}$, then
we will write $V$ instead of $\mathbf{V}$, and we will denote the associated
$(\varphi,\Gamma)$-module by $\mathbf{D}_{\rm rig}^{\dagger}(V)$. In
particular, we have $\mathbf{D}_{\rm
rig}^{\dagger}({{\mathbf{Q}}_{p}}(1))=\mathcal{R}_{E}(1)$.

If $x\in\mathrm{Spm}(A)$ with corresponding maximal ideal $\mathfrak{m}_{x}$
and the residue field $E_{x}=A/\mathfrak{m}_{x}$, then the specialization at
$x$ is a natural transformation from $\mathbf{D}_{{\rm rig},A}^{\dagger}$ to
$\mathbf{D}_{\rm rig}^{\dagger}(V)$, i.e. $\mathbf{D}_{{\rm
rig},A}^{\dagger}(V)_{x}=\mathbf{D}_{\rm rig}^{\dagger}(V_{x})$.

Note that not all $(\varphi,\Gamma)$-modules arise from $p$-adic
representations in this way, and in the case $A=E$, the essential image of
$\mathbf{D}_{\rm rig}^{\dagger}$ is the subcategory of étale (or slope zero)
$(\varphi,\Gamma)$-modules.

###  3.2. A brief introduction to $p$-adic Hodge Theory

In this subsection, we will give a summary of the contents of [Ber04], [BC09],
and [Ben21, §2.2].

First note that not all $G_{p}$-representations over $p$-adic fields are
Hodge-Tate, see [BC09, §2] for definitions and details. However, the ones that
we are interested in will be Hodge-Tate representations. Let $E$ be a finite
extension of ${{\mathbf{Q}}_{p}}$ and let $V$ be a representation of
$G_{\mathbf{Q}}$ over $E$ which is Hodge-Tate. It then has a Hodge-Tate
decomposition of the following form:

$V\otimes_{{\mathbf{Q}}_{p}}{{\mathbf{C}}_{p}}\cong\sum_{j\in{\mathbf{Z}}}{{\mathbf{C}}_{p}}(j)^{h_{j}},$

where $h_{j}$’s are non-negative integers such that $h_{j}=0$ for all but
finitely many $j$.

Recall that by our conventions on Hodge-Tate weights, integers $j$ for which
the corresponding $h_{j}$’s are non-zero are called Hodge-Tate weights of $V$.
In particular, the Hodge-Tate weight of ${{\mathbf{Q}}_{p}}(1)$ is $1$. We
will denote the multiset of Hodge-Tate weights of $V$ by ${\rm HT}(V)$.

######  Definition 3.3.

A filtered module $D$ is a finite dimensional $E$-vector space equipped with a
decreasing filtration $({\rm Fil}^{i}D)_{i\in{\mathbf{Z}}}$. A filtered
$(\varphi,N)$-module is a filtered module with a semilinear (w.r.t. the action
of $G_{p}$) bijective map $\varphi\mathrel{\mathop{\mathchar
58\relax}}D\rightarrow D$ and a $E$-linear monodromy operator $N$ with
$N\varphi=p\varphi N$. A filtered $\varphi$-module is a filtered
$(\varphi,N)$-module with $N=0$. We will denote the Tannakian categories of
these modules by $\mathbf{MF}_{E}$, $\mathbf{MF}_{E}^{\varphi,N}$, and
$\mathbf{MF}_{E}^{\varphi}$, respectively.

There are also filtered $(\varphi,N,G_{p})$-modules, see [Ben21] for the
precise definition. The category of such modules are denoted by
$\mathbf{MF}_{E}^{\varphi,N,G_{p}}$.

Fontaine’s formalism gives period rings $\mathbf{B}_{*}$ for $*\in\\{\rm
dR,pst,st,cris\\}$, and functors $\mathbf{D}_{*}\mathrel{\mathop{\mathchar
58\relax}}(\mathbf{B}_{*}\otimes_{{{\mathbf{Q}}_{p}}}{-})^{G_{p}}$ from the
category of $p$-adic representations over $E$ to the categories of “Filtered
modules”, “Filtered $(\varphi,N,G_{p})$-modules”, “Filtered
$(\varphi,N)$-modules” and “Filtered $\varphi$-modules” for $*={\rm
dR,pst,st,cris}$ respectively.

######  Definition 3.4.

We say that $V$ is $\mathbf{B}$-admissible if the dimension of $V$ is
preserved under these functors. We say that $V$ is de Rham, potentially
semistable, semistable, and crystalline if $V$ is $B_{\rm dR}$, $B_{\rm pst}$,
$B_{\rm st}$, $B_{\rm cris}$ admissible, respectively.

######  Theorem 3.5 (Berger).

A $(\varphi,\Gamma)$-module $\mathbf{D}$ is potentially semistable if and only
if it is de Rham. Let $*$ stand for ${\rm pst,st,cris}$, and $?$ for
$\mathbf{MF}_{E}^{\varphi,N,G_{p}}$, $\mathbf{MF}_{E}^{\varphi,N}$,
$\mathbf{MF}_{E}^{\varphi}$ respectively. The functors $\mathbf{D}_{*}$
factors from the category of $(\varphi,\Gamma)$-modules
$\mathbf{M}_{\mathcal{R}_{E}}^{\varphi,\Gamma}$ such that the following
diagrams commute:

${\mathbf{Rep}_{E}(G_{p})}$${\mathbf{M}_{\mathcal{R}_{E}}^{\varphi,\Gamma}}$${?}$$\scriptstyle{\mathbf{D}_{\rm
rig}^{\dagger}}$$\scriptstyle{\mathbf{D}_{*}}$$\scriptstyle{\exists!\mathscr{D}_{*}}$

Note that there is also a relation between de Rham filtration and the Hodge-
Tate weights. If $V$ is de Rham, then it is Hodge-Tate, and the jumps of the
filtration ${\rm Fil}^{i}\mathbf{D}_{\rm dR}(V)$ (i.e. the $i$’s for which
${\rm Fil}^{i}\neq{\rm Fil}^{i+1}$) are the additive inverses of the Hodge-
Tate weights of $V$, and the dimensions of the graded pieces ${\rm
Gr}^{i}(\mathbf{D}_{\rm dR}(V))\mathrel{\mathop{\mathchar 58\relax}}={\rm
Fil}^{i}\mathbf{D}_{\rm dR}(V)/{\rm Fil}^{i+1}\mathbf{D}_{\rm dR}(V)$ are the
multiplicities of the corresponding Hodge-Tate weights.

Let $V$ be a representation of $G_{\mathbf{Q}}$, which is de Rham at $p$. Let
$d^{+}(V)\mathrel{\mathop{\mathchar 58\relax}}=V^{c=1}$ denote the dimension
of the plus-eigenspace, where $c$ is any complex conjugation in
$G_{\mathbf{Q}}$. Let $\\#{\rm HT}^{+}(V)$ denote the number of positive
Hodge-Tate weights.

######  Definition 3.6.

We say that $V$ is critical if $d^{+}(V)=\\#{\rm HT}^{+}(V)$. If $V$ is
critical, then it is Panchishkin ordinary if there exists a
$G_{p}$-equivariant exact sequence

$0\rightarrow F^{+}(V)\rightarrow V\rightarrow F^{-}(V)\rightarrow 0$

with $\dim_{E}F^{+}(V)=d^{+}(V)$ and all Hodge-Tate weights of $F^{+}(V)$ are
positive.

######  Definition 3.7 (Deligne’s criticality condition).

We say that $s=n$ is a critical value of the $L$-series $L(V,s)$ associated to
the representation $V$ (or to the motive $M$ such that $p$-adic realization of
$M$ is $V$) if the product of the Euler factors at infinity

$L_{\infty}(V,s)L_{\infty}(V^{*}(1),-s)$

is holomorphic at $s=n$.

######  Remark 3.8.

Note that $s=n$ is critical (in the sense of Deligne) for $V$ if and only if
$V(n)$ is critical in the sense of Definition 3.6. First observe that $s=n$ is
critical for $V$ if and only if $s=0$ is critical for $V(n)$ (see our
conventions for $L$-functions attached to the twists of $V$ in the
introduction). Hence we claim that $s=0$ is critical for $V$ if and only if
$V$ is critical as in Definition 3.6. For the equivalence between these two
definitions see [CPR89, Lemma 2.3]. Note also that in op. cit. what we call
$V^{*}(-1)$ is denoted as $V^{*}$, hence Definition 3.7 coincides with [CPR89,
Definition 2.2].

The following definition by [Ber11] is a generalization of Panchishkin
conditions for the representations attached to the $p$-non-ordinary forms.

######  Definition 3.9 (Trianguline representations).

$V$ is called trianguline if there exists a finite extension $F$ of $E$ such
that $\mathbf{D}_{{\rm rig},F}^{\dagger}(V\otimes_{E}F)$ is a successive
extensions of rank 1 $(\varphi,\Gamma)$-modules as follows:

$0=\mathrel{\mathop{\mathchar 58\relax}}D_{0}\subseteq D_{1}\subseteq
D_{2}\subseteq\dots\subseteq D_{n}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}_{{\rm rig},F}(V\otimes_{E}F).$

##  4\. Eigencurve, Coleman Families and Triangulations

###  4.1. The Weight Space

Let $\Gamma$ be a profinite group, and let $H$ run through open subgroups of
$\Gamma$. Recall that we define the Iwasawa algebra $\Lambda_{E}(\Gamma)$ as

$\Lambda_{E}(\Gamma)=\mathcal{O}_{E}[[\Gamma]]\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{H}\mathcal{O}_{E}[\Gamma/H].$

The particular cases in which we will be most interested are
$\Gamma\cong{\mathbf{Z}}_{p}^{\times}$ or ${\mathbf{Z}}_{p}$. Note that we
have

$\Lambda_{E}({\mathbf{Z}}_{p}^{\times})\cong
\mathcal{O}_{E}[\Delta]\otimes\Lambda_{E}({{\mathbf{Z}}_{p}}),$

and $\Lambda_{E}({{\mathbf{Z}}_{p}})\cong\mathcal{O}_{E}[[T]]$, where the last
isomorphism is given by sending a topological generator
$\gamma\in{{\mathbf{Z}}_{p}}$ to $1+T$.

The weight space $\mathcal{W}$ is defined as the Hom-set of continuous group
homomorphism from ${\mathbf{Z}}_{p}^{\times}$ to the multiplicative group
scheme, i.e. if $R$ is a Banach $E$-algebra then

$\mathcal{W}(R)\mathrel{\mathop{\mathchar 58\relax}}={\mathrm{Hom}}_{\rm
cts,grp}({\mathbf{Z}}_{p}^{\times},R^{\times}).$

This definition is equivalent to saying that

$\mathcal{W}(R)={\mathrm{Hom}}_{{\rm cts},\mathcal{O}_{E}{\rm-
alg}}(\Lambda_{E}({\mathbf{Z}}_{p}^{\times}),R).$

Applying the functor of points idea, we can see $\mathcal{W}$ as the generic
fiber of ${\rm Spf}(\Lambda_{E}({\mathbf{Z}}_{p}^{\times}))$. Note that this
is not an affinoid space but it is a rigid analytic space: it is the disjoint
union of $p-1$ open balls of radius $1$. Each of these balls are admissibly
covered by affinoid disks $W_{n}$ defined in Section 2.2.

For every integer $k$, the character $\kappa(k+2)\mathrel{\mathop{\mathchar
58\relax}}z\mapsto z^{k}$ is an element of the weight space, thus we have an
injection ${\mathbf{Z}}\hookrightarrow\mathcal{W}$. Moreover, the integers
form a dense subset of the weight space. Thus we can identify ${\mathbf{Z}}$
with the subset of the weight space $\mathcal{W}$ via $\kappa$. Because of the
normalization identifying $k+2\in{\mathbf{Z}}$ with the $k$-power character,
from now on, we will denote the weights of modular forms (following also
[LZ16], [BLV21], [Bel12], etc.) by $k+2$ instead of $k$.

If we fix one of these components (i.e. a character
$\chi\mathrel{\mathop{\mathchar 58\relax}}({\mathbf{Z}}/p)^{\times}\rightarrow
R^{\times}$, without loss of generality we can assume it is the trivial
character) and call the component $\mathcal{W}_{\chi}$, it is an open ball of
radius $1$, hence it is admissibly covered by closed affinoid balls of radius
$p^{-1/n}$, each of which is isomorphic to

$W_{n}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Spm}\left(E\langle
X/p^{1/n}\rangle\right).$

Denote

$\mathcal{H}_{n}\mathrel{\mathop{\mathchar 58\relax}}=E\langle
X/p^{1/n}\rangle=\Gamma(W_{n},\mathcal{O}_{W_{n}}),$

so that we define the ring of locally analytic $E$-valued distributions on
${{\mathbf{Z}}_{p}}$ as

$\mathcal{H}_{E}({{\mathbf{Z}}_{p}})\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathcal{H}_{n}=\Gamma(\mathcal{W}_{\chi},\mathcal{O}_{\mathcal{W}_{\chi}}).$

We also define

$\mathcal{H}_{E}({\mathbf{Z}}_{p}^{\times})\cong
E[\Delta]\otimes\mathcal{H}_{E}({\mathbf{Z}}_{p}),$

and note that $\mathcal{H}_{E}({\mathbf{Z}}_{p})$ contains the Iwasawa algebra
$\Lambda_{E}({{\mathbf{Z}}_{p}})\otimes E$. We will denote
$\mathcal{H}_{E}({{\mathbf{Z}}_{p}})$ by $\mathcal{H}$. If $A$ is an affinoid
algebra over $E$, we will write $\mathcal{H}_{A}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{H}\widehat{\otimes}A$ and
$\Lambda_{A}\mathrel{\mathop{\mathchar 58\relax}}=\Lambda\widehat{\otimes}A$.

If $\Gamma$ is a quotient of $G_{\mathbf{Q}}$ or $G_{p}$ isomorphic to
${{\mathbf{Z}}_{p}}$ or ${\mathbf{Z}}_{p}^{\times}$, and
$\iota\mathrel{\mathop{\mathchar 58\relax}}\Gamma\rightarrow\Gamma$ is the
involution that maps $\gamma\in\Gamma$ to $\gamma^{-1}$, then
$\Lambda(\Gamma)^{\iota}$ will denote the rank one $\Lambda(\Gamma)$-module on
which $G_{\mathbf{Q}}$ acts via the inverse of the tautological character, see
[Nek06, §8] and [Ben14, §2.2 ] for details. This action naturally extends to
$\mathcal{H}_{A}$, and we denote this rank 1 $\mathcal{H}_{A}$-module by
$\mathcal{H}_{A}^{\iota}$.

######  Definition 4.1 (Nice affinoid neighborhood, [Bel12]).

Let $W=\mathrm{Spm}(R)$ be an admissible affinoid neighborhood of the weight
space $\mathcal{W}$, and $R^{0}$ denote the unit ball for the supremum norm
for $R$. Then $W$ is called nice if $R$ and the residue ring $R^{0}/pR^{0}$
are PID, and the subset ${\mathbf{Z}}\cap W$ is dense in $W$.

For example, any closed ball with center $k$ and radius $p^{-n}$ is nice. This
will be important when we discuss the perfectness of the Selmer complexes that
we are interested in.

###  4.2. Eigencurve and Coleman Families

In this subsection we will give the definitions of Coleman families, the big
representations attached to these families and state the result that these
representations are trianguline (due to Berger). We will also make a quick
review of the eigencurve for the sake of completeness, even though we will not
need this for this article, we will need this theory for the sequel article.

Let $U\subseteq\mathcal{W}$ be a wide open disk centered at a
$\kappa(k_{0})\in\mathcal{W}$ for some $k_{0}\geq 0$, i.e. a rigid analytic
space $U=B(k_{0},r)$ such that its $E$ valued points correspond to
$\\{x\in\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}\mathinner{\\!\left\lvert x-\kappa(k_{0})\right\rvert}<p^{-r}\\}$
for some $r$. Let $\Lambda_{U}$ denote the power bounded analytic functions on
$U$, i.e.

$\Lambda_{U}\mathrel{\mathop{\mathchar 58\relax}}=\\{f\in
E[[T]]\mathrel{\mathop{\mathchar 58\relax}}f\textnormal{ converges on
}U,\mathinner{\\!\left\lVert f\right\rVert}\leq 1\\}.$

Note that $\Lambda_{U}\simeq\mathcal{O}_{E}[[T]]$, where the isomorphism is
non-canonical (hence the notation $\Lambda_{U}$ is used). For $p\nmid N$, let
$\Gamma_{p}\mathrel{\mathop{\mathchar
58\relax}}=\Gamma_{1}(N)\cap\Gamma_{0}(p)$. The following definition is from
[LZ16, Definition 4.6.1] (also see [Och18, Theorem 3.8]).

