# On a parity result for the symmetric square of modular forms with congruent
residual representations

Jishnu Ray  Harish Chandra Research Institute, A CI of Homi Bhabha National
Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The parity of Selmer ranks for elliptic curves defined over the rational
numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been
studied by Shekhar. The proof of Shekhar relies on proving a parity result for
the $\lambda$-invariants of Selmer groups over the cyclotomic
$\mathbb{Z}_{p}$-extension $\mathbb{Q}_{\infty}$ of $\mathbb{Q}$. This has
been further generalized for elliptic curves with supersingular reduction at
$p$ by Hatley and for modular forms by Hatley–Lei. In this paper, we prove a
parity result for the $\lambda$-invariants of Selmer groups over
$\mathbb{Q}_{\infty}$ for the symmetric square representations associated to
two modular forms with congruent residual Galois representations. We treat
both the ordinary and the non-ordinary cases.

###### Key words and phrases:

Iwasawa theory, congruences, Selmer groups, symmetric square, modular forms

###### 2020 Mathematics Subject Classification:

Primary: 11R23, 11F11, 11R18 Secondary: 11F85

##  1\. Introduction

Suppose $K$ is a number field and fix an odd prime $p$. Let $K_{\infty}$ be
the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Let $E_{1}$ and $E_{2}$ be
two elliptic curves defined over $\mathbb{Q}$ both having good ordinary
reduction at the prime $p$. Let $\Sigma$ be a finite set of primes of $K$
containing the primes of bad reduction of $E_{1}$ and $E_{2}$, the infinite
primes and the primes above $p$. Suppose that $E_{1}$ and $E_{2}$ are
congruent at $p$, i.e. $E_{1}[p]\cong E_{2}[p]$ as representations of the
absolute Galois group
$G_{\mathbb{Q}}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Furthermore assume that $K=K(\mu_{p})$ and $E_{1}[p]$ is an irreducible
$G_{K}$-module (and hence $E_{2}[p]$ will also be an irreducible
$G_{K}$-module). Then, under the additional assumption that the Iwasawa
$\mu$-invariant of the Greenberg’s $p$-Selmer group
$\operatorname{Sel}_{p^{\infty}}(E_{1}/K_{\infty})$ vanishes (and hence also
for $\operatorname{Sel}_{p^{\infty}}(E_{2}/K_{\infty})$), Shekhar proved that

(1)
$\mathrm{corank}_{\mathcal{O}_{K}}\operatorname{Sel}_{p^{\infty}}(E_{1}/K)+|S_{E_{1}}|\equiv\mathrm{corank}_{\mathcal{O}_{K}}\operatorname{Sel}_{p^{\infty}}(E_{2}/K)+|S_{E_{2}}|\pmod{2}.$

Here $S_{E_{i}}$ is an explicitly determined subset of primes in $\Sigma$ (cf.
[She16, Theorem 1.1]). Assuming that the Shafarevich-Tate group
$\Sha(E_{i}/K)[p^{\infty}]$ is finite, this gives the parity of ranks of
elliptic curves with equivalent mod-$p$ Galois representations. If
$K\subset\mathbb{Q}(\mu_{p^{n}},m^{1/p^{n}})$ for $m,n\in\mathbb{Z}_{\geqslant
1}$ and $K$ is a Galois extension over $\mathbb{Q}$, then the $p$-parity
conjecture holds, i.e.

$\mathrm{corank}_{\mathcal{O}_{K}}\operatorname{Sel}_{p^{\infty}}(E_{i}/K)\equiv
r_{\mathrm{an}}(E_{i})\pmod{2}.$

Here $r_{\mathrm{an}}(E_{i})$ is the order of zero of the complex $L$-function
$L(E_{i}/K)$ at $1$ (see [She16, Conjecture 1 and Theorem 4.3]). This gives a
parity of analytic ranks for congruence mod-$p$ Galois representations.

Let $\lambda(E_{i}/K_{\infty})$ be the Iwasawa $\lambda$-invariant attached to
$\operatorname{Sel}_{p^{\infty}}(E_{i}/K_{\infty})$. The proof of (1) relies
primarily on proving the following:

(2)
$\lambda(E_{1}/K_{\infty})+|S_{E_{1}}|\equiv\lambda(E_{2}/K_{\infty})+|S_{E_{2}}|\pmod{2}$

and then using a result proved by Greenberg (cf. [Gre99, Prop. 3.10]), which
is

(3)
$\lambda(E_{i}/K_{\infty})\equiv\mathrm{corank}_{\mathcal{O}_{K}}\operatorname{Sel}_{p^{\infty}}(E_{i}/K)\pmod{2}.$

Shekhar’s result has been generalized to elliptic curves with supersingular
reduction at the prime $p$ [Hat17] and to modular forms [HL19, Theorem 5.7].
Although the result in [HL19] is written for modular forms non-ordinary at $p$
but essentially the same technique will also work for modular forms ordinary
at $p$ with signed Selmer groups replaced by classical $p$-Selmer group using
[EPW06, Theorem 4.3.4 (ii)]. The main goal in this paper is to prove an
analogue of (2) for the symmetric square representations associated to modular
forms ordinary at $p$. Let $f_{i}=\sum a_{n}(f_{i})q^{n}$ (for $i=1,2$) be two
normalized new cuspidal eigenforms of the same weight $k$, level $N_{i}$
(coprime to $p$) and character $\varepsilon_{i}$. Assume that they are non CM
and their residual representations are isomorphic and are irreducible as
$G_{\mathbb{Q}}$-modules. For any Dirichlet character $\psi$ of conductor
coprime to $p$, let $\mathbf{V}_{f_{i},\psi}$ be the symmetric square
representation associate to $f_{i}$ twisted by $\psi$. We can enlarge the
coefficient field and choose an extension $L$ over $\mathbb{Q}$ which contains
the image of $\psi$, and all the coefficients of $f_{1}$ and $f_{2}$. There is
a unique Galois stable lattice which is denoted by $\mathbf{T}_{f_{i},\psi}$
and let
$\mathbf{A}_{f_{i},\psi}=\mathbf{V}_{f_{i},\psi}/\mathbf{T}_{f_{i},\psi}.$ Let
$\mathfrak{P}$ be a prime of $L$ above $p$. Assume that $f_{i}$ is ordinary at
$\mathfrak{P}$. One can choose such a nontrivial even Dirichlet character
where it is known that the Selmer group
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$
is cotorsion (cf. [LZ19]). Once such choice is to choose $\psi$ satisfying the
conditions listed in [LZ19, Theorem C]. Also assume the vanishing of certain
Galois cohomology groups as in (inv) (see after definition 4.8). We prove the
following result on the parity of $\lambda$-invariance (cf. Theorem 3.2).

  * •

Suppose that the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
vanishes. Assume that for all primes $\ell\mid N_{i},a_{\ell}(f_{i})\not\equiv
0\pmod{\mathfrak{P}}$.

Then there exists some explicitly computable finite sets
$\mathcal{S}_{f_{1},\psi}$ and $\mathcal{S}_{f_{2},\psi}$ such that

$\lambda(\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{1},\psi}|\equiv\lambda(\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{2},\psi}|\pmod{2}.$

Note that we have the additional restriction $a_{\ell}(f_{i})\not\equiv
0\pmod{\mathfrak{P}}$ for all primes $\ell\mid N_{i}$. This is a condition
under which we know explicitly the form of the $p$-adic Galois representation
attached to the modular form $f_{i}$ by Deligne restricted to the
decomposition subgroup $G_{\mathbb{Q}_{\ell}}$. From this, we could compute
the associated symmetric square representation restricted to
$G_{\mathbb{Q}_{\ell}}$ (see Lemma 2.8 and Lemma 2.9). We don’t know how to
remove this assumption.

The analogue of (3) in our case will follow if the representations
$\mathbf{V}_{f_{1},\psi}$ and $\mathbf{V}_{f_{2},\psi}$ are self dual and then
we obtain a analogue of (1).

Suppose now that $f_{i}$ is non-ordinary at $\mathfrak{P}$ and
$a_{p}(f_{i})=0$. Let $\mathcal{S}$ denote the set of pairs
$\\{(+,-),(+,\bullet),(-,\bullet)\\}.$ For
$\mathfrak{S}=(\clubsuit,\spadesuit)\in\mathfrak{S}$, and $\psi$ satisfying
the conditions mentioned in [BLV21, page 3], Büyükboduk–Lei–Venkat defined
three signed Selmer groups
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}(\mu_{p^{\infty}}))$
which are conjecturally cotorsion. Below is the summary of our results.

  * •

The Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$
contains no proper $\Lambda$-submodule of finite index (Theorem 4.4).

  * •

The $\mu$-invariant of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
vanishes if and only if the $\mu$-invariant of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})$
vanishes. When these $\mu$-invariants are trivial, then the imprimitive signed
$\lambda$-invariants of
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
and
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})$
coincide (Theorem 4.12).

  * •

Assume that for all primes $\ell\mid N_{i},a_{\ell}(f_{i})\not\equiv
0\pmod{\mathfrak{P}}$. Also assume the vanishing of the $\mu$-invariants of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
and
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})$.
Then we have the congruence

$\lambda(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{1},\psi}|\equiv\lambda(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{2},\psi}|\pmod{2}$

where $\mathcal{S}_{f_{i},\psi}$ is the same set of primes as in the ordinary
case above (see Theorem 4.13).

The main inputs in this article are the computations done to make the set
$\mathcal{S}_{f_{i},\psi}$ as explicit as possible (i.e. Lemma 2.7, Lemma 2.8
and Lemma 2.9). In the non-ordinary setting, our main input is to use the
local condition at $p$ defining these signed Selmer groups over
$\mathbb{Q}(\mu_{p^{\infty}})$ in order to define the signed Selmer groups
over $\mathbb{Q}_{\infty}$ and the finite layers $\mathbb{Q}_{(n)}$ in such a
way that the usual control theorem holds (cf. Lemma 4.5). Note that the local
condition at $p$ over $\mathbb{Q}_{(n)}$ is defined here using the local
condition at $\mathbb{Q}(\mu_{p^{\infty}})$ via descending. This is unlike
Kobayashi’s approach [Kob03] where the local condition at the finite layers is
given first and then a direct limit was taken to define the signed Selmer
group at $\mathbb{Q}_{\infty}$ resulting in a more difficult control theorem
[Kob03, Theorem 9.3]. These two approaches give, in general, two different
Selmer groups at the layer $\mathbb{Q}_{(n)}$. Our approach solves the purpose
we are interested in. Further we take certain cyclotomic twists of signed
Selmer groups such that some global to local map defining appropriate Selmer
condition becomes surjective (cf. Lemma 4.6 and Lemma 4.7). Taking such twists
is a crucial part of the argument, without which the arguments fails. Finally,
we analyze the Pontryagin dual of the local condition at $p$ defining the
signed Selmer structures and prove a congruence result (Proposition 4.11).
Such an analysis of local condition at $p$ for signed Selmer groups is also
need in the proof of Lemma 4.7. All these combined efforts finally help to
prove our main theorems.

## Acknowledgement

We would like to thank Jeffrey Hatley, Antonio Lei, Aprameyo Pal and Dipendra
Prasad for numerous discussions and helpful comments. We gratefully
acknowledge support from the Inspire Research Grant, DST, Govt. of India.

