# On the signed Selmer groups for motives at non-ordinary primes in
$\mathbb{Z}_{{p}}^{2}$-extensions

Jishnu Ray  Harish Chandra Research Institute, A CI of Homi Bhabha National
Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India
<EMAIL_ADDRESS>and  Florian Sprung  School of Mathematical and
Statistical Sciences, Arizona State University Tempe, AZ 85287-1804, USA
<EMAIL_ADDRESS>
###### Abstract.

Generalizing the work of Kobayashi and the second author for elliptic curves
with supersingular reduction at the prime $p$, Büyükboduk and Lei constructed
multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_{p}$-extension of a
number field $F$ for more general non-ordinary motives. In particular, their
construction applies to abelian varieties over $F$ with good supersingular
reduction at all the primes of $F$ above $p$. In this article, we scrutinize
the case in which $F$ is imaginary quadratic, and prove a control theorem
(that generalizes Kim’s control theorem for elliptic curves) of multi-signed
Selmer groups of non-ordinary motives over the maximal abelian pro-$p$
extension of $F$ that is unramified outside $p$, which is the
$\mathbb{Z}_{p}^{2}$-extension of $F$. We apply it to derive a sufficient
condition when these multi-signed Selmer groups are cotorsion over the
corresponding two-variable Iwasawa algebra. Furthermore, we compare the
Iwasawa $\mu$-invariants of multi-signed Selmer groups over the
$\mathbb{Z}_{p}^{2}$-extension for two such representations which are
congruent modulo $p$.

###### Key words and phrases:

non-ordinary motives, signed Selmer groups and Coleman maps, control theorem,
$\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field,
$\mu$-invariants under congruences.

###### 2020 Mathematics Subject Classification:

Primary: 11R23, Secondary: 11G10, 11R20

##  1\. Introduction

The goals of this article are to

  1. (1)

establish cotorsion of certain Selmer groups associated to motives in the non-
ordinary case via a control theorem, and to

  2. (2)

prove a vanishing statement of $\mu$-invariants associated to motives that are
congruent in an appropriate sense.

###  1.1. The objects involved

Suppose $p$ is an odd prime, $F$ be an imaginary quadratic field in which $p$
splits into two primes ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$. Let
$F_{\mathrm{cyc}}$ and $F_{\infty}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension and the unique $\mathbb{Z}_{p}^{2}$-extension of
$F$; thus $F_{\infty}$ contains $F_{\mathrm{cyc}}$. Let
$\Omega=\operatorname{Gal}(F_{\infty}/F)$ and
$\Gamma=\operatorname{Gal}(F_{\mathrm{cyc}}/F)$. For $E$ an elliptic curve
over $\mathbb{Q}$ with supersingular reduction at $p$ and $a_{p}=0$, Kobayashi
[Kob03] constructed signed Selmer groups
$\operatorname{Sel}^{\pm}(E/\mathbb{Q}_{\mathrm{cyc}})$ which are cotorsion
over the Iwasawa algebra
$\mathbb{Z}_{p}[[\operatorname{Gal}(\mathbb{Q}_{\mathrm{cyc}}/\mathbb{Q})]]$.
For a construction that includes the general case, see [Spr12], where the
modified Selmer groups are called $\sharp,\flat$-signed Selmer groups over
$\mathbb{Q}_{\mathrm{cyc}}$. Generalizing the work of Kobayashi, Kim
constructed multi-signed Selmer groups (assuming $a_{p}=0$)
$\operatorname{Sel}^{\pm,\pm}(E/F_{\infty})$ over the
$\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$ which are also conjectured to be
cotorsion as $\mathbb{Z}_{p}[[\Omega]]$-modules [Kim14]. Out of these four
signed Selmer groups, only the ones with the same signs
$\operatorname{Sel}^{+,+}(E/F_{\infty})$
$\operatorname{Sel}^{-,-}(E/F_{\infty})$ are known to be cotorsion (see [LP19,
Remark 8.5]). In the general case [Spr16] (i.e. removing the $a_{p}=0$
condition), the second author has given a construction which shows that at
least one of the chromatic Selmer groups is cotorsion. The local conditions
defining these Selmer groups were then later reinterpreted in terms of Coleman
maps using the theory of Wach-modules which enables one to define signed
Selmer groups over $F_{\mathrm{cyc}}$ also for modular forms which are non-
ordinary at $p$ (see [LLZ10], [LLZ11]). Generalizing the work of Kobayashi,
Büyükboduk and Lei constructed signed Selmer groups over the cyclotomic
extension of any number field for motives $\mathcal{M}$ over $\mathbb{Q}$
which are non-ordinary at $p$. Let $\mathcal{M}_{p}$ be the $p$-adic
realization of the motive $\mathcal{M}$ with a $\mathbb{Z}_{p}$-lattice $T$
which is $G_{F}$-stable. Set $M^{*}=T^{*}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$
where $T^{*}=\operatorname{Hom}(T,\mathbb{Z}_{p}(1))$ is the Tate dual of $T$.
Upon fixing a Hodge-compatible basis of the Dieudonné module attached to $T$
(see Section 2.2.1), for each $\underline{I}$ varying over a certain set (see
[BL17, Definition 3.1]), Büyükboduk and Lei constructed the signed Selmer
group $\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})$ which they
conjecture to be $\mathbb{Z}_{p}[[\Gamma]]$-cotorsion. When $T$ is the Tate
module of an elliptic curve, their construction recovers Kobayashi’s signed
Selmer groups [BL17, p. 397]. The assumption that the signed Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})$ is cotorsion as a
$\mathbb{Z}_{p}[[\Gamma]]$-module can be found in many occasions throughout
literature, see e.g. results in [Pon20].

###  1.2. The control theorem

Suppose that the motive $\mathcal{M}$ is non-ordinary at both primes
${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$. Using the big dual exponential map
of Loeffler–Zerbes for $F_{\infty}$ constructed in [LZ14], one can construct
the multi-signed Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ for the non-ordinary
motive $\mathcal{M}$ over the $\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$. We
are mainly interested in finding a sufficient condition for this multi-signed
Selmer group to be cotorsion as a $\mathbb{Z}_{p}[[\Omega]]$-module that works
in this general setup. Note that in [Dio23], the author assumed that this
Selmer group is cotorsion while proving a two variable algebraic functional
equation for this Selmer group (see [Dio23, Theorem A]).

Our main result in this article is to show the following.

######  Corollary 1.1 (see Corollary 5.4).

If the Bloch-Kato Selmer group $\operatorname{Sel}_{\mathrm{BK}}(M^{*}/F)$ is
finite then the Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ is
$\mathbb{Z}_{p}[[\Omega]]$-cotorsion.

The key ingredient that goes into the proof of the above corollary is a
control theorem (similar to Kim’s for elliptic curves [Kim14]). We show that

######  Theorem 1.2 (see Theorem 5.1).

For all but finitely many $s\in\mathbb{Z}$, the kernel and the cokernel of the
restriction map

$\rho_{\vec{n}}:\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\vec{n}})\rightarrow\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\infty})^{\Omega_{\vec{n}}}$

are bounded as $\vec{n}$ varies.

Here, we have fixed generators $(\gamma_{1},\gamma_{2})$ of
$\Omega\cong\mathbf{Z}_{p}^{2}$ and for a choice of non-negative integers
$\vec{n}:=(n_{1},n_{2}),n_{i}\geqslant 0$, we have denoted by
$\Omega_{\vec{n}}$ the subgroup of $\Omega$ generated by
$(\gamma_{1}^{n_{1}},\gamma_{2}^{n_{2}})$, and the extension $F_{\vec{n}}$ is
the extension $F_{\infty}^{\Omega_{\vec{n}}}$. The Selmer group
$\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{*})$ is a twisted Selmer
group (twisting $M:=T\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ by the $s$-th power
of the cyclotomic character) defined in Section 3.1 for a certain indexing set
$\underline{I}^{c}$, cf. Remark 3.7.

###  1.3. Congruence and previous works

In the last part of this article, we study congruences of the signed Selmer
group $\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$. For elliptic
curves $E$ and $E^{\prime}$ over $\mathbb{Q}$ with good ordinary reduction at
$p$ such that $E[p]\cong E^{\prime}[p]$ as Galois modules, Greenberg and
Vatsal [GV00] initiated the study of such congruences in Iwasawa theory. They
showed that the $\mu$-invariant of the Pontryagin dual of
$\operatorname{Sel}_{p}(E/\mathbb{Q}_{\mathrm{cyc}})$ vanishes if and only if
the $\mu$-invariant of the Pontryagin dual of
$\operatorname{Sel}_{p}(E^{\prime}/\mathbb{Q}_{\mathrm{cyc}})$ vanishes. Kim
generalized this result for supersingular elliptic curves for signed Selmer
groups over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ (see
[Kim09]). It was further generalized by Ponsinet to the case of supersingular
abelian varieties and the multi-signed Selmer groups of Bübükboduk–Lei [BL17]
over the cyclotomic extension of a number field ([Pon20, Theorem 3.13]). Under
additional hypotheses, in [NMMdCS, Theorem 4.15] the result of [Kim09] was
generalized to a wider context for supersingular elliptic curves, while still
working over the cyclotomic extension, which was further generalized for
supersingular elliptic curves over the $\mathbb{Z}_{p}^{2}$-extension of an
imaginary quadratic field $F$ where $p$ splits under the assumption of
Conjecture A (see [Ham22, Corollary 4.6]). Conjecture A is a conjecture
regarding the vanishing of $\mu$-invariant of the dual fine
$p^{\infty}$-Selmer group over the cyclotomic $\mathbb{Z}_{p}$-extension of a
number field (see [CS05]).

###  1.4. Context and contrast of our result on congruence

In this article we generalize both the results [Pon20, Theorem 3.13] and
[Ham22, Corollary 4.6] mentioned above. In particular, we work in the setting
of supersingular abelian varieties and multi-signed Selmer groups of
Bübükboduk–Lei over the $\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$. We note
that in contrast to Hamidi’s result, our theorem stated below is not dependent
on Conjecture A.

Hamidi’s result (loc.cit) were stated for supersingular elliptic curves with
$a_{{\mathfrak{p}}}=a_{{\mathfrak{p}}^{c}}=0$ where ${\mathfrak{p}}$ and
${\mathfrak{p}}^{c}$ are two primes of $F$ above $p$ (see [Ham22, Hyp 1
(iv)])111The analogue of the assumption $a_{p}=0$ in the context of
supersingular abelian varieties is an assumption on the shape of the matrix of
Frobenius (on a basis of the associated Dieudonné module). This shape is a
specific block anti-diagonal form (see [LP20, Section 3.2.1]) – the abelian
varieties of the $GL(2)$-type provide such examples (see [LP20, Section
3.3]).. We remove these assumptions to arrive at the following result. Let
$\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ and
$\mathcal{X}_{\underline{I}}({M^{\prime}}^{*}/F_{\infty})$ be the Pontryagin
duals of the Selmer groups
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ and
$\operatorname{Sel}_{\underline{I}}({M^{\prime}}^{*}/F_{\infty})$, where
$M^{\prime}$ comes from a motive $\mathcal{M^{\prime}}$ with the same residual
representation as $\mathcal{M}$.

