Let $ p $ be an odd prime.
Let $ E $ be an elliptic curve defined over a quadratic imaginary field where $ p $ splits completely. Suppose $ E $ has supersingular reduction at primes above $ p $.
Under appropriate hypotheses, we extend the results of [17] to $ \bZ_p^{2} $-extensions.
We  define and study the fine double-signed residual Selmer groups in these settings.

We  prove that for two residually isomorphic elliptic curves, the vanishing of the signed $ \mu $-invariants of one elliptic curve implies the  vanishing of the signed $ \mu $-invariants of the other.
Finally, we  show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

§ INTRODUCTION

Consider two elliptic curves defined over $ \bQ $ with good ordinary reduction at prime $ p $ whose residual Galois representations are isomorphic.
Greenberg and Vatsal in [7] showed that the vanishing of the $ \mu $-invariant attached to the Pontryagin dual of the Selmer group over the cyclotomic $ \bZ_p $-extension  of one curve implies the vanishing of the $ \mu $-invariant for the other.
Their work uses an auxiliary Selmer group called nonprimitive dual Selmer group which has the same $ \mu $-invariant as the Pontryagin dual of the Selmer group.

For an elliptic curve $ E/\bQ $ with good supersingular reduction at $ p $, the Selmer group over the cyclotomic extension is no longer cotorsion.
Kobayashi in [12] defined the signed Selmer group over the cyclotomic $ \bZ_p $-extension using moderately stronger local conditions at $ p $ and showed that they are cotorsion over Iwasawa modules.

A similar approach has been taken in the study of elliptic curves with supersingular reduction for the signed Selmer groups.
The $ \mu $-invariants of the Pontryagin dual of the signed Selmer groups are referred to as the signed $ \mu $-invariants.
An analogue of Greenberg–Vatsal was proved by B. D. Kim in [11] for the signed $ \mu $-invariants over cyclotomic $ \bZ_p $-extension using nonprimitive dual Selmer groups.

Later, in [17] the authors used a new technique to work with residual representations of elliptic curves and they improved upon the results of Kim in [11].
They defined a new Selmer group called the fine residual signed Selmer group

and studied its structure as an Iwasawa module, in particular, the vanishing of the signed $ \mu $-invariants (cf. <cit.>).

In this paper, we follow the strategy of [17] to prove new results for the signed $ \mu $-invariants of $ \bZ_p^{2} $-extensions of quadratic imaginary fields.
However, the Iwasawa theory of $ \bZ_p^{2} $-extensions has additional technicalities compared to the cyclotomic  $ \bZ_p$-extensions.
We  define the fine signed residual Selmer groups in these settings (Definition <ref>).
In Proposition <ref> we show that these residual singed Selmer groups depend only on the isomorphism class of the residual Galois representation of elliptic curves.
Therefore, these groups provide a natural method to study congruences of elliptic curves.
We then relate the structure of the fine signed residual  Selmer groups to that of the signed Selmer groups as Iwasawa modules (Proposition <ref>).
Theorem <ref> relates the module structure of the fine signed residual Selmer groups to the vanishing of the signed $ \mu $-invariants.
We use this theorem to show that if two elliptic curves have isomorphic residual representations, then vanishing of the signed $\mu $-invariants for one implies the same for the other.
Furthermore, we relate these results to analogous results over cyclotomic extensions proved in [17] (Corollary <ref>).

Moreover, we prove in  that the Pontryagin dual of the signed Selmer groups and the classical Selmer group have no non-zero pseudo-null submodules (Theorem <ref>).

This is an important property for an Iwasawa module, since the structure of finitely generated modules over commutative Iwasawa algebras is known up to pseudo-isomorphism.
Hence, whenever such modules have no non-zero pseudo-null submodules we have a better understanding of their  structure.
To this this, we invoke Auslander–Buchsbaum–Serre formula and compute the depth of the signed Selmer groups by studying Galois cohomology of these groups.

This paper consists of five sections including this introductory section and it is organized as follows.

In section <ref>, we introduce preliminary definitions, notations, and assumptions.

Our goal in section <ref> is to prove some of the essential components of the  proof of our main results.

In section <ref>, we prove Theorem <ref> and record some important consequences of it.

Finally, in section <ref>  we use purely algebraic tools to compute the depth of signed Selmer groups and prove Theorem <ref> which states that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules.

§ PRELIMINARIES

Let $ E/L $ be an elliptic curve defined over a quadratic imaginary number field  $L/\bQ$.
Suppose $ p $ is an odd prime and let $ S_p $ denote the set of all primes of $ L $ over $ p $.
Denote the finite set of primes above $ p $ where $ E/L $ has supersingular reduction by $ \Sss \subseteq S_p $ and
let $ \Sbad $ denote the finite set of primes in $ L $ where $ E $ has bad reduction.
Let $ S $ be the disjoint union of the sets $ S_p $ and $ \Sbad $.
For any field extension $ \cL/L $, let $ S_p(\cL) $ be the set primes in $ \cL $ above the set $ S_p $.
Similarly, let $ S^{*}(\cL) $ for $ * \in \{ \emptyset, \ \T{bad}, \ \T{ss} \} $ denote the set of primes in $ \cL $ above the corresponding finite set $ S^{*} $.
Throughout this article we assume the following hypotheses, which we refer to as [Hyp1]Hyp 1.

* The prime $ p $ is odd.

* The number field $L/\bQ$ is a quadratic imaginary field extension.
Furthermore, assume that $ p $ splits completely in $ L/\bQ $. We denote the primes of $L$ over $ p $ by $\fp$ and $ \bar{\fp}$.
In particular, $ \fp \neq \bar{\fp} $ and $ L_\fp = L_\bfp = \bQ_p $.

*  The elliptic curve $E/L$ has good supersingular reduction at both primes above prime $ p $.
Therefore, the sets $  \Sss = S_p = \{ \fp, \bar{\fp} \} $.

* We have $ a_\fp=1+ N_{L/\bQ}(\fp) - \#\widetilde{E}_\fp(\cO_L/\fp)= a_\bfp = 0 $ where

[By Hasse's Inequality, this assumption is automatically satisfied for $ p \geq 5 $ and elliptic curves $ E $ defined over $ \bQ $, given that $ E $ has good supersingular reduction at $ p $.]

$ N_{L/\bQ} : L \to \bQ  $ is the usual norm map and $\widetilde{E}_\fp$ is the reduced curve modulo $ \fp $.

Let $ L_\cyc/ L $ denote the cyclotomic  $ \bZ_p $-extension of $ L $ and let $ L_\infty/ L $ denote the compositum of all $ \bZ_p $-extensions of $ L $.
Leopoldt's conjecture implies that $ \Gal(L_\infty/L) \cong \bZ_p^2 $.
Note that Leopoldt's conjecture is known for abelian extensions of rational numbers, and therefore it is not a hypothesis in this case (cf. Corollary 5.32 and Theorem 13.4 of [22]).
Let $ L^{S} $ be the maximal algebraic extension of $ L $ unramified outside of the primes inside $ S $ and $ G_{L}^{S} $ denote the Galois group of the extension $ L^{S}/L $. Note that $L_\infty \subset L^{S}$.

\begin{align*}
G_{\infty}^{S}:=\Gal(L^{S}/L_\infty),  \ G := \Gal(L_\infty/ L) \cong \bZ_p^{2}, \ \T{and }   \Gamma :=  \Gal(L_\cyc/ L) \cong \bZ_p.
\end{align*}

Given a prime $ w $ in $ S_p(L_\infty) $, by abuse of notation, let the prime below $  w$ in $ L_n $  be again denoted by $ w $.

It should be clear from the context whether the prime $ w $ is in $ L_n $ for some $ n\geq 0 $ or is in $L_\infty  $.
Finally, for an abelian group  $ M $ and an integer $ t \geq 1 $ we write $ M_t $ for the subgroup of elements of $ M $ that are annihilated by $ t $.
Moreover, for a prime $ p $, we define the $p$-primary torsion subgroup of $M$, which we denote by $ M_{p^{\infty}} $, as

\[
M_{p^{\infty}}:= \ds\cup_{i\geq 1} M_{p^{i}}.
\]

§.§ Plus and minus norm groups

Suppose $ K/\bQ_p $ is a finite unramified extension of $\bQ_p$.
Denote the cyclotomic (resp. the unramified) $ \bZ_p $-extension of $ K $ by $ K_\cyc $ (resp. $K^{\ur} $)
which is totally ramified.

For any integer $ r\geq 0 $, let $ K_r $ be the unique intermediate field extension of $ K_\cyc/K $ such that $ \Gal(K_r/K)= \bZ/p^{r}\bZ $ and for any integer $ l\geq 0 $, let $ K^{(l)} $ be the unique intermediate field extension of $ K^{\ur}/K $ such that $ \Gal(K^{l}/K)$ is isomorphic to $\bZ/p^{l}\bZ $.
Further, let $ K^{(l)}_r$ be $ K^{l} K_r $, so $ \Gal(K^{(l)}_r/K)$ is $\bZ/p^{l}\bZ \times \bZ/p^{r}\bZ  $ and

\begin{align*}%\label{eq:k-and-fk}
K^{\ur}_{\cyc} = K^{\ur}K_{\cyc} = \cup_{r,l\geq 0}K^{l} K_r,
\end{align*}

with $ \Gal(K^{\ur}_{\cyc}/K)$ equal to $\bZ_p^{2} $.

Let $ \mu_{p^{r}} $ denote the $ p^{r} $-th roots of unity and let $ \bK^{(l)}_r $ to be $ K^{(l)}(\mu_{p^{r+1}})  $ for $ l \geq 0 $ and $ r\geq -1 $.
For all $ l,r \geq 0 $, denote

\begin{align}\label{not:delta}
\Delta := \Gal(K(\mu_p)/K)= \Gal(\bK^{(0)}_{0}/K^{(0)}_{0})\cong \Gal(\bK^{l}_{r}/K^{l}_{r}).
\end{align}

In this article, with abuse of notation, we write
$ (\bK^{(l)}_r)^{\Delta} $ is equal to $K^{(l)}_{r} $.

In what follows, let $ \widehat{E}(\fm_\cL) $ by $ \widehat{E}(\cL) $ where $ \cL/\bQ_p $ is any local field over $ \bQ_p $ and $ \fm_\cL $ is the maximal ideal of the valuation ring  of $ \cL $.
Denote the ring of integers of $ \cL  $ by $\cO_\cL   $.

Suppose $ E/\bQ_p $ is an elliptic curve  and let $\widehat{E}  $ denote the formal group over $ \bZ_p $

[For details about formal groups of elliptic curves see <cit.>].

associated to the minimal model of $ E $ over $ \bQ_p $.

Consider the trace map

\begin{align*}
\Tr^{r}_{m+1} : {E}(\bK^{(l)}_r) &\to {E}(\bK^{(l)}_{m+1})\\
P &\mapsto \sum_{\sigma \in \Gal(\bK^{(l)}_r/\bK^{(l)}_{m+1})} \sigma(P)
\end{align*}

where $ l,m $, and $ r $ are non-negative integers.
Then Kobayashi [12] used these trace maps to define the plus and minus norm groups.

Following <cit.>,

define the plus and minus norm groups as

\begin{align*}
\widehat{E}^{\pm}(\bK^{(l)}_r) & := \left\{ P \in \widehat{E}(\bK^{(l)}_r) \mid  \Tr^{r}_{m+1} (P) \in  \widehat{E}(\bK^{(l)}_m), -1 \leq m \leq r-1 \T{ s.t. } (-1)^{m}= \pm 1 \right\},%\label{eq:traceEhat}
\\
E^{\pm}(\bK^{(l)}_r) & := \left\{ P \in E(\bK^{(l)}_r) \mid  \Tr^{r}_{m+1} (P) \in  E(\bK^{(l)}_m),  -1 \leq m \leq r-1 \T{ s.t. } (-1)^{m}= \pm 1\right\},\\
\widehat{E}^{\pm}(K^{(l)}_r) &:= \left( \widehat{E}^{\pm}(\bK^{(l)}_r) \right)^{\Delta}, \T{ and } E^{\pm}(K^{(l)}_r) := \left( E^{\pm}(\bK^{(l)}_r) \right)^{\Delta},
\end{align*}

\[
\widehat{E}^{\pm}(\bK^{(\ur)}_\cyc) := \bigcup_{r,l\geq 0} \widehat{E}^{\pm}(\bK^{(l)}_r), \T{ and } E^{\pm}(\bK^{\ur}_\cyc) := \bigcup_{r,l\geq 0} \widehat{E}^{\pm}(\bK^{(l)}_r).
\]

Suppose $ E/\bQ_p $ is an elliptic curve with good reduction at $ p $ and such that $ a_p=0 $.

Then, by Proposition 8.7 of [12], for all $ l,r \geq 0 $, $ \widehat{E}(\bK^{(l)}_{r}) $ has no non-trivial $ p $-torsion points.
This implies that

\begin{align}\label{eq:Ep=0}
E(\bK^{(l)}_{r})_{p}=E(K^{(l)}_{r})_{p} = \{ 0 \}.
\end{align}

§.§ Signed Kummer maps

Suppose $ L $ and $ S $ are as [Hyp1]Hyp 1.
For any $ n \geq 0 $, let $ L_n $ be the unique sub-extension of $ L_\infty/L $ such that

\begin{align}\label{def:Ln}
\Gal(L_n/L) \cong (\bZ/p^{n}\bZ)^{2}, \quad L= L_0 , \quad L_\infty := \ds\cup_{n\geq 0} L_n.
\end{align}

For simplicity, let $ G_{n}^{S}$ denote $\Gal(L^{S}/L_n) $.

Note that if  $ w|v $ is a prime of $  L_\infty $ not above $ p $, then $ L_{\infty,w} $ is the unique unramified $ \bZ_p $-extension (cf. for example <cit.>) of the local field $ L_v $.