##  2\. The symmetric square representation

Throughout we fix an embedding $\iota_{\infty}$ of a fixed algebraic closure
$\overline{\mathbb{Q}}$ of $\mathbb{Q}$ into $\mathbb{C}$ and also an
embedding $\iota_{\ell}$ of $\overline{\mathbb{Q}}$ into a fixed algebraic
closure $\overline{\mathbb{Q}}_{\ell}$ for every prime $\ell$. Let $f=\sum
a_{n}(f)q^{n}$ be a normalized new cuspidal eigenform of even weight
$k\geqslant 2$, level $N$, nebentypus $\varepsilon$, with coefficients in a
number field $L\subset\mathbb{C}$. Assume that $f$ is not of CM type. Let
$p\geqslant 5$ be a prime such that $p\nmid N$ and let $\mathfrak{P}$ be a
prime of the field $L$ above $p$. We assume that $f$ is ordinary at
$\mathfrak{P}$ (i.e. $v_{\mathfrak{P}}(a_{p}(f))=0$). Let $\alpha_{p}(f)$ be
the unique root of the Hecke polynomial at $p$ that lies in
$\mathcal{O}^{\times}_{L,\mathfrak{P}}.$ Let
$\omega_{p}:G_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$ be the $p$-adic
cyclotomic character and let $\psi$ be a Dirichlet character of conductor
$N_{\psi}$ coprime to $p$. Enlarging $L$ if necessary, we assume that $\psi$
takes values in $L^{\times}$.

######  Theorem 2.1 (Eichler, Shimura, Deligne, Mazur-Wiles, Wiles, etc.).

There exists a Galois representation $\rho_{f}:G_{\mathbb{Q}}\longrightarrow
GL_{2}(L_{\mathfrak{P}})$ such that

  1. (1)

For all primes $\ell\nmid Np$, $\rho_{f}$ is unramified with the
characteristic polynomial of the (arithmetic) Frobenius is given by
$\mathrm{trace}(\rho_{f}(\operatorname{Frob}_{\ell}))=a_{\ell}(f),$ and
$\det(\rho_{f}(\operatorname{Frob}_{\ell}))=\varepsilon(\ell)\omega_{p}(\operatorname{Frob}_{\ell})^{k-1}=\varepsilon(\ell)\ell^{k-1}.$
It follows by the Chebotarev Density Theorem that
$\det(\rho_{f})=\varepsilon\omega_{p}^{k-1}.$

  2. (2)

Let $G_{p}$ be the decomposition subgroup of $G_{\mathbb{Q}}$ at $p$. Then,

$\rho_{f}|_{G_{p}}\sim\begin{pmatrix}\lambda_{f}^{-1}\varepsilon\omega_{p}^{k-1}&*\\\
~{}&\lambda_{f}\end{pmatrix}$

where $\lambda_{f}$ is the unramified character such that
$\lambda_{f}(\operatorname{Frob}_{p})=\alpha_{p}(f).$

Let $V_{f}$ denote the representation space of $\rho_{f}$. Since
$G_{\mathbb{Q}}$ is compact, choose an $\mathcal{O}_{L_{\mathfrak{P}}}$\-
lattice $T_{f}$ which is invariant under $\rho_{f}$. Let

$\tilde{\rho}_{f}:G_{\mathbb{Q}}\longrightarrow
GL_{2}(\frac{\mathcal{O}_{L_{\mathfrak{P}}}}{\pi_{L}})$

be the residual representation attached to $\rho_{f}$.

Throughout we assume that $\tilde{\rho}_{f}$ is absolutely irreducible so that
the choice of the Galois stable lattice $T_{f}$ is unique. By part (2) of
Theorem 2.1, there exists a $G_{p}$-stable two step filtration

$V_{f}=\mathrm{Fil}^{0}V_{f}\supset\mathrm{Fil}^{1}V_{f}\supset
0=\mathrm{Fil}^{2}V_{f}$

such that the action of $G_{p}$ on the graded pieces
$\operatorname{Gr}^{i}V_{f}:=\mathrm{Fil}^{i}V_{f}/\mathrm{Fil}^{i+1}V_{f}$ is
given as follows. The $G_{p}$-action of $\operatorname{Gr}^{1}V_{f}$ (resp.
$\operatorname{Gr}^{0}V_{f}$) is given via the character
$\lambda_{f}^{-1}\varepsilon\omega_{p}^{k-1}$ (resp. $\lambda_{f}$). Hence the
action of $G_{p}$ on $\operatorname{Gr}^{0}V_{f}$ is unramified.

Now consider a basis $v_{1}$ of $\mathrm{Fil}^{1}V_{f}$ and expand it to a
basis $\\{v_{1},v_{2}\\}$ of $V_{f}$. Let $\mathbf{V}_{f}$ be the symmetric
square representation associate to $V_{f}$. A basis for $\mathbf{V}_{f}$ is
given by $\\{w_{i,j}\mid 1\leqslant i\leqslant j\leqslant 2\\}$ where
$w_{i,j}=v_{i}\otimes v_{j}+v_{j}\otimes v_{i}$. Let
$\mathrm{Fil}^{2}\mathbf{V}_{f}=\mathrm{span}\\{w_{1,1}\\}$ and
$\mathrm{Fil}^{1}\mathbf{V}_{f}=\mathrm{span}\\{w_{1,1},w_{1,2}\\}$. Then
$\mathbf{V}_{f}$ has a $3$-step $G_{p}$-stable filtration

$\mathbf{V}_{f}=\mathrm{Fil}^{0}\mathbf{V}_{f}\supset\mathrm{Fil}^{1}\mathbf{V}_{f}\supset\mathrm{Fil}^{2}\mathbf{V}_{f}\supset
0=\mathrm{Fil}^{3}\mathbf{V}_{f}$

such that the action of $G_{p}$ on the $1$-dimensional graded pieces
$\operatorname{Gr}^{2}\mathbf{V}_{f},\operatorname{Gr}^{1}\mathbf{V}_{f}$ and
$\operatorname{Gr}^{0}\mathbf{V}_{f}$ is given by the characters
$(\lambda_{f}^{-1}\varepsilon\omega_{p}^{k-1})^{2},\varepsilon\omega_{p}^{k-1}$
and $\lambda_{f}^{2}$ respectively.

###  2.1. The Greenberg Selmer group

Let $\mathbf{A}_{f}:=\mathbf{V}_{f}/\mathbf{T}_{f}$ where
$\mathbf{T}_{f}=\mathrm{Sym}^{2}T_{f}$. We write $\mathbf{V}_{f,\psi}$ for the
twisted representation $\mathbf{V}_{f}\otimes\psi$ with lattice
$\mathbf{T}_{f,\psi}$ and we denote the corresponding $p$-divisible Galois
module as $\mathbf{A}_{f,\psi}$. Let $\mathrm{Fil}^{1}\mathbf{A}_{f,\psi}$ be
the image of $\mathrm{Fil}^{1}\mathbf{V}_{f,\psi}$ under the canonical map
$\mathbf{V}_{f,\psi}\rightarrow\mathbf{A}_{f,\psi}$. Let $\Sigma$ be a finite
set of places of $\mathbb{Q}$ that contains $p$, primes that divide the level
$N$, primes that divide the conductor $N_{\psi}$ and $\infty$. We write
$\mathbb{Q}_{\Sigma}$ for the maximal extension of $\mathbb{Q}$ unramified
outside $\Sigma$. Let $\mathbb{Q}_{\infty}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ with Galois group $\Gamma$ and
Iwasawa algebra $\Lambda$. The $p$-primary Greenberg Selmer group
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$ is
defined as

$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty}):=\ker\Big{(}H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},\mathbf{A}_{f,\psi})\xrightarrow{\lambda_{f,\psi}}\prod_{\ell\in\Sigma}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})\Big{)}$

where $\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})$ is defined
as follows. If $\ell\neq p$,

$\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}):=\prod_{\eta\mid\ell}H^{1}(\mathbb{Q}_{\infty,\eta},\mathbf{A}_{f,\psi})$

where the product is over the finite set of primes $\eta$ of
$\mathbb{Q}_{\infty}$ lying over $\ell$. Let $\eta_{p}$ be the unique prime of
$\mathbb{Q}_{\infty}$ lying above $p$ and $I_{\eta_{p}}$ be the inertia
subgroup at $\eta_{p}$. Then

$\mathcal{H}_{p}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}):=H^{1}(\mathbb{Q}_{\infty,\eta_{p}},\mathbf{A}_{f,\psi})/\mathcal{L}_{\eta_{p}}$

with

$\mathcal{L}_{\eta_{p}}=\ker\Big{(}H^{1}(\mathbb{Q}_{\infty,\eta_{p}},\mathbf{A}_{f,\psi})\longrightarrow
H^{1}(I_{\eta_{p}},\mathbf{A}_{f,\psi}/\mathrm{Fil}^{1}\mathbf{A}_{f,\psi})\Big{)}.$

We make the following hypothesis throughout the article.

(Tor)
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$ is
a cotorsion $\Lambda$-module.

######  Remark 2.2.

Under various strict assumptions on the character $\psi$ listed in [LZ19,
Theorem C], (Tor) if known to be true by the works of Loeffler–Zerbes [LZ19]

######  Definition 2.3.

Let $\Sigma_{0}=\Sigma\setminus\\{p,\infty\\}$. The $\Sigma_{0}$-imprimitive
Selmer group is defined as

$\operatorname{Sel}_{p^{\infty}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})=\ker\Big{(}H^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},\mathbf{A}_{f,\psi})\xrightarrow{\lambda_{f,\psi}^{\Sigma_{0}}}\prod_{\ell\in\Sigma\setminus\Sigma_{0}}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})\Big{)}$

Let
$(\mathbf{T}_{f,\psi})^{*}:=\operatorname{Hom}(\mathbf{T}_{f,\psi},\mu_{p^{\infty}}).$
We make the following assumptions.

  1. (i)

$\psi$ is even.

  2. (ii)

(inv) The Galois cohomology groups $H^{0}(\mathbb{Q}_{p},\mathbf{A}_{f,\psi})$
and $H^{0}(\mathbb{Q}_{p},(\mathbf{T}_{f,\psi})^{*})$ are trivial.

Under these assumptions, the localization map $\lambda_{f,\psi}$ (and hence
also ${\lambda_{f,\psi}^{\Sigma_{0}}}$) is surjective (cf. [RSV23, Proposition
3.3]). It follows that

(4)
$\operatorname{Sel}_{p^{\infty}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})/\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})\cong\prod_{\ell\in\Sigma_{0}}\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}).$

The following result is well-known (cf. [RSV23, Lemma 3.5]).

######  Lemma 2.4.

If $\ell\neq p$, $\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})$
is cofinitely generated and cotorsion $\Lambda$-module with trivial
$\mu$-invariant.

Let $\lambda^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$ and
$\lambda(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$ be the $\lambda$-invariants
for the Selmer groups
$\operatorname{Sel}_{p^{\infty}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$
and $\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$
respectively. It follows that

(5)
$\lambda^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})=\lambda(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})+\sum_{\ell\in\Sigma_{0}}\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$

where the $\lambda$-invariant of
$\mathcal{H}_{\ell}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})$ is given by
$\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty}):=\sum_{\eta\mid\ell}\tau_{\eta}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$.