######  Definition 4.2 (Coleman family).

A Coleman Family $\mathcal{F}$ over a wide open $U\subseteq\mathcal{W}$ is a
power series

$\mathcal{F}=\sum_{n\geq 1}a_{n}(\mathcal{F})q^{n}\in\Lambda_{U}[[q]],$

where $a_{1}(\mathcal{F})=1$, and $a_{p}(\mathcal{F})$ is invertible in
$\Lambda_{U}[1/p]$ such that for all but finitely many weights $k\in
U\cap{\mathbf{N}}$, the specialization $\mathcal{F}_{k}\in
S_{k+2}(\Gamma_{p})$ is a $p$-stabilized and normalized Hecke eigenform, and
for all $k\in U\cap{\mathbf{N}}$, the slope $v_{p}(a_{p}(\mathcal{F}_{k}))$ is
constant.

######  Remark 4.3.

Although we gave the definition of a Coleman family over a wide open disk $U$,
we can also restrict $\mathcal{F}$ to an affinoid disk $X=\mathrm{Spm}(A)$
inside $U$ such that the same interpolation property holds. In that case, a
Coleman family is a power series in $A[[q]]$. In this article we will work
with Coleman families (and their associated representations) over an affinoid
algebra $A$.

######  Theorem 4.4.

Suppose $f_{\alpha}\in S_{k+2}(\Gamma_{p})$ is a Hecke eigenform which is not
$\theta$-critical. Then there is a unique Coleman family $\mathcal{F}$ over
some affinoid neighborhood $U$ of $k\in\mathcal{W}$ such that
$\mathcal{F}_{k}=f_{\alpha}$.

###### Proof.

See [LZ16, Theorem 4.6.4]. ∎

Now we give the definition (and the universal property) of the eigencurve,
which is invented by Coleman and Mazur in [CM98] and generalized by Buzzard in
[Buz07]. The following definition can be found in [Bel12, §2.1], also see
[Loe18, Definition 3.5] and [BB20, §5].

######  Definition 4.5 (Cuspidal Coleman-Mazur-Buzzard eigencurve).

The cuspidal eigencurve with tame level $N$
$\mathcal{C}^{0}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{C}^{0}_{N}$ is
a rigid analytic space of equidimension $1$, which is equipped with a weight
morphism $\kappa\mathrel{\mathop{\mathchar
58\relax}}=\kappa_{\mathcal{C}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}^{0}\to\mathcal{W}$ that is locally finite and flat, and
a universal eigenform $\mathcal{F}^{\rm
univ}\in\mathcal{O}_{\mathcal{C}^{0}}[[q]]$ such that for all affinoid $U$
with a weight morphism $w\mathrel{\mathop{\mathchar 58\relax}}U\to\mathcal{W}$
and any Coleman family $\mathcal{F}$ over $U$, there is a unique morphism
$\widetilde{w}\mathrel{\mathop{\mathchar 58\relax}}U\to\mathcal{C}^{0}$ such
that $\kappa\circ\widetilde{w}=w$ and
$\mathcal{F}=\widetilde{w}^{*}(\mathcal{F}^{\rm univ})$.

The eigencurve $\mathcal{C}$ is equipped with a finite flat weight map, which
parametrizes modular forms, and contains $\mathcal{C}^{0}$ as a closed subset.
$\mathcal{C}$ is admissibly covered by open affinoid subdomains of the form
$\mathcal{C}_{W,\nu}$, where $W=\mathrm{Spm}(A)\subseteq\mathcal{W}$, and
$\nu>0$, where $\mathcal{C}_{W,\nu}$ is the maximal spectrum of affinoid
subalgebra $\mathcal{H}\otimes R$ of
$\mathrm{End}_{R}(M^{\dagger}(\Gamma_{p},W)^{\leq\nu})$ (and only here
$\mathcal{H}$ denotes the Hecke algebra).

Note that $f_{\alpha}$ is $\theta$-critical if and only if the eigencurve is
not étale over $\mathcal{W}$ at $x\in\mathcal{C}$, where $x$ is the point
corresponding to $f_{\alpha}$ in the eigencurve.

###  4.3. Motives and Galois Representations Attached to Modular Forms

Let $f\in S_{k+2}(\Gamma_{1}(N),\varepsilon)$ be a normalised newform of
weight $k+2$ and level $\Gamma_{1}(N)$ with nebentype $\varepsilon$. Suppose
that $p\nmid N$. In [Sch90], Scholl constructs a motive $M(f)$ such that its
$p$-adic realization is the Deligne’s cohomological $G_{\mathbf{Q}}$
representation $V_{f}$, which is irreducible, crystalline at $p$ (since
$p\nmid N$) and the action of $\varphi$ on $\mathbf{D}_{\rm cris}(V_{f})$ is
given by the characteristic polynomial
$X^{2}-a_{p}(f)X+\varepsilon(p)p^{k+1}$. If $\chi$ is a Dirichlet character,
we have

$V_{f_{\chi}}\cong V_{f}\otimes\chi.$

Note that for $g\in\\{f_{\alpha},f_{\beta}\\}$ we have the isomorphisms of
Galois representations $V_{g}\cong V_{f}$, and $V_{g^{c}}\cong V_{f^{*}}$.
Note also that due to the Poincaré duality, we have the isomorphism:

(4.1) $V_{f}^{*}\cong V_{f}(k+1)\otimes\varepsilon^{-1}\cong V_{f^{*}}(k+1).$

The Hodge-Tate weights of $V_{f}$ are $0$ and $-1-k$. Note also that the
action of geometric Frobenius ${\rm Frob}^{-1}_{p}$ at $p$ on the $\ell$-adic
realizations (for $\ell\neq p$) is given by the characteristic polynomial
$X^{2}-a_{p}(f)X+\varepsilon(p)p^{k+1}$. Let $\alpha$ and $\beta$ denote the
roots of this characteristic polynomial, and let $E$ be a finite extension of
${{\mathbf{Q}}_{p}}$ which contains $\alpha$, $\beta$, and the Hecke field of
$f$.

We say that $f$ is ordinary at $p$ if $v_{p}(a_{p}(f))=0$, otherwise we say
$f$ is $p$-non-ordinary. In this article we are interested in the non-ordinary
case. This implies that $v_{p}(\alpha),v_{p}(\beta)>0$, therefore none of the
$p$-stabilizations of $f$ are at critical slope (i.e.
$v_{p}(\alpha),v_{p}(\beta)\neq k+1$). Thus they are not $\theta$-critical,
i.e. $V_{f}|_{G_{p}}$ does not split into the direct sum of two characters.
Indeed, in the non-ordinary case, the Panchishkin ordinarity condition for
$V_{f}^{*}$ fails (even though it satisfies the criticality condition in
Definition 3.6 since $d^{+}(V_{f}^{*})=1$ and $V_{f}^{*}$ has Hodge-Tate
weights $0$ and $k+1$) as $V_{f}$ is irreducible as a $G_{p}$ representation.

Let $\mathcal{F}$ be a Coleman family passing through $f_{\alpha}$ over a nice
affinoid subdomain $U=\mathrm{Spm}(A)$ of the weight space. Let $x_{0}$ be the
point corresponding to $f_{\alpha}$ in the eigencurve. Let $\mathcal{X}$ be
the preimage of $U$ under the weight map $\kappa$ of the eigencurve, and let
$\mathcal{X}^{\rm cl}$ denote the classical points of $\mathcal{X}$, i.e. the
points which are sent to non-negative integers under the weight map $\kappa$
of the eigencurve.

######  Theorem 4.6 (The big Galois representation attached to a Coleman
family).

There exists a free $A$-module $\mathbf{V}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}_{\mathcal{F}}$ of rank $2$ equipped with Galois action
$G_{{\mathbf{Q}},S}$ for some finite set of primes $S$ such that for each
$x\in\mathcal{X}^{\rm cl}$, the following holds:

  * •

$\mathbf{V}_{x}$ has Hodge-Tate weights $\\{-\kappa(x)-1,0\\}$,

  * •

$\mathbf{V}_{x}$ is crystalline (semistable if $p\mid N$ were the case) as a
$G_{p}$-representation,

  * •

There is an $\mathbf{\alpha}\in A$ with $\mathbf{\alpha}(x_{0})=\alpha$ such
that $\mathbf{D}_{\rm cris}(\mathbf{V}_{x})^{\varphi=\mathbf{\alpha}(x)}$ is
$1$-dimensional,

  * •

If $x\neq x_{0}$, then $\mathbf{D}_{\rm
cris}(\mathbf{V}_{x})^{\varphi=\mathbf{\alpha}(x)}\cap{\rm
Fil}^{\kappa(x)+1}(\mathbf{D}_{\rm st}(\mathbf{V}_{x}))=0$.

If the last condition holds also for $x_{0}$, then $f_{\alpha}$ is
non-$\theta$-critical, otherwise it is $\theta$-critical. In the former case,
$A_{\mathcal{X}}$ and $A$ will be isomorphic as rings, but the situation is
more complicated when $f_{\alpha}$ is $\theta$-critical. See [BB20, §3.3.2]
for a more detailed explanation. Since we are interested in the
non-$\theta$-critical case in this article, we will identify these rings and
their spectra inside the weight space.

The following theorem is due to [KPX14] (see also [BB20, §3.3]).

######  Theorem 4.7 (Triangulation).

In the non $\theta$-critical case, there exists a rank-one
$(\varphi,\Gamma)$-submodule $\mathbf{D}$ of $\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V})$ such that

  * •

$\mathbf{D}=\mathbf{D}_{\delta}$ for $\delta\mathrel{\mathop{\mathchar
58\relax}}{\mathbf{Q}}_{p}^{\times}\to{A}^{\times}$ such that
$\delta(p)=\alpha$ and $\delta(u)=1$ for $u\in{\mathbf{Z}}_{p}^{\times}$,

  * •

$\mathscr{D}_{\rm cris}(\mathbf{D}_{x})=\mathbf{D}_{\rm
cris}(\mathbf{V}_{x})^{\varphi=\mathbf{\alpha}(x)}$,

  * •

For all classical points $x\in\mathcal{X}^{\rm cl}$ (including $x_{0}$ in the
non-$\theta$-critical case), $\mathbf{D}_{x}$ is saturated in $\mathbf{D}_{\rm
rig}^{\dagger}(\mathbf{V}_{x})$.

##  5\. Selmer Complexes

###  5.1. Review of Galois Cohomology and Iwasawa Cohomology

Let $G$ be a Galois group of a Galois extension, and let $\mathbf{V}$ be a
$p$-adic representation of $G$ with coefficients in an affinoid algebra $A$
over $E$. When $A=E$, we will denote $\mathbf{V}$ by $V$. Let
$C^{\bullet}(G,\mathbf{V})$ denote the cochain complex of continuous
cohomology of $G$, and let $\mathbf{R\Gamma}(G,\mathbf{V})$ denote the
corresponding object in the derived category $\mathscr{D}(A)$ of complexes of
$A$-modules. We will denote the cohomologies of
$\mathbf{R\Gamma}(G,\mathbf{V})$ by $H^{i}(G,\mathbf{V})$. If $A=E$, and
$T\subseteq V$ is a $G$-stable $\mathcal{O}_{E}$-lattice, we can also define
$C^{\bullet}(G,T)$, $C^{\bullet}(G,V/T)$ and the corresponding derived objects
$\mathbf{R\Gamma}(G,T)$ and $\mathbf{R\Gamma}(G,V/T)$ in
$\mathscr{D}(\mathcal{O}_{E})$ and $\mathscr{D}(E/\mathcal{O}_{E})$
respectively in a similar manner.

When $K$ denotes a finite extension of ${\mathbf{Q}}$, ${{\mathbf{Q}}_{p}}$ or
${{\mathbf{Q}}_{\ell}}$, and $G=G_{K}$ is the absolute Galois group of $K$, we
will denote the complex $C^{\bullet}(G_{K},\mathbf{V})$ by
$C^{\bullet}(K,\mathbf{V})$, the corresponding derived object
$\mathbf{R\Gamma}(G_{K},\mathbf{V})$ in $\mathscr{D}(A)$ by
$\mathbf{R\Gamma}(K,\mathbf{V})$, and its cohomologies by
$H^{i}(K,\mathbf{V})$. When $A=E$ and $T\subseteq V$ is a $G$-stable
$\mathcal{O}_{E}$-lattice, we will denote $\mathbf{R\Gamma}(G_{K},T)$ and
$\mathbf{R\Gamma}(G_{K},V/T)$ by $\mathbf{R\Gamma}(K,T)$ and
$\mathbf{R\Gamma}(K,V/T)$, respectively.

From now on, let $\mathbf{V}$ be a representation of $G_{F}$ for a number
field $F$ over an affinoid algebra $A$. Let $S$ be a finite set of places of
$F$ containing the places $v$ for which $\mathbf{V}$ is ramified, places above
$\infty$ and $p$, and let $S_{f}=S-\\{v\mid\infty\\}$. If $F_{S}$ denotes the
maximal extension of $F$ unramified outside $S$, the Galois group
$\mathrm{Gal}(F_{S}/F)$ of this extension will be denoted as $G_{F,S}$. Note
that $\mathbf{V}$ is a representation of $G_{F,S}$. In this case we denote
$C^{\bullet}(G_{F,S},\mathbf{V})$ by $C^{\bullet}_{S}(F,\mathbf{V})$, and the
corresponding derived object in $\mathscr{D}(A)$ by
$\mathbf{R\Gamma}_{S}(F,\mathbf{V})$ or simply by
$\mathbf{R\Gamma}_{S}(\mathbf{V})$ when $F$ is clear from the context.

Let us state the definitions of Iwasawa theoretic versions of the complexes
above. Let $\overline{\mathbf{V}}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}\otimes_{A}\mathcal{H}_{A}^{\iota}$ denote the cyclotomic
deformation of $\mathbf{V}$. Then $\overline{\mathbf{V}}$ is also a
representation of $G_{F,S}$ (and of $G_{F_{v}}$ for each $v\in S_{f}$), thus
we can define the complexes

$\displaystyle C^{\bullet}_{{\rm Iw},S}(F,\mathbf{V})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=C^{\bullet}_{S}(F,\overline{\mathbf{V}}),$ $\displaystyle
C^{\bullet}_{{\rm Iw}}(F_{v},\mathbf{V})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=C^{\bullet}(F_{v},\overline{\mathbf{V}}){\rm\ for\ }v\in S_{f}.$

We denote the corresponding objects in the derived category
$\mathscr{D}(\mathcal{H}_{A})$ by $\mathbf{R\Gamma}_{{\rm
Iw},S}(F,\mathbf{V})$ and $\mathbf{R\Gamma}_{\rm Iw}(F_{v},\mathbf{V})$, and
their cohomologies by $H^{i}_{{\rm Iw},S}(F,\mathbf{V})$ and $H^{i}_{\rm
Iw}(F_{v},\mathbf{V})$, respectively. Note that by Shapiro Lemma, when $v\mid
p$, we have

$H^{i}_{{\rm
Iw}}(F_{v},\mathbf{V})\cong\varprojlim_{n}H^{i}(F_{v,n},\mathbf{V}),$

where the transition maps are given by corestriction. When $F={\mathbf{Q}}$,
we will denote $\mathbf{R\Gamma}_{{\rm Iw},S}(F,\mathbf{V})$ by
$\mathbf{R\Gamma}_{{\rm Iw},S}(\mathbf{V})$.

####  5.1.1. Cohomology of $\left(\varphi,\Gamma\right)$-Modules

######  Definition 5.1 (Fontaine-Herr complex).

Let $K$ be a finite extension of ${{\mathbf{Q}}_{p}}$, and let $\mathbf{D}$ be
a $(\varphi,\Gamma_{K})$-module over $\mathcal{R}_{A}$. We define the
Fontaine-Herr complex $C^{\bullet}_{\varphi,\gamma_{n}}(\mathbf{D})$, which is
concentrated at degrees [0,2], as follows:

$\left[\mathbf{D}^{\Delta}\xrightarrow{d_{0}}\mathbf{D}^{\Delta}\oplus\mathbf{D}^{\Delta}\xrightarrow{d_{1}}\mathbf{D}^{\Delta}\right],$

where $d_{0}(z)=(\gamma_{n}(z)-z,\varphi(z)-z)$ and
$d_{1}(x,y)=(\gamma_{n}(x)-x)+(y-\varphi(y))$.