Let $ \cL \in \{ K^{(l)}_{r}, \bK^{(l)}_{r}, L_{n} \} $ and $ \cG \in \{ G_{K^{(l)}_{r}}, G_{\bK^{(l)}_{r}}, G_n^{S}  \} $.
For any integer $ t \geq 0 $, there is a short exact sequence of $ \cG  $-modules

\[
0 \longrightarrow
E(\cL)/p^{t}E(\cL) \xrightarrow{\kappa^{p^{t}}_{\cL}}
H^{1}(\cG, E_{p^{t}}) \longrightarrow
H^{1}(\cG, E)_{p^{t}} \longrightarrow
\]

where the map  $ \kappa^{p^{t}}_{\cL} $ is the Kummer map for $ E_{p^{t}} $ over $ \cL $.
Taking the direct limit of the above sequence we obtain the following

\[
0 \longrightarrow
E(\cL) \otimes \bQ_p/\bZ_p \xrightarrow{\kappa^{p^{\infty}}_{\cL}}
H^{1}(\cG, E_{p^{\infty}}) \longrightarrow
H^{1}(\cG, E)_{p^{\infty}} \longrightarrow
\]

For $v \in S$, we define the local condition

\begin{align}\label{def:Jv(Ln)}
J_v(E_{p^{\infty}}/L_n) :=  \displaystyle\bigoplus_{w \mid v} H^1(L_{n,w},E)_{p^{\infty}}\cong
\displaystyle\bigoplus_{w \mid v} \frac{H^1 (L_{n,w},E_{p^\infty})}{E(L_{n,w})  \otimes \bQ_p/\bZ_p},
\end{align}

where the isomorphism is due to the Kummer map (cf. <cit.>).

The classical $ p^{\infty} $-Selmer group of the elliptic curve $ E $ over $ L_n $ is defined by the following sequence

\begin{equation}\label{def:Selmer with J_n}
0 \longrightarrow
\Sel(E_{p^{\infty}}/L_n) \rightarrow H^1(L^{S}/L_n, E_{p^\infty}) \xrightarrow{\lambda_n}  \displaystyle\bigoplus_{v \in S}J_v(E_{p^{\infty}}/L_n),
\end{equation}

where  $\lambda_n$ consists of restriction maps coming from Galois cohomology.

By definition of the plus/minus norm groups, there are inclusions

\[
E^{\pm}(L_{n,w}) \hooklongrightarrow E(L_{n,w}),\T{ and }  \widehat{E}^{\pm}(L_{n,w})  \hooklongrightarrow \widehat{E}(L_{n,w}),
\]

where  $ w $ is a prime in $ L_n $ above $ p $,  $ \fm_n $ is the maximal ideal of the ring of integers of $L_{n,w}(\mu_{p})  $.

By Lemma 3.4 of [17], the above maps remain injective after applying the functor $ - \ot \ \bZ/p^{t}$ for any $ t \geq 1 $

\[
\begin{tikzcd}
E^{\pm}(L_{n,w})/ p^{t} E^{\pm}(L_{n,w}) \arrow[r, hook] & E(L_{n,w})/ p^{t} E(L_{n,w}),
\end{tikzcd}
\]

where $ w $ is a prime in $ L_n $ above $ p $.
The Kummer map $ \kappa^{p^{t}}_{\cL} $ induces the following injective map

\begin{equation}\label{signedkummermaps}
\begin{tikzcd}
\kappa^{\pm,p^{t}}_{L_{n,w}} \ : \ E^{\pm}(L_{n,w})/ p^{t} E^{\pm}(L_{n,w}) \arrow[r, hook] & H^{1}(L_{n,w}, E_{p^{t}}).
\end{tikzcd}
\end{equation}

Similarly, for any $ n,t \geq 1  $, there is an injection

\[
\begin{tikzcd}
\kappa^{\pm,p^{t}}_{L_{n,w}} \ : \ \widehat{E}^{\pm}(L_{n,w})/ p^{t} \widehat{E}^{\pm}(L_{n,w}) \arrow[r, hook] & H^{1}(L_{n,w}, \widehat{E}_{p^{t}}).
\end{tikzcd}
\]

Refer to the maps $ \kappa^{\pm,p^{t}}_{L_{n,w}} $ as the signed Kummer maps  for $ E_{p^{t}} $ over $ L_{n,w} $.

We may pass the signed Kummer maps $ \kappa^{\pm,p^{t}}_{L_{n,w}} $ to the limit over $ t $ to obtain the exact sequences

\[
\begin{tikzcd}
0 \rightarrow
E^{\pm}(L_{n,w})\otimes \bQ_p/\bZ_p \xrightarrow{\kappa^{\pm,p^{\infty}}_{L_{n,w}}}
H^{1}(L_{n,w}, E_{p^{\infty}}) \rightarrow
H^{1}(L_{n,w}, E_{p^{\infty}}) /\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}} \rightarrow
\end{tikzcd}
\]

\[
\begin{tikzcd}
0 \rightarrow
\widehat{E}^{\pm}(L_{n,w})\otimes \bQ_p/\bZ_p \xrightarrow{\kappa^{\pm,p^{\infty}}_{L_{n,w}}}
H^{1}(L_{n,w}, \widehat{E}_{p^{\infty}}) \rightarrow
H^{1}(L_{n,w}, \widehat{E}_{p^{\infty}}) / \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}} \rightarrow
\end{tikzcd}
\]

§.§ Selmer group, signed Selmer groups, and fine signed residual Selmer groups

Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1 and let $ L_n/ L $ be as in (<ref>).

If $ w  $ belongs to $  \Sss(L_n) = S_p(L_n) $ then $ w| \fp $ or $ w| \bfp $, by [Hyp1]Hyp 1 part [allss](iii).

Denote a prime in $ L_n $ which lies over $ \fp $ (resp. $ \bfp $) by $ \fq $ (resp. $ \bfq $).

Similarly, if $  w  \in S_p(L_\infty) $ and $ w |\fp $ (resp. $  w |\bfp $), denote $ w $ by $ \fq $ (resp. $ \bfq $).
Finally, if $ \fq  $ (resp. $ \bfq  $) is in $ L_\infty  $, denote its restriction to $ L_n $ again by $ \fq $ (resp. $ \bfq $).
It should be clear from the context whether $ \fq  $ and $ \bfp $ represent primes in $ L_n $ or in $ L_\infty $.

Following [17], for $ v \in S $ define the local cohomological groups

\begin{equation}\label{eq:K_v pm}
\begin{dcases}
\bigoplus_{w|\fl} H^{1}(L_{n,w}, E_{p}) & \text{if}\ v=\fl \in \Sbad ,\\
\bigoplus_{\fq | \fp } H^{1}(L_{n,\fq}, E_{p}) / \Image \ \kappa^{\pm,p}_{L_{n,\fq}} & \text{if}\ v=\fp \in \Sss,\\
\bigoplus_{\bfq | \bfp } H^{1}(L_{n,\bfq}, E_{p}) / \Image \ \kappa^{\pm,p}_{L_{n,\bfq}} & \text{if}\ v=\bfp \in \Sss.
\end{dcases}
\end{equation}

\begin{equation}\label{eq:J_v pm}
\begin{dcases}
\bigoplus_{w|\fl} H^{1}(L_{n,w}, E_{p^{\infty}}) & \text{if}\ v=\fl \in \Sbad, \\
\bigoplus_{\fq | \fp } H^{1}(L_{n,\fq}, E_{p^{\infty}}) / \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,\fq}} & \text{if}\ v=\fp \in \Sss,\\
\bigoplus_{\bfq | \bfp } H^{1}(L_{n,\bfq}, E_{p^{\infty}}) / \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,\bfq}} & \text{if}\ v=\bfp \in \Sss.
\end{dcases}
\end{equation}

Here, when $ v \in \Sss $, the sign of $ ^{\pm}\widetilde{K}_v(E_{p}/L_n) $ (resp. $ J_v^{\pm}(E_{p^{\infty}}/L_n) $) agrees with the choice of the sign of the Kummer map $ \kappa^{\pm,p} $ (resp. $  \kappa^{\pm,p^{\infty}} $) in the direction of $ \fq|\fp $ or in the direction of $ \bfq|\bfp $.

For instance,

\begin{equation*}
^{+}\widetilde{K}_{\fp}(E_{p}/L_n)=\bigoplus_{\fq | \fp } H^{1}(L_{n,\fq}, E_{p}) / \Image \ \kappa^{+,p}_{L_{n,\fq}}.
\end{equation*}

When $ v \in \Sbad $, the sign choice of the sign does not matter.

Similarly, when $ v \in \Sbad $ then $ J_v^{\pm}(E_{p^{\infty}}/L_n) $ coincides with $ J_v(E_{p^{\infty}}/L_n) $ for any choice of the sign.
The following definition is the analogue of <cit.> over $ \bZ_p^{2} $-extensions.

Let $ n\geq 0 $
For every intermediate field $L_n $ in the tower $ L_\infty/L $,  let $ ^{\pm}\widetilde{K}_v(E_{p}/L_n) $ be as (<ref>) for any $ v \in S $.
Define the fine signed residual Selmer group $ \cR^{\pm/\pm}(E_p/L_n) $ of $ E_p $ over $ L_n  $ by:

\begin{align*}
\cR^{\pm/\pm} (E_p/L_n) := \ker \left( H^{1}(G^{S}_n,E_p) \to
\displaystyle\bigoplus_{v \in S}
{^{\pm}\widetilde{K}_v(E_{p}/L_n)} \right).
\end{align*}

The choice of the first sign of $ \cR^{\pm/\pm} (E_p/L_n) $  is in accordance with the choice of $ ^{+}\widetilde{K}_{\fp}(E_{p}/L_n) $ or $ ^{-}\widetilde{K}_{\fp}(E_{p}/L_n) $ and the second sign is in accordance with the choice of $ ^{+}\widetilde{K}_{\bfp}(E_{p}/L_n) $ or $ ^{-}\widetilde{K}_{\bfp}(E_{p}/L_n)$.

Here, we use the notation $ \cR^{\pm/\pm} (E_p/L_n) $ for convenience.
It unifies notation for any of the four possibilities: "$  \cR^{+/+} (E_p/L_n) $, or $ \cR^{+/-} (E_p/L_n) $, or $ \cR^{-/+} (E_p/L_n) $, or $ \cR^{-/-} (E_p/L_n) $".
Similar notation is used throughout the article for simplicity.

For example,

\begin{align*}
\cR^{+/-} (E_p/L_n)= \ker \left( H^{1}(G^{S}_n,E_p) \to
\bigoplus_{\fl \in \Sbad} {^{\pm}}\widetilde{K}_v(E_{p}/L_n)
\op {^{+}}\widetilde{K}_\fp(E_{p}/L_n)
\op {^{-}}\widetilde{K}_\bfp(E_{p}/L_n)  \right).
\end{align*}

For every $ n\geq 0$ let $ J_v^{\pm}(E_{p^{\infty}}/L_n) $ be as (<ref>) for any $ v \in S $.
Define the signed Selmer groups  $ \Sel^{\pm/\pm}(E_{p^{\infty}}/L_n) $ of $ E_{p^{\infty}} $ over the intermediate field $L_n $ in the tower $ L_\infty/L $ as follows

\begin{align*}
\Sel^{\pm/\pm} (E_{p^{\infty}}/L_n) := \ker \left( H^{1}(G^{S}_n,E_{p^{\infty}}) \to
\displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_n) \right).
\end{align*}

The first sign of $ \Sel^{\pm/\pm} (E_{p^{\infty}}/L_n) $  corresponds to the choice of $ J_\fp^{+}(E_{p^{\infty}}/L_n) $ or $ J_\fp^{-}(E_{p^{\infty}}/L_n) $ and
the second sign corresponds to the choice of $ J_\bfp^{+}(E_{p^{\infty}}/L_n) $ or $ J_\bfp^{-}(E_{p^{\infty}}/L_n) $.

Suppose $ S\pr $ is a finite set of primes containing set $ S $.
Then for any prime $ v \in S\pr \backslash S $, define the local condition $ J_v^{\pm}(E_{p^{\infty}}/L_n) $ or ${^{\pm}}\widetilde{K}_v(E_{p}/L_n)  $ the same way we did for the bad primes in $ \Sbad $.

We note that adding a finite set of primes to the set $ S $ does not change these Selmer groups (Chapter 1, Section 1.7 of [2]).

Let $ G_n := \Gal(L_n/L) $.
Then,  there is an exact sequence of $ \Lambda(G_n) $-modules (resp. $ \Omega(G_n) $-modules):

\begin{equation}\label{eq:signed Selmer with J_v pm over L_n}
0 \longrightarrow
\Sel^{\pm/\pm} (E_{p^{\infty}}/L_n) \longrightarrow H^1(G_n^{S}, E_{p^{\infty}}) \longrightarrow \displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_n)
\end{equation}

\begin{equation}\label{eq:signed residual Selmer with J_v pm over L_n}
0 \longrightarrow
\cR^{\pm/\pm} (E_p/L_n) \longrightarrow H^1(G_n^{S}, E_{p}) \longrightarrow  \displaystyle\bigoplus_{v \in S}
\end{equation}

By taking the direct limit over the intermediate field extensions $L_n$ of the exact sequences (<ref>), (<ref>), and (<ref>) we obtain

\begin{align}
0 \longrightarrow
\Sel(E_{p^{\infty}}/L_\infty) \longrightarrow H^1(G_{\infty}^{S}, E_{p^\infty}) \xrightarrow{\lambda_\infty}  \displaystyle\bigoplus_{v \in S}J_v(E_{p^{\infty}}/L_\infty),\label{eq:Selmer infty}\\
0 \longrightarrow
\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty) \longrightarrow H^1(G_{\infty}^{S}, E_{p^{\infty}}) \xrightarrow{\xi^{\pm/\pm}}  \displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_\infty),\label{eq:signed Selmer infty with J_v pm}\\
0 \longrightarrow
\cR^{\pm/\pm} (E_p/L_\infty) \longrightarrow H^1(G_{\infty}^{S}, E_{p}) \xrightarrow{\xi_p^{\pm/\pm}}  \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}. \label{eq:signed residual Selmer infty with k_v pm}
\end{align}

The above exact sequences   (<ref>), (<ref>), and (<ref>) have a natural $ G $-action.
This action gives these Selmer groups a module structure over Iwasawa algebras, which we now introduce.