######  Theorem 1.3 (see Theorem 6.3).

Let $\mathcal{M}$ and $\mathcal{M^{\prime}}$ have $G_{F}$-stable lattices $T$
and $T^{\prime}$ with congruent mod $p$ reduction that satisfy certain mostly
$p$-adic Hodge theoretic hypotheses (H.-T.), (Cryst.), (Tors.), (Fil), and
(Slopes) described in detail in the Preliminaries section. Suppose that
$\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ and
$\mathcal{X}_{\underline{I}}({M^{\prime}}^{*}/F_{\infty})$ have the same
corank as $\mathbb{Z}_{p}[[\Omega]]$-modules. Then the $\mu$-invariant of
$\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ vanishes if and only if the
$\mu$-invariant of $\mathcal{X}_{\underline{I}}({M^{\prime}}^{*}/F_{\infty})$
vanishes.

######  Corollary 1.4 (see Corollary 6.4).

Suppose that the above Selmer groups are both
$\mathbb{Z}_{p}[[\Omega]]$-cotorsion. Then their $\mu$-invariants either both
vanish, or they both don’t.

Note that the theorem above can be reinterpreted in terms of an Euler
characteristic, since the $\mu$-invariant of
$\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ is the $p$-adic valuation of
the Euler characteristic
$\chi\big{(}\Omega,\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})(p)\big{)}$
(see [How02, Corollary 1.7]).

Our original goal was to relate the $\Omega$-Euler characteristic
$\chi\big{(}\Omega,\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})\big{)}$ with
the $\Gamma$-Euler characteristic
$\chi\big{(}\Gamma,\mathcal{X}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})\big{)}$
generalizing [LS20b, Theorem 5.15]. It would be interesting if one can do
this.

###  1.5. Outline

In Section 2, we recall the preliminaries including the notion of Yager module
which will be an important tool in developing the theory of the two-variable
signed Coleman map, the construction of which is given in Section 3. Using the
signed Coleman map giving the local conditions at the primes of $F$ above $p$,
the signed Selmer group $\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$
is constructed in Section 3. Ranks of certain Iwasawa modules, including the
ranks of the images and the kernels of the signed two variable Coleman map,
are analyzed in Section 4. In Section 5, we prove a control theorem (Theorem
5.1) and use it to provide a sufficient condition for the signed Selmer group
over the $\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$ to be cotorsion (see
Corollary 5.4). Finally, in Section 6 we prove our theorem regarding the
$\mu$-invariants under congruence (see Theorem 6.3).

## Acknowledgments.

We thank Cédric Dion, Parham Hamidi, Antonio Lei, Filippo Nuccio and Gautier
Ponsinet for carefully answering various questions. The first author is
supported by the Inspire research grant, the second author by an NSF grant
(2001280).

##  2\. Preliminaries

The goal of this section is to fix some notations.

###  2.1. The setup

Let $p$ be an odd prime and $F$ be a number field unramified at $p$. Let
$\mathcal{M}$ be a motive over $F$ with coefficients in $\mathbb{Q}$ (see
[FPR94]). Let $\mathcal{M}_{p}$ be its $p$-adic realization and fix a
$G_{F}$\- stable $\mathbb{Z}_{p}$-lattice $T$ inside $\mathcal{M}_{p}$. Let
$g=\dim_{\mathbb{Q}_{p}}(\mathrm{Ind}_{F}^{\mathbb{Q}}\mathcal{M}_{p})$ and
$g_{+}=\dim_{\mathbb{Q}_{p}}(\mathrm{Ind}_{F}^{\mathbb{Q}}\mathcal{M}_{p})^{+}$
the dimension of the $+1$ eigenspace under the action of a fixed complex
conjugation on $\mathrm{Ind}_{F}^{\mathbb{Q}}\mathcal{M}_{p}$. We write
$g_{-}=g-g_{+}$ and
$g_{v}=\dim_{\mathbb{Q}_{p}}(\mathrm{Ind}_{F_{v}}^{\mathbb{Q}_{p}}\mathcal{M}_{p})$
where $v$ is a prime of $F$ above $p$. We then have $g=\sum_{v\mid p}g_{v}$.

######  Remark 2.1.

Suppose $T$ is of rank $d$. Then
$g_{v}=\operatorname{rank}(T)[F_{v}:\mathbb{Q}_{p}].$ Then
$g=\sum_{v|p}g_{v}=\sum_{v|p}\operatorname{rank}(T)[F_{v}:\mathbb{Q}_{p}]$. If
needed, one can also assume $g_{+}=g_{-}=\frac{g}{2}$ wherever necessary (see
hypothesis (H.P) of [LP17]).

Let $T^{*}=\operatorname{Hom}(T,\mathbb{Z}_{p}(1))$ be the Tate dual of $T$.
We set $M=T\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ and
$M^{*}=T^{*}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$.

For every prime $v$ of $F$ above $p$, we with the hypotheses on $\mathcal{M}$
as in [Pon20]:

(H.-T.) The Hodge–Tate weights of $\mathcal{M}_{p}$, as a
$G_{F_{v}}$-representation, are in $[0,1]$.

(Cryst.) The $G_{F_{v}}$-representation $\mathcal{M}_{p}$ is crystalline.

(Tors.) The cohomology groups $H^{0}(F_{v},T/pT)$ and $H^{2}(F_{v},T/pT)$ are
trivial.

We note that the hypotheses (Cryst.) and (H.-T.) are then also satisfied for
$\mathcal{M}^{*}$, which is the dual of $\mathcal{M}$. The hypothesis (Tors.)
is also satisfied by $T^{*}$, which is a $G_{F}$-stable
$\mathbb{Z}_{p}$-lattice inside the $p$-adic realization $\mathcal{M}_{p}^{*}$
of $\mathcal{M}^{*}$.

###  2.2. Dieudonné modules

For a prime $v$ of $F$ dividing $p$, let $\mathbb{D}_{\mathrm{cris},v}(T)$ be
the Dieudonné module associated to $T$ (see [Ber04, Defn. V.1.1]). It is a
free $\mathcal{O}_{F_{v}}$\- module of rank
$\dim_{\mathbb{Q}_{p}}(\mathcal{M}_{p})$ and is equipped with a filtration of
$\mathcal{O}_{F_{v}}$-modules
$(\mathrm{Fil}^{i}\mathbb{D}_{\mathrm{cris},v}(T))_{i\in\mathbb{Z}}$ and a
Frobenius operator ${\varphi}$. The filtration is given by

$\mathrm{Fil}^{i}\mathbb{D}_{\mathrm{cris},v}(T)=\begin{cases}0&\text{if
$i\geqslant 1$,}\\\ \mathbb{D}_{\mathrm{cris},v}(T)&\text{if
$i\leqslant-1$.}\end{cases}$

We set
$\mathbb{D}_{\mathrm{cris},v}(\mathcal{M}_{p}):=\mathbb{D}_{\mathrm{cris},v}(T)\otimes\mathbb{Q}_{p}$
and this is a Fontaine’s filtered ${\varphi}$-module associated to
$\mathcal{M}_{p}.$ We also make the following assumptions.

(Fil.) $\sum_{v\mid
p}\dim_{\mathbb{Q}_{p}}\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{cris},v}(T)\otimes\mathbb{Q}_{p}=g_{-}.$

(Slopes) The slopes of the Frobenius operator ${\varphi}$ on the Dieudonné
module $\mathbb{D}_{\mathrm{cris},v}(\mathcal{M}_{p})$ lie inside $(-1,0)$.

####  2.2.1. Hodge-compatible basis

We fix once and for all a Hodge-compatible basis of
$\mathbb{D}_{\mathrm{cris},v}(T)$ which is, by definition, a
$\mathbb{Z}_{p}$-basis $\\{u_{1},...,u_{g_{v}}\\}$ of
$\mathbb{D}_{\mathrm{cris},v}(T)$ such that $\\{u_{1},...,u_{d_{v}}\\}$ is a
basis for $\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{cris},v}(T)$ for some $d_{v}$.
The matrix of the Frobenius ${\varphi}$ with respect to this basis is of the
form

$C_{{\varphi},v}=C_{v}\left[\begin{array}[]{c|c}I_{d_{v}}&0\\\ \hline\cr
0&\frac{1}{p}I_{g_{v}-d_{v}}\end{array}\right]$

for some $C_{v}\in\mathrm{GL}_{g_{v}}(\mathbb{Z}_{p})$ and where $I_{n}$ is
the identity $n\times n$ matrix (see [BL17, Section 2.2]). We also note that
the Dieudonné module $\mathbb{D}_{\mathrm{cris},v}(T^{*})$ satisfies the
hypotheses (Fil.) and (Slopes).

######  Remark 2.2.

Suppose that $A$ is an abelian variety defined over $F$ having good
supersingular reduction at all the primes of $F$ dividing $p$. Let $T_{p}(A)$
be the Tate module of $A$ and $V_{p}(A):=T_{p}(A)\otimes\mathbb{Q}_{p}$. Then
the $G_{F}$-representation $V_{p}(A)$ and its Galois stable lattice $T_{p}(A)$
satisfy all the hypotheses (H.-T.), (Cryst.), (Tors.), (Fil.) and (Slopes)
(cf. [Pon20, Example 1.1]).

###  2.3. The Yager module

Let $L/\mathbb{Q}_{p}$ be a finite unramified extension. Let $L_{\infty}$ be
any unramified $p$-adic Lie extension of $L$ with Galois group $U$. For
$x\in\mathcal{O}_{L}$, define

$y_{L/\mathbb{Q}_{p}}(x):=\sum_{\tau\in\operatorname{Gal}(L/\mathbf{Q}_{p})}\tau(x)\cdot\tau^{-1}\in\mathcal{O}_{L}[\operatorname{Gal}(L/\mathbb{Q}_{p})].$

Let $S_{L/\mathbb{Q}_{p}}$ be the
sub-$\mathcal{O}_{L}[\operatorname{Gal}(L/\mathbb{Q}_{p})]$-module generated
by the image of $y_{L/\mathbb{Q}_{p}}$ in
$\mathcal{O}_{L}[\operatorname{Gal}(L/\mathbb{Q}_{p})]$. Then there is an
isomorphism of $\Lambda_{\mathcal{O}_{L}}(U)$-modules (cf. [LZ14, Section
3.2])

$y_{L_{\infty}/\mathbb{Q}_{p}}:\varprojlim_{\mathbb{Q}_{p}\subseteq L\subseteq
L_{\infty}}\mathcal{O}_{L}\xrightarrow{\cong}S_{L_{\infty}/\mathbb{Q}_{p}}:=\varprojlim_{\mathbb{Q}_{p}\subseteq
L\subseteq L_{\infty}}S_{L/\mathbb{Q}_{p}},$

where the inverse limit is taken with respect to the trace maps on the left
and the projection maps
$\operatorname{Gal}(L^{\prime}/\mathbb{Q}_{p})\to\operatorname{Gal}(L/\mathbb{Q}_{p})$
for $L\subseteq L^{\prime}$ on the right. By [LZ14, Proposition 3.2],
$\varprojlim_{\mathbb{Q}_{p}\subseteq L\subseteq L_{\infty}}\mathcal{O}_{L}$
is a free $\Lambda_{\mathcal{O}_{L}}(U)$-module of rank one, so that the Yager
module $S_{L_{\infty}/\mathbb{Q}_{p}}$ is also free of rank one over
$\Lambda_{\mathcal{O}_{L}}(U)$. The Yager module
$S_{L_{\infty}/\mathbb{Q}_{p}}$ comes equipped with a compact and Hausdorff
topology that coincides with the subspace topology from
$\Lambda_{\widehat{\mathcal{O}}_{L_{\infty}}}(U)$, where
$\widehat{\mathcal{O}}_{L_{\infty}}$ is the completion of the ring of integers
of $L_{\infty}$.