We denote the cohomologies of this complex by $H^{i}(K_{n},\mathbf{D})$. For
$n=0$, we will denote the complex by
$C^{\bullet}_{\varphi,\Gamma_{K}}(\mathbf{D})$, and the corresponding object
in the derived category $\mathscr{D}(A)$ is denoted by
$\mathbf{R\Gamma}(K,\mathbf{D})$ (or simply by $\mathbf{R\Gamma}(\mathbf{D})$,
when $K={{\mathbf{Q}}_{p}}$).

######  Definition 5.2 (Iwasawa cohomology).

Let $C^{\bullet}_{\psi}(\mathbf{D})$ be the following complex that is
concentrated in degrees $1$ and $2$:

$\mathbf{D}^{\Delta}\xrightarrow{\psi-1}\mathbf{D}^{\Delta},$

where $\psi$ is the unique continuous map (and a left inverse of $\varphi$)
$\psi\mathrel{\mathop{\mathchar 58\relax}}\mathcal{R}_{A}\to\mathcal{R}_{A}$
such that:

$\varphi(\psi(f(\pi)))\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{p}\sum_{\xi\in\mu_{p}}f(\xi(1+\pi)-1).$

We denote the corresponding object in the derived category
$\mathscr{D}(\mathcal{H}_{A})$ by $\mathbf{R\Gamma}_{\rm Iw}(K,\mathbf{D})$
(or simply by $\mathbf{R\Gamma}_{\rm Iw}(\mathbf{D})$, when
$K={{\mathbf{Q}}_{p}}$), and its cohomologies by $H^{i}_{\rm
Iw}(K,\mathbf{D})$.

The following theorem relates the Galois cohomologies of a
$G_{K}$-representation $\mathbf{V}$ and the cohomologies of its associated
$(\varphi,\Gamma)$-module (see [BB16, §2.2, §2.4]):

######  Proposition 5.3.

If $\mathbf{V}$ is a $G_{K}$ representation over $A$ then we have

$\displaystyle H^{i}(K,\mathbf{V})$ $\displaystyle\cong
H^{i}(K,\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})),$ $\displaystyle
H^{i}_{{\rm Iw}}(K,\mathbf{V})$ $\displaystyle\cong H^{i}_{{\rm
Iw}}(K,\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})).$

###  5.2. Selmer Groups

In this subsection we set $A=E$. Let $V$ be a $G_{F}$ representation over $E$,
let $T\subseteq V$ be a $G_{F}$-stable $\mathcal{O}_{E}$-lattice in $V$. Let
$S$ be as in previous subsections. We first define the local conditions
attached to $V$, i.e for each place $v\in S_{f}$, certain subspaces of
$H^{1}(F_{v},V)$.

######  Definition 5.4 (Local conditions).

For each $v\in S_{f}$ and $v\nmid p$, let

$H^{1}_{f}(F_{v},V)\mathrel{\mathop{\mathchar
58\relax}}=\ker(H^{1}(F_{v},V)\rightarrow H^{1}(F_{v}^{\rm ur},V)),$

where $F_{v}^{\rm ur}$ denotes the maximal unramified extension of $F_{v}$.
For $v\mid p$, let

$\displaystyle H^{1}_{\rm str}(F_{v},V)$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=0,$ $\displaystyle
H^{1}_{(p)}(F_{v},V)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=H^{1}(F_{v},V),$ $\displaystyle H^{1}_{f}(F_{v},V)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\ker(H^{1}(F_{v},V)\rightarrow
H^{1}(F_{v},V\otimes\mathbf{B}_{\mathrm{cris}}))$

be the strict, relaxed, and Bloch-Kato local conditions at $p$ attached to
$V$, respectively. For $v\in S_{f}$, we define $H^{1}_{f}(F_{v},T)$ as the
preimage of $H^{1}_{f}(F_{v},V)$ under $H^{1}(F_{v},T)\rightarrow
H^{1}(F_{v},V)$, and $H^{1}_{f}(F_{v},V/T)$ as the image of $H^{1}(F_{v},V)$
under $H^{1}(F_{v},V)\rightarrow H^{1}(F_{v},V/T)$. For strict and relaxed
conditions at $p$, we define

$\displaystyle H^{1}_{\rm str}(F_{v},T)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=0=\mathrel{\mathop{\mathchar 58\relax}}H^{1}_{\rm str}(F_{v},V/T),$
$\displaystyle H^{1}_{(p)}(F_{v},T)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=H^{1}(F_{v},T),$ $\displaystyle H^{1}_{(p)}(F_{v},V/T)$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=H^{1}(F_{v},V/T).$

######  Definition 5.5 (Selmer group).

Let $M\in\\{V,T,V/T\\}$. For $?\in\\{{\rm str},(p),f\\}$, we define
respectively the strict, relaxed, Bloch-Kato Selmer groups as follows:

$H^{1}_{?}(F,M)\mathrel{\mathop{\mathchar
58\relax}}=\ker\left(H^{1}(F,M)\longrightarrow\bigoplus_{\begin{subarray}{c}v\in
S_{f}\\\ v\nmid
p\end{subarray}}\frac{H^{1}(F_{v},M)}{H^{1}_{f}(F_{v},M)}\oplus\bigoplus_{\begin{subarray}{c}v\in
S_{f}\\\ v\mid
p\end{subarray}}\frac{H^{1}(F_{v},M)}{H^{1}_{?}(F_{v},M)}\right).$

Let $D$ be a $(\varphi,\Gamma_{K})$-module over the Robba ring
$\mathcal{R}_{E}$. Let $H^{1}_{f}(K,D)\subseteq H^{1}(K,D)$ be the subspace of
the crystalline extensions of $D$. Then $H^{1}_{f}(K,D)$ sits in the following
exact sequence (see [Ben21, Proposition 2.9.2]):

$0\rightarrow
H^{0}(K,D)\rightarrow\mathscr{D}_{\mathrm{cris}}(D)\xrightarrow{1-\varphi,{\rm
pr}}\mathscr{D}_{\mathrm{cris}}(D)\oplus t_{D}(K)\rightarrow
H^{1}_{f}(K,D)\rightarrow 0,$

where $t_{D}(K)\mathrel{\mathop{\mathchar
58\relax}}=\mathscr{D}_{\mathrm{dR}}(D)/{\mathrm{Fil}}^{0}\mathscr{D}_{\mathrm{dR}}(D)$
is called the tangent space of $D$. When $V$ is potentially semistable, then
we have

$H^{1}_{f}(K,V)=H^{1}_{f}(K,\mathbf{D}_{\rm rig}^{\dagger}(V)).$

###  5.3. Selmer Complexes

In this subsection we will give a brief summary of Nekovář’s Selmer Complexes,
details of which can be found in [Nek06]. We will here focus on the rings $R$
which are complete local Noetherian rings with finite residue field, and give
the generalizations to affinoid algebras in the following subsection.

Let $F$ be a finite extension of ${\mathbf{Q}}$, and let $S$ be a finite set
of primes as before.

######  Definition 5.6 (Selmer complex).

Let $X$ be an admissible $R[G_{F}]$-module ([Nek06, Definition 3.2.1]), and
for each $v\in S$, let $U_{v}^{+}(X)$ be a complex satisfying some finiteness
conditions (see [Nek06, §4.2] for the details), and let
$i_{v}^{+}\mathrel{\mathop{\mathchar 58\relax}}U_{v}^{+}(X)\rightarrow
C^{\bullet}(G_{v},X)$ be a morphism of complexes. The local conditions
attached to $X$ is the collection of these morphisms
$\Delta(X)\mathrel{\mathop{\mathchar 58\relax}}=\\{i_{v}^{+}\\}$. Let
$U_{S}^{+}(X)\mathrel{\mathop{\mathchar 58\relax}}=\bigoplus_{v\in
S_{f}}U_{v}^{+}(X)$.

The Selmer complex attached to the data $(G_{F},X,\Delta(X))$ is the following
mapping cone:

$\widetilde{C}_{f}^{\bullet}(F,X,\Delta)\mathrel{\mathop{\mathchar
58\relax}}={\rm Cone}\left(C^{\bullet}_{S}(X)\oplus
U_{S}^{+}(X)\xrightarrow{res_{S_{f}}-i_{S}^{+}}\bigoplus_{v\in
S_{f}}C^{\bullet}(G_{v},X)\right)[-1],$

where $i_{S}^{+}=\oplus i_{v}^{+}\mathrel{\mathop{\mathchar
58\relax}}U_{S_{f}}^{+}(X)\to\bigoplus_{v\in S_{f}}C^{\bullet}(G_{v},X)$. The
corresponding object in the derived category $\mathscr{D}_{\rm ft}^{b}(R)$
will be denoted by $\widetilde{\mathbf{R\Gamma}}_{f}(F,X,\Delta)$, or simply
by omitting $F$ and $\Delta$ when they are clear from the context.

We set

$U_{S}^{-}(F,X)\mathrel{\mathop{\mathchar 58\relax}}={\rm
Cone}\left(U_{S}^{+}(F,X)\to\bigoplus_{v\in
S_{f}}C^{\bullet}(F_{v},X)\right),$

and

$C^{\bullet}_{\rm c}(G_{F,S},X)\mathrel{\mathop{\mathchar 58\relax}}={\rm
Cone}\left(C^{\bullet}_{S}(F,X)\to\bigoplus_{v\in
S_{f}}C^{\bullet}(F_{v},X)\right)[-1].$

Then by definition of these complexes, we have the following exact triangles
(see [Nek06, 6.1.3.2]):

(Tr1) $\displaystyle U_{S}^{+}(F,X)\rightarrow\bigoplus_{v\in
S_{f}}\mathbf{R\Gamma}(F_{v},X)\rightarrow U_{S}^{-}(F,X) \rightarrow
U_{S}^{+}(F,X)[1],$ (Tr2) $\displaystyle
U_{S}^{+}(F,X)[-1]\rightarrow\mathbf{R\Gamma}_{\rm
c}(G_{F,S},X)\rightarrow\widetilde{\mathbf{R\Gamma}}_{f}(F,X) \rightarrow
U_{S}^{+}(F,X),$ (Tr3) $\displaystyle
U_{S}^{-}(F,X)[-1]\rightarrow\widetilde{\mathbf{R\Gamma}}_{f}(F,X)\rightarrow\mathbf{R\Gamma}_{S}(F,X)
\rightarrow U_{S}^{-}(F,X).$

In particular, we will focus on the case where the local conditions (called
Greenberg’s local conditions) are as follows: for $v\nmid p$ we will set

$U_{v}^{+}(X)\mathrel{\mathop{\mathchar
58\relax}}=C^{\bullet}(G_{v}/I_{v},X^{I_{v}}),$

and for $v\mid p$ we choose a certain subrepresentation $F^{+}_{v}X\subseteq
X$ and set $U_{v}^{+}(X)\mathrel{\mathop{\mathchar
58\relax}}=C^{\bullet}(G_{v},F^{+}_{v}X)$.

In the case where $F^{+}_{v}X=X$ (resp. $F^{+}_{v}X=0$) for all $v\mid p$ we
will denote the associated Selmer complex by
$\widetilde{C}^{\bullet}_{(p)}(X)$ (resp. $\widetilde{C}^{\bullet}_{\rm
str}(X)$).

######  Theorem 5.7 (Exact triangle of Selmer complexes).

Let $X_{1}$, $X_{2}$, $X_{3}$ be admissible $G_{F}$-representations forming a
split exact sequence as follows:

Suppose also that for each $v\mid p$, this exact sequence induce the following
exact sequences:

(5.1) $\displaystyle 0\longrightarrow
U_{v}^{+}(X_{1})\xrightarrow{U_{v}^{+}(f)}U_{v}^{+}(X_{2})\xrightarrow{U_{v}^{+}(g)}U_{v}^{+}(X_{3})\longrightarrow
0.$

Then we have the short exact sequence of the Selmer complexes:

$\displaystyle
0\longrightarrow\widetilde{C}^{\bullet}_{f}(F,X_{1},\Delta(X_{1}))\longrightarrow\widetilde{C}^{\bullet}_{f}(F,X_{2},\Delta(X_{2}))\longrightarrow\widetilde{C}^{\bullet}_{f}(F,X_{3},\Delta(X_{3}))\longrightarrow
0.$

Moreover, if the splitting of (LABEL:eq:split-exact-reps-nek) induces the
splitting of (5.1), then the coboundary maps in the long exact sequence coming
from the exact triangle above are $0$.

###### Proof.

We will follow the strategy that Nekovář used in the beginning of Chapter 11
of [Nek06], which lead to the definition of $p$-adic height pairing. First
note that the splitness of the exact sequence in (LABEL:eq:split-exact-reps-
nek) yield the exact sequence (5.1) also for $v\nmid p$. For each $v\in S$,
let $U_{v}^{-}(X)\mathrel{\mathop{\mathchar 58\relax}}={\rm
Cone}(-i_{v}^{+}\mathrel{\mathop{\mathchar 58\relax}}U_{v}^{+}(X)\rightarrow
C^{\bullet}(G_{v},X))$ as in [Nek06].

Then we have a short exact sequence

$\displaystyle 0\longrightarrow
U_{v}^{-}(X_{1})\xrightarrow{U_{v}^{-}(f)}U_{v}^{-}(X_{2})\xrightarrow{U_{v}^{-}(g)}U_{v}^{-}(X_{3})\longrightarrow
0$

obtained by applying 3x3 lemma to the following commutative diagram with exact
top and bottom rows:

${0}$${U_{v}^{+}(X_{1})}$${U_{v}^{+}(X_{2})}$${U_{v}^{+}(X_{3})}$${0}$${0}$${C^{\bullet}(G_{v},X_{1})}$${C^{\bullet}(G_{v},X_{2})}$${C^{\bullet}(G_{v},X_{3})}$${0}$${0}$${U_{v}^{-}(X_{1})}$${U_{v}^{-}(X_{2})}$${U_{v}^{-}(X_{3})}$${0.}$$\scriptstyle{U_{v}^{+}(f)}$$\scriptstyle{i_{v}^{+}}$$\scriptstyle{U_{v}^{+}(g)}$$\scriptstyle{i_{v}^{+}}$$\scriptstyle{i_{v}^{+}}$$\scriptstyle{{\rm
Cone}(i_{v}^{+})}$$\scriptstyle{{\rm Cone}(i_{v}^{+})}$$\scriptstyle{{\rm
Cone}(i_{v}^{+})}$

Hence we have the following exact sequence:

$\displaystyle 0\longrightarrow
U_{S}^{-}(X_{1})\xrightarrow{U_{S}^{-}(f)}U_{S}^{-}(X_{2})\xrightarrow{U_{S}^{-}(g)}U_{S}^{-}(X_{3})\longrightarrow
0.$

Recall the following exact sequence from [Nek06, 6.1.3.1] inducing (Tr3):

$\displaystyle 0\longrightarrow
U_{S}^{-}(X)[-1]\longrightarrow\widetilde{C}^{\bullet}_{f}(F,X,\Delta)\longrightarrow
C^{\bullet}_{S}(X)\longrightarrow 0.$

Combining these and applying 3x3 lemma to the following diagram yields the
desired exact sequence:

${0}$${U_{S}^{-}(X_{1})[-1]}$${U_{S}^{-}(X_{2})[-1]}$${U_{S}^{-}(X_{3})[-1]}$${0}$${0}$${\widetilde{C}^{\bullet}_{f}(X_{1})}$${\widetilde{C}^{\bullet}_{f}(X_{2})}$${\widetilde{C}^{\bullet}_{f}(X_{3})}$${0}$${0}$${C^{\bullet}_{S}(X_{1})}$${C^{\bullet}_{S}(X_{2})}$${C^{\bullet}_{S}(X_{3})}$${0.}$$\scriptstyle{U_{S}^{-}(f)}$$\scriptstyle{U_{S}^{-}(g)}$

Finally, for $v\mid p$ if the exact sequence (5.1) also splits, then we have
$U_{S}^{-}(X_{2})=U_{S}^{-}(X_{1})\oplus U_{S}^{-}(X_{3})$, hence the first
and third rows in the 3x3 diagram above split. Hence the second row induced by
the 3x3 lemma also becomes split exact, and the coboundary map in the long
exact sequence becomes $0$. ∎

Selmer complexes satisfy some duality theorems (via Grothendieck and Matlis
dualities), control theorems, etc. We will be mainly interested in Pottharst
style Selmer complexes, which generalizes the construction of Nekovář to the
local conditions at $p$ arising from $(\varphi,\Gamma)$-modules and
representations with coefficients in affinoid algebras. Thus we will state
these theorems in terms of Pottharst style Selmer complexes in the following
subsections.

