# On the corank of the fine Selmer group of an elliptic curve over a
$\mathbb{Z}_{p}$-extension

Anwesh Ray  Department of Mathematics
University of British Columbia
Vancouver BC, Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

Let $p$ be an odd prime and $F_{\infty}$ be a $\mathbb{Z}_{p}$-extension of a
number field $F$. Given an elliptic curve $E$ over $F$, we study the structure
of the _fine Selmer group_ over $F_{\infty}$. It is shown that under certain
conditions, the fine Selmer group is a cofinitely generated module over
$\mathbb{Z}_{p}$ and furthermore, we obtain an upper bound for its corank
(i.e., the $\lambda$-invariant), in terms of various local and global
invariants.

##  1\. Introduction

The variation of arithmetic objects in families of number fields of interest
is a central theme in number theory. Iwasawa studied the variation of class
numbers of number fields in certain infinite towers of number field
extensions. In greater detail, let $p$ be a prime number and $F$ be a number
field. Set $\mathbb{Z}_{p}$ to denote the ring of $p$-adic integers, defined
to be the inverse limit $\varprojlim_{n}\mathbb{Z}/p^{n}\mathbb{Z}$. A
_$\mathbb{Z}_{p}$ -tower_ consists of a family of Galois extensions
$\\{F_{n}\\}$ of $F$, satisfying the containments

$F=F_{0}\subset F_{1}\subset F_{2}\subset\dots\subset F_{n}\subset
F_{n+1}\subset\dots,$

and such that $\operatorname{Gal}(F_{n}/F)$ is isomorphic to
$\mathbb{Z}/p^{n}\mathbb{Z}$. The infinite Galois extension
$F_{\infty}\mathrel{\mathop{\mathchar 58\relax}}=\cup_{n\geq 1}F_{n}$ is
referred to a $\mathbb{Z}_{p}$-extension, since the Galois group
$\operatorname{Gal}(F_{\infty}/F)$ is isomorphic to $\mathbb{Z}_{p}$. Let
$e_{n}$ be the highest power of $p$ dividing the class number of $F_{n}$.
Iwasawa showed that there are integers $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and
$\nu\in\mathbb{Z}$ such that for all sufficiently large values of $n$, we have
that $e_{n}=p^{n}\mu+n\lambda+\nu$ (cf. [Iwa73] or [Was97, Chapter 13]). Let
$\mu_{p^{\infty}}$ be the $p$-primary roots of unity in $\bar{F}$; the
_cyclotomic $\mathbb{Z}_{p}$-extension of $F$_ is the unique
$\mathbb{Z}_{p}$-extension of $F$ contained in $F(\mu_{p^{\infty}})$. It is
conjectured by Iwasawa that for the cyclotomic $\mathbb{Z}_{p}$-extension of
$F$, the $\mu$-invariant vanishes. In other words, Iwasawa conjectured that
over the cyclotomic $\mathbb{Z}_{p}$-extension of $F$, the $p$-primary part of
the Hilbert class group is cofinitely generated as a $\mathbb{Z}_{p}$-module.
The corank of this $\mathbb{Z}_{p}$-module is the $\lambda$-invariant.

Mazur (cf. [Maz72]) initiated the Iwasawa theory of abelian varieties with
good ordinary reduction. The original motivation was to study the growth of
Selmer groups, Tate-Shafarevich groups and Mordell–Weil ranks of an abelian
variety in a $\mathbb{Z}_{p}$-tower of number fields. The Selmer groups in
question are defined by local conditions at each prime, defined in terms of
certain local Kummer maps. These Selmer groups capture various arithmetic
properties of the elliptic curves or abelian varieties in question. A closer
analogue with the Hilbert class groups holds for the _fine Selmer group_ ,
which is the subgroup defined by _strict_ local conditions at all primes (cf.
Definition 2.1). The fine Selmer group appeared in various guises in work of
Perrin-Riou (cf. [PR93, PR95]), Rubin (cf. [Rub00]) and Kato (cf. [K+04]).
Their algebraic properties were systematically studied by Coates and Sujatha
in [CS05], with an emphasis on the vanishing of the $\mu$-invariant. Indeed,
it is conjectured that the fine Selmer group of an elliptic curve over the
cyclotomic $\mathbb{Z}_{p}$-extension of a number field is a finitely
generated $\mathbb{Z}_{p}$-module (cf. Conjecture A in _loc. cit._). In recent
work (cf. [DRS22]), it it shown that this conjecture is indeed satisfied over
the cyclotomic $\mathbb{Z}_{p}$-extension, for various Galois representations
of interest, provided certain additional conditions are satisfied.

When the $\mu$-invariant is known to vanish, the $\mathbb{Z}_{p}$-corank of
the fine Selmer group is equal to the $\lambda$-invariant. The main focus of
this paper is to shed light on the $\lambda$-invariant of the fine Selmer
group of an elliptic curve over a general $\mathbb{Z}_{p}$-extension. For the
classical Selmer groups of elliptic curve with good ordinary reduction, the
$\lambda$-invariant over the cyclotomic $\mathbb{Z}_{p}$-extension can be
computed since there are efficient algorithms to compute the $p$-adic
L-function [Ste07]. This calculation is possible since it the _main
conjecture_ is known to hold in this context, thanks to the seminal work of
Skinner and Urban [SU14]. The $\mathbb{Z}_{p}$-corank of the fine Selmer group
is shrouded in mystery since there are not many satisfactory computational
tools to obtain a precise description of the characteristic ideal except for
certain trivial cases. It should be noted here that one possible techniques
used in obtaining some insight into the $\lambda$-invariant is via the _Euler
characteristic formula_ for the leading term of the characteristic series (cf.
[Wut07, section 6]). If the formula for the leading term is a $p$-adic unit,
then, under further hypotheses, the $\lambda$-invariant may be exactly
computed to be equal to $\operatorname{max}\\{0,r-1\\}$, where $r$ is the
Mordell-Weil rank of the elliptic curve in question. Such ideas are used in
studying distribution questions for the Iwasawa invariants of the fine Selmer
groups of elliptic curves (cf. [RS21]). However, this technique does not
provide any upper bound on the $\lambda$-invariant, when the leading term is
not a $p$-adic unit. The upper bounds obtained in this paper can be expressed
in terms of various local and global invariants. Theorem 4.6 describes and
upper bound which holds in generality. The formula is better understood in a
special case in which the global invariants in our formula are shown to
vanish, see Theorem 4.9. We provide a concrete example to better clarify the
main results of the paper.

##  2\. Preliminary Notions

In this section, we introduce relevant notation and recall the definition and
basic properties of fine Selmer groups associated to elliptic curves. Let $F$
be a number field. Throughout, $p$ will be an odd prime number. Fix an
algebraic closure $\bar{F}$ of $F$. For each prime $v$ of $F$, set $F_{v}$ to
denote the completion of $F$ at $v$, and let $\bar{F}_{v}$ be a choice of
algebraic closure of $F_{v}$. Fix an embedding
$\iota_{v}\mathrel{\mathop{\mathchar
58\relax}}\bar{F}\hookrightarrow\bar{F}_{v}$ for each prime $v$ of $F$. Given
a field $\mathcal{F}$ contained in $\bar{F}$, let
$\operatorname{G}_{\mathcal{F}}$ denote the absolute Galois group
$\operatorname{Gal}(\bar{F}/\mathcal{F})$.

Let $F_{\infty}$ be a _$\mathbb{Z}_{p}$ -extension_ of $F$, i.e., $F_{\infty}$
is an infinite Galois extension of $F$ for which
$\operatorname{Gal}(F_{\infty}/F)$ is topologically isomorphic to
$\mathbb{Z}_{p}$. For $n\in\mathbb{Z}_{\geq 1}$, we set $F_{n}$ to be the
extension of $F$ such that $[F_{n}\mathrel{\mathop{\mathchar
58\relax}}F]=p^{n}$. Note that $F_{n}$ is a Galois extension of $F$ and that
$\operatorname{Gal}(F_{n}/F)$ is isomorphic to $\mathbb{Z}/p^{n}\mathbb{Z}$.
Let $E_{/F}$ be an elliptic curve with good reduction at all primes $v|p$ of
$F$. Let $S$ be the set of primes $v$ of $F$ such that either $v|p$ or such
that $E$ has bad reduction at $v$. Let $S_{p}$ be the set of primes of $F$
that lie above $p$. Given a set of primes $\Sigma$ of $F$ and an algebraic
extension $\mathcal{F}/F$, let $\Sigma(\mathcal{F})$ be the set of primes of
$\mathcal{F}$ that lie above the set $\Sigma$. Given a prime $v$ of $F$, set
$v(\mathcal{F})$ to be the set of primes of $\mathcal{F}$ that lie above $v$.