The Iwasawa algebra of $ \cG $, denoted by $ \Lambda(\cG) $ (resp. $ \Omega(\cG) $), to be the completed group algebra of $ \cG $ over $ \bZ_p $ (resp. $ \bF_p $).
That is,

\[
\Lambda(\cG):=\bZ_{p}[[\cG]]=\varprojlim_{\cN \subset \cG} \bZ_{p}[\cG/\cN], \  (\T{\textit{resp.} } \Omega(\cG):=\bF_{p}[[\cG]]=\Lambda(\cG)/p\Lambda(\cG)),
\]

where $\cN$ runs over open normal sub-groups of $\cG$.

In this article, we deal with $\cG \cong \bZ_p^n$ from some $ n  \in \{1 , 2\} $.
When the group $  \cG = G \cong \bZ_p^2$  then $ n = 2 $  and when we have $   \cG = \Gamma \cong \bZ_p$ then  $ n=1 $.
The Iwasawa algebra $\Lambda(\cG)$ (resp. $ \Omega(\cG) $) is isomorphic to the ring of formal power series $\bZ_p[[T_1,\cdots,T_n]]$ (resp. $\bF_{p}[[T_1,\cdots,T_n]]$) with indeterminate variables $T_1,\cdots,T_n$.
Therefore $\Lambda(\cG)$ is a commutative, regular local ring of Krull dimension $n+1$.

Moreover, $\Lambda(\cG)$ admits a nice dimension theory for its finitely generated modules[For information on the ring-theoretic properties of Iwasawa algebras we suggest Ardakov's survey paper <cit.>.].

Note that (<ref>) and (<ref>) are exact sequences of $ \Lambda(G) $-modules and (<ref>) is an exact sequence of $ \Omega(G) $-modules.

Let $\mathfrak{X}(E_{p^{\infty}}/L_\infty)$ denote the Pontryagin dual of $\Sel(E_{p^{\infty}}/L_\infty) $.
It is important to note here that $\mathfrak{X}(E_{p^{\infty}}/L_\infty)$ is conjectured to have positive rank, and therefore it is not a torsion $ \Lambda(G) $-module.
This is shown over the cyclotomic $\bZ_p$-extensions (cf. <cit.>).
For a Galois extension with a pro-$ p $  Galois group without $ p $-torsion which contains the cyclotomic  $\bZ_p$-extension, this is conjectured to hold (cf.  [18]).

However, the Pontryagin dual of the signed Selmer group $\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)   $, which we denote by $\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)$, is conjectured to be torsion as a $ \Lambda(G) $-module. For a more generalized version of this see Conjecture 4.11 of [13].

We denote

\[
\mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty):=\mu_G(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)),
\]

and we call them the signed $ \mu $-invariants.

Suppose that $ M $ is a discrete cofinitely generated $\Lambda(\cG)$-module and let $ M\Pd $ denote its Pontryagin dual.
Then, there is an isomorphism

$ (M_p)\Pd \cong M\Pd / p M\Pd $

and the  inequality

\begin{align}\label{ineq:coranks}
\T{corank}_{\Lambda(\cG)} (M) \leq \T{corank}_{\Omega(\cG)} (M_p).
\end{align}

The inequality (<ref>) becomes an equality exactly when the $\mu$-invariant of $M\Pd$ is zero.

§ LOCAL AND GLOBAL COHOMOLOGY CALCULATIONS

Throughout this section, we assume that $ E/L $ is an elliptic curve satisfying [Hyp1]Hyp 1 and $ L_n/L $ is as defined in equation (<ref>) for any $ n \geq 0 $.
Let $ \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) $ denote the Pontryagin dual of the fine signed residual Selmer group $ \cR^{\pm/\pm}(E_{p}/L_\infty) $ defined in Definition <ref>.

Our goal in this article is to show that if $ E_1/L $ and $ E_2/L $ are two elliptic curves satisfying [Hyp1]Hyp 1 and they are such that their residual Galois representations $ (E_1)_{p} $ and $(E_2)_{p}  $ are isomorphic, then

\[
\mu_G^{\pm/\pm}((E_{1})_{p^{\infty}}/L_\infty) = 0 \iff
\mu_G^{\pm/\pm}((E_{2})_{p^{\infty}}/L_\infty) = 0
\]

given that $ \mathcal{Y}^{\pm/\pm}((E_{j})_{p}/L_\infty) $  is a torsion $ \Omega(G) $-module for $ j \in \{1,2\} $.

§.§ Comparison of the local and global cohomology groups

Let $ n\geq 0 $ and let $ L_n/L $ be as in  (<ref>).

For convenience, given a local field  $ \cK $, let $ G_{\cK} $ denote the Galois group of the extension $ \overline{\cK}/\cK $ where $ \overline{\cK} $ denotes the separable closure of $ \cK $.

Suppose $ \cG \in \{ G_n^{S}, G_{L_{n,w}} \} $ where $ w \in S(L_n) $.
There is a short exact sequence

\begin{equation}\label{eq:step-to-define-psi}
0 \longrightarrow
E(\cL)_{p^{\infty}}/p E(\cL)_{p^{\infty}} \longrightarrow
H^{1}(\cG, E_p) \xrightarrow{\psi_{\cG,n}}
H^{1}(\cG, E_{p^{\infty}})_p  \longrightarrow
\end{equation}

Recall that $ L_{n,w}= K^{(l)}_r $ for some $ l,r \geq 0 $. Equation (<ref>) tells us that  $ 	E(L_{n,w})_p = E(L_{n})_p =\{ 0 \}$.
This means that the first term in the short exact sequence (<ref>) vanishes.
Therefore for any $ n \geq 0 $,

the map $ \psi_{\cG,n}  $ is an isomorphism.
It remains to investigate $ \psi_{\cG,n}  $ when $  \cG= G_{L_{n,w}} $ for some  $ w \in \Sbad(L_n)  $.
Note that

$ \ker(\psi_{\cG,n})= E(L_{n,w})_{p^\infty}/p E(L_{n,w})_{p^\infty} $

has a finite $ \bF_p $-dimension

and by <cit.>

\[
\dim_{\bF_{p}}(\ker(\psi_{\cG,n}))=\dim_{\bF_{p}}(E(L_{n,w})_{p}) \leq \dim_{\bF_{p}}(E(\overline{L_{w}})_{p}) =2.
\]

Passing the exact sequence (<ref>) to direct limit, define the surjective map

\begin{equation}\label{eq:define-psi-infty}
\psi_{\cG,\infty} : H^{1}(\cG, E_p) \twoheadrightarrow H^{1}(\cG,E_{p^{\infty}})_p,
\end{equation}

where $ \cG \in \{ G_\infty^{S}, G_{L_{\infty,w}} \} $ for $ w \in S(L_\infty) $ and
$ G_\infty^{S} := \Gal(L_\infty^{S}/L_\infty) $.

Let $ \cG \in \{ G_\infty^{S}, G_{L_{\infty,w}} \} $ for $ w \in S(L_\infty) $ and let $ \psi_{\cG,\infty} $ be the map described in (<ref>).

*  If $ \cG \in \{ G_\infty^{S}, G_{L_{\infty,w}} \} $ for $ w \in S_p(L_\infty) $, then $ \psi_{\cG,\infty} $ is an isomorphism.

*  If $  \cG= G_{L_{\infty,w}} $ for  $ w \in \Sbad(L_\infty)  $, then
$ \dim_{\bF_{p}}(\ker(\psi_{\cG,\infty})) \leq 2 $.

The proof is similar to Proposition 4.1 in [17].

Suppose $ w \in S(L_\infty)  $ and assume $ w |\fp $ (resp. $ w|\bfp $).
Denote $ w$  by $ \fq $ (resp. by $ \bfq $) and the isomorphism  $ \psi_{\cG,\infty} $ where $ \cG = \Gal(\overline{L_{\infty,\fq}}/L_{\infty,\fq}) $ by $ \psi_{\fq,\infty} $ (resp. $ \psi_{\bfq,\infty} $).

Finally, when $  w \in \Sbad(L_\infty)$ then $L_{\infty,w} = L_{\cyc,w}  $ and $ \cG = \Gal(\overline{L_{\infty,w}}/L_{\infty,w}) $.
Denote the surjective map $ \psi_{\cG,\infty} $ by $  \psi_{w,\infty}$.

Let $  \fq $  (resp. $ \bfq $) be a prime in $ S_p(L_\infty) $ above $ \fp $ (resp. $ \bfp $). Then,	 the isomorphism $ \psi_{\fq,\infty} $ (resp. $ \psi_{\bfq,\infty}$) induces an isomorphism

\begin{equation*}\label{eq:define-psi-infty-fq}
\psi_{\fq,\infty}^{\pm} :
H^{1}(L_{\infty,\fq}, E_p)/ \Image \left(\kappa^{\pm,p}_{L_{\infty,\fq}}\right) \xrightarrow{\cong} \left(H^{1}(L_{\infty,\fq},E_{p^{\infty}})/ \Image \left(\kappa^{\pm,p^{\infty}}_{L_{\infty,\fq}}\right)\right)_p
\end{equation*}

\begin{equation*}\label{eq:define-psi-infty-bfq}
\psi_{\bfq,\infty}^{\pm} :
H^{1}(L_{\infty,\bfq}, E_p)/ \Image \left(\kappa^{\pm,p}_{L_{\infty,\bfq}}\right) \xrightarrow{\cong} \left(H^{1}(L_{\infty,\bfq},E_{p^{\infty}})/ \Image \left(\kappa^{\pm,p^{\infty}}_{L_{\infty,\bfq}}\right)\right)_p.
\end{equation*}

The proof is similar to <cit.> .

§.§ Fine signed residual Selmer group and residual representations

This section explains why the fine signed residual Selmer group only depends on the residual Galois representation $ E_p $.
To do this, we show that the local conditions $  ^{\pm}\widetilde{K}_v(E_{p}/L_\infty)$, which define the fine signed residual Selmer group (cf. Definition <ref>), only depend on the residual Galois representation (cf. Proposition <ref>).
Suppose $ \cL$ is an algebraic extension of $ \bQ_p $.
By Lemma 3.3 of [17], there exists an exact sequence

\[
\begin{tikzcd}\label{exact:lemma3.3-suj-fil}
0 \arrow[r] & \widehat{E}(\cL) \arrow[r] & E(\cL) \arrow[r] & D \arrow[r] & 0,
\end{tikzcd}
\]

where $D  $ is a finite group of order prime-to-$ p $.

In particular, for any $ n \geq 0 $ and any prime $ w \in \Sss(L_n)=S_p(L_n) $,

\[
\widehat{E}^{\pm}(L_{n,w}) \ot \bQ_p/\bZ_p \cong E^{\pm}(L_{n,w}) \ot \bQ_p/\bZ_p.
\]

If $ w $ is a prime in $ \Sss(L_\infty) $, then

$ \widehat{E}^{\pm}(L_{\infty,w}) \ot \bQ_p/\bZ_p \cong E^{\pm}(L_{\infty,w}) \ot \bQ_p/\bZ_p$.

Putting this together with equation (<ref>), for all $ n \geq 0 $

\begin{align}\label{eq:imkummermapiso}
\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}} (\Epinf^{\pm}(L_{n,w})) \cong
\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}} (\wEpinf^{\pm}(L_{n,w})).
\end{align}

Furthermore, the exact sequence (<ref>) induces an isomorphism for any $ n \geq 0 $ and any prime $ w \in \Sss(L_n)$,

$ H^{1}(L_{n,w},E_{p^{\infty}}) \cong H^{1}(L_{n,w}\widehat{E}_{p^{\infty}}) $.

The isomorphisms in the above equation and equation (<ref>) are compatible for all $ n \geq 0 $ and they induce the isomorphism

\begin{align}\label{eq:H1ofEandEhatiso}
H^{1}(L_{n,w},E_{p^{\infty}})/ \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}
\cong
H^{1}(L_{n,w},\widehat{E}_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}.
\end{align}

Note that for $ n  \geq 0 $ and  $  w \in \Sss(L_n)  $, then $L_{n,w} =  K^{(l)}_{r} $ for some $ l,r \geq 0 $.
Let $ \cO $ denote the ring of integers of $ K^{(l)}_{0} $.
By Honda theory (cf. [9]), there exists a formal group $ \cF_{\T{ss}} $ in $ \cO[[X]] $, known as the supersingular formal group of Honda type $ t^{2}+p $ which is an  Eisenstein polynomial.

Further, part (ii) and (iv) of [Hyp1]Hyp 1 implies that the formal group  $ \widehat{E} $

is also of Honda type $ t^{2}+p  $ (section 8.2 of [12]).
Therefore, there exists an

$ \cO $-isomorphism

\begin{align}\label{eq:expologiso}
\exp_{\widehat{E}} \circ \log_{\cF_{\T{ss}}} \ : \ \cF_{\T{ss}}(\fm) \to \widehat{E}(\fm)
\end{align}

where $ \fm $	is the maximal ideal of the valuation ring $ \cO $ of the local field $ L_{n,w} $.
Since the Honda type $ t^{2}+p $ is independent of the choice of the elliptic curve, the supersingular formal group  $ \cF_{\T{ss}} $, and hence the formal group $ \widehat{E}  $ by the above isomorphism, are  independent of the choice of elliptic curve $ E/L $, assuming $ E/L $ satisfies [Hyp1]Hyp 1.

