# Quasi-conical domains with embedded eigenvalues

David Krejčiřík  (D. Krejčiřík) Department of Mathematics
Faculty of Nuclear Sciences and Physical Engineering
Czech Technical University in Prague
Trojanova 13, 120 00, Prague, Czech Republic
E-mail<EMAIL_ADDRESS>and  Vladimir Lotoreichik  (V.
Lotoreichik) Department of Theoretical Physics
Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech
Republic
E-mail<EMAIL_ADDRESS>
###### Abstract.

The spectrum of the Dirichlet Laplacian on any quasi-conical open set
coincides with the non-negative semi-axis. We show that there is a connected
quasi-conical open set such that the respective Dirichlet Laplacian has a
positive (embedded) eigenvalue. This open set is constructed as the tower of
cubes of growing size connected by windows of vanishing size. Moreover, we
show that the sizes of the windows in this construction can be chosen so that
the absolutely continuous spectrum of the Dirichlet Laplacian is empty.

##  1\. Introduction

An open set $\Omega\subset\mathbb{R}^{d}$ with any $d\geq 2$ is said to be
_quasi-conical_ if it contains an arbitrarily large ball. We consider the
self-adjoint Dirichlet Laplacian $-\Delta_{\rm D}^{\Omega}$ on $\Omega$ acting
in the Hilbert space $L^{2}(\Omega)$. It is easy to see that

(1) $\sigma(-\Delta_{\rm D}^{\Omega})=[0,\infty)$

for any quasi-conical open set.

Sufficient conditions to ensure an absence of eigenvalues of the Dirichlet
Laplacian on quasi-conical sets are available in the literature. In
particular, Rellich proved in [17] that there are no embedded eigenvalues for
the Dirichlet Laplacian on the exterior of any compact set. Jones [13]
excluded embedded eigenvalues for certain other classes of quasi-conical open
sets such as domains conical at infinity. More recently, Bonnet-Ben Dhia _et
al._ [2] excluded the existence of eigenvalues in non-convex conical sectors.
(The critical case of the Robin Laplacian in a half-space is considered in
[3].)

On the other hand, it is easy to construct examples of disconnected quasi-
conical open sets for which there are (embedded) eigenvalues in $[0,\infty)$
(_e.g._ , take the union of any bounded set and its exterior). The question
whether there are examples of quasi-conical _domains_ (_i.e._  _connected_
open sets) for which there are eigenvalues seems not to have been considered
in the literature. Our goal is to fill in this gap. The main result of the
present paper reads as follows.

######  Theorem 1.

  * (i)

Given any positive number $\lambda$, there exists a connected quasi-conical
open set such that

$\lambda\in\sigma_{\mathrm{p}}(-\Delta_{\rm D}^{\Omega})\,.$

  * (ii)

The connected quasi-conical open set in (i) can be constructed so that

$\sigma_{\mathrm{ac}}(-\Delta_{\rm D}^{\Omega})=\varnothing\,.$

Our proof relies on an explicit construction, which we believe is of
independent interest. The desired connected quasi-conical open set is
constructed as the tower of cubes of growing size connected by windows of
vanishing size (see Figure 1). We show that the sizes of the windows
connecting the cubes can be chosen so small that we get an embedded eigenvalue
and that the absolutely continuous spectrum turns out to be empty. Our result
is partially of qualitative nature, because we have no explicit control on the
sizes of the windows, which ensure the desired property.

Figure 1. The disconnected quasi-conical open set $\Omega$ with places for the
small holes to be dug indicated by circles.

In order to show the existence of an embedded eigenvalue, we start from the
tower made of disjoint cubes. The respective Dirichlet Laplacian clearly has
an embedded eigenvalue. Then we start to subsequently open the windows
choosing their sizes so small that this embedded eigenvalue and the respective
eigenfunction do not change much (in a proper sense). In the course of this
construction we get a family of eigenfunctions weakly convergent to a
function, which turns out to be an eigenfunction corresponding to an embedded
eigenvalue for the tower of cubes with all the windows opened. In this
argument we rely on perturbation results for Dirichlet eigenvalues and
eigenfunctions under variation of the open set [4].

The absence of the absolutely continuous spectrum is shown via a perturbation
type argument. Clearly, for the disjoint cubes the absolutely continuous of
the Dirichlet Laplacian spectrum is empty. Adapting the technique of [12], we
show that provided the sizes of the windows are sufficiently small, a suitable
resolvent power difference between the Dirichlet Laplacian on a tower of cubes
with opened windows and the Dirichlet Laplacian on the disjoint cubes belongs
to the trace class. The claim follows from the stability of the absolutely
continuous under such a perturbation.

In the classification of Glazman [10] (see also [7, Section X.6.1]) there are
three types of open sets: _quasi-conical_ (_i.e._ presently considered),
_quasi-cylindrical_ (_i.e._ not quasi-conical but containing infinitely many
pairwise disjoint balls of the same radius), and _quasi-bounded_ (_i.e._
neither quasi-conical nor quasi-cylindrical). Quasi-cylindrical open sets with
embedded eigenvalues are constructed in a large number of papers. Behind these
constructions there is typically a symmetry based argument. For example, it
follows from the considerations in [8, 1] (see also [6]) that there is an
embedded eigenvalue in the essential spectrum of the Dirichlet Laplacian in an
infinite Swiss-cross waveguide. Other examples are given in [22, 21]. The
construction in the present paper for quasi-conical domains is significantly
different and not based on symmetry. The Dirichlet Laplacian on a quasi-
bounded open set typically has purely discrete spectrum, but not necessarily
if the set is not bounded.*** In passing, our present construction can be
considered as a dual geometry to the celebrated domain “rooms and passages”
due to Fraenkel [9], which exhibits an essential spectrum for the Neumann
Laplacian on a bounded domain.

To the best of our knowledge, as in the present situation of quasi-conical
domains, embedded eigenvalues for unbounded quasi-bounded open sets are not
studied in the literature.

The structure of the paper is as follows. First, in Section 2 we provide
preliminary material that is used throughout the paper. In Subsection 2.1 we
rigorously introduce the Dirichlet Laplacian on a quasi-conical open set and
evoke its spectral property (1). Next, in Subsection 2.2 we recall general
results on perturbation of eigenvalues, eigenfunctions, and the resolvent of
the Dirichlet Laplacian under variation of the open set. Further, in
Subsection 2.3 we recall the definitions of the singular values for a compact
operator and of the trace class. In Section 3 we prove Theorem 1 (i) on the
existence of embedded eigenvalues. Theorem 1 (ii) on the absence of the
absolutely continuous spectrum is proved in Section 4. The paper is
complemented by Section 5 with a discussion of a related open question.

##  2\. Preliminaries

###  2.1. Dirichlet Laplacian on a (quasi-conical) open set

Let $\Omega\subset{\mathbb{R}}^{d}$ ($d\geq 2$) be an open set. The self-
adjoint Dirichlet Laplacian $-\Delta_{\rm D}^{\Omega}$ in $L^{2}(\Omega)$ can
be rigorously introduced using the first representation theorem [18, Thm.
10.7] via the quadratic form

$\mathfrak{h}_{\rm D}^{\Omega}[u]:=\int_{\Omega}|\nabla
u|^{2}\mathrm{d}x\,,\qquad\mathop{\mathrm{dom}}\nolimits\mathfrak{h}_{\rm
D}^{\Omega}:=H^{1}_{0}(\Omega)\,.$

The operator $-\Delta_{\rm D}^{\Omega}$ can be characterised as

$-\Delta_{\rm D}u=-\Delta
u\,,\qquad\mathop{\mathrm{dom}}\nolimits(-\Delta_{\rm D})=\big{\\{}u\in
H^{1}_{0}(\Omega)\colon\Delta u\in L^{2}(\Omega)\big{\\}}\,.$

Since the Dirichlet Laplacian is a non-negative operator, we have
$\sigma(-\Delta_{\rm D}^{\Omega})\subseteq[0,\infty)$ for any open set
$\Omega$. Now, assume additionally that $\Omega$ is a quasi-conical open set
(recall that an open set is called quasi-conical if it contains a ball of
arbitrarily large radius). Then, using quasi-modes based on plane waves
modulated by suitable cut-off functions in the enlarging balls (see, _e.g._ ,
[7, Thm. X.6.5]), the opposite inclusion of (1) follows. In particular, the
spectrum of the Dirichlet Laplacian on any quasi-conical open set is purely
essential.

###  2.2. Dirichlet Laplacian on varying domains

In this subsection we recall some convergence results for the Dirichlet
Laplacian on varying domains contained in [4] (see also [5, 15, 20]). We begin
by introducing a type of convergence for a sequence of open sets.

######  Definition 2.1 ([4, Def. 1.1]).

If $\Omega_{n},\Omega\subset{\mathbb{R}}^{d}$ are such that

  * (i)

the weak limit points of every sequence $u_{n}\in H^{1}_{0}(\Omega_{n})$,
$n\in{\mathbb{N}}$, in $H^{1}({\mathbb{R}}^{d})$ are in $H^{1}_{0}(\Omega)$,

  * (ii)

for every $\varphi\in H^{1}_{0}(\Omega)$ there exist $\varphi_{n}\in
H^{1}_{0}(\Omega_{n})$ such that $\varphi_{n}\rightarrow\varphi$ in
$H^{1}({\mathbb{R}}^{d})$,

then we write $\Omega_{n}\rightarrow\Omega$ and say that $\Omega_{n}$
converges to $\Omega$ _in the sense of Mosco_.

In general, verifying the conditions of Definition 2.1 can be a difficult
problem. However, in the case that the sequence $\Omega_{n}$ approximates
$\Omega$ monotonously from inside or outside the convergence in the sense of
Mosco can be easily checked. In the present paper we only use the
approximation from outside. (For any set $S\subset{\mathbb{R}}^{d}$ we denote
by ${\rm int}(S)$ its interior.)

