Journal de Théorie des Nombresde Bordeaux 00 (XXXX), 000–000

Control Theorems for Fine Selmer Groups
par Debanjana Kundu et Meng Fai Lim
Résumé. Nous étudions la croissance du groupe de Selmer fin p-primaire, R(E/F 0 ),d’une courbe elliptique sur une sous-extension intermédiaire F 0 d’une extension
de Lie p-adique, L/F . Nous estimons le Zp -corank du noyau et du conoyau de0l’application de restriction rL/F 0 : R(E/F 0 ) → R(E/L)Gal(L/F ) avec F 0 une extension finie de F contenue dans L. Nous montrons également que la croissance desgroupes de Selmer fins dans ces sous-extensions intermédiaires est liée à la structure
du groupe de Selmer fin sur le niveau infini. En nous spécialisant dans les extensions
de Lie p-adiques classiques (éventuellement non commutatives), nous prouvons lafinitude du noyau et du conoyau et fournissons des estimations de croissance surleurs ordres.
Abstract. We study the growth of the p-primary fine Selmer group, R(E/F 0 ), of
an elliptic curve over an intermediate sub-extension F 0 of a p-adic Lie extension,
L/F . We estimate the Zp -corank of the kernel and cokernel of the restriction map
0
rL/F 0 : R(E/F 0 ) → R(E/L)Gal(L/F ) with F 0 a finite extension of F containedin L. We show that the growth of the fine Selmer groups in these intermediate
sub-extension is related to the structure of the fine Selmer group over the infinite
level. On specializing to certain (possibly non-commutative) p-adic Lie extensions,
we prove finiteness of the kernel and cokernel and provide growth estimates on theirorders.

1. Introduction
Iwasawa theory began as the study of ideal class groups over infinite towers of number
fields [16]. In his fundamental paper [32], Mazur developed an analogous theory to study
the growth of Selmer groups of Abelian varieties in Zp -extensions. He proved what is
nowadays called a “control theorem”, which we describe briefly here. Let A be an Abelianvariety defined over a number field, F , with potential good ordinary reduction at all primes
above p, and let L be a Zp -extension of F . For every intermediate sub-extension F 0 of L/F ,we have natural maps
0sL/F 0 : Sel(A/F 0 ) −→ Sel(A/L)Gal(L/F )on the Selmer groups induced by the restriction maps on cohomology. Mazur’s Control
Theorem asserts that the kernel and cokernel of sL/F 0 are finite and bounded independent
1991 Mathematics Subject Classification. 11R23.
Mots-clefs. control theorem, fine Selmer groups.