Let $\mathcal{F}$ be an extension of $F$ contained in $\bar{F}$. For a prime
$w$ in $\mathcal{F}$, let $\mathcal{F}_{w}$ be the completion of $\mathcal{F}$
at $w$. For ease of notation, $H^{i}(\mathcal{F}_{w},\cdot)$ (resp.
$H^{i}(\mathcal{F}_{w},\cdot)$) will be used as a shorthand for
$H^{i}(\operatorname{G}_{\mathcal{F}},\cdot)$ (resp.
$H^{i}(\operatorname{G}_{\mathcal{F}_{w}},\cdot)$). Given a prime $v$ of $F$,
we set $\mathcal{H}_{v}\left(\mathcal{F},E[p^{\infty}]\right)$ to denote the
following (possibly infinite) product

$\mathcal{H}_{v}\left(\mathcal{F},E[p^{\infty}]\right)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{w\in v(\mathcal{F})}H^{1}(\mathcal{F}_{w},E[p^{\infty}]).$

Note that all primes $v\notin S_{p}$ are unramified in any
$\mathbb{Z}_{p}$-extension $F_{\infty}$. Therefore, $F_{\infty}$ is contained
in $F_{S}$. Let $F_{S}$ denote the maximal extension of $F$ contained in
$\bar{F}$ in which all the primes $v\notin S$ are unramified. Assume that
$\mathcal{F}$ is contained in $F_{S}$. The $p$-primary fine Selmer group of
$E$ over $\mathcal{F}$ is defined as follows

(2.1) $R\left(E/\mathcal{F}\right)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(F_{S}/\mathcal{F},E[p^{\infty}])\rightarrow\bigoplus_{v\in
S}\mathcal{H}_{v}\left(\mathcal{F},E[p^{\infty}]\right)\right\\},$

where the map in question is the natural map induced by restriction to
$\mathcal{F}_{w}$ as $w$ ranges over the set $S(\mathcal{F})$.

We set $\Gamma$ to denote the Galois group $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}\left(F_{\infty}/F\right)$. The Iwasawa algebra
$\Lambda$ is the completed group algebra
$\Lambda=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. Let $\gamma$
be a topological generator of $\Gamma$, and set $T$ to denote $(\gamma-1)$. We
identify the Iwasawa algebra $\Lambda$ with the formal power series ring
$\mathbb{Z}_{p}\llbracket T\rrbracket$. A polynomial $f(T)\in\Lambda$ is said
to be _distinguished_ if it is a nonconstant monic polynomial and all of its
non-leading coefficients are divisible by $p$.

We introduce the Iwasawa invariants associated to a finitely generated module
over $\Lambda$. A map of $\Lambda$-modules $M_{1}\rightarrow M_{2}$ is said to
be a _pseudo-isomorphism_ if its kernel and cokernel are both finite. Let $M$
be a finitely generated module over $\Lambda$. According to the structure
theorem of $\Lambda$-modules [Was97, Chapter 13], there is a pseudoisomorphism
of the form

$M\longrightarrow\Lambda^{r}\oplus\left(\bigoplus_{i=1}^{s}\Lambda/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\Lambda/\left(f_{j}(T)^{\lambda_{j}}\right)\right),$

where $r\in\mathbb{Z}_{\geq 0}$ is the rank, $\mu_{i}$ are positive integers
and $f_{j}(T)$ are irreducible distinguished polynomials. The $\mu$ and
$\lambda$-invariants are defined as follows

$\mu(M)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{s}\mu_{i},\text{ and
}\lambda(M)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{j=1}^{t}\lambda_{j}\operatorname{deg}f_{j},$

where we set $\mu(M)=0$ (resp. $\lambda(M)=0$) if $s=0$ (resp. $t=0$). Denote
the $\mu$ (resp. $\lambda$-invariant) of $R(E/F_{\infty})^{\vee}$ by
$\mu(E/F_{\infty})$ (resp. $\lambda(E/F_{\infty})$). We note that when
$F_{\infty}/F$ is the cyclotomic $\mathbb{Z}_{p}$-extension, it is conjectured
that $R(E/F_{\infty})^{\vee}$ is a torsion $\Lambda$-module and
$\mu(E/F_{\infty})=0$, cf. [CS05, Conjecture A].

We are interested in the following question.

######  Question 2.1.

Under what conditions can it be shown that the dual fine Selmer group
$R(E/F_{\infty})^{\vee}$ is a finitely generated torsion $\Lambda$-module with
$\mu(E/F_{\infty})=0$? Furthermore, assuming that these conditions are
satisfied, how well can one characterize the $\lambda$-invariant
$\lambda(E/F_{\infty})$?

That $R(E/F_{\infty})^{\vee}$ is a finitely generated as a $\Lambda$-module is
well known, and follows from a standard argument that involves an application
of Nakayama’s lemma.

######  Lemma 2.2.

With respect to notation above, the following conditions are equivalent.

  1. (1)

$R(E/F_{\infty})^{\vee}$ is a torsion $\Lambda$-module with
$\mu(E/F_{\infty})=0$,

  2. (2)

$R(E/F_{\infty})[p]$ is finite.

Furthermore, if the above conditions are satisfied, then,

$\lambda(E/F_{\infty})\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E/F_{\infty})[p]\right).$

###### Proof.

Let $M$ be a finitely generated $\Lambda$-module. It is an easy consequence of
the structure theory that $M$ is a torsion $\Lambda$-module with $\mu(M)=0$ if
and only if $M$ is finitely generated as a $\mathbb{Z}_{p}$-module. Let $M$ be
the dual fine Selmer group $R(E/F_{\infty})^{\vee}$, we find therefore that
$M$ is a torsion $\Lambda$-module with $\mu(M)=0$ if and only if $M/pM$ is
finite. Note that the dual of $R(E/F_{\infty})[p]$ is $M/pM$, and hence the
two conditions above are equivalent. Furthermore, $\lambda(M)$ is equal to the
$\mathbb{Z}_{p}$-rank of $M$, and hence,
$\lambda(M)\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}M/pM$. Therefore, we
find that
$\lambda(E/F_{\infty})\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E/F_{\infty})[p]\right)$.
∎

Given a $\Lambda$-module $M$, we say that $M$ is cofinitely generated (resp.
cotorsion) as a $\Lambda$-module if the Pontryagin dual
$M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$ is
finitely generated (resp. torsion) as a $\Lambda$-module.

##  3\. The residual fine Selmer group

Recall that $F$ is a number field and $E$ is an elliptic curve defined over
$F$, and $p$ is an odd prime. Denote by $E[p]$ the $p$-torsion subgroup of
$E(\bar{F})$ and note that as an abelian group, $E[p]$ is isomorphic to
$\left(\mathbb{Z}/p\mathbb{Z}\right)^{2}$. We define an analogue of the fine
Selmer group associated with the residual representation on $E[p]$.

Recall that $S$ consists of the set of primes $v\nmid p$ at which $E$ has bad
reduction and the primes $v\mid p$. For $v\in S$, set
$\mathcal{H}_{v}(F_{\infty},E[p])$ to be the (possibly infinite) product
$\prod_{w\in v(F_{\infty})}H^{1}(F_{\infty,w},E[p])$. The _residual fine
Selmer group_ is defined as follows

$R(E[p]/F_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left\\{H^{1}(F_{S}/F_{\infty},E[p])\xrightarrow{\Phi}\bigoplus_{v\in
S}\mathcal{H}_{v}\left(F_{\infty},E[p]\right)\right\\}.$

The map $\Phi$ is the natural map obtained by restriction to $F_{\infty,w}$ as
$w$ ranges over the primes in $S(F_{\infty})$. Consider the Kummer sequence

$0\rightarrow E[p]\rightarrow E[p^{\infty}]\rightarrow
E[p^{\infty}]\rightarrow 0.$

Given a prime $v\in S$, let $h_{v}$ be the map

$h_{v}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{v}(F_{\infty},E[p])\rightarrow\mathcal{H}_{v}(F_{\infty},E[p^{\infty}])[p]$

which is the product of the natural maps induced from the Kummer sequence

$h_{w}\mathrel{\mathop{\mathchar 58\relax}}H^{1}(F_{\infty,w},E[p])\rightarrow
H^{1}(F_{\infty,w},E[p^{\infty}])[p],$

as $w$ ranges over $v(F_{\infty})$. Let $h$ be the direct sum of the maps
$h_{v}$, as $v$ ranges over $S$

$h\mathrel{\mathop{\mathchar 58\relax}}\bigoplus_{v\in
S}\mathcal{H}_{v}(F_{\infty},E[p])\longrightarrow\bigoplus_{v\in
S}\mathcal{H}_{v}(F_{\infty},E[p^{\infty}])[p].$

From the Kummer sequence, we have the following diagram

(3.1)
${0}$${R(E[p]/F_{\infty})}$${H^{1}(F_{S}/F_{\infty},E[p])}$${\operatorname{im}(\Phi)}$${0}$${0}$${R(E/F_{\infty})[p]}$${H^{1}(F_{S}/F_{\infty},E[p^{\infty}])[p]}$${\bigoplus_{v\in
S}\mathcal{H}_{v}(F_{\infty},E[p^{\infty}])[p].}$$\scriptstyle{\Psi}$$\scriptstyle{g}$$\scriptstyle{h^{\prime}}$

In the above diagram, the map $g$ is surjective with kernel isomorphic to
$\frac{H^{0}(F_{\infty},E[p^{\infty}])}{pH^{0}(F_{\infty},E[p^{\infty}])}$.
The vertical map $h^{\prime}$ is the restriction of the map $h$ to
$\operatorname{im}(\Phi)$. By an application of the Snake lemma, we have the
following exact sequence

(3.2)
$0\rightarrow\operatorname{ker}\Psi\rightarrow\operatorname{ker}g\rightarrow\operatorname{ker}h^{\prime}\rightarrow\operatorname{cok}\Psi\rightarrow
0.$

Note that for $v\in S$, the kernel of $h_{v}$ is the product of kernels of
$h_{w}$ as $w$ ranges over the primes of $F_{\infty}$ that lie above $v$.