###  2.2. Computing the parity of
$\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$

For a prime $\ell\in\Sigma_{0}$, let $\operatorname{Frob}_{\ell}$ denote the
arithmetic Frobenius automorphism in
$\operatorname{Gal}(\mathbb{Q}_{\ell}^{\mathrm{unr}}/\mathbb{Q}_{\ell})$. Let
$I_{\ell}$ be the inertia subgroup
$\operatorname{Gal}(\overline{\mathbb{Q}}_{l}/\mathbb{Q}_{\ell}^{\mathrm{unr}})$
of $G_{\mathbb{Q}_{\ell}}$, $(\mathbf{V}_{f,\psi})_{I_{\ell}}$ be the maximal
quotient of $\mathbf{V}_{f,\psi}$ on which $I_{\ell}$ acts trivially. Let
$k_{L}$ be the residue field of $L$ and let $x\mapsto\tilde{x}$ be the
reduction modulo $\mathfrak{P}$ map from $\mathcal{O}_{L}$ to $k_{L}$. The
following proposition of Greenberg–Vatsal explains how to compute
$\tau_{\eta}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$ (cf. [GV00, Proposition
2.4]).

######  Proposition 2.5.

Let $\ell\in\Sigma_{0}$ and write

$P_{\ell,f}(X)=\det(1-\operatorname{Frob}_{\ell}X|_{(\mathbf{V}_{f,\psi})_{I_{\ell}}})\in\mathcal{O}_{L}[X].$

Let $d_{\ell,f}$ denote the multiplicity of $X=\tilde{\ell}^{-1}$ as a root of
$\tilde{P}_{\ell,f}\in k_{L}[X].$ Then for each prime $\eta\mid\ell$, we have
$\tau_{\eta}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})=d_{\ell,f}$

######  Corollary 2.6.

For each $\ell\in\Sigma_{0}$, we have
$\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})\equiv
d_{\ell,f}\pmod{2}.$

###### Proof.

Let $s_{\ell}$ denote the number of primes $\eta$ of $\mathbb{Q}_{\infty}$
lying over $\ell$. That is, $s_{\ell}=[\Gamma:\Gamma_{\ell}]$ where
$\Gamma_{\ell}$ denotes the decomposition subgroup of $\Gamma$ for any such
$\eta$. It follows from [GV00, page 37 and Proposition 2.4] that
$\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})=s_{\ell}d_{\ell,f}$.
Since $s_{\ell}$ is a power of $p$, it is necessarily odd and hence the result
follows. ∎

Our next goal is to compute the parity of $d_{\ell,f}$ for each prime
$\ell\in\Sigma_{0}$.

######  Lemma 2.7.

If $\ell\nmid N$, then $d_{\ell,f}$ is odd if and only if $\tilde{\psi}$ is
unramified at $\ell$ and any of the following mutually disjoint conditions
(7), (10), (11) given below hold.

###### Proof.

If $\ell\nmid N$ but $\tilde{\psi}$ is ramified at $\ell$ then
$(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}=0$. This can be deduced easily from
[DT94, last paragraph of page 255].

Suppose $\ell\nmid N$ and $\tilde{\psi}$ is unramified at $\ell$, then
$\tilde{\mathbf{V}}_{f,\psi}$ is unramified at $\ell$ and hence
$(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}=\tilde{\mathbf{V}}_{f,\psi}.$ It is
well-known that (cf. [LZ19, Section 2.1])

$P_{\ell,f}(X)\equiv(1-\alpha_{\ell}^{2}\psi(\ell)X)(1-\beta_{\ell}^{2}\psi(\ell)X)(1-\alpha_{\ell}\beta_{\ell}\psi(\ell)X)\pmod{\mathfrak{P}}$

where $\alpha_{\ell}^{-1}$ and $\beta_{\ell}^{-1}$ are two roots of the Hecke
polynomial $1-a_{\ell}(f)X+\varepsilon(l)\ell^{k-1}X^{2}.$ Therefore,

$\displaystyle\tilde{P}_{\ell,f}(X)$
$\displaystyle=\Big{(}1-\psi(\ell)(\alpha_{\ell}^{2}+\beta_{\ell}^{2})X+\psi(\ell)^{2}\alpha_{\ell}^{2}\beta_{\ell}^{2}X^{2}\Big{)}^{\sim}\Big{(}1-\varepsilon(\ell)\psi(\ell)\ell^{k-1}X\Big{)}^{\sim}$
$\displaystyle=\Big{(}1-\big{(}\psi(\ell)a_{\ell}(f)^{2}-2\psi(\ell)\varepsilon(\ell)\ell^{k-1}\big{)}X+\psi(\ell)^{2}\varepsilon(\ell)^{2}\ell^{2k-2}X^{2}\Big{)}^{\sim}\Big{(}1-\varepsilon(\ell)\psi(\ell)\ell^{k-1}X\Big{)}^{\sim}.$

Let
$\tilde{g}(X)=\Big{(}1-\varepsilon(\ell)\psi(\ell)\ell^{k-1}X\Big{)}^{\sim}$
and $\tilde{h}(X)=\tilde{P}_{\ell,f}(X)/\tilde{g}(X)$. It follows that
$d_{\ell,f}=1$ if and only if either of the following two cases hold.

  1. (I)

$\tilde{\ell}^{-1}$ is a root of $\tilde{g}(X)$ and $\tilde{\ell}^{-1}$ is not
a root of $\tilde{h}(X)$,

  2. (II)

$\tilde{\ell}^{-1}$ is a simple root of $\tilde{h}(X)$ and $\tilde{\ell}^{-1}$
is not a root of $\tilde{g}(X).$

Case (I): $\tilde{\ell}^{-1}$ is a root of $\tilde{g}(X)$ if and only if

(6) $\varepsilon(\ell)\psi(\ell)\ell^{k-2}\equiv 1\pmod{\mathfrak{P}}.$

Also, $\tilde{\ell}^{-1}$ is not a root of $\tilde{h}(X)$ means
$\tilde{h}(\tilde{\ell}^{-1})\neq 0.$ Hence Case (I) holds if and only if

(7) $\text{equation }\eqref{eq:CaseIi}\text{ holds and }h(\ell^{-1})\not\equiv
0\pmod{\mathfrak{P}}.$

Case (II): We first find equivalent conditions when $\tilde{\ell}^{-1}$ is a
simple root of $\tilde{h}(X)$. Since the product of two roots of $h(X)$ is
$\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{2-2k}$,

$\tilde{\ell}^{-1}\text{ is a root if and only if
}\Big{(}\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{3-2k}\Big{)}^{\sim}\text{
is a root. }$

Therefore, $\tilde{\ell}^{-1}$ is a root of $\tilde{h}(X)$ if and only if

$\ell^{-1}+\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{3-2k}\equiv\big{(}\psi(\ell)a_{\ell}(f)^{2}-2\psi(\ell)\varepsilon(\ell)\ell^{k-1}\big{)}\big{(}\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{2-2k}\big{)}\pmod{\mathfrak{P}}.$

Simplifying, we get that $\tilde{\ell}^{-1}$ is a root of $\tilde{h}(X)$ if
and only if

(8)
$\psi(\ell)^{-1}a_{\ell}(f)^{2}\varepsilon(\ell)^{-2}\ell^{3-2k}-2\psi(\ell)^{-1}\varepsilon(\ell)^{-1}\ell^{2-k}-\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{4-2k}\equiv
1\pmod{\mathfrak{P}}.$

Hence, if $\tilde{l}^{-1}$ is a root of $\tilde{h}(X)$, it is a simple root if
and only if

(9)
$\ell^{-1}\not\equiv\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{3-2k}\pmod{\mathfrak{P}}\text{
i.e. }\psi(\ell)^{-2}\varepsilon(\ell)^{-2}\ell^{4-2k}\not\equiv
1\pmod{\mathfrak{P}}.$

Therefore, Case (II) holds if and only if

(10) $\text{equations }\eqref{eq: IIi}\text{ and }\eqref{eq: IIii}\text{ hold
and equation }\eqref{eq:CaseIi}\text{ does not hold.}$

This completes the case when $d_{\ell,f}=1$. Now, it is easy to see that
$d_{\ell,f}=3$ if and only if

(11) $\text{equations }\eqref{eq: IIi}\text{ and }\eqref{eq:CaseIi}\text{ hold
and equation }\eqref{eq: IIii}\text{ does not hold.}$

∎

Now we deal with the cases when $\ell\mid N$ and we make the following
hypothesis.

(Hyp) For all primes $\ell\mid N$, $a_{\ell}(f)\not\equiv
0\pmod{\mathfrak{P}}.$

It is known that $a_{\ell}(f)\not\equiv 0\pmod{\mathfrak{P}}$, if and only if
one of the following holds (cf. [MTT86, page 16, Section 12, Remark II]):

  * •

$\ell\mid\mid N$ and $\ell\nmid M$; or

  * •

$\operatorname{ord}_{\ell}(N)=\operatorname{ord}_{\ell}(M)$.

Here $M$ is the conductor of the nebentypus $\varepsilon$.

######  Lemma 2.8.

Suppose $\ell\mid\mid N$ and $\ell\nmid M$. Then $d_{\ell,f}$ is odd if and
only if $\tilde{\psi}$ is unramified at $\ell$ and $\ell\equiv
a_{\ell}(f)^{2}\psi(\ell)\pmod{\mathfrak{P}}.$

###### Proof.

In this case [Hid00, Theorem 3.26, 3(b)] gives

$\tilde{\rho}_{f}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\omega_{p}}\tilde{\chi}&\tilde{D}\\\
~{}&\tilde{\chi}\end{pmatrix},$

where $\tilde{\chi}$ is unramified such that
$\tilde{\chi}(\operatorname{Frob}_{\ell})=\tilde{a}_{\ell}(f)$ . Since the
residual representation attached to $(V_{f})_{I_{\ell}}$ is one dimensional
(cf. [HL19, Proof of Lemma 5.4]), the character $\tilde{D}$ must be ramified.
Therefore,

$\tilde{\mathbf{V}}_{f}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\omega_{p}}^{2}\tilde{\chi}^{2}&\tilde{\omega_{p}}\tilde{\chi}\tilde{D}&\tilde{D}^{2}\\\
~{}&\tilde{\omega_{p}}\tilde{\chi}^{2}&2\tilde{\chi}\tilde{D}\\\
~{}&~{}&\tilde{\chi}^{2}\end{pmatrix}\text{ and hence
}\tilde{\mathbf{V}}_{f,\psi}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\omega_{p}}^{2}\tilde{\chi}^{2}\tilde{\psi}&\tilde{\omega_{p}}\tilde{\chi}\tilde{D}&\tilde{D}^{2}\\\
~{}&\tilde{\omega_{p}}\tilde{\chi}^{2}\tilde{\psi}&2\tilde{\chi}\tilde{D}\\\
~{}&~{}&\tilde{\chi}^{2}\tilde{\psi}\end{pmatrix}.$

If $\tilde{\psi}$ is unramified at $\ell$, then the action of $I_{\ell}$ on on
$\tilde{\mathbf{V}}_{f,\psi}$ is via the matrix
$\begin{pmatrix}1&\tilde{D}&\tilde{D}^{2}\\\ ~{}&1&2\tilde{D}\\\
~{}&~{}&1\end{pmatrix}$ and hence $(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$
is one dimensional and the action of $\operatorname{Frob}_{\ell}$ on this
space is via $\tilde{\chi}^{2}\tilde{\psi}$. Thus
$\tilde{P}_{\ell,f}=(1-\tilde{a}_{\ell}(f)^{2}\psi(\ell)X)$. It follows that
$\tilde{\ell}^{-1}$ is a root of $\tilde{P}_{\ell,f}$ if and only if
$\ell\equiv a_{\ell}(f)^{2}\psi(\ell)\pmod{\mathfrak{P}}.$

If $\tilde{\psi}$ is ramified at $\ell$, then the action of $I_{\ell}$ on on
$\tilde{\mathbf{V}}_{f,\psi}$ is via the matrix
$\begin{pmatrix}\tilde{\psi}&\tilde{D}&\tilde{D}^{2}\\\
~{}&\tilde{\psi}&2\tilde{D}\\\ ~{}&~{}&\tilde{\psi}\end{pmatrix}$ and in this
case $(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}=0.$ ∎