######  Proposition 2.2 ([4, Prop. 7.4 and Thm. 7.5], [5, Sec. 5.4]).

Suppose that $\Omega_{n}\supset\Omega_{n+1}\supset\Omega$ for all
$n\in{\mathbb{N}}$, and that ${\rm
int}\,\big{(}\cap_{n\in{\mathbb{N}}}\Omega_{n}\big{)}=\Omega$. If
$H^{1}_{0}(\Omega)=H^{1}_{0}(\Omega\cup\Gamma)$, where

$\Gamma:=\bigcap_{n\in{\mathbb{N}}}\left(\overline{\bigcup_{k\geq
n}(\Omega_{k}\cap\partial\Omega)}\right)\subset\partial\Omega\,,$

then $\Omega_{n}\rightarrow\Omega$.

The convergence of the open sets in the sense of Mosco is convenient in the
study of convergence of the respective Dirichlet Laplacians.

######  Proposition 2.3 ([4, Cor. 4.7]).

Suppose that $\Omega_{n},\Omega$ are contained in a fixed bounded set and that
$\Omega_{n}\rightarrow\Omega$. Then

$\left\|(-\Delta_{\rm D}^{\Omega_{n}}+1)^{-1}-(-\Delta_{\rm
D}^{\Omega}+1)^{-1}\right\|\rightarrow 0\,,\qquad n\rightarrow\infty\,.$

This norm-resolvent convergence particularly implies the convergence of the
respective spectral projections.

###  2.3. Singular values and the trace class

In this subsection we recall the concepts of singular values and the trace
class. Further details can be found in the monographs [11, 19].

######  Definition 2.4.

For a compact operator ${\mathsf{K}}$ in a Hilbert space the eigenvalues
$\\{s_{k}({\mathsf{K}})\\}_{k\geq 1}$ of the self-adjoint non-negative compact
operator $({\mathsf{K}}^{*}{\mathsf{K}})^{1/2}$ enumerated in the non-
increasing order and repeated with multiplicities taken into account are
called the _singular values_ of ${\mathsf{K}}$.

For a compact operator ${\mathsf{K}}$ and a bounded closed operator
${\mathsf{B}}$ acting in a Hilbert space the products
${\mathsf{B}}{\mathsf{K}}$ and ${\mathsf{K}}{\mathsf{B}}$ are compact and by
[19, Thm. 1.6]

(2)
$s_{k}({\mathsf{B}}{\mathsf{K}})\leq\|{\mathsf{B}}\|\,s_{k}({\mathsf{K}})\quad\text{and}\quad
s_{k}({\mathsf{K}}{\mathsf{B}})\leq\|{\mathsf{B}}\|\,s_{k}({\mathsf{K}}).$

######  Definition 2.5.

A compact operator ${\mathsf{K}}$ is said to belong to the _trace class_ if
$\displaystyle\sum_{k=1}^{\infty}s_{k}({\mathsf{K}})<\infty$.

According to [19, Sec. 2], the trace class is a Banach space with the norm
given by

(3) $\|{\mathsf{K}}\|_{1}:=\sum_{k=1}^{\infty}s_{k}({\mathsf{K}}).$

##  3\. Embedded eigenvalues

In this section we prove part (i) of Theorem 1 on the existence of a connected
quasi-conical open set with an embedded eigenvalue. The argument is performed
via an explicit construction. For clarity, we divide the proof into several
steps.

#### Disconnected set

Let $Q_{a}(x)\subset{\mathbb{R}}^{d}$ denote an open cube with sides of half-
width $a>0$ centred at a point $x\in\mathbb{R}^{d}$. With a slight abuse of
notation we will occasionally interpret $t\in{\mathbb{R}}$ as
$(t,0,0,\dots,0)\in{\mathbb{R}}^{d}$. We aim at constructing a tower of cubes
of growing size, in which each next cube is placed “on the top” of the
previous one. To this aim, let us introduce the sequence of centres
$0<x_{1}<x_{2}<\dots<x_{n}<\dots$ such that the associated sequence of half-
widths $a_{1}:=x_{1}$, $a_{n}:=x_{n}-x_{n-1}-a_{n-1}$ ($n\geq 2$) satisfies
$a_{n}>0$ for all $n\in{\mathbb{N}}$ and $a_{n}\rightarrow\infty$ as
$n\rightarrow\infty$. At this stage we fix only $x_{1}>0$ while the sequence
$\\{x_{n}\\}_{n\geq 2}$ is not fixed and we will specify it in the course of
the construction. Let $\lambda_{1}$ be the lowest (simple) eigenvalue of
$-\Delta_{\rm D}^{Q_{a_{1}}(x_{1})}$. Let $\psi_{1}$ denote a corresponding
eigenfunction, normalised to $1$ in $L^{2}(Q_{a_{1}}(x_{1}))$.

Define a (disconnected) quasi-conical open set $\Omega$ as a disjoint union of
enlarging cubes:

(4)
$\Omega:=\bigcup_{n=1}^{\infty}\omega_{n}\qquad\mbox{with}\qquad\omega_{n}:=Q_{a_{n}}(x_{n})\,.$

Then (extending $\psi_{1}$ by zero to $\Omega$), it is easy to see that
$\psi_{1}$ is an eigenfunction of $-\Delta_{\rm D}^{\Omega}$ corresponding to
$\lambda$. Thus, we have constructed a disconnected quasi-conical open set
with an embedded eigenvalue.

To construct a connected open set based on $\Omega$, we dig small holes at the
places where the disjoint cubes $\omega_{n}$ touch (see Figure 1). A careful
attention is needed in order to ensure that this construction does not
eliminate the embedded eigenvalue.

#### 1st interconnection

Let us set $\Omega_{1}:=\omega_{1}$ and let us choose $x_{2}>x_{1}$ so that
$a_{2}=x_{2}-x_{1}-a_{1}\geq a_{1}+1$ and that $\lambda_{1}$ is a simple
eigenvalue of $-\Delta_{\rm D}^{\Omega_{1}\cup\omega_{2}}$. For any positive
$\delta_{1}<a_{1}$, let us connect the first two cubes $\Omega_{1}=\omega_{1}$
and $\omega_{2}$ by defining

(5) $\Omega_{2}:=\Omega_{1}\cup\omega_{2}\cup Q_{\delta_{1}}(x_{1}+a_{1})\,.$

It follows from Proposition 2.2 (with $\Gamma=\varnothing$) that $\Omega_{2}$
converges to $\Omega_{1}\cup\omega_{2}$ as $\delta_{1}\rightarrow 0$ in the
sense of Definition 2.1. Hence, Proposition 2.3 combined with [14, Sec.
IV.3.5] yields that given any (small) positive number $\varepsilon_{1}$, there
exist a positive half-width $\delta_{1}$, a simple eigenvalue $\lambda_{2}$ of
$-\Delta_{\rm D}^{\Omega_{2}}$ and a corresponding eigenfunction $\psi_{2}$,
normalised to $1$ in $L^{2}(\Omega_{2})$, such that

(6)
$|\lambda_{2}-\lambda_{1}|\leq\varepsilon_{1}\qquad\mbox{and}\qquad\|\psi_{2}-\psi_{1}\|_{L^{2}(\Omega_{2})}\leq\varepsilon_{1}\,.$

Using additionally the monotonicity of Dirichlet eigenvalues with respect to
inclusion of open sets, the first inequality in (6) implies that we can choose
$\delta_{1}$ so small that

(7) $\lambda_{1}-\varepsilon_{1}\leq\lambda_{2}\leq\lambda_{1}\,.$

Using the normalisations of $\psi_{1}$ and $\psi_{2}$, together with the fact
that $\psi_{1}=0$ in $\omega_{2}$, the second inequality in (6) yields

(8)
$\|\psi_{2}\|_{L^{2}(\Omega_{2}\setminus\Omega_{1})}\leq\varepsilon_{1}\qquad\mbox{and}\qquad\|\psi_{2}\|_{L^{2}(\Omega_{1})}\geq
1-\varepsilon_{1}\,.$

#### 2nd interconnection

Similarly, let us choose $x_{3}>x_{2}$ so that $a_{3}=x_{3}-x_{2}-a_{2}\geq
a_{2}+1$ and that $\lambda_{2}$ (constructed in the previous step) is a simple
eigenvalue of $-\Delta_{\rm D}^{\Omega_{2}\cup\omega_{3}}$. For any
$\delta_{2}<a_{2}$, let us connect $\Omega_{2}$ with the third cube
$\omega_{3}$ by defining

(9) $\Omega_{3}:=\Omega_{2}\cup\omega_{3}\cup Q_{\delta_{2}}(x_{2}+a_{2})\,.$

Arguing as before, given any positive number $\varepsilon_{2}$, there exist a
positive half-width $\delta_{2}$, a simple eigenvalue $\lambda_{3}$ of
$-\Delta_{\rm D}^{\Omega_{3}}$, and a corresponding eigenfunction $\psi_{3}$,
normalised to $1$ in $L^{2}(\Omega_{3})$, such that

(10)
$|\lambda_{3}-\lambda_{2}|\leq\varepsilon_{2}\qquad\mbox{and}\qquad\|\psi_{3}-\psi_{2}\|_{L^{2}(\Omega_{3})}\leq\varepsilon_{2}\,.$

Using the monotonicity of Dirichlet eigenvalues and (7), the first inequality
in (10) implies that we can choose $\delta_{2}$ so small that

(11)
$\lambda_{1}-\varepsilon_{1}-\varepsilon_{2}\leq\lambda_{3}\leq\lambda_{2}\leq\lambda_{1}\,.$

From the second inequality in (10) and the first inequality in (8), we deduce

(12)
$\|\psi_{3}\|_{L^{2}(\Omega_{3}\setminus\Omega_{1})}\leq\|\psi_{3}-\psi_{2}\|_{L^{2}(\Omega_{3}\setminus\Omega_{1})}+\|\psi_{2}\|_{L^{2}(\Omega_{3}\setminus\Omega_{1})}\leq\varepsilon_{2}+\varepsilon_{1}\,.$

Consequently,

(13)
$1=\|\psi_{3}\|_{L^{2}(\Omega_{3})}\leq\|\psi_{3}\|_{L^{2}(\Omega_{1})}+\|\psi_{3}\|_{L^{2}(\Omega_{3}\setminus\Omega_{1})}\leq\|\psi_{3}\|_{L^{2}(\Omega_{1})}+\varepsilon_{2}+\varepsilon_{1}\,.$

