T\otimes_{R}\Phi/R\cong V_{\alpha}/T_{\alpha}.$

Let $\gamma\in\Gamma$ be a topological generator, and set
$C_{\alpha}\coloneq\begin{cases}v_{p}(\alpha(\gamma)-\alpha^{-1}(\gamma))&\alpha\neq\alpha^{-1},\\\
0&\alpha=\alpha^{-1}.\end{cases}$

The first remarkable result from [CGLS22] is the following theorem
generalizing [How04, Theorem 2.2.10].

######  Theorem 3.0.1.

Suppose $\alpha\neq 1$ and that there is a Kolyvagin system
$\kappa_{\alpha}=\\{\kappa_{\alpha,n}\\}_{n\in\mathcal{N}}\in\mathbf{KS}(T_{\alpha},\mathcal{F}_{{\operatorname{ord}}},\mathcal{L}_{f})$
with $\kappa_{1}$ being non-torsion. Then
$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,T_{\alpha})$ has rank
one, and there is a finite $R$-module $M_{\alpha}$ such that

$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,A_{\alpha})\cong(\Phi/R)\oplus
M_{\alpha}\oplus M_{\alpha}$

with

${\operatorname{length}}_{R}(M_{\alpha})\leq{\operatorname{length}}_{R}(\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,T_{\alpha})/R\cdot\kappa_{\alpha,1})+E_{\alpha}$

for some constant $E_{\alpha}\in\mathbf{Z}_{\geq 0}$ depending only on
$C_{\alpha},T_{f}$, and ${\operatorname{rk}}_{\mathcal{O}}(R)$.

In [CGS23, section 6], the above result was generalized so that the error term
can be made independent of $\alpha$, at the expense of requiring $\alpha$ to
be ‘sufficiently close to $1$’. More precisely, they have

######  Theorem 3.0.2.

Assume $\alpha\equiv 1\pmod{\varpi^{m}}$. Then there exist non-negative
integers $\mathcal{M}$ and $\mathcal{E}$ depending only on $T_{p}E$ and
rank${}_{\mathbf{Z}_{p}}(R)$ such that if $m\geq\mathcal{M}$ and if there is a
Kolyvagin system
$\kappa_{\alpha}=\\{\kappa_{\alpha,n}\\}_{n\in\mathcal{N}}\in\mathbf{KS}(T_{\alpha},\mathcal{F}_{{\operatorname{ord}}},\mathcal{L}_{f})$
with $\kappa_{1}$ being non-torsion. Then
$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,T_{\alpha})$ has rank
one, and there is a finite $R$-module $M_{\alpha}$ such that

$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,A_{\alpha})\cong(\Phi/R)\oplus
M_{\alpha}\oplus M_{\alpha}$

with

${\operatorname{length}}_{R}(M_{\alpha})\leq{\operatorname{length}}_{R}(\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{ord}}}}(K,T_{\alpha})/R\cdot\kappa_{\alpha,1})+\mathcal{E}$

and the ‘error term’ $\mathcal{E}$ is independent of $m$.

We mention that Theorem 3.0.1 (in combination with our Theorem 1.5.1 and
Theorem 2.2.3) is sufficient for us to get a desired BSD formula in rank $1$,
but together with Theorem 3.0.2 it leads to a stronger result on the
anticyclotomic Main Conjecture, in the sense that the divisibility in [CGLS22,
Theorem 3.4.1] holds more generally without the need to invert $\gamma-1$ (see
[CGS23, Theorem 6.5.1]).

Thanks to the highly axiomatized arguments from [CGLS22], [CGS23], we only
need to explain how to get the error term $C_{1}$ in op. cit., as everything
else can be directly extended to our more general setting.

According to the proof of [CGLS22, Proposition 3.6], we only need to find one
non-trivial scalar matrix in
$\mathcal{O}^{\times}\cap{\operatorname{im}}(\rho_{f}|_{G_{K_{\infty}}})$.

######  Lemma 3.0.3.

The intersection
$U=\mathcal{O}^{\times}\cap{\operatorname{im}}(\rho_{f}|_{G_{K_{\infty}}})$
contains infinitely many scalar matrices. Here by abuse of notation
$\mathcal{O}^{\times}$ denotes $\mathcal{O}^{\times}\cdot I_{2}$, the
invertible scalar matrices in ${\operatorname{GL}}_{2}(\mathcal{O})$.

###### Proof.

Note that by the discussion above [CGLS22, Lemma 3.3.3], we have

$\rho_{f}(G_{\mathbf{Q}})=\rho_{f}(G_{K_{\infty}}).$

Since $\rho_{f}$ is a Hodge–Tate representation, Bogomolov in [Bog80] showed
the image (possibly after embedding ${\operatorname{GL}}_{2}(F)$ in
${\operatorname{GL}}_{2n}(\mathbf{Q}_{p})$ for some $n$) is open in its
algebraic envelope. Since $\rho_{f}$ is irreducible by [Rib77], so is its
natural projection in ${\operatorname{PGL}}_{2}(F)$. We also know
${\operatorname{im}}(\rho_{f})$ must be infinite. Actually, there will be
infinitely many determinants since $\det\rho_{f}=\chi^{k-1}$ from op. cit. Now
by the classification of all algebraic subgroups of
${\operatorname{GL}}_{2}(F)$ (see [NvdPT08], whose results work over any
algebraically closed field of characteristic $0$), we see that the algebraic
envelope of ${\operatorname{im}}(\rho_{f})$ must be $F^{\times}$ times a
minimal group (in Theorem $4$ of op. cit., only (2)(a)(b) have infinitely many
determinants, yet they are reducible), therefore it must contain infinitely
many scalar matrices. So does the image. ∎

######  Remark 3.0.4.

It is mentioned in [CGLS22] that $f$ has no CM by $K$ under their assumptions.
When $f$ has no CM, there is a more straightforward proof of this openness
result, using Momose’s results (see for example [Rib85]) generalizing the open
image results of Serre.

For $U=\mathcal{O}^{\times}\cap{\operatorname{im}}(\rho_{f})$ as in Lemma
3.0.3, let

$C_{1}\colonequals\min\\{v_{p}(u-1):u\in U\\}.$

Since $U$ is an open subgroup of $\mathcal{O}^{\times}$, $0\leq C_{1}<\infty$.

The rest of the proof of Theorem 3.0.1 and Theorem 3.0.2 can be directly
borrowed from the corresponding papers. We next explain how to apply Iwasawa
theory to prove the Main Conjectures. First we study a consequence of Theorem
3.0.1, Theorem 3.0.2.

######  Theorem 3.0.5.

Suppose there is a Kolyvagin system
$\kappa\in\mathbf{KS}(\mathbf{T},\mathcal{F}_{\Lambda},\mathscr{L}_{f})$ where
$\kappa_{1}$ is non-torsion. Then
$\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ has $\Lambda$-rank one,
and there is a finitely generated torsion $\Lambda$-module $M$ such that

  1. (i)

$\mathcal{X}\sim\Lambda\oplus M\oplus M$,

  2. (ii)

${\operatorname{Char}}_{\Lambda}(M)$ divides
${\operatorname{Char}}_{\Lambda}(\text{{H}}^{1}_{\mathcal{F}}(K,\mathbf{T})/\Lambda\kappa_{1})$
in $\Lambda$.

###### Proof.

With Theorem 3.0.1 and Theorem 3.0.2, this theorem follows exactly as in
[CGLS22] and [CGS23], using specializations at height one primes of $\Lambda$.
We mention that in the aforementioned papers the divisibilities hold only
after inverting $p$, which is sufficient to their applications, but the
computation works for the prime $p$ too, as is readily explained in [How04,
Theorem 2.2.10]. ∎

The next result allows us to construct a non-trivial Kolyvagin system.

######  Theorem 3.0.6.

There exists a Kolyvagin system
$\kappa^{\operatorname{Hg}}\in\mathbf{KS}(\mathbf{T},\mathcal{F}_{\Lambda},\mathscr{L}_{f})$
such that
$\kappa^{\operatorname{Hg}}_{1}\in\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$
is non-torsion.

###### Proof.

The construction of Generalized Heegner cycles and how they form a Kolyvagin
system are essentially given in [LV16, Section 4]. We explain how to modify
their construction to our situation, following [CGLS22]. As in op. cit., we do
not assume $p\nmid h_{K}$ or the ‘big image’ result. The first change appears
in [LV16, Theorem 2.4], where the vanishing of $\text{{H}}^{0}(K_{\infty},W)$
now follows from our Proposition 1.3.3. As in [CGLS22], among the axioms in
the Kolyvagin systems in [LV16, section 3.1], (H.1) is replaced by (h.1) and
(H.2) is not needed. Their Lemma 4.2 still works if we first replace $p$ with
a generator $\pi$ of $\mathfrak{p}$, but this then implies the original result
for $p$. Finally, as in [CGLS22, Theorem 4.1.1], the isomorphism

$\text{{H}}^{1}(K[n],\mathbf{T}/T_{n}\mathbf{T})^{\mathcal{G}(n)}\xleftarrow{\simeq}\text{{H}}^{1}(K,\mathbf{T}/I_{n}\mathbf{T})$

follows from the fact that the extensions $K[n]$ and
$\mathbf{Q}(\overline{\rho}_{f})$ are linearly disjoint, and that
$\text{{H}}^{0}(K_{\infty},\overline{\rho}_{f})=0$. The construction in [LV16,
Section 4] now gives a Kolyvagin system in our setting.

That the class is non-trivial follows from [CH18, Proposition 3.9, Theorem
4.9]. Indeed, the left hand side of the equation in Theorem 4.9 in op. cit. is
nonzero by Proposition 3.9 there, so the right hand side is nonzero. Therefore
the class $z_{f}\neq 0$. But this is just
${\operatorname{cores}}_{K[1]/K}\beta_{0}[1]$ in [LV16, Proposition 4.9].
Therefore from its construction it’s easy to see the first Kolyvagin class
$\kappa_{1}$ is non-trivial. That $\kappa_{1}$ is non-torsion follows from the
fact that $\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ is torsion
free (see for example [How04, Theorem 2.2.9] where the condition
$\text{{H}}^{0}(K_{\infty},\overline{\rho}_{f})=0$ follows from (h.1)). ∎

We are ready to prove the Iwasawa Main Conjectures similar to those in
[CGLS22].

######  Theorem 3.0.7.

Assume that $p=v\overline{v}$ splits in $K$ and
$\text{{H}}^{0}(K,\overline{\rho}_{f})=0$ Then the following statements hold:

  1. (IMC1)

Both $\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ and
$\mathcal{X}=\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,M_{f})^{\vee}$ have
$\Lambda$-rank one, and the equality

${\operatorname{Char}}_{\Lambda}(\mathcal{X}_{{\operatorname{tors}}})={\operatorname{Char}}_{\Lambda}(\text{{H}}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})/\Lambda\kappa_{\infty})^{2}$

holds in $\Lambda_{ac}$.

  2. (IMC2)

Both $\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,\mathbf{T})$ and
$\mathfrak{X}_{f}=\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{f})^{\vee}$
are $\Lambda$-torsion, and the equality

${\operatorname{Char}}_{\Lambda}(\mathfrak{X}_{f})\Lambda^{{\operatorname{nr}}}=(\mathcal{L}_{f})$

holds in $\Lambda^{{\operatorname{nr}}}$.

###### Proof.

We begin by noting that (IMC1) and (IMC2) are equivalent; in fact, an either
side divisibility in one of them is equivalent to a corresponding divisibility
in the other (see [CGLS22, Prop. 4.2.1] and [BCK21, Thm. 5.2]). Theorem 3.0.5
shows one divisibility of (IMC1), which translates to a divisibility of
(IMC2). Using the equality between the algebraic and analytic Iwasawa
invariants (see Theorem 1.5.1 and Theorem 2.2.3), the divisibility in (IMC2)
is turned into an equality. Hence (IMC1) also holds. ∎

######  Remark 3.0.8.

As is explained in the beginning of section 1.4, the above theorem still holds
without assuming $\text{{H}}^{0}(K,\overline{\rho}_{f})=0$.