Next we deal with the case
$\operatorname{ord}_{\ell}(N)=\operatorname{ord}_{\ell}(M)>0$. In this case,

$\tilde{\rho}_{f}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\chi}_{1}&~{}\\\
~{}&\tilde{\chi}_{2}\end{pmatrix}$

where $\tilde{\chi}_{2}$ is an unramified character such that
$\tilde{\chi}_{2}(\operatorname{Frob}_{\ell})=\tilde{a}_{\ell}(f)$ (cf.
[Hid00, Theorem 3.26(3a)]). The residual representation attached to
$(V_{f})_{I_{\ell}}$ is one dimensional (cf. [HL19, Proof of Lemma 5.4]) and
hence the character $\tilde{\chi}_{1}$ must be ramified. It follows that

$\tilde{\mathbf{V}}_{f}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\chi}_{1}^{2}&~{}&~{}\\\
~{}&\tilde{\chi}_{1}\tilde{\chi}_{2}&~{}\\\
~{}&~{}&\tilde{\chi}_{2}^{2}\end{pmatrix}\text{ and hence
}\tilde{\mathbf{V}}_{f,\psi}|_{G_{\ell}}\sim\begin{pmatrix}\tilde{\chi}_{1}^{2}\tilde{\psi}&~{}&~{}\\\
~{}&\tilde{\chi}_{1}\tilde{\chi}_{2}\tilde{\psi}&~{}\\\
~{}&~{}&\tilde{\chi}_{2}^{2}\tilde{\psi}\end{pmatrix}.$

######  Lemma 2.9.

Suppose $\operatorname{ord}_{\ell}(N)=\operatorname{ord}_{\ell}(M)>0$. The
$d_{\ell,f}$ is odd if and only if

  * •

$\varepsilon(\ell)\ell^{k-2}\psi(\ell)\equiv 1\pmod{\mathfrak{P}}$ if
$\tilde{\chi}_{1}\tilde{\psi}$ is unramified.

  * •

$a_{\ell}(f)^{2}\psi(\ell)\ell^{-1}\equiv 1\pmod{\mathfrak{P}}$ if
$\tilde{\chi}_{1}\tilde{\psi}$ and $\tilde{\chi}_{1}^{2}\tilde{\psi}$ are both
ramified.

  * •

${a}_{\ell}(f)^{-2}{\ell}^{2k-3}{\varepsilon}(\ell)^{2}{\psi}(\ell)\equiv
1\pmod{\mathfrak{P}}$ if $\tilde{\chi}_{1}\tilde{\psi}$ and $\tilde{\psi}$ are
ramified but $\tilde{\chi}_{1}^{2}\tilde{\psi}$ is unramified.

  * •

any one of equations (13) or (14) below holds and the other does not hold, if
$\tilde{\chi}_{1}\tilde{\psi}$ is ramified but
$\tilde{\chi}_{1}^{2}\tilde{\psi}$ and $\tilde{\psi}$ are unramified.

The bullets above exhaust all the possibilities that can occur when
$\operatorname{ord}_{\ell}(N)=\operatorname{ord}_{\ell}(M)>0$.

###### Proof.

Case 1: Suppose that the character $\tilde{\chi}_{1}\tilde{\psi}$ is
unramified. Since $\tilde{\chi}_{1}$ is ramified, it implies that
$\tilde{\chi}_{1}^{2}\tilde{\psi}$ is also ramified. The character
$\tilde{\psi}$ must also be ramified at $\ell$ because if not then
$\tilde{\chi_{1}}=(\tilde{\chi}_{1}\tilde{\psi})(\tilde{\psi}^{-1})$ becomes
unramified which is a contradiction. Then
$\tilde{\mathbf{V}}_{f,\psi}|_{I_{\ell}}\sim\begin{pmatrix}\tilde{\chi}_{1}^{2}\tilde{\psi}&~{}&~{}\\\
~{}&1&~{}\\\ ~{}&~{}&\tilde{\psi}\end{pmatrix}.$ It follows that the the space
$(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$ is one dimensional and the action
of $\operatorname{Frob}_{\ell}$ on this space is via
$\tilde{\chi}_{1}\tilde{\chi}_{2}\tilde{\psi}$. Therefore
$\tilde{P}_{\ell,f}=(1-\tilde{\varepsilon}(\ell)\tilde{\ell}^{k-1}\tilde{\psi}(\ell)X)$.
It follows that $d_{\ell,f}=1$ if and only if

(12) $\varepsilon(\ell)\ell^{k-2}\psi(\ell)\equiv 1\pmod{\mathfrak{P}}.$

Case 2: Suppose that both the characters $\tilde{\chi}_{1}\tilde{\psi}$ and
$\tilde{\chi}_{1}^{2}\tilde{\psi}$ are ramified. Then
$(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$ is nontrivial if and only if
$\tilde{\psi}$ is unramified at $\ell$; in this case the action of
$\operatorname{Frob}_{\ell}$ on the one-dimensional space
$(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$ is given by
$\tilde{\chi}_{2}^{2}\tilde{\psi}$. Therefore,
$\tilde{P}_{\ell,f}=(1-\tilde{a}_{\ell}(f)^{2}\tilde{\psi}(\ell)X)$. It
follows that $d_{\ell,f}=1$ if and only if

(13) $a_{\ell}(f)^{2}\psi(\ell)\ell^{-1}\equiv 1\pmod{\mathfrak{P}}.$

Case 3: Suppose that the character $\tilde{\chi}_{1}\tilde{\psi}$ is ramified
but $\tilde{\chi}_{1}^{2}\tilde{\psi}$ is unramified. Now there are two
subcases of this, which we deal separately.

Subcase (3i): $\tilde{\psi}$ is ramified at $\ell$. In this case
$\tilde{\mathbf{V}}_{f,\psi}|_{I_{\ell}}\sim\begin{pmatrix}1&~{}&~{}\\\
~{}&\tilde{\chi}_{1}\tilde{\psi}&~{}\\\ ~{}&~{}&\tilde{\psi}\end{pmatrix}.$ It
follows that the space $(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$ is one
dimensional and the action of $\operatorname{Frob}_{\ell}$ on this space is
via the character
$\tilde{\chi}_{1}^{2}\tilde{\psi}=\det(\tilde{\rho}_{f})^{2}\tilde{\chi}_{2}^{-2}\tilde{\psi}$.
It follows that
$\tilde{P}_{\ell,f}=\big{(}1-\tilde{a}_{\ell}(f)^{-2}\tilde{\ell}^{2k-2}\tilde{\varepsilon}(\ell)^{2}\tilde{\psi}(\ell)X\big{)}$.
Do $d_{\ell,f}=1$ if and only if

(14) ${a}_{\ell}(f)^{-2}{\ell}^{2k-3}{\varepsilon}(\ell)^{2}{\psi}(\ell)\equiv
1\pmod{\mathfrak{P}}.$

Subcase (3ii): $\tilde{\psi}$ is unramified at $\ell$. This means that
$\tilde{\chi}_{1}$ is ramified since we are under the assumption that
$\tilde{\chi}_{1}\tilde{\psi}$ is ramified. In this case
$\tilde{\mathbf{V}}_{f,\psi}|_{I_{\ell}}\sim\begin{pmatrix}1&~{}&~{}\\\
~{}&\tilde{\chi}_{1}&~{}\\\ ~{}&~{}&1\end{pmatrix}.$ It follows that the the
space $(\tilde{\mathbf{V}}_{f,\psi})_{I_{\ell}}$ is two dimensional and the
action of $\operatorname{Frob}_{\ell}$ on this space is via the diagonal
matrix $\begin{pmatrix}\tilde{\chi}_{1}^{2}\tilde{\psi}&~{}\\\
~{}&\tilde{\chi}_{2}^{2}\tilde{\psi}\end{pmatrix}.$ Therefore
$\tilde{P}_{\ell,f}=\big{(}1-\tilde{a}_{\ell}(f)^{-2}\tilde{\ell}^{2k-2}\tilde{\varepsilon}(\ell)^{2}\tilde{\psi}(\ell)X\big{)}\big{(}1-\tilde{a}_{\ell}(f)^{2}\tilde{\psi}(\ell)X\big{)}$.
Hence, $d_{\ell,f}=1$ if and only if any one of equations (13) or (14) holds
and the other does not hold. ∎

Summarizing, we have shown the following proposition.

######  Proposition 2.10.

We define $\mathcal{S}_{f,\psi}\subset\Sigma_{0}$ to be the subset consisting
of primes $\ell$ satisfying Lemma 2.7, Lemma 2.8 and Lemma 2.9 such that
$d_{\ell,f}$ is odd. Then

$\sum_{\ell\in\Sigma_{0}}\delta_{\ell}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})\equiv|\mathcal{S}_{f,\psi}|\pmod{2}.$

##  3\. Congruent modular forms

We consider two modular forms $f_{i}$ (for $i=1,2$) of level $N_{i}$ and
character $\varepsilon_{i}$ as in section 2. By enlarging $L$, if necessary,
we assume that $a_{n}(f_{i})\in L$ for all $n$. Similarly, enlarging $\Sigma$
if necessary, we assume that $\Sigma$ is a set of places of $\mathbb{Q}$ that
contains $p,\infty$, the primes dividing $N_{1}N_{2}$ and the primes dividing
the conductor $N_{\psi}$. We continue to assume that (Hyp) is true for both
$f_{1}$ and $f_{2}$. We further assume that the residual representations are
isomorphic, i.e.

$\tilde{\rho_{1}}\cong\tilde{\rho_{2}}.$

Under the above circumstances the following result is a work of
Ray–Sujatha–Vatsal (cf. [RSV23, Proposition 3.11]).

######  Proposition 3.1.

The $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
vanishes if and only if the $\mu$-invariant of
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})$
vanishes. Moreover, if these $\mu$-invariants are zero, then the
imprimitive-$\lambda$-invariants coincide, i.e.

