On the $\mu$ equals zero Conjecture for Fine Selmer groups in Iwasawa theory

Let $F$ be a number field and $p$ be an odd prime. The $p$-primary roots of unity, considered as a module over the absolute Galois group of $F$, is denoted by $\mu_{p^{\infty}}.$ Set $F_{n}$ to be the unique field contained in $F(\mu_{p^{\infty}})$ such that $[F_{n}\mathrel{\mathop{\mathchar 58\relax}}F]=p^{n}$. The cyclotomic $\mathbb{Z}_{p}$-extension $F_{cyc}$ of $F$ is the subextension of $F(\mu_{p^{\infty}})$ obtained as the union of the fields $F_{n}.$ In his seminal work [Iwa73], Iwasawa studied the growth of the $p$-part of the class groups over certain towers of number fields. In particular, for the cyclotomic $\mathbb{Z}_{p}$-extension of $F$, Iwasawa proved that there are invariants $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{Z}$ such that

where $\text{Cl}_{p}(F_{n})$ is the $p$-Sylow subgroup of the class group of $F_{n}$. The invariants $\mu$ and $\lambda$ are the Iwasawa invariants associated to the $\mathbb{Z}_{p}$-extension $F_{\operatorname{cyc}}/F$.

The genesis of Iwasawa theory arose from the study of these invariants for a broad class of $\mathbb{Z}_{p}$-extensions of a number field. Iwasawa formulated various conjectures about these invariants. Amongst them is a famous conjecture, henceforth referred to as Iwasawa’s $\mu=0$ conjecture, which asserts that the $\mu$-invariant of the cyclotomic $\mathbb{Z}_{p}$-extension $F_{\operatorname{\mathrm{cyc}}}/F$ of any number field $F$ is zero. Let $\mathcal{F}$ be the maximal abelian, unramified, pro-$p$ extension of $F_{\operatorname{\mathrm{cyc}}}$, let $X(F_{\operatorname{\mathrm{cyc}}})=\operatorname{Gal}(\mathcal{F}/F_{\operatorname{\mathrm{cyc}}})$ and let $\Gamma=\operatorname{Gal}(F_{\operatorname{\mathrm{cyc}}}/F)$. Then an equivalent formulation of this conjecture is the statement that $X(F_{\operatorname{\mathrm{cyc}}})$ is a finitely generated $\mathbb{Z}_{p}$-module. When $F$ is an abelian number field, the $\mu$-invariant is known to vanish by the work of Ferrero and Washington [FW79]. We remark that no general result in this direction is known for any broader class of non-abelian number fields.

In the modern language of Galois representations and arithmetic geometry, Iwasawa’s $\mu=0$ conjecture can be interpreted in terms of structural invariants associated to certain modules over the Iwasawa algebra of $\operatorname{Gal}(F_{\operatorname{cyc}}/F)$. These modules arise naturally in the context of the Galois representation associated to the Tate motive $\mu_{p^{\infty}}.$ Subsequently, Iwasawa theory evolved to cover the study of a range of modules over Iwasawa algebras that arise from other Galois representations. An important instance of this evolution is the work of Mazur [Maz72] who initiated the Iwasawa theory of elliptic curves and abelian varieties. In this pathbreaking work, Mazur studied the $p$-primary Selmer group of an elliptic curve (or abelian variety) $E_{/F}$ which has good, ordinary reduction at all primes dividing $p$ over $F_{\operatorname{cyc}}$, and defined analogous Iwasawa $\mu$ and $\lambda$-invariants in this context. The $\mu$-invariant of the $p$-primary Selmer group need not vanish. For example, for $E_{/\mathbb{Q}}=X_{0}(11)$, the $\mu$-invariant of the $5$-primary Selmer group is positive. Given the analogy between class groups and Selmer groups of elliptic curves, it is natural to frame an analogue of Iwasawa’s $\mu=0$ conjecture in this setting which we describe below.