For simplicity we denote $ \cF_{\T{ss}}(\fm) $ by $ \cF_{\T{ss}}(L_{n,w}) $ and define the plus minus norm groups $ \cF_{\T{ss}}^{\pm}(L_{n,w})  \subseteq \cF_{\T{ss}}(L_{n,w})  $ using the isomorphism (<ref>)

\[
\cF_{\T{ss}}^{\pm}(L_{n,w}):=\log_{\cF_{\T{ss}}}  \circ \exp_{\widehat{E}}(\widehat{E}^{\pm}(L_{n,w})).
\]

Moreover, the signed Kummer maps, defined for the formal groups in equation (<ref>), can be defined for the supersingular formal group $ \cF_{\T{ss}}(L_{n,w}) $

\[
\begin{tikzcd}
0 \rightarrow
\cF_{\T{ss}}^{\pm}(L_{n,w})\otimes \bQ_p/\bZ_p \xrightarrow{\kappa^{\pm,p^{\infty}}_{L_{n,w}}}
H^{1}(L_{n,w}, (\cF_{\T{ss}})_{p^{\infty}})
\rightarrow
H^{1}(L_{n,w}, (\cF_{\T{ss}})_{p^{\infty}}) / \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}\rightarrow
\end{tikzcd}
\]

The isomorphism (<ref>) is $ \cO_{K^{(l)}_{0}} $-linear and thus it commutes with $ G_{L_{n,w}} $-action.
Hence, there is an isomorphism

\begin{align}\label{eq:H1ofEhatandFssiso}
H^{1}(L_{n,w},(\cF_{\T{ss}})_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}
\cong
H^{1}(L_{n,w},\widehat{E}_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}.
\end{align}

Combining (<ref>) and (<ref>), for $ w \in \Sss(L_n)  $

\begin{align}\label{eq:H1ofEandFssiso}
H^{1}(L_{n,w},(\cF_{\T{ss}})_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}
\cong
H^{1}(L_{n,w},E_{p^{\infty}})/ \Image \ \kappa^{\pm,p^{\infty}}_{L_{n,w}}.
\end{align}

Since the left-hand side is independent of $ E $, so is the right-hand side.
This implies that $ J_\fp^{\pm}(E_{p^{\infty}}/L_n) $ and $J_\bfp^{\pm}(E_{p^{\infty}}/L_n) $ are independent of the choice of  $ E/L $ for any $ n \geq 0 $.
Passing to the direct limit we have the following isomorphism

\begin{align}\label{eq:H1ofEandFssiso-infty}
H^{1}(L_{\infty,w},(\cF_{\T{ss}})_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{\infty,w}}
\cong
H^{1}(L_{\infty,w},E_{p^{\infty}})/ \Image \ \kappa^{\pm,p^{\infty}}_{L_{\infty,w}},
\end{align}

where $ w \in \Sss(L_\infty)=S_p(L_\infty) $ which implies that the same is true for that the local conditions  $ J_\fp^{\pm}(E_{p^{\infty}}/L_\infty) $ and $J_\bfp^{\pm}(E_{p^{\infty}}/L_\infty) $.

Suppose $ E_1/L $ and $ E_2/L $ are two elliptic curves satisfying [Hyp1]Hyp 1.
Let $ j \in \{1, 2\} $ and let
$ 	S_j = \{\fp, \bfp \} \cup \Sbad_j  $
where $ \Sbad_j $ is the set of primes where $ E_j $ has bad reduction.
From now on, we shall now enlarge the set $ S $ by declaring

\[
S : = S_1 \cup S_2 = \{\fp, \bfp \} \cup \Sbad_1 \cup \Sbad_2.
\]

Recall from Remark <ref> that adding a finite set of primes to the set $ S_j $ does not change $ \cR^{\pm/\pm} ((E_j)_p/L_\infty)  $ and $ \Sel^{\pm/\pm} ((E_{j})_{p^{\infty}}/L_\infty) $.

Let  $E_1/L $ and $ E_2/L $ be two elliptic curves that satisfy [Hyp1]Hyp 1.
Moreover, suppose  $  (E_1)_p \cong (E_2)_p $ as $ \Gal(\overline{L}/L) $-modules.
Then, there is an isomorphism

\[
\cR^{\pm/\pm} ((E_1)_p/L_\infty) \cong \cR^{\pm/\pm} ((E_2)_p/L_\infty).
\]

Let $ j \in \{1, 2\} $ and $ S $ be as above.

Suppose $ v = \fp $. Then, for any $ \fq | \fp  $ in $ L_{\infty} $ by Corollary <ref> and isomorphism (<ref>)

\begin{align*}
H^{1}(L_{\infty,\fq}, (E_j)_p)/ \Image \left(\kappa^{\pm,p}_{L_{\infty,\fq}}\right)&\cong \left(H^{1}(L_{\infty,\fq},(E_j)_{p^{\infty}})/ \Image \left(\kappa^{\pm,p^{\infty}}_{L_{\infty,\fq}}\right)\right) \\
\left(H^{1}(L_{\infty,w},(\cF_{\T{ss}})_{p^{\infty}})/\Image \ \kappa^{\pm,p^{\infty}}_{L_{\infty,w}}\right)_p.
\end{align*}

The right-hand side is independent of the choice $ j \in \{ 1, 2\} $.

\[
H^{1}(L_{\infty,\fq}, (E_1)_p)/ \Image \left(\kappa^{\pm,p}_{L_{\infty,\fq}}\right)\cong
H^{1}(L_{\infty,\fq}, (E_2)_p)/ \Image \left(\kappa^{\pm,p}_{L_{\infty,\fq}}\right)
\]

which implies that
{^{\pm}\widetilde{K}_\fp((E_1)_{p}/L_\infty)} \cong {^{\pm}\widetilde{K}_\fp((E_2)_{p}/L_\infty)}.
Similar argument works for when $ v = \bfp  $.
Now let $ v \in S \backslash \{\fp, \bfp \}  $.
Since $(E_1)_p \cong (E_2)_p $, it follows that for any prime $ w |v $

\[
H^{1}(L_{n,w}, (E_1)_{p}) \cong H^{1}(L_{n,w}, (E_2)_{p})
\]

and thus
^{\pm}\widetilde{K}_v((E_1)_{p}/L_n) \cong ^{\pm}\widetilde{K}_v((E_2)_{p}/L_n).
This means that for any $v \in S   $ the local terms match, and therefore we have the result.

Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1. Then, there exists an injective map

\begin{align*}
\varphi^{\pm/\pm}: \cR^{\pm/\pm} (E_p/L_\infty) \hooklongrightarrow \Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_p
\end{align*}

such that the $\mathrm{coker}(\varphi^{\pm/\pm} )  $ is a cotorsion $\Omega(G) $-module. Therefore, the following equality holds

\begin{align*}
\mathrm{corank}_{\Omega(G)}(\cR^{\pm/\pm} (E_p/L_\infty))= \mathrm{corank}_{\Omega(G)}(\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_p).
\end{align*}

Let $ v \in S $ and suppose $ w |v $ is a prime in $ L_\infty $.

\begin{align*}
\begin{dcases}
\psi_{w,\infty}
& \text{if}\ w| \fl \in \Sbad,\\
\psi^{\pm}_{\fq,\infty}
& \text{if}\ w=\fq| \fp,\\
\psi^{\pm}_{\bfq,\infty}
& \text{if}\ w=\bfq| \bfp.
\end{dcases}
\end{align*}

The maps  $ \psi^{\pm}_{\fq,\infty} $ and $ \psi^{\pm}_{\bfq,\infty} $ are isomorphisms by Proposition <ref>-[prop:psi-infty-1](1) and the $ \bF_p $-dimension of $ \ker(\psi_{w,\infty})$ is less than or equal to $ 2 $ by Proposition <ref>-[prop:psi-infty-2](2).
Define the map

\begin{align}\label{eq:varphi{v}{pm}}
{\varphi_{v}^{\pm}} : {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} &\to \left(J_v^{\pm}(E_{p^{\infty}}/L_\infty)\right)_p\\
\varphi_{v}^{\pm} &: = \bigoplus_{w|v} \varphi_{w}^{\pm}. \nonumber
\end{align}

Therefore,  $ \varphi_{v}^{\pm} $ is an isomorphism  when $ v \in \{\fp,\bfp \} $.

Define the map $ \varphi^{\pm/\pm} $ using the following commutative diagram (cf. <cit.>)

\begin{equation}\label{eq:commutative-diagram-varphi{pm/pm}}
\begin{tikzcd}
0 \arrow[r] & \cR^{\pm/\pm} (E_p/L_\infty) \arrow[r] \arrow[d, "\varphi^{\pm/\pm}"']          & H^1(G_{\infty}^{S}, E_{p}) \arrow[r, "\xi_p^{\pm/\pm}"] \arrow[d, "\psi_{G_{\infty}^{S},\infty}"', "\cong"]    & \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} \arrow[d, "{\ds\bigoplus_{v \in S}\varphi_{v}^{\pm}}"] \\
0 \arrow[r] & \Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_p \arrow[r] & H^1(G_{\infty}^{S}, E_{p^{\infty}})_p \arrow[r, "\xi^{\pm/\pm}"'] & \displaystyle\bigoplus_{v \in S} \left(J_v^{\pm}(E_{p^{\infty}}/L_\infty)\right)_p.
\end{tikzcd}
\end{equation}

The middle vertical map is an isomorphism by Proposition <ref>-[prop:psi-infty-1](1).
This implies that  $ \varphi^{\pm/\pm} $ is injective and

\begin{align*}
\mathrm{coker}(\varphi^{\pm/\pm} ) \subseteq \ker(\bigoplus_{v \in S}\varphi_{v}^{\pm})
= \bigoplus_{v \in \Sbad}\ker(\varphi_{v}^{\pm}).
%\\ &= \bigoplus_{\fl \in \Sbad} \bigoplus_{w | \fl} \ker(\psi_{w,\infty}).
\end{align*}

Suppose $ v \in \Sbad $ and fix a prime $ w|v $ in $ L_\infty $.

Then we have

\begin{align*}
\ker(\varphi_{v}^{\pm}) = \mathrm{Ind}^{G}_{G_{w}}\ker(\psi_{w,\infty})
% &= \ker(\psi_{w,\infty}) \hat{\bigotimes_{\Lambda(G_{w}) }} \Lambda(G)
\cong \ker(\psi_{w,\infty}) \widehat{\otimes}_{\Omega(G_{w}) } \Omega(G),
\end{align*}

where $ G_w = \Gal(L_{\cyc,w},L_v) \cong \bZ_p $ is the decomposition group

of $ G $ at the prime $ w $ and $-\widehat{\otimes}_{\Omega(G_{w}) } -$ denotes the completed tensor product over $\Omega(G_{w})$.

Recall that the map $ \psi_{w,\infty} $

is surjective with finite kernel by Proposition <ref>-[prop:psi-infty-2](2).Thus $ \ker(\psi_{w,\infty}) $ is a cotorsion $\Omega(G_{w}) $-module.
As $\Omega(G)  $ is a flat $ \Omega(G_{w}) $-module (cf. <cit.>), we have $\bigoplus_{v \in \Sbad}\ker(\varphi_{v}^{\pm}) $ is cotorsion as a $\Omega(G) $-module.

§ SIGNED SELMER AND FINE RESIDUAL SELMER GROUPS AS IWASAWA MODULES

Here, we prove our main theorem.

Throughout this section, we assume that $ E/L $ is an elliptic curve satisfying the assumptions of [Hyp1]Hyp 1.

For any choice of the signs,
$ \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ is a torsion $ \Lambda(G) $-module.

If $ E$ is defined over $ \bQ $, then Lei and Sprung showed in [14] that the above conjecture holds (cf. proof of <cit.>).

* $ \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ is a torsion $ \Lambda(G) $-module.

Note that Conjecture <ref> implies [Hyp2]Hyp 2$ ^{\pm/\pm} $ for all choices of the signs.
In [17], to prove their main results, the authors assume (cf. <cit.>):

* $ \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\cyc)  $ is a torsion

$ \Lambda(\Gamma) $-module.

Here, the $ \Lambda(\Gamma) $-module $ \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\cyc)  $ denotes the Pontryagin dual of the Selmer groups $ \Sel^{\pm/\pm}(E_{p^{\infty}}/L_\cyc)  $.
We show later that [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc) implies [Hyp2]Hyp 2$ ^{\pm/\pm} $ (cf. Proposition <ref>).
If $ \Sel(E_{p^{\infty}}/L) = \Sel^{\pm/\pm}(E_{p^{\infty}}/L)  $ is finite then [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc) is automatically satisfied (cf. Remark 4.5 of [15]).

[Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc) (resp. [Hyp2]Hyp 2$ ^{\pm/\pm} $) holds if and only if

* $ H^{2}(G_{\cyc}^{S},E_{p^{\infty}})$ (resp. $H^{2}(G_{\infty}^{S},E_{p^{\infty}})$) vanishes; and

* The map $ \xi_{\cyc}^{\pm/\pm}  $ in the exact sequence (resp. the map $ \xi^{\pm/\pm}  $ defined in <ref>)

\begin{align}\label{eq:short-exact-selmer-cyc}
0 \rightarrow
\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc) \rightarrow H^1(G_{\cyc}^{S}, E_{p^{\infty}}) \xrightarrow{\xi_{\cyc}^{\pm/\pm}}  \displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_\cyc)
\end{align}

is surjective.

See Proposition 4.4 of [15] (resp. Proposition 4.12 of [13]).

§.§ Cassels–Poitou–Tate exact sequence for fine signed residual Selmer group

Here, we aim to produce an analogous statement to Remark <ref> for  signed fine residual Selmer groups.
More specifically, we would like to show that the Pontryagin dual of the signed fine residual Selmer group, $ \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) $, is a torsion $ \Omega(G) $-module exactly when the exact sequence (<ref>) is short exact.
In other words, exactly when the map $ \xi_{p}^{\pm/\pm} $ is surjective.