##  3\. The multi-signed Selmer groups over a $\mathbb{Z}_{p}^{2}$-tower

In this section, we recall some known results, mainly the construction of
certain Coleman maps and present a conjecture due to Büyükboduk–Lei concerning
the cotorsion of a Selmer group constructed from the Coleman map in the case
of a cyclotomic $\mathbb{Z}_{p}$-extension. We generalize their conjecture to
the case of a $\mathbb{Z}_{p}^{2}$-extension.

Let $F$ be imaginary quadratic, $F_{\infty}$ be the compositum of all
$\mathbb{Z}_{p}$-extensions of $F$, so that in particular
$F_{\mathrm{cyc}}\subset F_{\infty}$, where $F_{\mathrm{cyc}}$ is the
cyclotomic $\mathbb{Z}_{p}$-extension of $F$. We let
$\Omega:=\operatorname{Gal}(F_{\infty}/F)\cong\mathbb{Z}_{p}^{2}$. Suppose
that the prime $p$ splits into two primes ${\mathfrak{p}}$ and
${\mathfrak{p}}^{c}$ of $F$, where $c$ denotes complex conjugation. We will
write ${\mathfrak{q}}$ to denote either of these two primes. If $\mathfrak{a}$
is an ideal of $\mathcal{O}_{F}$, let $F(\mathfrak{a})$ be the ray class field
of $F$ of conductor $\mathfrak{a}$. If $n\geqslant 0$ is an integer, we will
write $\Omega_{n}:=\Omega^{p^{n}}$ and
$F_{n}:=F_{\infty}^{\Omega_{n}}=F(p^{n+1})^{\operatorname{Gal}(F(1)/F)}$. We
assume that $F(1)\bigcap F_{\infty}=F$, hence $\Omega\cong
G_{{\mathfrak{p}}}\times G_{{\mathfrak{p}}^{c}}$ where $G_{\mathfrak{q}}$ is
the Galois group of the extension $F({\mathfrak{q}}^{\infty})\bigcap
F_{\infty}/F$. We fix topological generators $\gamma_{\mathfrak{p}}$ and
$\gamma_{{\mathfrak{p}}^{c}}$ respectively for the groups $G_{\mathfrak{p}}$
and $G_{{\mathfrak{p}}^{c}}$. Let
$\Lambda(\Omega):=\mathbf{Z}_{p}[[\Omega]]\cong\mathbf{Z}_{p}[[\gamma_{\mathfrak{p}}-1,\gamma_{{\mathfrak{p}}^{c}}-1]]$
be the Iwasawa algebra of $\Omega$. More generally, for a choice of non-
negative integers $\vec{n}:=(n_{1},n_{2});n_{i}\geqslant 0$, denote by
$\Omega_{\vec{n}}$ the subgroup of $\Omega$ generated by
$(\gamma_{{\mathfrak{p}}}^{n_{1}},{\gamma_{{\mathfrak{p}}^{c}}}^{n_{2}})$, and
let $F_{\vec{n}}:=F_{\infty}^{\Omega_{\vec{n}}}$.

Write $\Omega_{\mathfrak{q}}$ for the decomposition group of ${\mathfrak{q}}$
in $\Omega$. Let $\Sigma$ be a finite set of primes of $F$ containing the
primes dividing $p$, the archimedean primes and the primes of ramification of
$M^{*}$. Let $F_{\Sigma}$ be the maximal extension of $F$ that is unramified
outside $\Sigma$. If $w$ is a place of $F_{\infty}$ above ${\mathfrak{q}}$,
then by local class field theory, the Galois group
$\operatorname{Gal}(F_{\infty,w}/F_{\mathfrak{q}})$ is isomorphic to
$\mathbb{Z}_{p}^{2}$. Hence the extension $F_{\infty,w}$ coincides with the
compositum of the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}_{p}$
and the unramified $\mathbb{Z}_{p}$-extension of $\mathbb{Q}_{p}$.

Let $L_{\infty}$ and $k_{\text{cyc}}$ be the unramified
$\mathbb{Z}_{p}$-extension and the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}_{p}$ respectively. Let $k_{\infty}$ be the compositum of
$L_{\infty}$ and $k_{\mathrm{cyc}}$. For $n\geqslant 0$, let $k_{n}$ and
$L_{n}$ be the subextensions of $k_{\mathrm{cyc}}$ and $L_{\infty}$
respectively such that $[k_{n}:\mathbb{Q}_{p}]=p^{n}$ and
$[L_{n}:\mathbb{Q}_{p}]=p^{n}$. We set
$\Omega_{p}:=\operatorname{Gal}(k_{\infty}/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}^{2}$,
$\Gamma_{\text{ur}}:=\operatorname{Gal}(L_{\infty}/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}$
and
$\Gamma_{\text{cyc}}:=\operatorname{Gal}(k_{\text{cyc}}/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}$.
Thus $\Omega_{p}\cong\Gamma_{\mathrm{ur}}\times\Gamma_{\mathrm{cyc}}$. For
$\vec{n}=(n_{1},n_{2});n_{i}\geqslant 0$, on fixing two topological generators
$(\gamma_{1},\gamma_{2})$ of $\Omega_{p}\cong\mathbb{Z}_{p}^{2}$, we will
write $\Omega_{p,\vec{n}}$ to denote the subgroup of $\Omega_{p}$ generated by
$(\gamma_{1}^{n_{1}},\gamma_{2}^{n_{2}})$. Choosing a topological generator
$\gamma_{2}$ of $\Gamma_{\mathrm{cyc}}$, we let
$\mathcal{H}(\Gamma_{\text{cyc}})$ be the set of power series

$\sum_{n\geqslant 0}c_{n}\cdot(\gamma_{2}-1)^{n}$

with coefficients in $\mathbb{Q}_{p}$ such that $\sum_{n\geqslant
0}c_{n}X^{n}$ converges on the open unit disk.

Fix a Hodge-compatible basis $\\{u_{1},...,u_{g_{\mathfrak{q}}}\\}$ of
$\mathbb{D}_{\mathrm{cris},{\mathfrak{q}}}(T)$. Let
$\\{Y_{L_{\infty}/\mathbb{Q}_{p}}\\}$ be a basis of the Yager module
$S_{L_{\infty}/\mathbb{Q}_{p}}$. Then Loeffler–Zerbes [LZ14] constructed a
two-variable big logarithm map

$\mathcal{L}_{T,{\mathfrak{q}}}^{\infty}:H^{1}_{\mathrm{Iw}}(k_{\infty},T)\to
Y_{L_{\infty}/\mathbb{Q}_{p}}\cdot\left(\mathcal{H}(\Gamma_{\text{cyc}})\widehat{\otimes}\Lambda(\Gamma_{\text{ur}})\right)\otimes_{\mathbb{Z}_{p}}\mathbb{D}_{\mathrm{cris},{\mathfrak{q}}}(T).$

which interpolates the Bloch–Kato dual exponential map [LZ14, Theorem 4.15].

Let $K$ be a finite unramified extension of $\mathbb{Q}_{p}$ and let
$K_{\mathrm{cyc}}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$
with Galois group $\Gamma_{\mathrm{cyc}}$. In [BL17], the authors show the
existence of one-variable Coleman maps

$\operatorname{Col}_{T,K,i}:H^{1}_{\mathrm{Iw}}(K_{\mathrm{cyc}},T)\to\mathcal{O}_{K}\otimes\Lambda(\Gamma_{\mathrm{cyc}})$

for $1\leqslant i\leqslant g_{v}$. Those maps are compatible with the
corestriction maps

$H^{1}_{\mathrm{Iw}}(L_{m,\mathrm{cyc}},T)\to
H^{1}_{\mathrm{Iw}}(L_{m-1,\mathrm{cyc}},T)$

and the trace maps
$\mathcal{O}_{L_{m}}\otimes\Lambda(\Gamma_{\mathrm{cyc}})\to\mathcal{O}_{L_{m-1}}\otimes\Lambda(\Gamma_{\mathrm{cyc}})$,
where $L_{m,\mathrm{cyc}}$ is the cyclotomic $\mathbb{Z}_{p}$-extension of
$L_{m}$. We define the two-variable Coleman maps by taking the inverse limit
of $\operatorname{Col}_{T,L_{m},i}$ as $L_{m}$ runs through the finite
extensions between $L_{\infty}$ and $\mathbb{Q}_{p}$. In order to get a family
of maps landing in $\Lambda(\Gamma_{p})$, we further compose it with the Yager
module $S_{L_{\infty}/\mathbb{Q}_{p}}$. More precisely, the two-variable
Coleman maps are defined by

(1)
$\displaystyle\operatorname{Col}_{T,i}^{\infty}:H^{1}_{\mathrm{Iw}}(k_{\infty},T)\cong\varprojlim_{L_{m}}H^{1}_{\mathrm{Iw}}(L_{m,cyc},T)$
$\displaystyle\to Y_{L_{\infty}/\mathbb{Q}_{p}}\cdot\Lambda(\Omega_{p})$
$\displaystyle(z_{m})$
$\displaystyle\mapsto(y_{L_{\infty}/\mathbb{Q}_{p}}\otimes
1)\circ(\varprojlim_{L_{m}}\operatorname{Col}_{T,L_{m},i}(z_{m})).$

We can identify $Y_{L_{\infty}/\mathbb{Q}_{p}}\cdot\Lambda(\Omega_{p})$ with
$\Lambda(\Omega_{p})$, and hence omit the basis
$Y_{L_{\infty}/\mathbb{Q}_{p}}$ from the notation and see the Coleman maps
$\operatorname{Col}_{T,i}^{\infty}$ as taking value in $\Lambda(\Omega_{p})$.
By combining [BL17, Theorem 2.13] and [LZ14, Theorem 4.7 (1)] and including
the prime ${\mathfrak{q}}$ in our notation, we have the Coleman maps

(2)
$\operatorname{Col}_{T,{\mathfrak{q}},i}^{\infty}:H^{1}(K_{\mathfrak{q}},T\otimes\Lambda(\Omega_{\mathfrak{q}})^{\iota})\rightarrow\mathbb{Z}_{p}[[\Omega_{p}]]$

for $i\in\\{1,...,g_{\mathfrak{q}}\\}$ such that

$\mathcal{L}_{T,{\mathfrak{q}}}^{\infty}=(u_{1},\ldots,u_{g_{\mathfrak{q}}})\cdot
M_{T,{\mathfrak{q}}}\cdot\begin{bmatrix}\operatorname{Col}_{T,{\mathfrak{q}},1}^{\infty}\\\
\vdots\\\
\operatorname{Col}_{T,{\mathfrak{q}},g_{\mathfrak{q}}}^{\infty}\end{bmatrix}.$

Here the matrix $M_{T,{\mathfrak{q}}}$ is defined in the following way. For
$n\geqslant 1$, as in [BL17], we can define

$C_{{\mathfrak{q}},n}=\left[\begin{array}[]{c|c}I_{d_{\mathfrak{q}}}&0\\\
\hline\cr
0&\Phi_{p^{n}}(1+X)I_{g_{\mathfrak{q}}-d_{\mathfrak{q}}}\end{array}\right]C_{\mathfrak{q}}^{-1}$

and

$M_{{\mathfrak{q}},n}=(C_{{\varphi},{\mathfrak{q}}})^{n+1}C_{{\mathfrak{q}},n}\cdots
C_{1},$

where $\Phi_{p^{n}}$ is the $p^{n}$-th cyclotomic polynomial and the matrices
$C_{{\varphi},{\mathfrak{q}}}$ and $C_{\mathfrak{q}}$ are as in Section 2.2.1.
By [BL17, Proposition 2.5] the sequence $(M_{{\mathfrak{q}},n})_{n\geqslant
1}$ converges to some $g_{\mathfrak{q}}\times g_{\mathfrak{q}}$ logarithmic
matrix $M_{T,{\mathfrak{q}}}$ with entries in
$\mathcal{H}(\Gamma_{\mathrm{cyc}})$.