###  5.4. Pottharst Style Selmer Complexes

In this subsection, we will review results from [Pot12], [Pot13], [Ben14],
[BB16] and [Ben21]. Pottharst’s Selmer Complexes are a direct generalization
of Selmer Complexes introduced by Nekovář in [Nek06] in a way that we are able
to work with local conditions coming from triangulations and to work with
coefficients in an affinoid algebra. In this subsection, for notational
simplicity and our purposes, we will take $F={\mathbf{Q}}$ and consider
$p$-adic representations $\mathbf{V}$ of $G_{{\mathbf{Q}},S}$ with
coefficients in an affinoid algebra $A$ over $E$ (including the case $A=E$).

Let $\mathbf{D}\subseteq\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})$ be a
$(\varphi,\Gamma)$-submodule. Let

$U_{p}^{+}(\mathbf{V},\mathbf{D})\mathrel{\mathop{\mathchar
58\relax}}=C^{\bullet}_{\varphi,\Gamma}(\mathbf{D})$

denote the local condition at $p$, which is the Fontaine-Herr complex defined
in Definition 5.1. If $\ell\neq p$, then $U_{\ell}^{+}(\mathbf{V},\mathbf{D})$
will denote the unramified local conditions

$U_{\ell}^{+}(\mathbf{V},\mathbf{D})\mathrel{\mathop{\mathchar
58\relax}}=\left[\mathbf{V}^{I_{\ell}}\xrightarrow{f_{\ell}-1}\mathbf{V}^{I_{\ell}}\right],$

which is a complex concentrated at degrees $[0,1]$.

Let $K_{p}(\mathbf{V})$ denote the total complex

${\rm Tot}\left(C^{\bullet}({{\mathbf{Q}}_{p}},\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}))\xrightarrow{\varphi-1}(C^{\bullet}({{\mathbf{Q}}_{p}},\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V})\right),$

and

$i_{p}^{+}\mathrel{\mathop{\mathchar
58\relax}}U_{p}^{+}(\mathbf{V},\mathbf{D})\to
C_{\varphi,\Gamma}^{\bullet}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}))\xrightarrow{q.is}K_{p}(V).$

We finally set

$K_{\ell}(\mathbf{V})\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{R\Gamma}({{\mathbf{Q}}_{\ell}},\mathbf{V})$

and

$K_{S}(\mathbf{V})\mathrel{\mathop{\mathchar 58\relax}}=\bigoplus_{v\in
S_{f}}K_{v}(\mathbf{V}),$

and also denote the corresponding objects to these complexes in the derived
category $\mathscr{D}(A)$ exactly the same.

######  Definition 5.8.

The Selmer complex $\mathbf{R\Gamma}(\mathbf{V},\mathbf{D})$ attached to the
data $(\mathbf{V},\mathbf{D})$ is defined as the mapping cone:

$\displaystyle{\rm
Cone}\left(\mathbf{R\Gamma}_{S}(\mathbf{V})\oplus\bigoplus_{v\in
S_{f}}U_{v}^{+}(\mathbf{V},\mathbf{D})\to K_{S}(\mathbf{V})\right)[-1].$

If we work with representations of $G_{F}$ for general number fields $F$, and
for each prime $\mathfrak{p}$ of $F$ lying above $p$, if we have a local
condition $\mathbf{D}_{\mathfrak{p}}$, and
$\mathbf{D}=(\mathbf{D}_{\mathfrak{p}})_{\mathfrak{p}}$, then we define the
corresponding Selmer complexes in a similar manner, and denote them by
$\mathbf{R\Gamma}(F,\mathbf{V},\mathbf{D})$.

Let $\widetilde{\mathbf{D}}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})/\mathbf{D}$, and for
$v\in S_{f}$, let

$U_{v}^{-}(\mathbf{V},\mathbf{D})\mathrel{\mathop{\mathchar 58\relax}}={\rm
Cone}\left(U_{v}^{+}(\mathbf{V},\mathbf{D})\xrightarrow{-i_{v}^{+}}K_{v}(\mathbf{V})\right).$

Then we have
$U_{p}^{-}(\mathbf{V},\mathbf{D})=\mathbf{R\Gamma}(\widetilde{\mathbf{D}})$
and the following commutative diagram of exact triangles with vertical arrows
being quasi-isomorphisms in the derived category $\mathscr{D}(A)$:

${U_{p}^{+}(\mathbf{V},\mathbf{D})}$${K_{p}(\mathbf{V})}$${U_{p}^{-}(\mathbf{V},\mathbf{D})}$${U_{p}^{+}(\mathbf{V},\mathbf{D})[1]}$${\mathbf{R\Gamma}(\mathbf{D})}$${\mathbf{R\Gamma}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}))}$${\mathbf{R\Gamma}(\widetilde{\mathbf{D}})}$${\mathbf{R\Gamma}(\mathbf{D})[1]}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Thus we have $H^{i}({{\mathbf{Q}}_{p}},\widetilde{\mathbf{D}})\cong
H^{i}(U_{p}^{-}(\mathbf{V},\mathbf{D}))$.

For the Iwasawa theoretic analogues of these Selmer complex, we define our
local conditions as follows: First, let us look at the case $A=E$. Let $T$ be
a $G_{\mathbf{Q}}$-stable $\mathcal{O}_{E}$-lattice in $V$. For any $\ell\neq
p$, we define $U_{{\rm Iw},\ell}^{+}(V,\mathbf{D})$ as

$\displaystyle U_{{\rm Iw},\ell}^{+}(V,\mathbf{D})=\mathbf{R\Gamma}_{f,{\rm
Iw}}({{\mathbf{Q}}_{\ell}},T)\otimes_{\Lambda}\mathcal{H},$

where $\mathbf{R\Gamma}_{f,{\rm Iw}}({{\mathbf{Q}}_{\ell}},T)$ is the
corresponding derived object of the following complex concentrated at degrees
$[0,1]$:

$\displaystyle\left[T^{I_{\ell}}\otimes\Lambda^{\iota}\xrightarrow{{\rm
Fr}_{\ell}-1}T^{I_{\ell}}\otimes\Lambda^{\iota}\right].$

For $v=p$, we define $U^{+}_{{\rm Iw},p}(V,\mathbf{D})$ as
$\mathbf{R\Gamma}_{\rm Iw}(\mathbf{D})$. We define $U^{+}_{{\rm
Iw},S}(V,\mathbf{D})$ as before, and for each $v\in S_{f}$, $K_{{\rm
Iw},v}(V)\mathrel{\mathop{\mathchar
58\relax}}=K_{v}(T\otimes_{\Lambda}\mathcal{H}^{\iota})\simeq\mathbf{R\Gamma}_{\rm
Iw}({\mathbf{Q}}_{v},V)$ Then we can define the Iwasawa theoretic Selmer
complex $\mathbf{R\Gamma}_{\rm Iw}(V,\mathbf{D})$ associated to this local
data as follows:

$\displaystyle{\rm Cone}\left(\left(\mathbf{R\Gamma}_{{\rm
Iw},S}(T)\otimes_{\Lambda}^{\mathbf{L}}\mathcal{H}\right)\oplus U_{{\rm
Iw},S}^{+}(V,\mathbf{D})\to\bigoplus_{v\in S_{f}}\mathbf{R\Gamma}_{\rm
Iw}({\mathbf{Q}}_{v},V)\right)[-1].$

Note that we could simply define the Iwasawa theoretic Selmer complex as

$\mathbf{R\Gamma}_{\rm Iw}(V,\mathbf{D})\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{R\Gamma}(\overline{V},\overline{\mathbf{D}}),$

where we recall that $\overline{V}=V\otimes\mathcal{H}^{\iota}$ and
$\overline{\mathbf{D}}=\mathbf{D}\otimes\mathcal{H}^{\iota}$.

If $\mathbf{V}$ is a $p$-adic representation over an affinoid algebra $A$, we
replace $T$ in the definition above by $\mathbf{V}$. In this case, the Iwasawa
Selmer complex we are interested in is the Selmer Complex
$\mathbf{R\Gamma}(\overline{\mathbf{V}},\overline{\mathbf{D}})$, where

$\overline{\mathbf{V}}=\mathbf{V}{\otimes}_{A}\mathcal{H}_{A}^{\iota}$

and

$\overline{\mathbf{D}}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}{\otimes}_{A}\mathcal{H}_{A}^{\iota}$

are the cyclotomic deformations of $\mathbf{V}$ and $\mathbf{D}$,
respectively. More explicitly, our local conditions are given as follows: For
$\ell\neq p$ we have

$\displaystyle U_{\ell,{\rm
Iw}}^{+}(\mathbf{V},\mathbf{D})=U_{\ell}^{+}(\overline{\mathbf{V}},\overline{\mathbf{D}})\mathrel{\mathop{\mathchar
58\relax}}=\left[(\mathbf{V}\otimes_{A}\Lambda_{A}^{\iota})^{I_{\ell}}\xrightarrow{\rm{Fr}_{\ell}-1}(\mathbf{V}\otimes_{A}\Lambda_{A}^{\iota})^{I_{\ell}}\right]\otimes^{\mathbf{L}}_{\Lambda_{A}}\mathcal{H}_{A},$

and at $p$ we have

$\displaystyle U_{p,{\rm
Iw}}^{+}(\mathbf{V},\mathbf{D})=U_{p}^{+}(\overline{\mathbf{V}},\overline{\mathbf{D}})\mathrel{\mathop{\mathchar
58\relax}}=\left[\mathbf{D}^{\Delta}\xrightarrow{\psi-1}\mathbf{D}^{\Delta}\right].$

We will also denote this Selmer complex by $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})$. Now we will state some useful properties of these
Selmer complexes.

######  Proposition 5.9 (Twisting property).

Let $\chi\mathrel{\mathop{\mathchar 58\relax}}=\left\langle\chi_{\rm
cyc}\right\rangle$ be the restriction of the cyclotomic character to
$\Gamma_{0}$, i.e. $\left\langle\chi_{\rm cyc}\right\rangle=\chi_{\rm
cyc}\omega^{-1}$. Then we have

$\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}(\chi),\mathbf{D}(\chi))\simeq\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})(\chi).$

###### Proof.

By the definition of Selmer complexes, it is sufficient to show that the
twisting property holds for $\mathbf{R\Gamma}_{{\rm Iw},S}(\mathbf{V})$,
$\mathbf{R\Gamma}_{\rm Iw}({\mathbf{Q}}_{v},\mathbf{V})$ and $U^{+}_{{\rm
Iw},v}(\mathbf{V},\mathbf{D})$ for each $v\in S_{f}$. If $\ell\neq p$, then

$U_{{\rm Iw},\ell}^{+}(\mathbf{V}(\chi),\mathbf{D}(\chi))=U_{{\rm
Iw},\ell}^{+}(\mathbf{V},\mathbf{D})(\chi)$

since $\chi$ is unramified for all primes $\ell\neq p$. Same argument goes for
$\mathbf{R\Gamma}_{\rm Iw}({{\mathbf{Q}}_{\ell}},V)$.

For the prime $p$, recall the definition of

$U_{{\rm Iw},p}^{+}(\mathbf{V},\mathbf{D})=\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{D}),$

and

$\mathbf{R\Gamma}_{\rm
Iw}({{\mathbf{Q}}_{p}},\mathbf{V})=\mathbf{R\Gamma}_{\rm Iw}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V})).$

Since $\Delta$ acts trivially on $\chi$, we can take $\chi$ outside of these
complexes.

Finally for $\mathbf{R\Gamma}_{{\rm Iw},S}(\mathbf{V})$, we will refer to
[Rub00, §VI Proposition 2.1(i)]. ∎

######  Theorem 5.10.

Let $\mathbf{V}_{1}$, $\mathbf{V}_{2}$, $\mathbf{V}_{3}$ be finite free
$G_{K}$-representations with coefficients in an affinoid algebra $A$ over a
$p$-adic field $E$ forming a split exact sequence as follows:

(5.2) $\displaystyle
0\longrightarrow\mathbf{V}_{1}\xlongrightarrow{f}\mathbf{V}_{2}\xlongrightarrow{g}\mathbf{V}_{3}\longrightarrow
0.$

For $i\in\\{1,2,3\\}$, let $\mathbf{D}_{i}$ be a $(\varphi,\Gamma)$-submodule
of $\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V}_{i})$ such that
$\mathbf{D}_{1}$ is the preimage of $\mathbf{D}_{2}$ under $\mathbf{D}_{{\rm
rig},A}^{\dagger}(f)$, and $\mathbf{D}_{3}$ is the image of $\mathbf{D}_{2}$
under $\mathbf{D}_{{\rm rig},A}^{\dagger}(g)$. In other words suppose there is
an exact sequence of the following form:

(5.3) $\displaystyle
0\longrightarrow\mathbf{D}_{1}\xrightarrow{\mathbf{D}_{{\rm
rig},A}^{\dagger}(f)}\mathbf{D}_{2}\xrightarrow{\mathbf{D}_{{\rm
rig},A}^{\dagger}(g)}\mathbf{D}_{3}\longrightarrow 0.$

Then we have the exact triangles of the Selmer complexes:

(5.4)
$\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{1},\mathbf{D}_{1})\longrightarrow$
$\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{2},\mathbf{D}_{2})\longrightarrow\mathbf{R\Gamma}_{?}(\mathbf{V}_{3},\mathbf{D}_{3}),$

where $?\in\\{\emptyset,{{\rm Iw}}\\}$. Moreover, if the splitting of (5.2)
induces the splitting of (5.3), then the coboundary maps in the long exact
sequences coming from the exact triangles above are $0$.

###### Proof.

This is essentially Theorem 5.7 adapted to this setting. ∎

######  Proposition 5.11 (Control theorems).

Let $\mathbf{V}$ and $\mathbf{D}$ be as before, $x\in\mathrm{Spm}(A)$ with
corresponding maximal ideal $\mathfrak{m}_{x}$ of $A$. Suppose that
$A/\mathfrak{m}_{x}\cong E$. Let ${{\rm sp}_{x,{\rm
Iw}}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{A}\rightarrow\mathcal{H}$ be induced by the
specialization map ${\rm sp}_{x}\mathrel{\mathop{\mathchar
58\relax}}A\rightarrow A/\mathfrak{m}_{x}$, and let
$V_{x}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}\otimes_{A}A/\mathfrak{m}_{x}$. Finally for an affinoid
algebra $B$, let $f\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow B$ a
morphism of affinoid algebras. Then we have:

(CTRL-1) $\displaystyle\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D})$
$\displaystyle\otimes_{\mathcal{H}_{A}}^{\mathbf{L}}A\simeq\mathbf{R\Gamma}(\mathbf{V},\mathbf{D}),$
(CTRL-2) $\displaystyle\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D})$
$\displaystyle\otimes_{{\rm sp}_{x,{\rm
Iw}}}^{\mathbf{L}}\mathcal{H}\simeq\mathbf{R\Gamma}_{\rm
Iw}(V_{x},\mathbf{D}_{x}),$ (CTRL-3)
$\displaystyle\mathbf{R\Gamma}(\mathbf{V},\mathbf{D})$
$\displaystyle\otimes_{A}^{\mathbf{L}}{A/\mathfrak{m}_{x}}\simeq\mathbf{R\Gamma}(V_{x},\mathbf{D}_{x}),$
(CTRL-4) $\displaystyle\mathbf{R\Gamma}(\mathbf{V},\mathbf{D})$
$\displaystyle\otimes_{f}^{\mathbf{L}}B\simeq\mathbf{R\Gamma}(\mathbf{V}\otimes_{f}B,\mathbf{D}\otimes_{f}B).$

###### Proof.

All of these situations are a special case of [Pot13, Theorem 1.12 - Situation
(3)]. Here it is crucial that the $p$-adic open unit ball
$W\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{W}_{\chi}$ where
$\chi=\omega^{j}$ for some $j\in({\mathbf{Z}}/p{\mathbf{Z}})^{\times}$ (recall
Section 2.2) is admissibly covered by affinoid subdomains (since
$\mathcal{H}=\Gamma(W,\mathcal{O}_{W})$). ∎

Now we state a general theorem on spectral sequences, which will help us to
compute the cohomologies above.

######  Proposition 5.12 (Tor spectral sequence).

Let $R$ be a ring, $M$ be an $R$-module, and $K^{\bullet}$ be a bounded
cochain complex of $R$-modules. Then there exists a page-2 spectral sequence

$E_{2}^{i,j}\mathrel{\mathop{\mathchar 58\relax}}{\rm
Tor}_{i}^{R}(H^{-j}(K^{\bullet}),M)\Rightarrow
H^{-i-j}(K^{\bullet}\otimes_{R}^{\mathbf{L}}M),$

where $d_{2}^{i,j}\mathrel{\mathop{\mathchar
58\relax}}E_{2}^{i+2,j-1}\rightarrow E_{2}^{i,j}$. If ${\rm
Tor}_{i}^{R}(H^{-j}(K^{\bullet}),M)=0$ for $i\geq 2$, then this spectral
sequence degenerates to the following exact sequences for each
$n\in{\mathbf{Z}}$:

$0\rightarrow H^{n}(K^{\bullet})\otimes_{R}M\rightarrow
H^{n}(K^{\bullet}\otimes_{R}^{\mathbf{L}}M)\rightarrow{\rm
Tor}_{1}^{R}(K^{\bullet},M)\rightarrow 0,$

i.e.

