2. Review of some commutative algebra

We collect certain commutative algebraic results that will be required for the discussion of the paper. Throughout this section, $\Lambda$ will always denote a regular local ring. (For our arithmetic purposes, we are usually concerned with regular local rings of the form $\mathcal{O}[[ T_1, T_2,..., T_r ]]$ , where $\mathcal{O}$ is some integral domain which is finite flat over $\mathbb{Z}_{p}$ .) It’s a standard fact that $\Lambda$ is therefore a unique factorisation domain (cf. [Reference Matsumura and Reid34, theorem 20·3]). In particular, it follows from [Reference Matsumura and Reid34, theorem 20·1] that every prime ideal of $\Lambda$ of height one is principal.

Recall that a finitely generated $\Lambda$ -module M is said to be torsion if for every element $m\in M$ , there exists $x\in \Lambda$ such that $x m=0$ . Equivalently, this is saying that $\textrm{Hom}_{\Lambda}(M,\Lambda)=0$ . The module M is said to be pseudo-null if the localisation $M_{\mathfrak{p}}$ of M at every prime ideal $\mathfrak{p}$ of height $\leq 1$ is trivial. The latter is equivalent to $\textrm{Ext}_{\Lambda}^i(M,\Lambda)=0$ for $i=0,1$ (for instance, see [Reference Matsumura and Reid34, chapter. 5] or [Reference Neukirch, Schmidt and Wingberg40, chapter. V, section 1]).

We now present a useful lemma (compare with [Reference Perrin-Riou44, section 1·3, lemme 4])

Proof. We begin proving assertion (i). In fact, we shall establish a slightly stronger assertion: namely, if z is an element of $\Lambda$ which is coprime to x, then $\Lambda/(x,z)$ is a torsion $\Omega$ -module. Since z is coprime to x, it does not lie in the ideal (x). Hence $z+(x)$ is a nonzero element of $\Omega$ , and it plainly annihilates $\Lambda/(x,z)$ . Therefore, this proves our first assertion.

For the proof of (ii), we shall write N for either M[x] or $M/x$ . Since the module N is annihilated by x, it may be viewed as a $\Omega$ -module. Consider the following spectral sequence

(1) \begin{equation} \textrm{Ext}^i_\Omega\big(N, \textrm{Ext}^j_\Lambda(\Omega, \Lambda)\big) \Longrightarrow \textrm{Ext}^{i+j}_\Lambda(N, \Lambda)\end{equation}

(cf. [Reference Weibel53, exercise 5·6·3]). From the $\Lambda$ -free resolution

\[ 0\longrightarrow \Lambda\longrightarrow \Lambda \longrightarrow \Omega \longrightarrow 0\]

of $\Omega$ , we see that

\[\textrm{Ext}^j_\Lambda(\Omega, \Lambda)=\left\{ \begin{array}{ll} \Omega, & \quad\mbox{if $j=1$,} \\[5pt] 0, & \quad\hbox{otherwise.} \\ \end{array} \right.\]

Therefore, the spectral sequence (1) degenerates yielding the isomorphism

\[ \textrm{Ext}^i_{\Omega}(N, \Omega) \cong \textrm{Ext}^{i+1}_{\Lambda}(N, \Lambda)\]

for $i\geq 0$ . In particular, we have

\[ \textrm{Hom}_{\Omega}(N, \Omega)=\textrm{Ext}^{1}_{\Lambda}(N, \Lambda)=0,\]

where the final zero follows from the assumption that M is pseudo-null over $\Lambda$ . This in turn implies that N is torsion over $\Omega$ and we have the second assertion.

The final assertion follows from [Reference Perrin-Riou44, section 1·3, lemme 2] or [Reference Lim27, corollary 4·13].

Now, let M be a finitely generated torsion $\Lambda$ -module. By [Reference Neukirch, Schmidt and Wingberg40, proposition 5·1·7], there is a pseudo-isomorphism

(2) \begin{equation}\varphi\,:\, M \longrightarrow \bigoplus_{i\in I}\Lambda/x_i^{n_i},\end{equation}

where I is a finite indexing set, each $x_i$ is a generator of a prime ideal of height one and $n_i$ is a non-negative integer. Note that the prime ideals $\Lambda x_i$ and integers $n_i$ are determined by the module M.

Proof. Let $P_1=\ker \varphi$ , $P_2=\textrm{coker}\,\varphi$ and $Q=\textrm{im}\,\varphi$ , where $\varphi$ is given as in (2). From which, we have the following two short exact sequences

\[ 0\longrightarrow P_1\longrightarrow M\longrightarrow Q \longrightarrow 0,\]

\[ 0\longrightarrow Q\longrightarrow \bigoplus_{i\in I}\Lambda/x_i^{n_i} \longrightarrow P_2\longrightarrow 0.\]

From which, we have

\[ P_1/x\longrightarrow M/x\longrightarrow Q/x \longrightarrow 0,\]

\[ P_2[x] \longrightarrow Q/x\longrightarrow \bigoplus_{i\in I}\Lambda/(x_i^{n_i},x) \longrightarrow P_2/x\longrightarrow 0.\]

By Lemma 2·1, the modules $P_1/x$ , $P_2[x]$ and $\bigoplus_{i\in I}\Lambda/(x_i^{n_i},x)$ are torsion over $\Omega$ . Putting these observations into the above two exact sequences, we see that so is $M/x$ .

3. Fine Selmer groups of abelian varieties

3·1. Fine Selmer groups

We begin reviewing the fine Selmer groups of abelian varieties following [Reference Coates and Sujatha3, Reference Coates and Sujatha4, Reference Hachimori12, Reference Lim28, Reference Lim and Murty31, Reference Wuthrich54–Reference Wuthrich56]. Fix an odd prime p. Let A be an abelian variety defined over a number field F. Let S be a finite set of primes of F containing the primes above p, the bad reduction primes of A and the infinite primes. Denote by $F_S$ the maximal algebraic extension of F which is unramified outside S. For every extension $\mathcal{L}$ of F contained in $F_S$ , we write $G_S(\mathcal{L})=\textrm{Gal}(F_S/\mathcal{L})$ , and denote by $S(\mathcal{L})$ the set of primes of $\mathcal{L}$ above S.

Let L be a finite extension of F contained in $F_S$ . The fine Selmer group of A over L is defined by

\[ R(A/L) =\ker\left(H^1(G_S(L),A[p^{\infty}])\longrightarrow \bigoplus_{v\in S(L)} H^1(L_v, A[p^{\infty}])\right).\]

We remark that the above definition is independent of the choice of S (see [Reference Lim and Murty31, lemma 4·1]). Just as the classical p-primary Selmer group, the fine Selmer group sits in the following analogous short exact sequence

(3)

where $\mathcal{M}(A/F)$ is the fine (p-)Mordell–Weil group and is the fine (p-) Tate–Shafarevich group in the sense of Wuthrich [Reference Wuthrich56]. The fine Mordell–Weil group $\mathcal{M}(A/F)$ is defined to be the subgroup of $A(F)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}$ consisting of those elements which are mapped to zero in $A(F_v)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}/\mathbb{Z}_{p}$ for all primes v above p. It can be shown that $\mathcal{M}(A/F)$ injects into $R(A/F)$ . The fine Tate–Shafarevich group is then defined to be the cokernel of this injection (see [Reference Wuthrich56, section 2] for details).