Our strategy is the same as <cit.>. We do this by studying the exact sequence (<ref>)
using Cassels–Poitou–Tate exact sequence (cf. <cit.> for details).
We begin by defining some new modules involved in the sequence.
Let $ n \geq 0 $ and $ w \in S(L_n) $. Define

\begin{align}\label{def:cW_w(E_p/L_n)}
\cW^{\pm}_w(E_{p}/L_n):=
\begin{dcases}
0 & \text{if}\ w \in \Sbad(L_n), \\
\Image \ \kappa^{\pm,p}_{L_{n,\fq}} & \text{if}\ w=\fq|\fp,\\
\Image \ \kappa^{\pm,p}_{L_{n,\bfq}} & \text{if}\ w=\bfq|\bfp,
\end{dcases}
\end{align}

where the choice of the sign, as usual, depends on the sign of the Kummer map $ \kappa^{\pm,p} $ in the direction of primes $ \fp $ and $  \bfp$.
For $ v \in S $ (cf. equation (<ref>))

\[
^{\pm}\widetilde{K}_v(E_{p}/L_n)=\bigoplus_{w|v} H^{1}(L_{n,w}, E_{p}) / \cW^{\pm}_w(E_{p}/L_n).
\]

Let $ {\cW^{\pm}_w(E_{p}/L_n)}^{\bot} \subset H^{1}(L_{n,w}, E_{p})  $ denote the orthogonal complement of the group $ {\cW^{\pm}_w(E_{p}/L_n)} $ with respect to the Tate local duality (cf. <cit.>).

Define $ \fR^{\pm/\pm} (E_p/L_n)  $, a $ \Omega(G_n) $-module, using the exact sequence

\begin{align*}%\label{def:fR{pm/pm} (E_p/L_n)}
0 \rightarrow \fR^{\pm/\pm} (E_p/L_n)  \rightarrow H^1(G_{n}^{S}, E_{p}) \rightarrow  \bigoplus_{w \in S(L_n)} H^{1}(L_{n,w}, E_{p}) / {\cW^{\pm}_w(E_{p}/L_n)}^{\bot}.
\end{align*}

For any $ n \geq 0 $, the Cassels–Poitou–Tate exact sequence gives

\begin{align}\label{eq: fine selmer Cassels--Poitou--Tate exact sequence}
0 \rightarrow \cR^{\pm/\pm} (E_p/L_n) \rightarrow H^1(G_{n}^{S}, E_{p})  \rightarrow \displaystyle\bigoplus_{w \in S(L_n)} H^{1}(L_{n,w}, E_{p}) / \cW^{\pm}_w(E_{p}/L_n) \\ \nonumber
\rightarrow   \fR^{\pm/\pm} (E_p/L_n)\Pd \rightarrow H^2(G_{n}^{S}, E_{p}) \rightarrow \displaystyle\bigoplus_{w \in S(L_n)}  H^{2}(L_{n,w}, E_{p})  \rightarrow 0.
\end{align}

The final zero is due to equation (<ref>) (cf. <cit.>).

\begin{align}
\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty}) := \varprojlim_{\mathrm{cores}} \fR^{\pm/\pm} (E_p/L_n) \subset H^{1}_{\mathrm{Iw}}(L,E_p) \label{eq:Iwasawacohomology}
\end{align}

where the Iwasawa cohomology module

$ H^{1}_{\mathrm{Iw}}(L,E_p) := \varprojlim_{\mathrm{cores}} H^{1}(G_n^{S}, E_p) $.

Since  $ H^{0}(G_{n}^{S}, E_p) $ vanishes by equation (<ref>), using Jannsen’s spectral sequence (cf. <cit.>), there is an isomorphism

\begin{align*}
H^{1}_{\mathrm{Iw}}(L,E_p) \cong \Hom_{\Omega(G)}\left(\Hom_{\bF_p}(H^{1}(G^{S}_{\infty}),\bF_p),\Omega(G) \right).
\end{align*}

The left-hand side is a torsion-free $\Omega(G)$-module and therefore so is the right-hand side.
In particular,  $\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})  $ is also torsion-free being a  $\Omega(G)$-submodule of $ 	H^{1}_{\mathrm{Iw}}(L,E_p) $.
Moreover (cf. <cit.>)

\begin{align*}
\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd = \Hom(\varprojlim_{\mathrm{cores}} \fR^{\pm/\pm} (E_p/L_n), \bQ_p/\bZ_p)
%	\\
%	&= \varinjlim_{\mathrm{cores}\Pd} (\Hom( \fR^{\pm/\pm} (E_p/L_n), \bQ_p/\bZ_p))
= \varinjlim_{\mathrm{cores}\Pd} \fR^{\pm/\pm} (E_p/L_n)\Pd.
\end{align*}

Taking the direct limit of the exact sequence (<ref>) yields

\begin{align}
0\rightarrow\cR^{\pm/\pm} (E_p/L_\infty) \rightarrow H^1(G_{\infty}^{S}, E_{p}) \xrightarrow{\xi_p^{\pm/\pm}} \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}\label{eq: fine selmer Cassels--Poitou--Tate exact sequence-}	\\
\rightarrow  \mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd \rightarrow H^2(G_{\infty}^{S}, E_{p}) \rightarrow \displaystyle\bigoplus_{w \in S(L_\infty)}  H^{2}(L_{\infty,w}, E_{p}) \rightarrow 0\nonumber.
\end{align}

Our plan now is to analyze this exact sequence.
In  [3], the authors introduce the $  \mu = 0 $-conjecture for the fine Selmer group (Conjecture A of [2]).
Its original formulation asserts that for an elliptic curve $ E $ over any number filed $ F $, the Pontryagin dual of the classical fine Selmer group over the cyclotomic extension $ F_\cyc $ is a finitely generated $ \bZ_p $-module.
Here, we use an equivalent cohomological version of Conjecture A (cf. <cit.>).

* Let $ \Gamma =  \Gal(L_\cyc/ L) $ and $ G_{\cyc}^{S} := \Gal(L^{S}/L_{\cyc})$. Then the $ \Omega(\Gamma) $-module $ H^2(G_{\cyc}^{S}, E_{p}) $ is trivial.

§.§ Main theorem

We prove the main theorem (cf. Theorem <ref>) of this article.
This is an analogue of <cit.>.

Let $ v \in S $ and $ w | v $ be a prime in $ L_\infty $.

*  If $ w \nmid p $, then the Pontryagin dual of $ H^{1}(L_{\infty,w}, E_{p^{\infty}}) $ has $ \mu $-invariant equal to zero.
This implies that

\begin{align*}
\mathrm{corank}_{\Lambda(G)}H^{1}(L_{\infty,w}, E_{p^{\infty}}) =  \mathrm{corank}_{\Omega(G)}H^{1}(L_{\infty,w}, E_{p}).
\end{align*}

*  Suppose [Conjecture A]Conjecture A holds. Then, the Pontryagin dual of $ H^{1}(G^{S}_{\infty}, E_{p^{\infty}}) $ has $ \mu $-invariant equal to zero. This implies that

\begin{align*}
\mathrm{corank}_{\Lambda(G)}H^{1}(G_{\infty}^{S}, E_{p^{\infty}}) =  \mathrm{corank}_{\Omega(G)}H^{1}(G_{\infty}^{S}, E_{p}).
\end{align*}

When $ w \nmid p  $, then $ L_{\infty,w}$ is equal to  $L_{\cyc,w} $ and the statement that $ H^{1}(L_{\infty,w}, E_{p^{\infty}})\Pd $ has $ \mu $-invariant equal to zero is proven in <cit.>.
By Proposition <ref>, there is an isomorphism

$ (H^{1}(G_{\infty}^{S}, E_{p^{\infty}}))_p \cong H^{1}(G_{\infty}^{S}, E_{p}) $

and equation (<ref>) implies

\begin{align*}
\mathrm{corank}_{\Lambda(G)}H^{1}(L_{\infty,w}, E_{p^{\infty}}) =  \mathrm{corank}_{\Omega(G)}H^{1}(L_{\infty,w}, E_{p}).
\end{align*}

For part [lem:mu-inv-H1-2](2), Lemma 5.6 in [15] tells us

$ H^{1}(G_{\infty}^{S}, E_{p^{\infty}})^{H} \cong H^{1}(G_{\cyc}^{S}, E_{p^{\infty}}) $

where $ H=\Gal(L_\infty,L_{\cyc})\cong \bZ_p $. Then $ \mu_\Gamma(H^{1}(G_{\cyc}^{S}, E_{p^{\infty}})\Pd)=0 $ by <cit.>.
Let $ M=H^{1}(G_{\infty}^{S}, E_{p^{\infty}})\Pd $.
Note that $  \mu_\Gamma(M_H)$ vanishes
where $ M_H $ denotes

the $ H $-coinvariant of $ \Lambda(G) $-module $ M $.
Then, a version of the topological Nakayama's lemma (cf. <cit.>) implies that $  \mu_G(M)=0$.
The equality about the coranks is implied by Proposition <ref> and equation (<ref>).

Thus, given [Conjecture A]Conjecture A, the last two modules in the exact sequence (<ref>) vanish.

Therefore, to show equality (<ref>), it is enough to show that

\begin{align*}
\mathrm{corank}_{\Omega(G)}(H^1(G_{\infty}^{S}, E_{p})) = \mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}).
\end{align*}

Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1 and [Hyp2]Hyp 2$ ^{\pm/\pm} $.
Moreover, suppose  [Conjecture A]Conjecture A holds.

\begin{align}\label{eq:cR andmathscrR have the same corank}
\mathrm{corank}_{\Omega(G)}(\cR^{\pm/\pm} (E_p/L_\infty)) = \mathrm{corank}_{\Omega(G)}(\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd).
\end{align}

By Theorem 3.2 and 4.1 of [18],

$ \mathrm{corank}_{\Lambda(G)}(H^1(G_{\infty}^{S}, E_{p^{\infty}}))=[L:\bQ]=2. $

Putting this together with Lemma <ref>-[lem:mu-inv-H1-2](2) we get

\begin{align}\label{eq:coranks-the-same-1}
\mathrm{corank}_{\Lambda(G)}(H^1(G_{\infty}^{S}, E_{p^{\infty}}))=\mathrm{corank}_{\Omega(G)}(H^1(G_{\infty}^{S}, E_{p}))=2.
\end{align}

For a prime  $ w \nmid p $ in $ L_\infty $ Proposition 2 in [6] implies that

$ \Lambda(G) $-corank of $ H^{1}(L_{\infty,w},E_{p^{\infty}}) $ is zero.

This means that for $ v \nmid p $, the $ \Lambda(G) $-corank of $ J_v^{\pm}(E_{p^{\infty}}/L_\infty) $ is zero.
Hence, we have

\begin{align}\label{eq:coranks-the-same-2}
\mathrm{corank}_{\Lambda(G)}(\displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_\infty))
=\mathrm{corank}_{\Lambda(G)}(\displaystyle\bigoplus_{v \in S_p} J_v^{\pm}(E_{p^{\infty}}/L_\infty)).
\end{align}

Furthermore, by Lemma <ref>-[lem:mu-inv-H1-2](2) we know

\begin{align*}
\mathrm{corank}_{\Lambda(G)}( H^{1}(L_{\infty,w},E_{p^{\infty}}))&=
\mathrm{corank}_{\Omega(G)}H^{1}(L_{\infty,w}, E_{p})\\
=\mathrm{corank}_{\Omega(G)}( ^{\pm}\widetilde{K}_v(E_{p}/L_\infty))&=0\\
\implies
\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S}   {}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))
&=\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S_p}    {}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))
\end{align*}

For $ v | p $, recall that the map $ \varphi_{v}^{\pm} $ (cf. (<ref>)) gives an isomorphism

between $ (J_v^{\pm}(E_{p^{\infty}}/L_\infty))_p  $ and $ {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} $.

$ \Omega(G) $-corank of $(J_v^{\pm}(E_{p^{\infty}}/L_\infty))_p
is equal to
\mathrm{corank}_{\Omega(G)}( {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} ). $

Moreover, for $ v | p $ the local condition $ J_v^{\pm}(E_{p^{\infty}}/L_\infty) $ is a free $ \Lambda(G) $-module (cf. <cit.>) and

\[
\mathrm{corank}_{\Lambda(G)}(\displaystyle\bigoplus_{v \in S_p} J_v^{\pm}(E_{p^{\infty}}/L_\infty))=\mathrm{corank}_{\Lambda(G)}(H^1(G_{\infty}^{S}, E_{p^{\infty}})=2.
\]

This means that $ J_v^{\pm}(E_{p^{\infty}}/L_\infty) $ has $ \mu $-invariant zero.
By equation (<ref>),

\begin{align*}
\mathrm{corank}_{\Lambda(G)}(J_v^{\pm}(E_{p^{\infty}}/L_\infty))
\mathrm{corank}_{\Omega(G)}((J_v^{\pm}(E_{p^{\infty}}/L_\infty))_p)\\
\mathrm{corank}_{\Omega(G)}( {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} ).
\end{align*}

This, together with equation (<ref>) gives us that

\begin{align*}%\label{eq:coranks-the-same-3}
\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S}   {}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))
=\mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S_p}    {}^{\pm}\widetilde{K}_v(E_{p}/L_\infty))=2.
\end{align*}

Comparing the above equation and equation (<ref>) yields

\begin{align}\label{coranks-to-show-residual is surjective}
\mathrm{corank}_{\Omega(G)}(H^1(G_{\infty}^{S}, E_{p})) = \mathrm{corank}_{\Omega(G)}(\displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)})=2.
\end{align}

The exact sequence (<ref>) implies that

\begin{align*}
\mathrm{corank}_{\Omega(G)}(\cR^{\pm/\pm} (E_p/L_\infty)) = \mathrm{corank}_{\Omega(G)}(\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd).
\end{align*}

We are now ready to give our main theorem which is the analogue of <cit.> in our setting.
It describes a criterion for the vanishing of the signed $ \mu $-invariants based completely on the structure of the fine singed residual Selmer groups as Iwasawa modules.

Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1 and [Hyp2]Hyp 2$ ^{\pm/\pm} $.
Furthermore, suppose  [Conjecture A]Conjecture A holds.
Then the following statements are equivalent:

*  $\mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) = \cR^{\pm/\pm}(E_{p}/L_\infty)\Pd $ is $ \Omega(G) $-torsion.

The $ \mu $-invariant $ \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty):=\mu_G(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) $ vanishes.