######  Proposition 3.1.

With respect to notation above, assume that the following conditions are
satisfied

  1. (1)

$\operatorname{ker}(h)$ is finite,

  2. (2)

$R(E[p]/F_{\infty})$ is finite.

Then, the fine Selmer group $R(E/F_{\infty})$ is a cotorsion $\Lambda$-module
with $\mu(E/F_{\infty})=0$. Furthermore, we have that

$\lambda(E/F_{\infty})\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E[p]/F_{\infty})\right)+\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(\operatorname{ker}(h)\right).$

###### Proof.

First, we show that $R(E/F_{\infty})$ is a cotorsion $\Lambda$-module and
$\mu(E/F_{\infty})=0$. According to Lemma 2.2, $R(E/F_{\infty})$ is a
cotorsion $\Lambda$-module with $\mu=0$ if and only if $R(E/F_{\infty})[p]$ is
finite. Consider the map

$\Psi\mathrel{\mathop{\mathchar 58\relax}}R(E[p]/F_{\infty})\rightarrow
R(E/F_{\infty})[p],$

(cf. (3.1)) and we observe that it suffices to show that
$\operatorname{ker}\Psi$ and $\operatorname{cok}\Psi$ are finite. Referring to
the exact sequence (3.2), we deduce that $\operatorname{ker}\Psi$ is finite
since $\operatorname{ker}(g)$ is finite. Since it is assumed that
$\operatorname{ker}(h)$ is finite, it follows that $\operatorname{cok}\Psi$ is
finite as well. Therefore, we have shown that $R(E/F_{\infty})[p]$ is finite,
and therefore deduce that $R(E/F_{\infty})$ is a cotorsion $\Lambda$-module
and $\mu(E/F_{\infty})=0$.

Note that it also follows from Lemma 2.2 that
$\lambda(E/F_{\infty})\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E/F_{\infty})[p]\right)$.
Hence, we arrive at the bounds

$\begin{split}\lambda(E/F_{\infty})&\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E/F_{\infty})[p]\right)\\\
&\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E[p]/F_{\infty})\right)+\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(\operatorname{cok}\Psi\right)\\\
&\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(R(E[p]/F_{\infty})\right)+\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(\operatorname{ker}(h)\right),\end{split}$

and thus, this completes the proof. ∎

######  Definition 3.2.

Let $S_{0}$ be the set of primes $v\in S$ at which _both_ of the following
conditions are satisfied

  1. (1)

$E$ has bad reduction at $v$.

  2. (2)

If $p\geq 5$ and $\mu_{p}$ is contained in $F_{v}$, then, $E$ has split
multiplicative reduction at $v$.

######  Assumption 3.3.

Assume that each prime $v\in S_{0}\cup S_{p}$ is finitely decomposed in
$F_{\infty}$.

Since all primes of $F$ are finitely decomposed in its cyclotomic
$\mathbb{Z}_{p}$-extension, the above assumption is always satisfies when
$F_{\infty}$ is the cyclotomic $\mathbb{Z}_{p}$-extension of $F$. Note that
since $E$ is stipulated to have good reduction at all primes $v\in S$, we find
that $S_{0}$ is a subset of $S\backslash S_{p}$. Given $v\in S_{p}$, set
$\delta_{v}\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}&2\text{ if
}E(F_{v})[p]\neq 0;\\\ &0\text{ if }E(F_{v})[p]=0.\end{cases}$ For any prime
$v\in S\cup S_{p}$, let $g_{v}$ be the number of primes $w\in v(F_{\infty})$.

######  Lemma 3.4.

It follows from Assumption 3.3 that $\operatorname{ker}(h)$ is finite, and
moreover,

$\operatorname{dim}\left(\operatorname{ker}(h)\right)\leq\sum_{v\in
S_{0}}2g_{v}+\sum_{v\in S_{p}}\delta_{v}g_{v}.$

###### Proof.

Let $v\in S$ and let $w$ be a prime of $F_{\infty}$ that lies above $v$. From
the Kummer sequence, we find that the kernel of $h_{w}$ is identified with
$E(F_{\infty,w})[p^{\infty}]\otimes\mathbb{Z}/p\mathbb{Z}$. Note that in
particular, the dimension of $\operatorname{ker}h_{w}$ is at most $2$. There
are three cases to consider.

  1. (1)

If $v\in S_{0}$, then,
$\operatorname{dim}\left(\operatorname{ker}(h_{w})\right)$ is at most $2$.
Thus, $\operatorname{dim}\left(\operatorname{ker}(h_{v})\right)$ is at most
$2g_{v}$.

  2. (2)

If $v\in S\backslash(S_{0}\cup S_{p})$, then, $\mu_{p}$ is contained in
$F_{v}$ and $E$ has either nonsplit multiplicative reduction or additive
reduction at $v$. Since $v\nmid p$, it is unramified in any
$\mathbb{Z}_{p}$-extension, in particular, unramified in $F_{\infty}$. Thus,
the elliptic curve $E$ has either non-split multiplicative reduction or
additive reduction at every prime $w$ of $F_{\infty}$ that lies above $v$.
Then, it follows from [HM99, Proposition 5.1. (iii)] that
$E(F_{\infty,w})[p^{\infty}]=0$ for all primes $w\in v(F_{\infty})$. As a
result, the kernel of $h_{v}$ is $0$ in this case.

  3. (3)

Finally, consider the case when $v\in S_{p}$. There are two subcases to
consider.

    1. (a)

First, assume that $E(F_{v})[p]=0$. Then, since $F_{\infty,w}/F_{v}$ is a
pro-$p$ extension, it follows that $E(F_{\infty,w})[p^{\infty}]=0$ (cf.
[NSW13, Proposition 1.6.12]). Therefore, the kernel of $h_{v}$ is $0$ in this
case.

    2. (b)

On the other hand, if $E(F_{v})[p]\neq 0$, then, we do still have the bound
$\operatorname{dim}\left(\operatorname{ker}h_{w}\right)\leq 2$ for each prime
$w$ of $F_{\infty}$ that lies above $w$.

From the above case decomposition, we find that
$\operatorname{dim}h\leq\sum_{v\in S_{0}}2g_{v}+\sum_{v\in
S_{p}}\delta_{v}g_{v}$. Since $g_{v}$ is assumed to be finite for each $v\in
S_{0}\cup S_{p}$, the sum is finite.

∎

##  4\. Main results and their proofs

In this section, we state and prove the main results of the paper. We
illustrate the bounds obtained via a concrete example at the end of the
section.

Given an algebraic extension $\mathcal{F}$, let $H(\mathcal{F})$ (resp.
$A(\mathcal{F})$) be the maximal abelian unramified extension of $F$ such that
$\operatorname{Gal}(H(\mathcal{F})/\mathcal{F})$ (resp.
$\operatorname{Gal}(A(\mathcal{F})/\mathcal{F})$) is an elementary $p$-group
(resp. pro-$p$ group). Let $\mathcal{F}(E[p])$ be the extension of
$\mathcal{F}$ generated by $E[p]$. In other words, the field
$\mathcal{F}(E[p])$ is the field extension of $\mathcal{F}$ which is fixed by
the kernel of $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathcal{F}}\rightarrow\operatorname{Aut}(E[p])$.
Let $K_{n}$ be the extension $F_{n}(E[p])$ and set
$L_{n}\mathrel{\mathop{\mathchar 58\relax}}=H(K_{n})$. Set $K_{\infty}$ to
denote $F_{\infty}(E[p])$ and set $L_{\infty}$ (resp. $\mathcal{L}_{\infty}$)
to denote the extension $H(K_{\infty})$ (resp. $A(K_{\infty})$). We let
$X\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(L_{\infty}/K_{\infty})$ and
$\mathfrak{X}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathcal{L}_{\infty}/K_{\infty})$. We set $K$ to
denote the extension $F(F[p])$, and note that $K_{\infty}$ is a
$\mathbb{Z}_{p}$-extension of $K$. Let
$\Lambda=\mathbb{Z}_{p}\llbracket\operatorname{Gal}(K_{\infty}/K)\rrbracket$
denote the associated Iwasawa algebra, and note that $\mathfrak{X}$ is a
module over $\Lambda$. It is in fact known that $\mathfrak{X}$ is finitely
generated module over $\Lambda$ (cf. [Was97, Chapter 13]). By construction,
$X=\mathfrak{X}/p\mathfrak{X}$ is a module over
$\Omega\mathrel{\mathop{\mathchar 58\relax}}=\Lambda/p$.