$\lambda^{\Sigma_{0}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})=\lambda^{\Sigma_{0}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty}).$

######  Theorem 3.2.

Assume the vanishing of the $\mu$-invariants as in proposition 3.1. We have
the congruence

$\lambda(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})+|\mathcal{S}_{f_{1},\psi}|\equiv\lambda(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})+|\mathcal{S}_{f_{2},\psi}|\pmod{2}.$

###### Proof.

Proposition 3.1 and (5) give

$\lambda(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})+\sum_{\ell\in\Sigma_{0}}\delta_{\ell}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})=\lambda(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})+\sum_{\ell\in\Sigma_{0}}\delta_{\ell}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty}).$

The result now follows from proposition 2.10. ∎

######  Remark 3.3.

Furthermore, if the representation $\mathbf{V}_{f_{i},\psi}$ is self-dual,
i.e. one can identify $\mathbf{V}_{f_{i},\psi}$ with
$(\mathbf{V}_{f_{i},\psi})^{\vee}(1)$ and the local condition defining the
Selmer group
$\operatorname{Sel}_{p^{\infty}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{(n)})$ at
the prime above $p$ is its own orthogonal complement under the local Tate-
pairing (cf. [Fla90, eq. (8)]), then the proof of [Gre99, Proposotion 3.10]
shows that

$\mathrm{corank}_{\mathcal{O}_{L}}\operatorname{Sel}_{p^{\infty}}(\mathbf{V}_{f_{i},\psi}/\mathbb{Q})\equiv\lambda(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})\pmod{2}.$

Here $\mathbb{Q}_{(n)}$ is the subfield of $\mathbb{Q}_{\infty}$ of degree
$p^{n}$ over $\mathbb{Q}$. Note that Greenberg’s proof uses Cassels-Tate
pairing, but for our proof we should replace it by the generalized pairing of
Flach [Fla90, Theorem 2] which needs this self-duality assumption (cf. [Fla90,
eq. (18) and the following paragraph]). This gives

$\mathrm{corank}_{\mathcal{O}_{L}}\operatorname{Sel}_{p^{\infty}}(\mathbf{V}_{f_{1},\psi}/\mathbb{Q})+|\mathcal{S}_{f_{1},\psi}|\equiv\mathrm{corank}_{\mathcal{O}_{L}}\operatorname{Sel}_{p^{\infty}}(\mathbf{V}_{f_{2},\psi}/\mathbb{Q})+|\mathcal{S}_{f_{2},\psi}|\pmod{2}.$

##  4\. The non-ordinary case

We recall the setup as in [BLV21] with some notational changes to match with
section 2.1. Let $f$ is a normalized, cuspidal, eigen-newform of weight $k$
(in [BLV21] it is $k+2$), level $N$ and nebentypus $\varepsilon$. We also
assume that $p\nmid N$ and $p\geqslant k$ is an odd prime such that
$a_{p}(f)=0$. As before let $\Sigma$ be a finite set of places of $\mathbb{Q}$
that contains $p$, primes that divide the level $N$, primes that divide the
conductor $N_{\psi}$ and $\infty$. We write $\pm\alpha$ for the roots of the
Hecke polynomial $X^{2}+\varepsilon(p)p^{k-1}$ of $f$ at $p$. Let
$L/\mathbb{Q}$ be a number field containing the Hecke field
$\mathbb{Q}(\\{a_{n}(f)\\}_{n\geqslant 1})$ of $f$ as well as $\alpha^{2}$ and
the image of a Dirichlet character $\psi$ of conductor $N_{\psi}$ coprime to
$Np$. Assume that $\psi$ satisfies all the conditions mentioned in [BLV21,
page 3] (our $\psi$ is their $\chi^{-1}$). Let $\mathfrak{P}$ be a prime of
$L$ above $p$ and let $\mathcal{O}$ be the ring of integers of the completion
$L_{\mathfrak{P}}$. Let us put
$V_{f}^{*}=\operatorname{Hom}(V_{f},L_{\mathfrak{P}})$ and we endow it with
the contragredient Galois action. We set
$\mathbf{M}_{f,\psi^{-1}}:=\mathrm{Sym}^{2}\text{ }T_{f}^{*}(1+\psi^{-1})$.
Let
$\Gamma_{0}=\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q})$
so that $\Gamma_{0}=\Delta\times\Gamma$ where $\Delta$ is a finite group of
order $p-1$ and $\Gamma\cong\mathbb{Z}_{p}$. The hypothesis $a_{p}(f)=0$ gives
a $G_{\mathbb{Q}_{p}}$-equivariant decomposition

$\mathbf{M}_{f,\psi^{-1}}=M_{1,f,\psi^{-1}}\oplus M_{2,f,\psi^{-1}}$

(cf. [BLV21, page 5 and page 25]) and exploiting this decomposition,
Büyükboduk–Lei–Venkat defined three signed Coleman maps

$\operatorname{Col}^{\clubsuit}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),\mathbf{M}_{f,\psi^{-1}})\rightarrow\Lambda_{\mathcal{O}}(\Gamma_{0})$

for $\clubsuit\in\\{+,-,\bullet\\}$ (see [BLV21, Section 4.2]). The kernels of
these maps are used to define certain local Selmer conditions at $p$ which
leads to the following definition of doubly signed Selmer groups (cf. [BLV21,
Defn. 4.4.1]). Let us set
$\mathbf{M}_{f,\psi^{-1}}^{\vee}(1):=(\mathbf{M}_{f,\psi^{-1}})^{\vee}(1)$.

######  Definition 4.1.

Let $\mathcal{S}$ denote the set of pairs
$\\{(+,-),(+,\bullet),(-,\bullet)\\}.$ For
$\mathfrak{S}=(\clubsuit,\spadesuit)\in\mathfrak{S}$, we define the discrete
Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)/\mathbb{Q}(\mu_{p^{\infty}}))$
as the kernel of the restriction map

$\displaystyle
H^{1}(\mathbb{Q}(\mu_{p^{\infty}}),\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))\rightarrow\prod_{v\mid
p}\frac{H^{1}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))}{H^{1}_{\mathfrak{S}}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))}\times\prod_{v\nmid
p}\frac{H^{1}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))}{H^{1}_{\mathrm{un}}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))},$

where $v$ runs through all primes of $\mathbb{Q}(\mu_{p^{\infty}})$ and for
$v\mid p$, the local condition
$H^{1}_{\mathfrak{S}}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))$
is the orthogonal complement of
$\ker\big{(}\operatorname{Col}^{\clubsuit}\big{)}\cap\ker\big{(}\operatorname{Col}^{\spadesuit}\big{)}$
under the local Tate pairing.

######  Remark 4.2.

We have taken $\psi^{-1}$ in the definition of $\mathbf{M}_{f,\psi^{-1}}$
because
$\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)=\mathbf{T}_{f,\psi}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}=\mathbf{A}_{f,\psi}$
(cf. [LZ19, Notation 3.2.4]) which coincides with the notation we fixed in
section 2.1.

Here is a conjecture on the cotorsioness of these Selmer groups made in
[BLV21, Conjecture 4.4.3]

######  Conjecture 4.3.

For every $\mathfrak{S}\in\mathcal{S}$, and every character $\eta$ of
$\Delta$, the $\eta$-isotypic component
$e_{\eta}\operatorname{Sel}_{\mathfrak{S}}(\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)/\mathbb{Q}(\mu_{p^{\infty}}))$
is $\Lambda$-cotorsion.

Some evidence for this conjecture is also provided (see [BLV21, Theorem B,
(ii)]).

###  4.1. The cyclotomic and finite level.

Recall that $\Gamma$ is the Galois group of the cyclotomic extension
$\mathbb{Q}_{\infty}$ over $\mathbb{Q}$ and
$\Gamma_{n}=\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q}_{(n)})$ where
$\mathbb{Q}_{(n)}$ is the extension over $\mathbb{Q}$ such that
$[\mathbb{Q}_{(n)}:\mathbb{Q}]=p^{n}$. For $v\mid p$, we set

$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)):=H^{1}_{\mathfrak{S}}(\mathbb{Q}(\mu_{p^{\infty}})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))^{\Delta}$

and

$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{(n)})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)):=H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{M}_{f,\psi^{-1}}^{\vee}(1))^{\Gamma_{n}}$

and we define the corresponding Selmer groups
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)/\mathbb{Q}_{\infty})$
and
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{M}_{f,\psi^{-1}}^{\vee}(1)/\mathbb{Q}_{(n)})$
with these local conditions.

######  Theorem 4.4.

The Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$
contains no proper $\Lambda$-submodule of finite index.

Before we prove this theorem we have to prove some preliminary lemmas.

For $s\in\mathbb{Z}$, we can take the cyclotomic twist
$\mathbf{A}_{f,\psi,s}:=\mathbf{A}_{f,\psi}\otimes(\omega|_{\Gamma})^{s}$
where $\omega|_{\Gamma}:\Gamma\rightarrow 1+p\mathbb{Z}_{p}$ is an isomorphism
and one can define the corresponding Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})$
just as before. For the prime $v$ above $p$ one uses the
$G_{\mathbb{Q}_{p}}$-invariant submodule

$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})):=H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}))\otimes(\omega|_{\Gamma})^{s}.$

As a $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{\infty})$-module
$\mathbf{A}_{f,\psi,s}=\mathbf{A}_{f,\psi}$ and thus
$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})=H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})\otimes(\omega|_{\Gamma})^{s}$.
For a prime $v$,
$H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})=H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})\otimes(\omega|_{\Gamma})^{s}$.
For the finite level $\mathbb{Q}_{(n)}$, we can similarly define the twisted
Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{(n)})$ as
above with the local condition at $p$ defined as
$H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})^{\Gamma_{n}}$. Thus we
remark that for $K=\mathbb{Q}_{\infty}$ or $\mathbb{Q}_{(n)}$,
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/K)\cong\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi}/K)\otimes(\omega|_{\Gamma})^{s}$
as $\Lambda$-modules.

Let
$\mathbf{M}_{f,\psi,-s}:=\mathbf{M}_{f,\psi}\otimes(\omega|_{\Gamma})^{-s}$.
Let
$\mathbf{J}_{f,\psi^{-1},-s}=\mathbf{M}_{f,\psi,-s}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\cong(\mathbf{T}_{f,\psi,s})^{*}$
We begin with a "control theorem" for these signed Selmer groups.

######  Lemma 4.5.

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

$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{(n)})\rightarrow\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{\infty})^{\Gamma_{n}}$

are finite of bounded orders as $n$ varies.

###### Proof.

Consider the commutative diagram

(15)
${0}$${\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{(n)})}$${H^{1}(\mathbb{Q}_{(n)},\mathbf{J}_{f,\psi^{-1},-s})}$${\bigoplus_{v}\mathcal{P}_{v}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{(n)})}$${0}$${\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{\infty})^{\Gamma_{n}}}$${H^{1}(\mathbb{Q}_{\infty},\mathbf{J}_{f,\psi^{-1},-s})^{\Gamma_{n}}}$${\bigoplus_{v}\mathcal{P}_{v}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{\infty})^{\Gamma_{n}}}$$\scriptstyle{\rho_{{n}}}$$\scriptstyle{h}$$\scriptstyle{\lambda}$$\scriptstyle{\oplus_{v}\Xi_{{n,v}}}$

Here $\mathcal{P}_{v}(\mathbf{J}_{f,\psi^{-1},-s},-)$ is the local term at
place $v$ defining the corresponding signed Selmer group. By hypothesis (inv)
and the inflation restriction exact sequence the map $h$ is an isomorphism.
Next we analyse the kernel and cokernel of the map $\Xi_{n,v}$.

For the prime $v$ above $p$, consider the commutative diagram

(16)
${0}$${H^{1}_{\mathfrak{S}}((\mathbb{Q}_{(n)})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}$${H^{1}((\mathbb{Q}_{(n)})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}$${\frac{H^{1}((\mathbb{Q}_{(n)})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}{H^{1}_{\mathfrak{S}}((\mathbb{Q}_{(n)})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}}$${0}$${H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{J}_{f,\psi^{-1},-s}))^{\Gamma_{n}}}$${H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{J}_{f,\psi^{-1},-s}))^{\Gamma_{n}}}$${\left(\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}{H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{J}_{f,\psi^{-1},-s}))}\right)^{\Gamma_{n}}}$$\scriptstyle{m}$$\scriptstyle{g}$$\scriptstyle{\Xi_{n,v}}$

The map $m$ is an isomorphism by definition. The central map $g$ is an
isomorphism by inflation restriction exact sequence and (inv). Therefore, the
map $\Xi_{n,v}$ is injective.

For the prime $v\nmid p$, it is a usual known argument (see for example
[Pon20, page 1645, proof of Lemma 2.3] or [HL19, second paragraph of page
1275]) that the map $\Xi_{n,v}$ has finite kernel of bounded orders as $n$
varies. ∎

######  Lemma 4.6.

Suppose that the Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t}/\mathbb{Q}_{\infty})$
is $\Lambda$-cotorsion for some fixed $t\in\mathbb{Z}$. Then for all but
finitely many $s$, the map

$H^{1}(\mathbb{Q},\mathbf{A}_{f,\psi,s})\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q})$

is surjective, and for all $s\in\mathbb{Z}$, the map

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})$

is surjective.