That is,

(14) $\|\psi_{3}\|_{L^{2}(\Omega_{1})}\geq
1-\varepsilon_{1}-\varepsilon_{2}\,.$

####  $n$th interconnection

Continuing by induction, given any natural number $n\geq 1$, we choose
$x_{n+1}>x_{n}$ so that $a_{n+1}=x_{n+1}-x_{n}-a_{n}\geq a_{n}+1$ and
$\lambda_{n}$ (constructed in the $(n-1)$st step) is a simple eigenvalue of
$-\Delta_{\rm D}^{\Omega_{n}\cup\omega_{n+1}}$. For any $\delta_{n}<a_{n}$,
let us connect $\Omega_{n}$ with the $(n+1)$th cube $\omega_{n+1}$ by defining

(15) $\Omega_{n+1}:=\Omega_{n}\cup\omega_{n+1}\cup
Q_{\delta_{n}}\big{(}x_{n}+a_{n}\big{)}\,.$

As before given any positive number $\varepsilon_{n}$, there exist a positive
half-width $\delta_{n}$, a simple eigenvalue $\lambda_{n+1}$ of $-\Delta_{\rm
D}^{\Omega_{n+1}}$ and a corresponding eigenfunction $\psi_{n+1}$, normalised
to $1$ in $L^{2}(\Omega_{n+1})$, such that

(16)
$\lambda_{1}-(\varepsilon_{1}+\dots+\varepsilon_{n})\leq\lambda_{n+1}\leq\lambda_{n}\leq\lambda_{1}\qquad\mbox{and}\qquad\|\psi_{n+1}\|_{L^{2}(\Omega_{1})}\geq
1-(\varepsilon_{1}+\dots+\varepsilon_{n})\,.$

Given any positive number $\varepsilon$, and choosing
$\varepsilon_{k}:=2^{-k}\varepsilon$ for every $k\geq 1$, we get uniform
estimates

(17)
$\lambda_{1}-\varepsilon\leq\lambda_{n+1}\leq\lambda_{1}\qquad\mbox{and}\qquad\|\psi_{n+1}\|_{L^{2}(\Omega_{1})}\geq
1-\varepsilon\,.$

#### Connected set

By using the procedure of connecting the individual cubes described above, we
define a connected open set $\Omega^{\prime}$ by the limit

(18) $\Omega^{\prime}:=\lim_{n\to\infty}\Omega_{n}\,.$

It remains to show that the embedded eigenvalue “survives” this limit. After
the $n$th interconnection, we have the eigenvalue equation

(19) $\forall\phi\in
H_{0}^{1}(\Omega_{n+1})\,,\qquad\big{(}\nabla\phi,\nabla\psi_{n+1}\big{)}_{L^{2}(\Omega_{n+1})}=\lambda_{n+1}\,\big{(}\phi,\psi_{n+1}\big{)}_{L^{2}(\Omega_{n+1})}\,.$

As usual, extending $\psi_{n+1}$ and $\phi$ by zero to the whole
$\Omega^{\prime}$, the inner product can be replaced by that of
$L^{2}(\Omega^{\prime})$. Choosing $\phi:=\psi_{n+1}$ for the test function in
(19) and using the normalisation of $\psi_{n+1}$, we obtain

(20)
$\|\nabla\psi_{n+1}\|_{L^{2}(\Omega^{\prime})}^{2}\leq\lambda_{n+1}\leq\lambda_{1}\qquad\mbox{and}\qquad\|\psi_{n+1}\|_{L^{2}(\Omega^{\prime})}^{2}=1\,.$

Here the uniform bound to $\lambda_{n+1}$ follows from (17). It follows that
$\\{\psi_{n+1}\\}_{n=0}^{\infty}$ is a bounded sequence in
$H_{0}^{1}(\Omega^{\prime})$. Consequently, there exists a subsequence
$\\{\psi_{n_{j}+1}\\}_{j=0}^{\infty}$ which is weakly converging to some
$\psi_{\infty}$ in $H_{0}^{1}(\Omega^{\prime})$. In particular,
$\\{\psi_{n_{j}+1}\\}_{j=0}^{\infty}$ is also weakly converging to
$\psi_{\infty}$ in $L^{2}(\Omega^{\prime})$. Inserting this subsequence into
(19) and sending $j$ to infinity, we obtain

(21) $\forall\phi\in
H_{0}^{1}(\Omega^{\prime})\,,\qquad\big{(}\nabla\phi,\nabla\psi_{\infty}\big{)}_{L^{2}(\Omega^{\prime})}=\lambda_{\infty}\,\big{(}\phi,\psi_{\infty}\big{)}_{L^{2}(\Omega^{\prime})}\,,$

where the limit

(22) $\lambda_{\infty}:=\lim_{n\to\infty}\lambda_{n}$

is well defined because $\\{\lambda_{n}\\}_{n=1}^{\infty}$ is monotonously
decreasing by construction. Consequently, $\psi_{\infty}$ belongs to the
domain of the operator $-\Delta_{\rm D}^{\Omega^{\prime}}$ and solves the
equation $-\Delta_{\rm
D}^{\Omega^{\prime}}\psi_{\infty}=\lambda_{\infty}\psi_{\infty}$ with
$\lambda_{1}-\varepsilon\leq\lambda_{\infty}\leq\lambda_{1}$. It remains to
ensure that $\psi_{\infty}$ is not identically equal to zero. However, this
follows from the uniform estimate (17) on the norm of $\psi_{n+1}$ in the
$n$-independent bounded set $\Omega_{1}$, provided that $\varepsilon<1$.

We remark that by rescaling the domain $\Omega^{\prime}$ by a constant factor
$\alpha>0$ we get the domain $\alpha\Omega^{\prime}$ and the respective
Dirichlet Laplacian $-\Delta_{\rm D}^{\alpha\Omega^{\prime}}$ has an embedded
eigenvalue $\alpha^{-2}\lambda_{\infty}$. Hence, we can construct a connected
quasi-conical open set with an embedded eigenvalue at any prescribed position.

######  Remark 3.1.

The same type of examples of quasi-conical domains with embedded eigenvalues
can be given for other cell domains than cubes. Moreover, the interconnections
obtained by digging the holes for flat boundaries can be replaced for general
cell domains by using thin tubes as connections.

##  4\. Absence of the absolutely continuous spectrum

In this section we prove part (ii) of Theorem 1 by showing that by choosing
the sizes of the windows $\\{\delta_{n}\\}_{n\geq 1}$ sufficiently small the
absolutely continuous spectrum of the operator $-\Delta_{\rm
D}^{\Omega^{\prime}}$ becomes empty. In order to prove this result we adapt
the technique developed in [12]. As in Section 3, we subsequently open the
windows between the cubes. The half-widths of the windows
$\\{\delta_{n}\\}_{n\geq 1}$ will be determined so as to satisfy
simultaneously the conditions in Section 3 and some new additional conditions.

Let the sequence of the bounded domains $\\{\Omega_{n}\\}_{n\in{\mathbb{N}}}$
be constructed as in Section 3. We intend to modify the sizes of the windows
$\\{\delta_{n}\\}_{n\geq 1}$ in order to satisfy some additional assumptions.
The choice of the sequence $\\{x_{n}\\}_{n\in{\mathbb{N}}}$ in Section 3
easily adjusts to possibly different choices of the sizes of the holes. It is
not hard to see that in the construction in Section 3 on the $n$th
interconnection we are allowed to take any
$\delta_{n}\in(0,\delta_{n}^{\star})$ for some sufficiently small
$\delta_{n}^{\star}>0$, which depends on the values of the parameters chosen
on the previous steps of the construction. For the convenience of the reader
we split the remaining argument into two steps.

#### Step 1: modified sizes of the windows.

Without loss of generality we can assume that $\delta_{n}\leq a_{1}$ for all
$n\in{\mathbb{N}}$. Since $\Omega_{n+1}\rightarrow\Omega_{n}\cup\omega_{n+1}$
as $\delta_{n}\rightarrow 0$ in the sense of Definition 2.1, applying
Proposition 2.3 we conclude that for any $\varepsilon>0$ there exists
$\delta_{n}\in(0,\delta_{n}^{\star})$ such that

(23) $\left\|(-\Delta_{\rm D}^{\Omega_{n+1}}+1)^{-1}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-1}\right\|\leq\varepsilon\,.$

Let the power $\ell\in{\mathbb{N}}$ be fixed. By purely algebraic
manipulations the resolvent power difference of the Dirichlet Laplacians on
$\Omega_{n+1}$ and $\Omega_{n}\cup\omega_{n+1}$ acting in the same Hilbert
space $L^{2}(\Omega_{n}\cup\omega_{n+1})$ can be represented by

(24) $\displaystyle(-\Delta_{\rm D}^{\Omega_{n+1}}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-\ell}=$
$\displaystyle\quad\\!=\\!\sum_{k=1}^{\ell}(-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-(\ell-k)}\left[(-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-1}\\!-\\!(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-1}\right](-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-(k-1)}\,.$

We conclude from the Weyl spectral asymptotics for the Dirichlet Laplacian on
a bounded open set (see _e.g._ [18, Thm. 12.14]) that for any
$k\in\\{1,2,\dots,\ell\\}$ at least one of the two operators $(-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-(\ell-k)}$ and $(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-k-1}$ belongs to the trace class provided
that $\ell$ is chosen to be sufficiently large. The trace norm of this trace
class resolvent power can be estimated from above by a constant $C_{n}>0$
independent of $k$ and $\delta_{n}$ while the operator norm of the other
resolvent power can be estimated from above by $1$ thanks to non-negativity of
the Dirichlet Laplacian. Let us comment on the possibility to get an upper
bound on this trace norm independent of $\delta_{n}$. Indeed the operator
$-\Delta^{\Omega_{n}\cup\omega_{n+1}}_{\rm D}$ is independent of $\delta_{n}$
and hence the trace norm of the corresponding resolvent power (provided it
belongs to the trace class) is also independent of $\delta_{n}$. The trace
norm of the resolvent power corresponding to $-\Delta^{\Omega_{n+1}}_{\rm D}$
(provided it belongs to the trace class) can be expressed as

(25) $\|(-\Delta^{\Omega_{n+1}}_{\rm
D}+1)^{-(\ell-k)}\|_{1}=\sum_{m=1}^{\infty}\frac{1}{\big{[}1+\lambda_{m}(-\Delta^{\Omega_{n+1}}_{\rm
D})\big{]}^{\ell-k}}$

where $\\{\lambda_{m}(-\Delta_{\rm D}^{\Omega_{n+1}})\\}_{m\geq 1}$ are the
eigenvalues $-\Delta_{\rm D}^{\Omega_{n+1}}$ enumerated in the non-decreasing
way and repeated with multiplicities taken into account. It is easy to see by
the min-max principle that the eigenvalues of $-\Delta_{\rm D}^{\Omega_{n+1}}$
are non-increasing functions of $\delta_{n}$. Hence, the trace norm in (25) is
an increasing function of $\delta_{n}$ and therefore it can be estimated from
above by the respective trace norm for $\delta_{n}=a_{1}$.