The map  $ \xi_p^{\pm/\pm} $, described in diagram (<ref>), is surjective.

To see (<ref>) and (<ref>) are equivalent,
recall from Proposition <ref>

\begin{align*}
\mathrm{rank}_{\Omega(G)}(\mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty))= \mathrm{rank}_{\Omega(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) / p \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty));\\
0=\mathrm{rank}_{\Lambda(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) \leq 	\mathrm{rank}_{\Omega(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) / p \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty))
\end{align*}

with equality exactly when  $ \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ vanishes.
Therefore, given (<ref>) we have

\begin{align*}
\mathrm{rank}_{\Lambda(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) = 	\mathrm{rank}_{\Omega(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) / p \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) = 0
\end{align*}

which implies $ \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ is zero.
On the other hand, if (<ref>) holds, then

\begin{align*}
0=\mathrm{rank}_{\Lambda(G)}(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty)) =\mathrm{rank}_{\Omega(G)}(\mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty))
\end{align*}

and so  (<ref>) $ \iff $ (<ref>).
Now let us show (<ref>) $ \implies$ (<ref>).

The discrete $ \Omega(G) $-module $ H^2(G_{\infty}^{S}, E_{p}) $ vanishes by Proposition 4.7 [17].
By Proposition <ref>

\begin{align*}
0=\mathrm{corank}_{\Omega(G)}(\cR^{\pm/\pm} (E_p/L_\infty)) = \mathrm{corank}_{\Omega(G)}(\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd).
\end{align*}

This means that $ \mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd $ is a cotorsion $ \Omega(G) $-module.
However, being a  $\Omega(G)$-submodule of a cotorsion-free module $ 	H^{1}_{\mathrm{Iw}}(L,E_p) $, the module $\mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})  $ is also torsion-free (equation (<ref>)).
As a result, $ \mathscr{R}^{\pm/\pm}(E_p/ L_{\infty})\Pd $ vanishes.
Therefore, the map  $ \xi_p^{\pm/\pm} $ (cf. the exact sequence (<ref>)) is surjective.
Finally to see (<ref>) $ \implies$ (<ref>), suppose  the map $ \xi_p^{\pm/\pm} $ is surjective and hence we have

\begin{equation}\label{eq:short-exact-fine-selmer-infty}
0 \rightarrow \cR^{\pm/\pm} (E_p/L_\infty) \rightarrow H^1(G_{\infty}^{S}, E_{p}) \xrightarrow{\xi_p^{\pm/\pm}} \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)} \rightarrow 0.
\end{equation}

Taking  Pontryagin duals of the above sequence gives

\begin{equation*}
0 \rightarrow
\displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}\Pd
\xrightarrow{{\xi_p^{\pm/\pm}}\Pd}
H^1(G_{\infty}^{S}, E_{p})\Pd  \rightarrow
\mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) \rightarrow
\end{equation*}

In the proof of Proposition <ref>, we proved that (cf. equation (<ref>))
the first two terms in the above short exact sequence have the same $ \Omega(G) $-rank which is equal to $ 2 $.
So, $ \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) $ is a torsion $ \Omega(G)  $-module.

We record the following important corollary of Theorem <ref>.

Let  $E_1/L $ and $ E_2/L $ be two elliptic curves that satisfy [Hyp1]Hyp 1 and [Hyp2]Hyp 2$ ^{\pm/\pm} $.
Suppose they have isomorphic residual Galois representations and [Conjecture A]Conjecture A is satisfied for either $E_1/L $ or $ E_2/L $ (and hence both).

$ 	\mu_G^{\pm/\pm}((E_1)_{p^{\infty}}/L_\infty) =0 $ if and only if $ \mu_G^{\pm/\pm}((E_2)_{p^{\infty}}/L_\infty) =0. $

Without loss of generality assume $ \mu_G^{\pm/\pm}((E_1)_{p^{\infty}}/L_\infty) $  vanishes.

Note that [Conjecture A]Conjecture A for an elliptic curve $ E/L $ only depends on the isomorphism class of residual Galois representation $ E_p $.

By Theorem <ref>, the module
$ \mathcal{Y}^{\pm/\pm}((E_1)_{p}/L_\infty)$ is $ \Omega(G) $-torsion.
Proposition <ref> gives

\[
\cR^{\pm/\pm} ((E_1)_p/L_\infty) \cong \cR^{\pm/\pm} ((E_2)_p/L_\infty).
\]

Thus, the module $ \mathcal{Y}^{\pm/\pm}((E_2)_{p}/L_\infty)$ is also $ \Omega(G) $-torsion.
Again by  Theorem <ref>, this implies that $ \mu_G^{\pm/\pm}((E_2)_{p^{\infty}}/L_\infty) $ is equal to zero.

§.§ Comparison with the cyclotomic level

Let $ H$ denote the Galois group $\Gal(L_\infty/ L_{\cyc}) $ and consider the following commutative diagram

\begin{equation}\label{eq:commutative-diagram-R{pm/pm}}
\begin{tikzcd}
0 \arrow[r] & \cR^{\pm/\pm} (E_p/L_\cyc) \arrow[r] \arrow[d, "\alpha_p^{\pm/\pm}"']          & H^1(G_{\cyc}^{S}, E_{p}) \arrow[r, "\xi_{p,\cyc}^{\pm/\pm}"] \arrow[d, "\beta_p"']    & \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\cyc)} \arrow[d, "\widetilde{\gamma}^{\pm/\pm}_p:={\ds\bigoplus_{v \in S}\widetilde{\gamma}_{v}^{\pm}}"] \\
0 \arrow[r] & \cR^{\pm/\pm} (E_p/L_\infty)^{H} \arrow[r] & H^1(G_{\infty}^{S}, E_{p})^{H} \arrow[r, "\xi_{p}^{\pm/\pm,H}"'] & \displaystyle\bigoplus_{v \in S} {^{\pm}\widetilde{K}_v(E_{p}/L_\infty)}^{H}.
\end{tikzcd}
\end{equation}

All the vertical maps in the diagram (<ref>) are isomorphisms.
In particular,

$ \cR^{\pm/\pm} (E_p/L_\infty)^{H}  \cong \cR^{\pm/\pm} (E_p/L_\cyc) $.

The map $ \beta_p $  is an isomorphism by <cit.>.
Let us show that the map $ \widetilde{\gamma}^{\pm/\pm}_p:={\bigoplus_{v \in S}\widetilde{\gamma}_{v}^{\pm}} $ is an isomorphism.
For any $ v \in S \backslash S_p $, the map $ \widetilde{\gamma}_{v}^{\pm}  $ is  the identity map (cf. proof of Lemma 5.10 in [15]).
Let us assume $ v  = \fp $
and let $ A:= J^{\pm}_\fp (E_{p^{\infty}}/L_{\infty}) $.
Note that, by Corollary <ref>, $ A_p \cong {}^{\pm}\widetilde{K}_\fp(E_{p}/L_\infty) $ and by Lemma 5.10 of [15], $ A^{H} \cong J^{\pm}_\fp (E_{p^{\infty}}/L_{\cyc}) $.
Furthermore, by <cit.>, there is the following isomorphism

\begin{align*}
\psi_{\fq,\cyc}^{\pm} :	{}^{\pm}\widetilde{K}_\fp(E_{p}/L_\cyc)
\xrightarrow{\cong}
%(A^{H})_p =
\left(J^{\pm}_\fp (E_{p^{\infty}}/L_{\cyc})\right)_p
%	= H^{1}(E_{p}/L_{\cyc,\fq})/\Image \ \kappa^{\pm,p}_{L_{\cyc,\fq}}
\end{align*}

Using the isomorphism $ (A^{H})_p \cong (A_p)^{H} $ along with the above map, we see

\begin{equation*}
\widetilde{\gamma}_{\fp}^{\pm} : {}^{\pm}\widetilde{K}_\fp(E_{p}/L_\cyc) \xRightarrow[\psi_{\fq,\cyc}^{\pm}]{\cong} \left(J^{\pm}_\fp (E_{p^{\infty}}/L_{\cyc})\right)_p \xRightarrow{\cong} (A^{H})_p \cong (A_p)^{H} \xRightarrow{\cong} {}^{\pm}\widetilde{K}_\fp(E_{p}/L_\infty)^{H}
\end{equation*}

is an isomorphism.
The case where $   v  = \bfp$ is similar.

Finally, the snake lemma implies that  $ \alpha_p^{\pm/\pm} $ is also an isomorphism.

We record the analogue of Theorem <ref> in the cyclotomic setting.
Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1 and [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc).
Then the following are equivalent:

*  $\mathcal{Y}^{\pm/\pm}(E_{p}/L_\cyc) = \cR^{\pm/\pm}(E_{p}/L_\cyc)\Pd $ is $ \Omega(\Gamma) $-torsion.

The signed cyclic $ \mu $-invariant $ \mu_\Gamma^{\pm/\pm}(E_{p^{\infty}}/L_\cyc):=\mu_\Gamma(\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\cyc)) $ vanishes.

*  The map  $ \xi_{p,\cyc}^{\pm/\pm} $ in diagram (<ref>) is surjective and [Conjecture A]Conjecture A holds.

Suppose $ E/L $ satisfies [Hyp1]Hyp 1 and [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc).
Assume one of the equivalent statements in Theorem <ref> is satisfied. Then, the top and bottom exact sequences in diagram (<ref>) are short exact.

By Theorem <ref>, the map $\xi_{p,\cyc}^{\pm/\pm}  $ is surjective and hence the top row in diagram (<ref>) is short exact.
Since the map $ \widetilde{\gamma}^{\pm/\pm}_p \circ \xi_{p,\cyc}^{\pm/\pm}$ is surjective, the map $ \xi_{p}^{\pm/\pm,H} $ is also surjective.

We now combine Theorem <ref> and Proposition <ref>.

Suppose $ E/L $ satisfies [Hyp1]Hyp 1 and [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc).
Furthermore, assume one of the equivalent statements in Theorem <ref> is satisfied.
Then, the equivalent statements in Theorem <ref> are satisfied.

[Conjecture A]Conjecture A is satisfied by part <ref> of Theorem <ref>.
Proposition <ref> implies that [Hyp2]Hyp 2$ ^{\pm/\pm} $ holds.
Using Proposition <ref>

\[
\mathcal{Y}^{\pm/\pm}(E_{p}/L_\cyc) =\cR^{\pm/\pm}(E_{p}/L_\cyc)\Pd\cong (\cR^{\pm/\pm}(E_{p}/L_\infty)^{H})\Pd \cong \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty)_H.
\]

Suppose $ \mathcal{Y}^{\pm/\pm}(E_{p}/L_\cyc) $ which is isomorphic to $  \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty)_H $ is torsion as a $ \Omega(\Gamma) $-module.
Then, we can apply <cit.> to get that the module $ \mathcal{Y}^{\pm/\pm}(E_{p}/L_\infty) $ is torsion as a $ \Omega(G) $-module.

With the same assumption as Theorem <ref>,

\begin{align*}
H^{1}(H,\cR^{\pm/\pm} (E_p/L_\infty))=0.
\end{align*}

By Theorem <ref>, we have the short exact sequence (<ref>).

Taking the long exact $ H $-Galois cohomology of this sequence yields
$  \Coker(\xi_{p}^{\pm/\pm,H})$ is equal to $ H^{1}(H,\cR^{\pm/\pm} (E_p/L_\infty)) .$

By Corollary <ref>, the map $ \xi_{p}^{\pm/\pm,H} $ is surjective which proves the claim.

Suppose $ E/L $ is an elliptic curve satisfying [Hyp1]Hyp 1 and [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc).
Furthermore, assume the $ \Omega(\Gamma) $-module $ H^{1}(H,\cR^{\pm/\pm} (E_p/L_\infty)) $ vanishes.
Then, the converse of Theorem <ref> holds too.
That is, all the statements in Theorem <ref> and Theorem <ref> are equivalent.

Suppose the map  $ \xi_p^{\pm/\pm} $ is surjective and [Conjecture A]Conjecture A holds.
Since $ H^{1}(H,\cR^{\pm/\pm} (E_p/L_\infty)) $ vanishes, a similar argument as in  Corollary <ref> shows that the map $ \xi_{p}^{\pm/\pm,H} $ is surjective.
Using diagram (<ref>), the map $\xi_{p}^{\pm/\pm,H} \circ \beta_p  $ is surjective, and hence that the map $ \xi_{p,\cyc}^{\pm/\pm} $ is also surjective.

Theorem <ref> implies that if $ E/L $ satisfies  [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc), then

\begin{align}\label{eq:mu-cyc-implies-mu-infty}
\mu_\Gamma^{\pm/\pm}(E_{p^{\infty}}/L_\cyc) = 0 \implies  \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty)=0.
\end{align}

Moreover, if $ H^{1}(H,\cR^{\pm/\pm} (E_p/L_\infty)) $ vanishes and  [Conjecture A]Conjecture A holds then the two sides of the statement (<ref>) become equivalent by Corollary <ref>.

§ PSEUDO-NULL SUBMODULES

Let $ E/L $ be an elliptic curve satisfying [Hyp1]Hyp 1.
We show that under some mild assumptions,
the $ \Lambda(G) $-modules the $ \Lambda(G) $-modules $ \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ and $ \mathfrak{X}(E_{p^{\infty}}/L_\infty) $ have no non-trivial pseudo-null submodules 	(Definition 5.1.4 of [16] for the definition of pseudo-null submodules).

The Iwasawa algebra $\Lambda(G)$ is a Noetherian regular local commutative ring and hence a Cohen–Macaulay ring.

In particular, the depth of $ \Lambda(G) $ coincides with its Krull dimension (cf. Chapter 17 of [5].).
By the Auslander–Buchsbaum–Serre theorem, the global dimension of $ \Lambda(G) $ is finite and it coincides with its Krull dimension.