######  Remark 3.1.

By assumption, $F_{\infty}\cap F(1)=F$, which happens if $p$ does not divide
$h_{F}$, the class number of $F$. For a prime ${\mathfrak{q}}$ of $F$ over
$p$, ${\mathfrak{q}}$ does not split in $F_{\mathrm{cyc}}$ and every prime
above $p$ in the anticyclotomic extension of $F$ is totally ramified. Hence
the assumption $F_{\infty}\cap F(1)=F$ implies that there is a unique prime
above ${\mathfrak{q}}$ in $F_{\infty}$. However, for the definition of the
multi-signed Selmer groups given below, we do not need the assumption
$F_{\infty}\cap F(1)=F$.

Denote by $p^{t}$ the number of primes above ${\mathfrak{p}}$ and
${\mathfrak{p}}^{c}$ in $F_{\infty}$. Fix a choice of coset representatives
$\gamma_{1},\ldots,\gamma_{p^{t}}$ and $\delta_{1},\ldots,\delta_{p^{t}}$ for
$\Omega/\Omega_{\mathfrak{p}}$ and $\Omega/\Omega_{{\mathfrak{p}}^{c}}$
respectively. Since $p$ splits in $F$, we can identify $F_{\mathfrak{q}}$ with
$\mathbb{Q}_{p}$ and $\Omega_{\mathfrak{q}}$ with $\Omega_{p}$. Consider the
“semi-local" decomposition coming from Shapiro’s lemma

$H^{1}(F_{\mathfrak{p}},T\otimes\Lambda(\Omega)^{\iota})=\bigoplus_{j=1}^{p^{t}}H^{1}(F_{\mathfrak{p}},T\otimes\Lambda(\Omega_{\mathfrak{p}})^{\iota})\cdot\gamma_{j}\cong\bigoplus_{v|{\mathfrak{p}}}H^{1}_{\text{Iw}}(F_{\infty,v},T),$

where $v$ runs through the primes above ${\mathfrak{p}}$ in $F_{\infty},$ and
$\iota:\Lambda(\Omega)\rightarrow\Lambda(\Omega)$ is the involution obtained
by sending $g\in\Omega$ to $g^{-1}$. By choosing a Hodge-compatible basis of
$\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}}(T)$, define the Coleman map for $T$
at ${\mathfrak{p}}$ by

$\displaystyle\operatorname{Col}_{T,{\mathfrak{p}},i}^{k_{\infty}}:H^{1}(F_{\mathfrak{p}},T\otimes\Lambda(\Omega)^{\iota})$
$\displaystyle\to\Lambda(\Omega)$ $\displaystyle
x=\sum_{j=1}^{p^{t}}x_{j}\cdot\gamma_{j}$
$\displaystyle\mapsto\sum_{j=1}^{p^{t}}\operatorname{Col}_{T,{\mathfrak{p}},i}^{\infty}(x_{j})\cdot\gamma_{j}$

for all $1\leqslant i\leqslant g_{\mathfrak{p}}$.

Let
$\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}=\oplus_{j=1}^{p^{t}}\mathcal{L}_{T,{\mathfrak{p}}}^{\infty}\cdot\gamma_{j}$.
Define $\operatorname{Col}_{T,{\mathfrak{p}}^{c},i}^{k_{\infty}}$ and
$\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}$ in an analogous way. Let
$I_{\mathfrak{q}}$ denote a subset of $\\{1,\ldots,g_{\mathfrak{q}}\\}$ and
let

$\displaystyle\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty}:H^{1}(F_{\mathfrak{q}},T\otimes\Lambda(\Omega)^{\iota})$
$\displaystyle\to\bigoplus_{i=1}^{|I_{\mathfrak{q}}|}\Lambda(\Omega)$
$\displaystyle z$
$\displaystyle\mapsto(\operatorname{Col}^{k_{\infty}}_{T,{\mathfrak{q}},i}(z))_{i\in
I_{\mathfrak{q}}}.$

Tate’s local pairing induces a pairing

(3)
$\bigoplus_{w|{\mathfrak{q}}}H^{1}_{\mathrm{Iw}}(F_{\infty,w},T)\times\bigoplus_{w|{\mathfrak{q}}}H^{1}(F_{\infty,w},M^{*})\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$

for all places $w$ of $F_{\infty}$ above ${\mathfrak{q}}$. We define
$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})\subseteq\bigoplus_{v|{\mathfrak{q}}}H^{1}(F_{\infty,v},M^{*})$
as the orthogonal complement of
$\ker\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty}$ under the previous
pairing.

Let $K^{\prime}$ be a finite extension of $\mathbb{Q}_{p}$. We define

$H^{1}_{\mathrm{un}}(K^{\prime},\mathcal{M}_{p}^{*})=\ker\Big{(}H^{1}(K^{\prime},\mathcal{M}_{p}^{*})\rightarrow
H^{1}(K_{\mathrm{ur}},\mathcal{M}_{p}^{*})\Big{)}$

where $K_{\mathrm{ur}}$ is the maximal unramified extension of $K^{\prime}$.
Let $H^{1}_{\mathrm{ur}}(K,M^{*})$ be the image of
$H^{1}_{\mathrm{ur}}(K,\mathcal{M}_{p}^{*})$ under the natural map

$H^{1}_{\mathrm{ur}}(K,\mathcal{M}_{p}^{*})\rightarrow H^{1}(K,M^{*}).$

For an infinite extension $L$ of $\mathbb{Q}_{p}$, we define

$H^{1}_{{\mathrm{ur}}}(L,M^{*})=\varinjlim_{K^{\prime}}H^{1}_{\mathrm{ur}}(K^{\prime},M^{*})$

where $K^{\prime}$ runs through all the finite subextensions of $L$ and the
limit is taken with respect to restriction maps.

Let $\Sigma^{\prime}$ be the set of primes of $F_{\infty}$ above $\Sigma$. Let
$\mathcal{I}_{p}$ be the set of tuples
$\underline{I}=(I_{\mathfrak{p}},I_{{\mathfrak{p}}^{c}})$ such that
$I_{\mathfrak{q}}$ is a subset of $\\{1,...,g_{\mathfrak{q}}\\}$ such that
$|I_{\mathfrak{p}}|+|I_{{\mathfrak{p}}^{c}}|=g_{-}$.

######  Definition 3.2.

For any $\underline{I}$ as above, set

$\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty})=\prod_{w\in\Sigma^{\prime},w\nmid
p}\frac{H^{1}(F_{\infty,w},M^{*})}{H^{1}_{\mathrm{ur}}(F_{\infty,w},M^{*})}\times\prod_{{\mathfrak{q}}\mid
p}\frac{\bigoplus_{w\mid{\mathfrak{q}}}H^{1}(F_{\infty,w},M^{*})}{H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})}.$

Then the multi-signed Selmer group over the $\mathbb{Z}_{p}^{2}$-extension
$F_{\infty}$ is defined as

$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})=\ker\Big{(}H^{1}(F_{\Sigma}/F_{\infty},M^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty})\Big{)}.$

Our construction generalizes the construction of the multi-signed Selmer
groups for the cyclotomic $\mathbb{Z}_{p}$-extension $F_{\mathrm{cyc}}$ given
in [Pon20, Section 1.5 and Section 1.6] (originally constructed by
Büyükboduk–Lei [BL17]). For a prime ${\mathfrak{q}}$ of $F$ above $p$, we have
the signed Coleman map for the cyclotomic extension

$\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\mathrm{cyc}}:H^{1}_{\mathrm{Iw}}(F_{\mathfrak{q}},T)\rightarrow\bigoplus_{i=1}^{|I_{\mathfrak{q}}|}\mathbb{Z}_{p}[[\operatorname{Gal}(F_{\mathrm{cyc}}/F)]]$

which give the local condition
$H^{1}_{I_{\mathfrak{q}}}(F_{\mathrm{cyc},{\mathfrak{q}}},M^{*})$. This
enables the author to define the signed Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})$ (see [Pon20,
Definition 1.7]) (Ponsinet uses $F_{\infty}$ to denote the cyclotomic
$\mathbb{Z}_{p}$-extension and uses $\operatorname{Col}_{T,I_{\mathfrak{q}}}$
in place of $\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\mathrm{cyc}}$; for
convenience in this article we have changed the notations of Ponsinet a bit.)

We can also define the signed Selmer groups at finite layers. We now assume
that $F_{\infty}\cap F(1)=F$ and hence, by abuse of notation, we will let
${\mathfrak{q}}$ denote the unique prime at $F_{\infty}$ over the prime
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$ of $F$.

We define
$H^{1}_{I_{\mathfrak{q}}}(F_{\vec{n},{\mathfrak{q}}},M^{*}):=H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})^{\Omega_{\vec{n}}}$,
just as in [Pon20, Definition 1.5]. Defining the local condition at a finite
layer by using the local condition at an infinite tower and then descending to
the finite level is also crucially exploited in the work of
Burungale–Büyükboduk–Lei (see [BBL22, Remark 4.9]). This approach also appears
in many other works of Lei, for example see [Lei23, Definition 5.2 and Remark
5.4]. We will take this approach since it will have several advantages as we
will see in the control theorem later (see Theorem 5.1). Having defined the
analogous local condition
$\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\vec{n}})$ as before using
$H^{1}_{I_{\mathfrak{q}}}(F_{\vec{n},{\mathfrak{q}}},M^{*})$, we can define
the Selmer group

$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\vec{n}}):=\ker\Big{(}H^{1}(F_{\Sigma}/F_{\vec{n}},M^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\vec{n}})\Big{)}.$

Recall that $\vec{n}=(n_{1},n_{2})$ and if $n_{1}=n_{2}=n$, then we will
simply write $\vec{n}$ as $n$. Note that, since
$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})$ is a discrete
$\Omega_{p}$-module, by [NSW08, Proposition 1.1.8],

$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})=\varinjlim_{n}H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})^{\Omega_{p,n}}=\varinjlim_{n}H^{1}_{I_{\mathfrak{q}}}(F_{n,{\mathfrak{q}}},M^{*}).$

Since taking direct limits is an exact functor, this gives

$\varinjlim_{n}\frac{H^{1}(F_{n,{\mathfrak{q}}},M^{*})}{H^{1}_{I_{\mathfrak{q}}}(F_{n,{\mathfrak{q}}},M^{*})}\cong\frac{\varinjlim_{n}H^{1}(F_{n,{\mathfrak{q}}},M^{*})}{\varinjlim_{n}H^{1}_{I_{\mathfrak{q}}}(F_{n,{\mathfrak{q}}},M^{*})}\cong\frac{H^{1}(F_{\infty,{\mathfrak{q}}},M^{*})}{H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})}.$

Furthermore,
$\varinjlim_{n}H^{1}_{\mathrm{ur}}(F_{n,v},M^{*})=H^{1}_{\mathrm{ur}}(F_{\infty,v},M^{*})$
when $v\nmid p$. Hence we obtain the following compatibility between multi-
signed Selmer groups at finite levels $F_{n}$ and the infinite
$\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$. We have

$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\cong\varinjlim_{n}\operatorname{Sel}_{\underline{I}}(M^{*}/F_{n})$

as $\Lambda[[\Omega]]$-modules.