Combining the estimate (23) with the property of singular values in (2) and
the definition of the trace norm in (3) we derive from (24) the bound

(26) $\left\|(-\Delta_{\rm D}^{\Omega_{n+1}}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-\ell}\right\|_{1}\leq
C_{n}\ell\varepsilon\,,\qquad n\in{\mathbb{N}}\,.$

It follows from (23) and (26) that by choosing $\varepsilon>0$ sufficiently
small we can find a sufficiently small $\delta_{n}\in(0,\delta_{n}^{\star})$
so that

(27a) $\left\|(-\Delta_{\rm D}^{\Omega_{n+1}}+1)^{-1}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-1}\right\|\leq 2^{-n},\qquad
n\in{\mathbb{N}}\,,\\\ $ (27b) $\left\|(-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-\ell}\right\|_{1}\leq 2^{-n}\,,\qquad
n\in{\mathbb{N}}\,.$

Thus, we have selected the sequence $\\{\delta_{n}\\}_{n\geq 1}$ which fits
the construction in Section 3 and simultaneously the above two estimates hold.

#### Step 2: trace class resolvent power difference

Consider the sequence of quasi-conical unbounded domains

$\displaystyle\Omega_{1}^{\prime}$ $\displaystyle:=\Omega,$
$\displaystyle\Omega_{2}^{\prime}$ $\displaystyle:=\Omega\cup
Q_{\delta_{1}}(x_{1}+a_{1}),$ $\displaystyle\Omega_{3}^{\prime}$
$\displaystyle:=\Omega\cup Q_{\delta_{1}}(x_{1}+a_{1})\cup
Q_{\delta_{2}}(x_{2}+a_{2}),$ $\displaystyle\vdots$
$\displaystyle\Omega_{n}^{\prime}$
$\displaystyle:=\Omega\cup\Big{(}\bigcup_{k=1}^{n-1}Q_{\delta_{k}}(x_{k}+a_{k})\Big{)},$
$\displaystyle\vdots$

The Dirichlet Laplacian on $\Omega_{n}^{\prime}$ can be decomposed into the
orthogonal sum

$-\Delta_{\rm D}^{\Omega_{n}^{\prime}}=(-\Delta_{\rm
D}^{\Omega_{n}})\oplus(-\Delta_{\rm
D}^{\Omega\setminus\overline{\Omega_{n}}})$

with respect to the decomposition $L^{2}(\Omega)=L^{2}(\Omega_{n})\oplus
L^{2}(\Omega\setminus\overline{\Omega_{n}})$. In view of this decomposition we
obtain the following representation for the resolvent difference of the
Dirichlet Laplacians on $\Omega_{n}^{\prime}$ and $\Omega_{n+1}^{\prime}$:

$(-\Delta_{\rm D}^{\Omega_{n+1}^{\prime}}+1)^{-1}-(-\Delta_{\rm
D}^{\Omega_{n}^{\prime}}+1)^{-1}=\left((-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-1}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-1}\right)\oplus 0.$

Hence, it follows from (27a) that the sequence ${\mathbb{N}}\ni
n\mapsto(-\Delta_{\rm D}^{\Omega_{n}^{\prime}}+1)^{-1}$ is a Cauchy sequence
in the operator norm. Thus, there exists an operator ${\mathsf{T}}$ in the
Hilbert space $L^{2}(\Omega)$ such that

(28) $\left\|(-\Delta_{\rm
D}^{\Omega_{n}^{\prime}}+1)^{-1}-{\mathsf{T}}\right\|\rightarrow 0\,,\qquad
n\rightarrow\infty\,.$

On the other hand the sequence of operators $\\{-\Delta_{\rm
D}^{\Omega_{n}^{\prime}}\\}_{n\geq 1}$ is decreasing in the form sense. In
view of the identity $\cup_{n\geq
1}H^{1}_{0}(\Omega_{n}^{\prime})=H^{1}_{0}(\Omega^{\prime})$
($\Omega^{\prime}=\cup_{n\geq 1}\Omega_{n}^{\prime}$) we get by [16, Thm.
S.16] that $-\Delta_{\rm D}^{\Omega_{n}^{\prime}}$ converges to $-\Delta_{\rm
D}^{\Omega^{\prime}}$ in the strong resolvent sense. Thanks to (28) we obtain
that ${\mathsf{T}}=(-\Delta_{\rm D}^{\Omega^{\prime}}+1)^{-1}$ and that
$-\Delta_{\rm D}^{\Omega_{n}^{\prime}}$ converges to $-\Delta_{\rm
D}^{\Omega^{\prime}}$ in the norm resolvent sense.

Analogously it follows from (27b) that the sequence of resolvent powers
${\mathbb{N}}\ni n\mapsto(-\Delta_{\rm D}^{\Omega_{n}^{\prime}}+1)^{-\ell}$ is
a Cauchy sequence in the trace norm. Hence, it has a limit
${\mathsf{T}}^{\prime}$ in the trace norm. Since the trace norm is stronger
than the operator norm, we obtain that $(-\Delta_{\rm
D}^{\Omega_{n}^{\prime}}+1)^{-\ell}$ converges to ${\mathsf{T}}^{\prime}$ also
in the operator norm as $n\rightarrow\infty$. By the uniqueness of the limit
we get that ${\mathsf{T}}^{\prime}=(-\Delta_{\rm
D}^{\Omega^{\prime}}+1)^{-\ell}$. Therefore, we obtain

$\displaystyle\left\|(-\Delta_{\rm D}^{\Omega}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega^{\prime}}+1)^{-\ell}\right\|_{1}$
$\displaystyle\leq\sum_{n=1}^{\infty}\left\|(-\Delta_{\rm
D}^{\Omega_{n}^{\prime}}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega_{n+1}^{\prime}}+1)^{-\ell}\right\|_{1}$
$\displaystyle=\sum_{n=1}^{\infty}\left\|(-\Delta_{\rm
D}^{\Omega_{n+1}}+1)^{-\ell}-(-\Delta_{\rm
D}^{\Omega_{n}\cup\omega_{n+1}}+1)^{-\ell}\right\|_{1}\leq\sum_{n=1}^{\infty}2^{-n}<\infty.$

Thus, the resolvent power difference $(-\Delta_{\rm
D}^{\Omega}+1)^{-\ell}-(-\Delta_{\rm D}^{\Omega^{\prime}}+1)^{-\ell}$ is a
trace class operator. Hence, by [14, Thm. X.4.12, Rem. X.4.13] we have

$\sigma_{\rm ac}(-\Delta_{\rm D}^{\Omega^{\prime}})=\sigma_{\rm
ac}(-\Delta_{\rm D}^{\Omega})=\varnothing.$

In the course of this construction we have also satisfied all the assumption
in Section 3. Hence, the operator $-\Delta_{\rm D}^{\Omega^{\prime}}$
possesses also a positive embedded eigenvalue $\lambda_{\infty}$.

##  5\. Discussion

It remains an open question whether it is possible to select the sizes of the
windows so small that the singular continuous spectrum of the operator
$-\Delta_{\rm D}^{\Omega^{\prime}}$ is empty. If it is possible to do, then we
would get the Dirichlet Laplacian on a connected quasi-conical open set having
only purely point spectrum, which densely fills the non-negative semi-axis.

### Acknowledgement

D.K. was supported by the EXPRO grant No. 20-17749X of the Czech Science
Foundation. V.L. was supported by the grant No. 21-07129S of the Czech Science
Foundation.

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# On fine Selmer groups and signed Selmer groups of elliptic modular forms

Antonio Lei ID Département de Mathématiques et de Statistique
Université Laval, Pavillion Alexandre-Vachon
1045 Avenue de la Médecine
Québec, QC
Canada G1V 0A6<EMAIL_ADDRESS>and  Meng Fai Lim ID School of
Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences
Central China Normal University
Wuhan
430079
P.R.China<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be an elliptic modular form and $p$ an odd prime that is coprime to
the level of $f$. We study the link between divisors of the characteristic
ideal of the $p$-primary fine Selmer group of $f$ over the cyclotomic
$\mathbb{Z}_{p}$ extension of $\mathbb{Q}$ and the greatest common divisor of
signed Selmer groups attached to $f$ defined using the theory of Wach modules.
One of the key ingredients of our proof is a generalization of a result of
Wingberg on the structure of fine Selmer groups of abelian varieties with
supersingular reduction at $p$ to the context of modular forms.

###### Key words and phrases:

Elliptic modular forms, fine Selmer groups, signed Selmer groups

###### 2020 Mathematics Subject Classification:

11R23 (primary); 11F11, 11R18 (secondary)

The authors' research is partially supported by: the NSERC Discovery Grants
Program RGPIN-2020-04259 and RGPAS-2020-00096 (Lei), the National Natural
Science Foundation of China under Grant No. 11771164 and the Fundamental
Research Funds for the Central Universities of CCNU under grant CCNU20TD002
(Lim).