Let $ H$ be $\Gal(L_\infty,L_{\cyc}) $ and consider the following diagram of $ \Lambda(\Gamma) $-modules:

\begin{equation}\label{eq:fundumetal diagram for Selmer}
\begin{tikzcd}
0 \arrow[r] &  \Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc) \arrow[r] \arrow[d, "\alpha^{\pm/\pm}"', "\cong"]          & H^1(G_{\cyc}^{S}, E_{p^{\infty}}) \arrow[r, "\xi_\cyc^{\pm/\pm}" ] \arrow[d, "\beta"', "\cong"]    & \displaystyle\bigoplus_{v \in S} J_v^{\pm}(E_{p^{\infty}}/L_\cyc) \arrow[d, "{\gamma}^{\pm/\pm}:={\ds\bigoplus_{v \in S}{\gamma}_{v}^{\pm}}", "\cong"'] \\
0 \arrow[r] & \Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)^{H} \arrow[r] & H^1(G_{\infty}^{S}, E_{p^{\infty}})^{H} \arrow[r, "\xi^{\pm/\pm,H}"] & \displaystyle\bigoplus_{v \in S} \left(J_v^{\pm}(E_{p^{\infty}}/L_\infty)\right)^{H}.
\end{tikzcd}
\end{equation}

The maps $\beta $ and ${\gamma}^{\pm/\pm}  $ are isomorphisms by Lemma 5.6, 5.8, and 5.10 of [15].
By snake lemma, the map $ \alpha^{\pm/\pm} $ is also an isomorphism and hence, all the vertical maps in the diagram (<ref>) are isomorphisms.

Suppose $ E/L $ is an elliptic curve over quadratic number field $ L $ satisfying [Hyp1]Hyp 1.
If [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc) holds then

\begin{align}\label{eq:mathfrak{X}{pm/pm}(infty)-H iso mathfrak{X}{pm/pm}(cyc)}
\mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\cyc) \cong \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H,
\end{align}

where $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H  $ denotes the of $ H $-coinvariants of $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)$.
Moreover, [Hyp2]Hyp 2$ ^{\pm/\pm} $ holds and  the group	$ H^{1}(H,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)) $ vanishes.

By Proposition <ref>, the top row of the diagram (<ref>) becomes a short exact sequence.
A diagram chase gives that, since the map ${\gamma}^{\pm/\pm}  \circ \xi_\cyc^{\pm/\pm}  $ is surjective, the map $ \xi^{\pm/\pm,H} $ is also surjective and hence, the diagram (<ref>) becomes an isomorphism of two short exact sequences.
Taking the Pontryagin dual of the isomorphism $ \alpha^{\pm/\pm} $ gives the first claim.

By [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc), $ \Lambda(\Gamma) $-module $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H $ is finitely generated and torsion.
This implies that $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ is a finitely generated torsion $ \Lambda(G) $-module (cf. <cit.>) which is [Hyp2]Hyp 2$ ^{\pm/\pm} $.
By Proposition <ref>, this implies that the map $\xi^{\pm/\pm}   $ in (<ref>) is surjective.

Taking long exact $ H $-cohomology of the short exact sequence  (<ref>) , we get the bottom short exact sequence in the diagram (<ref>).
In particular, the map $\xi^{\pm/\pm,H}$  is surjective which implies

$ 	\Coker(\xi^{\pm/\pm,H}) =H^{1}(H,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty))   = 0. $

Assuming [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc) holds, the cohomological group
$ H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)) $ vanishes if and only if $ \xi_{\cyc}^{\pm/\pm,\Gamma}$ is surjective.

By Proportion <ref>, $ H^{1}(H,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty))  $ vanishes.
The inflation-restriction exact sequence implies that

\begin{align*}
H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)) = H^{1}(G,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)).
\end{align*}

Let $ G^{S}:=\Gal(L^{S}/L) $ and consider the following commutative diagram

\begin{equation}\label{eq:fundumetal diagram for finite Selmer}
\begin{tikzcd}
0 \arrow[r] &  \Sel (E_{p^{\infty}}/L) \arrow[r] \arrow[d, "\alpha^{\pm/\pm}_{\cyc}"']          & H^1(G^{S}, E_{p^{\infty}}) \arrow[r, "\lambda_0" ] \arrow[d, "\beta_{\cyc}"']    & \displaystyle\bigoplus_{v \in S} J_v(E_{p^{\infty}}/L) \arrow[d, "{\gamma}_{\cyc}^{\pm/\pm}:={\ds\bigoplus_{v \in S}{\gamma}_{\cyc,v}^{\pm}}"]\\
0 \arrow[r] & \Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)^{\Gamma} \arrow[r] & H^1(G_{\cyc}^{S}, E_{p^{\infty}})^{\Gamma} \arrow[r, "\xi_{\cyc}^{\pm/\pm,\Gamma}"] & \displaystyle\bigoplus_{v \in S} \left(J_v^{\pm}(E_{p^{\infty}}/L_\cyc)\right)^{\Gamma}.
\end{tikzcd}
\end{equation}

The map $ \lambda_{0} $ is as in the exact sequence (<ref>).
Note that at $  L_0 = L $, the classical $ p^{\infty} $-Selmer and the signed Selmer groups coincide.

Recall that [Hyp2(cyc)]Hyp 2$ ^{\pm/\pm} $(cyc)  implies the map $\xi^{\pm/\pm}   $ is surjective by Proposition <ref>.
Taking long exact $ \Gamma $-cohomology  of the short exact sequence (<ref>) yields that

$ \Coker(\xi_{\cyc}^{\pm/\pm,\Gamma} ) = H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)) $.

The map $ \beta_{\cyc} $ in diagram (<ref>) is an isomorphism.

The Hochschild–Serre spectral sequence gives

\begin{align*}
0 \rightarrow H^{1}(\Gamma , E_{p^{\infty}}(L_{\cyc}))    \rightarrow H^1(G^{S}, E_{p^{\infty}}) \xrightarrow{\beta_{\cyc}} H^{1}(G_{\cyc}^{S},E_{p^{\infty}})^{\Gamma} \rightarrow  0
\end{align*}

The last term is zero as $\Gamma  $ has $ p $-cohomological dimension.
Moreover, equation (<ref>) gives that $E_{p^{\infty}}(L_{\cyc})  $ is zero and so
$ H^{1}(\Gamma , E_{p^{\infty}}(L_{\cyc})) $ vanishes.

If $ \Sel(E_{p^{\infty}}/L) $ is finite, the map $ \xi_{\cyc}^{\pm/\pm,\Gamma}$ is surjective and hence

\[
H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)) = H^{1}(G,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty)) = 0.
\]

Let $ \gamma $ and $ h $ be topological generators of the groups $ \Gamma $ and $ H $ respectively.
Since $ G \cong H \times \Gamma$, there is an isomorphism

\begin{align*}
\Lambda(G)=\bZ_p[[G]]\cong \bZ_p[[H \times \Gamma]] \xrightarrow{\cong} \bZ_{p}[[T_1,T_2]]
\end{align*}

where $ h-1 $ (resp. $ \gamma-1 $)  is mapped to the indeterminate variable $ T_1 $ (resp. $ T_2 $ ) and we extend $ \bZ_p $-linearly.

In what follows we use $ \Lambda(H) $ (resp. $ \Lambda(\Gamma) $) and $ \bZ_p[[T_1]] $ (resp. $  \bZ_p[[T_2]] $) interchangeably.

Suppose $ R $ is a ring and let $ M $ be an $ R $-module.
A sequence of elements $ f_1,\cdots,f_k $ of $ R $ is called an $ M $-regular sequence if the following conditions hold:

* The element $ f_i $ is a non-zero divisor on $ M/(f_1,\cdots,f_{i-1})M $ for each $ i=1,\cdots,k $;

* the module $ M/(f_1,\cdots,f_k)M $ is not zero.

If $ I $ is an ideal of the ring $ R $ and $ f_1,\cdots,f_k \in I $ then we call $ f_{1},…,f_{k} $ an $ M $-regular sequence in $ I $.
Moreover, if $ M=R $, we call $ f_{1},\cdots,f_{k} $ a regular sequence in $ I $.

Using the above definition we can define the depth of a finitely generated module over an Iwasawa algebra $ \Lambda $.

Let $ I $ be an ideal of $ \Lambda $ and suppose $ M $ is a finitely generated $ \Lambda $-module such that $ IM \neq M $.
Then the $ I $-depth of $ M $, denoted by $ \depth_I(M) $, is the maximal length of a $ M $-regular sequence in $ I $.
The depth of $ M $ as an $  \Lambda$-module, denoted by $ \depth(M) $, is defined to be

\[
\depth(M) := \depth_\fm(M)
\]

where $ \fm $ is the maximal ideal of $ \Lambda $.

Suppose $E/L  $ is an elliptic curve satisfying [Hyp1]Hyp 1 and $ \Sel(E_{p^{\infty}}/L) $ is finite.
Then the following assertions hold.

*   $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ has no non-trivial pseudo-null $ \Lambda(G) $-submodule.

*   $ \mathfrak{X}(E_{p^{\infty}}/L_\infty) $ has no non-trivial pseudo-null $ \Lambda(G) $-submodule.

For part (<ref>), note that  for a finitely generated $ \Lambda(G) $-module $ M $

$$ \depth(M) \leq \T{Krulldim}(M) \leq \T{Krulldim}(\Lambda(G)) =3. $$

Since the module $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ is $ \Lambda(G) $-torsion then  $ \T{depth}(\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)) \leq 2$.
If the depth of $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)$ is two, then the Auslander–Buchsbaum formula (cf. Chapter 19.3 of [5]) will imply that the projective dimension of $  \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ is equal to one.
By Proposition 3.10 of [21], any module of projective dimension at most one has no non-trivial non-trivial pseudo-null submodule.
By Proposition <ref>, $ H{^{1}(H,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty))}\Pd  $ vanishes, and

\begin{align}\label{lem:Tor{1} is MT1}
0=H{^{1}(H,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty))}\Pd \nonumber
H_{1}(H,\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty))\\
=\Tor_{1}^{\Lambda(H)}(\bZ_p,\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty))
&= \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)[T_1]
% = \{ x \in \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) | (T_1)\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)=0\}
\end{align}

where $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)[T_1] $ is the set of elements in $  \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)$ that are annihilated by $ T_1 $.
Therefore, the element $ T_1 $ is a non-zero divisor on $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $.

Proposition <ref> also tells us that

$ 	\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H \cong \mathfrak{X}^{\pm/\pm}(E_{p^{\infty}}/L_\cyc)  $

as $ \Lambda(\Gamma) $-modules.
Note that $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H  $ is non-zero, otherwise, by Nakayama's lemma $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $  is trivial as  $ T_1 \in  \fm_G $.

Arguing as above, we see

\begin{align*}
H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc))\Pd
\cong
H_1(\Gamma,\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H)
%						&= \Tor_{1}^{\Lambda(\Gamma)}(\bZ_p,\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H)\\
= (\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H)[T_2].
\end{align*}

By Remark <ref>,  the group $ H^{1}(\Gamma,\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\cyc)) $ vanishes.
Hence, $ T_2 $ is a non-zero divisor on  $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)_H  $.
Note that using Nakayama's lemma  we can see that the $ \Lambda(G) $-module
$ 	\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)/(T_1,T_2)\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ is non-trivial.
Therefor, the set $ \{T_1,T_2\} $ is an $ \fm_{G} $-sequence on $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $.
Since $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) $ is  $ \Lambda(G) $-torsion, then  $ \T{depth}(\mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty))$ is two.

Similar to the first part, to show part (<ref>), it suffices to show that $ \mathfrak{X} (E_{p^{\infty}}/L_\infty) $ has projective dimension at most one.

Consider the short exact sequence:

\begin{align*}
0 \longrightarrow
\Sel^{\pm/\pm} (E_{p^{\infty}}/L_\infty) \longrightarrow \Sel (E_{p^{\infty}}/L_\infty) \longrightarrow \bigoplus_{v \in \{\fp,\bfp\}} J_v^{\pm}(E_{p^{\infty}}/L_\infty)
\longrightarrow 0.
\end{align*}

The last map is exact by Proposition <ref>.
Let $ 	U:= \bigoplus_{v \in \{\fp,\bfp\}} J_v^{\pm}(E_{p^{\infty}}/L_\infty)\Pd $, then
taking the Pontryagin dual of the above sequence yields

\begin{align}\label{eq:signed Selmer and Selmer infty with J_v pm}
0 \longrightarrow
U \longrightarrow \mathfrak{X} (E_{p^{\infty}}/L_\infty) \longrightarrow  \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)
\longrightarrow 0.
\end{align}

By Corollary 3.9 of [13], the $ \Lambda(G) $-module $ U $ is free of rank $ 2 $ and hence it has  projective dimension zero.

By part (<ref>), the depth of $ \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty)  $ is equal to two, and so by the Auslander–Buchsbaum–Serre formula it has projective dimension one.
Now, applying Lemma  10.109.9 of [20] to the short exact sequence (<ref>) shows that the projective dimension of $ \mathfrak{X} (E_{p^{\infty}}/L_\infty) $ is at most one.

Let us end this section by relating the signed $ \mu $-invariant $ \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $ to
the $ \mu $-invariant of the torsion $ \Lambda(G) $-submodule of the Pontryagin dual of the Selmer group.
Suppose $ M $ is a finitely generated $ \Lambda(G) $-module and  let $ T_{\Lambda(G) }(M) $ denote the $ \Lambda(G)  $-torsion submodule of $ M $.
Note that

\begin{align*}
T_{\Lambda(G)}(M) = \ker\left( M \to M \ot_{\Lambda(G)} K \right),
\end{align*}

where $ K = \T{Frac}(\Lambda(G)) $ is the fraction field of $ \Lambda(G) $.
With abuse of terminology, by the $ \mu$-invariant of $ M $, we mean the $ \mu $-invariant of the $ \Lambda(G)  $-torsion submodule of $ M $.

Suppose $E/L  $ is an elliptic curve satisfying [Hyp1]Hyp 1 and [Hyp2]Hyp 2$ ^{\pm/\pm} $.
Then, the $ \mu $-invariant of

$\mathfrak{X}(E_{p^{\infty}}/L_\infty)  $ is bounded by the signed $ \mu $-invariant $ \mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty) $.