Let $F_\infty$ be a (not necessarily cyclotomic) $\mathbb{Z}_{p}$ -extension of F, whose Galois group $\textrm{Gal}(F_\infty/F)$ will be denoted by $\Gamma$ . If $\Gamma_n$ denotes the unique subgroup of $\Gamma$ of index $p^n$ , we write $F_n$ for the fixed field of $\Gamma_n$ . The fine Selmer group of A over $F_\infty$ is defined to be $R(A/F_\infty) = \displaystyle \mathop{\varinjlim}\limits_n R(A/F_n)$ which comes naturally equipped with a $\mathbb{Z}_{p}[[ \Gamma ]]$ -module structure. The $\mathbb{Z}_{p}[[ \Gamma ]]$ -modules $\mathcal{M}(A/F_\infty)$ and are similarly defined by taking limit of the corresponding objects over the intermediate subfields. We shall write $Y(A/F_\infty)$ for the Pontryagin dual of $R(A/F_\infty)$ . Upon taking direct limit of the sequence (3) and following up by taking Pontryagin dual, we obtain

(4)

It is not difficult to verify that the modules occurring in the exact sequence are finitely generated over $\mathbb{Z}_{p}[[ \Gamma ]]$ (for instance, see [Reference Lim28, Lemma 3·2]). In fact, one expects more (see [Reference Lim28, Reference Perrin–Riou45, Reference Wuthrich54]).

Conjecture 3·1 (Conjecture Y) Let A be an abelian variety defined over a number field F and $F_\infty$ a $\mathbb{Z}_{p}$ -extension of F. Then $Y(A/F_\infty)$ is torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ .

Remark 3·2. (1) When $F_\infty$ is the cyclotomic $\mathbb{Z}_{p}$ -extension, the above conjecture is a consequence of a conjecture of Mazur [Reference Mazur36] and Schneider [Reference Schneider51] on the structure of the classical Selmer groups. The latter is known when A is an elliptic curve over $\mathbb{Q}$ with good reduction at p and F is an abelian extension of $\mathbb{Q}$ (see [Reference Kato18, Reference Kobayashi22]).

(2) Suppose that E is an elliptic curve defined over $\mathbb{Q}$ with complex multiplication given by a imaginary quadratic field K at which p split completely in $K/\mathbb{Q}$ . Let $K_\infty$ be the $\mathbb{Z}_{p}$ -extension of K which is unramified outside one of the primes of K above p. Then the validity of Conjecture Y is a consequence of a result of Coates (see [Reference de Shalit9, chapter IV, corollary 1·8]; also see the recent papers [Reference Choi, Kezuka and Li1, Reference Kezuka19]).

(3) Suppose that E is an elliptic curve defined over $\mathbb{Q}$ , and $K^{\textrm{ac}}$ the anti-cyclotomic $\mathbb{Z}_{p}$ -extension of an imaginary quadratic field K. Then the $\mathbb{Z}_{p}[[ \textrm{Gal}(K^{\textrm{ac}}/K) ]]$ -torsionness of $Y(A/K^{\textrm{ac}})$ is known in many cases (for instance, see [Reference Longo and Vigni33, Reference Pollack and Weston47]).

We record two results taken from [Reference Lim28] which serve as support for Conjecture Y, and will play some role in the subsequent discussion of the paper.

Theorem 3·3. Let A be an abelian variety defined over a number field F. Let $F_{\infty}$ be a $\mathbb{Z}_{p}$ -extension of F. If $Y(A/F)$ is finite, then $Y(A/F_\infty)$ is torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ .

Proof. See [Reference Lim28, corollary 3·5].

Proposition 3·4. Let A be an abelian variety defined over a number field F, and $F_{\infty}$ a $\mathbb{Z}_{p}$ -extension of F. Suppose that is finite for every n. Then is a cotorsion $\mathbb{Z}_{p}[[ \Gamma ]]$ -module.

Proof. See [Reference Lim28, proposition 4·1].

We now record a corollary of the preceding proposition.

Corollary 3·5 Let A be an abelian variety defined over a number field F, and $F_{\infty}$ a $\mathbb{Z}_{p}$ -extension of F. Suppose that the following statements are valid.

1. (a) is finite for every n.

2. (b) $A(F_\infty)$ is a finitely generated abelian group.

Then $Y(A/F_{\infty})$ is a torsion $\mathbb{Z}_{p}[[ \Gamma ]]$ -module.

Proof. From Proposition 3·4, we see that is torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ under hypothesis (a). Hypothesis (b) tells us that $\mathcal{M}(A/F_\infty)^\vee$ is torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ . The conclusion follows from these and the short exact sequence (4).

3·2. Connection with the pseudo-nullity conjecture of Coates–Sujatha

We now study the relation between our Conjecture Y and the Conjecture B of Coates–Sujatha [Reference Coates and Sujatha4, conjecture B]. As a start, we recall their conjecture, which for now is stated for $\mathbb{Z}_{p}^2$ -extensions; see Conjecture 7·4 below for the general version.

Conjecture 3·6 (Conjecture B). Let $L_{\infty}$ be a $\mathbb{Z}_{p}^2$ -extension of F which contains $F^{\textrm{cyc}}$ . Then $Y(A/L_\infty)$ is pseudo-null over $\mathbb{Z}_{p}[[ G ]]$ , where $G=\textrm{Gal}(L_\infty/F)$ .

Remark 3·7. In [Reference Coates and Sujatha4], they formulated their conjecture under the extra assumption which is their so-called Conjecture A (see [Reference Coates and Sujatha4, conjecture A]). In this paper, we do not require this extra hypothesis, and so the above formulation suffices.

Retaining the above notation, we denote by $\Phi(L_\infty/F)$ the set of all $\mathbb{Z}_{p}$ -extensions of F contained in $L_\infty$ . For each $\mathcal{L}\in \Phi(L_\infty/F)$ , write $\Gamma_{\mathcal{L}}=\textrm{Gal}(\mathcal{L}/F)$ and $H_\mathcal{L}=\textrm{Gal}(L_\infty/\mathcal{L})$ . Fix a topological generator $h_{\mathcal{L}}$ of $H_\mathcal{L}$ . Then $h_{\mathcal{L}}-1$ generates a prime ideal of $\mathbb{Z}_{p}[[ G ]]$ of height one with

\[ \mathbb{Z}_{p}[[ G ]]/(h_{\mathcal{L}}-1) \cong \mathbb{Z}_{p}[[ \Gamma_\mathcal{L} ]].\]

We can now establish the following observation as mentioned in the introduction.