##  1\. Introduction

Let $p$ be a fixed odd prime number and $f$ a fixed normalized eigen-cuspform
of level $N$ and weight $k\geq 2$, with $p\nmid N$. Let $K$ denote the
completion of the Hecke field of $f$ at a prime above $p$ and write
$\mathcal{O}$ for its ring of integers. Let $V_{f}$ denote the $K$-adic
$G_{\mathbb{Q}}$-representation of $f$ defined by Deligne [8]. We fix a
Galois-stable $\mathcal{O}$-lattice $T_{f}$ inside $V_{f}$ and write
$A_{f}=V_{f}/T_{f}$. We write ${\overline{f}}$ for the conjugate modular form
of $f$ and we write $V_{\overline{f}}$, $T_{\overline{f}}$ and
$A_{\overline{f}}$ for the corresponding $G_{\mathbb{Q}}$-modules attached to
${\overline{f}}$ with $T_{\overline{f}}$ chosen to be
$\operatorname{Hom}_{\mathcal{O}}(T_{f},\mathcal{O})(1-k)$, where $M(j)$
denotes the $j$-th Tate twist of a $G_{\mathbb{Q}}$-module $M$.

Let $\mathbb{Q}_{\operatorname{cyc}}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and let $\Gamma$ denote the Galois
group
$\operatorname{Gal}\left(\mathbb{Q}_{\operatorname{cyc}}/\mathbb{Q}\right)$.
The Iwasawa algebra $\Lambda=\mathcal{O}\llbracket\Gamma\rrbracket$ is defined
to be $\varprojlim\mathcal{O}[\Gamma/\Gamma^{p^{n}}]$, where the connecting
maps are projections. After fixing a topological generator $\gamma$ of
$\Gamma$, there is an isomorphism of rings $\Lambda\cong\mathcal{O}\llbracket
X\rrbracket$, sending $\gamma$ to $X+1$. Given a $\Lambda$-module $M$, denote
its Pontryagin dual by $M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\mathcal{O}}\left(M,K/\mathcal{O}\right)$. We
write $M^{\iota}$ for the $\Lambda$-module which is $M$ as an
$\mathcal{O}$-module equipped with a $\Gamma$-action given by
$\gamma\cdot_{\iota}m=\gamma^{-1}m$. Finally, if
$F\in\Lambda=\mathcal{O}\llbracket X\rrbracket$, we write $F^{\iota}$ for the
power series $F(\frac{1}{1+X}-1)$.

For $g=f$ or ${\overline{f}}$ and $j\in\mathbb{Z}$, let
$\operatorname{Sel}_{0}(A_{g}(j)/\mathbb{Q}_{\operatorname{cyc}})$ denote the
fine Selmer group of $A_{g}(j)$ over $\mathbb{Q}_{\operatorname{cyc}}$ (whose
definition will be reviewed in §2.1). The fine Selmer groups of abelian
varieties were first systematically studied by Coates and Sujatha in [7], and
a little later by Wuthrich [34]. Various conjectures on the structure of these
groups have been formulated and they are still wide open to this date. See
also [15, 20, 21, 22, 23] for some recent results on the topic. In the context
of modular forms, the fine Selmer groups have been studied in [1, 11, 12, 10].

It has been proved by Kato [13] that
$\operatorname{Sel}_{0}(A_{g}(j)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ is a
finitely generated torsion $\Lambda$-module. Let $\mathfrak{F}_{g,j}$ denote a
choice of characteristic element (i.e. a generator of the characteristic ideal
$\mathrm{Char}_{\Lambda}\operatorname{Sel}_{0}(A_{g}(j)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$).
The main goal of the present article is to study the divisors of
$\mathfrak{F}_{f,i}$ and $\mathfrak{F}_{{\overline{f}},k-i}$ for a fixed
integer $i$. These specific twists are considered due to the perfect pairing

$T_{f}(i)\times A_{\overline{f}}(k-i)\rightarrow\mu_{p^{\infty}}$

of $G_{\mathbb{Q}}$-modules.

We are able to relate the divisors of $\mathfrak{F}_{f,i}$ and
$\mathfrak{F}_{{\overline{f}},k-i}$ to the greatest common divisors of the
signed Selmer groups attached to ${\overline{f}}(k-i)$ defined in [18, 19] by
the theory of Wach modules (see §2.3 where the definition of these groups are
reviewed). Let us write $\mathfrak{F}_{{\overline{f}},k-i}^{\sharp}$ and
$\mathfrak{F}_{{\overline{f}},k-i}^{\flat}$ for a choice of characteristic
elements of the Pontryagin duals of these signed Selmer groups. Under a mild
hypothesis on the local representations at bad primes and a hypothesis on the
validity of Kato's Iwasawa main conjecture for ${\overline{f}}(k-i)$ (labelled
(H0) and (H-IMC) respectively in the main text), we show that if $F$ is an
irreducible element of $\Lambda$ that is outside a certain explicit set of
linear factors, $F^{\iota}\nmid\mathfrak{F}_{f,i}$ and
$F\nmid\mathfrak{F}_{{\overline{f}},k-i}$ if and only if
$F\nmid\gcd\left(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat}\right)$.
(See Theorem 4.4 for the precise statement of this result.)

In the case where $T_{f}$ is the Tate module of an elliptic curve
$E/\mathbb{Q}$ with good supersingular reduction at $p$, a similar result has
been proved in [20]. One of the key ingredients of the proof given in [20] is
a link between the fine Selmer group of $E$ over
$\mathbb{Q}_{\operatorname{cyc}}$ and the maximal $\Lambda$-torsion submodule
of the Pontryagin dual of the $p$-primary Selmer group of $E$ over
$\mathbb{Q}_{\operatorname{cyc}}$, which was proved by Wingberg [33] (see also
[24], where an alternative proof is given). In the present paper, we prove
Theorem 4.4 by first establishing analogues of Wingberg's result in the
context of modular forms (see Theorems 3.1 and 3.4). In [33], Wingberg worked
with Selmer groups defined using flat cohomology, whereas Matar worked with
Selmer groups defined using Galois cohomology in [24] instead. In the present
paper, the latter definition is used. Our proof is very different from the
ones employed in both [33] and [24]. We make use of Nekovář's spectral
sequence, which seems to give a somewhat simpler and more general proof than
the previous proofs available in the literature. We hope that these results
may be of independent interest.

##  2\. Definitions of Selmer groups and related objects

###  2.1. Fine Selmer groups

Let $L_{n}$ be the unique subextension of $\mathbb{Q}_{\operatorname{cyc}}$
such that $[L_{n}\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=p^{n}$.
Given an algebraic extension $L$ of $\mathbb{Q}$, we write $S(L)$ for the
primes of $L$ lying above $pN$ as well as the archimedean primes. We write
$G_{S}(L)$ for the Galois group of the maximal algebraic extension of $L$ that
is unramified outside $S(L)$.

For $g\in\\{f,{\overline{f}}\\}$ and $j\in\mathbb{Z}$, we define the fine
Selmer group of $A_{g}(j)$ over $L$ to be

$\operatorname{Sel}_{0}(A_{g}(j)/L)=\ker\left(H^{1}(G_{S}(L),A_{g}(j))\rightarrow\prod_{v\in
S(L)}H^{1}(L_{v},A_{g}(j))\right).$

Recall that
$\operatorname{Sel}_{0}(A_{g}(j)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ is
torsion over $\Lambda$ and we write $\mathfrak{F}_{g,j}$ for a choice of a
generator of the characteristic ideal
$\mathrm{Char}_{\Lambda}\operatorname{Sel}_{0}(A_{g}(j)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$.

The classical Selmer group of $A_{g}(j)$ over $L$ is defined to be

$\operatorname{Sel}(A_{g}(j)/L)=\ker\left(H^{1}(G_{S}(L),A_{g}(j))\rightarrow\prod_{v\in
S(L)}\frac{H^{1}(L_{v},A_{g}(j))}{H^{1}_{\mathrm{f}}(L_{v},A_{g}(j))}\right),$

where $H^{1}_{\mathrm{f}}(L_{v},A_{g}(j))$ is defined as in [4, §3].

###  2.2. Wach modules and signed Coleman maps

We recall the definition of signed Coleman maps from [18, 19], which
generalize those studied in [14, 32] in the context of elliptic curves with
supersingular reduction at $p$. Here, we do not require $p$ to be a non-
ordinary prime for $f$. The only requirement is that $p$ is coprime to $N$ so
that the representation $V_{f}$ is crystalline at $p$.

We shall write for $g\in\\{f,{\overline{f}}\\}$, $j\in\mathbb{Z}$ and
$m\in\\{1,2\\}$,

$\mathfrak{H}^{m}_{g,j}=\varprojlim_{n}H^{m}(L_{n},T_{g}(j)),\quad\mathbb{H}^{m}_{g,j}=\varprojlim
H^{m}(L_{n,p},T_{g}(j)),$

where the connecting maps are corestrictions and we have abused notation by
writing $p$ for the unique prime of $L_{n}$ above $p$. We write
$\mathrm{loc}_{p}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{H}^{1}_{g,j}\rightarrow\mathbb{H}^{1}_{g,j}$ for the
localization map.

Let $T=T_{g}(k-1)$, where $g\in\\{f,{\overline{f}}\\}$. We write
$\mathbb{N}(T)$ for the Wach module of $T$. See [3, §II.1] for the precise
definition of $\mathbb{N}(T)$. Roughly speaking, it is a filtered
$\varphi$-module over the ring $\mathcal{O}\llbracket\pi\rrbracket$, where
$\pi$ is an element in the ring of of Witt vectors of
$\displaystyle\varprojlim_{x\mapsto x^{p}}\mathbb{C}_{p}$, given by
$[(1,\zeta_{p},\zeta_{p^{2}},...)]-1$ and $\zeta_{p^{n}}$ is a primitive
$p^{n}$-th root of unity in $\mathbb{C}_{p}$ such that
$\zeta_{p^{n+1}}^{p}=\zeta_{p^{n}}$. One may regard $\pi$ as a formal variable
equipped with an action of $\varphi$ and
$\Gamma_{0}=\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})$
given by $\varphi(\pi)=(1+\pi)^{p}-1$ and
$\sigma(\pi)=(1+\pi)^{\chi_{\operatorname{cyc}}(\sigma)}-1$ for
$\sigma\in\Gamma_{0}$, where $\chi_{\operatorname{cyc}}$ is the $p$-adic
cyclotomic character. Let
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T)=\varprojlim_{n}H^{1}(\mathbb{Q}_{p}(\mu_{p^{n}}),T)$,
where the connecting maps are corestrictions. Berger proved in [2, Appendix A]
that there is an isomorphism of
$\mathcal{O}\llbracket\Gamma_{0}\rrbracket$-modules

$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T)\cong\mathbb{N}(T)^{\psi=1},$

where $\psi$ is a canonical left inverse of $\varphi$ and the superscript
$\psi=1$ denotes the kernel of $\psi-1$.