######  Definition 4.1.

Denote by $H^{S}(K_{\infty})$ the maximal extension of $K_{\infty}$, which is
contained in $H(K_{\infty})$, in which the primes of $S(K_{\infty})$ are
completely split. Set $\mathfrak{Y}$ to denote
$\operatorname{Gal}(H^{S}(K_{\infty})/K_{\infty})$, and set
$Y\mathrel{\mathop{\mathchar 58\relax}}=\mathfrak{Y}/p\mathfrak{Y}$.

Set $G$ to denote the Galois group
$\operatorname{Gal}(K_{\infty}/F_{\infty})$. From the inflation-restriction
sequence, we have the following exact sequence

$0\rightarrow
H^{1}(G,E(K_{\infty})[p])\xrightarrow{\operatorname{inf}}H^{1}(F_{S}/F_{\infty},E[p])\xrightarrow{\operatorname{res}}H^{1}(K_{S}/K_{\infty},E[p]).$

Note that the action of $\operatorname{Gal}(K_{S}/K_{\infty})$ on $E[p]$ is
the trivial action, and therefore, we find that

$H^{1}(K_{S}/K_{\infty},E[p])=\operatorname{Hom}\left(\operatorname{Gal}(K_{S}/K_{\infty}),\mathbb{Z}/p\mathbb{Z}\right)^{2}.$

The residual fine Selmer group $R(E[p]/F_{\infty})$ is a subgroup of
$H^{1}(F_{S}/F_{\infty},E[p])$. Let $\mathcal{Z}$ denote
$\operatorname{inf}^{-1}(R(E[p]/F_{\infty}))$ and set
$\mathcal{Z}_{1}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{res}(Y)$.
Note that $\mathcal{Z}_{1}$ can be identified with a subgroup of
$\operatorname{Hom}(Y,\mathbb{Z}/p\mathbb{Z})^{2}$. In this way, we have an
exact sequence

(4.1) $0\rightarrow\mathcal{Z}\rightarrow
R(E[p]/F_{\infty})\rightarrow\operatorname{Hom}(Y,\mathbb{Z}/p\mathbb{Z})^{2},$

where the image of the rightmost map is $\mathcal{Z}_{1}$. Since $G$ is
finite, it is clear that $\mathcal{Z}$ is finite as well. The following
assumption is a special case of Iwasawa’s classical $\mu=0$ conjecture.

######  Assumption 4.2.

Assume that $\mathfrak{Y}$ is torsion $\Lambda$-module and that
$\mu(\mathfrak{Y})=0$. Equivalently, assume that $\mathfrak{Y}$ is finitely
generated as a $\mathbb{Z}_{p}$-module.

######  Lemma 4.3.

With respect to notation above, the following conditions are equivalent.

  1. (1)

Assumption 4.2 is satisfied,

  2. (2)

$Y$ is finite.

Furthermore, if the above equivalent conditions are satisfied, then,
$\lambda(\mathfrak{Y})\leq\operatorname{rank}_{\mathbb{Z}_{p}}\mathfrak{Y}$.

###### Proof.

It is an easy consequence of the structure theory of finitely generated
Iwasawa modules that Assumption 4.2 is satisfied if and only if $\mathfrak{Y}$
is finitely generated as a $\mathbb{Z}_{p}$-module and that
$\lambda(\mathfrak{Y})=\operatorname{rank}_{\mathbb{Z}_{p}}\mathfrak{Y}$. Note
that

$\operatorname{rank}_{\mathbb{Z}_{p}}\mathfrak{Y}\leq\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}\left(\mathfrak{Y}/p\mathfrak{Y}\right)=\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}Y,$

and the result follows. ∎

In many situations, it is shown that certain Iwasawa modules that arise from
class groups do not contain any non-zero finite $\Lambda$-submodules. This
property, if it holds, implies that the dimension of $Y$ would equal the
$\lambda$-invariant of $\mathfrak{Y}$, as the following result shows.

######  Proposition 4.4.

With respect to notation above, suppose that Assumption 4.2 holds. Then the
following conditions are equivalent

  1. (1)

$\mathfrak{Y}$ does not contain any non-zero finite $\Lambda$-submodules,

  2. (2)

$\lambda(\mathfrak{Y})=\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}Y$.

###### Proof.

It follows from Lemma 4.3 that
$\operatorname{rank}_{\mathbb{Z}_{p}}\mathfrak{Y}$ is finite and

$\mathfrak{Y}=\mathbb{Z}_{p}^{\lambda(\mathfrak{Y})}\oplus A,$

where $A$ is a finite abelian $p$-group. Note that $A$ is the $p$-primary
torsion subgroup of $\mathfrak{Y}$, hence, $A$ is a finite $\Lambda$-submodule
of $\mathfrak{Y}$. Therefore, condition (1) is equivalent to the vanishing of
$A$. On the other hand,
$\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}(Y)=\lambda(\mathfrak{Y})+\operatorname{dim}_{\mathbb{Z}/p\mathbb{Z}}(A)$.
Therefore, (1) and (2) are equivalent. ∎

######  Proposition 4.5.

With respect to notation above, suppose that Assumption 4.2 holds. Then,
$R(E[p]/F_{\infty})$ is finite and

$\operatorname{dim}R(E[p]/F_{\infty})\leq
2\operatorname{dim}(Y)+\operatorname{dim}(\mathcal{Z}).$

###### Proof.

It follows from Lemma 4.3 that $Y$ is finite. The result follows from the
exact sequence (4.1). ∎

######  Theorem 4.6.

Let $E$ be an elliptic curve over a number field $F$ and let $p$ be an odd
prime such that $E$ has good reduction at all primes that lie above $p$. Let
$F_{\infty}/F$ be a $\mathbb{Z}_{p}$-extension of $F$. Suppose that
Assumptions 3.3 and 4.2 hold. Let $Y$ be as in Definition 4.1. Then, the
following assertions hold

  1. (1)

$R(E/F_{\infty})$ is a cotorsion module over $\Lambda$ with
$\mu(E/F_{\infty})=0$,

  2. (2)

$Y$ is finite and the following bound on the $\lambda$-invariant holds

$\lambda(E/F_{\infty})\leq
2\operatorname{dim}(Y)+\operatorname{dim}(\mathcal{Z})+\sum_{v\in
S_{0}}2g_{v}+\sum_{v\in S_{p}}\delta_{v}g_{v}.$

###### Proof.

Recall that Proposition 3.1 asserts that if

  1. (1)

$\operatorname{ker}(h)$ is finite,

  2. (2)

$R(E[p]/F_{\infty})$ is finite,

then, $R(E/F_{\infty})$ is a cotorsion $\Lambda$-module with
$\mu(E/F_{\infty})=0$. Furthermore, we have that

$\lambda(E/F_{\infty})\leq\operatorname{dim}R(E[p]/F_{\infty})+\operatorname{dim}(\operatorname{ker}(h)).$

Lemma 3.4 states that $\operatorname{ker}(h)$ is finite and
$\operatorname{dim}(\operatorname{ker}(h))$ is bounded above by $\sum_{v\in
S_{0}}2g_{v}+\sum_{v\in S_{p}}\delta_{v}g_{v}$. The result follows therefore
from Proposition 4.5. ∎

We now show that $Y$ and $\mathcal{Z}$ can be shown to be trivial provided
further conditions are satisfied. Let $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$
be the _residual representation_ on $E[p]$.

######  Lemma 4.7.

Suppose that the image of $\bar{\rho}$ is a non-solvable subgroup of
$\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$. Then,
$\bar{\rho}_{|\operatorname{G}_{F_{\infty}}}$ is irreducible.

###### Proof.

Note that if $\bar{\rho}_{|\operatorname{G}_{F_{\infty}}}$ is reducible, then,
its image is solvable. Since $F_{\infty}$ is an abelian extension of $F$, we
find that the image of $\bar{\rho}$ must also be solvable. ∎

######  Lemma 4.8.

The following assertions are satisfied.