Proposition 3·8. Let A be an abelian variety defined over a number field F, and $L_{\infty}$ a $\mathbb{Z}_{p}^2$ -extension of F which contains $F^{\textrm{cyc}}$ . Suppose that Conjecture B is valid for A over $L_\infty$ , or in other words, $Y(A/L_\infty)$ is pseudo-null over $\mathbb{Z}_{p}[[ G ]]$ . Then $Y(A/\mathcal{L})$ is a torsion $\mathbb{Z}_{p}[[ \textrm{Gal}(\mathcal{L}/F) ]]$ -module for every $\mathcal{L}\in\Phi(L_\infty/F)$ .

Proof. From the following commutative diagram

we see that $\ker s$ is contained in $H^1(H_\mathcal{L},A(L_\infty))$ which is cofinitely generated over $\mathbb{Z}_{p}$ . Upon taking Pontryagin dual, we obtain a map

\[ Y(A/F_\infty)_{H_\mathcal{L}}\longrightarrow Y(A/\mathcal{L}), \]

whose cokernel is finitely generated over $\mathbb{Z}_{p}$ . Therefore, for a given $\mathcal{L}\in\Phi(L_\infty/F)$ , whenever $Y(A/L_\infty)_{H_\mathcal{L}}$ is torsion over $\mathbb{Z}_{p}[[ \Gamma_{\mathcal{L}} ]]$ , so is $Y(A/\mathcal{L})$ .

Now, in view of the hypothesis that $Y(A/L_\infty)$ is pseudo-null over $\mathbb{Z}_{p}[[ G ]]$ , we may apply Lemma 2·1(ii) to conclude that $Y(A/F_\infty)_{H_\mathcal{L}}$ is torsion over $\mathbb{Z}_{p}[[ \Gamma_{\mathcal{L}} ]]$ . Combining this with the assertion in the previous paragraph, we have the conclusion.

We now state and prove the following.

Theorem 3·9. Let A be an abelian variety defined over a number field F, and $L_{\infty}$ a $\mathbb{Z}_{p}^2$ -extension of F which contains $F^{\textrm{cyc}}$ . Suppose that $Y(A/F^{\textrm{cyc}})$ is torsion over $\mathbb{Z}_{p}[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ . Then for all but finitely many $\mathcal{L}\in\Phi(L_\infty/F)$ , Conjecture Y is valid for $Y(A/\mathcal{L})$ or, in other words, $Y(A/\mathcal{L})$ is torsion over $\mathbb{Z}_{p}[[ \Gamma_\mathcal{L} ]]$ .

Proof. By [Reference Lim27, proposition 7·2], it follows from the $\mathbb{Z}_{p}[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ -torsionness of $Y(A/F^{\textrm{cyc}})$ that $Y(A/L_\infty)$ is torsion over $\mathbb{Z}_{p}[[ G ]]$ . The structure theorem of $\mathbb{Z}_{p}[[ G ]]$ -modules (cf. [Reference Neukirch, Schmidt and Wingberg40, proposition 5·1·7]) then implies that there is a pseudo-isomorphism

\[ Y(A/L_\infty) \sim \bigoplus_{i\in I} \mathbb{Z}_{p}[[ G ]]/Q_i^{n_i}\]

of $\mathbb{Z}_{p}[[ G ]]$ -modules, where I is some finite indexing set and each $Q_i$ is a (principal) prime ideal of $\mathbb{Z}_{p}[[ G ]]$ of height one. Since there are only finitely many $Q_i$ ’s, the element $h_{\mathcal{L}}-1$ is coprime to these $Q_i$ ’s for all but finitely many $\mathcal{L}\in\Phi(L_\infty/F)$ . For each of such element $h_{\mathcal{L}}-1$ , it then follows from Lemma 2·2 that $Y(A/L_\infty)_{H_\mathcal{L}}=Y(A/L_\infty)/(h_\mathcal{L}-1)$ is torsion over $\mathbb{Z}_{p}[[ \Gamma_{\mathcal{L}} ]]$ . By a similar argument to that in Proposition 3·8, we see that $Y(A/\mathcal{L})$ is torsion over $\mathbb{Z}_{p}[[ \Gamma_{\mathcal{L}} ]]$ for every such $\mathcal{L}$ . This yields the required conclusion of the theorem.

We record one case, where we can obtain many cases of validity of Conjecture Y.

Corollary 3·10. Let E be an elliptic curve defined over $\mathbb{Q}$ and F a finite abelian extension of $\mathbb{Q}$ . Let $L_{\infty}$ be a $\mathbb{Z}_{p}^2$ -extension of F which contains $F^{\textrm{cyc}}$ . Then for all but finitely many $\mathcal{L}\in\Phi(L_\infty/F)$ , $Y(E/\mathcal{L})$ is cotorsion over $\mathbb{Z}_{p}[[ \Gamma_\mathcal{L} ]]$ .

Proof. A well-known theorem of Kato [Reference Kato18] (also see [Reference Kobayashi22]) asserts that $R(E/F^{\textrm{cyc}})$ is cotorsion over $\mathbb{Z}_{p}[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ . The corollary is now an immediate consequence of this and Theorem 3·9.

We also refer readers to Section 6 for examples, where F is not necessarily abelian over $\mathbb{Q}$ but Theorem 3·9 applies.

4. Fine Selmer groups of elliptic modular forms

As before, p will denote a fixed odd prime. Let f be a normalised new cuspidal modular eigenform of even weight $k\geq 2$ , level N and nebentypus $\epsilon$ . Let $\mathcal{K}_f$ be the number field obtained by adjoining all the Fourier coefficients of f to $\mathbb{Q}$ . Throughout, we shall fix a prime $\mathfrak{p}$ of $\mathcal{K}_f$ above p, and let $V_f$ denote the corresponding two-dimensional $\mathcal{K}_{f,\mathfrak{p}}$ -linear Galois representation attached to f in the sense of Deligne [Reference Deligne8]. Writing $\mathcal{O}=\mathcal{O}_{\mathcal{K}_{f,\mathfrak{p}}}$ for the ring of integers of $\mathcal{K}_{f,\mathfrak{p}}$ , we fix a $\textrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ -stable $\mathcal{O}$ -lattice $T_f$ in $V_f$ . We then set $A_f = V_f/T_f$ . Note that $A_f$ is isomorphic to $\mathcal{K}_{f,\mathfrak{p}}/\mathcal{O}\oplus \mathcal{K}_{f,\mathfrak{p}}/\mathcal{O}$ as $\mathcal{O}$ -modules.

Let F be a finite extension of $\mathbb{Q}$ . Denote by S a finite set of primes of F containing those dividing pN and all the infinite primes. Let $F_\infty$ be a $\mathbb{Z}_{p}$ -extension of F. Following [Reference Jha16, Reference Jha and Sujatha17], we define the fine Selmer group of $A_f$ over $F_\infty$ to be $\displaystyle \mathop{\varinjlim}\limits_nR(A_f/F_n)$ , where $F_n$ is the intermediate subfield of $F_\infty/F$ with $|F_n:F|=p^n$ and $R(A_f/F_n)$ is defined by

\[ R(A_f/F_n) =\ker\left(H^1(G_S(F_n),A_f)\longrightarrow \bigoplus_{v\in S(F_n)} H^1(F_{n,v}, A_f)\right).\]

The Pontryagin dual of $R(A_f/F_\infty)$ is then denoted by $Y(A_f/F_\infty)$ . As before, one can similarly show that $Y(A_f/F_\infty)$ is finitely generated over $\mathcal{O}[[ \Gamma ]]$ . The following conjecture is the natural analogue of Conjecture Y for modular forms.