We recall from [18, 19] that for $\bullet\in\\{\sharp,\flat\\}$, on choosing
an appropriate basis of $\mathbb{N}(T_{g}(k-1))$, one can define signed
Coleman maps

$\mathrm{Col}_{g}^{\bullet}\mathrel{\mathop{\mathchar
58\relax}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T_{g}(k-1))\rightarrow\mathcal{O}\llbracket\Gamma_{0}\rrbracket,$

decomposing Perrin-Riou's big logarithm map defined in [27].

We define the twisted Coleman maps

$\mathrm{Col}_{g,j}^{\bullet}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{H}^{1}_{g,j}\rightarrow\Lambda$

as follows. We have the $\mathcal{O}$-isomorphism

$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T_{g}(j))\stackrel{{\scriptstyle
e^{k-1-j}}}{{\longrightarrow}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T_{g}(k-1)),$

where $e$ is a basis of the $G_{\mathbb{Q}_{p}}$-representation
$\mathbb{Z}_{p}(1)$, as given in [28, §A.4]. Let $\operatorname{Tw}$ be the
$\mathcal{O}$-linear automorphism on
$\mathcal{O}\llbracket\Gamma_{0}\rrbracket$ sending $\sigma\in\Gamma_{0}$ to
$\chi_{\operatorname{cyc}}(\sigma)\sigma$. We can define a $\Lambda$-morphism

$\operatorname{Tw}^{i-k+1}\circ\ \mathrm{Col}_{g}^{\bullet}\circ
e^{k-1-j}\mathrel{\mathop{\mathchar
58\relax}}H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T_{g}(j))\rightarrow\mathcal{O}\llbracket\Gamma_{0}\rrbracket$

and obtain $\mathrm{Col}_{g,j}^{\bullet}$ on taking the trivial isotypic
component of $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p})/\mathbb{Q}_{p})$.

###  2.3. Signed Selmer groups and main conjectures

Let $H_{\bullet}^{1}(\mathbb{Q}_{\operatorname{cyc},p},A_{\overline{f}}(k-i))$
denote the orthogonal complement of $\ker\mathrm{Col}_{f,i}^{\bullet}$ under
the local Tate pairing

$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p}(\mu_{p^{\infty}}),T_{f}(i))\times
H^{1}(\mathbb{Q}_{\operatorname{cyc},p},A_{\overline{f}}(k-i))\rightarrow\mathbb{Q}_{p}/\mathbb{Z}_{p}.$

We define the signed Selmer groups
$\operatorname{Sel}^{\bullet}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})$
as

$\ker\left(H^{1}(G_{S}(\mathbb{Q}_{\operatorname{cyc}}),A_{\overline{f}}(k-i))\rightarrow\frac{H^{1}(\mathbb{Q}_{\operatorname{cyc},p},A_{\overline{f}}(k-i))}{H_{\bullet}^{1}(\mathbb{Q}_{\operatorname{cyc},p},A_{\overline{f}}(k-i))}\times\prod_{v\nmid
p}\frac{H^{1}(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i))}{H_{\mathrm{f}}^{1}(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i))}\right).$

Note that when $A_{\overline{f}}(k-i)$ is given by $E[p^{\infty}]$ for some
elliptic curve $E/\mathbb{Q}$ with good supersingular reduction at $p$ or when
$a_{p}({\overline{f}})=0$, one may choose an appropriate basis of the Wach
module so that these Selmer groups coincide with the ones studied in [14, 32,
16] (see [18, §§5.2-5.4]).

Let $\mathbf{z}_{f,i}$ be Kato's zeta element inside
$H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{f}(i))\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$
as defined in [13]. It is conjectured that $\mathbf{z}_{f,i}\in
H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{f}(i))$ (see Conjecture
12.10 of op. cit.). Assuming this (which we shall do for the rest of the
article), recall from [18, Eqn. (61)] the Poitou–Tate exact sequence

(2.1)
$0\rightarrow\frac{\mathfrak{H}^{1}_{f,i}}{\Lambda\mathbf{z}_{f,i}}\rightarrow\frac{\mathrm{Image}\left(\mathrm{Col}_{f,i}^{\bullet}\right)}{(L_{p}^{\bullet})}\rightarrow\operatorname{Sel}^{\bullet}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\rightarrow\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\rightarrow
0,$

where
$L_{p}^{\bullet}=\mathrm{Col}_{f,i}^{\bullet}\circ\mathrm{loc}_{p}(\mathbf{z}_{f,i})$
is the signed $p$-adic $L$-function associated to ${\overline{f}}(k-i)$. If
$L_{p}^{\bullet}\neq 0$, then
$\operatorname{Sel}^{\bullet}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
is a finitely generated torsion $\Lambda$-module. We write
$\mathfrak{F}_{{\overline{f}},k-i}^{\bullet}$ for a characteristic element of
this module.

Kato [13] proved that the first and fourth terms of (2.1) are
$\Lambda$-torsion. He further formulated the following Iwasawa main
conjecture:

(2.2)
$\mathrm{Char}_{\Lambda}\mathfrak{H}^{1}_{f,i}/\Lambda\mathbf{z}_{f,i}=\mathrm{Char}_{\Lambda}\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}=(\mathfrak{F}_{{\overline{f}},k-i}).$

It follows from (2.1) that it is equivalent to

(2.3)
$\mathrm{Char}_{\Lambda}\operatorname{Sel}^{\bullet}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}=\mathrm{Char}_{\Lambda}\mathrm{Image}\left(\mathrm{Col}_{f,i}^{\bullet}\right)/(L_{p}^{\bullet}),$

provided that $L_{p}^{\bullet}\neq 0$.

For the rest of the article, we shall assume that the following hypothesis
holds:

(H-IMC) Kato's Iwasawa main conjecture (2.2) holds and that both
$L_{p}^{\sharp}$ and $L_{p}^{\flat}$ are non-zero.

######  Remark 2.1.

We say a few words on the hypothesis (H-IMC). In his seminal work [13], Kato
established that the inclusion ``$\subseteq$'' in (2.2) holds after tensoring
by $\mathbb{Q}_{p}$ under the assumption that the image of the representation
$T_{f}|_{G_{\mathbb{Q}(\mu_{p^{\infty}})}}$ contains a copy of
$\mathrm{SL}_{2}(\mathbb{Z}_{p})$, provided that certain local terms vanish
(see in particular Theorem 12.5(4) of op. cit.). Note that in [13, Conjecture
12.10], Kato's main conjecture is formulated in terms of étale cohomology
groups. Kobayashi and Kurihara showed that it can be recast in terms of Galois
cohomology groups and fine Selmer groups (see for example [14, Proposition
7.1]). The reverse inclusion of (2.2) has been established in the monumental
work of Skinner-Urban [31] for a $p$-ordinary modular form (under certain
hypotheses, see Theorem 1 of op. cit. for details). In the non-ordinary form,
there have been several recent breakthroughs in this reverse direction (for
instance, see [5, 6, 9]). Finally, the work [18] has supplied many sufficient
conditions for the non-vanishing of $L_{p}^{\sharp}$ and $L_{p}^{\flat}$ (see
in particular Corollary 3.29 and Proposition 3.39 in op. cit.). In view of
these developments, it seems reasonable to assume that the hypothesis (H-IMC),
on which our main results are reliant, holds.

###  2.4. Images of signed Coleman maps

We review an explicit description of the images of these Coleman maps. Recall
that $\gamma$ is a fixed topological generator of $\Gamma$ giving the
identification $\Lambda=\mathcal{O}\llbracket X\rrbracket$ via $X=\gamma-1$.
Let $u=\chi_{\operatorname{cyc}}(\gamma)$. It was proved in [19, §5A]111In
loc. cit., it is assumed that $f$ is non-ordinary at $p$, meaning that
$a_{p}(f)$ is a non-unit in $\mathcal{O}$. But the same calculations still
apply to the ordinary case. The only difference is that the linear relations
describing the images of the Coleman maps will be slightly different. See
Remark 1.10 in [19]. that if $\eta$ is a character on $\Delta$, then there
exist constants $c_{\eta,j}\in K$ such that

$e_{\eta}\mathrm{Image}(\mathrm{Col}_{f}^{\sharp}\oplus\mathrm{Col}_{f}^{\flat})\otimes_{\mathcal{O}}K=\\{(F,G)\in\Lambda^{2}\mathrel{\mathop{\mathchar
58\relax}}F(u^{j}-1)=c_{\eta,j}G(u^{j}-1),0\leq j\leq k-2\\}.$

In particular, this tells us that there is an isomorphism of $\Lambda$-modules

$\frac{\Lambda^{2}\otimes_{\mathcal{O}}K}{e_{\eta}\mathrm{Image}(\mathrm{Col}_{f}^{\sharp}\oplus\mathrm{Col}_{f}^{\flat})\otimes_{\mathcal{O}}K}\cong\frac{\Lambda\otimes_{\mathcal{O}}K}{\prod_{j=0}^{k-2}(X-u^{j}+1)},$

where $e_{\eta}$ denotes the idempotent attached to $\eta$. Integrally, we
have:

######  Lemma 2.2.

There is a pseudo-isomorphism of $\Lambda$-modules

$\frac{\Lambda^{2}}{e_{\eta}\mathrm{Image}(\mathrm{Col}_{f}^{\sharp}\oplus\mathrm{Col}_{f}^{\flat})}\sim\frac{\Lambda}{\prod_{j=0}^{k-2}(X-u^{j}+1)}.$

###### Proof.

It is enough to show that the $\mu$-invariant of the quotient on the left-hand
side vanishes. For simplicity, let us write

$C=e_{\eta}\mathrm{Image}(\mathrm{Col}_{f}^{\sharp}\oplus\mathrm{Col}_{f}^{\flat}),\quad\text{and}\quad
C_{\bullet}=e_{\eta}\mathrm{Image}\mathrm{Col}_{f}^{\bullet}$

for $\bullet\in\\{\sharp,\flat\\}$.