##  4\. Proof of the $p$-converse and the $p$-part of BSD formula

In this section, we discuss several applications of our results in the
previous sections, whose prove are direct generalizations of those in [CGLS22,
§ 5].

###  4.1. The $p$-converse theorem

######  Theorem 4.1.1.

Let $A/\mathbf{Q}$ be a simple RM abelian variety associated to a newform $f$
and $\mathfrak{P}\mid p>2$ a prime ideal of its endomorphism ring
$\mathcal{O}={\operatorname{End}}(A)$ of good reduction such that
$\rho_{A,\mathfrak{P}}$ is reducible.

Then

${\operatorname{corank}}_{\mathcal{O}}{\operatorname{Sel}}_{\mathfrak{P}^{\infty}}(A/\mathbf{Q})=r\in\\{0,1\\}\implies{\operatorname{rk}}_{\mathcal{O}}A(\mathbf{Q})=r_{\operatorname{an}}(f)=r,$

and $\Sha(A/\mathbf{Q})[\mathfrak{P}^{\infty}]$ is finite.

###### Proof.

The proof of [CGLS22, Theorem 5.2.1] works if one replaces the reference
[Mon96, Theorem 1.5] there by [Nek18, Theorem C] and replaces [CGLS22,
Corollary 4.2.3] by our Theorem 3.0.7 (IMC1). ∎

The $p$-converse theorem has the following applications to Goldfeld’s
conjecture in quadratic twist families with a $3$-isogeny where $3$ is a prime
of good or bad multiplicative reduction. We thank Ari Shnidman for
communicating and explaining them to us.

######  Remark 4.1.2.

  1. (i)

Better proportions of quadratic twists of (algebraic and analytic) rank $1$ in
[BKOS19, Theorem 2.5] for a fixed elliptic curve, in particular a lower bound
of $\frac{3}{4}\cdot\frac{5}{12}=\frac{5}{16}=31.25\,\%$ in the most
advantageous cases (for example, when it’s the curve having Cremona label
$19a3$). Note that the proportion claimed in [CGLS22, Remark 5.2.4] is only
for a subfamily (corresponding to good reduction and with their assumptions on
the isogeny character) of the twists. Indeed, for the twists $E_{d}$ of the
curve $E=19a3$, the results in loc. cit. only apply when $d\equiv 2\pmod{3}$
whereas our result also covers the cases when $d\equiv 1\pmod{3}$, covering at
least $\frac{3}{8}+\frac{3}{8}=\frac{3}{4}$ of the proportion of the
$3$-Selmer rank $1$ twists. The twists with $d\equiv 0\pmod{3}$ correspond to
bad additive reduction and are not covered by our $p$-converse theorems.

  2. (ii)

Our result seems not to improve the bounds for rank $0$ in [BKOS19] in the
situation of (i), but it allows one to quantify how often the BSD rank
conjecture holds. Namely, their results give us that at least $\frac{1}{4}$ of
the twists have $3$-Selmer rank $0$, hence at least
$\frac{3}{4}\cdot\frac{1}{4}=\frac{3}{16}$ of them have analytic rank $0$.
Together with (i), $\frac{3}{16}+\frac{5}{16}=\frac{1}{2}$ of the twists
satisfy the BSD rank conjecture.

  3. (iii)

Similar improvements apply to quadratic twist families of abelian varieties
$A$ of ${\operatorname{GL}}_{2}$-type over $\mathbf{Q}$ with a $3$-isogeny, as
in [BFS23, Theorem 1.4, Corollary 7.2] and [Shn21, Theorem 1.5]. For example,
in the context of the latter result (where the abelian varieties all have good
reduction at $3$), our Theorem 4.1.1 gives an unconditional lower bound of
$\frac{5}{16}$ for the proportion of twists with rank $1$ over the RM ring.

###  4.2. The $p$-part of BSD formula

######  Theorem 4.2.1.

Let $E/\mathbf{Q}$ be an elliptic curve and $p>2$ a prime of good reduction.
Assume that $E$ admits a cyclic $p$-isogeny with kernel
$C=\mathbf{F}_{p}(\varphi)$ for some character $\varphi\colon
G_{\mathbf{Q}}\to\mathbf{F}_{p}^{\times}$ (equivalently, $E[p]$ is reducible).
Assume that $r_{\operatorname{an}}(E)\in\\{0,1\\}$.

Then the $p$-part of the BSD formula holds for $E/\mathbf{Q}$, i.e.,

${\operatorname{ord}}_{p}\Big{(}\frac{L^{*}(E/\mathbf{Q},1)}{\Omega_{E}\cdot{\operatorname{Reg}}_{E/\mathbf{Q}}}\Big{)}={\operatorname{ord}}_{p}\Big{(}\frac{{\operatorname{Tam}}(E/\mathbf{Q})\\#\Sha(E/\mathbf{Q})[p^{\infty}]}{(\\#E(\mathbf{Q})_{\operatorname{tors}})^{2}}\Big{)}.$

Here, $L^{*}(E/\mathbf{Q},1)$ denotes the leading Taylor coefficient of
$L(E/\mathbf{Q},s)$ at $s=1$.

###### Proof.

This follows from [CGLS22, Theorem 5.3.1], where the use of Theorem 4.2.2 in
op. cit. is replaced by our Theorem 3.0.7 (IMC2). We remark that the
assumption that the character is either ramified at $p$ and even, or
unramified at $p$ and odd coming from [GV00] is now removed by [CGS23] in both
the rank $0$ and rank $1$ formulae, and the assumption that
$\varphi|_{G_{p}}\neq\mathbf{1},\omega$ is removed by us.

Note that to make the arguments in [CGLS22, Theorem 5.3.1] work, we need an
anticyclotomic control theorem which allows torsion submodules. This is done
in Appendix B. The formula in [CGLS22, Theorem 5.1.1] then becomes

$\\#\mathbf{Z}_{p}/f_{E}(0)=\\#\Sha(E/K)[p^{\infty}]\cdot\Bigg{(}\frac{\\#(\mathbf{Z}_{p}/(\frac{1-a_{p}+p}{p})\cdot\log_{\omega_{E}}P)}{[E(K)_{/{\operatorname{tors}}}:\mathbf{Z}\cdot
P]_{p}\cdot\\#(E(K)_{{\operatorname{tors}}})_{p}}\Bigg{)}^{2}\times\prod_{w}c_{w}(E/K)_{p},$

where $x_{p}\coloneq p^{{\operatorname{ord}}_{p}(x)}$ denotes the $p$-part of
a rational number $x$. We remark that in the control theorem from [JSW17],
their anticyclotomic Selmer group is actually the ‘Greenberg Selmer group’ for
$f$ (which agrees with the unramified Selmer group in [CGLS22]). However,
since $\ker({\operatorname{res}}_{M_{f}})$ is finite by Lemma 1.3.6, these two
Selmer groups generate the same characteristic ideals.

Together with the above formula coming from the control theorem and the fact
that $E(K)[p^{\infty}]=E(\mathbf{Q})[p^{\infty}]\oplus
E^{K}(\mathbf{Q})[p^{\infty}]$ (because $p>2$), equation (5.7) in [CGLS22]
becomes

$\displaystyle{\operatorname{ord}}_{p}\Big{(}\frac{L^{\prime}(E,1)}{{\operatorname{Reg}}(E/\mathbf{Q})\cdot\Omega_{E}\cdot\prod_{l}c_{l}(E/\mathbf{Q})}\Big{)}-{\operatorname{ord}}_{p}(\\#\Sha(E/\mathbf{Q}))+2{\operatorname{ord}}_{p}(E(\mathbf{Q})_{\operatorname{tors}})$
$\displaystyle=-\Big{(}{\operatorname{ord}}_{p}\Big{(}\frac{L(E^{K},1)}{\Omega_{E^{K}}\cdot\prod_{l}c_{l}(E^{K}/\mathbf{Q})}\Big{)}-{\operatorname{ord}}_{p}(\\#\Sha(E^{K}/\mathbf{Q}))+2{\operatorname{ord}}_{p}(E^{K}(\mathbf{Q})_{\operatorname{tors}})\Big{)}$

where the right-hand side vanishes by [CGLS22, Theorem 5.1.4], a result of
Greenberg–Vatsal. ∎

##  5\. Proof of the Iwasawa Main Conjecture with multiplicative reduction

Let $f\in S_{2}(\Gamma_{0}(N))$ be a newform of weight $2$ with coefficient
field $\mathbf{Q}(f)$. Let $F$ be a finite extension of the completion of
$\mathbf{Q}(f)$ at a chosen prime above $p\neq 2$ and let $\mathcal{O}$ be its
ring of integers. Let $\mathfrak{p}$ be the maximal ideal of $\mathcal{O}$ and
let $\mathbf{F}=\mathcal{O}/\mathfrak{p}$ be the residue field of
$\mathcal{O}$. Let $\Lambda=\mathcal{O}\llbracket T\rrbracket$ and
$\Lambda^{{\operatorname{nr}}}=\Lambda\hat{\otimes}_{\mathbf{Z}_{p}}\mathbf{Z}_{p}^{{\operatorname{nr}}}$.
Let $\overline{\rho}_{f}$ be the residual representation attached to $f$ and
assume $\overline{\rho}_{f}$ is reducible. Then there are characters
$\varphi,\psi$ such that there is an exact sequence

(5.1) $0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f}\to\mathbf{F}(\psi)\to 0.$

Further assume that $p\mid\mid N$, so $f$ has bad multiplicative reduction at
$p$. In this section, we prove the Iwasawa Main Conjectures for such forms
using Hida families, following the ideas in [Ski14].

As in the good reduction case, define the unramified Selmer group for $f$ by

$\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{f})=\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{f})\xrightarrow{{\operatorname{res}}_{{\operatorname{nr}},f}}\prod_{w\in\Sigma,w\nmid
p}\text{{H}}^{1}(I_{w},M_{f})^{G_{w}/I_{w}}\times\text{{H}}^{1}(I_{\overline{v}},M_{f})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}}.$

For a finite set $\Sigma$ of places of $K$ which contains $\infty$ and the
primes above divisors of $N$, and such that the finite places in $\Sigma$ are
all split in $K$, let $S=\Sigma\setminus\\{v,\bar{v},\infty\\}$ and define the
$S$-imprimitive Selmer group for $f$ by

$\text{{H}}^{1}_{\mathcal{F}^{S}_{{\operatorname{nr}}}}(K,M_{f})=\ker\Big{\\{}\text{{H}}^{1}(K^{\Sigma}/K,M_{f})\xrightarrow{{\operatorname{res}}^{S}_{{\operatorname{nr}},f}}\text{{H}}^{1}(I_{\overline{v}},M_{f})^{G_{\overline{v}}/I_{\overline{v}}}\Big{\\}}.$

Let
$\mathfrak{X}_{f}:=\text{{H}}^{1}_{\mathcal{F}_{{\operatorname{nr}}}}(K,M_{f})^{\vee}$
and
$\mathfrak{X}_{f}^{S}:=\text{{H}}^{1}_{\mathcal{F}^{S}_{{\operatorname{nr}}}}(K,M_{f})^{\vee}$
denote the dual Selmer groups.

According to [Ski14, Section 3.1], replacing $F$ by a finite extension if
necessary, for each integer $m>0$ we get

  1. (a)

a newform $f_{m}\in S_{k_{m}}(\Gamma_{0}(N))$ with $\mathbf{Q}(f_{m})\subset
F,\ k_{m}>2$ and $k_{m}\equiv 2\pmod{p-1}$ and such that
$a_{p}(f_{m})\in\mathcal{O}^{\times}$;

  2. (b)

an equality of ideals
$(\mathcal{L}^{S}_{f},\mathfrak{p}^{m})=(\mathcal{L}^{S}_{f_{m}},\mathfrak{p}^{m})\subseteq\Lambda^{{\operatorname{nr}}}$;

  3. (c)

$\overline{\rho}_{f}\cong\overline{\rho}_{f_{m}}$ is reducible. In particular,
the characters appearing in their semisimplifications are the same (both
globally and locally);

  4. (d)

$\mathfrak{X}^{S}_{f}(f_{m})$ is a torsion $\Lambda$-module and
Char$(\mathfrak{X}^{S}_{fm})=(\mathcal{L}^{S}_{f_{m}})\subseteq\Lambda^{{\operatorname{nr}}}$;

  5. (e)

$\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}$ has no nonzero finite
$\Lambda$-submodules, so
Fitt$(\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}})={\operatorname{Char}}(\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}})$.
Here $\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}$ denotes the
$\Lambda$-free part of $\mathfrak{X}^{S}_{f_{m}}$ and Fitt denotes the
$\Lambda$-Fitting ideal. The same is true for $f$.