We now present natural analogue of Proposition 3·8 and Theorem 3·9 for the fine Selmer group of a modular form. As a start, we recall the following analogue of Conjecture B which was first studied by Jha [Reference Jha16].

As in Section 3, denote by $\Phi(L_\infty/F)$ the set of all $\mathbb{Z}_{p}$ -extensions of F contained in $L_\infty$ . For each $\mathcal{L}\in \Phi(L_\infty/F)$ , write $\Gamma_{\mathcal{L}}=\textrm{Gal}(\mathcal{L}/F)$ and $H_\mathcal{L}=\textrm{Gal}(L_\infty/\mathcal{L})$ . From which, a similar argument to that in Proposition 3·8 yields the following.

Similarly, we can establish the following by a similar argument to that in Theorem 3·9.

Combining the above with Kato’ result [Reference Kato18], we have the following analogue of Corollary 3·10.

We end the section by establishing a control theorem for the fine Selmer groups of elliptic modular forms, which is the analogue to that in [Reference Lim28, theorem 3·3] proved for abelian varieties. From now on, we let $S_{fd}$ denote the set of primes in S which do not split completely in $F_\infty/F$ . We shall also write $W(L) = W^{\textrm{Gal}(F_S/L)}$ for any $F \subseteq L \subseteq F_S$ . Similarly, for each $v \in S$ , we write $W(L) = W^{\textrm{Gal}(\bar{F}_v/L)}$ for any $F_v \subseteq L \subseteq \bar{F}_v$ .

Theorem 4·6. Let f be a normalized new cuspidal modular eigenform of even weight $k\geq 2$ , level N and nebentypus $\epsilon$ . Write $A_f$ for the Galois module attached to f defined as above. Let $F_{\infty}$ be a $\mathbb{Z}_{p}$ -extension of F with $F_n$ being the intermediate subfield satisfying $|F_n\,:\,F|=p^n$ . Suppose that either of the following statements is valid.

1. (i) For every prime $v\in S_{fd}$ and $v_n$ of $F_n$ dividing v, then $H^0\left(F_{n,v_n}, A_f\right)$ is finite.

2. (ii) $K_{f,\mathfrak{p}}\cap \mathbb{Q}_{p}(\mu_{p^\infty})=\mathbb{Q}_{p}$ and $H^0(F_v, A_f)$ is finite for every prime $v\in S_{fd}$ .

Then the restriction map

\[r_n\,:\, R(A_f/F_n) \longrightarrow R(A_f/F_{\infty})^{\Gamma_n}\]

has finite kernel and cokernel which are bounded independently of n.

Before proving Theorem 4·6, we establish the following preliminary lemma.

Proof. Suppose that hypothesis (a) is valid. Since M is plainly torsion as a $\mathcal{O}[[ \Gamma_n ]]$ -module, one therefore has

(5) \begin{equation} 0 = \textrm{rank}_{\mathcal{O}[[ \Gamma_n ]]}(M) = \textrm{rank}_{\mathcal{O}}\big(M_{\Gamma_n}\big)- \textrm{rank}_{\mathcal{O}}\big(M^{\Gamma_n}\big), \end{equation}

where the second equality follows from [Reference Neukirch, Schmidt and Wingberg40, proposition 5·3·20]. Combining these observations, we see that each $M^{\Gamma_n}$ is finite. This in turn implies that $M^{\Gamma_n}$ is contained in $M[p^\infty]$ . But since M is finitely generated over $\mathcal{O}$ , the latter is finite, and hence we conclude that $M^{\Gamma_n}$ is finite with order bounded independently of n. Finally, since $\Gamma_n\cong \mathbb{Z}_{p}$ , we have the identification $M^{\Gamma_n}\cong H_1(\Gamma_n,M)$ , thus proving the lemma under the validity of the hypothesis (a).

Now suppose that hypothesis (b) is valid. Identify $\mathcal{O}[[ \Gamma ]]$ with $\mathcal{O}[[ T ]]$ under a choice of generator of $\Gamma$ . Since $M_{\Gamma}$ is finite, we see that T has to be coprime to the characteristic polynomial of M. Since $K_{f,\mathfrak{p}}\cap \mathbb{Q}_{p}(\mu_{p^\infty})=\mathbb{Q}_{p}$ , every other cyclotomic polynomial is irreducible over $\mathcal{O}[[ T ]]$ . Since such a polynomial has degree $\geq p-1$ , it has to be coprime to the characteristic polynomial of M. Consequently, $M_{\Gamma_n}$ is finite for every n. We are therefore in the situation of hypothesis (a), and so the conclusion follows from the above discussion.

We can now give the proof of Theorem 4·6.

Proof of Theorem 4·6. Consider the following commutative diagram

with exact rows. Since $\Gamma_n$ has p-cohomological dimension 1, the restriction-inflation sequence tells us that $h_n$ is surjective and that $\ker h_n = H^1\big(\Gamma_n, A_f(F_\infty)\big)$ . It therefore remains to show the finiteness and boundness of $\ker h_n$ and $\ker g_{n}$ .

We begin by showing the finiteness and boundness of $\ker g_{n}$ . For each $v_n\in S(F_n)$ , fix a prime of $F_\infty$ above $v_n$ which is denoted by $w_n$ , and write v for the prime of F below $v_n$ . Write $\Gamma_{w_n}$ for the decomposition group of $w_n$ in $\Gamma$ . By the Shapiro’s lemma and the restriction-inflation sequence, we have

\[ \ker\left(\bigoplus_{v_n\in S(F_n)} g_{n,v_n} \right) = \bigoplus_{v_n\in S(F_n)} H^1\left(\Gamma_{w_n}, A_f(F_{\infty,v_n})\right).\]