2

Debanjana Kundu, Meng Fai Lim

of F 0 . The Control Theorem has subsequently been generalized to general p-adic Lie extensions by Greenberg [11]. Such a Control Theorem has great importance in Iwasawa
theory. In [32], Mazur conjectured that the Selmer group Sel(A/F cyc ) is cotorsion over
Zp JΓK, where Γ = Gal(F cyc /F ) and F cyc is the cyclotomic Zp -extension of F . The first
theoretical evidence towards this conjecture was provided by Mazur himself; using the Control Theorem, he verified the conjecture when Sel(A/F ) is finite. Till date, this conjecture
is known only when E is an elliptic curve over Q and F is an Abelian extension of Q; see[18, 38]. The Selmer group Sel(A/F cyc ) is known to be related to a p-adic L-function via
the main conjecture. Therefore, Mazur’s Control Theorem opens up a channel to extract
information on Sel(A/F ) from the said main conjecture which provides an invaluable approach towards the study of the Birch and Swinnerton-Dyer Conjecture (for instance, see[18, 38, 45]). The Control Theorem connects the Selmer groups at the finite layers with
the Selmer group over the infinite tower, thereby allowing one to deduce properties of thisarithmetic object over the infinite tower from those at the finite layers, and vice versa.
Recently, there has been an interest in the study the fine Selmer group (see [6, 17, 23,
25, 27, 30, 36, 49, 51]). This is a subgroup of the classical Selmer group obtained by
imposing stronger vanishing conditions at primes above p (see §5 for its definition). In [6],Coates and Sujatha initiated a systematic study of the fine Selmer group and postulated
conjectures on its structure over a p-adic Lie extension. In this paper, we prove Control
Theorems for fine Selmer groups of elliptic curves in general p-adic Lie extensions. This
allows us to deduce properties of the fine Selmer group over the infinite tower from thoseat the finite layers, and vice versa. We remark that the (classical) Control Theorem canbe proven only when the Abelian variety has potential good ordinary reduction at primes
above p (see [10, p. 51]); however, our results do not require this hypothesis.
First, we establish estimates on the Zp -coranks of the kernel and cokernel of the restriction maps
0rL/F 0 : R(E/F 0 ) −→ R(E/L)Gal(L/F )for a p-adic Lie extension L/F . Using these, we show how the module theoretic structure ofR(E/L) determines the growth of Zp -coranks of R(E/F 0 ) in intermediate sub-fields F 0 . To
obtain sharper results, we specialize to three cases of p-adic Lie extensions: Zdp -extensions,multi-false-Tate extensions, and the trivializing extension obtained by adjoining to F all
the p-power division points of the elliptic curve, E. In each of these cases, we show (underappropriate assumptions) that the kernel and cokernel of the restriction map are finite,and establish growth estimates for their orders.
For a Zdp -extension, Control Theorems have been studied in [39, 49, 51, 27], often withadditional hypotheses. Our results however, are more general and can provide precise
growth estimates for the kernel and cokernel of rL/F 0 . Further, we prove that if R(E/F ) is
finite, then R(E/F∞ ) is cotorsion over Zp JGK, where G = Gal(F∞ /F ) ∼= Zdp . This provides
the impetus to conjecture the following.
Conjecture (Conjecture Yd ). Let F∞ be a p-adic extension of F with G = Gal(F∞ /F ) ∼=Zdp . Then R(E/F∞ )∨ is a torsion Zp JGK-module.

3

When d = 1, this is conjectured by the second named author in [27]. Currently, we
refrain from formulating a more general conjecture, such as over a non-commutative padic extension not containing the cyclotomic Zp -extension, as we feel there is insufficientevidence towards the same.
Control Theorems for fine Selmer groups over non-commutative extensions (e.g. (multi)
false-Tate extensions and trivializing extensions) have not been recorded in the literature.
The key step of controlling the growth of the cokernel requires careful analysis of the local
restriction maps at primes above p. We emphasize that the method of proof differs fromthat of Greenberg in [11], where he developed a Lie algebraic approach to attack suchproblems. The main reason for requiring a different approach is that we want to estimate
the growth of cohomology groups of open subgroups of a p-adic Lie group. However, open
subgroups share the same Lie algebra, therefore the cohomology of the Lie algebra cannot
distinguish the cohomology groups of the subgroups. Our analysis is therefore significantly
different, intricate, and in fact, more effective than the case of the (classical) Selmer group.
Our results show that the (conjectured) structure of the fine Selmer group at the infinite
level has bearing on the growth of the size of R(E/Fn )[pn ]. Finally, we remark that
our argument uses an improvement of Tate’s Lemma (see Lemma 2.3) which should beinteresting in its own right, and should potentially have further applications.
It would seem that some of our results may be extended to fine Selmer groups attachedto Abelian varieties, modular forms or even broader classes of Galois representations ofinterest. Our case of representations arising from elliptic curves can therefore be seen as afirst step in this line of study.
We now give an outline of the paper. In §2, 3, we record algebraic facts required
throughout this article. In §4, we estimate the growth of cohomology groups of the pdivision points of an elliptic curve in a p-adic Lie extension of a local field. In §5, we provea Control Theorem which studies the growth of the Zp -coranks of the kernel and cokernelof the restrictions maps in a general p-adic Lie extension. We also study an analogue where
we vary the cyclotomic Zp -extensions of the intermediate sub-extensions. In §6, we prove
more precise versions of the Control Theorem in special cases and establish (with growthestimates) the finiteness of the kernel and cokernel of the restriction map. In §7, we provide
numerical examples to illustrate our otherwise abstract results.
2. Some Basic Estimates on Cohomology
In this section, we record estimates on the cohomology groups which are required
throughout the article. For an Abelian group, M , let M [pj ] denote the subgroup of M
consisting of elements of M annihilated by pj . Write M [p∞ ] for ∪j≥1 M [pj ]. If M is a discrete p-primary Abelian group or a compact pro-p Abelian group, we define its Pontryagin
dual, M ∨ = Homcont (M, Qp /Zp ). For a profinite group, G, and a G-module, M , let M G
be the subgroup of M consisting of elements fixed by G and MG be the largest quotientof M on which G acts trivially. If M is a discrete G-module, we write H i (G, M ) for thei-th Galois cohomology group of G with coefficients in M .