(a) and (c) follow from Hida theory. Similarly as in op. cit., (b) comes from
a $2$-variable $p$-adic $L$-function (adding a ‘weight’ variable) for the Hida
families constructed in [Cas20, Theorem 2.11] which works for Eisenstein
primes. (d) follows from Theorem 3.0.7 (IMC2) for good ordinary forms $f_{m}$
in the previous sections, the surjectivity of global-to-local maps (Remark
1.2.3 and Remark 1.4.2) and the equality of local factors appearing in the
definition of the imprimitive Selmer groups and $p$-adic $L$-functions. (e)
comes from basic properties of Fitting ideals.

######  Lemma 5.0.1.

The $\mu$-invariants of $\mathfrak{X}^{S}_{f}$ and $\mathfrak{X}^{S}_{f_{m}}$
are $0$.

###### Proof.

This is essentially proved in Theorem 1.4.1 since the proof only relies on the
extension (5.1). Note that the characters can be arbitrary. ∎

By (d) above, for every $f_{m}$, we also have an extension

$0\to\mathbf{F}(\varphi)\to\overline{\rho}_{f_{m}}\to\mathbf{F}(\psi)\to 0.$

Theorem 1.5.1 then implies that
$\lambda(\mathfrak{X}^{S}_{f})=\lambda(\mathfrak{X}^{S}_{f_{m}})$. On the
other hand, since the $\mu$-invariants are $0$,
$\lambda(\mathfrak{X}^{S}_{f})$ (resp. $\lambda(\mathfrak{X}^{S}_{f_{m}})$)
determines $\mathfrak{X}^{S}_{f,{\operatorname{free}}}/\mathfrak{p}^{m}$
(resp. $\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}/\mathfrak{p}^{m}$).
Thus there is an equality of ideals

$({\operatorname{Fitt}}(\mathfrak{X}^{S}_{f,{\operatorname{free}}}),\mathfrak{p}^{m})=({\operatorname{Fitt}}(\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}),\mathfrak{p}^{m}).$

Then following the arguments in [Ski14, Section 3.1], we have the following
result.

######  Lemma 5.0.2.

${\operatorname{Char}}(\mathfrak{X}^{S}_{f})=(\mathcal{L}^{S}_{f})\subseteq\Lambda^{{\operatorname{nr}}}$.

###### Proof.

We have

$\displaystyle({\operatorname{Char}}(\mathfrak{X}^{S}_{f}),\mathfrak{p}^{m})$
$\displaystyle=({\operatorname{Char}}(\mathfrak{X}^{S}_{f,{\operatorname{free}}}),\mathfrak{p}^{m})$
$\displaystyle=({\operatorname{Fitt}}(\mathfrak{X}^{S}_{f,{\operatorname{free}}}),\mathfrak{p}^{m})$
$\displaystyle=({\operatorname{Fitt}}(\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}),\mathfrak{p}^{m})$
$\displaystyle=({\operatorname{Char}}(\mathfrak{X}^{S}_{f_{m},{\operatorname{free}}}),\mathfrak{p}^{m})$
$\displaystyle=({\operatorname{Char}}(\mathfrak{X}^{S}_{f_{m}}),\mathfrak{p}^{m})$
$\displaystyle=(\mathcal{L}^{S}_{f_{m}},\mathfrak{p}^{m})$
$\displaystyle=(\mathcal{L}^{S}_{f},\mathfrak{p}^{m}).$

The result then follows from taking limit with respect to $m$ as in [Ski14,
section 3.1]. ∎

######  Theorem 5.0.3.

${\operatorname{Char}}(\mathfrak{X}_{f})=(\mathcal{L}_{f})\subseteq\Lambda^{{\operatorname{nr}}}.$

###### Proof.

Similar to the good reduction case, this result follows from the comparison
between primitive and imprimitive Selmer groups (see Theorem 1.5.1, whose
proof still works in the multiplicative reduction setting) and the comparison
between primitive and imprimitive $p$-adic $L$-functions (see Theorem 2.2.2).
We mention that the same arguments as in [Kri16, Prop. 37] (which has an
assumption $p\nmid N$) still work since Kriz’s computation of the congruence
is with the $p$-depletion which is identified with the
$((p^{2})^{(0)})$-stabilization, which allows $p$ in the level. ∎

Similar as in the good reduction case, we also obtain a $p$-converse theorem
for multiplicative reduction.

######  Corollary 5.0.4.

Let $A/\mathbf{Q}$ be a simple RM abelian variety associated to a newform $f$
and $\mathfrak{P}\mid p>2$ a prime ideal of its endomorphism ring
$\mathcal{O}={\operatorname{End}}(A)$ of multiplicative reduction such that
$\rho_{A,\mathfrak{P}}$ is reducible.

Then

${\operatorname{corank}}_{\mathcal{O}}{\operatorname{Sel}}_{\mathfrak{P}^{\infty}}(A/\mathbf{Q})=r\in\\{0,1\\}\implies{\operatorname{rk}}_{\mathcal{O}}A(\mathbf{Q})=r_{\operatorname{an}}(f)=r,$

and $\Sha(A/\mathbf{Q})[\mathfrak{P}^{\infty}]$ is finite.

##  Appendix A A Kolyvagin system argument

As in [How04], by a coefficient ring $R$ we mean a complete, Noetherian, local
ring with finite residue field of characteristic $p$. If $L$ is a perfect
field, we denote by $\mathbf{Mod}_{R,L}$ the category of finitely generated
$R$-modules equipped with continuous, linear actions of $G_{L}$, assumed to be
unramified outside of a finite set of primes in the case where $L$ is a global
field.

In this subsection, $R^{(k)}$ is assumed to be an Artinian principal idea ring
of length $k$. Denote its residue field by $\mathbf{F}$. Let $T^{(k)}$ be an
object of $\mathbf{Mod}_{R^{(k)},G_{K}}$. Fix a generator $\pi^{(k)}$ of the
maximal ideal $\mathfrak{m}^{(k)}$ of $R^{(k)}$. We will ignore the indexes
and simply write $\pi$ and $\mathfrak{m}$ when the $k$ are clear from the
contexts. Let $T^{(k)*}={\operatorname{Hom}}(T^{(k)},R(1))$ and fix a Selmer
structure $\mathcal{F}$ on $T^{(k)}$. Let $\mathcal{F}^{*}$ denote the dual
Selmer structure on $T^{(k)*}$. In all what follows we assume that
$(T^{(k)},\mathcal{F})$ and $(T^{(k)*},\mathcal{F}^{*})$ satisfy the
hypotheses $\text{{H}}.0$ and $\text{{H}}.3$–$\text{{H}}.5$ in op. cit. § 1.3.
In particular, we do _not_ assume $\text{{H}}^{0}(K,\overline{\rho}_{f})=0$
(assumption (h1) in [CGLS22, § 3.2]), and $\text{{H}}.2$ is not necessary for
our purpose.

The main result of this appendix is Theorem A.1.1, which follows from Lemma
A.1.4 and the arguments in [How04].

We will make the following assumption about stabilizing $\text{{H}}^{0}$
groups throughout this section:

######  Assumption A.0.1.

There is an absolute constant $N$ such that $\text{{H}}^{0}(K,T^{(k)})\cong
R/\pi^{k}$ when $k\leq N$, and $\text{{H}}^{0}(K,T^{(k)})\cong R/\pi^{N}$ for
all $k>N$.

Note that by a version of Proposition 1.3.4 (i) in the case where
$\text{{H}}^{0}(K,\overline{\rho}_{f})\neq 0$, $\text{{H}}^{0}(K,V_{f}/T_{f})$
is still finite (since $\text{{H}}^{0}(K,V_{f})=0$ because $\rho_{f}$ is
irreducible), this assumption is satisfied when $T^{(k)}=T_{f}/\pi^{k}T_{f}$
where $\pi$ is the uniformizer of $R$. For the exposition, we often denote
$\text{{H}}^{i}(K,-)$ by $\text{{H}}^{i}(-)$.

Following [How04, Lemma 1.3.3] and [MR04, Lemma 3.5.4], for $i\leq k$, we
would like to compare
$\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})$ and
$\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}[\mathfrak{m}^{i}])$ with
$\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)})[\mathfrak{m}^{i}]$. Note that the
first two groups are isomorphic (see [How04, Remark 1.1.4]), we will use
$T^{(k)}/\mathfrak{m}^{i}T^{(k)}$ and $T^{(k)}[\mathfrak{m}^{i}]$
interchangeably without further explanation in cohomology groups. In [How04],
the vanishing of several $\text{{H}}^{0}$ groups forces the natural maps
between the above groups to be isomorphisms. Since we allow non-trivial
$\text{{H}}^{0}$ groups, we need to study the kernels and cokernels more
carefully.

######  Lemma A.0.2.

Let $k\geq i$. The injection $\pi^{k-i}\colon
T^{(k)}/\mathfrak{m}^{i}T^{(k)}\hookrightarrow T^{(k)}$ induces morphisms

$\displaystyle[\pi^{k-i}]\colon$
$\displaystyle(\text{{H}}^{1}(K,T^{(k)}[\mathfrak{m}^{i}])\stackrel{{\scriptstyle\sim}}{{\longrightarrow}})\
\text{{H}}^{1}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})\to\text{{H}}^{1}(K,T^{(k)})[\mathfrak{m}^{i}]$
$\displaystyle[\pi^{k-i}]\colon$
$\displaystyle(\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}[\mathfrak{m}^{i}])\stackrel{{\scriptstyle\sim}}{{\longrightarrow}})\
\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})\to\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)})[\mathfrak{m}^{i}]$

with finite kernel $K^{(i)}$ and cokernel $C^{(i)}$ and finite kernel
$K_{\mathcal{F}}^{(i)}$ and cokernel $C_{\mathcal{F}}^{(i)}$, respectively.

One has
$|K^{(i)}|=|K_{\mathcal{F}}^{(i)}|=|C^{(i)}|=\frac{|\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})||\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{k-i}T^{(k)})|}{|\text{{H}}^{0}(K,T^{(k)})|}\geq|C_{\mathcal{F}}^{(k)}|$.

###### Proof.

We want to point it out that this is essentially already proved in [MR04,
Lemma 3.5.4]. Cohomology of the following two exact sequences

  1. (i)

$0\to
T^{(k)}/\mathfrak{m}^{i}T^{(k)}\xrightarrow{\pi^{k-i}}T^{(k)}\xrightarrow{p_{1}}T^{(k)}/\mathfrak{m}^{k-i}T^{(k)}\to
0$

  2. (ii)

$0\to
T^{(k)}/\mathfrak{m}^{k-i}T^{(k)}\xrightarrow{\pi^{i}}T^{(k)}\xrightarrow{p_{2}}T^{(k)}/\mathfrak{m}^{i}T^{(k)}\to
0$

gives
$K^{(i)}\simeq\frac{\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{k-i}T^{(k)})}{p_{1}(\text{{H}}^{0}(K,T^{(k)}))}$
and
$C^{(i)}\simeq\frac{\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})}{p_{2}(\text{{H}}^{0}(K,T^{(k)}))}$.
(In particular, they are finite, so we can speak about their order.) But
$|p_{1}(\text{{H}}^{0}(K,T^{(k)}))|=\frac{|\text{{H}}^{0}(T^{(k)})|}{|\text{{H}}^{0}(T^{(k)}/\mathfrak{m}^{i}T^{(k)})|}$
and
$|p_{2}(\text{{H}}^{0}(K,T^{(k)}))|=\frac{|\text{{H}}^{0}(T^{(k)})|}{|\text{{H}}^{0}(T^{(k)}/\mathfrak{m}^{k-i}T^{(k)})|}$,
so
$|K^{(i)}|=|C^{(i)}|=\frac{|\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{i}T^{(k)})||\text{{H}}^{0}(K,T^{(k)}/\mathfrak{m}^{k-i}T^{(k)})|}{|\text{{H}}^{0}(K,T^{(k)})|}$.