  1. (1)

Suppose that _any one_ (or both) of the following conditions for $\bar{\rho}$
is satisfied

    1. (a)

$\bar{\rho}$ is irreducible when restricted to
$\operatorname{G}_{F_{\infty}}$,

    2. (b)

the image of $\bar{\rho}$ has cardinality coprime to $p$.

Then, we have that $\mathcal{Z}=0$.

  2. (2)

Suppose that both of the following conditions are satisifed

    1. (a)

there is only one prime $v$ of $K$ that lies above $p$ which ramifies in
$K_{\infty}$ and $v$ is totally ramified in $K_{\infty}$,

    2. (b)

$p$ does not divide the class number of $K$.

Then, it follows that $Y=0$.

###### Proof.

We first prove part (1). Since $\bar{\rho}$ is assumed to satisfy one (or
both) of two conditions, we consider the conditions one at a time. First,
assume that $\bar{\rho}$ is irreducible when restricted to
$\operatorname{G}_{F_{\infty}}$. Then, the vanishing of $\mathcal{Z}$ follows
from [PS21, Lemma 2.2]. Next, assume that the image of $\bar{\rho}$ has
cardinality coprime to $p$. Since
$E[p]\simeq\left(\mathbb{Z}/p\mathbb{Z}\right)^{2}$ is a $p$-group, and $G$
has cardinality coprime to $p$, it follows that $H^{1}(G,E[p])=0$. As a
consequence, we deduce that $\mathcal{Z}=0$.

Part (2) follows from [Was97, Proposition 13.22]. ∎

######  Theorem 4.9.

Let $E$ be an elliptic curve over a number field $F$ and let $p$ be an odd
prime such that $E$ has good reduction at all primes in $S_{p}$. Let
$F_{\infty}/F$ be a $\mathbb{Z}_{p}$-extension of $F$. With respect to
notation above, assume that Assumption 3.3 is satisfied. Furthermore, suppose
that the following conditions are satisfied

  1. (1)

the image of $\bar{\rho}$ is non-solvable or has cardinality coprime to $p$,

  2. (2)

there is only one prime of $K$ which ramifies in $K_{\infty}$ and this prime
is totally ramified.

  3. (3)

Suppose that $A(K)=0$.

Then, we find that $R(E/F_{\infty})$ is a cotorsion module over $\Lambda$ with
$\mu(E/F_{\infty})=0$, and

$\lambda(E/F_{\infty})\leq\sum_{v\in S_{0}}2g_{v}+\sum_{v\in
S_{p}}\delta_{v}g_{v}.$

In particular, if $S_{0}=\emptyset$ and $E(F_{v})[p]=0$ for all primes $v\in
S_{p}$, then, $\lambda(E/F_{\infty})=0$ for all $\mathbb{Z}_{p}$-extensions
$F_{\infty}/F$.

###### Proof.

Note that Lemma 4.8 implies that $Y=0$ and hence, Assumption 4.2 is
automatically satisfied. The result follows directly form Theorem 4.6 and
Lemma 4.8. ∎

An Example: We illustrate the above results through an example. Let us
consider the elliptic curve $E$ with Cremona label 11a2 and let $p=5$. The
residual representation $\bar{\rho}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$
is a direct sum of two characters $\bar{\rho}=\bar{\chi}\oplus 1$, where
$\bar{\chi}$ is a mod-$p$ cyclotomic character.

  1. (1)

First, consider the case when $F=\mathbb{Q}$ and
$F_{\infty}=\mathbb{Q}_{\infty}$ is the cyclotomic $\mathbb{Z}_{p}$-extension
of $\mathbb{Q}$. Note that $K=\mathbb{Q}(E[p])$ is the number field
$\mathbb{Q}(\mu_{p})$, and $K_{\infty}$ is $\mathbb{Q}(\mu_{p^{\infty}})$, the
cyclotomic $\mathbb{Z}_{p}$-extension of $K$. The set of primes $S$ is equal
to $\\{11,p\\}$ and $S_{0}=\\{11\\}$. All primes of $\mathbb{Q}$ are finitely
decomposed in $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_{p}$-extension
of $\mathbb{Q}$, and hence, Assumption 3.3 is satisfied. We verify the
conditions of Theorem 4.9.

     * •

Clearly the image of $\bar{\rho}$ has cardinality coprime to $p$.

     * •

Let $\eta_{p}$ be the prime of $K$ that lies above $p$. Note that $\eta_{p}$
is the principal ideal generated by $(1-e^{2\pi i/p})$ and that
$p\mathcal{O}_{K}=\eta_{p}^{p-1}$. The prime $\eta_{p}$ is totally ramified in
$K_{\infty}=\mathbb{Q}(\mu_{p^{\infty}})$.

     * •

The prime $p=5$ is a regular prime, i.e., $p$ does not divide the class number
of $K=\mathbb{Q}(\mu_{p})$ (the smallest irregular prime is $37$).

We compute $g_{11}$, i.e, the number of primes of $\mathbb{Q}_{\infty}$ that
lie above $11$. Since $5^{2}\nmid(11^{4}-1)$, it follows that $g_{11}=1$. On
the other hand, $p$ is totally ramified in $\mathbb{Q}_{\infty}$, hence,
$g_{p}=1$. Theorem 4.9 then implies that $R(E/F_{\infty})$ is a cotorsion
module over $\Lambda$ with $\mu(E/\mathbb{Q}_{\infty})=0$, and

$\lambda(E/\mathbb{Q}_{\infty})\leq\sum_{v\in S_{0}}2g_{v}+\sum_{v\in
S_{p}}\delta_{v}g_{v}\leq 4.$

  2. (2)

Next, suppose that $F=\mathbb{Q}(\mu_{p})$. Note that $K=F$ and that there are
$p-1=4$ independent $\mathbb{Z}_{p}$-extensions of $F$. Let $F_{\infty}/F$ be
any $\mathbb{Z}_{p}$-extension in which $11$ is finitely decomposed and in
which $\eta_{p}$ is totally ramified. Thus in particular, Assumption 3.3
holds. Note that $K_{\infty}$ is equal to $F_{\infty}$. The same arguments as
in part (1) above imply that the assumptions of Theorem 4.9 are satisfied.
Theorem 4.9 then implies that $R(E/F_{\infty})$ is a cotorsion module over
$\Lambda$ with $\mu(E/F_{\infty})=0$, and

$\lambda(E/F_{\infty})\leq\sum_{v\in S_{0}}2g_{v}+\sum_{v\in
S_{p}}\delta_{v}g_{v}=\sum_{v|11}2g_{v}+\delta_{\eta_{p}}.$

When $F_{\infty}$ is taken to be the cyclotomic $\mathbb{Z}_{p}$-extension of
$F$, then Assumption 3.3 is clearly satisfied. In this case, we find that $11$
is totally inert in $F_{\infty}$, and therefore, we find that
$\lambda(E/F_{\infty})\leq 4$.

## References

  * [CS05] John Coates and Ramdorai Sujatha.  Fine selmer groups of elliptic curves over p-adic lie extensions.  Mathematische Annalen, 331(4):809–839, 2005.
  * [DRS22] Shaunak V Deo, Anwesh Ray, and R Sujatha.  On the $\mu$ equals zero conjecture for the fine selmer group in iwasawa theory.  arXiv preprint arXiv:2202.09937, 2022.
  * [HM99] Yoshitaka Hachimori and Kazuo Matsuno.  An analogue of kida’s formula for the selmer groups of elliptic curves.  Journal of Algebraic Geometry, 8(3):581–601, 1999.
  * [Iwa73] Kenkichi Iwasawa.  On $\mathbb{Z}_{\ell}$-extensions of algebraic number fields.  Annals of Mathematics, pages 246–326, 1973.
  * [K+04] Kazuya Kato et al.  p-adic hodge theory and values of zeta functions of modular forms.  Astérisque, 295:117–290, 2004.
  * [Maz72] Barry Mazur.  Rational points of abelian varieties with values in towers of number fields.  Inventiones mathematicae, 18(3):183–266, 1972.
  * [NSW13] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg.  Cohomology of number fields, volume 323.  Springer Science & Business Media, 2013.
  * [PR93] Bernadette Perrin-Riou.  Fonctions l $p$-adiques d’une courbe elliptique et points rationnels.  In Annales de l’institut Fourier, volume 43, pages 945–995, 1993\.
  * [PR95] Bernadette Perrin-Riou.  Fonctions L p-adiques des représentations p-adiques.  Number 94. Société mathématique de France, 1995.
  * [PS21] Dipendra Prasad and Sudhanshu Shekhar.  Relating the tate–shafarevich group of an elliptic curve with the class group.  Pacific Journal of Mathematics, 312(1):203–218, 2021.
  * [RS21] Anwesh Ray and R Sujatha.  Arithmetic statistics for the fine selmer group in iwasawa theory.  arXiv preprint arXiv:2112.13335, 2021.
  * [Rub00] Karl Rubin.  Euler systems.  Number 147. Princeton University Press, 2000.
  * [Ste07] William Stein.  Sage mathematics software.  http://www. sagemath. org/, 2007.
  * [SU14] Christopher Skinner and Eric Urban.  The iwasawa main conjectures for gl2.  Inventiones mathematicae, 195(1):1–277, 2014.
  * [Was97] Lawrence C Washington.  Introduction to cyclotomic fields, volume 83.  Springer Science & Business Media, 1997.
  * [Wut07] Christian Wuthrich.  Iwasawa theory of the fine selmer group.  Journal of Algebraic Geometry, 16(1):83–108, 2007.