If v is a prime of F below $w_n$ such that v splits completely in $F_\infty/F$ , then $\Gamma_{w_n}=0$ and so one has $H^1\left(\Gamma_{w_n}, A_f\left(F_{\infty,w_n}\right)\right) =0$ . Thus, it remains to consider the primes $v\in S$ which do not split completely in $F_\infty/F$ . Since S is a finite set, the number of such possibly nonzero summands $\bigoplus H^1\left(\Gamma_{w_n}, A_f\left(F_{\infty,w_n}\right)\right)$ is therefore finite and bounded independently of n. Hence it remains to show that each $H^1\left(\Gamma_{w_n}, A_f\left(F_{\infty,w_n}\right)\right)$ is finite and bounded independently for those primes lying above v which do not decompose completely in $F_\infty/F$ . For this, one just needs to verify that either (a) or (b) of Lemma 4·8 is valid. We note that hypothesis (a) is a direct consequence of hypothesis (i) of the theorem. It remains to show that hypothesis (ii) of our theorem yields (b) of the said lemma. For this, it suffices to show that $\textrm{corank}_{\mathcal{O}}\left(A_f\left(F_{\infty,w_n}\right)\right)<p-1$ . We first consider the case when v does not divide p. In this setting, $F_{\infty,w_n}$ is the cyclotomic $\mathbb{Z}_{p}$ -extension of $F_v$ which is unramified. Since v divides N, $A_f$ cannot be an unramified $\textrm{Gal}(\bar{F}_v/F_v)$ -module and so $A_f$ cannot be realised over $F_{\infty,w_n}$ . Therefore, we must have $\textrm{corank}_{\mathcal{O}}\left(A_f\left(F_{\infty,w_n}\right)\right)\leq 1$ . As the prime p is assumed to be odd, this in turn implies that $\textrm{corank}_{\mathcal{O}}\left(A_f\left(F_{\infty,w_n}\right)\right)<p-1$ . Now suppose that v divides p. It is well-known that $F_v(A_f)$ is a p-adic Lie extension of $F_v$ of dimension at least 2 (see [Reference Ribet49]). It follows from this that $A_f$ cannot be realised over $F_{\infty,w_n}$ , and so we have $\textrm{corank}_{\mathcal{O}}\left(A_f\left(F_{\infty,w_n}\right)\right)\leq 1$ .

We now show that $\ker h_n = H^1(\Gamma_n, A_f(F_\infty))$ is finite and bounded independent of n. Now since $F_\infty/F$ is a $\mathbb{Z}_{p}$ -extension, it must have at least one prime v above p which is ramified in $F_\infty/F$ . In view of the discussion in the preceding paragraph, we have $H^0(F_{n,w},A_f)$ being finite, where w is a prime of $F_\infty$ above v. This in turn implies that $H^0(G_S(F_n), A_f)$ is finite for every n. The desired conclusion is now a consequence of Lemma 4·8.

Theorem 4·6 has the following natural corollary.

5. Hida deformations

Let us briefly introduce certain notion and facts arising from the work of Hida [Reference Hida15]. Denote by $\Gamma'$ the group of diamond operators for the tower of modular curves $\{Y_1(p^r)\}$ . There is a natural identification of $\Gamma'$ with $1+p\mathbb{Z}_{p}$ which we denote by $\kappa:\Gamma'\stackrel{\sim}{\longrightarrow}1+p\mathbb{Z}_{p}$ . For an integer N coprime to p, we write $h^{\textrm{ord}}_{\mathcal{F}}$ for the quotient of the universal ordinary Hecke algebra with conductor N corresponding to an ordinary $\mathbb{Z}_{p}[[ \Gamma' ]]$ -adic eigenform $\mathcal{F}$ . The ring $h^{\textrm{ord}}_{\mathcal{F}}$ is a local integral domain which is finite flat over $\mathbb{Z}_{p}[[ \Gamma' ]]$ . In [Reference Hida15], Hida constructed an irreducible representation

\[ \rho\,:\,\textrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\longrightarrow \textrm{Aut}_{h^{\textrm{ord}}_{\mathcal{F}}}(\mathcal{T}_{\mathcal{F}})\]

which is unramified outside Np, and where $\mathcal{T}_{\mathcal{F}}$ is a finitely generated torsion-free module of generic rank two over $h^{\textrm{ord}}_{\mathcal{F}}$ . We now impose two standing assumptions on the pair $\left(\mathcal{T}_{\mathcal{F}}, h^{\textrm{ord}}_{\mathcal{F}}\right)$ .

1. (H1) The ring $h^{\textrm{ord}}_{\mathcal{F}}$ is isomorphic to $\mathcal{O}[[ \Gamma' ]]$ for the ring of integers $\mathcal{O}$ of a finite extension of $\mathbb{Q}_{p}$ .

2. (H2) Denote by $\mathfrak{m}$ the maximal ideal of $h^{\textrm{ord}}_{\mathcal{F}}$ . The residual representation $\bar{\rho}\longrightarrow \textrm{Aut}_{h^{\textrm{ord}}_{\mathcal{F}}/\mathfrak{m}}(\mathcal{T}_{\mathcal{F}}/\mathfrak{m} \mathcal{T}_{\mathcal{F}})$ is absolutely irreducible as a $\textrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ -module.

Under $\textbf{(H2)}$ , the module $\mathcal{T}_{\mathcal{F}}$ is free over $h^{\textrm{ord}}_{\mathcal{F}}$ (see [Reference Mazur and Tilouine38, section 2, corollary 6]). Let $\mathfrak{X}_{\textrm{arith}}\left(h^{\textrm{ord}}_{\mathcal{F}}\right)$ be the set consisting of $\mathbb{Z}_{p}$ -algebra homomorphism $\lambda:h^{\textrm{ord}}_{\mathcal{F}}\longrightarrow \bar{\mathbb{Q}}_p$ which satisfies the property that there exists an open subgroup U of $\Gamma'$ and a non-negative integer w such that $\lambda(u) = \kappa^w(u)$ for every $u\in U$ . For each of such $\lambda$ , we shall write $w_\lambda$ for the integer w appearing the above definition. We also write $P_\lambda$ for the kernel of $\lambda$ which is a principal prime ideal of $h^{\textrm{ord}}_{\mathcal{F}}$ of height one. In fact, identifying $h^{\textrm{ord}}_{\mathcal{F}}\cong \mathcal{O}[[ T ]]$ (recall that we are assuming $\textbf{(H1)}$ ), the prime ideal $P_\lambda$ can be viewed as lying above the prime ideal of $\mathbb{Z}_{p}[[ T ]]$ generated by $(1+p)^{w_\lambda}-(1+T)$ . We shall write $p_\lambda$ for a generator of the prime ideal $P_\lambda$ .

By Hida theory, for each $\lambda\in\mathfrak{X}_{\textrm{arith}}\left(h^{\textrm{ord}}_{\mathcal{F}}\right)$ , there exists a normalised cuspidal eigenform $f_{\lambda}$ of weight $w_\lambda+2$ such that $\mathcal{T}_{\mathcal{F}}/P_\lambda\cong T_{f_{\lambda}}$ , where $T_{f_\lambda}$ is the lattice of the Galois representation attached to $f_{\lambda}$ as in the sense of Deligne. Here $T_{f_\lambda}$ is a free $\mathcal{O}_\lambda$ -module of rank two, where $\mathcal{O}_{\lambda}= h^{\textrm{ord}}_{\mathcal{F}}/P_{\lambda}$ .

From now on, to simplify notation, we sometimes write $\mathcal{T}$ for $\mathcal{T}_{\mathcal{F}}$ and $\mathcal{R}$ for $h^{\textrm{ord}}_{\mathcal{F}}$ . Recall that $\mathcal{R}\cong\mathcal{O}[[ \Gamma' ]]$ by our standing assumption $\textbf{(H1)}$ . Set $\mathcal{A} = \mathcal{T}\otimes_{\mathcal{R}}(\mathcal{R}^{\vee})$ . The next lemma is left to the reader as an exercise.