###### Proof.

See [Sta23, Example 061Z]. Note that in Stacks, homological notation is used
and the roles of $i$ and $j$ were swapped, hence our differential maps $d_{r}$
are of degree $(-r,r-1)$ contrary to the case in Stacks. Also note that the
actual result is on computing the homologies of of a chain complex
$K_{\bullet}$, here we used the re-indexing convention by setting
$K^{i}\mathrel{\mathop{\mathchar 58\relax}}=K_{-i}$. When ${\rm
Tor}_{i}(K^{\bullet},M)=0$ for $i\geq 2$, then we have
$E_{\infty}^{i,j}=E_{2}^{i,j}$, therefore we get the short exact sequence
above for each $n\in{\mathbf{Z}}$. ∎

We will now focus on the case $A=E$ and state structure theorem of
coadmissible $\mathcal{H}$-modules (in the sense of [Pot12] and [ST03]), and
the duality theorems. Recall that a coadmissible $\mathcal{H}$-module $M$ is
the inverse limit of finitely generated $\mathcal{H}_{n}$-modules $M_{n}$ such
that $M_{n}\otimes_{\mathcal{H}_{n}}\mathcal{H}_{n-1}\cong M_{n-1}$, where the
isomorphism is induced by the transition maps. According to [Pot12, Remark
1.2], conjecturally these theorems hold also for coadmissible
$\mathcal{H}_{A}$-modules. For our purposes, these versions are sufficient
thanks to the control theorems.

######  Proposition 5.13 (Structure theorem, [Pot12] Proposition 1.1, also
[Ben14] Proposition 3).

Let $M$ be a coadmissible $\mathcal{H}$-module. Then the torsion submodule
$M_{\rm tors}$ of $M$ is coadmissible, $M/M_{\rm tors}$ is finitely generated
free $\mathcal{H}$-module. Moreover, $M_{\rm tors}$ is isomorphic to
$\prod_{\alpha\in I}\mathcal{H}/{\mathfrak{p}_{\alpha}^{n_{\alpha}}}$, where
each $\mathfrak{p}_{\alpha}$ is a prime in $\mathcal{H}_{n}[1/p]$ for some
$n\in{\mathbf{N}}$, and for each $n\in{\mathbf{N}}$ there are finitely many
prime ideals $\mathfrak{p}_{\alpha}$ with $n_{\alpha}=0$.

Let $\mathcal{K}$ be the field of fractions of $\mathcal{H}$, and let

$\omega^{\bullet}\mathrel{\mathop{\mathchar 58\relax}}={\rm
Cone}\left[\mathcal{K}\rightarrow\mathcal{K}/\mathcal{H}\right][-1]$

be the dualizing complex for $\mathcal{H}$. For a coadmissible
$\mathcal{H}$-module $M$, let

$\mathscr{D}(M)\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{\mathcal{H}}([M],\omega^{\bullet}),$

and let $\mathscr{D}^{i}(M)$ denote $H^{i}(\mathscr{D}(M))$. Then we have the
following theorem (see [Pot12, §1], also [Ben14, §1.6])

######  Proposition 5.14 (Dualizing complex).

We have

$\displaystyle\mathscr{D}^{0}(M)={\rm Hom}_{\mathcal{H}}(M/M_{\rm
tors},\mathcal{H}),$ $\displaystyle\mathscr{D}^{1}(M)=\mathscr{D}^{1}(V_{\rm
tors})=\prod_{\alpha\in I}{\mathfrak{p}_{\alpha}^{-n_{\alpha}}}/\mathcal{H},$

and $\mathscr{D}^{i}(M)=0$ for $i\neq 0,1$.

######  Proposition 5.15 (Grothendieck duality).

Let $V$ be a $p$-adic representation over $E$, for a $(\varphi,\Gamma)$-module
$\mathbf{D}\subseteq\mathbf{D}_{\rm rig}^{\dagger}(V)$, let
$\mathbf{D}^{\perp}\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{\mathcal{R}_{E}}(\mathbf{D}_{\rm
rig}^{\dagger}(V)/\mathbf{D},\mathcal{R}_{E}(1))$. Then we have

(5.5) $\displaystyle\mathscr{D}\mathbf{R\Gamma}_{\rm
Iw}(V,\mathbf{D})\simeq\mathbf{R\Gamma}_{\rm
Iw}(V^{*}(1),\mathbf{D}^{\perp})^{\iota}[3],$

and we get the following (split) exact sequence:

$\displaystyle 0\rightarrow\mathscr{D}^{1}H^{4-i}_{{\rm
Iw}}(V,\mathbf{D})\rightarrow H^{i}_{{\rm
Iw}}(V^{*}(1),\mathbf{D}^{\perp})^{\iota}\rightarrow\mathscr{D}^{0}H^{3-i}_{{\rm
Iw}}(V,\mathbf{D})\rightarrow 0.$

###### Proof.

See [Pot12, Theorem 4.1]. ∎

We also give the analogous duality statement for Selmer complexes of
representations over affinoid algebras (see [Ben21, Theorem 3.1.9] and [Pot13,
Theorem 1.16]):

######  Proposition 5.16 (Duality over affinoids).

Let $\mathbf{V}$ be a $p$-adic representation over an affinoid algebra $A$,
for a $(\varphi,\Gamma)$-module $\mathbf{D}\subseteq\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V})$, let
$\mathbf{D}^{\perp}\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{\mathcal{R}_{A}}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V})/\mathbf{D},\mathcal{R}_{A}(1))$. Let
$\mathscr{D}_{\rm parf}^{[a,b]}(A)$ denote the category of perfect complexes
of $A$-modules with perfect amplitude $[a,b]$ (see next section for details).
Suppose that the local conditions attached to
$\mathbf{R\Gamma}(\mathbf{V},\mathbf{D})$ and
$\mathbf{R\Gamma}(\mathbf{V}^{*}(1),\mathbf{D}^{\perp})$ lie in the category
$\mathscr{D}_{\rm parf}^{[0,2]}(A)$. Then these complexes lie in
$\mathscr{D}_{\rm parf}^{[0,3]}(A)$, and we have the following duality in
$\mathscr{D}_{\rm parf}^{[0,3]}(A)$:

(5.6) $\displaystyle\mathbf{R}{\rm
Hom}_{A}(\mathbf{R\Gamma}(\mathbf{V},\mathbf{D}),A)\simeq\mathbf{R\Gamma}(\mathbf{V}^{*}(1),\mathbf{D}^{\perp})[3].$

######  Corollary 5.17 (Pottharst).

For $i\neq 1,2,3$ we have

$H^{i}_{{\rm Iw}}(V,\mathbf{D})=0,$

and

$H^{3}_{{\rm
Iw}}(V,\mathbf{D})=(T^{*}(1)^{\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})})^{*}\otimes_{\Lambda}\mathcal{H}.$

When the local conditions are (strict) ordinary (in the sense of [Pot12]), we
have

${\rm rank}_{\mathcal{H}}H^{1}_{\rm Iw}(V,\mathbf{D})={\rm
rank}_{\mathcal{H}}H^{2}_{\rm Iw}(V,\mathbf{D}).$

###### Proof.

See [Pot12, Corollary 4.2.] and [Ben14, Proposition 6] ∎

######  Corollary 5.18.

Let $V$ be an absolutely irreducible representation of
$G_{{\mathbf{Q}},S}=\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}})$ over $E$, and
set $H_{{\mathbf{Q}},S}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})$, so that
$\Gamma_{0}=G_{{\mathbf{Q}},S}/H_{{\mathbf{Q}},S}$. If $V^{*}(1)$ is not a
character of $\Gamma_{0}$, then $H^{3}_{\rm Iw}(V,\mathbf{D})=0$.

###### Proof.

We will show that $(V^{*}(1))^{H}_{{\mathbf{Q}},S}=0$. First note that
$V^{*}(1)$ is also absolutely irreducible. Since $H_{{\mathbf{Q}},S}$ is a
normal subgroup of $G_{{\mathbf{Q}},S}$, $V^{*}(1)^{H_{{\mathbf{Q}},S}}$ is a
subset of $V^{*}(1)$ which is stable under the action of $G_{{\mathbf{Q}},S}$.
By irreducibility of $V^{*}(1)$, either
$V^{*}(1)^{H_{{\mathbf{Q}},S}}=V^{*}(1)$ or $V^{*}(1)^{H_{{\mathbf{Q}},S}}=0$.
Suppose the former scenario, note that this representation factors through
$\Gamma_{0}\cong{{\mathbf{Z}}_{p}}$, which is an abelian group. By Schur’s
lemma, nontrivial irreducible representations of an abelian group over an
algebraically closed field must be a character of that group. This gives us
$V^{*}(1)\otimes_{E}\overline{E}\cong\overline{E}$ as vector spaces, which
gives that $V^{*}(1)$ is also a $1$-dimensional representation. By our
assumption this is not the case, thus we get
$V^{*}(1)^{H_{{\mathbf{Q}},S}}=0$, which implies that
$T^{*}(1)^{H_{{\mathbf{Q}},S}}\subseteq V^{*}(1)^{H_{{\mathbf{Q}},S}}=0$.
Therefore the results follows by Proposition 5.17. ∎

We finally relate the Selmer complex associated to $(V,\mathbf{D})$ to the
Bloch-Kato Selmer group under certain conditions:

######  Proposition 5.19.

Let $V$ be a $p$-adic representation with coefficients in $E$, and
$\mathbf{D}\subseteq\mathbf{D}_{\rm cris}(V)$ be a crystalline submodule
coming from the triangulation of $V$. Suppose that $V$ and $\mathbf{D}$ enjoy
the following properties:

  * •

${\rm Fil}^{0}\mathbf{D}=0$,

  * •

${\rm Fil}^{0}\mathbf{D}_{\rm cris}(V)/\mathbf{D}=\mathbf{D}_{\rm
cris}(V)/\mathbf{D}$,

  * •

$(\mathbf{D}_{\rm cris}(V)/\mathbf{D})^{\varphi=1}=0$,

  * •

$\mathbf{D}/(p\varphi-1)\mathbf{D}=0$.

Then the Bloch-Kato local condition coincides with the local condition at $p$
and the first cohomology of the Selmer Complex gives the Bloch-Kato Selmer
group. More precisely,

$H^{1}_{f}({{\mathbf{Q}}_{p}},V)=H^{1}(\mathbf{D}),{\textrm{ and
}}H^{1}_{f}({\mathbf{Q}},V)=H^{1}(V,\mathbf{D}).$

###### Proof.

See [Ben14, Proposition 6 (v)] and [Pot12, Theorem 4.1 (4)]. ∎

###  5.5. Perfectness of Selmer Complexes

In this subsection we will prove that under certain conditions, the Selmer
complexes that we are interested in are perfect complexes with perfect
amplitude $[1,2]$.

######  Definition 5.20 (Perfect complex).

We say a cochain complex $K^{\bullet}$ of $R$-modules is perfect if
$K^{\bullet}$ is quasi isomorphic to a bounded complex consisting of finite
projective $R$-modules. We denote the derived category of perfect complexes of
$R$-modules by $\mathscr{D}_{\rm parf}(R)$. If $K^{\bullet}$ can be
represented by finite projective $R$-modules concentrated at degrees $[a,b]$,
then we write $K^{\bullet}\in\mathscr{D}_{\rm parf}^{[a,b]}(R)$.

As usual let $\mathbf{V}$ denote a $p$-adic representation of
$G_{{\mathbf{Q}},S}$ with coefficients in an affinoid algebra $A$ over a
finite extension $E$ of ${{\mathbf{Q}}_{p}}$, and let
$\mathbf{D}\subseteq\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})$ be a
$(\varphi,\Gamma)$-module. We first state the main theorem of this section:

######  Theorem 5.21.

Assume that the following conditions hold:

  1. (i)

$H^{3}_{\rm Iw}(\mathbf{V},\mathbf{D})=0$.

  2. (ii)

For each prime ideal $\mathfrak{p}$ of
$\Lambda_{A}[1/p]\mathrel{\mathop{\mathchar
58\relax}}=\Lambda[1/p]\widehat{\otimes}A$, we have
$(\mathbf{V}\otimes_{A}(\Lambda_{A}[1/p]^{\iota})/\mathfrak{p})^{G_{{\mathbf{Q}},S}}=0$,

  3. (iii)

For each $\ell\in S_{f}$, $\ell\neq p$, $H^{1}(I_{\ell},\mathbf{V})$ is
projective as $A$-module.

Then $\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D})\in\mathscr{D}_{\rm
parf}^{[1,2]}(\mathcal{H}_{A})$.

######  Remark 5.22.

In [BCS23, Proposition 4.46] the perfectness of Nekovář’s Selmer complexes of
$R$-modules with perfect-amplitude $[1,2]$ is established, where $R$ is a
complete Noetherian local ring with finite residue field of characteristic
$p$. In Theorem 5.21, we generalize this theorem to the Pottharst style Selmer
complexes of $\mathcal{H}_{A}$-modules. The key difference in our theorem is
that $\mathcal{H}_{A}$ is neither a Noetherian nor a local ring.

Now we state several lemmas that we will use for the proof of Theorem 5.21.
For $\ell\neq p$ and $\ell\in S_{f}$, let $I_{\ell}^{w}$ denote the wild
inertia subgroup of $I_{\ell}$, and let $t_{\ell}$ be a topological generator
of $I_{\ell}/I_{\ell}^{w}$.

######  Lemma 5.23.

The following are valid:

  * •

$V^{I_{\ell}^{w}}$ is finite projective submodule of $V$,

  * •

$\mathbf{R\Gamma}(I_{\ell},V)\in\mathscr{D}_{\rm parf}^{[0,1]}(A)$, and can be
represented by the following perfect complex:

$\left[V^{I_{\ell}^{w}}\xrightarrow{t_{\ell}-1}V^{I_{\ell}^{w}}\right],$

  * •

If $f\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow R$ is any ring
homomorphism, then
$V^{I_{\ell}^{w}}\otimes_{A}R\cong(V\otimes_{A}R)^{I_{\ell}^{w}}$, and
$\mathbf{R\Gamma}(I_{\ell},V)\otimes_{A}^{\mathbf{L}}R\simeq\mathbf{R\Gamma}(I_{\ell},V\otimes_{A}R)$.
In particular,
$\mathbf{R\Gamma}(I_{\ell},V)\otimes_{A}^{\mathbf{L}}\mathcal{H}_{A}^{\iota}\simeq\mathbf{R\Gamma}_{\rm
Iw}(I_{\ell},V)$.

###### Proof.

See [BCS23, Lemma 4.21]. The only difference is that in the setting of
[BCS23], $V$ is a representation over a complete local Noetherian ring with
finite residue field of characteristic $p$. The key point to the proof is to
show that the image of the wild inertia subgroup $I_{\ell}^{w}$ in
$\mathrm{Aut}_{A}(V)$ is finite. However this is also true for representations
over affinoid algebras (see (55) in [Ben21, §3.1.6]). The rest of the proof is
exactly the same. ∎

######  Lemma 5.24 (Perfectness of Selmer complexes).

Suppose that the condition (iii) in Theorem 5.21 holds. Then the Iwasawa
theoretic analytic Selmer complex $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})$ lies in
$\mathscr{D}_{\rm{parf}}^{[0,3]}(\mathcal{H}_{A})$.

###### Proof.

By [Pot13, Theorem 1.11], it is sufficient to check that for each $v\in
S_{f}$, $U_{{\rm Iw},v}^{+}(\mathbf{V},\mathbf{D})\in\mathscr{D}_{\rm
parf}^{[0,2]}(\mathcal{H}_{A})$. For $v=p$ we have $U_{{\rm
Iw},p}^{+}(\mathbf{V},\mathbf{D})=\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{D})\in\mathscr{D}_{\rm parf}^{[0,2]}(\mathcal{H}_{A})$ by [KPX14,
Theorem 4.4.6]. Thus it is sufficient to show that for all $\ell\neq p$ in
$S_{f}$, $U_{{\rm
Iw},\ell}^{+}(\mathbf{V},\mathbf{D})\in\mathscr{D}_{\rm{parf}}^{[0,1]}(\mathcal{H}_{A})$.