Consider the tautological short exact sequence

$0\rightarrow C\rightarrow\Lambda^{2}\rightarrow\Lambda^{2}/C\rightarrow 0.$

This gives the exact sequence

$0\rightarrow\left(\Lambda^{2}/C\right)[\varpi]\rightarrow
C/\varpi\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}\Lambda^{2}/\varpi,$

where $\varpi$ is a fixed uniformizer of $\mathcal{O}$. Similarly, we have the
exact sequences

$0\rightarrow\left(\Lambda/C_{\bullet}\right)[\varpi]\rightarrow
C_{\bullet}/\varpi\stackrel{{\scriptstyle\Phi_{\bullet}}}{{\longrightarrow}}\Lambda/\varpi,$

for $\bullet\in\\{\sharp,\flat\\}$.

Recall from [19, Theorem 5.10] that the $\mu$-invariants of
$\Lambda/C_{\bullet}$ are zero. Therefore,
$\ker\Phi_{\bullet}=(\Lambda/C_{\bullet})[\varpi]$ are finite for both choices
of $\bullet$. Note that $C\subset C_{\sharp}\oplus C_{\flat}$ and
$\Phi=(\Phi_{\sharp}\oplus\Phi_{\flat})|_{C/\varpi}$ by definitions. It
implies that

$\ker\Phi\subset\ker\Phi_{\sharp}\oplus\ker\Phi_{\flat}.$

Hence, $\ker\Phi=\left(\Lambda^{2}/C\right)[\varpi]$ is finite. In particular,
the $\mu$-invariant of $\Lambda^{2}/C$ is zero, which finishes the proof of
the lemma. ∎

######  Remark 2.3.

In the case where $T_{f}$ is the $p$-adic Tate module of an elliptic curve
with good supersingular reduction at $p$, we can in fact describe the set $C$
explicitly. See [17, Proposition 2.2].

It follows from Lemma 2.2 that there is an exact sequence

(2.4)
$0\rightarrow\mathbb{H}^{1}_{f,i}\rightarrow\Lambda^{2}\rightarrow\mathcal{C}_{k,i}\rightarrow
0,$

where the first map is given by
$\mathrm{Col}_{f,i}^{\sharp}\oplus\mathrm{Col}_{f,i}^{\flat}$ and
$\mathcal{C}_{k,i}$ is a $\Lambda$-module which is pseudo-isomorphic to
$\Lambda/\eta_{k,i}$, with $\eta_{k,i}$ being the image of
$\prod_{j=0}^{k-2}(X-u^{j}+1)$ under the twisting map
$\operatorname{Tw}^{i-k+1}$.

##  3\. A generalization of a result of Wingberg

Let $\mathcal{A}$ be an abelian variety over $\mathbb{Q}$ with supersingular
reduction at $p$. Wingberg proved that there is a pseudo-isomorphism of
$\Lambda$-modules

$\left(\operatorname{Sel}(\mathcal{A}[p^{\infty}]/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)_{{\Lambda-\mathrm{tor}}}\sim\left(\operatorname{Sel}_{0}(\mathcal{A}^{t}[p^{\infty}]/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)^{\iota},$

where $\mathcal{A}^{t}$ is the dual abelian variety of $\mathcal{A}$ (see [33,
Corollary 2.5]). Here, $M_{\Lambda-\mathrm{tor}}$ denotes the maximal torsion
submodule of a $\Lambda$-module $M$,
$\operatorname{Sel}(\mathcal{A}[p^{\infty}]/\mathbb{Q}_{\operatorname{cyc}})$
is the $p$-primary Selmer group of $\mathcal{A}$ over
$\mathbb{Q}_{\operatorname{cyc}}$ and
$\operatorname{Sel}_{0}(\mathcal{A}^{t}[p^{\infty}]/\mathbb{Q}_{\operatorname{cyc}})$
is the $p$-primary fine Selmer group of $\mathcal{A}^{t}$ over
$\mathbb{Q}_{\operatorname{cyc}}$ defined in a similar manner as the fine
Selmer groups for $A_{g}(j)$ in §2.1 above. We prove the following analogue of
Wingberg's result in the context of modular forms.

######  Theorem 3.1.

We have a pseudo-isomorphism of $\Lambda$-modules

$\left(H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{g}(j))^{\vee}\right)_{\Lambda-\mathrm{tor}}\sim\left(\mathfrak{H}^{2}_{g,j}\right)^{\iota},$

where $g\in\\{f,{\overline{f}}\\}$ and $j\in\mathbb{Z}$.

###### Proof.

From the low degree terms of Nekovář's spectral sequence (see [25, Lemma
9.1.5])

$\operatorname{Ext}_{\Lambda}^{r}(H^{3-s}_{\operatorname{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{g}(j)),\Lambda)\Rightarrow
H^{3-r-s}(\mathbb{Q}_{\operatorname{cyc}},A_{g}(j))^{\vee},$

we obtain the following exact sequence of $\Lambda$-modules:

$0\longrightarrow\operatorname{Ext}_{\Lambda}^{1}(H^{2}_{\operatorname{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{g}(j)),\Lambda)\longrightarrow
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{g}(j))^{\vee}\longrightarrow\operatorname{Hom}_{\Lambda}(H^{1}_{\operatorname{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{g}(j)),\Lambda).$

Since
$\operatorname{Hom}_{\Lambda}(H^{1}_{\operatorname{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{g}(j)),\Lambda)$
is a reflexive $\Lambda$-module by [26, Corollary 5.1.3], it is
$\Lambda$-torsion-free. Hence, it follows from the above exact sequence that
there is an isomorphism

$\left(H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{g}(j))^{\vee}\right)_{\Lambda-\mathrm{tor}}\cong\operatorname{Ext}_{\Lambda}^{1}(H^{2}_{\operatorname{Iw}}(\mathbb{Q}_{\operatorname{cyc}},T_{g}(j)),\Lambda)$

of $\Lambda$-modules. By [26, Proposition 5.5.13], the latter is pseudo-
isomorphic to $\left(\mathfrak{H}^{2}_{g,j}\right)^{\iota}$, which concludes
the proof of the theorem. ∎

When $f$ is non-ordinary at $p$, we shall establish a direct analog of
Wingberg's theorem on the level of Selmer groups under the following
hypothesis on the local representation $A_{\overline{f}}$ (see Theorem 3.4
below).

(H0) For all $v\in S(\mathbb{Q}_{\operatorname{cyc}})$, the group
$H^{0}(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i))$ is finite.

######  Remark 3.2.

Note that when $v|p$, the group
$H^{0}(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i))$ is always
finite by [10, Lemma 3.3]. See also §5 of op. cit. where sufficient conditions
and explicit examples of the finiteness of
$H^{0}(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i))$ for $v|N$ are
studied.

######  Lemma 3.3.

Under (H0), we have a pseudo-isomorphism of $\Lambda$-modules

$\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\sim\mathfrak{H}^{2}_{{\overline{f}},k-i}.$

###### Proof.

Let $n\geq 0$ be an integer. By the Poitou–Tate exact sequence (see for
example [28, §A.3.1]), we have the following exact sequence:

$\displaystyle\bigoplus_{v\in
S(L_{n})}H^{0}\left(L_{n,v},A_{\overline{f}}(k-i)\right)\longrightarrow
H^{2}\left(G_{S}(L_{n}),T_{f}(i)\right)^{\vee}$
$\displaystyle\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}$
$\displaystyle H^{1}\left(G_{S}(L_{n}),A_{\overline{f}}(k-i)\right)$
$\displaystyle\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\bigoplus_{v\in
S(L_{n})}H^{1}\left(L_{n,v},A_{\overline{f}}(k-i)\right).$

By definition,

$\mathrm{Image}(\delta)=\ker(\tau)=\operatorname{Sel}_{0}\left(A_{\overline{f}}(k-i)/L_{n}\right).$

This gives the exact sequence

$\bigoplus_{v\in
S(L_{n})}H^{0}\left(L_{n,v},A_{\overline{f}}(k-i)\right)\longrightarrow
H^{2}\left(G_{S}(L_{n}),T_{f}(i)\right)^{\vee}\longrightarrow\operatorname{Sel}_{0}\left(A_{\overline{f}}(k-i)/L_{n}\right)\longrightarrow
0.$

By the local Tate duality, we have

$H^{0}(L_{n,v},A_{\overline{f}}(k-i))^{\vee}\cong H^{2}(L_{n,v},T_{f}(i))$

for all $v\in S$. Thus, after taking Pontryagin duals and inverse limits, we
obtain

(3.1)
$0\longrightarrow\operatorname{Sel}_{0}\left(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}}\right)^{\vee}\longrightarrow
H^{2}_{\operatorname{Iw}}\left(G_{S}(\mathbb{Q}_{\operatorname{cyc}}),T_{f}(i)\right)\longrightarrow\bigoplus_{v\in
S(\mathbb{Q}_{\operatorname{cyc}})}H^{0}\left(\mathbb{Q}_{\operatorname{cyc},v},A_{\overline{f}}(k-i)\right)^{\vee}.$

Therefore, the lemma follows from (H0). ∎

######  Theorem 3.4.

Suppose that $f$ is non-ordinary at $p$ and (H0) is valid. We have a pseudo-
isomorphism of $\Lambda$-modules

$\left(\operatorname{Sel}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)_{\Lambda-\mathrm{tor}}\sim\left(\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)^{\iota}.$

###### Proof.

Consider the defining sequence of the Selmer group

$0\longrightarrow\operatorname{Sel}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})\longrightarrow
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{f}(i))\longrightarrow\bigoplus_{w\in
S(\mathbb{Q}_{\operatorname{cyc}})}\frac{H^{1}(\mathbb{Q}_{\mathrm{cyc},w},A_{f}(i))}{H^{1}_{\mathrm{f}}(\mathbb{Q}_{\mathrm{cyc},w},A_{f}(i))}.$

By [4, Proposition 3.8], there is an isomorphism

$\varprojlim
H^{1}(L_{n,w_{n}},T_{\overline{f}}(k-i))\cong\left(\frac{H^{1}(\mathbb{Q}_{\mathrm{cyc},w},A_{f}(i))}{H^{1}_{\mathrm{f}}(\mathbb{Q}_{\mathrm{cyc},w},A_{f}(i))}\right)^{\vee},$

where $w_{n}\in S(L_{n})$ such that $w_{n}$ lies below $w$ and $w_{n+1}$. When
$w\nmid p$, this is zero by [13, §17.10]. When $w|p$, this is also zero by
[29, Theorem 0.6] under the hypothesis that $f$ is non-ordinary at $p$. Hence,
we have an isomorphism

$\operatorname{Sel}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})\cong
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{f}(i))$

of $\Lambda$-modules. Combining this with Theorem 3.1, we obtain a pseudo-
isomorphism

$\left(\operatorname{Sel}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)_{\Lambda-\mathrm{tor}}\sim\left(\mathfrak{H}^{2}_{f,i}\right)^{\iota}$

of $\Lambda$-modules. But the latter is pseudo-isomorphic to
$\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ by
Lemma 3.3 under (H0). This concludes the proof of the theorem. ∎

##  4\. Comparison of characteristic elements

###  4.1. Preliminary lemmas

We prove several preliminary lemmas that will be used in the proofs of
Theorems 4.3 and 4.4 below.