# The “SPectrogram Analysis and Cataloguing Environment” (SPACE) Labelling
Tool

C. K. Louis 1,∗, C. M. Jackman 1, S. W. Mangham 2, K. D. Smith 1, E. P.
O’Dwyer 1, A. Empey 1, B. Cecconi 3, P. Zarka 3, S. Maloney 1

###### Abstract

The SPectrogram Analysis and Cataloguing Environment (SPACE) tool is an
interactive python tool designed to label radio emission features of interest
in a time-frequency map (called “dynamic spectrum”). The program uses
Matplotlib’s Polygon Selector widget to allow a user to select and edit an
undefined number of vertices on top of the dynamic spectrum before closing the
shape (polygon). Multiple polygons may be drawn on any spectrum, and the
feature name along with the coordinates for each polygon vertex are saved into
a “.json” file as per the “Time-Frequency Catalogue” (TFCat) format along with
other data such as the feature id, observer name, and data units. This paper
describes the first official stable release (version 2.0) of the tool.

##  1 Keywords:

Python tool, Labelling tool, Planetary radio emissions, Solar radio emissions,
Jupiter, Saturn, Earth, Sun

##  2 Introduction

Non-thermal planetary radio emissions are produced by out-of-equilibrium
populations of charged particles in planetary magnetospheres, and are observed
at almost all strongly magnetized planets in our solar system: the Earth,
Jupiter, Saturn, Uranus and Neptune. The radio emissions can be divided into
different classes, such as plasma waves, electromagnetic radio waves or
electrostatic radio waves. It is highly desirable to select these distinct
classes, which can often have characteristic frequency ranges, morphologies,
or polarizations. Once catalogues of different emission types have been built
up, that can enable large statistical studies unveiling both the average and
extreme behaviour of planetary radio emissions.

There have been many long-running planetary spacecraft which have returned
huge volumes of radio data and we have only scratched the surface of its
analysis. For example, the Cassini mission at Saturn spent 13 years studying
the kronian system, revealing several components to its radio spectrum (Lamy
et al., 2008; Ye et al., 2011; Lamy, 2017; Taubenschuss et al., 2011).
Furthermore, the Wind spacecraft has spent almost two decades observing
terrestrial (and solar) radio emissions from a range of vantage points near
Earth (Waters et al., 2021; Fogg et al., 2022; Bonnin et al., 2008).

Significant efforts have been made in recent years to classify radio emissions
from Jupiter, where the non-thermal radio emission is composed of half a dozen
components (Louis et al., 2021c). These components overlap themselves in time
and frequency, making automatic detection non-trivial. Therefore, manually
cataloguing them is mandatory to be able to study them independently. Previous
catalogues have been using square boxes to define the time and frequency
intervals containing the radio signal (such as Leblanc, 2020a, e, b, c, d),
but to be able to automatically disentangle the emissions when using the
catalogue, it is then needed to construct catalogues with polygon vertices and
with distinct labels. This was done by a few authors, primarily using tools
built in IDL to construct such catalogues of Jupiter radio emissions Marques
et al. (2017); Zarka et al. (2021). Once catalogues are built they can
comprise training sets which form the basis of supervised machine learning
approaches to classify larger samples of unseen data.

Here we present a Python user interface tool to allow the drawing of polygons
around features in dynamic spectra. Section 3 describes the package, and an
example of the use of this package. Section 4 summarises the version history
of the code. Section 5 gives examples of the application of the catalogues
produced by the tool. Section 6 presents some future avenues to continue to
improve the tool.

##  3 The SPACE package

The SPectrogram Analysis and Cataloguing Environment (SPACE) tool is an
interactive python tool designed to label radio emission features of interest
in a time-frequency map (called a “dynamic spectrum”). The program enables
users to create and edit the vertices of a polygon on the dynamic spectrum
plot, before naming and saving it as a ’feature’ in a catalogue for future
analysis.

###  3.1 Installation

The code is available on an open-source GitHub repository (Louis et al.,
2022), with full installation instructions present on the repository page, and
packed with the tool. It can easily be downloaded and installed using git and
pip, with all prerequisites included in the provided requirements.txt file.

###  3.2 Usage

Once installed using pip, the space labelling tool is available as a system-
wide command spacelabel. It can then be used to view and label spacecraft
observational data, by providing an input file in HDF5 or NASA CDF format (see
section 3.2.3 for specifics), and a time window to view. An example use is
shown in Figure 1.

Users may select any number of the measurement types present in the file (e.g.
polarisation, flux and/or power), and view them all tiled on the screen. For
example, Figure 1a displays Cassini Flux and Polarization radio data, while
Figure 1b only display Juno Flux radio data. Users can then click to select
polygonal regions of the observation to label as features (top panel of Figure
1a). To close the polygon, users should click on the first drawn vertex of the
polygon. Once done, a window pops up and ask to name the drawn feature
appropriately (see top panel of Figure 1a). Features labelled in one view
(e.g. intensity) appear simultaneously on the other views once they have been
named (see bottom panel of Figure 1a), allowing users to easily see how a
feature presents in multiple measurement types.

Once a region has been labelled, a user can pan their viewing window back and
forth through the time range within the dataset by clicking on the Prev or
Next buttons (see Figure 1), with an overlap applied between each view in
order to facilitate labelling features that lie on the edge of a window. Once
finished, the labelled regions can be saved as a TFCat (Time-Frequency
Catalogue) formatted JSON111https://www.json.org/ file (Cecconi et al., 2022,
this issue) by clicking on the Save button, and used later. If a user re-opens
the same data file, or another data file with the same naming structure (e.g.
observations_20180601_v02.cdf and observations_20180602_v02.cdf) saved
features from previous sessions will be pre-loaded (see Figure 1).

Figure 1: Caption on next page

Figure 1: Examples of plots from the SPACE labelling Tool.
Panel (a) displays Cassini/RPWS (Radio and Plasma Waves Science Gurnett et
al., 2004) data (Lamy et al., 2008, 2009). The two panels show Intensity and
Polarization data, respectively. At the top right of the top panel one can see
a polygon that has just been drawn, with the window for naming the feature
appearing at the top left of the graphics window. Other features have already
been labelled, and appear in both intensity and polarisation views, with their
names overlaid.
The data displays in panel (b) are the estimated flux density (Louis et al.,
2021a, c) from by Juno/Waves measurements (Kurth et al., 2017), with the Louis
et al. (2021b) catalogue overlaid.
Panel (c) displays observations of Polar/PWI instrument (Gurnett et al.,
1995). The horizontal dashed-white line shows an example of the use of the
-horizontal_line option. The variable dashed-white line show that the tool is
also able to read a 1D table from the CDF file (provided that this has been
specified in the configuration file)

Full usage documentation is available on the GitHub repository for the code
(Louis et al., 2022).

####  3.2.1 Procedure

When the code first opens a datafile, it compares the columns within to a
selection of pre-made (and user-creatable) ’configuration’ files for each type
of input file (e.g. CDF, HDF5). Each describes a file in terms of the column
names within it, and provides metadata for use in the tool - units and display
names, and scaling factors that can be applied to change data stored in one
unit into data viewed in another. If the data are in an unspecified format,
the code will prompt the user to create a configuration file (see section
3.2.3). Alternatively, if their data fit multiple configuration files, they
will be prompted to select which they want to use (see section 3.2.2).

Once a configuration has been determined, the code parses the observations and
may re-bin the time into a coarser resolution in order to improve performance,
taking the average of measurements in the new larger bins. It can also rescale
the frequency data into evenly-spaced logarithmic bins between the minimum and
maximum bins in the original data, as some data files have non-monotonic bin
structures. When altering the frequency bins, measurements are logarithmically
interpolated between the readings on the previous scale. Default adjustments
can be defined in configuration files for file types, and may be over-ridden
by command-line arguments to the tool. The parsed and adjusted data are then
re-saved as a compressed HDF5 file, reducing both the size of the data and
time to access.