4Debanjana Kundu, Meng Fai Lim

Lemma 2.1. Let G be a pro-p group and M be a discrete
G-module which is cofinitely1generated over Zp . If h1 (G) = dimZ/pZ H G, Z/pZ is finite, then
dimZ/pZ H 1 (G, M ) [p] ≤ h1 (G) corankZp (M ) + ordp M/Mdiv .

If h2 (G) = dimZ/pZ H 2 G, Z/pZ is finite, then
dimZ/pZ H 2 (G, M ) [p] ≤ h2 (G) corankZp (M ) + ordp M/Mdiv .
Proof. The first inequality is proven in [29, Lemma 3.2]. The second inequality is provensimilarly.
When M is a finite G-module, we have the following sharper conclusion.
Lemma 2.2. Let G be a pro-p groupdiscrete G-module which is a
 and M be a(finite)1
finite p-group. If h1 (G) = dimZ/pZ H G, Z/pZ
is finite, then H 1 (G, M ) is finite
withordp H 1 (G, M ) ≤ h1 (G) ordp |M | .

If h2 (G) = dimZ/pZ H 2 G, Z/pZ is finite, then H 2 (G, M ) is finite withordp H 2 (G, M ) ≤ h2 (G) ordp |M | .
Proof. This follows from a standard dévissage argument and noting that the only simple
discrete G-module is Z/pZ with trivial G-action (cf. [34, Corollary 1.6.13]).

If F is a finite extension of Q or Qp which contains a primitive root of unity, andΓ = Gal(F (µp∞ )/F ), then Tate’s Lemma (see [46, p. 526]) asserts that H 1 (Γ, Qp /Zp (i)) = 0
for i 6= 0. Observe that H 0 (Γ, Qp /Zp (i)) is finite by virtue of i 6= 0. Motivated by thisobservation, we prove the following lemma which can be thought of as a generalization of
Tate’s Lemma. We note that our argument is different from the classical proof.
Lemma 2.3. Let G ∼= Zp and M be a discrete, p-divisible G-module, which is cofinitely
generated over Zp . If M G is finite, then H 1 (G, M ) = 0.
Proof. Set U = M ∨ . This is a compact Zp JGK-module which is finitely generated over Zp .
Thus, it is torsion over Zp JGK, and
0 = rankZp JGn K (U ) = rankZp UG − rankZp U G .

This combined with the hypothesis that UG = (M G )∨ is finite, yields that U G is finite.
Since M is p-divisible, U is Zp -torsion free. Since U G is finite, it follows that U G = 0. But,
H 1 (G, M )∨ ∼= (MG )∨ ∼= U G = 0,
where the first isomorphism follows from [34, Proposition 1.7.7(i)].