###### Proof.

Since the Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t}/\mathbb{Q}_{\infty})$
is $\Lambda$-cotorsion, then for all but finitely many $u\in\mathbb{Z}$,
$(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t}/\mathbb{Q}_{\infty})\otimes\omega^{u})^{\Gamma_{n}}=\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},u-t}/\mathbb{Q}_{\infty})^{\Gamma_{n}}$
is finite for every $n$. Since $t$ is fixed, it means that for all but
finitely many $s\in\mathbb{Z}$,
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{\infty})^{\Gamma_{n}}$
is finite for every $n$. Thus by Lemma 4.5, possibly avoiding another finite
number of $s\in\mathbb{Z}$,
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-s}/\mathbb{Q}_{(n)})$
is finite for every $n$. For such an $s$ and any $n$, [Gre99, Proposition
4.13] shows that the cokernel of the map

(17)
$H^{1}(\mathbb{Q}_{(n)},\mathbf{A}_{f,\psi,s})\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{(n)})$

is the Pontryagin dual of
$H^{0}(\mathbb{Q}_{(n)},\mathbf{J}_{f,\psi^{-1},-s}).$ Recall that
$\mathbf{J}_{f,\psi^{-1},-s}=(\mathbf{T}_{f,\psi,s})^{*}.$ By (inv) we have
$H^{0}(\mathbb{Q},\mathbf{J}_{f,\psi^{-1},0})=0$. Since
$\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ is pro-$p$,
$H^{0}(\mathbb{Q}_{\infty},\mathbf{J}_{f,\psi^{-1},0})$ is trivial. Moreover,
as $\mathbf{J}_{f,\psi^{-1},-s}\cong\mathbf{J}_{f,\psi^{-1},0}$ as
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{\infty})$-modules, hence
$H^{0}(\mathbb{Q},\mathbf{J}_{f,\psi^{-1},-s})=0$ and
$H^{0}(\mathbb{Q}_{(n)},\mathbf{J}_{f,\psi^{-1},-s})=0$. Therefore the map in
(17) is surjective for any $n$. The lemma then follows by passing to the
direct limit relative to the restriction maps and noting that, for any $s$,
$\mathbf{A}_{f,\psi,s}\cong\mathbf{A}_{f,\psi}$ as
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}_{\infty})$-modules. ∎

######  Lemma 4.7.

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

$\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q})\rightarrow\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})^{\Gamma}$

is surjective.

###### Proof.

Let $v$ be a prime above $p$. Since $a_{p}(f_{i})=0$, we have the
decomposition $\mathbf{M}_{f_{i},\psi^{-1}}=M_{1,f_{i},\psi^{-1}}\oplus
M_{2,f_{i},\psi^{-1}}$ as $G_{\mathbb{Q}_{p}}$-modules where
$M_{j,f_{i},\psi^{-1}}$ is of rank $j$ (cf. [BLV21, Corollary 4.1.2]). In
[BLV21], Büyükboduk–Lei–Venkat defined the Coleman maps
$\operatorname{Col}_{f_{i}}^{\pm}$ from the rank $2$ lattice
$M_{2,f_{i},\psi^{-1}}$ generalizing methods of [LLZ11] and [LLZ10] while the
Coleman map $\operatorname{Col}_{f_{i}}^{\bullet}$ was defined using the rank
$1$ lattice $M_{1,f_{i},\psi^{-1}}$ (cf. [BLV21, Lemma 4.1.5 and definition
4.2.1]). More precisely, they show the existence of Coleman maps

$\displaystyle\overline{\operatorname{Col}}^{\pm}_{f_{i}}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),M_{2,f_{i},\psi^{-1}})$
$\displaystyle\rightarrow\Lambda_{\mathcal{O}}(\Gamma_{0})$
$\displaystyle\overline{\operatorname{Col}}^{\bullet}_{f_{i}}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),M_{1,f_{i},\psi^{-1}})$
$\displaystyle\rightarrow\Lambda_{\mathcal{O}}(\Gamma_{0})$

such that
$\operatorname{Col}^{?}_{f_{i}}=\overline{\operatorname{Col}}^{?}_{f_{i}}\circ\mathrm{Proj}_{{f_{i}},?}.$
Here
$\mathrm{Proj}_{{f_{i}},?}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),\mathbf{M}_{f,\psi^{-1}})\rightarrow
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),M_{j,f_{i},\psi^{-1}})$
is the natural projection map where $j=2$ if $?=+\text{ or }-$, and $j=1$ if
$?=\bullet$.

Recall that $\mathcal{S}$ was the set of pairs
$\\{(+,-),(+,\bullet),(-,\bullet)\\}.$ If
$\mathfrak{S}=(\clubsuit,\spadesuit)\in\mathcal{S}$, then define

$\displaystyle\operatorname{Col}_{f_{i}}^{\mathfrak{S}}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),\mathbf{M}_{f_{i},\psi^{-1}})$
$\displaystyle\rightarrow\Lambda_{\mathcal{O}}(\Gamma_{0})^{\oplus 2}$
$\displaystyle z$
$\displaystyle\mapsto\operatorname{Col}_{f_{i}}^{\clubsuit}(z)\oplus\operatorname{Col}_{f_{i}}^{\spadesuit}(z).$

The Pontryagin dual of
$H^{1}_{\mathfrak{S}}((\mathbb{Q}(\mu_{p^{\infty}}))_{v},\mathbf{A}_{f_{i},\psi})$
is isomorphic to $\operatorname{Im}\operatorname{Col}_{f_{i}}^{\mathfrak{S}}$
which is contained in a free $\mathbb{Z}_{p}[[\Gamma_{0}]]$-module. Therefore,
the Pontryagin dual of
$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})$ is
contained in a free $\mathbb{Z}_{p}[[\Gamma]]$-module. and hence
$(H^{1}_{\mathfrak{S}}(\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})_{\Gamma}=0$.
This implies that
$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})_{\Gamma}$
is trivial. Therefore, we have an exact sequence

$0\rightarrow
H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})^{\Gamma}\rightarrow
H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})^{\Gamma}\rightarrow\Big{(}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})}{H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})}\Big{)}^{\Gamma}\rightarrow
0.$

By (inv), $H^{1}(\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})^{\Gamma}\cong
H^{1}(\mathbb{Q}_{p},\mathbf{A}_{f,\psi,s})$ and by definition
$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})^{\Gamma}\cong
H^{1}_{\mathfrak{S}}(\mathbb{Q}_{p},\mathbf{A}_{f,\psi,s})$. This gives a
surjection

$\frac{H^{1}(\mathbb{Q}_{p},\mathbf{A}_{f,\psi,s})}{H^{1}_{\mathfrak{S}}(\mathbb{Q}_{p},\mathbf{A}_{f,\psi,s})}\rightarrow\Big{(}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})}{H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi,s})}\Big{)}^{\Gamma}.$

For the primes not above $p$, the proof is standard (cf. [Pon20, Lemma 2.5]).
∎

Proof of Theorem 4.4: Note that it will be sufficient to show Theorem 4.4 for
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})$
for some $s$. Fix some $s$ satisfying Lemma 4.6 and Lemma 4.7. Using Poitou-
Tate exact sequence [PR95, Proposition A.3.2],
$H^{2}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})$ injects into
$\varinjlim\oplus H^{2}((\mathbb{Q}_{(n)})_{v},\mathbf{A}_{f,\psi,s}).$ By
local Tate-duality, this is isomorphic to $\varprojlim\oplus
H^{0}((\mathbb{Q}_{(n)})_{v},\mathbf{M}_{-s}).$ But this is zero (arguments as
in [HL19, Corollary 2.8]). Using Hochschild-Serre spectral sequence, as in
[HL19, Proposition 3.2], one can deduce that
$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})$ has no nontrivial submodule
of finite index. This implies that

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})_{\Gamma}=0.$

Lemma 4.6 and Lemma 4.7 gives that for all but finitely many $s\in\mathbb{Z}$
the map

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})^{\Gamma}\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})^{\Gamma}$

is surjective. Again by Lemma 4.6 we have the exact sequence

$0\rightarrow\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})\rightarrow
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})\rightarrow
0.$

On taking $\Gamma$-coinvariants we obtain the following long exact sequence.

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})^{\Gamma}\rightarrow\bigoplus_{v}\mathcal{P}_{v}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})^{\Gamma}\rightarrow\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})_{\Gamma}\rightarrow
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})_{\Gamma}.$

The proof follows by noting that the first map is surjective, hence the last
map is injective, but
$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi,s})_{\Gamma}=0$. Therefore,
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi,s}/\mathbb{Q}_{\infty})_{\Gamma}$
must be trivial. ∎

From the following exact sequence

$0\rightarrow\mathbf{A}_{f,\psi}[\mathfrak{P}]\rightarrow\mathbf{A}_{f,\psi}\rightarrow\mathbf{A}_{f,\psi}\rightarrow
0$

we obtain that the sequence

$0\rightarrow
H^{0}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})/\mathfrak{P}\rightarrow
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}[\mathfrak{P}])\rightarrow
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})[\mathfrak{P}]\rightarrow 0.$

As $H^{0}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})=0$ by (inv), we obtain

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}[\mathfrak{P}])\cong
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})[\mathfrak{P}].$

The same proof also gives that, for a prime $v\mid p$,

$H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])\cong
H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})[\mathfrak{P}].$

Define

$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}]):=H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})[\mathfrak{P}].$

It is also well-known that
$H^{1}_{\mathrm{un}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])=H^{1}_{\mathrm{un}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})[\mathfrak{P}]$
for primes $v\nmid p$ and $v\not\in\Sigma.$

######  Definition 4.8.

The $\Sigma_{0}$-imprimitive signed Selmer group
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})$
for the Galois module $\mathbf{A}_{f,\psi}$ is define as the kernel of the
following map

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi})\rightarrow\prod_{v\mid
p}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})}{H_{\mathfrak{S}}^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})}\times\prod_{v\in\Sigma\backslash\Sigma_{0}}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})}{H^{1}_{\mathrm{un}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi})}.$

The $\Sigma_{0}$-imprimitive signed Selmer group
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$
for the Galois module $\mathbf{A}_{f,\psi}[\mathfrak{P}]$ is define as the
kernel of the following map

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f,\psi}[\mathfrak{P}])\rightarrow\prod_{v\mid
p}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])}{H_{\mathfrak{S}}^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])}\times\prod_{v\in\Sigma\backslash\Sigma_{0}}\frac{H^{1}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])}{H^{1}_{\mathrm{un}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}[\mathfrak{P}])}.$

By the discussion before definition 4.8, it follows that the local conditions
defining the Selmer groups
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]$
and
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$
are the same. Hence there is a $\Lambda$-module isomorphism

(18)
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]\cong\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f,\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty}).$

######  Lemma 4.9.

Assume that $\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ and
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t})$ are cotorsion
$\Lambda$-modules for some fixed $t\in\mathbb{Z}$. Then the Selmer group
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ is also
$\Lambda$-cotorsion and

$\mu(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi}))=\mu(\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})).$

###### Proof.

The Pontryagin dual of
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ is a
quotient of $\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ and hence
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ is
cotorsion. As the Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t})$ is cotorsion,
by Lemma 4.6, the global to local map defining the Selmer group
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ is surjective. Hence
the global to local map defining the Selmer group
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ is also
surjective. Also,
$H^{1}_{\mathrm{un}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f,\psi}))=0$ for
$v\nmid p$ (cf. [PR95, A.2.4]). Therefore, (4) holds where the Selmer groups
are replaced by the corresponding signed versions. The proof now follows from
Lemma 2.4. ∎

######  Remark 4.10.

Assume that $\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ and
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{J}_{f,\psi^{-1},-t})$ are cotorsion
$\Lambda$-modules for some fixed $t\in\mathbb{Z}$. Then the imprimitive signed
Selmer group
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f,\psi})$ contains
no proper $\Lambda$-submodule of finite index. The proof is the same as the
proof of Theorem 4.4 and hence omitted.