The following conjecture has been made by Büyükboduk and Lei [BL17, Remark
3.57].

######  Conjecture 3.3.

For any $\underline{I}\in\mathcal{I}_{p}$, the signed Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})$ is cotorsion over
the Iwasawa algebra
$\mathbb{Z}_{p}[[\operatorname{Gal}(F_{\mathrm{cyc}}/F)]]$.

######  Remark 3.4.

When the base field $F$ is $\mathbb{Q}$ and $\mathcal{M}$ is the Tate module
of a supersingular elliptic curve with $a_{p}=0$, the signed Selmer groups
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\mathrm{cyc}})$ for
$\underline{I}\in\mathcal{I}_{p}$ coincides with Kobayashi’s plus and minus
Selmer groups (see [BL17, Appendix 4]). Kobayashi’s plus and minus Selmer
groups are known to be cortorsion [Kob03] and hence conjecture 3.3 holds.
Moreover, conjecture 3.3 also holds for at least one of the chromatic Selmer
groups in the case of $p$-supersingular elliptic curves with $a_{p}\neq 0$ and
eigenforms which are non-ordinary at $p$. (see [Spr12, Proposition 6.14,
Theorem 7.14] and [LLZ10, Theorem 6.5]).

Generalizing Conjecture 3.3 above in the case of
$\mathbb{Z}_{p}^{2}$-extension seems reasonable to make.

######  Conjecture 3.5.

For any $\underline{I}\in\mathcal{I}_{p}$, the Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ is
$\Lambda(\Omega)$-cotorsion.

######  Remark 3.6.

Suppose that $\mathcal{M}$ is the Tate module of a supersingular elliptic
curve. Then the two-variable Coleman maps we defined over the
$\mathbb{Z}_{p}^{2}$-extension $F_{\infty}$ are the same as as the
$\sharp/\flat$-Coleman maps in [LS20a, Section 5].

For $\bullet,\circ\in\\{\sharp,\flat\\}$, one can construct signed $p$-adic
$L$-functions $\mathfrak{L}_{p}^{\bullet,\circ}(E/F)$. Under the hypothesis
that $\mathfrak{L}_{p}^{\bullet,\circ}(E/F)$ is nonzero, [Spr16, Proposition
2.23] or [CÇSS18, Theorem 3.7] shows that the signed Selmer group
$\operatorname{Sel}^{\bullet,\circ}(E/F_{\infty})$ is
$\Lambda(\Omega)$-cotorsion. The $\sharp/\flat$-Selmer groups generalise Kim’s
$\pm/\pm$-Selmer groups [Kim14], who assumed $a_{p}=0$. The Selmer groups for
$\underline{I}=(1,1)$ and $\underline{I}=(2,2)$ correspond to $+/+$ and $-/-$
Selmer groups and these are known to be cotorsion (see [LP19, Remark 8.5])
giving partial results toward conjecture 3.5.

######  Remark 3.7.

Recall that $\mathcal{I}_{p}$ was the set of tuples
$\underline{I}=(I_{\mathfrak{p}},I_{{\mathfrak{p}}^{c}})$ such that
$I_{\mathfrak{q}}$ is a subset of $\\{1,...,g_{\mathfrak{q}}\\}$ for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$ satisfying
$|I_{\mathfrak{p}}|+|I_{{\mathfrak{p}}^{c}}|=g_{-}$. Denote by
$I_{\mathfrak{q}}^{c}$ the complement of the subset $I_{\mathfrak{q}}$. Then
$\underline{I}^{c}=(I_{\mathfrak{p}}^{c},I_{{\mathfrak{p}}^{c}}^{c})$
satisfies $|I_{\mathfrak{p}}^{c}|+|I_{{\mathfrak{p}}^{c}}^{c}|=g-g_{-}=g_{+}$.
One can also define Coleman maps
$\operatorname{Col}_{T^{*},I_{\mathfrak{q}}}^{\infty}$ and construct the
signed Selmer groups $\operatorname{Sel}_{\underline{I}^{c}}(M/F_{\infty})$
for $M$ (for details, see [Dio23, Lemma 3.14]). Also note that Conjecture 3.5
is expected to hold for
$\operatorname{Sel}_{\underline{I}^{c}}(M/F_{\infty})$. Under certain
conditions, for supersingular abelian varities of $GL(2)$-type, there is also
a functional equation relating the characteristic ideal of the dual of
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ with that of
$\operatorname{Sel}_{\underline{I}^{c}}(M/F_{\infty})$ (see [Dio23, Theorem
A]).

###  3.1. Twisted Selmer groups

We follow the notations as in [Pon20, discussion after Remark 1.10]. For
$s\in\mathbb{Z}$, set $M_{s}^{*}=M^{*}\otimes\chi^{s}$ where $\chi$ is the
cyclotomic character. Note that the cyclotomic character factors through the
cyclotomic extension of $F$. We have $M_{s}^{*}=M^{*}$ as a
$\operatorname{Gal}(\overline{F}/F_{\infty})$-module and hence
$H^{1}(F_{\infty},M_{s}^{*})=H^{1}(F_{\infty},M^{*})\otimes\chi^{s}$.

For every prime $v$ of $F$, we also have
$H^{1}(F_{v,\infty},M_{s}^{*})=H^{1}(F_{v,\infty},M^{*})\otimes\chi^{s}$ and
$H^{0}(F_{v,\infty},M_{s}^{*})=0$. At any prime ${\mathfrak{q}}$ of $F$
dividing $p$, we set

$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M_{s}^{*})=H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})\otimes\chi^{s}.$

We can now define multi-signed Selmer groups with these twisted local
conditions. As in [Pon20, p. 1643], we also note that
$\operatorname{Sel}_{\underline{I}}(M_{s}^{*}/F_{\infty})\cong\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\otimes\chi^{s}$
as $\Lambda[[\Omega]]$-modules.

##  4\. Ranks of Iwasawa modules

Recall that ${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$, put
$H=\operatorname{Gal}(F_{\infty}/F_{\mathrm{cyc}})$, and
$\Gamma=\Omega/H=\operatorname{Gal}(F_{\mathrm{cyc}}/F)$.

######  Proposition 4.1.

For ${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$, we have

  1. (1)

$\operatorname{rank}_{\mathbb{Z}_{p}[[\Gamma]]}H^{1}_{\mathrm{Iw}}(F_{\mathrm{cyc},{\mathfrak{q}}},T)=g_{\mathfrak{q}}$

  2. (2)

The torsion sub-$\mathbb{Z}_{p}[[\Gamma]]$-module of
$H^{1}_{\mathrm{Iw}}(F_{\mathrm{cyc},{\mathfrak{q}}},T)$ is isomorphic to
$T^{G_{F_{\mathrm{cyc}}}}$.

  3. (3)

$\operatorname{rank}_{\mathbb{Z}_{p}[[\Omega_{p}]]}H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)=g_{\mathfrak{q}}$

###### Proof.

(1) and (2) is due to Perrin-Riou (see [PR95, A.2]).

Note that, by [BL21, p. 23] and the proof of Lemma 2.16 of [BL21], we have

$H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)\cong\big{(}N_{L_{\infty}}(T)^{\psi=1}\big{)}^{\Delta}\cong\big{(}N_{\mathbb{Q}_{p}}(T)^{\psi=1}\big{)}^{\Delta}\widehat{\otimes}S_{L_{\infty}/\mathbb{Q}_{p}}\cong
H^{1}_{\mathrm{Iw}}(F_{\mathrm{cyc},{\mathfrak{q}}},T)\widehat{\otimes}S_{L_{\infty}/\mathbb{Q}_{p}},$

where $\Delta=\operatorname{Gal}(L_{\infty}(\mu_{p^{\infty}})/k_{\infty})$ is
of order $p-1$, $N_{L_{\infty}}(T)$ and $N_{\mathbb{Q}_{p}}(T)$ are Wach
modules as in (loc.cit) and $\psi$ (à la Fontaine) is the left inverse of the
Frobenius ${\varphi}$-operator on the respective Wach modules. Finally, note
that the Yager module is a free module of rank one over
$\Lambda_{\mathbb{Q}_{p}}(U)$ (cf. section 2.3). This proves (3). ∎

######  Lemma 4.2.

The kernel and the image of $\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty}$
are $\mathbb{Z}_{p}[[\Omega_{p}]]$-modules of rank
$g_{\mathfrak{q}}-|I_{\mathfrak{q}}|$ and $|I_{\mathfrak{q}}|$ respectively.

###### Proof.

By the proof of Lemma 2.16 of [BL21],

$\operatorname{Im}(\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty})\cong\operatorname{Im}(\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\mathrm{cyc}})\widehat{\otimes}S_{L_{\infty}/\mathbb{Q}_{p}.}$

and
$\operatorname{Im}(\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\mathrm{cyc}})$ is
of rank $|I_{\mathfrak{q}}|$ contained in a free
$\mathbb{Z}_{p}[[\Gamma_{\mathrm{cyc}}]]$-module of finite index (see [Pon20,
Lemma 1.2]). Hence
$\operatorname{Im}(\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty})$ is of
rank $|I_{\mathfrak{q}}|$ contained in a free
$\mathbb{Z}_{p}[[\Omega_{p}]]$-module of finite index. ∎

##  5\. A Kim-type control theorem and a mini-control theorem

We prove two results, a Kim-type control theorem, generalizing B.D. Kim’s
theory for elliptic curves, and a corollary, which is a mini-control theorem
that shows that the finitenes of a Selmer group at base level implies
cotorsion of the same Selmer group in our towers of number fields. The first
theorem below (Theorem 5.1) generalizes a control theorem of Ponsinet which
was over the cyclotomic extension (see [Pon20, Lemma 2.3]).

######  Theorem 5.1.

For all but finitely many $s\in\mathbb{Z}$, the kernel and the cokernel of the
restriction map

$\rho_{\vec{n}}:\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\vec{n}})\rightarrow\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\infty})^{\Omega_{\vec{n}}}$

are bounded as $\vec{n}$ varies.