To pass to Selmer groups, consider the following diagram (omitting $K$ in the
cohomology groups) with the “map defining the Selmer groups” the global-to-
local map:
${0}$${K_{\mathcal{F}}^{(i)}}$${K^{(i)}}$${\ker(f)}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{i}])}$${\text{{H}}^{1}(T^{(k)}[\mathfrak{m}^{i}])}$${{\operatorname{im}}(\text{map\
defining\
}\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{i}]))}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{i}]}$${\text{{H}}^{1}(T^{(k)})[\mathfrak{m}^{i}]}$${{\operatorname{im}}(\text{map\
defining\
}\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}))[\mathfrak{m}^{i}]}$${0}$${C_{\mathcal{F}}^{(i)}}$${C^{(i)}}$${{\operatorname{coker}}(f)}$$\scriptstyle{f_{\mathcal{F}}}$$\scriptstyle{f_{1}}$$\scriptstyle{f}$
What is proved in [MR04, Lemma 3.5.4] is equivalent to $\ker(f)=0$. Thus by
the snake lemma $K_{\mathcal{F}}^{(i)}=K^{(i)}$ and
$C_{\mathcal{F}}^{(i)}\hookrightarrow C^{(i)}$. The claim follows. ∎

######  Remark A.0.3.

In the situations considered by the aforementioned authors (i.e., if (h1)
holds), we have $K^{(i)}=C^{(i)}=0$. Then it easily follows that
$K_{\mathcal{F}}^{(i)}=C_{\mathcal{F}}^{(i)}=0$, too. In our situation, three
of them are equal and can be described explicitly, but $C_{\mathcal{F}}^{(i)}$
is only bounded from above. It seems very hard to determine
$C_{\mathcal{F}}^{(i)}$ completely, but as we will see eventually, it is
enough to control its growth using induction.

###  A.1. Proof of a structure theorem of finite level Selmer groups

To ease notation, let $(T,\mathcal{F},\mathcal{L})$ denote the Selmer triple
$(T_{\alpha},\mathcal{F}_{{\operatorname{ord}}},\mathcal{L}_{f})$, and let
$\rho=\rho_{\alpha}$. For any $k\geq 0$, let

$R^{(k)}=R/\mathfrak{m}^{k}R,\ T^{(k)}=T/\mathfrak{m}^{k}T,\
\mathcal{L}^{(k)}=\\{\ell\in\mathcal{L}:I_{\ell}\subset
p^{k}\mathbf{Z}_{p}\\},$

and let $\mathcal{N}^{(k)}$ be the set of square-free products of primes
$\ell\in\mathcal{L}^{(k)}$.

Recall that we have the self-duality $T\simeq T^{*}$ and $(T^{(k)})^{*}\simeq
T^{(k)}$.

######  Theorem A.1.1.

For every $n\in\mathcal{N}^{(k)}$ where $k\gg 0$, there is an $R^{(k)}$-module
$M^{(k)}(n)$ and an integer $\varepsilon$ such that there is a pseudo-
isomorphism with kernel and cokernel bounded independent of $k$ (which can be
computed explicitly):

(A.1)
$\text{{H}}^{1}_{\mathcal{F}(n)}(K,T^{(k)})\sim(R^{(k)})^{\varepsilon}\oplus
M^{(k)}(n)\oplus M^{(k)}(n).$

Moreover, $\varepsilon$ can be taken to be $\varepsilon\in\\{0,1\\}$ and is
independent of all $k$ for which $\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)})$ is
not free, and $n$.

###### Proof.

We follow the ideas in [How04, sections 1.4–1.5]. As in Theorem 1.4.2 in op.
cit., we still define

$V_{s}=\frac{\mathcal{H}[\mathfrak{m}^{s}]}{\mathfrak{m}\mathcal{H}[\mathfrak{m}^{s+1}]},\quad
W_{s}=\frac{\mathcal{H}[\mathfrak{m}]}{\mathfrak{m}^{s}\mathcal{H}[\mathfrak{m}^{s+1}]},$

where $\mathcal{H}=\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)})$ is a finitely
generated $R^{(k)}$-module and $1\leq s\leq k$.

We also define

$P_{s}=\frac{\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}/\mathfrak{m}^{s}T^{(k)})}{{\operatorname{proj}}(\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}/\mathfrak{m}^{s+1}T^{(k)}))},\quad
Q_{s}=\frac{\text{{H}}^{1}_{\mathcal{F}^{*}}(K,(T^{(k)})^{*}[\mathfrak{m}])}{\pi^{s}\text{{H}}^{1}_{\mathcal{F}^{*}}(K,(T^{(k)})^{*}[\mathfrak{m}^{s+1}])}.$

Here ${\operatorname{proj}}$ denotes the canonical projection and $Q_{s}$ can
be identified with
$\frac{\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}[\mathfrak{m}])}{\pi^{s}\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)}[\mathfrak{m}^{s+1}])}$
by self-duality. To further ease notation, let
$\mathcal{H}(T[\mathfrak{m}^{i}])$ denote
$\text{{H}}_{\mathcal{F}}^{1}(K,T^{(k)}[\mathfrak{m}^{i}])$. We may then
identity $P_{s}$ with
$\frac{\mathcal{H}(T[\mathfrak{m}^{s}])}{\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])}$
and rewrite $Q_{s}$ as
$\frac{\mathcal{H}(T[\mathfrak{m}])}{\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}])}$.
Note that these isomorphisms may not be canonical, but as will be clear
momentarily, we only care about the sizes of these groups.

The Cassels–Tate pairing (see [How04, Proposition 1.4.1, Theorem 1.4.2]) gives
a nondegenerate pairing of finite dimensional $R^{(k)}/\mathfrak{m}$-vector
spaces

$(\ ,\ )_{s,1}\colon P_{s}\times Q_{s}\to R^{(k)}[\mathfrak{m}].$

However, in our case, $P_{s}$ (resp. $Q_{s}$) is not necessarily isomorphic to
$V_{s}$ (resp. $W_{s}$). Moreover, $K_{s}\colonequals\ker(P_{s}\to Q_{s})$ is
not the same as $P_{s-1}$. What the arguments in [How04, Theorem 1.4.2] show
for our case is that for each $s<k$, $\frac{P_{s}}{K_{s}}$ is even dimensional
as $R^{(k)}/\mathfrak{m}$-vector spaces. We claim that $\frac{V_{s}}{V_{s-1}}$
(which is no longer identified with $\frac{P_{s}}{K_{s}}$) is also even
dimensional for all but maybe two $s$’s bounded (independent of $k$). This
will be proved in the following series of lemmas, where we compare the
dimensions of $\frac{P_{s}}{K_{s}}$ and $\frac{V_{s}}{V_{s-1}}$ using direct
computations.

More precisely, we will assume $\mathcal{H}$ is not free as an
$R^{(k)}$-module (otherwise it automatically has the desired structure (A.1)
with $M^{(k)}(n)=0$ and $\varepsilon=1$), and show that
$\frac{V_{s}}{V_{s-1}}$ is even dimensional for all $1\leq s<k$ except at $N$
and some $1\leq N_{0}\leq N+1$ independent of $s$ to be defined later, where
$N$ is given as in Assumption A.0.1. When $N=N_{0}$, all
$\frac{V_{s}}{V_{s-1}}$ will be even. See Lemma A.1.4 for the precise
statement. Then by the structure theorem for finitely generated
$R^{(k)}$-modules, there is a finitely generated $R^{(k)}$-module $M^{(k)}$
such that

$\mathcal{H}\cong(R^{(k)})^{\varepsilon}\oplus M^{(k)}\oplus M^{(k)}\oplus
E^{(k)},$

where the “error term” $E^{(k)}=E^{(k_{1})}\oplus E^{(k_{2})}$ is a direct sum
of two (possibly isomorphic) cyclic $R^{(k)}$-modules of lengths between $1$
and $N+1$. Since this proof (and those of the following lemmas) still works if
we replace $\mathcal{F}$ by any $\mathcal{F}(n)$, we also get

$\text{{H}}_{\mathcal{F}(n)}^{1}(K,T^{(k)})\cong(R^{(k)})^{\varepsilon_{k}}\oplus
M^{(k)}(n)\oplus M^{(k)}(n)\oplus E^{(k)}(n)$

for some $R^{(k)}$-module $M^{(k)}$ where $E^{(k)}(n)$ again consists of two
cyclic summands of lengths between $1$ and $N+1$.

That $\varepsilon$ is independent of $n\in\mathcal{N}$ is due to the “parity
lemma” of [How04, Lemma 1.5.3], which says
$\rho(n)\colonequals\text{dim}_{R^{(k)}/\mathfrak{m}}\text{{H}}^{1}_{\mathcal{F}(n)}(K,\bar{T})\pmod{2}$
is independent of $n\in\mathcal{N}$. The same argument can be modified to work
in our situation. Indeed, we will see that
$\text{{H}}^{1}_{\mathcal{F}(n)}(K,\bar{T})$ and
$\text{{H}}^{1}_{\mathcal{F}(n)}(K,T^{(k)})[\mathfrak{m}]$ will have the same
size if $\text{{H}}^{1}_{\mathcal{F}(n)}(K,T^{(k)})$ is not free, and differ
by $\oplus\mathbf{F}$ otherwise. Then if
$\text{{H}}^{1}_{\mathcal{F}(n)}(K,T^{(k)})$ is not free, we have

$\varepsilon+2\text{dim}_{R^{(k)}/\mathfrak{m}}M(n)^{(k)}[\mathfrak{m}]+E^{(k)}(n)[\mathfrak{m}]=\text{dim}_{R^{(k)}/\mathfrak{m}}\text{{H}}_{\mathcal{F}(n)}^{1}(K,T^{(k)})[\mathfrak{m}]=\rho(n),$

so $\varepsilon\equiv\rho(n)\pmod{2}$, and if
$\text{{H}}^{1}_{\mathcal{F}(n)}(K,T^{(k)})$ is free,
$\varepsilon\equiv\rho(n)+1\pmod{2}$. ∎

######  Lemma A.1.2.

For a fixed $k\gg 0$ and $s<k$,

$K_{s}\colonequals\ker(P_{s}\to
Q_{s})\simeq\frac{\mathcal{H}(T[\mathfrak{m}^{s}])[\mathfrak{m}^{s-1}]}{\mathfrak{m}(\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}^{s}])}.$

###### Proof.

Recall that $\ker(P_{s}\to Q_{s})$ fits in the exact sequence

$0\to\ker(P_{s}\to
Q_{s})\to\frac{\mathcal{H}(T[\mathfrak{m}^{s}])}{\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])}\xrightarrow{\mathfrak{m}^{s-1}}\frac{\mathcal{H}(T[\mathfrak{m}])}{\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}])},$

so we can write $\ker(P_{s}\to
Q_{s})=\frac{(\mathfrak{m}^{s-1})^{-1}(\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}]))\cap\mathcal{H}(T[\mathfrak{m}^{s}])}{\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])}$.
We claim that the (well-defined) natural map

$\mathcal{H}(T[\mathfrak{m}^{s}])[\mathfrak{m}^{s-1}]\to\frac{(\mathfrak{m}^{s-1})^{-1}(\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}]))\cap\mathcal{H}(T[\mathfrak{m}^{s}])}{\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])}$

induced by quotient after inclusion is surjective.
Indeed, let
$c\in(\mathfrak{m}^{s-1})^{-1}(\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}]))\cap\mathcal{H}(T[\mathfrak{m}^{s}])$,
then $\mathfrak{m}^{s-1}c=\mathfrak{m}^{s}c_{0}$ for some
$c_{0}\in\mathfrak{m}^{s}\mathcal{H}(T[\mathfrak{m}^{s+1}])$. Let
$a=\mathfrak{m}c_{0}\in\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])\subset\mathcal{H}(T[\mathfrak{m}^{s}])$,
then $\mathfrak{m}^{s-1}(c-a)=0$ so
$c-a\in\mathcal{H}(T[\mathfrak{m}^{s}])[\mathfrak{m}^{s-1}]$ and $c-a$ maps to
the class of $c$.