Let $E$ be an elliptic curve defined over a number field $F$ and $p$ be a prime above which $E$ has potential good reduction. Set $\Gamma\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(F_{\operatorname{cyc}}/F)$ and let $\Lambda$ denote the corresponding Iwasawa algebra, which is defined to be the completed group algebra $\Lambda\mathrel{\mathop{\mathchar 58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. Then, the $p$-primary fine Selmer group consists of the Galois cohomology classes in $H^{1}\left(\operatorname{Gal}(\bar{F}/F_{\operatorname{cyc}}),E[p^{\infty}]\right)$ that are trivial at all primes above $p$ and the primes at which $E$ has bad reduction. The fine Selmer group is believed to be cotorsion as an Iwasawa module and this is known to be true for elliptic curves over $\mathbb{Q}$ (see [CS05]). In contrast, the usual Selmer group need not satisfy this condition unless the elliptic curve is assumed to have ordinary reduction at all primes above $p$.

The properties of the fine Selmer group were systematically studied by Coates and the third named author who showed that the Iwasawa $\mu$-invariant of the fine Selmer group is related to the classical Iwasawa $\mu$-invariant of the cyclotomic $\mathbb{Z}_{p}$-extension of the number field $F(E[p])$ cut out by the $p$-torsion in $E$. We refer to [CS05, Corollary 3.5] for the precise statement of this result. This relationship shows that the vanishing of the $\mu$-invariant of the fine Selmer group is a consequence of Iwasawa’s classical $\mu=0$ conjecture. This led Coates and the third named author to formulate the $\mu=0$ conjecture for the fine Selmer group of an elliptic curve (see [CS05, Conjecture A]), as the analogue of Iwasawa’s $\mu=0$ conjecture for Galois representations associated to elliptic curves.

Iwasawa theory of $p$-adic Galois representations is currently a central theme in number theory. The fine Selmer group can be defined in the more general context of $p$-adic Galois representations (see Definition 2.1 and Definition 2.2). We remark that the $\mu=0$ conjecture can be formulated in this broader framework (see Conjecture 2.6 and [LS18, Conjecture A, p.74]). It has interesting applications to the study of Selmer groups associated to Galois representations of interest. The goal of this article is to study the $\mu=0$ conjecture in this broader setting. The main results formulate explicit conditions under which the $\mu$-invariant vanishes for the fine Selmer groups associated to important classes of $p$-adic Galois representations. The cases of interest are the Galois representations arising from elliptic curves defined over number fields, Hecke eigencuspforms of weight at least $2$, Tate twists of Artin representations and Tate twists of the adjoint representations associated to modular Galois representations.

In the study of Galois representations, the theme of $p$-adic variation plays a key role. More precisely, the deformation theory of Galois representations, introduced and developed by Mazur in [Maz89], allows us to study $p$-adic families of lifts of a fixed mod-$p$ Galois representation. Hida [Hid00] studied the relationship between certain modular deformation rings defined over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and the Iwasawa theory of the classical Selmer group of the adjoint representation of a modular form. In this article, the $\mu=0$ conjecture for the fine Selmer group of the first Tate twist of the adjoint representation of a $p$-adic Galois representation $\rho$ is related to unobstructedness of the residual representation of $\rho$ (see Definition 4.1). Note that the notion of unobstructedness appears naturally in Galois deformation theory in connection with the structure of the universal deformation ring (in the sense of Mazur). This provides a new approach in investigating the fine Selmer group. It is our belief that the linkage established here will help advance the current state of art in both the Iwasawa theory of the fine Selmer group as well as deformation theory of Galois representations. In particular, our methods provide a large class of new examples of fine Selmer groups of Galois representations for which the $\mu$-invariant vanishes.

Let us now briefly outline the results proved in this article. Section 2 is preliminary in nature. In section 3, we establish a criterion for the vanishing of the $\mu$-invariant of the fine Selmer group of a Galois representation (see Theorem 3.7). To be precise, we prove that the $\mu$-invariant of the fine Selmer group of a $p$-adic Galois representation $\rho$ vanishes if the second global Galois cohomology group of the first Tate twist of the dual of the residual representation of $\rho$ vanishes. Thus, this criterion is determined purely in terms of the residual representation. Theorem 3.8 gives explicit conditions for the vanishing of the $\mu$-invariant of the fine Selmer group, which we specialize to the case of elliptic curves in Theorem 3.9.