###### Proof.

By definition, we have the following commutative diagram:

(4)
${0}$${\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\vec{n}})}$${H^{1}(F_{\Sigma}/F_{\vec{n}},M_{s})}$${\mathcal{P}_{\Sigma,\underline{I}^{c}}(M_{s}/F_{\vec{n}})}$${0}$${\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{\infty})^{\Omega_{\vec{n}}}}$${H^{1}(F_{\Sigma}/F_{\infty},M_{s})^{\Omega_{\vec{n}}}}$${\mathcal{P}_{\Sigma,\underline{I}^{c}}(M_{s}/F_{\infty})^{\Omega_{\vec{n}}}}$$\scriptstyle{\rho_{\vec{n}}}$$\scriptstyle{h}$$\scriptstyle{\lambda}$$\scriptstyle{\Xi_{\vec{n}}}$

By the assumption (Tors), the fact that
$\operatorname{Gal}(F_{\infty,{\mathfrak{q}}}/F_{\mathfrak{q}})$ is a pro-$p$
group along with the orbit-stabilizer theorem implies that
$H^{0}(F_{\infty,{\mathfrak{q}}},M_{s})=0$ and hence $\ker(h)=0$ and
$\operatorname{coker}(h)=0$. Thus, we have that $\ker({\rho_{\vec{n}}})=0$ and
$\operatorname{coker}({\rho_{\vec{n}}})=\ker(\Xi_{\vec{n}})\cap\operatorname{Im}(\lambda)$
by the snake lemma. Thus, it suffices to prove the theorem for
$\ker(\Xi_{\vec{n}})$, which we now study one prime $v$ at a time. We show
that the $v$-component is almost always $0$, and then analyze the
$v$-component when it is not. To this end, we consider the commutative diagram

(5)
${0}$${H^{1}_{*}(F_{\vec{n},v},M_{s})}$${H^{1}(F_{\vec{n},v},M_{s})}$${\frac{H^{1}(F_{\vec{n},v},M_{s})}{H^{1}_{*}(F_{\vec{n},v},M_{s})}}$${0}$${H^{1}_{*}(F_{\infty,v},M_{s})^{\Omega_{\vec{n}}}}$${H^{1}(F_{\infty,v},M_{s})^{\Omega_{\vec{n}}}}$${\left(\frac{H^{1}(F_{\infty,v},M_{s})}{H^{1}_{*}(F_{\infty,v},M_{s})}\right)^{\Omega_{\vec{n}}},}$$\scriptstyle{\ell}$$\scriptstyle{g}$$\scriptstyle{p_{v}}$

where
$H^{1}_{*}(F_{\vec{n},v},M_{s})=\begin{cases}H^{1}_{I_{v}}(F_{\vec{n},v},M_{s})\text{
if }v={\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\},\text{ and}\\\
H^{1}_{\mathrm{ur}}(F_{\vec{n},v},M_{s})\text{ if not.}\end{cases}$

The case $v={\mathfrak{q}}$. When $v|p$, both $\ell$ and $g$ are isomorphisms;
$\ell$ is an isomorphism by definition, whereas $g$ is an isomorphism by the
inflation-restriction exact sequence and (Tors). Thus, the snake lemma implies
that $\ker(p_{v})=0$.

The case of archimedean $v$. Here, we have $\operatorname{coker}(\ell)=0$
[PR00, Sect A2.4], and also $\ker(g)=0$, since $v$ splits completely.

The case $v\nmid p$. Since $\operatorname{coker}(\ell)=0$, (this follows from
the hypothesis (Tors) and using inflation-restriction) we know that $\ker(g)$
surjects onto $\ker(p_{v})$ by the snake lemma. Now
$\ker(g)=H^{1}\big{(}F_{\infty,v}/F_{\vec{n},v},M_{s}^{G_{F_{\infty},v}}\big{)}$.
Since $v\nmid p$, $F_{\infty,v}$ is the unique unramified
$\mathbb{Z}_{p}$-extension of $F_{v}$. Hence
$\operatorname{Gal}(F_{\infty,v}/F_{\vec{n},v})$ is topologically generated by
a single element $\gamma_{n}$. For all but finitely many $s\in\mathbb{Z}$,
$M_{s}^{G_{F_{\vec{n},v}}}$ is finite for every $\vec{n}$; using Lemma 5.2
below, this implies that the order of $\ker(g)$ is bounded by that of
$M_{s}^{G_{F_{\infty},v}}/(M_{s}^{G_{F_{\infty},v}})_{\mathrm{div}}$.

∎

######  Lemma 5.2.

Suppose that $N$ is a $\mathbf{Z}_{p}$-module with an action of
$\Gamma\cong\mathbf{Z}_{p}$ so that $N^{\Gamma}$ is finite. Then the order of
$H^{1}(\Gamma,N)$ is bounded by that of $N/N_{\mathrm{div}}$.

###### Proof.

Fix a topological generator $\gamma$ of $\Gamma$. Since $\Gamma$ is pro-$p$
cyclic, we know that $H^{1}(\Gamma,N)\cong N/(\gamma-1)N$. Since
$N^{\Gamma}=\ker(N\xrightarrow{\gamma-1}N)$ is finite, $(\gamma-1)N$ must
contain the maximal divisible subgroup $N_{\mathrm{div}}$. ∎

######  Proposition 5.3.

Suppose that the signed Selmer group
$\operatorname{Sel}_{\underline{I}^{{c}}}(M/F_{\infty})$ is
$\Lambda(\Omega)$-cotorsion. Then the defining sequence

$0\rightarrow\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\rightarrow
H^{1}(F_{\Sigma}/F_{\infty},M^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty})\rightarrow
0$

is short exact.

###### Proof.

Suppose $\operatorname{Sel}_{\underline{I}^{{c}}}(M/F_{\infty})$ is cotorsion
as a $\Lambda(\Omega)$-module. Then for all but finitely many
$s\in\mathbb{Z}$,
$\big{(}\operatorname{Sel}_{\underline{I}^{{c}}}(M/F_{\infty})\otimes\chi^{s}\big{)}^{\Omega_{{n}}}\cong\big{(}\operatorname{Sel}_{\underline{I}^{{c}}}(M_{s}/F_{\infty})\big{)}^{\Omega_{{n}}}$
is finite for every ${n}$. Therefore, by Theorem 5.1, for all by finitely many
$s$, $\operatorname{Sel}_{\underline{I}^{c}}(M_{s}/F_{{n}})$ is finite for
every ${n}$. Hence for those $s$ and for all ${n}$, by [Gre99, Proposition
4.13], we obtain that the cokernel of the map

$f_{{n},s}:H^{1}(F_{\Sigma}/F_{{n}},M_{-s}^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M_{-s}^{*}/F_{{n}})$

is the Pontryagin dual of $H^{0}(F_{{n}},M_{s})$ (note that the local
components $H^{1}_{I_{\mathfrak{q}}}(F_{n,{\mathfrak{q}}},M^{*}_{-s})$ and
$H^{1}_{\mathrm{ur}}(F_{n,v},M^{*}_{-s})$ for $v\nmid p$ of
$\mathcal{P}_{\Sigma,\underline{I}}(M_{-s}^{*}/F_{{n}})$ are divisible groups
by [Dio23, Lemma 3.16] and [Pon20, Lemma 1.6]). By hypothesis (Tors),
$H^{0}(F,M)=0$ and this implies that $H^{0}(F_{\infty},M)=0$. Also $M_{s}\cong
M$ as a $\operatorname{Gal}(\overline{F}/F_{\mathrm{cyc}})$-module and hence
as a $\operatorname{Gal}(\overline{F}/F_{\infty})$-module. Therefore,
$H^{0}(F_{\infty},M_{s})=0$ and hence $H^{0}(F_{{n}},M_{s})=0$ for any ${n}$.
This gives that the maps $f_{{n},s}$ are surjective for all ${n}$ and for all
but finitely many $s\in\mathbb{Z}$. Passing to the direct limit of relative to
restriction maps, the surjectivity of $f_{{n},s}$ implies the surjectivity of
the following map:

$f_{\infty,s}:H^{1}(F_{\Sigma}/F_{\infty},M_{-s}^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M_{-s}^{*}/F_{\infty}).$

Twisting the map $f_{\infty,s}$ by $\chi_{|_{\Gamma}}^{-s}$ we obtain the map

$f_{\infty}:H^{1}(F_{\Sigma}/F_{\infty},M^{*})\rightarrow\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty}),$

which is hence a surjection. ∎

######  Corollary 5.4.

If the Bloch-Kato Selmer group $\operatorname{Sel}_{\mathrm{BK}}(M^{*}/F)$ is
finite then the Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ is
$\Lambda[[\Omega]]$-cotorsion.

###### Proof.

By [Pon20, Lemma 1.11],
$\operatorname{Sel}_{\mathrm{BK}}(M^{*}/F)=\operatorname{Sel}_{\underline{I}}(M^{*}/F)$.
By the control theorem 5.1,
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})^{\Omega}$ is finite and
hence by Nakayama’s lemma (see [BH97]) we obtain that
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ is
$\Lambda[[\Omega]]$-cotorsion. ∎

##  6\. Preservation of vanishing of $\mu$-invariants under congruences

The goal of this subsection is to deduce the vanishing of $\mu$-invariant
associated to a motive $\mathcal{M}^{\prime}$ from that of the motive
$\mathcal{M}$ in the case that their residual Galois representations are
isomorphic. We first remind the reader of the definition of $\mu$-invariants,
then state the theorem, before embarking on its proof.

###  6.1. Some facts about $\mu$-invariants

We recall Howson’s treatment of $\mu$-invariants in [How02].

Let $G$ be a pro-$p$ $p$-adic Lie group without $p$-torsion.

######  Definition 6.1.

[How02, equation 33] For a finitely generated $\mathbb{Z}_{p}[[G]]$-module
$N$, its $\mu$-invariant is

$\mu(N)=\sum_{i\geqslant
0}\operatorname{rank}_{\mathbb{F}_{p}[[G]]}\Big{(}p^{i}\big{(}N(p)\big{)}/p^{i+1}\Big{)},$

where $N(p)$ is the submodule of $N$ consisting of its elements annihilated by
some power of $p$.

Denote by $N[p]$ the $p$-torsion of $N$.

######  Lemma 6.2.

For $N$ as in the above definition, we have

$\operatorname{rank}_{\mathbb{Z}_{p}[[G]]}(N)=\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N/pN)-\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N[p]).$

Further, $\mu(N)=0$ is equivalent to

$\operatorname{rank}_{\mathbb{Z}_{p}[[G]]}(N)=\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N/pN).$

###### Proof.

The first statement is [How02, Cor. 1.10].

For the second statement, first suppose $\mu(N)=0$. Then in particular,

$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N(p)/pN(p))=0.$

But $\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N(p)/pN(p))=0$ is equivalent to
$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N[p])=0$ by [How02, equations 42
and 43].

Conversely, suppose $\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N[p])=0$. We
have $(p^{i}N)[p]\subseteq N[p],$ so that
$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}((p^{i}N)[p]))\leqslant\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N[p])=0.$
But we have again by [How02, equations 42 and 43] that
$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}((p^{i}N)[p])=0$ is equivalent to

$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(p^{i}N(p)/p^{i+1}N(p))=0.$

Thus, $\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(N[p])=0$ implies
$\operatorname{rank}_{\mathbb{F}_{p}[[G]]}(p^{i}N(p)/p^{i+1}N(p))=0$ for all
$i$, which is the same as $\mu(N)=0$. ∎

###  6.2. The Theorem

Let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ be motives with $G_{F}$-stable
$\mathbb{Z}_{p}$-lattices $T$ and $T^{\prime}$ inside the the $p$-adic
realizations satisfying the hypotheses (H.-T.), (Cryst.), (Tors.), (Fil.),
(Slopes). In this section, we make the following assumption.

(Cong.) $T/pT\cong T^{\prime}/pT^{\prime}$ as $G_{F}$-representations.

Let $\mathcal{X}:=\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ be the
Pontryagin dual of the Selmer group
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ and
$\mathcal{X}^{\prime}$ that of
$\operatorname{Sel}_{\underline{I}}(M^{\prime*}/F_{\infty})$. Recall that
$\Omega=\mathbf{Z}_{p}^{2}$.