######  Lemma 4.1.

Assume that (H0) holds. If $F$ is an irreducible element of $\Lambda$ such
that $F\nmid\mathfrak{F}_{{\overline{f}},k-i}$, then we have a psuedo-
isomorphism

$(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i})[F^{\infty}]\sim\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee,\iota}[F^{\infty}]$

of $\Lambda$-modules. In particular,
$(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i})[F]$ is finite if and only if
$F^{\iota}\nmid\mathfrak{F}_{f,i}$.

###### Proof.

We have the Poitou–Tate exact sequence

$0\rightarrow\mathfrak{H}^{1}_{f,i}\rightarrow\mathbb{H}^{1}_{f,i}\rightarrow
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{\overline{f}}(k-i))^{\vee}\rightarrow\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\rightarrow
0.$

This gives the following short exact sequence

$0\rightarrow(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i})\rightarrow
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{\overline{f}}(k-i))^{\vee}\rightarrow\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\rightarrow
0,$

which in turn induces the following exact sequence:

(4.1)
$0\rightarrow(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i})[F^{\infty}]\rightarrow
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{\overline{f}}(k-i))^{\vee}[F^{\infty}]\rightarrow\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}[F^{\infty}].$

By assumption,
$\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}[F^{\infty}]$
is finite. Therefore, we obtain a psuedo-isomorphism

(4.2) $(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i})[F^{\infty}]\sim
H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{\overline{f}}(k-i))^{\vee}[F^{\infty}].$

By Theorem 3.1 and Lemma 3.3, we have the pseudo-isomorphism of
$\Lambda$-modules

$H^{1}(\mathbb{Q}_{\operatorname{cyc}},A_{\overline{f}}(k-i))^{\vee}[F^{\infty}]\sim\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee,\iota}[F^{\infty}].$

Combining these two pseudo-isomorphisms finishes the proof of the lemma. ∎

######  Lemma 4.2.

Assume that (H-IMC) holds. Let $F$ be an irreducible element of $\Lambda$ such
that $F\nmid\eta_{k,i}$. Recall that $\mathrm{loc}_{p}$ denotes the
localization map from $\mathfrak{H}^{1}_{f,i}$ to $\mathbb{H}^{1}_{f,i}$. Then
one has a pseudo-isomorphism

$\left(\Lambda^{2}/\Lambda(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})\right)[F^{\infty}]\sim\left(\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\right)[F^{\infty}]$

of $\Lambda$-modules. Hence, it follows that
$F\nmid\gcd(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})$
if and only if
$\left(\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\right)[F]$ is
finite.

###### Proof.

By (2.4) and (2.3), we have the following exact sequence

$0\rightarrow\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\rightarrow\Lambda^{2}/\Lambda(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})\rightarrow\Lambda/\eta_{k,i}\rightarrow
0.$

This gives the exact sequence

$0\rightarrow\left(\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\right)[F^{\infty}]\rightarrow\left(\Lambda^{2}/\Lambda(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})\right)[F^{\infty}]\rightarrow\Lambda/\eta_{k,i}[F^{\infty}].$

The last term is finite since $F\nmid\eta_{k,i}$. Hence, the result follows. ∎

###  4.2. Comparison between characteristic ideals of fine Selmer groups and
signed Selmer groups

The goal of this section is to prove a generalization of [20, Theorem 1.2]
(see Theorem 4.4 below). We shall do so via the following intermediate result:

######  Theorem 4.3.

Assume that (H-IMC) and (H0) hold. Let $F$ be an irreducible element of
$\Lambda$ such that $F\nmid\eta_{k,i}$ and that
$F\nmid\mathfrak{F}_{{\overline{f}},k-i}$. There is a pseudo-isomorphism of
$\Lambda$-modules

$\left(\Lambda^{2}/\Lambda(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})\right)[F^{\infty}]\sim\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee,\iota}[F^{\infty}].$

In particular, $F^{\iota}\nmid\mathfrak{F}_{f,i}$ if and only if
$F\nmid\gcd\left(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat}\right)$.

###### Proof.

By Lemma 4.2, we have a pseudo-isomorphism

$\left(\Lambda^{2}/\Lambda(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat})\right)[F^{\infty}]\sim\left(\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\right)[F^{\infty}]$

of $\Lambda$-modules. Consider the short exact sequence

$0\rightarrow\mathfrak{H}^{1}_{f,i}/\Lambda\mathbf{z}_{f,i}\rightarrow\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\rightarrow\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i}\rightarrow
0.$

Since $\mathfrak{H}^{1}_{f,i}/\Lambda\mathbf{z}_{f,i}$ is a torsion
$\Lambda$-module with
$\mathrm{Char}_{\Lambda}\left(\mathfrak{H}^{1}_{f,i}/\Lambda\mathbf{z}_{f,i}\right)=\mathfrak{F}_{{\overline{f}},k-i}$
not divisible by $F$, we have the following pseudo-isomorphism of
$\Lambda$-modules

$\left(\mathbb{H}^{1}_{f,i}/(\mathrm{loc}_{p}(\mathbf{z}_{f,i}))\right)[F^{\infty}]\sim\left(\mathbb{H}^{1}_{f,i}/\mathfrak{H}^{1}_{f,i}\right)[F^{\infty}].$

By Lemma 4.1, the latter is pseudo-isomorphic to
$\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee,\iota}[F^{\infty}]$.
On combining these pseudo-isomorphisms, the theorem follows. ∎

We can now state and prove the main result of the article.

######  Theorem 4.4.

Assume that (H-IMC) and (H0) hold. Let $F$ be an irreducible element of
$\Lambda$ such that $F\nmid\eta_{k,i}$. Then
$F^{\iota}\nmid\mathfrak{F}_{f,i}$ and
$F\nmid\mathfrak{F}_{{\overline{f}},k-i}$ if and only if
$F\nmid\gcd\left(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat}\right)$.

###### Proof.

Theorem 4.3 tells us that if $F^{\iota}\nmid\mathfrak{F}_{f,i}$ and
$F\nmid\mathfrak{F}_{{\overline{f}},k-i}$, then
$F\nmid\gcd\left(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat}\right)$.
This proves the "only if" implication.

Conversely, suppose that
$F\nmid\gcd\left(\mathfrak{F}_{{\overline{f}},k-i}^{\sharp},\mathfrak{F}_{{\overline{f}},k-i}^{\flat}\right)$.
From the inclusion

$\operatorname{Sel}_{0}(A_{\overline{f}}(k-i))\subset\operatorname{Sel}^{\bullet}(A_{\overline{f}}(k-i)),$

we have $F\nmid\mathfrak{F}_{{\overline{f}},k-i}$. The remaining assertion
that $F^{\iota}\nmid\mathfrak{F}_{f,i}$ now follows from Theorem 4.3. ∎

######  Remark 4.5.

Let $E/\mathbb{Q}$ be an elliptic curve with good supersingular reduction at
$p$. In [20], it is stated in the proof of Proposition 3.1 that by [33,
Corollary 2.5], we have the equality

$\mathrm{Char}_{\Lambda}\left(\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}\right)_{\Lambda-\mathrm{tor}}=\mathrm{Char}_{\Lambda}\operatorname{Sel}_{0}(E/\mathbb{Q}_{\operatorname{cyc}})^{\vee}.$

However, one of the two Selmer groups should be twisted by $\iota$ in order
for the equality to hold. As such, for the rest of the proof to go through,
the additional hypothesis that the irreducible element $f$ (not to be confused
with the notation for a modular form in the present article) satisfies
$(f)=(f^{\iota})$ is required. Consequently, the statement of Theorem 1.2 in
op. cit. should also be modified.

Specializing our Theorem 4.4 to the case $F=\varpi$, where $\varpi$ is a
uniformizer of $\mathcal{O}$, we may deduce the following:

######  Corollary 4.6.

Assume that (H-IMC) and (H0) hold. Then the following statements are
equivalent.

(a) The $\mu$-invariants of
$\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ and
$\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
are zero.

(b) At least one of
$\operatorname{Sel}^{\sharp}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
and
$\operatorname{Sel}^{\flat}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
has trivial $\mu$-invariant.

######  Remark 4.7.

Recall from [7, Conjecture A], [12, Conjecture A] and [1, Conjecture 1.2] that
the $\mu$-invariants of
$\operatorname{Sel}_{0}(A_{f}(i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$ and
$\operatorname{Sel}_{0}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
are predicted to always vanish. Corollary 4.6 gives an alternative formulation
of these conjectures in terms of signed Selmer groups. In fact, numerical
calculations carried out by Pollack (see [30, §7] and
http://math.bu.edu/people/rpollack/Data/data.html) suggest that the
$\mu$-invariants of both
$\operatorname{Sel}^{\sharp}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
and
$\operatorname{Sel}^{\flat}(A_{\overline{f}}(k-i)/\mathbb{Q}_{\operatorname{cyc}})^{\vee}$
are zero when $A_{\overline{f}}(k-i)$ is given by $E[p^{\infty}]$ for some
elliptic curve $E/\mathbb{Q}$ with good supersingular reduction at $p$.

## acknowledgement

We would like to thank the anonymous referee for providing various helpful
comments that have helped us improve the exposition of this article.

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