5

3. Modules over the Iwasawa Algebra
Throughout, p denotes a fixed prime. We record algebraic facts required in this article.
3.1. Uniform pro-p groups. In this subsection, we lay out some facts regarding a uniform pro-p group. For further background on these groups, we refer the reader to [8].
nnFor a finitely generated pro-p group, G, we write Gp = hg p | g ∈ Gi, i.e., the groupnngenerated by the pn -th-powers of elements in G. We also write G{p } = {g p | g ∈ G}, i.e.,the set consisting of the pn -th-powers of elements in G. The pro-p group G is said to be
powerful if G/Gp (resp. G/G4 ) is Abelian for odd p (resp. p = 2). Here (·) denotes theclosure with respect to the topology of the pro-p group. The lower p-series of G is definedby P1 (G) = G, and
Pn+1 (G) = Pn (G)p [Pn (G), G], for i ≥ 1.
nn
By [8, Theorem 3.6], if G is powerful, then Gp = G{p } = Pn+1 (G). Further, the p-powermap·p
Pn (G)/Pn+1 (G) −→ Pn+1 (G)/Pn+2 (G)is surjective for each n ≥ 1. If the p-power maps are isomorphisms for all n ≥ 1, we say
G is uniformly powerful (abbrv. uniform). Then, [G : P2 (G)] = [Pi (G) : Pn+1 (G)] forevery n ≥ 1. Consequently, [G : Pn+1 (G)] = pnd , where d = dim G (see [8, Definition 4.1]).
A well-known result of Lazard (cf. [8, Corollary 8.34]) asserts that a compact p-adic Liegroup always contains an open normal uniform subgroup. Therefore, one can always reduce
consideration for a general compact p-adic Lie group to the case of a uniform group, whichnn
we will do throughout the paper. In particular, for a uniform group, we have Gp = G{p } ,which will be utilized without further mention.
We record the following lemma which will also be applied frequently without mention.
Lemma 3.1. Let G be a uniform group and N be a closed normal subgroup of G suchnthat R := G/N is uniform. Then N is also uniform. Furthermore, writing Nn = N p ,nnppGn = G , and Rn = R , we have Nn = Gn ∩ N and Gn /Nn ∼= Rn .
Proof. Using [8, Proposition 4.31], N is a uniform group. Clearly, one has Nn ⊆ Gn ∩ N .
nConversely, let x ∈ Gn ∩ N . Then x = y p for some y ∈ G; hence, the coset yN is atorsion element in R = G/N . Since R is assumed to be uniform, it has no p-torsion (cf. [8,nTheorem 4.5]); so yN = N or y ∈ N . Hence, x = y p ∈ Nn . This proves the first equality.
The second follows from the observation thatnn
Gn /Nn = Gn /(Gn ∩ N ) ∼= Gn N/N = Gp N/N = (G/N )p = Rn .

3.2. Torsion modules and pseudo-null modules. For a compact p-adic Lie group, G,
its Iwasawa algebra is the completed group algebra of G over Zp . It is given by
Zp JGK = lim Zp [G/U ],←−U

where U runs over the open normal subgroups of G and the inverse limit is taken withrespect to the canonical projection maps.

6

Debanjana Kundu, Meng Fai Lim

When G is pro-p and has no p-torsion, it is well-known that Zp JGK is an Auslander
regular ring (cf. [47, Theorem 3.26]; for the definition of Auslander regular rings, see [47,
Definition 3.3]). Furthermore, the ring Zp JGK has no zero divisors (cf. [35]), and therefore,
admits a skew field, Q(G), which is flat over Zp JGK (see [9, Chapters 6 and 10] or [20,Chapter 4, §9 and §10]). If M is a finitely generated Zp JGK-module, define the Zp JGK-rankof M asrankZp JGK M = dimQ(G) Q(G) ⊗Zp JGK M.
A Zp JGK-module, M , is torsion if rankZp JGK M = 0. Equivalently, M is torsion if andonly if HomZp JGK (M, Zp JGK) = 0 (cf. [24, Lemma 2.2.1]). A torsion Zp JGK-module, M , ispseudo-null if Ext1Zp JGK (M, Zp JGK) = 0. Finally, we record a result which will be frequently
used in our discussion.
Proposition 3.2 (Harris [13]). Let G be a d-dimensional uniform group. Let M be a
finitely generated Zp JGK-module. Then,
rankZp (MGn ) = rankZp JGK (M )pdn + O p(d−1)n .
3.3. µG -invariant. In this subsection, G will denote a uniform group. Therefore, bothZp JGK and Fp JGK are Auslander regular rings with no zero divisors. For a finitely generated
Zp JGK-module, M , it follows from [15, Proposition 1.11] (or [47, Theorem 3.40]) that there
is a Zp JGK-homomorphism
ϕ : M [p∞ ] −→

s
M
i=1

Zp JGK/π αi ,whose kernel and cokernel are pseudo-null Zp JGK-modules. Furthermore, the integers ssXand αi are uniquely determined. We define the µG -invariant, µG (M ) =
αi .
i=1

Lemma 3.3. Let M be a finitely generated Zp JGK-module. Suppose there is a Zp JGKhomomorphismsM∞
ϕ : M [p ] −→Zp JGK/pαi ,i=1whose kernel and cokernel are pseudo-null Zp JGK-modules. Then,µG (M/pn ) = n rankZp JGK (M ) +sX

min{n, αi }for n ≥ 1.

i=1

In particular, M is Zp JGK-torsion with trivial µG -invariant if and only if µG (M/p) = 0.
Proof. The first equality is [24, Lemma 2.4.1]; the last assertion follows immediately.