We now compute the kernel of the above map. By definition, the kernel is

$\displaystyle\\{\text{classes\ of\
}c\in\mathcal{H}(T[\mathfrak{m}^{s}]):\mathfrak{m}^{s-1}c=0,c=\mathfrak{m}a,a\in\mathcal{H}(T[\mathfrak{m}^{s+1}])\\}$
$\displaystyle=$ $\displaystyle\\{\text{classes\ of\
c}\in\mathcal{H}(T[\mathfrak{m}^{s}]):\mathfrak{m}^{s-1}c=0,c=\mathfrak{m}a,\mathfrak{m}^{s}a=0,a\in\mathcal{H}(T[\mathfrak{m}^{s+1}])\\}$
$\displaystyle=$ $\displaystyle\\{\text{classes\ of\
c}\in\mathcal{H}(T[\mathfrak{m}^{s}]):\mathfrak{m}^{s-1}c=0,c=\mathfrak{m}a,a\in\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}^{s}]\\}$
$\displaystyle=$
$\displaystyle\mathfrak{m}(\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}^{s}]),$

hence the lemma. ∎

######  Lemma A.1.3.

For a fixed $k\gg 0$ and all $s<k$,

$\frac{|\frac{V_{s}}{V_{s-1}}|}{|\frac{P_{s}}{K_{s}}|}=(|K_{\mathcal{F}}^{(s)}|)^{-2}\times\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}\times\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|\times\Bigg{(}\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{s}|}/|C_{s-1}^{s}|\Bigg{)}^{-1}\times|K_{\mathcal{F}}^{(s-1)}|\times|K_{\mathcal{F}}^{(s+1)}|,$

where $K_{s}^{s+1},C_{s}^{s+1}$ (and similarly $K_{s-1}^{s},C_{s-1}^{s}$) are
the kernel and cokernel of the morphism

$\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)}[\mathfrak{m}^{s}])\to\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)})[\mathfrak{m}^{s}].$

###### Proof.

Denote by $N(P_{s}):=\mathcal{H}(T[\mathfrak{m}^{s}])$ and
$D(P_{s}):=\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])$ the numerator and
denominator of $P_{s}$. Similarly define
$N(K_{s}),D(K_{s}),N(V_{s-1}),D(V_{s-1}),N(V_{s}),D(V_{s})$. We will study the
following $4$ maps separately.

$N(P_{s})\to N(V_{s})$: This is the map

$0\to
K_{\mathcal{F}}^{(s)}\to\mathcal{H}(T[\mathfrak{m}^{s}])\xrightarrow{N_{P\to
V}}\mathcal{H}(T)[\mathfrak{m}^{s}]\to C_{\mathcal{F}}^{(s)}\to 0.$

So
$\frac{|N(P_{s})|}{|N(V_{s})|}=\frac{|K_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s)}|}$.

$D(P_{s})\to D(V_{s})$: This is the map

$\mathfrak{m}\mathcal{H}(T[\mathfrak{m}^{s+1}])\xrightarrow{D_{P\to
V}}\mathfrak{m}(\mathcal{H}(T)[\mathfrak{m}^{s+1}]).$

To study the kernel and cokernel of this map, we study the following diagram
${\bar{K}}$${K_{\mathcal{F}}^{(s+1)}}$${\ker(D_{P\to
V})}$${0}$${\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}]}$${\mathcal{H}(T[\mathfrak{m}^{s+1}])}$${{\operatorname{im}}(\mathfrak{m}_{(1)})}$${0}$${0}$${\mathcal{H}(T)[\mathfrak{m}]}$${\mathcal{H}(T)[\mathfrak{m}^{s+1}]}$${{\operatorname{im}}(\mathfrak{m}_{(2)})}$${0}$${\bar{C}}$${C_{\mathcal{F}}^{(s+1)}}$${{\operatorname{coker}}{D_{P\to
V}}}$$\scriptstyle{\mathfrak{m}_{(1)}}$$\scriptstyle{D_{P\to
V}}$$\scriptstyle{\mathfrak{m}_{(2)}}$
From the snake lemma, we see that
$\frac{|D(P_{s})|}{|D(V_{s})|}=\frac{|\ker(D_{P\to
V})|}{|{\operatorname{coker}}(D_{P\to
V})|}=\frac{|K_{\mathcal{F}}^{(s+1)}|}{|C_{\mathcal{F}}^{(s+1)}|}\frac{|\bar{C}|}{|\bar{K}|}$.

$N(K_{s})\to N(V_{s-1})$: This is the map

$\mathcal{H}(T[\mathfrak{m}^{s}])[\mathfrak{m}^{s-1}]\xrightarrow{N_{K\to
V}}\mathcal{H}(T)[\mathfrak{m}^{s-1}],$

To study the kernel and cokernel of this map, we need to consider the
following diagram with $2$ familiar maps:
${0}$${K_{s-1}^{s}}$${\mathcal{H}(T[\mathfrak{m}^{s}][\mathfrak{m}^{s-1}])}$${\mathcal{H}(T[\mathfrak{m}^{s}])[\mathfrak{m}^{s-1}]}$${C_{s-1}^{s}}$${0}$${0}$${K_{\mathcal{F}}^{(s-1)}}$${\mathcal{H}(T)[\mathfrak{m}^{s-1}]}$${\mathcal{H}(T)[\mathfrak{m}^{s-1}]}$${C_{\mathcal{F}}^{(s-1)}}$${0}$$\scriptstyle{=}$$\scriptstyle{N_{K\to
V}}$
Comparing the dimensions, we see that
$\frac{|N(K_{s})|}{|N(V_{s-1})|}=\frac{|K_{\mathcal{F}}^{(s-1)}|}{|K_{s-1}^{s}|}\frac{|C_{s-1}^{s}|}{|C_{\mathcal{F}}^{(s-1)}|}$.

$D(K_{s})\to D(V_{s-1})$: This is the map

$\mathfrak{m}(\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}^{s}])\xrightarrow{D_{K\to
V}}\mathfrak{m}(\mathcal{H}(T)[\mathfrak{m}^{s}]).$

To study the kernel and cokernel of this map, we study the following diagram
${\bar{K}}$${\hat{K}}$${\ker(D_{K\to
V})}$${0}$${\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}]}$${\mathcal{H}(T[\mathfrak{m}^{s+1}])[\mathfrak{m}^{s}]}$${{\operatorname{im}}(\mathfrak{m}_{(1)})}$${0}$${0}$${\mathcal{H}(T)[\mathfrak{m}]}$${\mathcal{H}(T)[\mathfrak{m}^{s+1}][\mathfrak{m}^{s}]}$${{\operatorname{im}}(\mathfrak{m}_{(2)})}$${0}$${\bar{C}}$${\hat{C}}$${{\operatorname{coker}}{D_{K\to
V}}}$$\scriptstyle{\mathfrak{m}_{(1)}}$$\scriptstyle{D_{K\to
V}}$$\scriptstyle{\mathfrak{m}_{(2)}}$
Note that the first column also appeared in the second case and the middle
column is already studied in the previous case (except we had $s-1$ in place
of $s$): we have
$\frac{|\hat{K}|}{|\hat{C}|}=\frac{|K_{\mathcal{F}}^{(s)}|}{|K_{s}^{s+1}|}\frac{|C_{s}^{s+1}|}{|C_{\mathcal{F}}^{(s)}|}$.
Now the snake lemma tells us that
$\frac{|D(K_{s})|}{|D(V_{s-1})|}=\frac{|\ker(D_{K\to
V})|}{|{\operatorname{coker}}(D_{K\to
V})|}=\frac{|\hat{K}|}{|\hat{C}|}\frac{|\bar{C}|}{|\bar{K}|}=\frac{|K_{\mathcal{F}}^{(s)}|}{|K_{s}^{s+1}|}\frac{|C_{s}^{s+1}|}{|C_{\mathcal{F}}^{(s)}|}\frac{|\bar{C}|}{|\bar{K}|}$.

Now

$\displaystyle\frac{|\frac{V_{s}}{V_{s-1}}|}{|\frac{P_{s}}{K_{s}}|}=$
$\displaystyle\frac{|N(V_{s})|}{|N(P_{s})|}\frac{|D(V_{s-1})|}{|D(K_{s})|}\frac{|D(P_{s})|}{|D(V_{s})|}\frac{|N(K_{s})|}{|N(V_{s-1})|}$
$\displaystyle=$
$\displaystyle\frac{|C_{\mathcal{F}}^{(s)}|}{|K_{\mathcal{F}}^{(s)}|}\frac{|K_{s}^{s+1}|}{|K_{\mathcal{F}}^{(s)}|}\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{s}^{s+1}|}\frac{|\bar{K}|}{|\bar{C}|}\frac{|K_{\mathcal{F}}^{(s+1)}|}{|C_{\mathcal{F}}^{(s+1)}|}\frac{|\bar{C}|}{|\bar{K}|}\frac{|K_{\mathcal{F}}^{(s-1)}|}{|K_{s-1}^{s}|}\frac{|C_{s-1}^{s}|}{|C_{\mathcal{F}}^{(s-1)}|}$
$\displaystyle=$
$\displaystyle(|K_{\mathcal{F}}^{(s)}|)^{-2}\times\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}\times\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|\times(\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{s}|}/|C_{s-1}^{s}|)^{-1}\times|K_{\mathcal{F}}^{(s-1)}|\times|K_{\mathcal{F}}^{(s+1)}|.$

∎

Despite that we do not have full understanding of each
$C_{\mathcal{F}}^{(i)}$, we are able to determine the parity of the dimension
of the right hand side by comparing these cokernels inductively, provided that
$k$ is sufficiently large.

######  Lemma A.1.4.

For a fixed $k\gg 0$, if $\text{{H}}^{1}_{\mathcal{F}}(K,T^{(k)})$ is not
$R^{(k)}$-free, then there is an $N_{0}$ with $1\leq N_{0}\leq N+1$ such that
$\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}\times\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|\times\Bigg{(}\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{s}|}/|C_{s-1}^{s}|\Bigg{)}^{-1}\times|K_{\mathcal{F}}^{(s-1)}|\times|K_{\mathcal{F}}^{(s+1)}|$
is an even power of $|\mathbf{F}|$ when $s\neq N_{0},N$. When $s=N_{0}$ or $N$
when $N\neq N_{0}$, the product is an odd power of $|\mathbf{F}|$, and when
$s=N_{0}=N$, the product is an even power. Consequently, by Lemma A.1.3,
$\frac{V_{s}}{V_{s-1}}$ is even-dimensional for all $s\neq N_{0},N$. When
$N\neq N_{0}$, it is odd-dimensional for $s=N,N_{0}$. When $N=N_{0}$, it is
even-dimensional for $s=N=N_{0}$ and hence for all $1\leq s<k$.

###### Proof.

Taking cohomology of the short exact sequence

$0\to T^{(k)}[\mathfrak{m}^{s}]\to
T^{(k)}[\mathfrak{m}^{s+1}]\xrightarrow{\mathfrak{m}^{s}}T^{(k)}[\mathfrak{m}]\to
0$

yields an exact sequence (omitting $K$ from the cohomology groups)

$0\to\text{{H}}^{0}(T^{(k)}[\mathfrak{m}])/\mathfrak{m}^{s}\to\text{{H}}^{1}(T^{(k)}[\mathfrak{m}^{s}])\to\text{{H}}^{1}(T^{(k)}[\mathfrak{m}^{s+1}])\to\text{{H}}^{1}(T^{(k)}[\mathfrak{m}])\to\ldots$

Note that $T^{(k)}[\mathfrak{m}^{s}]\simeq T^{(k)}/\mathfrak{m}^{s}\simeq
T^{(s)}/\mathfrak{m}^{s}\simeq T^{(s)}$, we have
$|\text{{H}}^{0}(T^{(k)}[\mathfrak{m}])/\mathfrak{m}^{s}|=\frac{|\text{{H}}^{0}(T^{(k)}[\mathfrak{m}])||\text{{H}}^{0}(T^{(k)}[\mathfrak{m}^{s}])|}{|\text{{H}}^{0}(T^{(k)}[\mathfrak{m}^{s+1}])|}=\frac{|\text{{H}}^{0}(\bar{T})||\text{{H}}^{0}(T^{(s)})|}{|\text{{H}}^{0}(T^{(s+1)})|}$.