Section 4 establishes the linkage between deformation theory of Galois representations and the vanishing of the $\mu$-invariant of the fine Selmer group. We study the fine Selmer group of the first Tate twist of the $n^{2}$-dimensional adjoint Galois representation $\operatorname{Ad}\rho$ associated to an $n$-dimensional Galois representation $\rho$. An important class of such Galois representations arises naturally from automorphic forms. More precisely, Theorem 4.3 shows that the $\mu$-invariant vanishes for the fine Selmer group of $\operatorname{Ad}\rho(1)$ if $\rho$ is unobstructed. This concept was initially introduced by Mazur in [Maz89], and it was shown by Weston in [Wes04] that Galois representations arising from modular forms are unobstructed under some explicit conditions.

Let $f$ be a normalized Hecke newform of weight $k\geq 2$. Given a prime $\mathfrak{p}$ in the field of Fourier coefficients of $f$, let $\rho_{f,\mathfrak{p}}$ be the associated $2$-dimensional $\mathfrak{p}$-adic Galois representation. Theorem 4.6 shows that the $\mu$-invariant of the fine Selmer group of $\operatorname{Ad}\rho_{f,\mathfrak{p}}(1)$ vanishes for a set of primes $\mathfrak{p}$ of Dirichlet density $1$. Moreover, if $k>2$, then it is shown to vanish for all but finitely many primes $\mathfrak{p}$. Furthermore, if the level of $f$ is squarefree, then there is an explicit set of primes $\mathfrak{p}$ outside of which the associated $\mu$-invariant is $0$. When $k=2$ and $f$ has trivial nebentypus, our results imply that the $\mu$-invariant of the fine Selmer group of $\operatorname{Sym}^{2}\rho_{f,\mathfrak{p}}$ vanishes for a set of primes $\mathfrak{p}$ of Dirichlet density $1$. We also prove a similar result in the setting of Hilbert modular forms (see Theorem 4.11). On the other hand, we prove that if $E$ is a rational elliptic curve with squarefree conductor, then there are infinitely many primes $p$ such that the $\mu$-invariant of the fine Selmer group of $\operatorname{Ad}^{0}\rho_{E,p}$ vanishes (see Theorem 4.12). Here, $\rho_{E,p}$ is the $2$-dimensional $p$-adic Galois representation attached to $E$ and $\operatorname{Ad}^{0}\rho_{E,p}$ is the subrepresentation of $\operatorname{Ad}\rho_{E,p}$ consisting of matrices with trace $0$.

We then turn our attention to Artin representations. Note that we can realize an Artin representation as a $p$-adic Galois representation for all primes $p$. We prove that if $\rho$ is an Artin representation such that $H^{0}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),\rho)=0$, then the $\mu$-invariants of the fine Selmer groups of the first Tate twists of the $p$-adic realizations of the dual of $\rho$ vanish for all but finitely many primes $p$ (see Theorem 4.14).

We also connect the $\mu=0$ conjecture with the notion of neatness introduced by Mazur in [Maz89] to prove that the $\mu=0$ conjecture is true for first Tate twist of the adjoint representation of a $p$-adic Galois representation $\rho$ whose residual representation is neat (see Corollary 4.19). Leveraging a construction of Mazur and Böckle, we end section 4 with explicit families of odd and even $S_{3}$-representations such that the $\mu=0$-conjecture holds for the fine Selmer groups of first Tate twists of all their characteristic $0$ lifts.

In section 5, we study the vanishing of the $\mu$-invariant of the fine Selmer group associated to $2$-dimensional residually dihedral Galois representations. The conditions are stated purely in terms of the residual representation and the set of primes at which the characteristic zero representation ramifies. In Theorem 5.6 we give explicit conditions for the vanishing of the $\mu$-invariant and obtain a large class of examples for which our results apply. Finally, we remark that our results are illustrated with concrete numerical examples.