Now suppose $f_{i}$ (for $i=1,2$) are modular forms of level $N_{i}$ and
character $\varepsilon_{i}$ as in section 3 with isomorphic residual
representations $\tilde{\rho_{1}}\cong\tilde{\rho_{2}}$. Suppose that
$a_{p}(f_{i})=0$ for both $i=1,2$. Let $\Sigma$ be a set of places of
$\mathbb{Q}$ that contains $p,\infty$, the primes dividing $N_{1}N_{2}$ and
the primes dividing the conductor $N_{\psi}$ and as before let
$\Sigma_{0}=\Sigma\backslash\\{p,\infty\\}.$

######  Proposition 4.11.

We have the following isomorphism as $\Lambda$-modules.

$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{1},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})\cong\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{2},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty}).$

###### Proof.

Clearly, we have the isomorphisms

$H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f_{1},\psi}[\mathfrak{P}])\cong
H^{1}(\mathbb{Q}_{\infty},\mathbf{A}_{f_{2},\psi}[\mathfrak{P}])\text{ and
}\mathcal{P}_{v}(\mathbf{A}_{f_{1},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})\cong\mathcal{P}_{v}(\mathbf{A}_{f_{2},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$

for primes $v\nmid p$.

Since
$\mathbf{T}_{f_{1},\psi}/\mathfrak{P}\mathbf{T}_{f_{1},\psi}\cong\mathbf{T}_{f_{2},\psi}/\mathfrak{P}\mathbf{T}_{f_{2},\psi}$,
this gives, by duality, the congruence

$M_{1,f_{1},\psi^{-1}}/\mathfrak{P}M_{1,f_{1},\psi^{-1}}\cong
M_{1,f_{2},\psi^{-1}}/\mathfrak{P}M_{1,f_{2},\psi^{-1}}\text{ and
}M_{2,f_{1},\psi^{-1}}/\mathfrak{P}M_{2,f_{1},\psi^{-1}}\cong
M_{2,f_{2},\psi^{-1}}/\mathfrak{P}M_{2,f_{2},\psi^{-1}}.$

By the theory of Wach modules, these congruence implies that the Iwasawa
cohomologies
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),M_{j,f_{1},\psi^{-1}})$
and
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),M_{j,f_{2},\psi^{-1}})$
are congruent modulo $\mathfrak{P}$ for $j=1,2$ (cf. [Pon20, Section 3.1]).
Therefore, we have the following congruence of the images of signed Coleman
maps defined in the proof of Lemma 4.7.

(19)
$\displaystyle\operatorname{Im}(\overline{\operatorname{Col}}_{f_{1}}^{?})/\mathfrak{P}\operatorname{Im}(\overline{\operatorname{Col}}_{f_{1}}^{?})$
$\displaystyle\cong\operatorname{Im}(\overline{\operatorname{Col}}_{f_{2}}^{?})/\mathfrak{P}\operatorname{Im}(\overline{\operatorname{Col}}_{f_{2}}^{?})\text{
for }?\in\\{\pm,\bullet\\}.$

Recall that $\mathcal{S}$ was the set of pairs
$\\{(+,-),(+,\bullet),(-,\bullet)\\}.$ If
$\mathfrak{S}=(\clubsuit,\spadesuit)\in\mathcal{S}$, then the following
Coleman map was defined in the proof of Lemma 4.7.

$\displaystyle\operatorname{Col}_{f_{i}}^{\mathfrak{S}}:H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),\mathbf{M}_{f_{i},\psi^{-1}})$
$\displaystyle\rightarrow\Lambda_{\mathcal{O}}(\Gamma_{0})^{\oplus 2}$
$\displaystyle z$
$\displaystyle\mapsto\operatorname{Col}_{f_{i}}^{\clubsuit}(z)\oplus\operatorname{Col}_{f_{i}}^{\spadesuit}(z).$

The Pontryagin dual of
$H^{1}_{\mathfrak{S}}((\mathbb{Q}(\mu_{p^{\infty}}))_{v},\mathbf{A}_{f_{i},\psi}[\mathfrak{P}])$
is isomorphic to
$\operatorname{Im}\operatorname{Col}_{f_{i}}^{\mathfrak{S}}/\mathfrak{P}\operatorname{Im}\operatorname{Col}_{f_{i}}^{\mathfrak{S}}$.
Therefore, using (19), we deduce that

$H^{1}_{\mathfrak{S}}((\mathbb{Q}(\mu_{p^{\infty}}))_{v},\mathbf{A}_{f_{1},\psi}[\mathfrak{P}])\cong
H^{1}_{\mathfrak{S}}((\mathbb{Q}(\mu_{p^{\infty}}))_{v},\mathbf{A}_{f_{2},\psi}[\mathfrak{P}]).$

Taking $\Delta$-invariance, one obtains

$H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f_{1},\psi}[\mathfrak{P}])\cong
H^{1}_{\mathfrak{S}}((\mathbb{Q}_{\infty})_{v},\mathbf{A}_{f_{2},\psi}[\mathfrak{P}])$

and this completes the proof of the proposition. ∎

######  Theorem 4.12.

Assume that the hypothesis of Lemma 4.9 is true for both $f_{1}$ and $f_{2}$.
Then the $\mu$-invariant of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})$
vanishes if and only if the $\mu$-invariant of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})$
vanishes. When these $\mu$-invariants are trivial, then the imprimitive signed
$\lambda$-invariants coincide, i.e.

$\lambda(\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty}))=\lambda(\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})).$

###### Proof.

By Lemma 4.9, the $\mu$-invariant of
$\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$
vanishes if and only if
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$
is cofinitely generated as an $\mathcal{O}$-module which is true if and only
if
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]$
is finite. By (18),
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]$
is finite if and only if
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{i},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$
is finite. Then Proposition 4.11 gives that
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{1},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$
is finite if and only if
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{2},\psi}[\mathfrak{P}]/\mathbb{Q}_{\infty})$
is finite. Since the imprimitive Selmer group
$\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$
has no proper $\Lambda$-submodule of finite index, one case easily show that

$\lambda(\operatorname{Sel}^{\Sigma_{0}}_{\mathfrak{S}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty}))=\dim_{\mathbb{F}_{p}}\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}].$

But
$\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]\cong\operatorname{Sel}_{\mathfrak{S}}^{\Sigma_{0}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty})[\mathfrak{P}]$
and hence their $\mathbb{F}_{p}$-dimensions coincide. ∎

Theorem 4.12 yields the following theorem. The proof is the same as in the
ordinary case, since it only involves computing the $\lambda$-invariants of
$\mathcal{P}_{v}(\mathbf{A}_{f_{i},\psi}/\mathbb{Q}_{\infty})$ for primes $v$
that lie above places of $\Sigma_{0}$ and hence different from $p$.

######  Theorem 4.13.

Assume the vanishing of the $\mu$-invariants as in Theorem 4.12. We have the
congruence

$\lambda(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{1},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{1},\psi}|\equiv\lambda(\operatorname{Sel}_{\mathfrak{S}}(\mathbf{A}_{f_{2},\psi}/\mathbb{Q}_{\infty}))+|\mathcal{S}_{f_{2},\psi}|\pmod{2}$

where $\mathcal{S}_{f_{i},\psi}$ are the set of primes in $\Sigma$ defined as
in proposition 2.10.

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# Temporally Extended Successor Representations

Matthew J. Sargent
Department of Computer Science
University College London
<EMAIL_ADDRESS>
&Peter J. Bentley
Department of Computer Science
University College London
<EMAIL_ADDRESS>
Caswell Barry
Department of Cell and Developmental Biology
University College London
<EMAIL_ADDRESS>
&William de Cothi††footnotemark:
Department of Cell and Developmental Biology
University College London
<EMAIL_ADDRESS>
Corresponding Author. https://github.com/mjsargentEqual Contribution

###### Abstract

We present a temporally extended variation of the successor representation,
which we term _t-SR_. t-SR captures the expected state transition dynamics of
temporally extended actions by constructing successor representations over
primitive action repeats. This form of temporal abstraction does not learn a
top-down hierarchy of pertinent task structures, but rather a bottom-up
composition of coupled actions and action repetitions. This lessens the amount
of decisions required in control without learning a hierarchical policy. As
such, t-SR directly considers the time horizon of temporally extended action
sequences without the need for predefined or domain-specific options. We show
that in environments with dynamic reward structure, t-SR is able to leverage
both the flexibility of the successor representation and the abstraction
afforded by temporally extended actions. Thus, in a series of sparsely
rewarded gridworld environments, t-SR optimally adapts learnt policies far
faster than comparable value-based, model-free reinforcement learning methods.
We also show that the manner in which t-SR learns to solve these tasks
requires the learnt policy to be sampled consistently less often than non-
temporally extended policies.

Keywords:

reinforcement learning; temporal abstraction; successor representation

### Acknowledgements

Matthew J. Sargent was funded by an EPSRC Doctoral Training Partnership
Research Studentship (EP/R513143/1). Caswell Barry was funded by a Wellcome
Senior Research Fellowship (212281/Z/18/Z). Matthew J. Sargent would like to
thank Augustine N. Mavor-Parker for his assistance in data visualisation.

##  1 Introduction

In reinforcement learning (RL), agents must solve sequential decision making
tasks to maximise some notion of expected cumulative reward, known as the
value function. Often the solutions to these tasks share elements in terms of
their structure - different tasks can be decomposed into sequences of commonly
shared sub-tasks. These sub-tasks link the timescales of primitive actions
that take place at every step, and that of the broader abstract task,
facilitating planning over these extended timescales. As such, _temporal
abstraction_ is the process by which an agent learns the temporal structure of
a task in a way that can both reduce cognitive load and improve generalisation
across tasks with shared structure.

The canonical approach to address this form of abstraction is the Options
framework [1]. Agents using Options seek to learn a set of policies associated
with distinct sub-tasks, as well as the initiation and termination conditions
for these policies. While this successfully quantifies the temporal
abstraction problem, without predefined handcrafted domain-specific Options
this process can be more complex than learning simple value functions [2].

Rather than learning a specific policy associated with each sub-task, an
alternative approach to temporal abstraction is to directly consider the time
horizon of the actions taken in tandem with the policy for selecting them.
This approach of temporally extending primitive actions by repeating them has
been successfully applied to model-free RL; an agent’s action space can be
augmented with temporally extended actions [3], or an additional policy can be
learnt over the space of possible action repeats [4, 5]. This is in contrast
to the typical usage of fixed action repetition [6, 7] in popular RL
environments such as the Atari Arcade Learning Environment [8], where action
repetition is a fixed property of the environment rather than a process
determined by the agent.

Given the success of temporally extended model-free methods, we set out to
investigate whether temporally extending the successor representation (SR)
could improve generalisation in a similar way. The SR linearly decomposes the
problem of learning the value function into independently learning two
components: a matrix of expected discounted state occupancies, and a vector of
state-dependant expected rewards [9]. Thus, the SR is able to improve upon the
generalisation of model-free methods in tasks where the transition or reward
structures changes [10].