We assume $k>2N+1$ and we consider several cases depending on how big $s$ is
and how close it is to $k$. When $1\leq s<N$, the above quotient is trivial by
computing dimensions. When $s\geq N$,
$|\text{{H}}^{0}(T^{(s)})|=|\text{{H}}^{0}(T^{(s+1)})|$ so the above quotient
is (non-canonically) $\text{{H}}^{0}(K,\bar{T})\cong\mathbf{F}$.

Now consider the diagram
${K^{(s)}}$${K^{(s+1)}}$${K^{(1)}}$${0}$${\frac{|\text{{H}}^{0}(\bar{T})||\text{{H}}^{0}(T^{(s)})|}{|\text{{H}}^{0}(T^{(s+1)})|}}$${\text{{H}}^{1}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}(T^{(k)}[\mathfrak{m}^{s+1}])}$${\text{{H}}^{1}(T^{(k)}[\mathfrak{m}])}$${0}$${\text{{H}}^{1}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}(T^{(k)})[\mathfrak{m}^{s+1}]}$${\text{{H}}^{1}(T^{(k)})[\mathfrak{m}]}$${C^{(s)}}$${C^{(s+1)}}$${C^{(1)}}$$\scriptstyle{f_{1}^{s}}$$\scriptstyle{f_{1}^{s+1}}$$\scriptstyle{f_{1}^{1}}$
When $1\leq s<N$ (so $k-s>N$), by Lemma A.0.2 and Assumption A.0.1,
$|K^{(s)}|=|C^{(s)}|=\frac{|\text{{H}}^{0}(K,T^{(s)})||\text{{H}}^{0}(K,T^{(k-s)})|}{|\text{{H}}^{0}(K,T^{(k)})|}=|\text{{H}}^{0}(K,T^{(s)})|=|\mathbf{F}|^{s}$
and $|K^{(s+1)}|=|C^{(s+1)}|=|\mathbf{F}|^{s+1}$. Note also that
$K^{(1)}=C^{(1)}=\mathbf{F}$.

To pass to the Selmer groups, we have the following diagram (noting
$K_{\mathcal{F}}^{(i)}=K^{(i)}$ for all $i$):
${R/\pi^{s}}$${R/\pi^{s+1}}$${\mathbf{F}}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s+1}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s+1}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]}$${C_{\mathcal{F}}^{(s)}}$${C_{\mathcal{F}}^{(s+1)}}$${C_{\mathcal{F}}^{(1)}(\subset
C^{(1)}\simeq\mathbf{F})}$$\scriptstyle{f_{\mathcal{F}}^{s}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{f_{\mathcal{F}}^{s+1}}$$\scriptstyle{f_{\mathcal{F}}^{1}}$$\scriptstyle{\mathfrak{m}^{s}_{(2)}}$
By the structure theorem of finitely generated $R^{(k)}$-modules, the map
$\mathfrak{m}^{s}_{(2)}$ is surjective if and only if
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ has a summand with height less than or
equal to $s$. Let $N_{0}\colonequals\min\\{s\leq
k:\mathfrak{m}^{s}_{(2)}\text{is not surjective}\\}$. Then it is clear that if
$s\geq N_{0}$, $\mathfrak{m}^{s}_{(2)}$ will also be non-surjective.

Note that when $N_{0}=k$, $\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ must be a
free $R^{(k)}$-module. We claim that if $N_{0}\neq k$, then $N_{0}\leq N+1$.
Namely, the lowest height of all non-free summands of
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ must have height less than or equal to
$N$. In other words, assuming $\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ is not a
free $R^{(k)}$-module, as soon as $s\geq N+1$, $\mathfrak{m}^{s}_{(2)}$ will
never be surjective.

Assume by contradiction that $N_{0}>N+1$. Then for all $1\leq s\leq N+1$,
$\mathfrak{m}^{s}_{(2)}$ is surjective. Consider the following diagram when
$s<N$
${0}$${R/\pi^{s}}$${R/\pi^{s+1}}$${K_{s}(\subset\mathbf{F})}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s+1}])}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(1)})}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s+1}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]}$${0}$${0}$${C_{\mathcal{F}}^{(s)}}$${C_{\mathcal{F}}^{(s+1)}}$${C_{s}(\subset
C^{(1)}\simeq\mathbf{F})}$${0}$$\scriptstyle{f_{\mathcal{F}}^{s}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{f_{\mathcal{F}}^{s+1}}$$\scriptstyle{f_{\mathcal{F}}^{1^{\prime}}}$$\scriptstyle{\mathfrak{m}^{s}_{(2)}}$
Since $K_{s}$ contains ${\operatorname{coker}}(R/\pi^{s}\to R/\pi^{s+1})$,
$K_{s}$ is at least $\mathbf{F}$, so $K_{s}=\mathbf{F}$. Similarly, when $s=N$
or $N+1$, the first row becomes

$0\to\text{{H}}^{0}(\bar{T})\cong\mathbf{F}\to\text{{H}}^{0}(T^{(k-s)})\cong
R/\pi^{N}\to\text{{H}}^{0}(T^{(k-s-1)})\cong R/\pi^{N}\to K_{s},$

and $K_{s}$ must be $\mathbf{F}$. Therefore the snake lemma shows that
$C_{\mathcal{F}}^{(s)}\to C_{\mathcal{F}}^{(s+1)}$ is injective with cokernel
$C_{s}$. We consider $2$ subcases.

Case I: $\mathfrak{m}^{s}_{(1)}$ is never surjective for $1\leq s\leq N+1$. In
this case, consider the rightmost vertical map $f^{1^{\prime}}_{\mathcal{F}}$.
If we replace its domain ${\operatorname{im}}(\mathfrak{m}^{s}_{(1)})$ by
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])$ which is strictly bigger
in dimension, since the codomain is fixed and the kernel cannot be any bigger,
its cokernel must change from $C_{s}$ to something strictly smaller. Therefore
$C_{s}\neq 0$ and $C_{s}=\mathbf{F}$. Then for all $1\leq s\leq N+1$, the
dimension of $C_{\mathcal{F}}^{(s)}$ increases whenever $s$ increases, so
$|C_{\mathcal{F}}^{(N+2)}|\geq|\mathbf{F}^{N+1}|>|K_{\mathcal{F}}^{(N+2)}|$.
But $C_{\mathcal{F}}^{(s)}\subset K_{\mathcal{F}}^{(s)}$ by Lemma A.0.2,
contradiction.

Case II: $\mathfrak{m}^{s}_{(1)}$ is surjective for some $1\leq s\leq N+1$. In
this case, the rightmost vertical map $f_{\mathcal{F}}^{1^{\prime}}$ agrees
with the map
$f_{\mathcal{F}}^{1}\colon\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])\to\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]$
and is independent of $s$. Therefore if for one such $s$ the cokernel of
$f_{\mathcal{F}}^{1^{\prime}}$ (so also that of $f_{\mathcal{F}}^{1}$) is
$\mathbf{F}$, then the cokernel is $\mathbf{F}$ for all such $s$. Therefore
$C_{\mathcal{F}}^{(s+1)}$ also has one more dimension than
$C_{\mathcal{F}}^{(s)}$ when $\mathfrak{m}^{s}_{(1)}$ is surjective. But for
other $s$, the same is true as in Case I, so again $|C_{\mathcal{F}}^{(N+2)}|$
would be too big. Thus the cokernel of $f_{\mathcal{F}}^{1}$ must be $0$. This
in turn means all $\mathfrak{m}^{s}_{(2)}$ must be surjective because
otherwise if we replace $\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]$
by ${\operatorname{im}}(\mathfrak{m}^{s}_{(2)})$, the cokernel of
$f_{\mathcal{F}}^{1}$ must be smaller (since $f_{\mathcal{F}}^{1}$ has fixed
domain and kernel), which is impossible. But in this case $N_{0}=k$ and
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ is free.

So far we showed that either $\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})$ is free
or it has a summand with height lower than $N+1$, and as a consequence we may
assume $\mathfrak{m}^{s}_{(2)}$ is not surjective for all $s>N$ which is
independent of $k$. Moreover, in this case $f_{\mathcal{F}}^{1}$ will have
kernel and cokernel both equal to $\mathbf{F}$, so in what follows, we will
keep this assumption.

We first study the dimensions appearing in this lemma when $k>s>N$. By Lemma
A.0.2 and Assumption A.0.1,
$|K^{(s)}|=|C^{(s)}|=\frac{|\text{{H}}^{0}(K,T^{(s)})||\text{{H}}^{0}(K,T^{(k-s)}|}{|\text{{H}}^{0}(K,T^{(k)})|}=|\text{{H}}^{0}(K,T^{(k-s)})|$
and $|K^{(s+1)}|=|C^{(s+1)}|=|\text{{H}}^{0}(K,T^{(k-s-1)})|$. Note also that
$K^{(1)}=C^{(1)}=\mathbf{F}$.

To pass to the Selmer groups, we have the following diagram(noting
$K_{\mathcal{F}}^{(i)}=K^{(i)}$ for all $i$):
${0}$${\text{{H}}^{0}(\bar{T})=\mathbf{F}}$${\text{{H}}^{0}(T^{(k-s)})}$${\text{{H}}^{0}(T^{(k-s-1)})}$${\mathbf{F}}$${0}$${\mathbf{F}}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s+1}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s+1}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]}$${C_{\mathcal{F}}^{(s)}}$${C_{\mathcal{F}}^{(s+1)}}$${C_{\mathcal{F}}^{(1)}(\subset
C^{(1)}\simeq\mathbf{F})}$$\scriptstyle{f_{\mathcal{F}}^{s}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{f_{\mathcal{F}}^{s+1}}$$\scriptstyle{f_{\mathcal{F}}^{1}}$$\scriptstyle{\mathfrak{m}^{s}_{(2)}}$
To apply snake lemma, we consider the following diagram where the first two
rows may or may not be changed depending on whether $\mathfrak{m}^{s}_{(1)}$
is surjective:
${0}$${\mathbf{F}}$${\text{{H}}^{0}(T^{(k-s)})}$${\text{{H}}^{0}(T^{(k-s-1)})}$${K(\subset\mathbf{F})}$${0}$${\mathbf{F}}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s+1}])}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(1)})}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s+1}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]}$${C_{\mathcal{F}}^{(s)}}$${C_{\mathcal{F}}^{(s+1)}}$${C_{\mathcal{F}}^{(1)}(\subset
C^{(1)}\simeq\mathbf{F})}$$\scriptstyle{f_{\mathcal{F}}^{s}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{f_{\mathcal{F}}^{s+1}}$$\scriptstyle{f_{\mathcal{F}}^{1^{\prime}}}$$\scriptstyle{\mathfrak{m}^{s}_{(2)}}$
In the rightmost column, if we replace
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]$ by
${\operatorname{im}}(\mathfrak{m}^{s}_{(2)})$ which is strictly smaller, the
only possibility is $C_{\mathcal{F}}^{(1)}$ gets strictly smaller because the
domain and kernel of $f_{\mathcal{F}}^{1^{\prime}}$ are determined by other
columns(by snake lemma) which do not change. This in turn means
$C_{\mathcal{F}}^{(1)}$ must be $\mathbf{F}$ to begin with, so if we do
consider ${\operatorname{im}}(\mathfrak{m}^{s}_{(2)})$ we get the following
diagram:
${0}$${\mathbf{F}}$${\text{{H}}^{0}(T^{(k-s)})}$${\text{{H}}^{0}(T^{(k-s-1)})}$${K(\subset\mathbf{F})}$${0}$${\mathbf{F}}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}^{s+1}])}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(1)})}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}^{s+1}]}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(2)})}$${0}$${C_{\mathcal{F}}^{(s)}}$${C_{\mathcal{F}}^{(s+1)}}$${0}$$\scriptstyle{f_{\mathcal{F}}^{s}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{f_{\mathcal{F}}^{s+1}}$$\scriptstyle{f_{\mathcal{F}}^{1^{\prime\prime}}}$$\scriptstyle{\mathfrak{m}^{s}_{(2)}}$

When $k=s+1$, we have a special diagram
${0}$${K^{\prime}(\subset\mathbf{F})}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)}[\mathfrak{m}^{s}])}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)})}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(1)})}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)})[\mathfrak{m}^{s}]}$${\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)})}$${{\operatorname{im}}(\mathfrak{m}^{s}_{(2^{\prime})})}$${0}$${C_{s}^{s+1}}$${0}$${0}$$\scriptstyle{f_{\mathcal{F}}^{s^{\prime}}}$$\scriptstyle{\mathfrak{m}^{s}_{(1)}}$$\scriptstyle{=}$$\scriptstyle{f_{\mathcal{F}}^{1^{\prime\prime\prime}}}$$\scriptstyle{\mathfrak{m}^{s}_{(2^{\prime})}}$
from which we see $C_{s}^{s+1}=K^{\prime}$ by the snake lemma.