Throughout, $p$ is an odd prime number and $F$ is a number field. Set $S_{p}$ to be the set of primes of $F$ above $p$. Let $S$ be a finite set of primes of $F$ and assume that $S$ contains $S_{p}$ and all archimedean primes of $F$. Throughout, fix an algebraic closure $\bar{F}$ of $F$. Let $F_{S}$ be the maximal extension of $F$ contained in $\bar{F}$ in which all primes $v\notin S$ are unramified. Set $\operatorname{G}_{F,S}$ to be the Galois group $\operatorname{Gal}(F_{S}/F)$, identified with the maximal quotient of $\operatorname{Gal}(\bar{F}/F)$ in which the primes $v\notin S$ are unramified. Given a field $L\subset F_{S}$, we set

For a number field $L$ contained in $F_{S}$, set

where $w$ runs through the primes of $L$ above $v$. The fine Selmer group associated with a Galois module is obtained by imposing the strict local condition at each prime that lies above the primes in $S$.

Note that, in general, the definition of $\mathcal{R}_{p^{\infty}}(M/\mathcal{L})$ depends on the choice of $S$. However, if $\mathcal{L}$ contains the cyclotomic $\mathbb{Z}_{p}$-extension of $F$, then $\mathcal{R}_{p^{\infty}}(M/\mathcal{L})$ does not depend on the choice of $S$ (see [SW18, Lemma 3.2]).

For every integer $n\geq 1$, let $\mu_{p^{n}}$ be the group of $p^{n}$-th roots of unity contained in $\bar{F}$ and $\mu_{p^{\infty}}\mathrel{\mathop{\mathchar 58\relax}}=\cup_{n\geq 1}\mu_{p^{n}}$. Let $F(\mu_{p^{n}})$ (resp. $F(\mu_{p^{\infty}})$) be the field extension of $F$ generated by $\mu_{p^{n}}$ (resp. $\mu_{p^{\infty}}$). The cyclotomic $\mathbb{Z}_{p}$-extension of $F$ is the unique $\mathbb{Z}_{p}$-extension of $F$ contained in $F(\mu_{p^{\infty}})$, and is denoted by $F_{\operatorname{cyc}}$. Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and $\mathcal{O}$ be its valuation ring. Given an integral Galois representation, $\rho\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$, let $\mathbf{T}_{\rho}$ be the underlying $\mathcal{O}$-lattice on which $\operatorname{G}_{F,S}$ acts by $\rho$ and set $\mathbf{A}(\rho)\mathrel{\mathop{\mathchar 58\relax}}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}\left(K/\mathcal{O}\right)$.

Thus [SW18, Lemma 3.2] implies that the definition of $\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$ does not depend on the choice of $S$.

Set $\Gamma\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(F_{\operatorname{cyc}}/F)$ and let

be the associated Iwasawa algebra. Given a finitely generated $\Lambda$-module $\mathfrak{M}$, there is a map of $\Lambda$-modules

with finite kernel and cokernel. Here, $\alpha=\operatorname{rank}_{\Lambda}\mathfrak{M}$ and each power series $f_{j}(T)$ is a distinguished polynomial. In other words, $f_{j}(T)\in\Lambda$ is a monic polynomial all of whose non-leading coefficients are non-units in $\mathcal{O}$. The $\mu$-invariant is the sum $\sum_{i}\mu_{i}$. Given an integral Galois representation

set $\mu^{\operatorname{fn}}(\rho)=\mu^{\operatorname{fn}}(\rho/F_{\operatorname{cyc}})$ to be the $\mu$-invariant of $Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$, the Pontryagin dual of the fine Selmer group $\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$. The following is Conjecture A in [CS05].

The following result provides evidence for the above conjecture.