Recall the definition of $U_{{\rm Iw},\ell}^{+}$:

$U_{{\rm Iw},\ell}^{+}(\mathbf{V},\mathbf{D})\mathrel{\mathop{\mathchar
58\relax}}=\left[(\mathbf{V}\otimes_{A}\mathcal{H}_{A})^{I_{\ell}}\xrightarrow{{\rm
Frob}_{\ell}-1}(\mathbf{V}\otimes_{A}\mathcal{H}_{A})^{I_{\ell}}\right].$

Since $I_{\ell}$ acts trivially on $\mathcal{H}_{A}^{\iota}$, by [BCS23,
Proposition 4.24 (3)] (note that this is also true in our setting thanks to
5.23) we have:

$U_{{\rm
Iw},\ell}^{+}(\mathbf{V},\mathbf{D})\simeq\left[\mathbf{V}^{I_{\ell}}\otimes_{A}\mathcal{H}_{A}\xrightarrow{{\rm
Frob}_{\ell}-1}\mathbf{V}^{I_{\ell}}\otimes_{A}\mathcal{H}_{A}\right]\simeq\left[\mathbf{V}^{I_{\ell}}\xrightarrow{{\rm
Frob}_{\ell}-1}\mathbf{V}^{I_{\ell}}\right]\otimes_{A}\mathcal{H}_{A},$

where the last isomorphism follows from the definition of tensor product of
complexes.

Now we claim that $\mathcal{H}_{A}$ is (faithfully) flat over $A$. By the
base-change property of faithfully flatness, it is sufficient to show that
$\mathcal{H}$ is faithfully flat over $E$. Notice that
$\Lambda_{E}\cong\mathcal{O}_{E}[[T]]$ is faithfully flat over
$\mathcal{O}_{E}$, hence $\Lambda_{E}[1/p]\mathrel{\mathop{\mathchar
58\relax}}=\Lambda_{E}\otimes_{\mathcal{O}_{E}}E$ is faithfully flat over $E$.
Since we know that $\mathcal{H}=\mathcal{H}_{E}$ is faithfully flat over
$\Lambda_{E}[1/p]$, we obtain the result. Therefore,

$U_{{\rm Iw},\ell}^{+}(\mathbf{V},\mathbf{D})\simeq
U_{\ell}^{+}(\mathbf{V},\mathbf{D})\otimes^{\mathbf{L}}_{A}\mathcal{H}_{A}.$

By our assumption, thanks to [Ben21, 3.1.7],
$U_{\ell}^{+}(\mathbf{V},\mathbf{D})$ is a perfect complex of $A$-modules with
perfect amplitude $[0,1]$, therefore
$U_{\ell}^{+}(\mathbf{V},\mathbf{D})\otimes^{\mathbf{L}}_{A}\mathcal{H}_{A}$
lies in $\mathscr{D}_{\rm parf}^{[0,1]}(\mathcal{H}_{A})$. ∎

######  Lemma 5.25.

If the affinoid algebra $A$ is a PID, then $H^{1}(I_{\ell},\mathbf{V})$ is
projective, and Lemma 5.24 holds true.

###### Proof.

Since $A$ is a Dedekind domain, every submodule of projective $A$-module is
projective. Hence $H^{0}(I_{\ell},\mathbf{V})$ and
$(t_{\ell}-1)\mathbf{V}^{I_{\ell}^{w}}$ are projective $A$-modules. Now we
will use the arguments of [Ben21, Theorem 3.1.7 (ii)] to show that
$H^{1}(I_{\ell},\mathbf{V})$ is projective. Consider the following exact
sequence:

(5.7) $\displaystyle
0\rightarrow\mathbf{V}^{I_{\ell}}\rightarrow\mathbf{V}^{I_{\ell}^{w}}\xrightarrow{t_{\ell}-1}(t_{\ell}-1)\mathbf{V}^{I_{\ell}^{w}}\rightarrow
0.$

This sequence splits since $(t_{\ell}-1)\mathbf{V}^{I_{\ell}^{w}}$ is
projective. Dualizing (5.7) we get

(5.8) $\displaystyle
0\rightarrow(t_{\ell}-1)\mathbf{V}^{*}(1)^{I_{\ell}^{w}}\rightarrow\mathbf{V}^{*}(1)^{I_{\ell}^{w}}\rightarrow(\mathbf{V}^{I_{\ell}})^{*}(1)\rightarrow
0,$

which splits again since $(\mathbf{V}^{I_{\ell}})^{*}(1)$ is projective. Hence
we get

$H^{1}(I_{\ell},\mathbf{V}^{*}(1))(1)\cong\mathbf{V}^{*}(1)^{I_{\ell}^{w}}/(t_{\ell}-1)\mathbf{V}^{*}(1)^{I_{\ell}^{w}}\cong(\mathbf{V}^{I_{\ell}})^{*}(1).$

Hence (replacing the roles of $\mathbf{V}$ and $\mathbf{V}^{*}(1)$), we
conclude that $H^{1}(I_{\ell},\mathbf{V})$ is projective. ∎

We will prove that our Selmer complexes can be represented by a complex of
projectives concentrated at degrees $[1,2]$. In the following definitions and
lemmas, let $R$ denote a ring and $K^{\bullet}$ denote a complex of
$R$-modules in $\mathscr{D}(R)$.

######  Definition 5.26.

We say that a complex $K^{\bullet}\in\mathscr{D}(R)$ has finite ${\rm Tor}$
dimension with ${\rm Tor}$ amplitude $[a,b]$ if for every $R$-module $M$, we
have $H^{i}(K^{\bullet}\otimes^{\mathbf{L}}_{R}M)=0$ if $i\notin[a,b]$.

######  Lemma 5.27.

$K^{\bullet}$ is perfect iff $K^{\bullet}$ is pseudo-coherent and has finite
${\rm Tor}$ dimension. Moreover if one of these properties holds and
$K^{\bullet}$ has ${\rm Tor}$ amplitude $[a,b]$, then
$K^{\bullet}\in\mathscr{D}_{\rm parf}^{[a,b]}(R)$.

###### Proof.

See [Sta23, Lemma 0658]. ∎

######  Lemma 5.28.

Let $K^{\bullet}\in\mathscr{D}_{\rm parf}^{[a,b]}(R)$ such that
$H^{b}(K^{\bullet})=0$. Then $K^{\bullet}\in\mathscr{D}_{\rm
parf}^{[a,b-1]}(R)$.

###### Proof.

Let $K^{\bullet}\simeq\left[P^{a}\rightarrow\dots\rightarrow P^{b}\right]$
(with $P^{i}$ finite projective for $i\in[a,b]$) such that
$H^{b}(K^{\bullet})=0$. Then the map $d^{b-1}\mathrel{\mathop{\mathchar
58\relax}}P^{b-1}\rightarrow P^{b}$ is surjective. Note that by definition,

$K^{\bullet}\otimes^{\mathbf{L}}_{R}M\simeq\left[P^{a}\otimes_{R}M\xrightarrow{d^{a}\otimes
1_{M}}\dots\xrightarrow{d^{b-1}\otimes 1_{M}}P^{b}\otimes_{R}M\right],$

therefore the right-exactness of ${-}\otimes_{R}M$ implies that
$d^{b-1}\otimes 1_{M}\mathrel{\mathop{\mathchar
58\relax}}P^{b-1}\otimes_{R}M\rightarrow P^{b}\otimes_{R}M$ is surjective.
This completes the proof. ∎

######  Lemma 5.29.

Let

$K^{\bullet}\rightarrow L^{\bullet}\rightarrow M^{\bullet}\rightarrow
K^{\bullet}[1]$

be an exact triangle in $\mathscr{D}(R)$. If any two of
$\\{K^{\bullet},L^{\bullet},M^{\bullet}\\}$ are perfect, so is the third. If
$L^{\bullet}$ has ${\rm Tor}$ amplitude $[a+1,b+1]$ and $M^{\bullet}$ has
${\rm Tor}$ amplitude $[a,b]$, then $K^{\bullet}$ has ${\rm Tor}$ amplitude
$[a+1,b+1]$.

###### Proof.

See [Sta23, Lemma 066R-(3)] and [Sta23, Lemma 0655]. ∎

######  Lemma 5.30.

Let $R\rightarrow R^{\prime}$ be a ring map. If $K^{\bullet}$ has ${\rm Tor}$
amplitude $[a,b]$ as a complex of $R$-modules, then
$K^{\bullet}\otimes_{R}^{\mathbf{L}}R^{\prime}$ has ${\rm Tor}$ amplitude
$[a,b]$ as a complex of $R^{\prime}$-modules. If $R\rightarrow R^{\prime}$ is
faithfully flat, and if $K^{\bullet}\otimes_{R}R^{\prime}\in\mathscr{D}_{\rm
parf}(R^{\prime})$, then $K^{\bullet}\in\mathscr{D}_{\rm parf}(R)$.

###### Proof.

See [Sta23, Lemma 068T] and [Sta23, Lemma 066L]. ∎

###### Proof of Theorem 5.21.

It remains to show that $\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D})$ has
the perfect amplitude $[1,2]$. If we can show that $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})$ has ${\rm Tor}$ amplitude in $[1,2]$, then we are
done by Lemma 5.27. Suppose that we showed $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})$ has ${\rm Tor}$ amplitude in $[1,3]$. Then
combining with our running assumption that $H^{3}_{\rm
Iw}(\mathbf{V},\mathbf{D})=0$, we are done by Lemma 5.28.

Consider the following exact triangle (see [HP14, §7.2]):

$\displaystyle\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})\longrightarrow\mathbf{R\Gamma}_{(p),{\rm
Iw}}({\mathbf{Q}},\mathbf{V})\longrightarrow\mathbf{R\Gamma}_{\rm
Iw}(\widetilde{\mathbf{D}}),$

where $\widetilde{\mathbf{D}}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}_{{\rm rig},A}^{\dagger}(\mathbf{V})/\mathbf{D}$ and
$\mathbf{R\Gamma}_{(p),{\rm Iw}}({\mathbf{Q}},\mathbf{V})$ denotes the
$p$-relaxed Selmer complex. We know from [KPX14, Theorem 4.4.6] that

$\mathbf{R\Gamma}_{\rm Iw}(\widetilde{\mathbf{D}})\in\mathscr{D}_{\rm
parf}^{[0,2]}(\mathcal{H}_{A}).$

Thus by Lemma 5.29 it suffices to show that

$\mathbf{R\Gamma}_{(p),{\rm Iw}}({\mathbf{Q}},\mathbf{V})\in\mathscr{D}_{\rm
parf}^{[1,3]}(\mathcal{H}_{A}).$

Let

$K^{\bullet}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{R\Gamma}_{(p)}({\mathbf{Q}},\mathbf{V}\otimes_{A}\Lambda_{A}[1/p]^{\iota}),$

then we have

$K^{\bullet}\otimes_{\Lambda_{A}[1/p]}\mathcal{H}_{A}\simeq\mathbf{R\Gamma}_{(p),{\rm
Iw}}({\mathbf{Q}},\mathbf{V}).$

Note that $\mathcal{H}$ is faithfully flat over $\Lambda[1/p]$, therefore
$\mathcal{H}_{A}$ is faithfully flat over $\Lambda_{A}[1/p]$ by the base-
change property of faithfully flat maps. Hence by Lemma 5.30, $K^{\bullet}$ is
perfect and we reduce the problem to showing that
$H^{0}(K^{\bullet}\otimes_{\Lambda_{A}[1/p]}^{\mathbf{L}}M)=0$ for all $M$.

Note that $\Lambda_{A}[1/p]$ is a Noetherian ring, therefore the rest of the
argumentation is reduced to the what is shown in [BCS23, §4.5].

Since $K^{\bullet}$ is perfect $K^{\bullet}\simeq\left[P^{0}\rightarrow
P^{1}\rightarrow P^{2}\rightarrow P^{3}\right]$, where $P^{i}$’s are all
projective. Then we have

$H^{0}(K^{\bullet}\otimes^{\mathbf{L}}M)\mathrel{\mathop{\mathchar
58\relax}}=H^{0}([P^{0}\otimes M\rightarrow P^{1}\otimes M])\cong{\rm
Tor}_{1}^{\Lambda[1/p]}(P^{1}/P^{0},M).$

Note that the last isomorphism is not true in general, however our assumption
ii ensures that $H^{0}(K^{\bullet})=0$, and in this case this isomorphism
holds. To show that for any $M$ the module ${\rm
Tor}_{1}^{\Lambda_{A}[1/p]}(P^{1}/P^{0},M)=0$, we show for any prime
$\mathfrak{p}$ of $\Lambda_{A}[1/p]$, the localization at $\mathfrak{p}$ is 0,
i.e.

$\left({\rm Tor}_{1}^{\Lambda[1/p]}(P^{1}/P^{0},M)\right)_{\mathfrak{p}}={\rm
Tor}_{1}^{(\Lambda_{A}[1/p])_{\mathfrak{p}}}((P^{1}/P^{0})_{\mathfrak{p}},M_{\mathfrak{p}})=0.$

If we show that $(P^{1}/P^{0})_{\mathfrak{p}}$ is flat over
$\Lambda_{A}[1/p]_{\mathfrak{p}}$, then the result follows. This can be shown
by using the local criterion for flatness: Let $X\mathrel{\mathop{\mathchar
58\relax}}=(P^{1}/P^{0})_{\mathfrak{p}}$. Note that $X$ is finitely generated
and $X/\mathfrak{p}$ is flat over
$(\Lambda_{A}[1/p])_{\mathfrak{p}}/\mathfrak{p}$, thus we only need to show
that ${\rm
Tor}_{1}^{\Lambda[1/p]_{\mathfrak{p}}}((\Lambda_{A}[1/p]_{\mathfrak{p}})/\mathfrak{p},X)=0$.
Note that $K^{\bullet}\simeq[P^{0}\rightarrow P^{1}\rightarrow P^{2}
\rightarrow P^{3}]$ gives
$K^{\bullet}_{\mathfrak{p}}\simeq[P^{0}_{\mathfrak{p}}\rightarrow
P^{1}_{\mathfrak{p}}\rightarrow P^{2}_{\mathfrak{p}} \rightarrow
P^{3}_{\mathfrak{p}}]$. Hence

${\rm
Tor}_{1}^{(\Lambda_{A}[1/p])_{\mathfrak{p}}}((\Lambda_{A}[1/p]_{\mathfrak{p}})/\mathfrak{p},X)=H^{0}(K^{\bullet}_{\mathfrak{p}}\otimes^{\mathbf{L}}(\Lambda_{A}[1/p]_{\mathfrak{p}})/\mathfrak{p}).$

By the definition of $K^{\bullet}$, the last term is

$H^{0}_{(p)}({\mathbf{Q}},\mathbf{V}\otimes(\Lambda_{A}[1/p])_{\mathfrak{p}}^{\iota}/\mathfrak{p})\cong
H^{0}_{(p)}({\mathbf{Q}},\mathbf{V}\otimes\Lambda_{A}[1/p]^{\iota}/\mathfrak{p})\otimes_{\Lambda_{A}[1/p]}(\Lambda_{A}[1/p])_{\mathfrak{p}},$

and

$H^{0}_{(p)}({\mathbf{Q}},\mathbf{V}\otimes\Lambda_{A}[1/p]^{\iota}/\mathfrak{p})\subseteq(\mathbf{V}\otimes(\Lambda_{A}[1/p])_{\mathfrak{p}}^{\iota}/\mathfrak{p})^{G_{{\mathbf{Q}},S}}.$

By our assumption ii, we get
$(\mathbf{V}\otimes\Lambda_{A}[1/p]/\mathfrak{p})^{G_{{\mathbf{Q}},S}}=0$,
hence the result follows. ∎

######  Remark 5.31.

If the following duality theorem in $\mathscr{D}_{\rm coad}(\mathcal{H})$ for
Iwasawa theoretic Selmer complexes attached to representations $V$ over $E$

$\mathbf{R\Gamma}_{\rm
Iw}(V^{*}(1),\mathbf{D}^{\perp})\simeq\mathbf{R\Gamma}_{\rm
Iw}(V,\mathbf{D})^{*}[-3]$

could be generalized to the Selmer complexes attached to representations
$\mathbf{(}V)$ over $A$ lying in $\mathscr{D}_{\rm
parf}^{[0,3]}(\mathcal{H}_{A})$, then if the assumptions i and iii of Theorem
5.21 hold for $(\mathbf{V},\mathbf{D})$ and
$(\mathbf{V}^{*}(1),\mathbf{D}^{\perp})$, we can still show that

$\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D}),\ \mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}^{\perp})\in\mathscr{D}_{\rm
parf}^{[1,2]}(\mathcal{H}_{A})$

by arguing as in [Nek06, Proposition 9.7.4-(ii)].