Proposition 3.4. For G a d-dimensional uniform group, and M a finitely generated
Fp JGK-module,
ordp |MGn | = µG (M )pdn + O p(d−1)n .

7Proof. See [37, Théorème 2.1] or [24, Proposition 2.5.1].



4. Local Considerations
We now estimate the growth of the first cohomology group of the p-division points of anelliptic curve in a p-adic Lie extension of a local field. For an extension L ofa local field, and
a Gal(L/L)-module, M , we write H i (L, M ) for the cohomology group H i Gal(L/L), M .
Throughout, K is a local field and K∞ is a uniformp- adic Lie extension, i.e., it is a pro-p
extension of K with Galois group G = Gal K∞ /K which is assumed to be uniform. Wenwrite Gn = Gp .
4.1. Local field over Q` with ` 6= p. Let K be a finite extension of Q` , where ` 6= p.
By a classical result of Iwasawa (cf. [34, Theorem 7.5.3]), if K∞ /K is a uniform p-adic
Lie extension, we know that Gal(K∞ /K) ∼= Zp or Zp o Zp . Further, Iwasawa’s resultshows that the Zp o Zp -extension of K is unique. We now estimate the growth of the firstcohomology group in both cases.
Proposition 4.1. Let E be an elliptic curve defined over K, and K∞ /K be a uniform
extension of K. Then, the group H 1 Gn , E(K∞ )[p∞ ] is finite for every n. Furthermore,
the following assertions are true.
(a) If Gal(K∞ /K) ∼= Zp , then H 1 Gn , E(K∞ )[p∞ ] = O(1).
 (b) If Gal(K∞ /K) ∼= Zp o Zp , then dimZ/pZ H 1 Gn , E(K∞ )[p∞ ] [p] = O(1) and
ordp H 1 Gn , E(K∞ )[p∞ ] = O(n).
Proof. (a) This is proven in [27, Lemma 3.4].
(b) The first estimate follows from Lemma 2.1. For the second estimate, note that by
Iwasawa’s result (cf. [34, Theorem 7.5.3]), K∞ has no non-trivial p-extension, so
H 1 (K∞ , E[p∞ ]) = 0. By the inflation-restriction sequence, we obtain that

H 1 Gn , E(K∞ )[p∞ ] ∼= H 1 (Kn , E[p∞ ]).
Here Kn refers to the field fixed by Gn . Since ` 6= p, the latter
is isomorphic
∨1∞∞
to H (Kn , E)[p ]; this in turn is isomorphic to E(Kn )[p ]by Tate-duality(cf. [33, Chap. I, Corollary 3.4]). It is now easy to see that this is finite and∨= O(n).
ordp E(Kn )[p∞ ]
4.2. Local field over Qp . We now consider the situation when K is a finite extension of
Qp , and E is an elliptic curve defined over K. We begin with the following easy observation.
Proposition 4.2. Let K∞ be a uniform p-adic Lie extension extension of K of dimension
d. Suppose E(K∞ )[p∞ ] is finite. Then, the group H 1 (Gn , E(K∞ )[p∞ ]) is finite andordp H 1 Gn , E (K∞ ) [p∞ ] = O(1).

8Debanjana Kundu, Meng Fai Lim

Proof. This follows from Lemma 2.2.

Proposition 4.3. Let K∞ be a Zdp -extension of K. Then H 1 Gn , E(K∞ )[p∞ ] is finiteand
ordp H 1 Gn , E(K∞ )[p∞ ] = O(n).
Proof. Observe that H 0 Gn , E (K∞ ) [p∞ ] = E(Kn )[p∞ ] is finite because for any elliptic
curve over a local field the torsion subgroup is finite, see [44, Proposition VII.6.3]. The
finiteness of H 1 Gn , E (K∞ ) [p∞ ] is then essentially a consequence of this and [42, Chap.
IV, Theorem 1]. (One may also consult p. 106 in op. cit., where they obtain a resultfor k[G], where k is a field. But the same discussion carries over if k is replaced by
Zp .) In view of this finiteness observation, we may apply Lemma 2.1 to conclude that1∞
dimZ/pZ H Gn , E(K∞ )[p ] [p] is bounded independent of n. On the other hand, by [7,Theorem 2.8] (or [22, Lemma 2.1.1]) there
exists a constant c independent of n such that