We remark that the representation $\bar{\rho}$ in the statement of Theorem 2.5 is reducible if $E$ admits a rational $p$-isogeny. In particular, if $E(\mathbb{Q})[p]\neq 0$, then the residual representation is of the form $\bar{\rho}=\left({\begin{array}[]{cc}1&\ast\\ 0&\bar{\chi}_{p}\\ \end{array}}\right)$, where $\bar{\chi}_{p}$ is the mod-$p$ cyclotomic character. Although the $\mu=0$ conjecture for fine Selmer groups was originally stated in [CS05] for the $2$-dimensional Galois representations associated to elliptic curves, the following generalized conjecture is expected to hold (cf. [LS18, Conjecture A, p.74]):

In this section, we shall discuss various criteria for the vanishing of the $\mu$-invariant of the fine Selmer group associated to a Galois representation. As in the previous section, let $p$ be an odd prime, $F$ be a number field and $S$ be a finite set of primes containing all archimedean primes and the primes above $p$. Let $\mathcal{O}$ be the valuation ring of a finite extension $K/\mathbb{Q}_{p}$ with uniformizer $\varpi$ and let $\mathbb{F}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{O}/\varpi$ be its residue field. Fix an integral Galois representation $\rho\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{n}(\mathcal{O})$ on an $\mathcal{O}$-lattice $\mathbf{T}_{\rho}$. Let $\bar{\rho}$ denote the residual representation $\rho\mod{\varpi}$ and set $\mathbf{V}_{\bar{\rho}}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}\mathbb{F}$ to be the underlying vector space on which $\operatorname{G}_{F,S}$ acts via $\bar{\rho}$. Setting $\mathbf{V}_{\rho}=\mathbf{T}_{\rho}\otimes_{\mathcal{O}}K$, recall that $\mathbf{A}(\rho)=\mathbf{V}_{\rho}/\mathbf{T}_{\rho}$. We identify $\mathbf{V}_{\bar{\rho}}$ with $\mathbf{A}(\rho)[\varpi]$. Denote by $\rho^{\wedge}$ the dual of the representation $\rho$ and by $\chi_{p}$ be the $p$-adic cyclotomic character. We set $\rho^{*}$ to denote the representation $\rho^{\wedge}\otimes\chi_{p}$. Recall that $\bar{\chi}_{p}$ is the mod-$p$ cyclotomic character, and $\bar{\rho}^{\wedge}$ is the dual representation to $\bar{\rho}$. Identify $\bar{\rho}^{*}$ with $\bar{\rho}^{\wedge}\otimes\bar{\chi}_{p}$. We note that $\mathbf{T}_{\rho^{*}}$ is isomorphic to the twist $\mathbf{T}_{{\rho}^{\wedge}}\otimes_{\mathcal{O}}\mathcal{O}(\chi_{p})$. Therefore, to summarize, $\rho$ and $\rho^{*}$ are integral representations acting on the $\mathcal{O}$-modules $\mathbf{T}_{\rho}$ and $\mathbf{T}_{\rho^{*}}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Hom}\left(\mathbf{T}_{\rho},\mathcal{O}(\chi_{p})\right)$, respectively. The representations $\bar{\rho}$ and $\bar{\rho}^{*}$ act on the $\mathbb{F}$-vector spaces $\mathbf{V}_{\bar{\rho}}$ and $\mathbf{V}_{\bar{\rho}^{*}}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Hom}\left(\mathbf{V}_{\bar{\rho}},\mathbb{F}(\bar{\chi}_{p})\right)$, respectively. We also note that $\mathbf{A}(\rho)$ is a $p$-divisible $K/\mathcal{O}$-module which is identified with $\mathbf{T}_{\rho}\otimes_{\mathcal{O}}K/\mathcal{O}$.

We refer the reader to [PR00, Appendix B] for a list of cases where the Weak Leopoldt Conjecture is known.

We prove a criterion which shows that the vanishing of $\mu^{\operatorname{fn}}(\rho)$ depends only on the residual representation $\bar{\rho}$. In the special case when $\rho$ is the Galois representation on the $p$-adic Tate module of an elliptic curve, a brief sketch of the proof is given in [Suj10, Proposition 4.6].