In particular when $A=E$, the following assumption

$H^{3}_{\rm Iw}(V,\mathbf{D})=0=H^{3}_{\rm Iw}(V^{*}(1),\mathbf{D}^{\perp})$

is sufficient to show that

$\mathbf{R\Gamma}_{\rm Iw}(V,\mathbf{D}),\ \mathbf{R\Gamma}_{\rm
Iw}(V^{*}(1),\mathbf{D}^{\perp})\in\mathscr{D}_{\rm
parf}^{[1,2]}(\mathcal{H})$

######  Corollary 5.32.

Assume that

$H^{1}_{\rm Iw}(\mathbf{V},\mathbf{D})=0=H^{1}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}^{\perp}).$

We have

$\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D})\simeq\left[P^{1}\xrightarrow{f}P^{2}\right],$

where $P^{1}$ and $P^{2}$ are finite projective $\mathcal{H}_{A}$-modules of
same rank. Then

$\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}))=\det(f)=\mathrel{\mathop{\mathchar 58\relax}}{\rm
Fitt}_{0}(H^{2}_{\rm Iw}(\mathbf{V},\mathbf{D})).$

In particular when $A=E$ is a finite extension of ${{\mathbf{Q}}_{p}}$, we
have

$\det(\mathbf{R\Gamma}_{\rm Iw}(V,\mathbf{D}))={\rm
char}_{\mathcal{H}}(H^{2}_{\rm Iw}(V,\mathbf{D})).$

######  Definition 5.33 (Algebraic $p$-adic $L$-function).

With the above motivation, we define the module of algebraic $p$-adic
$L$-functions attached to $(\mathbf{V},\mathbf{D})$ as
$\det(\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V},\mathbf{D}))$.

##  6\. Rankin-Selberg Products and Beilinson-Flach Elements

###  6.1. Rankin-Selberg Products and Adjoint Motives

Let $f\in S_{k_{f}+2}(N_{f},\varepsilon_{f})$ and $g\in
S_{k_{g}+2}(N_{g},\varepsilon_{g})$ be cuspidal normalized newforms of weights
$k_{f}+2$ and $k_{g}+2$, levels $\Gamma_{1}(N_{f})$ and $\Gamma_{1}(N_{g})$
with nebentypes $\varepsilon_{f}$ and $\varepsilon_{g}$ respectively. Let
$\psi$ be a finite order Dirichlet character with conductor $N_{\psi}$. We
will assume that $p\nmid N_{f}N_{g}N_{\psi}$. Let $\alpha_{f}$ and $\beta_{f}$
(resp. $\alpha_{g}$ and $\beta_{g}$) be the roots of the Hecke polynomial
$X^{2}-a_{p}(f)X+\varepsilon_{f}p^{k_{f}+1}$ (resp.
$X^{2}-a_{p}(g)X+\varepsilon_{g}p^{k_{g}+1}$). We will always assume that
$\alpha_{f}\neq\beta_{f}$ and $\alpha_{g}\neq\beta_{g}$ (called $p$-regularity
hypothesis), which is conjecturally true and known to be true when $f$ (and
$g$) is of weight $2$.

The imprimitive Rankin-Selberg $L$-function is defined as

$L(f,g,\psi,s)\mathrel{\mathop{\mathchar
58\relax}}=L_{(N_{f}N_{g}N_{\psi})}(\varepsilon_{f}\varepsilon_{g}\psi^{2},2s-2-k_{f}-k_{g})\sum_{n\geq
1}\frac{a_{n}(f)a_{n}(g)\psi(n)}{n^{-s}},$

where for $S\in{\mathbf{N}}^{+}$ and a character $\chi$, $L_{S}(\chi,s)$
denotes the Dirichlet $L$-series without Euler factors at primes dividing $S$
(see [AL21, §2] and [Loe18, §2.1] for further details). Note that this
impritimitve $L$-function differs from the primitive $L$-function attached to
$f\otimes g\otimes\psi$ up to finitely many Euler factors.

Recall that $V_{f}$ is the $2$-dimensional (cohomological) Galois
representation attached to $f$ with coefficients in $E$, which is a finite
extension of ${{\mathbf{Q}}_{p}}$ and which contains the Hecke field $K_{f}$
and $U_{p}$-eigenvalues $\alpha_{f}$ and $\beta_{f}$. Let $T_{f}\subseteq
V_{f}$ denote a $G_{\mathbf{Q}}$ stable rank $2$ $\mathcal{O}_{E}$-lattice
inside $V_{f}$. Let ${\mathrm{Sym}}^{2}f$ denote the symmetric-square motive,
whose $p$-adic realization is a 3-dimensional subrepresentation
${\mathrm{Sym}}^{2}(V_{f})$ of $V_{f}\otimes V_{f}$ consisting of symmetric
tensors. Following [LZ19, Proposition 2.1.4] we can define the imprimitive
$L$-function for ${\mathrm{Sym}}^{2}f$ as

$L^{\rm imp}({\mathrm{Sym}}^{2}f,\psi,s)\mathrel{\mathop{\mathchar
58\relax}}=\frac{L(f,f,\psi,s)}{L_{(}N_{f}N_{\psi})(\varepsilon\psi,s-k-1)}.$

Note that the representation $V_{f}\otimes_{E}V_{f}\otimes\psi$ splits as

$V_{f}\otimes V_{f}\otimes\psi\cong{\rm
Sym}^{2}(V_{f})(\psi)\oplus\wedge^{2}V_{f}(\psi),$

where

$\wedge^{2}V_{f}\mathrel{\mathop{\mathchar
58\relax}}=\det(V_{f})\cong\varepsilon_{f}\chi_{\rm cyc}^{-1-k}.$

We also define

${\mathrm{Ad}}(V_{f})\mathrel{\mathop{\mathchar 58\relax}}=V_{f}\otimes
V_{f}^{*},$

and

${{\rm Ad}^{0}}(V_{f})\mathrel{\mathop{\mathchar
58\relax}}=\ker\left({\mathrm{Ad}}(V_{f})\xrightarrow{\rm Tr}E\right).$

Then we have the splitting of representations

${\mathrm{Ad}}(V_{f})(\psi)\cong{{\rm Ad}^{0}}(V_{f})(\psi)\oplus E(\psi).$

Note that we have

${{\rm
Ad}^{0}}(V_{f})\cong{\mathrm{Sym}}^{2}V_{f}(k_{f}+1)\otimes\varepsilon^{-1}.$

This follows using the splittings above and (4.1), see also [Das16, §1.3].

Let us see that ${\mathrm{Ad}}(V_{f})(\psi)$ (or equivalently, $V_{f}\otimes
V_{f}\otimes\psi$) does not have any critical values, by applying the
criticality criteria given in Definition 3.6.

To see this, note that $d^{+}(V_{f})=1$. Recall that we defined
$d^{+}(V)\mathrel{\mathop{\mathchar 58\relax}}=\dim V^{c=1}$ just before
Definition 3.6. Hence for all $n\in{\mathbf{Z}}$, we have

$\displaystyle d^{+}({\mathrm{Ad}}(V_{f})(\psi+n))=2,$ $\displaystyle
d^{+}(E(\psi+n))=\frac{1+(-1)^{n}\psi(-1)}{2},$ $\displaystyle d^{+}({{\rm
Ad}^{0}}(V_{f})(\psi+n))=d^{+}({\mathrm{Ad}}(V_{f})(\psi+n))-d^{+}(E(\psi+n))=\frac{3-(-1)^{n}\psi(-1)}{2}.$

Let us also compute the Hodge-Tate weights of twists of these motives: $V_{f}$
has Hodge-Tate weights $\mathrm{HT}(V_{f})=\\{-1-k_{f},0\\}$, hence we get

$\displaystyle\mathrm{HT}({\mathrm{Ad}}(V_{f})(\psi+n))=\\{-1-k_{f}+n,n,n,1+k_{f}+n\\},$
$\displaystyle\mathrm{HT}(E(\psi+n))=\\{n\\},$ $\displaystyle\mathrm{HT}({{\rm
Ad}^{0}}(V_{f})(\psi+n))=\\{-1-k_{f}+n,n,1+k_{f}+n\\}.$

From this computation, one sees that there are no critical values for
${\mathrm{Ad}}(V_{f})(\psi)$. We can also compute the critical values for
${{\rm Ad}^{0}}(V_{f})(\psi)$ as

$\\{n\in{\mathbf{Z}}\mathrel{\mathop{\mathchar 58\relax}}\psi+n\textnormal{ is
even and }-k_{f}-1<n\leq 0\\}\cup\\{n\in{\mathbf{Z}}\mathrel{\mathop{\mathchar
58\relax}}\psi+n\textnormal{ is odd and }0<n\leq 1+k_{f}\\},$

and the critical values for $E(\psi)$ as

$\\{n\in{\mathbf{Z}}\mathrel{\mathop{\mathchar 58\relax}}\psi+n\textnormal{ is
even and }n\geq 1\\}\cup\\{n\in{\mathbf{Z}}\mathrel{\mathop{\mathchar
58\relax}}\psi+n\textnormal{ is odd and }n\leq 0\\}.$

Note that the critical values for ${{\rm Ad}^{0}}(V_{f})(\psi)$ and $E(\psi)$
do not coincide, which reflects the fact that there are no critical values for
${\mathrm{Ad}}(V_{f})(\psi)$.

Hence it is not trivial to establish $p$-adic Artin formalism for the $p$-adic
$L$-functions of these representations. Therefore one needs to work with
$p$-adic $L$-functions attached to the deformations of these representations,
i.e. replace $f$ by a suitable family of $p$-adic modular forms passing
through $f$. In this article we are interested in the case where $f$ is
$p$-non-ordinary, hence we will use Coleman families (see Definition 4.2) to
interpolate $f$.

From now on, let $f$ be $p$-non-ordinary, so both of its $p$-stabilizations
$f_{\alpha_{f}}$ and $f_{\beta_{f}}$ are non-$\theta$-critical. We fix a
$p$-stabilization $f_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}=f_{\alpha_{f}}$. Since $f$ is a cusp form, $f_{\alpha}$ corresponds
to a point $x_{0}\in\mathcal{C}^{0}_{N}$ such that
$\kappa(x_{0})=k_{f}\in\mathcal{W}$, where $\kappa\mathrel{\mathop{\mathchar
58\relax}}\mathcal{C}^{0}_{N}\rightarrow\mathcal{W}$ is the weight map.
Moreover, since $f_{\alpha}$ is non critical, $\kappa$ is étale at $x_{0}$, so
a nice affinoid neighborhood of $k_{f}\in\mathcal{W}$ can be identified with a
nice affinoid neighborhood $U=\mathrm{Spm}(A)$ of $x_{0}$ in the eigencurve.

Let $\mathcal{F}$ be the unique Coleman family over $U$ passing through
$f_{\alpha}$. Note that in this case the coefficient field $E$ of $V_{f}$ is
isomorphic to $A/\mathfrak{m}_{x_{0}}$, where $\mathfrak{m}_{x_{0}}$ is the
maximal ideal of $A$ corresponding to the point $x_{0}\in U$. Let
$\mathbf{V}_{\mathcal{F}}$ denote the big $G_{\mathbf{Q}}$ representation
(with coefficients in $A$) attached to $\mathcal{F}$ by Theorem 4.6. In
particular we have

$(\mathbf{V}_{\mathcal{F}})_{x_{0}}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}_{\mathcal{F}}\otimes_{A}A/\mathfrak{m}_{x_{0}}\cong
V_{f}.$

If $\Delta\mathrel{\mathop{\mathchar 58\relax}}U\rightarrow U\times_{E}U$
denotes the diagonal map, then the corresponding map of affinoid algebras
$\Delta^{*}\mathrel{\mathop{\mathchar
58\relax}}A\widehat{\otimes}_{E}A\rightarrow A$ is given by the multiplication
map sending $a_{1}\otimes a_{2}$ to $a_{1}a_{2}$. This map pulls back the
representation
$\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}$ to

$\Delta^{*}(\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}})\cong\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{A}\mathbf{V}_{\mathcal{F}}.$

Thus we have the following decomposition of representations in families:

$\Delta^{*}(\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}})(\psi)\cong{\mathrm{Sym}}^{2}(\mathbf{V}_{\mathcal{F}})\oplus\wedge^{2}\mathbf{V}_{\mathcal{F}}(\psi),$

where

$\wedge^{2}\mathbf{V}_{\mathcal{F}}=\varepsilon_{\mathcal{F}}\chi_{\rm
cyc}^{-1}\mathbf{k}_{\mathcal{F}}^{-1}.$

Here $\varepsilon_{\mathcal{F}}$ is the nebentype of $\mathcal{F}$ and
$\mathbf{k}_{\mathcal{F}}$ is the weight character of $\mathcal{F}$ (with
values in $A$) such that each character in $U\subseteq\mathcal{W}$ factors
through $\mathbf{k}_{\mathcal{F}}$.

Let

$\displaystyle\mathbf{V}_{\mathcal{F}}^{*}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{A}(\mathbf{V}_{\mathcal{F}},A)\cong\mathbf{V}_{\mathcal{F}^{c}}(\mathbf{k}_{\mathcal{F}}+1),$
$\displaystyle{\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{A}(\mathbf{V}_{\mathcal{F}},\mathbf{V}_{\mathcal{F}})\cong\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{A}\mathbf{V}_{\mathcal{F}}^{*},$
$\displaystyle{{\rm Ad}^{0}}(\mathbf{V}_{\mathcal{F}})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\ker({\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})\xrightarrow{\rm{Tr}}A).$

Note that we have the decomposition

$\Delta^{*}(\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*})(\psi)\cong{\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})(\psi)\cong{{\rm
Ad}^{0}}(\mathbf{V}_{\mathcal{F}})(\psi)\oplus A(\psi).$

The following theorem on factorization of $p$-adic $L$-functions, which
generalizes the main theorem of Dasgupta in [Das16] is due to Arlandini and
Loeffler in [AL21, Theorem A, Theorem B].

######  Theorem 6.1 (Arlandini-Loeffler).

Let

$L_{p}^{\rm geom}(\mathcal{F},\mathcal{G})\mathrel{\mathop{\mathchar
58\relax}}=L_{p}^{\rm
geom}(\mathcal{F},\mathcal{G})(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},\mathbf{j})$

denote the three variable geometric Rankin-Selberg $p$-adic $L$-function
defined in [LZ16, §9]. There exists a unique two variable meromorphic $p$-adic
$L$-function

$L_{p}^{\rm imp}({\mathrm{Sym}}^{2}(\mathbf{V}_{\mathcal{F}}))$

interpolating the critical values of $L^{\rm imp}({\mathrm{Sym}}^{2}(f),s)$
(see [AL21, Theorem A] for the exact interpolation properties) such that the
following factorization formula holds for all $(\sigma_{1},\sigma_{2})\in
U\times\mathcal{W}$:

$\displaystyle L_{p}^{\rm
geom}(\mathcal{F},\mathcal{F})(\sigma_{1},\sigma_{1},\sigma_{2})=L_{p}^{\rm
imp}({\mathrm{Sym}}^{2}(\mathbf{V}_{\mathcal{F}}))(\sigma_{1},\sigma_{2})L_{p}(\varepsilon_{f},\sigma_{2}-\sigma_{1}-1).$

In the following theorem [LZ16, Theorem 9.3.2.] of Loeffler and Zerbes, the
interpolation property of $L_{p}^{\rm geom}$ is given as follows:

######  Theorem 6.2 (Interpolation property of $L_{p}^{\rm geom}$).

For Coleman families $\mathcal{F}$ and $\mathcal{G}$ over (nice affinoid
domains) $U_{\mathcal{F}}$ and $U_{\mathcal{G}}$, the geometric $p$-adic
$L$-function $L_{p}^{\rm geom}$ has the following interpolation property: If
$(k,k^{\prime},j)\in
U_{\mathcal{F}}\times_{E}U_{\mathcal{G}}\times_{E}\mathcal{W}\cap{\mathbf{N}}^{3}$
with $k\geq 0$, $k^{\prime}\geq-1$, and $\frac{k+k^{\prime}+1}{2}\leq j\leq
k$, and if $\mathcal{F}_{k}$ and $\mathcal{G}_{k^{\prime}}$ are the
$p$-stabilizations of the newforms $f_{k}$ and $g_{k^{\prime}}$ with $U_{p}$
eigenvalues $\alpha_{f_{k}}$ and $\alpha_{g_{k^{\prime}}}$ respectively, then
we have

$\displaystyle L_{p}^{\rm
geom}(\mathcal{F},\mathcal{G})(k,k^{\prime},j)=\frac{E(f_{k},g_{k^{\prime}},1+j)}{E(f_{k})E^{*}(f_{k})}\frac{j!(j-k^{\prime}-1)!}{\pi^{2j-k^{\prime}+1}(-1)^{k-k^{\prime}}2^{2j+2+k-k^{\prime}}\langle
f_{k},f_{k}\rangle}L(f_{k},g_{k^{\prime}},1+j),$

where

$\displaystyle E(f_{k})$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\left(1-\frac{\beta_{f_{k}}}{p\alpha_{f_{k}}}\right),$
$\displaystyle E^{*}(f_{k})$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\left(1-\frac{\beta_{f_{k}}}{\alpha_{f_{k}}}\right),$
$\displaystyle E(f_{k},g_{k^{\prime}},1+j)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\left(1-\frac{p^{j}}{\alpha_{f_{k}}\alpha_{g_{k^{\prime}}}}\right)\left(1-\frac{p^{j}}{\alpha_{f_{k}}\beta_{g_{k^{\prime}}}}\right)\left(1-\frac{\beta_{f_{k}}\alpha_{g_{k^{\prime}}}}{p^{1+j}}\right)\left(1-\frac{\beta_{f_{k}}\beta_{g_{k^{\prime}}}}{p^{1+j}}\right).$

######  Remark 6.3.

We should note that Arlandini and Loeffler’s $p$-adic $L$-function of
${\mathrm{Sym}}^{2}\mathcal{F}$ is defined only at integral weights in the
cyclotomic variable, and the analogous $p$-adic $L$-function for the twists of
${\mathrm{Sym}}^{2}\mathcal{F}$ by Dirichlet characters $\psi$ is not defined.
Therefore they do not have a factorization result for $p$-adic $L$-function
attached to $\mathcal{F}\otimes\mathcal{F}\otimes\psi$ for a non-trivial
character $\psi$. However (as I learned from Loeffler by a personal
conversation at Iwasawa 2023 conference) the twisted version of the above
theorem is expected to follow applying the same methods. Therefore we will
state below the twisted version of the above theorem as a conjecture, and
assume it as an input for our results.