The pre-processed data are then displayed to the screen using MatPlotLib
(Hunter, 2007), in a window the size of the user’s initial time range. The
dynamic range of the data is constrained to improve visibility of features;
displaying, by default, the 5th-95th quantiles of the signal for each
measurement (to prevent anomalously low or high values overly-compressing the
ranges of interest, reducing the ability to discern features). GUI buttons
allow users to pan between time windows of equivalent size - with each window
will overlap the previous window by 25% in the direction of travel. Features
can then be defined by drawing polygons on the time window using the
MatPlotLib Polygon Selector widget.

####  3.2.2 Options

The user must specify the path to the file they want to visualise (or ’first’
file in a collection, e.g. of CDF files), along with the start and end dates
of their initial viewing window, in ISO year-month-day hour:minute:second
format (e.g. 2018-06-12 18:00:01). However, there are further options
available:

  * •

-f FREQUENCY_BINS: Rescales the data to this many evenly-spaced logarithmic bins. Overrides any default set in the configuration files.

  * •

-t MINIMUM_TIME_BIN: Rebins the data to time bins of this size, if it is currently more finely binned. The bin size need to be given in second

  * •

-s SPACECRAFT: Specifies the name of the spacecraft configuration file to use, if multiple describe the datafile the user has provided.

  * •

-frac_dyn_range FRAC_MIN FRAC_MAX: Defines the dynamic range of the colour bar in the visualisation, as a fraction of the distribution of values in the data file. This must be numbers between $0$ and $1$. Default values are $0.05$ and $0.95$ (the 5th-95th quantiles of the displayed signal)

  * •

-cmap CMAP: The name of the color map that will be used for the intensity plot (by default: viridis)

  * •

-g VALUES_1 ... VALUES_N: Draws horizontal line(s) on the visualisation at these specified frequencies to aid in interpretation of the plot. Values must be in the same units as the data. Lines can be toggled using check boxes.

  * •

\--not_verbose: If not_verbose is called, the debug log will not be printed.
By default: verbose mode

####  3.2.3 Input Formats

The code is designed to cope with input files in a variety of file formats and
column formats by use of configuration files, several of which are pre-
provided. HDF5 input files require at least three datasets, corresponding to
observation time (floats, in MJD), frequency range (floats, in any arbitrary
unit) and at least one measurement, stored with frequency as the rows and
observation as the columns. The names and units for each measurement (in LaTeX
form) must be provided in a configuration file, in easily-editable JSON
format. The appropriate configuration files are automatically-selected by the
code from those available - making it easy to work with HDF5 files from a
variety of collaborators with arbitrary naming schemes.

CDF files in NASA format are more structured, and can be read in either singly
or as a collection, combining all files in the directory matching the naming
scheme [...]_YYYMMDD_[...].cdf into a single pre-processed data file. As with
the HDF5 files, CDF files must contain a frequency attribute (floats, in any
arbitrary unit) and a time attribute (either in TT_2000 or CDF_EPOCH format,
which is parsed using Astropy, Price-Whelan et al., 2018) and at least one
measurement, stored with frequency as the rows and observation as the columns.

The code can easily be expanded to ingest other file formats (see section
3.3.2).

###  3.3 Structure

####  3.3.1 ’Model-View-Presenter’ Architecture

The code is designed using a standard object-oriented ’Model-View-Presenter’
architecture, with strong separation between the data input and management,
and the visualisation. This allows for easy development of both new input
file-types (see section 3.3.2) and pre-processing options, and alternative GUI
front-ends and settings (see section 6 for suggested development building on
top of this flexibility). A generic ’Presenter’ controls the logic of the
program, and feeds data from the data models to the selected GUI view, and
requests from the GUI for changes to the data models. Either the ’View’ or
’Model’ can be easily interchanged as long as they conform to the API expected
by the ’Presenter’.

Full development documentation is available on the GitHub repository for the
code (Louis et al., 2022).

####  3.3.2 Addition of new Input Formats

New input formats can be easily added by extending the base DataSet class
included in the code. A developer only needs to define the routines for
inputting the data from file; the code will then handle pre-processing and
data access.

##  4 History of the code

The first version of the labelling code was developed in IDL by P. Zarka. It
allowed users to read data from an IDL saveset (sav format), draw polygons
around features of interest and label them. However, this IDL version had to
be adapted to each new dataset. This code has been used to build many
catalogues based on different observers (such as the Nançay Decameter Array
(NDA) ground-based radio telescope (Marques et al., 2017), or the Cassini
(Zarka et al., 2021) or Juno (Louis et al., 2021c, b) spacecraft).

The second version of the labelling tool was written in Python and was the
first to be officially released (Empey et al., 2021). This version allowed to
automatically read any dataset in sav or cdf format, based on the information
requested from users from the terminal. The other main improvements compared
to the previous version were the number of vertices in the polygons
(unlimited) and the possibility to modify the vertices position during the
polygon drawing (using the Matplotlib’s Polygon Selector widget), as well as
the production of the catalogue directly in TFCat format.

The current version (Louis et al., 2022) brings a large number of
improvements, both in terms of architecture, usability and ergonomics, which
are described in the previous sections.

##  5 Applications

Once a catalogue has been produced, it can also be displayed using the SPACE
labelling Tool (see Figure 1 or the Autoplot Software (Faden et al., 2010).
For an example, the reader is invited to visit the web page
https://doi.org/10.25935/nhb2-wy29 where an autoplot template file is given to
display the Juno data (Louis et al., 2021a) and the Louis et al. (2021b)
catalogue overlaid in autoplot. See Cecconi et al. (2022, this issue) for more
information about the display of a Catalogue using Autoplot. The catalogue can
be used to study the different components of the radio emission spectrum, e.g.
as done by Louis et al. (2021c), where the data can then be automatically
selected using the catalogue via a mask or an inverse-mask. In the case
presented in Figure 1, not every type of emission is labelled, but in each
frequency range (kilometric or hectometric) only one radio component remains.
We can then study the components one by one (e.g. their latitudinal
distribution, as in Louis et al. (2021c), their distribution as a function of
observer’s or Sun’s longitude, as in Zarka et al. (2021),or their distribution
versus observer’s longitude and satellite (Io) phase as in Marques et al.
(2017)).

With the SPACE labelling Tool, we are also providing some useful routines to
use the catalogue222https://github.com/elodwyer1/Functions-for-SPACE-
Labelling-Tool.

These catalogues can also then be used to train machine learning algorithm to
detect automatically the radio emissions in past (Cassini, NDA) or future
observation (such as Juno, JUICE, NDA).

##  6 Limitations & Future Work

The code is ready for distribution and use, but has some technical
limitations. Potential works to address those limitations, and avenues of
future development, are:

  * •

Performance: The MatPlotLib-based front-end can struggle when provided with
especially high resolutions of data, or over large time windows. Rebinning
features exist to mitigate this, but ideally the front-end would be re-
implemented in a more performant framework (e.g. Plotly Inc. (2015)).

  * •

Scalability: The code loads all the data provided into memory at launch,
limiting its applicability for large datasets. Whilst this can be mitigated by
the feature to allow appending to TFCat files created by data files sharing
filename formats, a ’deferred load’ approach would be better. This would be
best accomplished using the Dask and XArray libraries (Dask Development Team,
2016; Hoyer and Hamman, 2017).

  * •

Configurations: The code depends heavily on pre-written configuration files,
and can prompt users to create missing ones - but does not yet contain a
’wizard’ or automatic walkthrough to aid users in creating them.

  * •

Catalogue integration: The modular format of the code would make it possible
to create ’dataset’ types that access and download data directly from online
catalogues, maintaining local caches.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

C. K. Louis and C. M. Jackman led the development of the SPACE Labelling Tool.
C. K. Louis wrote the paper and worked on the development of the code. P.
Zarka developed the first IDL version of the code. A. Empey developed the
first python version of the code. S. Maloney worked on the development of the
second version of the code. S. W. Mangham developed the current version of the
code. E. P. O’Dwyer tested the different versions of the code and give inputs
to the developers of the code. K. D. Smith added features to the latest
version of the code. B. Cecconi led the TFCat format of the catalogues. All
the co-authors have contributed to the writing of the paper.

## Funding

C.K. Louis’, C.M. Jackman’s, E. P. O’Dwyer’s and A. Empey’s work at the Dublin
Institute for Advanced Studies was funded by Science Foundation Ireland Grant
18/FRL/6199. S. Mangham’s work at the University of Southampton Research
Software Group was funded by a grant via the Alan Turing Institute. K.D.
Smith’s work at the Dublin Institute for Advanced Studies was funded by a 2022
SCOSTEP/PRESTO Grant.