The proof of the above theorem is provided later in this section, following Lemma 3.5. The vanishing of $H^{2}(F_{S}/F_{\operatorname{cyc}},\mathbf{V}_{{\bar{\rho}}^{*}})$ is studied in greater detail in this article. Given a $\operatorname{G}_{F,S}$-module $M$, the Iwasawa cohomology group $H^{i}_{\operatorname{Iw}}\left(F_{S}/F_{\operatorname{cyc}},M\right)$ is defined to be the inverse limit $\varprojlim_{n}H^{i}\left(F_{S}/F_{n},M\right)$ with respect to corestriction maps. Given a $\Lambda$-module $M$, we denote the Pontryagin dual by $M^{\vee}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$. Recall that the fine Selmer group of $\mathbf{A}(\rho)$ over $F_{\operatorname{cyc}}$ is $\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})$, and $Y(\mathbf{A}(\rho)/F_{\operatorname{cyc}})\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{R}_{p^{\infty}}(\mathbf{A}(\rho)/F_{\operatorname{cyc}})^{\vee}$.

From the Poitou-Tate sequence associated to $\mathbf{A}(\rho)$, we obtain the following exact sequences (see [PR00, Section 1.3.1] for more details):

We now give a proof of Theorem 3.2. Denote the mod-$p$ Iwasawa algebra by $\Omega\mathrel{\mathop{\mathchar 58\relax}}=\Lambda/\varpi\Lambda$.

The following criterion will be used in establishing new cases of the $\mu=0$ conjecture for the fine Selmer group.

In the remainder of this section, we outline sufficient conditions for the vanishing of $H^{2}(F_{S}/F,\mathbf{V}_{\bar{\rho}^{*}})$. We shall illustrate these conditions for Galois representations that arise from elliptic curves. In section 5, it is shown that these conditions are satisfied for a number of examples of dihedral Galois representations of interest. Given $i\geq 0$, we let ${}^{i}_{S}(\mathbf{V}_{\bar{\rho}})$ be defined as follows

Set $L$ to be the Galois extension of $F$ cut out by $\bar{\rho}$. In other words, it is the extension of $F$ fixed by the kernel of $\bar{\rho}$. Set $H_{L}$ to be the mod-$p$ class group $H_{L}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Cl}(L)\otimes\mathbb{F}_{p}$. Denote by $H_{L}^{\prime}$ the maximal quotient of $H_{L}$ such that the primes of $L$ that lie above $S$ are split in the corresponding subextension of the Hilbert class field of $L$. Note that $H_{L}^{\prime}$ is a stable with respect to the natural action of $G=\operatorname{Gal}(L/F)$ on $H_{L}$. Given a prime $v$ of $F$, set $\operatorname{G}_{v}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(\bar{F}_{v}/F_{v})$, which is viewed as a subgroup of $\operatorname{Gal}(\bar{F}/F)$ after fixing an embedding $\bar{F}\to\bar{F_{v}}$. Set $\rho_{|v}$ (resp. $\bar{\rho}_{|v}$) to denote the restriction of $\rho$ (resp. $\bar{\rho}$) to $\operatorname{G}_{v}$.

Let $E$ be an elliptic curve defined over a number field $F$ and let $p$ be an odd prime number. Let $\rho\mathrel{\mathop{\mathchar 58\relax}}\operatorname{G}_{F,S}\rightarrow\operatorname{GL}_{2}(\mathbb{Z}_{p})$ be the Galois representation on the $p$-adic Tate module of $E$ and let $\bar{\rho}$ be its residual representation. Let $G$ denote the Galois group $\operatorname{Gal}(L/F)$. For a prime $v$ of $F$, denote by $F_{v}$ the completion of $F$ at $v$ and by $\mathbb{F}_{v}$ the residue field of $F_{v}$.

We note here that for elliptic curves $E_{/\mathbb{Q}}$ with Mordell Weil rank $0$ and the Tate-Shafarevich group $\Sh(E/\mathbb{Q})[p]=0$, the upper bound given by [PS21, Theorem 4.2] provides sufficient conditions for the vanishing of $\operatorname{Hom}_{G}\left(H_{L},E[p]\right)$. To illustrate this, consider the following example.