Our aim is to compare $|C_{s}^{s+1}|$ with
$|C_{\mathcal{F}}^{(s)}/C_{\mathcal{F}}^{(s+1)}|=|K|$. Depending on whether
$k-s\leq N$ there are $2$ different possibilities:
Case i: $s>N$ and $k-s\leq N$. In this case $\text{{H}}^{0}(T^{(k-s)})\cong
R/\pi^{k-s}$ and $\text{{H}}^{0}(T^{(k-s-1)})\cong R/\pi^{k-s-1}$ so from
snake lemma we get an exact sequence

$0\to K\to C_{\mathcal{F}}^{(s)}\to C_{\mathcal{F}}^{(s+1)}\to 0.$

Notice that the two $\mathfrak{m}^{s}_{(1)}$ in the two diagrams are indeed
the same map because the rows are identical, since
$T^{(k)}[\mathfrak{m}^{i}]\simeq T^{(i)}$ for any $i\leq k$. Now if
$\mathfrak{m}^{s}_{(1)}$ is surjective, then
$K=K^{\prime}=K^{(1)}=\mathbf{F}$. If if $\mathfrak{m}^{s}_{(1)}$ is not
surjective, then in the last column if we replace
${\operatorname{im}}(\mathfrak{m}^{s}_{(1)})$ by
$\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])(=\text{{H}}^{1}_{\mathcal{F}}(T^{(s+1)}[\mathfrak{m}]))$
which is strictly bigger, it must be that $K$ (and $K^{\prime}$) becomes
strictly bigger because the codomain is fixed and the cokernel cannot be
smaller. This in turn means $K=K^{\prime}=0$.

Thus in case i, we always have $K=K^{\prime}$ and
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=0$.
Moreover, in case i, $K_{\mathcal{F}}^{(s-1)}\cong R/\pi^{n-s+1}$ and
$K_{\mathcal{F}}^{(s+1)}\cong R/\pi^{n-s-1}$ so
$|K_{\mathcal{F}}^{(s-1)}||K_{\mathcal{F}}^{(s+1)}|$ is an even power of
$\mathbf{F}$.
Case ii: $s>N$ and $k-s>N$. In this case
$\text{{H}}^{0}(T^{(k-s)})=\text{{H}}^{0}(T^{(k-s-1)})\cong R/\pi^{N}$, so we
have an exact row

$0\to\mathbf{F}\to R/\pi^{N}\to R/\pi^{N}\ \to K(\subset\mathbf{F}),$

which forces $K=\mathbf{F}$. Furthermore, by the snake lemma this means
$C_{\mathcal{F}}^{(s)}=C_{\mathcal{F}}^{(s+1)}$. This also implies
$\mathfrak{m}^{s}_{(1)}$ is surjective, so
$C_{s}^{s+1}=K^{\prime}=\mathbf{F}$.

Thus in case ii, $K=K^{\prime}=\mathbf{F}$ and
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=|\mathbf{F}|$.

Finally, note that $K_{s}^{s+1}=\mathbf{F}$ for all $s\geq N$ and
$K_{s-1}^{s}=0$ if $s<N$ by Lemma A.0.2 and Assumption A.0.1.

To sum up, when $k$ is at least $2N+2$, as $s$ grows from $N+2$ (so $s-1>N$)
to $k$, $k-s$ is first greater than $N$. For such $s$,
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=|\mathbf{F}|=\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{(s)}|}/|C_{s-1}^{s}|$,
and $K_{\mathcal{F}}^{(s-1)}\cong R/\pi^{N}=K_{\mathcal{F}}^{(s+1)}$; when
$s=k-N$,
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=0$
and
$\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{(s)}|}/|C_{s-1}^{s}|\cong\mathbf{F}$,
and $K_{\mathcal{F}}^{(s-1)}\cong R/\pi^{N}$, $K_{\mathcal{F}}^{(s+1)}\cong
R/\pi^{N-1}$; when $s>k-N$,
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=1=\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{(s)}|}/|C_{s-1}^{s}|$,
and $K_{\mathcal{F}}^{(s-1)}\cong R/\pi^{k-s+1}$ and
$K_{\mathcal{F}}^{(s+1)}\cong R/\pi^{k-s-1}$. In all cases,

$\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}\times\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|\times\Bigg{(}\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{s}|}/|C_{s-1}^{s}|\Bigg{)}^{-1}\times|K_{\mathcal{F}}^{(s-1)}|\times|K_{\mathcal{F}}^{(s+1)}|$

is an even power of $|\mathbf{F}|$.

Now the cases left are when $s\leq N$. When $\mathfrak{m}^{s}_{(2)}$ is not
surjective, then similarly as in Case II, $\mathfrak{m}^{s}_{(1)}$ must be
surjective. Then for such $s$, $C_{s}^{s+1}=|\mathbf{F}|$ if $s=N$ and
$C_{s}^{s+1}=0$ if $s<N$. Also $C_{\mathcal{F}}^{s}\simeq
C_{\mathcal{F}}^{s+1}$, so
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=|\mathbf{F}|^{-1}$
if $s=N$ and
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=1$ if
$s<N$.

When $\mathfrak{m}^{s}_{(2)}$ is surjective, in fact, $\mathfrak{m}^{s}_{(1)}$
must be surjective, too. Note that for $s\leq N$ we have the following
sequence

$0\to\mathbf{F}\to{\operatorname{im}}(\mathfrak{m}^{s}_{(1)})\xrightarrow{f^{1^{\prime}}_{\mathcal{F}}}\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]\to
C_{s}(\subset C^{(1)}\simeq\mathbf{F})\to 0.$

If $\mathfrak{m}^{S}_{(1)}$ is not surjective, then enlarging
${\operatorname{im}}(\mathfrak{m}^{s}_{(1)})$ would shrink $C_{s}$, yielding
an exact sequence

$0\to\mathbf{F}\to\text{{H}}^{1}_{\mathcal{F}}(T^{(k)}[\mathfrak{m}])\xrightarrow{f^{1}_{\mathcal{F}}}\text{{H}}^{1}_{\mathcal{F}}(T^{(k)})[\mathfrak{m}]\to
0,$

which would mean $\mathfrak{m}^{s}_{(2)}$ are always surjective, a
contradiction.

So as long as $\mathfrak{m}^{s}_{(s)}$ is surjective, $C_{\mathcal{F}}^{s}$ is
increasing steadily. And if $s+1\leq N$, then $C_{s}^{s+1}=0$ by Lemma A.0.2.
So
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=|\mathbf{F}|^{-1}$
if $s<N$. When $s=N$, since $\mathfrak{m}^{N}_{(1)}$ is surjective,
$C_{N}^{(N+1)}=K^{\prime}\cong\mathbf{F}$. So
$\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|=1$.

We summarize the behavior of the terms appearing in the lemma in the following
table:

| $\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|$ | $\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{(s)}|}/|C_{s-1}^{s}|$ | $|K_{\mathcal{F}}^{(s+1)}|$ | $|K_{\mathcal{F}}^{(s-1)}|$ | $\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}$
---|---|---|---|---|---
$N<S<k-N$ | $|\mathbf{F}|$ | $(s-1>N):|\mathbf{F}|$ | $|\mathbf{F}|^{N}$ | $|\mathbf{F}|^{N}$ | $1$
$(s-1=N\geq N_{0}):|\mathbf{F}|$
$(s-1=N<N_{0}):1$
$S=N<N_{0}$ | $1$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{N}$ | $|\mathbf{F}|^{N-1}$ | $|\mathbf{F}|$
$S=N=N_{0}$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{N}$ | $|\mathbf{F}|^{N-1}$ | $|\mathbf{F}|$
$S=N>N_{0}$ | $|\mathbf{F}|^{-1}$ | $1$ | $|\mathbf{F}|^{N}$ | $|\mathbf{F}|^{N-1}$ | $|\mathbf{F}|$
$N_{0}<S<N$ | $1$ | $1$ | $|\mathbf{F}|^{s+1}$ | $|\mathbf{F}|^{s-1}$ | $1$
$S=N_{0}<N$ | $1$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{s+1}$ | $|\mathbf{F}|^{s-1}$ | $1$
$S<N_{0}\leq N$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{-1}$ | $|\mathbf{F}|^{s+1}$ | $|\mathbf{F}|^{s-1}$ | $1$

We’re now able to determine the parity of the length of
$\frac{|K_{s}^{s+1}|}{|K_{s-1}^{s}|}\times\frac{|C_{\mathcal{F}}^{(s)}|}{|C_{\mathcal{F}}^{(s+1)}|}/|C_{s}^{s+1}|\times(\frac{|C_{\mathcal{F}}^{(s-1)}|}{|C_{\mathcal{F}}^{s}|}/|C_{s-1}^{s}|)^{-1}\times|K_{\mathcal{F}}^{(s-1)}|\times|K_{\mathcal{F}}^{(s+1)}|$
as a power of $|\mathbf{F}|$ via direct computation in different cases,
depending on the relation between $N_{0}$ and $N$.

Case 1: $N_{0}=N+1$. In this case, when $s=N_{0}$ and $s=N$ we get an odd
dimension, and even dimensions for all other $s\leq N+1$.

Case 2: $N_{0}=N$. In this case, when get even dimensions for all $s\leq N+1$.

Case 3: $N_{0}=N-1$. Same as in Case 1.

Case 4: $1<N_{0}<N-1$. Same as in Case 1.

Case 5: $N_{0}=1$. Same as in Case 1. ∎

##  Appendix B An anticyclotomic control theorem

In this section we follow [JSW17, section 3] and [Gre99, sections 3 and 4] to
prove an anticyclotomic control theorem with torsion, i.e., allowing
$\text{{H}}^{0}(K,W)\neq 0$. We use the notations from [JSW17] and will also
explain the difference between theirs and ours.

First, we make all the assumptions in [JSW17, section 3.1] except (irredK) and
(HT) (which fails if the weight of $f$ is higher than $2$). For our
application to BSD formula in weight $2$ we may still assume (HT), but we will
also explain how to remove it for higher weights.

The first result we need from op. cit. is the following:

######  Proposition B.0.1.

One has

$\\#\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,W)=\\#{}_{{\mathrm{BK}\,}}(W/K)\cdot(\\#\delta_{v})^{2},$

where
$\delta_{v}={\operatorname{coker}}\\{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{BK}\,}}}(K,T)\xrightarrow{{\operatorname{loc}}_{v}}\text{{H}}^{1}_{f}(K,W)/\text{{H}}^{1}(K_{v},T)_{{\operatorname{tors}}}\\}$.
In particular, $\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,W)$ has
finite order.

###### Proof.

See [JSW17, Proposition 3.2.1]. Note that by Remark 3.2.2 in op. cit., we do
not need (irredK) to get $W^{*}=T^{\tau}$.