######  Theorem 6.3.

Suppose that $\mathcal{X}$ and $\mathcal{X}^{\prime}$ have the same
$\mathbf{Z}_{p}[[\Omega]]$-rank $r$. Then $\mu(\mathcal{X})=0$ if and only if
$\mu(\mathcal{X^{\prime}})=0$.

Let us first sketch the proof. For the sake of readability, we let
$\operatorname{Sel}(M^{*})$ denote
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})$ and
$\operatorname{Sel}(M^{\prime*})$ denote
$\operatorname{Sel}_{\underline{I}}(M^{\prime*}/F_{\infty})$. We introduce
auxiliary Selmer groups
$\operatorname{Sel}^{\Sigma_{0}}(M^{*}[p])=\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}[p]/F_{\infty})$
(and similarly $\operatorname{Sel}^{\Sigma_{0}}(M^{\prime*}[p])$), where the
local condition is changed at certain ramified primes ($\Sigma$ stands for
‘set,’ and the subscript $0$ indicates ramification.)

The main idea is to set up two exact sequences and compare them via an
isomorphism $(I)$ of $\mathbf{Z}_{p}[[\Omega]]$-modules:

(6)
${0}$${\operatorname{Sel}(M^{*})[p]}$${\operatorname{Sel}^{\Sigma_{0}}(M^{*}[p])}$(cotorsion
$\mathbb{F}_{p}[[\Omega]]$-module)${0}$${\operatorname{Sel}(M^{\prime*})[p]}$${\operatorname{Sel}^{\Sigma_{0}}(M^{\prime*}[p])}$(cotorsion
$\mathbb{F}_{p}[[\Omega]]$-module)$\scriptstyle{(I)}$

Recall that $\mathcal{X}$ denotes the Pontryagin dual of
$\operatorname{Sel}(M^{*})$. Similarly, let $\mathcal{Y}$ and
$\mathcal{Y^{\prime}}$ be the Pontryagin duals of
$\operatorname{Sel}^{\Sigma_{0}}(M^{*}[p])$ and
$\operatorname{Sel}^{\Sigma_{0}}(M^{\prime*}[p])$. Dualizing the first exact
sequence gives

(7)
$\operatorname{rank}_{\mathbb{F}_{p}[[\Omega]]}\big{(}\mathcal{Y}\big{)}=\operatorname{rank}_{\mathbb{F}_{p}[[\Omega]]}\big{(}\mathcal{X}/p\mathcal{X}\big{)}.$

Recall we denote by $r$ the $\mathbf{Z}_{p}[[\Omega]]$-rank of $\mathcal{X}$.
If $\mu(\mathcal{X})=0$, then by Lemma 6.2,

$\operatorname{rank}_{\mathbb{F}_{p}[[\Omega]]}\big{(}\mathcal{X}/p\mathcal{X}\big{)}=r$

and hence by (7), $\mathcal{Y}$ has $\mathbb{F}_{p}[[\Omega]]$-rank $r$ as
well.

We now use (I) to reverse our steps: By (I), $\mathcal{Y^{\prime}}$ has
$\mathbb{F}_{p}[[\Omega]]$-rank $r$ as well. Dualizing the second exact
sequence results in an analogue of (7), which gives us

$\operatorname{rank}_{\mathbb{F}_{p}[[\Omega]]}\big{(}\mathcal{X^{\prime}}/p\mathcal{X^{\prime}}\big{)}=r.$

Since $\mathcal{X^{\prime}}$ was assumed to have rank $r$, we get

$\operatorname{rank}_{\mathbb{Z}_{p}[[\Omega]]}\big{(}\mathcal{X^{\prime}}\big{)}=\operatorname{rank}_{\mathbb{F}_{p}[[\Omega]]}\big{(}\mathcal{X^{\prime}}/p\mathcal{X^{\prime}}\big{)}=r.$

Again by Lemma 6.2, $\mu(\mathcal{X^{\prime}})=0$.

######  Corollary 6.4.

Suppose that $\mathcal{X}$ and $\mathcal{X}^{\prime}$ are
$\mathbf{Z}_{p}[[\Omega]]$-torsion. Then $\mu(\mathcal{X})=0$ if and only if
$\mu(\mathcal{X^{\prime}})=0$.

###### Proof.

This is the theorem with $r=0$. ∎

###  6.3. The auxiliary Selmer group, the isomorphism (I), and the exact
sequences (6)

Recall that what we need to do to make the proof work is to define the
auxiliary Selmer group, establish the isomorphism (I), and prove the exactness
in the diagram 6. To this end, we generalize Ponsinet’s construction, cf.
[Pon20, Section 3.3].

####  6.3.1. The auxiliary Selmer group

Let $\Sigma_{0}\subset\Sigma$ containing all the primes of ramification of
$M^{*}$ but neither the primes of $F$ dividing $p$ nor the archimedean places.
Let $\Sigma_{0}^{\prime}$ be the primes of $F_{\infty}$ above $\Sigma_{0}$.
Recall that ${\mathfrak{q}}$ is the unique prime above $p$ in $F_{\infty}.$
From [MR04, Lemma 3.5.3] and proofs analogous to [Pon20, Lemma 3.5], we know
that $H^{1}(F_{\infty,{\mathfrak{q}}},M^{*}[p])\cong
H^{1}(F_{\infty,{\mathfrak{q}}},M^{*})[p],$ so that

$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})[p]\subset
H^{1}(F_{\infty,{\mathfrak{q}}},M^{*}[p]).$

We set, just as Ponsinet did,

$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*}[p]):=H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*})[p].$

Thus, the following local condition is well-defined:

$\mathcal{P}_{\Sigma\backslash\Sigma_{0},\underline{I}}(M^{*}[p]/F_{\infty}):=\prod_{w\in\Sigma^{\prime}\backslash\Sigma_{0}^{\prime},w\nmid
p}H^{1}(F_{w,\mathrm{ur}},M^{*}[p])\times\prod_{{\mathfrak{q}}\mid
p}\frac{\bigoplus_{w\mid{\mathfrak{q}}}H^{1}(F_{\infty,w},M^{*}[p])}{H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*}[p])}.$

######  Definition 6.5.

We define the $\Sigma_{0}$-non-primitive $\underline{I}$-Selmer group of
$M^{*}[p]$ over $F_{\infty}$ as

$\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}[p]/F_{\infty}):=\ker\Big{(}H^{1}(F_{\Sigma}/F_{\infty},M^{*}[p])\rightarrow\mathcal{P}_{\Sigma\backslash\Sigma_{0},\underline{I}}(M^{*}[p]/F_{\infty})\Big{)}.$

We also use the abbreviated notation
$\operatorname{Sel}^{\Sigma_{0}}(M^{*}[p])$.

####  6.3.2. The isomorphism (I)

We would like to prove that as $\mathbb{F}_{p}[[\Omega]]$-modules,

$\operatorname{Sel}^{\Sigma_{0}}(M^{*}[p])\cong\operatorname{Sel}^{\Sigma_{0}}({M^{\prime}}^{*}[p]).$

under the hypothesis (Cong.). But (Cong.) implies that
$M^{*}[p]\cong{M^{\prime}}^{*}[p]$ as $G_{F}$-representations. Hence

$\displaystyle H^{1}(F_{\Sigma}/F_{\infty},M^{*}[p])$ $\displaystyle\cong
H^{1}(F_{\Sigma}/F_{\infty},{M^{\prime}}^{*}[p])\text{ and}$ $\displaystyle
H^{1}(F_{w,\mathrm{ur}},M^{*}[p])$ $\displaystyle\cong
H^{1}(F_{w,\mathrm{ur}},{M^{\prime}}^{*}[p]).$

Thus, all that remains is proving that

$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*}[p])\cong
H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},{M^{\prime}}^{*}[p]).$

We prove that their duals are isomorphic: The Tate-pairing (3) gives that the
Pontryagin dual of
$H^{1}_{I_{\mathfrak{q}}}(F_{\infty,{\mathfrak{q}}},M^{*}[p])$ is
$\operatorname{Im}(\operatorname{Col}_{T,I_{\mathfrak{q}}}^{\infty})/p.$ Thus,
our assertion follows from the two lemmas:

######  Lemma 6.6.

Assume (Cong.) holds. Then as $\mathbb{Z}_{p}[[\Omega_{p}]]$-modules,
$H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)/p\cong
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T^{\prime})/p.$

######  Lemma 6.7.

Assume (Cong.) holds. If $z\in
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)$ and $z^{\prime}\in
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T^{\prime})$ have the same image
under the isomorphism given in the above lemma, then the Coleman maps
$\operatorname{Col}_{T,i}^{\infty}(z)$ and
$\operatorname{Col}_{T^{\prime},i}^{\infty}(z^{\prime})$ are congruent modulo
$p$.

See subsection 6.3.4 for their proofs.

####  6.3.3. The exact sequences (6

) We prove exactness of the top sequence, as the second sequence is exact for
the same reasons. Recall that we let $\Sigma_{0}\subset\Sigma$ contain all the
primes of ramification of $M^{*}$ but neither the primes of $F$ dividing $p$
nor the archimedean places. Let $\Sigma_{0}^{\prime}$ be the primes of
$F_{\infty}$ above $\Sigma_{0}$.

We would like to compare the auxiliary Selmer group
$\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}[p]/F_{\infty})$ to the
($p$-torsion of the) $\Sigma_{0}$-non-primitive $\underline{I}$-Selmer group ,
which we define by

$\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})=\ker\Big{(}H^{1}(F_{\Sigma}/F_{\infty},M^{*})\rightarrow\mathcal{P}_{\Sigma\backslash\Sigma_{0},\underline{I}}(M^{*}/F_{\infty})\Big{)}.$

Comparing their local conditions gives a map

$\theta:\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\xrightarrow{\theta}\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty}).$

The map appearing in the exact sequence (6) is the $p$-torsion part of the map
$\theta$ (which we will denote by $\theta_{p}$, composed with the isomorphism
of the following lemma:

######  Lemma 6.8.

We have the isomorphism

(8)
$\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})[p]\cong\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}[p]/F_{\infty}).$

###### Proof.

We match the global term and the local conditions of the Selmer groups in
question, i.e. this isomorphism reduces to

  1. (1)

$H^{1}(F_{\Sigma}/F_{\infty},M^{*}[p])\cong
H^{1}(F_{\Sigma}/F_{\infty},M^{*})[p],$

  2. (2)

$H^{1}(F_{\infty,{\mathfrak{q}}},M^{*}[p])\cong
H^{1}(F_{\infty,{\mathfrak{q}}},M^{*})[p]$,

  3. (3)

$H^{1}(F_{w,\mathrm{ur}},M^{*}[p])\cong H^{1}(F_{w,\mathrm{ur}},M^{*})[p]$ for
any non-archimedean prime $w\in\Sigma_{0}^{\prime}$ not dividing $p$ or a
prime of ramification of $M^{*}$, where $F_{w,\mathrm{ur}}$ is the maximal
unramified extension of $F_{w}$.

But these isomorphisms are induced by the exact sequence

$0\rightarrow M^{*}[p]\rightarrow M^{*}\xrightarrow{p}M^{*}\rightarrow 0$

of $G_{F_{\infty}}$-modules, analogously to [Pon20, Lemma 3.5], see also
[MR04, Lemma 3.5.3].