Let F be a finite extension of $\mathbb{Q}$ . Denote by S a finite set of primes of F containing those dividing pN and all the infinite primes. For a $\mathbb{Z}_{p}$ -extension $F_\infty$ of F, the fine Selmer group $R(\mathcal{A}/F_\infty)$ of $\mathcal{A}$ over $F_\infty$ is defined to be $\displaystyle \mathop{\varinjlim}\limits_n R(\mathcal{A}/F_n)$ , where $F_n$ is the intermediate subfield of $F_\infty/F$ with $|F_n:F|=p^n$ and $R(\mathcal{A}/F_n)$ is given by

\[ R(\mathcal{A}/F_n) =\ker\left(H^1(G_S(F_n),\mathcal{A})\longrightarrow \bigoplus_{v_n\in S(F_n)} H^1(F_{n,v_n}, \mathcal{A})\right).\]

We can now state the following analogue of Conjecture Y for $\mathcal{A}$ .

We now prove a “hortizontal” analogue of Theorems 3·9 and 4·4. In the subsequent discussion, we write $S'_{cd}$ for the set of primes of S which does not divide p and split completely in $F_\infty/F$ .

Proof. Consider the following commutative diagram

with exact rows and vertical maps induced by the following short exact sequence

\[ 0 \longrightarrow A_{f_\eta}\longrightarrow \mathcal{A}\stackrel{p_\eta}{\longrightarrow} \mathcal{A}\longrightarrow 0. \]

We shall first show that the kernel and cokernel of $r_\eta$ are cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ . To start off, we see that $h_\eta$ is surjective with $\ker h_\eta = \mathcal{A}(F_\infty)/P_\eta$ . Since $\mathcal{A}(F_\infty)/P_\eta$ is cofinitely generated over $\mathcal{O}_\eta$ , it is cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ . It therefore remains to show that $\ker l$ is cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ . For this, we decompose $l=\oplus_{w\in S(F_\infty)}l_w =\oplus_{v\in S}(\oplus_{w|v}l_w)$ and show that $\ker\big(\oplus_{w|v}l_w\big)$ is cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ for each $v\in S$ . For each w, we have $\ker l_w = \mathcal{A}(F_{\infty,w})/P_\eta$ . Now if v is finitely decomposed in $F_\infty$ , then the sum $\oplus_{w|v}$ is finite, and so $\ker\big(\oplus_{w|v}l_w\big)$ is cofinitely generated over $\mathcal{O}_\eta$ for these v’s. Now suppose that v splits completely in $F_\infty$ . As noted in Remark 4·7, $\mathcal{A}(F_{\infty,w})[p_\eta] = A_{f_\eta}(F_{\infty,w})$ is finite for all v dividing p. For those primes not dividing p, the finiteness follows from assumption (a). Consequently, $\mathcal{A}(F_{\infty,w})/P_\eta$ is finite by Lemma 2·1(iii) and is hence annihilated by some powers of p. This power of p annihilates $\ker\big(\oplus_{w|v}l_w\big)$ . Therefore, we also have $\ker\big(\oplus_{w|v}l_w\big)$ being cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ for these primes. Hence we have shown that the kernel and cokernel of $r_\eta$ are cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ .

Combining this observation with hypothesis (b) of the theorem, we see that $R(\mathcal{A}/F_\infty)[p_\eta]$ is cotorsion over $\mathcal{O}_\eta[[ \Gamma ]]$ . It then follows from Lemma 2·1(iii) that $R(\mathcal{A}/F_\infty)$ is cotorsion over $\mathcal{R}[[ \Gamma ]]$ . This proves the first assertion of the theorem.

In view of $\textbf{(H1)}$ , the ring $\mathcal{R}[[ \Gamma ]]$ is isomorphic to $\mathcal{O}[[ W, T ]]$ , where $1+W$ (resp., $1+T$ ) corresponds to a topological generator of $\Gamma$ (resp., a topological generator of $\Gamma'$ ). By the structure theorem (cf. [Reference Neukirch, Schmidt and Wingberg40, proposition 5·1·7]), we then have a pseudo-isomorphism

\[ Y(\mathcal{A}/F_\infty)^\vee \sim \bigoplus_{i\in I} \mathcal{R}[[ \Gamma ]]/Q_i^{n_i}\]

of $\mathcal{R}[[ \Gamma ]]$ -modules, where I is some finite indexing set and each $Q_i$ is a principal prime ideal of $\mathcal{R}[[ \Gamma ]]$ of height one. Since there are only finitely many $Q_i$ ’s, for all but finitely many $\lambda\in\mathfrak{X}_{\textrm{arith}}\left(h^{\textrm{ord}}_{\mathcal{F}}\right)$ , we have $Y(\mathcal{A}/F_\infty)/P_\lambda$ being torsion over $\mathcal{O}_{\lambda}[[ \Gamma ]]$ . From the above discussion, we see that

\[ Y(\mathcal{A}/F_\infty)/P_\lambda \longrightarrow Y(A_{f_\lambda}/F_\infty) \]

has cokernel which is torsion over $\mathcal{O}_\lambda[[ \Gamma ]]$ . It then follows that $Y\left(A_{f_{\lambda}}/F_\infty\right)$ is torsion over $\mathcal{O}_\lambda[[ \Gamma ]]$ for these $\lambda$ ’s.

The finiteness condition $H^0(F_v, A_{f_{\eta}})$ is known to hold in several cases (see [Reference Hatley, Kundu, Lei and Ray14, section 5·2]). In particular, if $\eta$ comes from an elliptic curve, then this is always true, and so we have the following.

Comparing Theorems 4·3 and 5·3, one is naturally led to the following question.

To the best knowledge of the author, the structure of $R(\mathcal{A}/F_\infty)$ does not seem to be well-understood. Even in the context of a cyclotomic $\mathbb{Z}_{p}$ -extension, it is still an open question whether $R(\mathcal{A}/F_\infty)^\vee$ is finitely generated over $\mathcal{R}$ (see [Reference Jha and Sujatha17, conjecture 1]). There are some recent studies on this structure by Lei–Palvannan [Reference Lei and Palvannan25] in this direction. However, to the best knowledge of the author, the subject of the fine Selmer group of a Hida deformation over a general $\mathbb{Z}_{p}$ -extension does not seem to be covered in the literature. We can only hope to revisit this subject in a future study.

We end the section by specialising to the case of an imaginary quadratic field, where we explain how a conjecture of Mazur implies the various Conjecture Y’s. As a start, we recall the said conjecture of Mazur [Reference Mazur37].

6. Examples

We give some examples to illustrate our results.