######  Conjecture 6.1 (Arlandini-Loeffler, twisted version).

For a finite order Dirichlet character $\psi$, there exists a unique two
variable meromorphic $p$-adic $L$-function

$L_{p}^{\rm imp}({\mathrm{Sym}}^{2}(\mathbf{V}_{\mathcal{F}}),\psi)$

interpolating the critical values of $L^{\rm
imp}({\mathrm{Sym}}^{2}(f),\psi,s)$ such that the following factorization
formula holds for all $(\sigma_{1},\sigma_{2})\in U\times\mathcal{W}$:

$\displaystyle L_{p}^{\rm
geom}(\mathcal{F},\mathcal{F}_{\psi})(\sigma_{1},\sigma_{1},\sigma_{2})=L_{p}^{\rm
imp}({\mathrm{Sym}}^{2}(\mathbf{V}_{\mathcal{F}}),\psi)(\sigma_{1},\sigma_{2})L_{p}(\varepsilon_{f}\psi,\sigma_{2}-\sigma_{1}-1),$

where $\mathcal{F}_{\psi}$ denotes the twist $\mathcal{F}\otimes\psi$ of
$\mathcal{F}$ by $\psi$.

###  6.2. Beilinson-Flach Elements

In this section, we introduce the Beilinson-Flach elements and Euler system
coming from these elements, following [LZ16] and [BLV21]. We will not give the
definition of these elements and how they are constructed (interested reader
may look at op. cit. also the survey article [BCD+14] and [BDR15a, BDR15b],
and [LLZ14] for the construction of Euler systems coming from these elements).
However we present here the explicit reciprocity law ([LZ16, Theorem 7.1.5])
which relates these elements to Rankin-Selberg $p$-adic $L$-functions via
Perrin-Riou’s regulator (big logarithm) map introduced in [PR95]. We will
explain how to use this to bound the Bloch-Kato Selmer group attached to
${{\rm Ad}^{0}}(V_{f})$.

Let $\mathcal{F}$ and $\mathcal{G}$ be Coleman families over nice affinoid
domains $U_{\mathcal{F}}$ and $U_{\mathcal{G}}$ passing through
$f_{\alpha_{f}}$ and $g_{\alpha_{g}}$ respectively. Let $c>1$ be an integer
coprime to $6pN_{f}N_{g}$, and let $m\geq 1$ be an integer. Then there exist
cohomology classes

${}_{c}\mathcal{BF}^{[\mathcal{F},\mathcal{G}]}_{m,1}\in
H^{1}({\mathbf{Q}}(\mu(m)),\mathbf{V}_{\mathcal{F}}^{*}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{G}}^{*}\widehat{\otimes}_{E}\mathcal{H}(\Gamma)^{\iota}),$

called Beilinson-Flach elements, which interpolates the elements

${}_{c}\mathcal{BF}^{[k,k^{\prime},j]}_{m,1}\in
H^{1}({\mathbf{Q}}(\mu(m)),V_{\mathcal{F}_{k}}^{*}\widehat{\otimes}V_{\mathcal{G}_{k^{\prime}}}^{*}(-j))$

up to a nonzero factor (if $p^{j}\neq
a_{p}(\mathcal{F}_{k})a_{p}(\mathcal{G}_{k^{\prime}})$) for

$(k,k^{\prime},j)\in
U_{\mathcal{F}}\times_{E}U_{\mathcal{G}}\times_{E}\mathcal{W}\cap{\mathbf{N}}^{3}$

with $0\leq j\leq{\rm min}(k,k^{\prime})$. See [LZ16, Definition 3.5.2,
Theorem 5.4.2] for precise details.

The following Explicit Reciprocity Law is due to [LZ16, Theorem 7.1.5]:

######  Theorem 6.4 (Loeffler-Zerbes).

For $p$-stabilizations $f_{\alpha}$ and $g_{\alpha}$ of $f$ and $g$
respectively, let $\mathcal{F}$ and $\mathcal{G}$ denote the Coleman Families
passing through $f_{\alpha}$ and $g_{\alpha}$ respectively. Then we have

$\displaystyle\left\langle\mathcal{L}(_{c}\mathcal{BF}^{[\mathcal{F},\mathcal{G}]}_{1,1}),\eta_{\mathcal{F}}\otimes\omega_{\mathcal{G}}\right\rangle=\nu_{c}(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},1+\mathbf{j})(-1)^{1+\mathbf{j}}\lambda_{N}(\mathcal{F})^{-1}L_{p}^{\rm
geom}(\mathcal{F},\mathcal{G})(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},1+\mathbf{j}),$

where

$\nu_{c}(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},\mathbf{j})\mathrel{\mathop{\mathchar
58\relax}}=(c^{2}-c^{-\mathbf{k}_{\mathcal{F}}-\mathbf{k}_{\mathcal{G}}+2\mathbf{j}-2}\varepsilon_{\mathcal{F}}(c)^{-1}\varepsilon_{\mathcal{G}}(c)^{-1}),$

$\mathbf{k}_{\mathcal{G}}$ and $\mathbf{k}_{\mathcal{G}}$ are weights of
$\mathcal{F}$ and $\mathcal{G}$ respectively, $\lambda_{N}(\mathcal{F})$ is
the Atkin-Lehner pseudo-eigenvalue (see [AL78] and [KLZ17, Proposition 10.1.2]
for the definition and properties), $\eta_{\mathcal{F}}$ and
$\omega_{\mathcal{G}}$ are given as in [LZ16, §6.4], and $\mathcal{L}$ is
Perrin-Riou’s regulator map.

######  Corollary 6.5.

Let $\mathcal{F}$ and $\widetilde{\mathcal{F}}$ be Coleman families (over the
same nice affinoid neighborhood) passing through the $p$-stabilizations
$f_{\alpha_{f}}$ and $f_{\beta_{f}}$ of $f$, respectively. Suppose that the
product

$L_{p}^{\rm imp}({\rm
Sym}^{2}(\mathbf{V}_{\mathcal{F}}),\varepsilon^{-1}\psi^{-1})(k_{f},2+k_{f}-j)L_{p}(\psi^{-1},1-j)$

of $p$-adic $L$-functions does not vanish for $j$ with $-k_{f}-1<j\leq 0$ and
$\psi(-1)=(-1)^{j}$. Then there exists a $c$ coprime to $6pN_{f}N_{\psi}$ such
that the specializations of

${}_{c}\mathcal{BF}^{[\mathcal{F},\mathcal{F}^{c}_{\psi^{-1}}]}_{1,1}\
\textnormal{and}\
_{c}\mathcal{BF}^{[\mathcal{F},\widetilde{\mathcal{F}}^{c}_{\psi^{-1}}]}_{1,1}$

at $(k_{f},k_{f},1-j+k_{f})$ are nonzero, where $-k_{f}-1<j\leq 0$ and
$\psi(-1)=(-1)^{j}$.

###### Proof.

Using the factorization of Arlandini and Loeffler (twisted version, see
Conjecture 6.1), we have

$L_{p}^{\rm
geom}(\mathcal{F},\mathcal{F}^{c}_{\psi^{-1}})(k_{f},k_{f},2+k_{f}-j)=L_{p}^{\rm
imp}({\rm
Sym}^{2}(\mathbf{V}_{\mathcal{F}}),\varepsilon^{-1}\psi^{-1})(k_{f},2+k_{f}-j)L_{p}(\psi^{-1},1-j).$

Note that setting $\mathcal{G}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{F}^{c}_{\psi^{-1}}$ gives us
$\mathbf{k}_{\mathcal{G}}=\mathbf{k}_{\mathcal{F}}$ and
$\varepsilon_{\mathcal{G}}=\varepsilon_{\mathcal{F}}^{-1}\psi^{-2}(c)$, thus

$\nu_{c}(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},1+\mathbf{j})=(c^{2}-c^{-2\mathbf{k}_{\mathcal{F}}+2\mathbf{j}}\psi(c)^{2}).$

Specializing
$\nu_{c}(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},1+\mathbf{j})$ at
$(\mathbf{k}_{\mathcal{F}},\mathbf{k}_{\mathcal{G}},\mathbf{j})=(k_{f},k_{f},1-j+k_{f})$
gives

$\nu_{c}(k_{f},k_{f},1-j+k_{f})=(c^{2}-c^{2-2j}\psi(c)^{2}),$

thus we can choose a $c$ such that this is nonzero when $-k_{f}-1<j\leq 0$ and
$\psi(-1)=(-1)^{j}$. Recall that those $j$’s are critical for ${{\rm
Ad}^{0}}(V_{f})(\psi)$. Combining with our assumption, the RHS of the explicit
reciprocity law (Theorem 6.4) is nonzero modulo $(k_{f},k_{f},1-j+k_{f})$.
This shows that the specialization of
${}_{c}\mathcal{BF}^{[\mathcal{F},\mathcal{F}^{c}_{\psi^{-1}}]}_{1,1}$ at
$(k_{f},k_{f},1-j+k_{f})$, which is an element of
$H^{1}({\mathbf{Q}}(\mu(m)),{\mathrm{Ad}}(V_{f}^{*})(\psi+j))$, must also be
nonzero.

For
${}_{c}\mathcal{BF}^{[\mathcal{F},\widetilde{\mathcal{F}}^{c}_{\psi^{-1}}]}_{1,1}$,
applying the explicit reciprocity law (with
$\mathcal{G}=\widetilde{\mathcal{F}}^{c}_{\psi^{-1}}$) concludes the proof, as
we have $L_{p}^{\rm
geom}(\mathcal{F},\widetilde{\mathcal{F}}^{c}_{\psi^{-1}})=L_{p}^{\rm
geom}(\mathcal{F},\mathcal{F}^{c}_{\psi^{-1}})$ (see [BLLV19, Proposition
3.6.3]).

∎

For the sake of notational simplicity, let $V_{\mathrm{Ad}}$ and $V_{{\rm
Ad}^{0}}$ denote ${\mathrm{Ad}}(V_{f})(\psi+j)$ and ${{\rm
Ad}^{0}}(V_{f})(\psi+j)$, and let $T_{\mathrm{Ad}}$ and $T_{{\rm Ad}^{0}}$ be
$\mathcal{O}_{E}$-lattices inside $V_{\mathrm{Ad}}$ and $V_{{\rm Ad}^{0}}$,
respectively. Our aim is to show that the Bloch-Kato Selmer groups
$H^{1}_{f}({\mathbf{Q}},V_{{{\rm Ad}^{0}}})$ and
$H^{1}_{f}({\mathbf{Q}},V_{{{\rm Ad}^{0}}}^{*}(1))$ vanish. To be able to
prove this result, we will utilize the Beilinson-Flach Euler system.

######  Lemma 6.6.

Let $\psi+j$ be even, and $j$ be in the critical range of ${{\rm
Ad}^{0}}(V_{f})(\psi)$. Suppose that the hypotheses of Corollary 6.5 hold
true, and in addition suppose that the following hold true:

  * •

$p\geq 7$ and $E\cong{{\mathbf{Q}}_{p}}$,

  * •

${\rm Im}(G_{\mathbf{Q}}\rightarrow{\rm Aut}(V_{f}))$ contains a conjugate of
${\rm SL}_{2}({{\mathbf{Z}}_{p}})$,

  * •

There exists $u\in({\mathbf{Z}}/N_{f}N_{\psi})^{\times}$ such that
$\psi(u)\neq\pm 1\ ({\rm mod\ \mathfrak{P}})$ and
$\varepsilon_{f}^{-1}\psi(u)$ is a square modulo $\mathfrak{P}$, where
$\mathfrak{P}$ is the prime of $E$.

Let $\mathcal{P}_{\psi}$ be the set of primes $\ell\nmid pNN_{\psi}$ such that

  * •

$\ell\equiv$ 1 (${\rm mod}$ $p$),

  * •

$T_{{\rm Ad}^{0}}/({\rm Frob}_{\ell}-1)$ is a free $\mathcal{O}$-module of
rank $1$,

  * •

${\rm Frob}_{\ell}-1$ is bijective on $\mathcal{O}_{E}(\psi+j)$.

Let $\mathcal{R}_{\psi}$ be the set of square free products of primes in
$\mathcal{P}_{\psi}$. For each $r\in\mathcal{R}_{\psi}$, there exist two
cohomology classes

$d_{r}^{\alpha,\alpha},d_{r}^{\alpha,\beta}\in
H^{1}_{(p)}({\mathbf{Q}}(r),V_{{{\rm Ad}^{0}}}),$

where ${\mathbf{Q}}(r)$ is the maximal $p$-extension of ${\mathbf{Q}}$ in
${\mathbf{Q}}(\mu(r))$, satisfying the following properties:

  1. (i)

There exists a constant $D$ independent of $r$ such that

$Dd_{r}^{\alpha,\alpha}\,,\,Dd_{r}^{\alpha,\beta}\in
H^{1}({\mathbf{Q}}(r),T_{{{\rm Ad}^{0}}})\,.$

  2. (ii)

For $r\ell\in\mathcal{R}_{\psi}$ and $\mu\in\\{\alpha,\beta\\}$ we have

$\textup{cor}_{{\mathbf{Q}}(r\ell)/{\mathbf{Q}}(r)}\left(d_{r\ell}^{\alpha,\mu}\right)=P_{\ell}(\ell^{j-1}{\rm
Frob}_{\ell}^{-1})\cdot d_{r}^{\alpha,\mu},$

where $P_{\ell}(X)$ is the Euler polynomial for ${{\rm
Ad}^{0}}(V_{f}^{*})(\psi^{-1})$ at $\ell$ and ${\rm Frob}_{\ell}^{-1}$ is the
geometric Frobenius.

  3. (iii)

The classes $d_{1}^{\alpha,\alpha},d_{1}^{\alpha,\beta}\in
H^{1}_{(p)}({\mathbf{Q}},V_{{{\rm Ad}^{0}}})$ are linearly independent.

###### Proof.

This theorem essentially [BLV21, Theorem 3.2.12], [LZ16, Theorem 8.1.4], and
[Loe18, Theorem 5.3.3] adapted to our setting. We will follow the proofs of
these theorems: Let $\widetilde{c}_{r}^{\alpha,\alpha}$ and
$\widetilde{c}_{r}^{\alpha,\beta}$ be the specializations of Beilinson-Flach
elements

${}_{c}{\mathcal{BF}_{r,1}^{[\mathcal{F},\mathcal{F}^{c}_{\psi^{-1}}]}}\
\textnormal{and}\
_{c}{\mathcal{BF}_{r,1}^{[\mathcal{F},\widetilde{\mathcal{F}}^{c}_{\psi^{-1}}]}}$

at $(k_{f},k_{f},1-j+k_{f})$, respectively, where $\mathcal{F}$,