Noting that $ K $ is a flat $ \Lambda(G) $-module
Thus, when we apply the functor $ - \ot_{\Lambda(G)} K $
to the short exact sequence (<ref>), we obtain the following diagram:

\[	\begin{tikzcd}
0 \arrow[r] &  U \arrow[r] \arrow[d]          &\mathfrak{X} (E_{p^{\infty}}/L_\infty)  \arrow[r] \arrow[d]    & \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) \arrow[d] \arrow[r] & 0\\
0 \arrow[r] &  U \ot K \arrow[r]         &\mathfrak{X} (E_{p^{\infty}}/L_\infty) \ot K  \arrow[r]     & \mathfrak{X}^{\pm/\pm} (E_{p^{\infty}}/L_\infty) \ot K \arrow[r] & 0.
\end{tikzcd}\]

The snake lemma implies the following exact sequence  of $ \Lambda(G) $-modules:

\begin{align}\label{eq:bounding_mu_invariants}
\begin{tikzcd}
0 \rightarrow
\rightarrow   \mathfrak{X}^{\pm/\pm}(\mathsf{E}_{p^{\infty}}/L_\infty)
\rightarrow
U \ot_{\Lambda(G)} K/\Lambda(G)
\\
\rightarrow
\mathfrak{X}(\mathsf{E}_{p^{\infty}}/L_\infty) \ot_{\Lambda(G)} K/\Lambda(G)
\rightarrow
\end{tikzcd}
\end{align}

The exact sequence (<ref>) implies that the $ \Lambda(G) $-module $ 			T_{\Lambda(G)}(\mathfrak{X}(\mathsf{E}_{p^{\infty}}/L_\infty))  $ embeds into $ \mathfrak{X}^{\pm/\pm}(\mathsf{E}_{p^{\infty}}/L_\infty) $.
This means that

\[
\mu_G\left(T_{\Lambda(G)}(\mathfrak{X}(\mathsf{E}_{p^{\infty}}/L_\infty))\right) \leq	\mu_G^{\pm/\pm}(E_{p^{\infty}}/L_\infty).
\]

A similar argument shows this result for the cyclotomic $ \bZ_p $-extensions.
Also, note that Proposition <ref> does not assume  [Conjecture A]Conjecture A.

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# DeepTracks: Geopositioning Maritime Vehicles in Video Acquired from a Moving
Platform

Jianli Wei, Guanyu Xu, Alper Yilmaz, Senior Member, IEEE Photogrammetric
Computer Vision Lab., The Ohio State University, Columbus, OH, USA
{wei.909, xu.2186<EMAIL_ADDRESS>
###### Abstract

Geopositioning and tracking a moving boat at sea is a very challenging
problem, requiring boat detection, matching and estimating its GPS location
from imagery with no common features. The problem can be stated as follows:
given imagery from a camera mounted on a moving platform with known GPS
location as the only valid sensor, we predict the geoposition of a target boat
visible in images. Our solution uses recent ML algorithms, the camera-scene
geometry and Bayesian filtering. The proposed pipeline first detects and
tracks the target boat’s location in the image with the strategy of tracking
by detection. This image location is then converted to geoposition to the
local sea coordinates referenced to the camera GPS location using plane
projective geometry. Finally, target boat local coordinates are transformed to
global GPS coordinates to estimate the geoposition. To achieve a smooth
geotrajectory, we apply unscented Kalman filter (UKF) which implicitly
overcomes small detection errors in the early stages of the pipeline. We
tested the performance of our approach using GPS ground truth and show the
accuracy and speed of the estimated geopositions. Our code is publicly
available at https://github.com/JianliWei1995/AI-Track-at-Sea.

###### Index Terms:

Object Tracking by Detection, Deep Learning, Geopositioning, Plane Projective
Geometry, Unscented Kalman Filter

##  I Introduction

Tracking boats in images is a hard problem especially in oblique viewpoints.
Some challenges include partial and full occlusions, small target size
especially when the boats are close to horizon line and the area within the
image where the boats occur. Adding to these challenges, geopositioning the
tracked boat complicates the problem further. This paper proposes a systems
approach to geopositioning a moving boat using image sequence acquired from
another moving platform.

While there is an influx of papers on object detection and tracking by
detection, to the best of our knowledge there is no published literature on
geopositioning tracked objects especially for maritime scenarios. By and
large, one can apply any object detection and tracking algorithm for image
sequences of boats on sea and there are several surveys in the field on their
performances. In this paper, we use existing methods, such as [1] and [2], to
perform this task. Given the detected set of boats, without the loss of
generality, we track the boat that has ground truth GPS information to assess
the developed algorithm.

Estimating the position of the tracked boat requires establishing absolute
geometry between the camera and the scene which in this case is the sea. There
are two bottlenecks in solving this problem. The first of these is the lack of
training data to generate a network solution as in [3] where a deep learning
framework is used to estimate camera pose. The second problem is the lack of
features in the limited sea area that defines the view geometry. In order to
mitigate these shortcomings, we model the visible sea region as a plane and
estimate the plane projective relation [4] between the camera and the sea.
While the earth, hence the sea, has a curvature piece-wise planarity
assumption is shown to work well our experiments. The geopositioning of the
boat is followed by application of the UKF to ensure the predicted GPS
location produces a smooth sea trajectory.

The contributions of this paper to geopositioning of boats from image
sequences include: 1) being the first paper on geopositioning of sea targets
in monocular image sequences; 2) provide a goal oriented neural network
modeling the view geometry without traditional learning mechanisms 3) an end
to end system composed of detection, tracking and geopositioning modules.

The rest of this paper is organized as follows. Section II reviews related
work in object detection, tracking and geopositioning. Section III introduces
the problem and discusses the details of the proposed approach. Section IV
demonstrates the performance and Section V concludes the paper.

##  II Related Work

Object detection is fundamental problem and active area of research in image
processing and computer vision. There have been numerous publications on it
over the past few decades, such as two-stage R-CNN series [5, 6, 7] and one-
stage ”you only look once” (YOLO) series [8, 9, 10, 11, 12]. These one stage
models with pre-trained weights perform well enough for the object detection
sub-task in our approach.

Object tracking has also been an active area of research over the past
decades. Researchers have proposed numerous methods to track single or
multiple objects [13, 14]. In this paper, among alternatives, we choose to
apply a tracking by detection (TBD), that associates detected objects across
consequent images. As common practice in TBD, we introduce appearance and
distance constraints to ensure correct association. The appearance and
distance constraints respectively consider the width of the boat and the
proximity of the detected tracked object in consecutive frames.

In this paper, typical to many other practical implementations of object
tracking methods, the association in the image space for tracking, and
geopositioning the tracked boat in the object space use Bayesian filtering. In
particular, the tracking step is accompanied by a Kalman Filter [15, 16] to
model random noise in the object detection step. The geopositioning that
generate geo-trajectories uses Unscented Kalman Filter (UKF), which is chosen
to model non Gaussian noise that occur due to projective transformation of the
detected boat location. The UKF uses the speed and direction as object state
and models their change over time.

##  III Methodology

Geopositioning an object tracked in a monocular image sequence introduces a
number of challenges that include precise tracking of the boat in the image
plane (pixel errors translate to large geoposition errors); defining a
geometric relation between image and GPS coordinate systems; an end-to-end
system that resolves this problem; reduction of geoposition errors due to
large oblique view point.

To address these challenges, the proposed end-to-end system uses a cascade of
neural networks that is composed of several modules: the coordinate
transformation module, detection module, classification module, tracking
module and post processing module illustrated in Figure 1. The first module is
a linear Multilayer Perceptron (MLP) that performs the geometric
transformation between the image and the world reference frames. This is
followed by an boat detection module which also contributes to the generation
the MLP. The method proceeds by verification of the detected boat using a CNN
classifier, and Kalman filtering in both the image and world coordinates to
generate tracks and the geoposition of the target boat.

Figure 1: Flow diagram showing the relation between the geometric, detection,
classification, tracking and geopositioning modules.

###  III-A Coordinate transformation

Obtaining position in the world frame from a monocular camera is not a trivial
task. First and foremost, the developed approach should include a methodology
to transfer the boat location in the video to GPS location in world reference
frame. In order to achieve this goal, we introduce a linear MLP based geometry
model that transfers pixel location to absolute sea coordinates which is
modeled as piece-wise plane. In this model, the MLP parameters are estimated
using the training set provided with the dataset which contains the image
position $(x,y)$ of a boat and its corresponding GPS position $(lat,lon)$.
Using this quadruplet $<(x,y),(lat,lon)>$, we set the north and east
directions of the camera as x and y axes of the quasi sea-plane that overlap
with the GPS coordinates in western hemisphere. Since, the variation of
latitude and longitude as the boat sails remain infinitesimally small over
time, we convert the latitude and longitude differences to meters on the sea-
plane. The boats in the image are detected using a trained version of the YOLO
object detector from which the bottom center of the boat’s bounding box that
lies on the sea surface is used as the boat location. This and its
corresponding GPS location is then used to compute the plane-projective
transformation that estimates the homography matrix H between the two
coordinates:

$T_{world}*X_{world}=H*T_{local}*X_{local}$ (1)

where $T_{world}$ and $T_{local}$ are normalization matrices that serve as
preconditioning transformations to decrease condition number of H, $X_{world}$
and $X_{local}$ pairs are respectively homogeneous world geometry coordinates
and homogeneous local pixel coordinates. The resulting H matrix along with the
lens distortion parameters of the camera are converted into a linear MLP as
shown in Figure 2.

Figure 2: Linear MLP modeling the geometric transformation between the camera
and world reference frames.

###  III-B Object detection and tracking

The detection of the boats in the video sequences uses YOLO [12], which is
arguably one of the most advanced object detection algorithms that can detect
multiple objects in real time. In our implementation, we utilized YOLO to
detect only ‘boats’. Given the detected boats in consecutive frames, tracking
is accomplished by a two stage Kalman filtering operation. The first stage in
Kalman filtering operates on the consecutive images that ensures consistent
boat trajectory and reduce object detection errors that are assumed Gaussian.
It uses the output from the object detector as the observations used to
correct the boat state ($X_{local}$,$\hat{X}_{local}$ ). In particular, we use
the boat position (x,y) and the boat width (w) as observations in the image
frame. Besides, an additional VGG-16 [17] based boat classifier trained using
the training data improves methods robustness by assisting the Kalman filter
and ensures proximal and similar boat is associated to the boat in the
previous frame. We compute a detection score by convexly combining the
classification network probability $P_{c}$ and Kalman filter distance
distribution $P_{K}$:

$\displaystyle P_{K}=e^{-\frac{d^{2}}{\sigma^{2}}}$ (2) $\displaystyle
P=\alpha*P_{C}+(1-\alpha)*P_{K}$ (3)

where d is predicted target boat distance between two continuous frames,
$\sigma$ is the distance standard derivation, $\alpha$ is the combination
parameter and P is the combined probability factor for the tracked target
boat.

Figure 3: Predicted raw geo-trajectory and processed geo-trajectory using UKF
in the world frame.

The second stage Kalman filter operates in the world frame to generate
consistent latitude longitude information. For both stages, the boat’s states
are composed of positions $X_{local}$ and $X_{world}$ as well as the speeds
$\hat{X}_{local}$ and $\hat{X}_{world}$ in respective coordinate systems. This
Bayesian filtering is necessary to reduce problems related to the weak
geometric relation between the image and world frames. The weak geometry is
due to small support area where the quasi-collinear boat locations in the
training dataset occur and that only provides observations close to the
horizon. Hence, any small error, such as 1 pixel, in the image contributes to
meters of error in the world frame. This error can be reduced to a certain
degree by the second stage Kalman filtering which provides smooth boat
trajectories. In Figure 3, we show the predicted boat trajectory in the world
frame before/after the application of the second stage UKF.

##  IV Experiments

In the training process, we trained a boat classifier for classifying target
boat by appearance only and use the GPS positions in the training set to
estimate the geometric relation modeled as a linear MLP. The experiments of
the proposed system uses the dataset provided by the Office of Naval Research.

###  IV-A Dataset

The dataset used for evaluation of the proposed approach includes monocular
image sequence that contains GPS of a target boat and geoposition of the
camera. The sequence contains multiple boats in motion within the field of
camera view, including the target boat which the proposed approach seeks to
distinguish, track and estimate a geoposition. Target boat GPS data provided
every 1 to 5 seconds is used as ground truth. The camera used to acquire the
sequences is on a moving platform and has an associated GPS information.
Dataset is available at
https://drive.google.com/drive/folders/1Eq7afvav49OmWo5iNSKL7fEJk-TGNp0V.

###  IV-B Implementation details

In our experiment, for target boat tracking, we introduce a probability
threshold, such that we considered it as our target boat and the algorithm
continues tracking if combined probability P exceeded the threshold. We preset
the threshold $P_{thr}$=0.51, combination coefficient $\alpha=0.5$ and
distance standard derivation $\sigma=10$. No matter classifier probability or
Kalman filter approaches to 1, the target boat would be detected and tracked.
When the boat is recognized by our classifier and the Kalman filter is over
$P_{thr}$, we can ensure the reappeared boat is the target boat.

###  IV-C Results

Figure 4: (a) Boat trajectory on the video frame, ‘S’ and ‘E’ indicates where
the boat start and end its trajectory. (b) Predicted boat trajectory in the
world frame plotted against its ground truth with position denoted as blue
dot.

In Figure 4, we show the detected and tracked boat location in the image plane
4 and the predicted boat location in the world frame plotted against the
ground truth GPS position provided in the dataset 4.

##  V Conclusions and Future Work

Our AI and monocular camera based approach aims to estimate target boat gps
location at sea from the video. We noticed that the Kalman filter was not
capable to cope with some extreme conditions and tracking module shifts the
target to other neighboring boats. In the future, we will tackle this problem
and proposed a multi-boats track at sea that could predict multiple boats GPS
locations simultaneously.

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