## Data Availability Statement

The code of the SPACE labelling Tool is open-source and freely available on
github (Louis et al., 2022). The Cassini/RPWS dataset displayed in Figure 1a,
produced by Lamy et al. (2008) is available at
https://doi.org/10.25935/ZKXB-6C84 (Lamy et al., 2009). The Juno/Waves dataset
displayed in Figure 1b, produced by Louis et al. (2021c), is accessible at
https://doi.org/10.25935/6jg4-mk86 (Louis et al., 2021a), and the catalogue
can be download at https://doi.org/10.25935/nhb2-wy29 (Louis et al., 2021b) .
The Polar/PWI dataset displayed in Figure 1c is accessible through the CDAWeb
at https://cdaweb.gsfc.nasa.gov/pub/data/polar/pwi/.

## References

  * Bonnin et al. (2008) Bonnin, X., Hoang, S., and Maksimovic, M. (2008).  The directivity of solar type III bursts at hectometer and kilometer wavelengths: Wind-Ulysses observations.  _Astronomy and Astrophysic_ 489, 419–427.  10.1051/0004-6361:200809777
  * Dask Development Team (2016) Dask Development Team (2016).  _Dask: Library for dynamic task scheduling_
  * Empey et al. (2021) [Dataset] Empey, A., Louis, C. K., and Jackman, C. M. (2021).  SPACE Labelling Tool (version 1.1.0) [Code].  10.5281/zenodo.5636922
  * Faden et al. (2010) Faden, J. B., Weigel, R. S., Merka, J., and W., F. R. H. (2010).  Autoplot: a browser for scientific data on the web.  _Earth. Sci. Inform._ 3, 41–49.  10.1007/s12145-010-0049-0
  * Fogg et al. (2022) Fogg, A. R., Jackman, C. M., Waters, J. E., Bonnin, X., Lamy, L., Cecconi, B., et al. (2022).  Wind/WAVES Observations of Auroral Kilometric Radiation: Automated Burst Detection and Terrestrial Solar Wind - Magnetosphere Coupling Effects.  _Journal of Geophysical Research (Space Physics)_ 127, e30209.  10.1029/2021JA030209
  * Gurnett et al. (2004) Gurnett, D. A., Kurth, W. S., Kirchner, D. L., Hospodarsky, G. B., Averkamp, T. F., Zarka, P., et al. (2004).  The Cassini Radio and Plasma Wave Investigation.  _Space Science Reviews_ 114, 395–463.  10.1007/s11214-004-1434-0
  * Gurnett et al. (1995) Gurnett, D. A., Persoon, A. M., Randall, R. F., Odem, D. L., Remington, S. L., Averkamp, T. F., et al. (1995).  The Polar Plasma Wave Instrument.  _Space Science Reviews_ 71, 597–622.  10.1007/BF00751343
  * Hoyer and Hamman (2017) Hoyer, S. and Hamman, J. (2017).  xarray: N-D labeled arrays and datasets in Python.  _Journal of Open Research Software_ 5\.  10.5334/jors.148
  * Hunter (2007) Hunter, J. D. (2007).  Matplotlib: A 2d graphics environment.  _Computing in Science & Engineering_ 9, 90–95.  10.1109/MCSE.2007.55
  * Inc. (2015) [Dataset] Inc., P. T. (2015).  Collaborative data science
  * Kurth et al. (2017) Kurth, W. S., Hospodarsky, G. B., Kirchner, D. L., Mokrzycki, B. T., Averkamp, T. F., Robison, W. T., et al. (2017).  The Juno Waves Investigation.  _Space Science Reviews_ 213, 347–392.  10.1007/s11214-017-0396-y
  * Lamy et al. (2009) [Dataset] Lamy, L., , Cecconi, B., and Zarka, P. (2009).  Cassini/RPWS/HFR LESIA/Kronos SKR Data Collection (Version 1.0) [Data set].  10.25935/ZKXB-6C84
  * Lamy (2017) Lamy, L. (2017).  The Saturnian kilometric radiation before the Cassini Grand Finale.  In _Planetary Radio Emissions VIII_ , eds. G. Fischer, G. Mann, M. Panchenko, and P. Zarka. 171–190.  10.1553/PRE8s171
  * Lamy et al. (2008) Lamy, L., Zarka, P., Cecconi, B., Prangé, R., Kurth, W. S., and Gurnett, D. A. (2008).  Saturn kilometric radiation: Average and statistical properties.  _Journal of Geophysical Research (Space Physics)_ 113, A07201.  10.1029/2007JA012900
  * Leblanc (2020a) [Dataset] Leblanc, Y. e. a. (2020a).  A catalogue of Jovian decametric radio observations from January 1978 to December 1979 (Digitised version) [Data set].  10.25935/GXZF-ZT33
  * Leblanc (2020b) [Dataset] Leblanc, Y. e. a. (2020b).  A catalogue of Jovian decametric radio observations from January 1982 to December 1984 (Digitised version) [Data set].  10.25935/403B-VA51
  * Leblanc (2020c) [Dataset] Leblanc, Y. e. a. (2020c).  A catalogue of Jovian decametric radio observations from January 1985 to December 1987 (Digitised version) [Data set].  10.25935/gh59-py87
  * Leblanc (2020d) [Dataset] Leblanc, Y. e. a. (2020d).  A catalogue of Jovian decametric radio observations from January 1988 to December 1990 (Digitised version) [Data set].  10.25935/cmqb-jj10
  * Leblanc (2020e) [Dataset] Leblanc, Y. e. a. (2020e).  A catalogue of Jovian radio observations from January 1980 to December 1981 (Digitised version) [Data set].  10.25935/RAS0-ER93
  * Louis et al. (2022) [Dataset] Louis, C. K., Jackman, C. M., Mangham, S. W., Smith, K. D., O’Dwyer, E., Empey, A., et al. (2022).  SPACE Labelling Tool Version 2.0.0 (v2.0.0) [Code].  10.5281/zenodo.6886528
  * Louis et al. (2021a) [Dataset] Louis, C. K., Zarka, P., and Cecconi, B. (2021a).  Juno/Waves estimated flux density Collection (Version 1.0).  10.25935/6jg4-mk86
  * Louis et al. (2021b) [Dataset] Louis, C. K., Zarka, P., Cecconi, B., and Kurth, W. S. (2021b).  Catalogue of Jupiter radio emissions identified in the Juno/Waves observations (Version 1.0).  10.25935/nhb2-wy29
  * Louis et al. (2021c) Louis, C. K., Zarka, P., Dabidin, K., Lampson, P. A., Magalhães, F. P., Boudouma, A., et al. (2021c).  Latitudinal Beaming of Jupiter’s Radio Emissions From Juno/Waves Flux Density Measurements.  _Journal of Geophysical Research (Space Physics)_ 126, e29435.  10.1029/2021JA029435
  * Marques et al. (2017) Marques, M. S., Zarka, P., Echer, E., Ryabov, V. B., Alves, M. V., Denis, L., et al. (2017).  Statistical analysis of 26 yr of observations of decametric radio emissions from Jupiter.  _Astronomy & Astrophysics_ 604, A17.  10.1051/0004-6361/201630025
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B., Günther, H., Lim, P., Crawford, S., Conseil, S., et al. (2018).  The astropy project: Building an open-science project and status of the v2. 0 core package.  _The Astronomical Journal_ 156, 123
  * Taubenschuss et al. (2011) Taubenschuss, U., Leisner, J. S., Fischer, G., Gurnett, D. A., and Nemec, F. (2011).  Saturnian Low-Frequency Drifting Radio Bursts: Statistical Properties and Polarization.  In _Planetary, Solar and Heliospheric Radio Emissions (PRE VII)_ , eds. H. O. Rucker, W. S. Kurth, P. Louarn, and G. Fischer. 115–123
  * Waters et al. (2021) Waters, J. E., Jackman, C. M., Lamy, L., Cecconi, B., Whiter, D. K., Bonnin, X., et al. (2021).  Empirical Selection of Auroral Kilometric Radiation During a Multipoint Remote Observation With Wind and Cassini.  _Journal of Geophysical Research (Space Physics)_ 126, e29425.  10.1029/2021JA029425
  * Ye et al. (2011) Ye, S. Y., Fischer, G., Menietti, J. D., Wang, Z., Gurnett, D. A., and Kurth, W. S. (2011).  An Overview of Saturn Narrowband Radio Emissions Observed by Cassini RPWS.  In _Planetary, Solar and Heliospheric Radio Emissions (PRE VII)_ , eds. H. O. Rucker, W. S. Kurth, P. Louarn, and G. Fischer. 99–113
  * Zarka et al. (2021) Zarka, P., Magalhães, F. P., Marques, M. S., Louis, C. K., Echer, E., Lamy, L., et al. (2021).  Jupiter’s Auroral Radio Emissions Observed by Cassini: Rotational Versus Solar Wind Control, and Components Identification.  _Journal of Geophysical Research (Space Physics)_ 126, e29780.  10.1029/2021JA029780

## Figure captions