∎

######  Lemma 6.9.

As a $\mathbb{Z}_{p}[[\Omega]]$-module, the Pontryagin dual of
$\operatorname{coker}(\theta)$ is torsion.

###### Proof.

By Proposition 5.3, we have the following fundamental diagram.

(9)
${0}$${\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})}$${H^{1}(F_{\Sigma}/F_{\infty},M^{*})}$${\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty})}$${0}$${\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})}$${H^{1}(F_{\Sigma}/F_{\infty},M^{*})}$${\mathcal{P}_{\Sigma\backslash\Sigma_{0},\underline{I}}(M^{*}/F_{\infty}).}$$\scriptstyle{\theta}$$\scriptstyle{=}$$\scriptstyle{\delta}$$\scriptstyle{h_{\Sigma_{0}}}$

By the snake lemma we have an injection
$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\hookrightarrow\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})$
with cokernel isomorphic to
$\ker(h_{\Sigma_{0}})\cap\operatorname{Im}(\delta)$, where $h_{\Sigma_{0}}$ is
the projection map on local conditions. (Note that
$\mathcal{P}_{\Sigma\backslash\Sigma_{0},\underline{I}}(M^{*}/F_{\infty})$ has
the same local components as
$\mathcal{P}_{\Sigma,\underline{I}}(M^{*}/F_{\infty})$ but removing the places
in $\Sigma_{0}$.) Now,

$\ker(h_{\Sigma_{0}})=\prod_{w\in\Sigma_{0}^{\prime}}\frac{H^{1}(F_{\infty,w},M^{*})}{H^{1}_{\mathrm{ur}}(F_{\infty,w},M^{*})}.$

Since $w\in\Sigma_{0}^{\prime}$, $w$ is not above $p$ and hence $F_{\infty,w}$
is the unique unramified $\mathbb{Z}_{p}$-extension of $F_{v}$ where $v$ is a
prime of $F$ below $w$ (see [LS20b, Section 5.2]). In this case
$H^{1}(F_{\infty,w},M^{*})$ is a cotorsion $\mathbb{Z}_{p}[[\Gamma]]$-module
(see [Gre89, Proposition 2]) and $H^{1}_{\mathrm{ur}}(F_{\infty,w},M^{*})$ is
trivial (see [PR95, Section A.2.4]). Hence $\ker(h_{\Sigma_{0}})$ is a
cotorsion $\mathbf{Z}_{p}[[\Omega]]$-module and therefore it is so for
$\ker(h_{\Sigma_{0}})\cap\operatorname{Im}(\delta)=\operatorname{coker}(\theta)$.
Hence the conclusion.

∎

Multiply the nonzero terms in the following exact sequence by $p$:

$0\rightarrow\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})\xrightarrow{\theta}\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})\rightarrow\operatorname{coker}(\theta)\rightarrow
0,$

Applying the snake lemma to this multiplication by $p$, we get the exact
sequence

$0\rightarrow\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})[p]\xrightarrow{\theta_{p}}\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}/F_{\infty})[p]\rightarrow\operatorname{coker}(\theta)[p]\rightarrow\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})/p.$

We can quotient out the first nonzero term and its image under $\theta_{p}$ to
see that $\operatorname{coker}(\theta_{p})$ injects into
$\operatorname{coker}(\theta)[p]$. Thus, the Pontryagin dual of
$\operatorname{coker}(\theta_{p})$ is a quotient of the dual of
$\operatorname{coker}(\theta)[p]$, which is $\mathbb{F}_{p}[[\Omega]]$-torsion
by the previous lemma. Hence the Pontryagin dual of
$\operatorname{coker}(\theta_{p})$ is also $\mathbb{F}_{p}[[\Omega]]$-torsion.
Thus, combining $\theta_{p}$ with the isomorphism (8), we obtain an injection

$\operatorname{Sel}_{\underline{I}}(M^{*}/F_{\infty})[p]\hookrightarrow\operatorname{Sel}_{\underline{I}}^{\Sigma_{0}}(M^{*}[p]/F_{\infty})$

with cokernel a cotorsion $\mathbb{F}_{p}[[\Omega]]$-module; this is exactly
what was required.

####  6.3.4. Congruence lemmas in Iwasawa algebras

Recall that ${\mathfrak{q}}$ is the unique prime above $p$ in $F_{\infty}.$ We
complete the outstanding proof of the two last lemmas in subsection 6.3.2,
which may be of independent interest.

######  Lemma 6.10.

Assume (Cong.) holds. Then as $\mathbb{Z}_{p}[[\Omega_{p}]]$-modules,
$H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)/p\cong
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T^{\prime})/p.$

###### Proof.

Let $\mathbf{A}_{\mathbb{Q}_{p}}^{+}$ be the ring $\mathbb{Z}_{p}[[\pi]]$
equipped with a semilinear action of the Frobenius $\varphi$ acting as the
absolute Frobenius on $\mathbb{Z}_{p}$ and on the uniformizer $\pi$ as
$\varphi(\pi)=(\pi+1)^{p}-1$ with a Galois action given by
$g(\pi)=(\pi+1)^{\chi(g)}-1$ where
$g\in\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})$.
Then the (cyclotomic) Wach module $N_{\mathbb{Q}_{p}}(T)$ is equipped with a
filtration

$\mathrm{Fil}^{i}(N_{\mathbb{Q}_{p}}(T))=\\{x\in
N_{\mathbb{Q}_{p}}(T),\varphi(x)\in(\varphi(\pi)/\pi)^{i}N_{\mathbb{Q}_{p}}(T)\\}.$

Since the Hodge-Tate weights of $T$ and $T^{\prime}$ are in $[0,1]$, under
(Cong.), from [Ber04, Théorème IV.1.1], we deduce that

(10) $N_{\mathbb{Q}_{p}}(T)/pN_{\mathbb{Q}_{p}}(T)\cong
N_{\mathbb{Q}_{p}}(T^{\prime})/pN_{\mathbb{Q}_{p}}(T^{\prime})$

as $\mathbf{A}_{\mathbb{Q}_{p}}^{+}$-modules and the isomorphism is compatible
with the filtration, the Galois action and the action of the Frobenius
operator $\varphi$. Since
$H^{1}_{\mathrm{Iw}}(F_{{\mathfrak{q}}}(\mu_{p^{\infty}}),T)\cong
N_{\mathbb{Q}_{p}}(T)^{\psi=1}$ as Galois modules, we have the congruence of
the first Iwasawa cohomology groups

$H^{1}_{\mathrm{Iw}}(F_{{\mathfrak{q}}}(\mu_{p^{\infty}}),T)/p\cong
H^{1}_{\mathrm{Iw}}(F_{{\mathfrak{q}}}(\mu_{p^{\infty}}),T^{\prime})/p.$

The proof follows by tensoring with the Yager module (cf. Section 2.3) and
taking the $\Delta$-invariance of

$H^{1}_{\mathrm{Iw}}(L_{\infty}(\mu_{p^{\infty}}),T)\cong
N_{L_{\infty}}(T)^{\psi=1}\cong
N_{\mathbb{Q}_{p}}(T)^{\psi=1}\widehat{\otimes}S_{L_{\infty}/\mathbb{Q}_{p}}\cong
H^{1}_{\mathrm{Iw}}(F_{\mathfrak{q}}(\mu_{p^{\infty}}),T)\widehat{\otimes}S_{L_{\infty}/\mathbb{Q}_{p}}.$

∎

In the last lemma, we use the identification
$\mathds{D}_{\mathrm{cris},{\mathfrak{q}}}(T)=N_{\mathbb{Q}_{p}}(T)/pN_{\mathbb{Q}_{p}}(T)$
and fix Hodge-compatible bases (see Section 2.2.1) of
$\mathds{D}_{\mathrm{cris},{\mathfrak{q}}}(T)$ and
$\mathds{D}_{\mathrm{cris},{\mathfrak{q}}}(T^{\prime})$ such that their images
are the same under the isomorphism (10).

######  Lemma 6.11.

Assume (Cong.) holds. If $z\in
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)$ and $z^{\prime}\in
H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T^{\prime})$ have the same image
under the isomorphism given in Lemma 6.10, then the Coleman maps
$\operatorname{Col}_{T,i}^{\infty}(z)$ and
$\operatorname{Col}_{T^{\prime},i}^{\infty}(z^{\prime})$ are congruent modulo
$p$.

###### Proof.

Recall that $L_{m}$ is the unramified $\mathbb{Z}_{p}$-extension of degree
$p^{m}$ over $\mathbb{Q}_{p}$ and $L_{m,cyc}$ is the cyclotomic
$\mathbb{Z}_{p}$-extension of $L_{m}$. We identify
$H^{1}_{\mathrm{Iw}}(F_{\infty,{\mathfrak{q}}},T)$ with
$H^{1}_{\mathrm{Iw}}(k_{\infty},T)\cong\varprojlim_{L_{m}}H^{1}_{\mathrm{Iw}}(L_{m,cyc},T)$
and note that we have the following commutative diagram.

${H^{1}_{\mathrm{Iw}}(k_{\infty},T)/p}$${H^{1}_{\mathrm{Iw}}(L_{m,cyc},T)/p}$${H^{1}_{\mathrm{Iw}}(k_{\infty},T^{\prime})/p}$${H^{1}_{\mathrm{Iw}}(L_{m,cyc},T^{\prime})/p}$
$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The horizontal arrows are the natural projection maps. The left vertical arrow
is the isomorphism from Lemma 6.10. The right vertical arrow is an isomorphism
which follows from [Pon20, Eq. (18) in Section 3.2]. Let
$z=\varprojlim_{L_{m}}z_{m}$ where $z_{m}\in H^{1}_{\mathrm{Iw}}(L_{m,cyc},T)$
and correspondingly define $z_{m}^{\prime}$. Our hypotheses imply that $z_{m}$
and $z_{m}^{\prime}$ have the same image under the isomorphism given by the
right vertical arrow of the commutative square given above.

From (1), recall that

(11)
$\operatorname{Col}_{T,i}^{\infty}((z_{m}))=(y_{L_{\infty}/\mathbb{Q}_{p}}\otimes
1)\circ(\varprojlim_{L_{m}}\operatorname{Col}_{T,L_{m},i}(z_{m})).$

By [Pon20, Proposition 3.4], $\operatorname{Col}_{T,L_{m},i}(z_{m})$ and
$\operatorname{Col}_{T^{\prime},L_{m},i}(z_{m}^{\prime})$ are congruent modulo
$p$. The result follows by taking inverse limits and tensoring with the Yager
module using (11). ∎

######  Remark 6.12.

Note that if we had assumed that
$\mathcal{X}_{\underline{I}^{c}}(M/F_{\infty})$ is a torsion
$\Lambda(\Omega)$-module, then the map $\delta$ will be a surjection (see
Proposition 5.3). So it poses a natural question of when can we relate the
torsionness of the module $\mathcal{X}_{\underline{I}^{c}}(M/F_{\infty})$ with
that of $\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$. The result is the
following.

If the abelian variety is polarized then
$\mathcal{X}_{\underline{I}^{c}}(M/F_{\infty})$ is a torsion
$\Lambda(\Omega)$-module if and only if
$\mathcal{X}_{\underline{I}}(M^{*}/F_{\infty})$ is a torsion
$\Lambda(\Omega)$-module [Dio23, Corollary 5.13].

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