1. (i) Let E be the elliptic curve $y^2 + y = x^3 -x$ . Let $p=5$ . It is well-known that $\textrm{Sel}(E/\mathbb{Q}(\mu_{5^{\infty}})) =0$ (cf. [Reference Coates and Sujatha5, theorem 5·4]), where $\textrm{Sel}(E/\mathbb{Q}(\mu_{5^{\infty}}))$ is the classical Selmer group. Let F be a finite Galois p-extension of $\mathbb{Q}(\mu_5)$ . Then by [Reference Hachimori and Matsuno13, corollary 3·4], $\textrm{Sel}(E/F^{\textrm{cyc}})$ is cotorsion over $\mathbb{Z}_{p}[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ . Since the fine Selmer group $R(E/F^{\textrm{cyc}})$ is contained in $\textrm{Sel}(E/F^{\textrm{cyc}})$ , this in turn implies that $R(E/F^{\textrm{cyc}})$ is cotorsion over $\mathbb{Z}_5[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ . Therefore, Theorem 3·9 applies. In other words, for any given $\mathbb{Z}_5^2$ -extension $L_\infty$ of F containing $F^{\textrm{cyc}}$ , for all but finitely many $\mathbb{Z}_5$ -extension $F_\infty$ of F contained in $L_\infty$ , we have that $R(E/F_\infty)$ is cotorsion over $\mathbb{Z}_5[[ \textrm{Gal}(F_\infty/F) ]]$ .

   In view of Conjecture Y, we expect that $R(E/F_\infty)$ is cotorsion over $\mathbb{Z}_5[[ \textrm{Gal}(F_\infty/F) ]]$ for all $\mathbb{Z}_5$ -extensions of F. But this seems to be out of reach in general. (Note that Proposition 3·8 cannot apply here, since we do not have a good way of determining pseudo-nullity at our current state of knowledge.)

   We however describe how one may obtain the absolute validity of Conjecture Y for the elliptic curve E in question over certain classes of 5-extensions of $\mathbb{Q}(\mu_5)$ . Let F be a finite Galois 5-extension of $\mathbb{Q}(\mu_5)$ such that every ramified prime v of $F/\mathbb{Q}(\mu_5)$ outside 5 is neither a split multiplicative reduction prime of E nor a good reduction prime of E with $E(\mathbb{Q}(\mu_5)_v)[5]\neq 0$ . By Kida’s formula for elliptic curves (cf. [Reference Hachimori and Matsuno13, theorem 3·1]), $\textrm{Sel}(E/F^{\textrm{cyc}})$ is finite which in turn implies that $R(E/F^{\textrm{cyc}})$ is finite. It then follows from [Reference Lim28, theorem 3·3] that so is $R(E/F)$ . Applying Corollary 3·3, we see that Conjecture Y holds for every $\mathbb{Z}_5$ -extension of F. Examples of such F’s are $\mathbb{Q}(\mu_{5^n}, 5^{-5^m})$ .

2. (ii) The above discussion can also be applied to the elliptic curve $E: y^2 + xy = x^3 -x-1$ and $p=7$ . Indeed, in this case, one has $\textrm{Sel}(E/\mathbb{Q}(\mu_{7^{\infty}})) =0$ (cf. [Reference Coates and Sujatha5, theorem 5·31]). For more examples where the discussion in (i) also applies, we refer readers to the tables in [Reference Dokchitser and Dokchitser10] (basically look for those with $\mathcal{L}_E(\sigma)$ being a unit).

3. (iii) Even if $\textrm{Sel}(E/\mathbb{Q}(\mu_{p^\infty}))\neq 0$ , there are many numerical examples (for instance, see [Reference Dokchitser and Dokchitser10, Reference Pollack46]), where the group $\textrm{Sel}(E/\mathbb{Q}(\mu_{p^\infty}))$ can be shown to be cofinitely generated over $\mathbb{Z}_{p}$ . By virtue of [Reference Hachimori and Matsuno13, corollary 3·4], $\textrm{Sel}(E/F^{\textrm{cyc}})$ is cofinitely generated over $\mathbb{Z}_{p}$ for every finite Galois p-extension F of $\mathbb{Q}(\mu_p)$ which in turn implies that $R(E/F)$ is cofinitely generated over $\mathbb{Z}_{p}[[ \textrm{Gal}(F^{\textrm{cyc}}/F) ]]$ . Hence we can at least apply Theorem 3·9 for these examples. We mention that the data in [Reference Pollack46] also consists of elliptic curves with supersingular reduction at p, where the plus-minus Selmer groups in the sense of Kobayashi [Reference Kobayashi22] have been verified to be cofinitely generated over $\mathbb{Z}_{p}$ . As the fine Selmer group is contained in either of the plus-minus Selmer groups, the fine Selmer group is also cofinitely generated over $\mathbb{Z}_{p}$ . This cofinite generation of fine Selmer group is preserved under p-base change of fields (for instance, see the proof of [Reference Lim and Murty31, theorem 5·5]). Hence Theorem 3·9 applies for these examples.

4. (iv) Let E be any elliptic curve in 49a and $F=\mathbb{Q}(E[7])$ . Take $p=7$ . We shall show that Conjecture Y is valid for every $\mathbb{Z}_7$ -extension of F. Indeed, this is plainly true if the $\mathbb{Z}_7$ -extension is the cyclotomic $\mathbb{Z}_7$ -extension by [Reference Rasmussen and Tamagawa48, corollary 8] and [Reference Coates and Sujatha4, corollary 3·6]. For a non-cyclotomic $\mathbb{Z}_7$ -extension $F_\infty$ of F, the compositum of $F_\infty$ and $F^{\textrm{cyc}}$ is a $\mathbb{Z}_7^2$ -extension of F. By [Reference Lim26, section 4, example (a)], $Y(E/L_\infty)$ is pseudo-null. Therefore, the torsionness of $Y(E/F_\infty)$ follows from this and Proposition 3·8. For more of such examples, we refer readers to [Reference Lim26, section 4, example (a)] and [Reference Rasmussen and Tamagawa48, table 2].

5. (v) Let E be any elliptic curve in 32a and $F=\mathbb{Q}(\sqrt{-43})$ . Take $p=3$ . It has been verified by Lei-Palvannan that $R(E/L_\infty)$ is pseudo-null (see [Reference Lei and Palvannan24, section 8·4]), where $L_\infty$ is the $\mathbb{Z}_3^2$ -extension of F. Therefore, we may apply Theorem 3·8 to obtain the validity of Conjecture Y for every $\mathbb{Z}_3$ -extension of F. [Reference Lei and Palvannan24, table 2] provides more examples, where Proposition 3·8 can be applied too.

6. (vi) Let E be the elliptic curve 79a1 of Cremona’s tables given by

   \[y^2 + xy +y =x^3 +x^2 -2x.\]
   Let $p=3$ and $F=\mathbb{Q}(\mu_3)$ . It has been computed by Wuthrich (see [Reference Dokchitser and Dokchitser10, p. 253]) that E(F) has $\mathbb{Z}$ -rank 1 and the Tate–Sharafevich group is finite. Under these observations, it follows from a similar argument to that in [Reference Coates and McConnell2, theorem 12] that one obtains $H^2(G_S(F),E[3^\infty]) = 0$ . By [Reference Hachimori12, lemma 3·2], this in turn implies that $R(E/F)$ is finite. Let $F_\infty$ be any $\mathbb{Z}_3$ -extension of F. Then Theorem 3·3 applies to yield the torsionness of $R(E/F_\infty)$ . As noted in [Reference Jha16, p. 362], the residual representation E[Reference Coates and Sujatha3] is irreducible. Let $\mathcal{F}$ be the Hida family associated to E. Applying Corollary 5·4, we see that for all but finitely many $\lambda\in\mathfrak{X}_{\textrm{arith}}\left(h^{\textrm{ord}}_{\mathcal{F}}\right)$ , $R\left(A_{f_{\lambda}}/F_\infty\right)$ is cotorsion over $\mathcal{O}_\lambda[[ \textrm{Gal}(F_\infty/F) ]]$ .