We show that a temporally extended variant of the SR, which we term _t-SR_ ,
performs comparably to temporally-extended model-free approaches in static,
unchanging environments. Further, t-SR exhibits dramatically improved
performance over temporally-extended model-free methods when in environments
with dynamic reward structure - specifically we assess reward revaluation.

##  2 Preliminaries

Learnt Action Repetition In RL, a given task is characterised by a Markov
Decision Process (MDP)
$\mathcal{M}=\langle\mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R}\rangle$.
Here, the state space $\mathcal{S}$ contains all the states the agent can
occupy and the action space $\mathcal{A}$ describes the allowable actions in
each state. The transition probabilities $\mathcal{P}$ then define the
probability
$p(s^{\prime}|s,a)\dot{=}\text{Pr}(S_{t}=s^{\prime}|S_{t-1}=s,A_{t-1}=a)$ of
transitioning from state $s$ to state $s^{\prime}$ having taken action $a$,
with $\mathcal{R}$ being the rewards
$r(s,a)\dot{=}\mathbb{E}\left[R_{t}|S_{t-1}=s,A_{t-1}=a\right]$ that the agent
receives after doing so. The agent is tasked with learning a policy $\pi(a|s)$
which specifies the probability of choosing each action given the current
state. Specifically, for a given finite episodic MDP, the agent’s task is to
learn an optimal policy $\pi^{*}(a|s)$ that maximises expected cumulative
discounted reward across an episode.

Conventionally, these actions are single-step primitive actions - the agent
resamples its policy at $s^{\prime}$ to determine the next action. Recent work
has extended this action selection policy with a _holding-time_ (or _skip_)
for the action [4, 5]. This is the number of times the chosen action will be
repeated before the action selection policy is queried again. Agents are
therefore equipped with both an action selection policy
$\pi^{a}:\mathcal{S}\rightarrow\mathcal{A}$, and an action repetition policy
$\pi^{j}:\mathcal{S}\rightarrow\mathcal{J}$. Here we follow the notation of
[5] and let $j$ be an action repetition number, with $\mathcal{J}$ being the
set of allowable action repetitions.

The action repetition policy $\pi^{j}$ can be learnt using the same methods
used to learn a standard policy. Here, we use the $Q$-learning style update
rule for learnt action repetitions derived in [5] due to its relevance to
constructing SRs [11, 9], but methods for learning repetition policies through
policy gradients also exist [4]. An optimal $\pi^{j*}$ can be obtained by
greedily choosing from the optimal $Q^{\pi j*}(s,j|a)$, with the action-
repetition values drawn from transitions made in the environment according to:

$Q^{\pi
j}(s,j|a)=\mathbb{E}^{\pi}\left[\sum_{k=0}^{j-1}\gamma^{k}r_{t+k}+\gamma^{j}Q^{\pi}(s_{t+j},a*_{t_{j}})|s_{t}=s,a,j)\right]$
(1)

where $a*$ = $\text{argmax}_{a}\left[Q^{\pi}(s_{t+j},a)\right]$ and
$\gamma\in[0,1]$ represents a temporal discount factor that exponentially
downweights distal rewards.

Successor Representations The SR was originally introduced as a method to
improve generalisation in model-free RL [9], which it achieves by decomposing
the value function into expected state occupancies and state-rewards. The SR
itself, is therefore defined as the expected discounted future occupancy of
state $s^{\prime}$ along a trajectory initiated in state $s$ by taking action
$a$:

$M^{\pi}(s,a,s^{\prime})=\mathbb{E}^{\pi}\left[\sum_{t=0}^{\infty}\gamma^{t}\mathbb{1}(s=s^{\prime})|s_{0}=s,a_{0}=a\right]$
(2)

where $\mathbb{1}$ is the indicator function. State-action values for a given
policy can then be obtained by linearly combining $M^{\pi}$ with the rewards
associated with each state, which can be learnt separately via supervised
methods. Thus, the SR allows for rapid reward revaluation by reusing $M^{\pi}$
\- the learnt expected state dynamics for a given policy.

##  3 Temporally Extended Successor Representations

Expected dynamics can be temporally extended in the same fashion as value
functions. Consider the matrix formulation of the SR:
$\mathbf{M^{\pi}}=\sum^{\infty}_{t=0}\left(\gamma\mathbf{T}^{\pi}\right)^{t}$,
where $\mathbf{T}^{\pi}$ is the expected one-step transition function between
states following policy $\pi$. To specify a SR that considers action
repetition, let $\mathbf{T}_{a}$ be the expected transition conditioned on
action $a$ such that $\mathbf{T}^{\pi}=\mathbb{E}^{\pi}(\mathbf{T}_{a})$. The
temporally extended successor representation (t-SR) for an action repetition
of $j$ can then be obtained by:

$\mathbf{M}^{\pi
j}=\mathbb{E}^{\pi}\left[\sum_{k=0}^{j}\left(\gamma\mathbf{T}_{a}\right)^{k}\right]+{\mathbb{E}^{\pi}\left[(\gamma\mathbf{T}_{a})^{j}\right]}\left(\mathbb{I}-\gamma\mathbf{T}^{\pi}\right)^{-1}$
(3)

where the right most expression is derived from the converging infinite series
(Neumann series) of the standard SR. Compared to the SR, these representations
skew according to the action space, reflecting the extended transitions
instigated by repetition of the same action - this leads to a markedly
different representation of distal state than altering the discount factor
$\gamma$ [12] (Fig 1).

To use the SR for determining action repetition, an action is first sampled
from $\pi^{a}$ such that $M(s,a,s^{\prime},j)$ now has two of its arguments
specified: $s$ and $a$. The action repeat is then the $\max_{j}$ of
$M(s,a,s^{\prime},j)W$, where $W$ is the learnt reward vector. The baseline SR
is learnt through temporal difference learning with state occupancies acting
as cummulant rewards, while the reward weights $W$ are learnt through online
supervised learning based on the expected rewards and actual rewards obtained.
Thus, the t-SR can be learnt using using a typical temporal difference
learning rule:

$\displaystyle M^{\pi j}(s,a,s^{\prime},j)\xleftarrow{\alpha}M^{\pi
j}(s,a,s^{\prime},j)+\mathbb{E}^{\pi}\left[\sum_{k=0}^{j-1}\gamma^{k}\mathbb{1}(s_{k}=s)+\gamma^{j}M^{\pi}(s_{t+j},a*,s^{\prime})-M^{\pi
j}(s,a,s^{\prime},j)\right]$ (4)

Figure 1: Comparison of SRs for different $\gamma$ and action repetition $j$.
These SRs are taken under a random walk policy in a 10x10 gridworld with an
action space consisting of the cardinal directions. Each figure consists of a
single column of the SRs, projected over the gridworld. Projecting the column
over the grid therefore shows the expected likelihood of having started in a
state and moving to the state $i$ under the random policy. These figures show
the different temporal effects of changing $\gamma$ and varying the action
repeat $j$. Increasing action repetition leads to representations that are
heavily skewed in the directions of actions.

##  4 Experiments

To assess the reward revaluation capability of the t-SR, we used three tabular
gridworld environments which each had two possible goal locations (Fig 2).
Agents learn a policy over 10000 episodes with the reward fixed in one of
these locations. The reward is then moved to the other possible location, and
the agents must relearn their policy over an additional 10000 episodes. The
environments are sparsely rewarded, with $+1$ reward for reaching the terminal
goal location, $-1$ for entering one of black _lava_ squares (which terminate
the episode), and $0$ reward otherwise.

(a)

(b)

(c)

Figure 2: Three gridworld environments were used in the reward revaluation
task (a) Junction (b) Junction Hard (c) Junction Very Hard. Red kites indicate
starting locations. White squares are occupiable states and black squared are
terminal lava states yielding a $-1$ reward. Green hexagons are potential
terminal goal states and yield a $+1$ reward. A single goal is present in each
environment at a time.

Five agents were used. A basic Q-learning agent [11], an SR agent[9], a
temporally extended _skip Q-learning_ agent [5], a control SR agent with
random temporally extended exploration similar to [13], and t-SR. In all cases
$\alpha=1$ was used for value and SR updates, $\gamma=0.99$ , and
$\alpha_{r}=1$ was used for updating SR reward vectors. All agents followed an
$\epsilon-$greedy exploration policy with $\epsilon=0.05$. The skip Q and t-SR
agents selected both actions and action repetitions in an $\epsilon$-greedy
fashion, and used a $j_{max}$ of 7. The SR agent with random temporally
extended exploration sampled action repetition randomly with probability
$\epsilon$ during training. All results were evaluated under the greedy policy
using the same 50 seeds.

##  5 Results

Reporting the average episode return for each of the five agents, we see that
t-SR relearns an optimal policy to the new goal in fewer episodes than any
other agent used (Fig 3). While initially learning at the same rate as skip
Q-learning before the reward changes location, t-SR is subsequently able to
alter its policy much more rapidly after the change. Note that for the hardest
environment, the agents without temporal action repetition policies did not on
average find the optimal policy in $10000$ episodes.

To assess the rate of policy change after the reward changes, we calculated
the average total variation between policies after each episode of learning
(Fig 4). That is,
$TV_{\pi}(t)=\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}\frac{1}{2}||\pi(s)_{t-1}-\pi(s)_{t}||_{1}$.
We see that for the SR-based agents, the policy changes sharply and over a
much shorter number of episodes compared to the purely model-free agents -
which continue to update their policies over a protracted amount of time.

Finally, to assess the optimality of the trajectories taken we report
statistics on the average path length to reach to goal (Fig 5 top). From the
time to convergence to the minimum number of steps needed to solve the
environment, we see that t-SR performs reward revaluation approximately 3.5
times faster in the hardest environment than the next fastest agent. Further,
t-SR consistently queries its policy a fewer number of times throughout
learning (Fig 5 bottom).

Figure 3: Average episode return under the greedy policy for the three
environments measured against the number of episodes trained under the
$\epsilon$-greedy policy. Results are plotted as means and standard errors
over 50 seeds.

Figure 4: Average total variation between policies at successive episodes for
the three environments measured against the number of episodes trained under
the $\epsilon$-greedy policy. Results are plotted as means and standard errors
over 50 seeds.

Figure 5: Average transitions (top row) and decisions (bottom row) performed
by agents the three environments measured against the number of episodes
trained under the $\epsilon$-greedy policy. _Decisions_ refers to how many
times the agent queries its action policy during an episode. Results are
plotted as means and standard errors over 50 seeds.

##  6 Discussion and Future Work

The success of an agent in performing reward revaluation lies in the ability
to discover the new reward. This requires effectively unlearning the current
policy being followed. The total variation graphs in Fig 4 show that, as
expected, policies used by SR-based agents change significantly in a short
number of episodes compared to purely model-free value based approaches. The
improved performance of t-SR over SR or SR with random temporally extended
exploration is likely due to the coherent policies used in the episodes
immediately after the reward switch. The temporally extended dynamics learnt
by t-SR will favour moving significant distances in the state space when
compared to SR. This greatly increases the likelihood of following a sequence
of actions leading to the new goal. This is also true of skip Q-learning, but
these model-free agents are marred by policies that will consistently lead to
the old reward location for an extended number of episodes following the
reward switch, greatly decreasing the likelihood of finding the new reward. We
have proposed a new variation of the SR, the t-SR, which shows improved reward
revaluation performance compared to SR or other comparable temporally extended
methods Our next step is to extend this framework to the function
approximation domain using successor features [14]. We also are currently
developing a form of macro action discovery using the t-SR.

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