We mention that in the residually reducible setting, $\text{{H}}^{1}_{f}(K,T)$
may no longer be a free $\mathcal{O}$-module. In particular, their
$\delta_{v}$ fits into the exact sequence

(B.1)
$0\to\text{{H}}^{1}_{f}(K,T)_{{\operatorname{tors}}}\to\text{{H}}^{1}_{f}(K,T)\xrightarrow{{\operatorname{loc}}_{v}}\text{{H}}^{1}_{f}(K_{v},T)/\text{{H}}^{1}_{f}(K_{v},T)_{{\operatorname{tors}}}\to\delta_{v}\to
0,$

where $\ker({\operatorname{loc}}_{v})$ may not be $0$. This observation will
be helpful for our application to modular abelian varieties in section B.3. ∎

The main result of this appendix is the following anticyclotomic control
theorem which allows torsion.

######  Theorem B.0.2 (Anticyclotomic control theorem).

The $\Lambda$-module is $\Lambda$-torsion, and if
$f^{\Sigma}_{{\mathrm{ac}\,}}(T)$ is a generator of its characteristic
$\Lambda$-ideal ${\operatorname{Char}}(X^{\Sigma}_{{\mathrm{ac}\,}}(M_{f}))$,
then

$\\#\mathcal{O}/f^{\Sigma}_{{\mathrm{ac}\,}}(0)=\frac{\\#\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,W)\cdot
C^{\Sigma}(W)}{\\#\text{{H}}^{0}(K,W)\cdot\\#\text{{H}}^{0}(K,W)^{\vee}},$

where

$C^{\Sigma}(W)=\\#\text{{H}}^{0}(K_{v},W)\cdot\\#\text{{H}}^{0}(K_{\overline{v}},W)\cdot\prod_{w\in
S_{p}\setminus\Sigma,w\
split}\\#\text{{H}}^{1}_{{\operatorname{nr}}}(K_{w},W)\cdot\prod_{w\in\Sigma}\\#\text{{H}}^{1}(K_{w},W).$

We also define $\mathcal{P}={\displaystyle\prod_{w\in
S\setminus\Sigma}\frac{\text{{H}}^{1}(K_{w},W)}{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K_{w},W)}}$
and
$\mathcal{G}={\operatorname{im}}\\{\text{{H}}^{1}(K^{S}/K,W)\xrightarrow{{\operatorname{loc}}_{S}}\mathcal{P}\\}.$

We remark that their $S$ and $\Sigma$ are different from ours in section $1$.
Their choice of $v,\overline{v}$ is also different from ours. For our
application, the $\Sigma$ above would be taken as our
$\Sigma-\\{v,\overline{v},\infty\\}$ in the earlier sections and their $S$
will be chosen to be our $\Sigma\cup\\{v,\overline{v}\\}$ so that
$X^{\Sigma}_{{\mathrm{ac}\,}}(M_{f})$ is the dual of the imprimitive Greenberg
Selmer group $\text{{H}}^{1}_{\mathcal{F}^{S}_{Gr}}(K,M_{f})$. In particular,
$\mathcal{P}=\frac{\text{{H}}^{1}(K_{v},W)}{\text{{H}}^{1}(K_{v},W)_{{\mathrm{div}\,}}}\times\text{{H}}^{1}(K_{\overline{v}},W)$.
In the rest of this appendix, we explain how to adapt the proofs in [JSW17,
section 3] to our setting, following [Gre99].

We begin with recalling [Gre99, Lemma 4.2] which says

(B.2)
$\\#\mathcal{O}/f^{\Sigma}_{{\mathrm{ac}\,}}(0)\sim\frac{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}}{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})_{\Gamma}},$

provided
$\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}$ is
finite. For now we assume this finiteness result so we have the above
computation of $\\#\mathcal{O}/f^{\Sigma}_{{\mathrm{ac}\,}}(0)$. We will first
study the numerator.

###  B.1. Computing
$\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}$

As in [JSW17, section 3.3.9], there is a commutative diagram
${0}$${\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,W)}$${\text{{H}}^{1}(K^{S}/K,W)}$${\mathcal{P}}$${0}$${\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}}$${\text{{H}}^{1}(K^{S}/K,M_{f})^{\Gamma}}$${\mathcal{P}(M_{f})^{\Gamma}}$$\scriptstyle{s}$$\scriptstyle{{\operatorname{loc}}_{S}}$$\scriptstyle{h}$$\scriptstyle{r}$
where $\mathcal{P}(M_{f})={\displaystyle\prod_{w\in
S\setminus\Sigma}\frac{\text{{H}}^{1}(K_{w},M_{f})}{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K_{w},M_{f})}}$.
In loc. cit., the first row is also exact on the right because
${\operatorname{loc}}_{S}$ is surjective, but this surjectivity may fail for
us as we do not assume (irredK). Therefore, we would need to work with a
similar commutative diagram
${0}$${\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,W)}$${\text{{H}}^{1}(K^{S}/K,W)}$${\mathcal{G}}$${0}$${0}$${\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}}$${\text{{H}}^{1}(K^{S}/K,M_{f})^{\Gamma}}$${\mathcal{G}(M_{f})^{\Gamma}}$$\scriptstyle{s}$$\scriptstyle{{\operatorname{loc}}_{S}}$$\scriptstyle{h}$$\scriptstyle{g}$
where
$\mathcal{G}(M_{f})={\operatorname{im}}\\{\text{{H}}^{1}(K^{S}/K,M_{f})\xrightarrow{{\operatorname{loc}}_{S}}\mathcal{P}(M_{f})\\}$.

As in loc. cit., ${\operatorname{coker}}(h)=0$ since
$\text{{H}}^{1}(K^{S}/K,M_{f})^{\Gamma}=\text{{H}}^{1}(K^{S}/K,M_{f})[\gamma-1]$
and there is an exact sequence

$0\to W\to M_{f}\xrightarrow{\gamma-1}M_{f}\to 0.$

As $\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,W)$ is finite from
Proposition B.0.1, snake lemma then yields

(B.3)
$\frac{\\#\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,M_{f})^{\Gamma}}{\\#\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K,W)}=\frac{\\#{\operatorname{coker}}(s)}{\\#\ker(s)}=\frac{\\#\ker(g)}{\\#\ker(h)},$

where $\ker(h)=\text{{H}}^{0}(K,W)$ which no longer vanishes in our case. We
will see shortly that $\ker(g)$ (and hence ${\operatorname{coker}}(s)$) is
finite, which then gives the finiteness of
$\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})^{\Gamma}$ we
needed. It remains to compute $\ker(g)$.

###  B.2. Computing $\ker(g)$

We first claim that $\mathcal{G}(M_{f})=\mathcal{P}(M_{f})$, or in other
words, the map
$\text{{H}}^{1}(K^{S}/K,M_{f})\xrightarrow{{\operatorname{loc}}_{S}}\mathcal{P}(M_{f})\coloneq{\displaystyle\prod_{w\in
S\setminus\Sigma}\frac{\text{{H}}^{1}(K_{w},M_{f})}{\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}}(K_{w},M_{f})}}=\text{{H}}^{1}(K_{\overline{v}},M_{f})$
is surjective. This is nothing but the global-to-local map defining the
imprimitive Greenberg Selmer group, so it is surjective by Remark 1.4.2.

We now prove a lemma.

######  Lemma B.2.1.

$\text{{H}}^{1}(K^{\Sigma}/K,M_{f})$ is almost divisible, i.e., its Pontryagin
dual has no non-trivial finite submodules.

###### Proof.

Similarly as in Proposition 1.2.5, this follows from [Gre89, Proposition 3,
Proposition 5]. Note that by Lemma 1.3.5, $\text{{H}}^{2}(K^{\Sigma}/K,M_{f})$
must be cotorsion. Thus it’s $0$ and the result follows. ∎

From the proof of Lemma 1.3.5, we see that the $\mathcal{O}$-corank of
$\text{{H}}^{1}(K_{\overline{v}},W)$ is $2$. Since
${\operatorname{corank}}_{\mathcal{O}}\mathcal{P}={\operatorname{corank}}_{\mathcal{O}}\text{{H}}^{1}(K_{\overline{v}},W)$,
the same is true for $\mathcal{P}$. The arguments after Proposition 4.12 in
[Gre99] then shows that $\text{{H}}^{1}(K^{\Sigma}/K,M_{f})_{\Gamma}=0$. Now
Theorem 4.7 in op. cit. gives the equation

$\\#\ker(g)=\\#\ker(r)\cdot\\#\text{{H}}^{1}_{\mathcal{F}_{{\mathrm{ac}\,}}^{\Sigma}}(K,M_{f})_{\Gamma}\cdot\\#(\mathcal{P}/\mathcal{G}).$

In our case the surjectivity of the map
$(\mathcal{P}\to\mathcal{P}(M_{f})^{\Gamma})=\big{(}\frac{\text{{H}}^{1}(K_{v},W)}{\text{{H}}^{1}(K_{v},W)_{{\mathrm{div}\,}}}\times\text{{H}}^{1}(K_{\overline{v}},W)\to\text{{H}}^{1}(K_{\overline{v}},M_{f})^{\Gamma}\big{)}$
follows from the $p$-cohomological dimension of
$G(K_{\infty,\overline{v}}/K_{\overline{v}})$ being $1$.

Now by Cassels’ Theorem (Proposition 4.13 in op. cit.), since
$\text{{H}}^{1}_{\mathcal{F}_{ac}}(K,W)$ is finite by Proposition B.0.1 and
$\text{{H}}^{0}(K_{w},W)$ is finite by Proposition 1.3.4, we have
$\mathcal{P}/\mathcal{G}\cong\text{{H}}^{0}(K,W)^{\vee}$.

Combining this result with (B.2), (B.3) and the computation of $\ker(r)$ in
[JSW17, Proposition 3.3.7] (the proof of case 3(b) is the only place where
(HT) is used, but by Proposition 1.3.4, we already have the finiteness of
$M_{f}^{G_{K_{v}}}$ so (HT) is no longer needed), Theorem B.0.2 follows.

###  B.3. Applications to newforms and modular abelian varieties $A_{f}$

Most part of the computations are already done in [JSW17, section 3.5].

As is already noted in Proposition B.0.1, in our residually reducible setting,
$A_{f}(K)\otimes_{\mathbf{Z}[f]}\mathcal{O}=\text{{H}}^{1}_{f}(K,T)$ may no
longer be a free $\mathcal{O}$-module, and we need to replace $A_{f}(K)$ and
$\text{{H}}^{1}_{f}(K,T)$ in loc. cit. by $A_{f}(K)_{/{\operatorname{tors}}}$
and $\text{{H}}^{1}_{f}(K,T)_{/{\operatorname{tors}}}$ respectively. In
particular, from the sequence (B.1), equation (3.5.a) in op. cit. becomes

$\\#\delta_{v}=\\#[A_{f}(K_{v})_{/{\operatorname{tors}}}\otimes_{\mathbf{Z}[f]\otimes\mathbf{Z}_{p}}\mathcal{O}:A_{f}(K)_{/{\operatorname{tors}}}\otimes_{\mathbf{Z}[f]}\mathcal{O}].$

This means in our application to elliptic curves, the above index becomes
$[E(K)_{/{\operatorname{tors}}}:\mathbf{Z}\cdot P]_{p}$, which does not
include the torsion of $E(K)$. Indeed, as we will see immediately,
$E(K)_{{\operatorname{tors}}}$ has already been considered separately in
section B.1 and section B.2.

It remains to compute the torsion terms $\text{{H}}^{0}(K,W)$ and
$\text{{H}}^{0}(K,W)^{\vee}$. But these are identified with
$(A_{f}(K)\otimes_{\mathbf{Z}[f]}\mathcal{O})[\mathfrak{p}^{\infty}]$ and its
dual respectively, both having size
$\\#((A_{f}(K)\otimes_{\mathbf{Z}[f]}\mathcal{O})_{{\operatorname{tors}}})_{\mathfrak{p}}$.
When $A_{f}$ is an elliptic curve, $\mathbf{Z}[f]=\mathbf{Z}$,
$\mathcal{O}=\mathbf{Z}_{p}$ and the size is
$\\#(E(K)\otimes\mathbf{Z}_{p})[p^{\infty}]=\\#(E(K)[p^{\infty}])=(\\#E(K)_{{\operatorname{tors}}})_{p}$.

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