7. Non-commutative speculation and further remarks

In this final section, we formulate an extension of Conjecture Y for a (possibly non-commutative) p-adic Lie extension. To simplify the discussion and for better clarity of the ideas behind, we shall restrict our attention to the context of the paper. However, it should be evident that much of the discussion here carries over to broader classes of Galois representations.

We shall let $\bar{A}$ denote either one of the following objects: $A[p^\infty], A_f, \mathcal{A}$ , and $\bar{R}$ the corresponding coefficients rings of these objects; namely, resp., $\mathbb{Z}_{p}, \mathcal{O}, \mathcal{R}$ . Note that in the context of $\mathcal{A}$ , we still work under the hypotheses $\textbf{(H1)}$ and $\textbf{(H2)}$ . Let $F_\infty$ be a p-adic Lie extension of F such that $G:=\textrm{Gal}(F_\infty/F)$ is pro-p with no p-torsion and that $F_\infty/F$ is unramified outside a finite set of primes of F. The fine Selmer group of $\bar{A}$ over $F_\infty$ , which is defined similarly as before, now has the structure of a $\bar{R}[[ G ]]$ -module. We note that the ring $\bar{R}[[ G ]]$ is Auslander regular (cf. [Reference Venjakob52, theorem 3·26]; also see [Reference Lim27, theorem A·1]) and has no zero divisors (cf. [Reference Neumann41]). Therefore, there is a well-defined notion of torsion $\bar{R}[[ G ]]$ -modules and pseudo-null $\bar{R}[[ G ]]$ -modules. For our purpose, a $\bar{R}[[ G ]]$ -module M is said to be torsion (resp., pseudo-null) if $\textrm{Ext}_{\bar{R}[[ G ]]}^i(M,\bar{R}[[ G ]])=0$ for $i=0$ (resp., $i=0,1$ ). The following is then the natural generalization of Conjecture Y for a p-adic Lie extension.

For an abelian variety, the above conjecture was formulated in [Reference Kundu and Lim23] for a $\mathbb{Z}_{p}^d$ -extension of F.

A similar argument to that in Theorem 3·9 yields the following.

We finally end by providing a conceptual explanation on why the postulation of Conjecture Y over a general p-adic Lie extension is plausible. To give this explanation, we now recall the generalised Conjecture B of Coates–Sujatha [Reference Coates and Sujatha4] and Jha [Reference Jha16].

Proof. We first make the following remark. Let G be a compact p-adic Lie group and M a $\bar{R}[[ G ]]$ -module. For every open subgroup $G_0$ of G, we then have

\[\textrm{Ext}^i_{\bar{R}[[ G ]]}\left(M,\bar{R}[[ G ]]\right) \cong \textrm{Ext}^i_{\bar{R}[[ G_0 ]]}\left(M,\bar{R}[[ G_0 ]]\right). \]

(cf. [Reference Neukirch, Schmidt and Wingberg40, proposition 5·4·17]). Therefore, the question of M being torsion (resp., pseudo-null) over $\bar{R}[[ G ]]$ is equivalent to M being torsion (resp., pseudo-null) over $\bar{R}[[ G_0 ]]$ .

Now let $F_\infty$ be a p-adic Lie extension of F of dimension $>1$ . If $F_\infty$ contains $F^{\textrm{cyc}}$ , then by the assumption of the proposition, $Y(\bar{A}/F_\infty)$ is pseudo-null over $\bar{R}[[ G ]]$ , and hence is also torsion over $\bar{R}[[ G ]]$ . Suppose that $F^{\textrm{cyc}}$ is not contained in $F_\infty$ . Then $F^{\textrm{cyc}}\cap F_\infty$ is a finite extension of F. In view of the the remark in the first paragraph, we see that neither the hypothesis nor the conclusion is affected if we replace F by $F^{\textrm{cyc}}\cap F_\infty$ . Hence, upon relabelling, we might as well assume that $F=F^{\textrm{cyc}}\cap F_\infty$ . Therefore, writing $L_\infty =F^{\textrm{cyc}} \cdot F_\infty$ , we have $\mathcal{G}:=\textrm{Gal}(L_\infty/F)\cong Z\times G$ , where $Z\cong\textrm{Gal}(F^{\textrm{cyc}}/F)$ . Let z be a topological generator of Z. Then for every $\bar{R}[[ \mathcal{G} ]]$ -module M, we have $M_Z = M/(z-1)$ which can be viewed as a $\bar{R}[[ G ]]$ -module. As Z is central in $\mathcal{G}$ , $M_Z$ may also be viewed as a $\bar{R}[[ \mathcal{G} ]]$ -module. In particular,

\[0\longrightarrow \bar{R}[[ \mathcal{G} ]]\longrightarrow \bar{R}[[ \mathcal{G} ]] \longrightarrow \bar{R}[[ G ]]\longrightarrow 0\]

is a $\bar{R}[[ \mathcal{G} ]]$ -free resolution of the $\bar{R}[[ \mathcal{G} ]]$ -module $\bar{R}[[ G ]]$ . Via a spectral sequence argument similar to that in Lemma 2·1, we obtain

\[ \textrm{Ext}^i_{\bar{R}[[ G ]]}(Y(A/L_\infty)_{Z},\bar{R}[[ G ]]) \cong \textrm{Ext}^{i+1}_{\bar{R}[[ \mathcal{G} ]]}(Y(A/L_\infty)_{Z},\bar{R}[[ \mathcal{G} ]]). \]

In particular, one has

\[ \textrm{Hom}_{\bar{R}[[ G ]]}(Y(\bar{A}/L_\infty)_{Z},\bar{R}[[ G ]]) \cong \textrm{Ext}^{1}_{\bar{R}[[ \mathcal{G} ]]}(Y(\bar{A}/L_\infty)_{Z},\bar{R}[[ \mathcal{G} ]]). \]

Since $Y(\bar{A}/L_\infty)$ is pseudo-null over $\bar{R}[[ \mathcal{G} ]]$ by hypothesis, the latter is zero. From which, we see that $Y(\bar{A}/L_\infty)_{Z}$ is torsion over $\bar{R}[[ G ]]$ . On the other hand, a descent argument as in Theorem 3·9 yields a map

\[ Y(\bar{A}/L_\infty)_Z\longrightarrow Y(\bar{A}/F_\infty), \]

whose cokernel is a quotient of $H^1(Z, \bar{A}(L_\infty))^\vee$ , where the latter is plainly finitely generated over $\bar{R}$ . It then follows that $Y(\bar{A}/F_\infty)$ is torsion over $\bar{R}[[ G ]]$ as required.