∞
dn+c
1
p
annihilates H Gn , E(K∞ )[p ] . The assertion follows from these observations. Remark 4.4. If Gal(K∞ /K) = Zp , it is possible to prove the following better estimate(see [27, Lemma 3.4])

ordp H 1 Gn , E (K∞ ) [p∞ ] = O(1).
√√Let K∞ = K µp∞ , p∞ α1 , . . . , p∞ αd−1 , where α1 , . . . , αd−1 ∈ K × whose image in
K × /(K × )p are linearly independent over Fp . This is a multi-false-Tate extension of a local
field, K.

Proposition 4.5. Let K∞ be a multi-false-Tate extension of a local field of dimension≥ 2. Suppose E is an elliptic curve
defined over K with potential good reduction. Then,1∞the group H Gn , E(K∞ )[p ] is finite with

ordp H 1 Gn , E(K∞ )[p∞ ] = O(1).
Proof. Since E has potential good reduction at K, we see that E(K∞ )[p∞ ] is finite by aresult of Kubo-Taguchi [19, Theorem 1.1]. The claim follows from this observation and
Proposition 4.2.


If E has split multiplicative reduction, then H 1 Gn , E(K∞ )[p∞ ] can be infinite. Thisis a well-known fact. However, for a lack of proper reference, we supply an argument here.
Proposition 4.6. Let E be an elliptic curve defined over K with split multiplicative reduc√ ∞tion. Suppose that K contains a primitive p-th root of unity. Let K∞ = K µp∞ , p αsuch that E[p∞ ] is not realized over K∞ . Then the group H 1 G, E[p∞ ] is infinite, where
G = Gal(K∞ /K).
Proof. From the theory of Tate curves, we have a short exact sequence
0 −→ µp∞ −→ E[p∞ ] −→ Qp /Zp −→ 0

9of Gal(K/K)-modules. Taking Gal(K/K∞ )-invariance, we obtain the exact sequence,f

0 −→ µp∞ −→ E(K∞ )[p∞ ] −→ Qp /Zp .
Since E[p∞ ] is not realized over K∞ , the Zp -corank of E(K∞ )[p∞ ] is at most one. The
above sequence implies that the image of f , B (say), is finite. Thus, we have a short exact
sequence,f

0 −→ µp∞ −→ E(K∞ )[p∞ ] −→ B −→ 0of G-modules. Taking G-cohomology, we obtain the following exact sequence
B G −→ H 1 (G, µp∞ ) −→ H 1 (G, E(K∞ )[p∞ ]).
Since B is finite, the proposition will follow once we show that H 1 (G, µp∞ ) is infinite. For
this, we appeal to the inflation-restriction sequence; there exists a short exact sequence
0 −→ H 1 (Γ, µp∞ ) −→ H 1 (G, µp∞ ) −→ H 1 (H, µp∞ )Γ −→ 0,
where Γ = Gal(K cyc /K). By Kummer theory, H ∼= Tp µp∞ as Γ-modules. Thus,
H 1 (H, µp∞ )Γ ∼= Hom(Tp µp∞ , µp∞ )Γ ∼= (Qp /Zp )Γ = Qp /Zp ,which is an infinite group. We have therefore established the proposition.



In Proposition 4.8, we will see that one can still say something on the growth of the Zp corank. Now consider the case of K∞ = K(E[p∞ ]), i.e., the extension obtained by adjoining
all the p-power division points of the elliptic curve, E. Here, we have no assumption on
the reduction type of E.
Proposition 4.7. Let E/K be an elliptic curve such that K/Qp is finite, and let K∞ =n
K(E[p∞ ]). Suppose G= Gal(K∞ /K) is uniform. Writing Gn = Gp , the cohomologygroup H 1 Gn , E[p∞ ] is finite with p-power order O(n).
Proof. By base changing, we assume that E[p] is rational over K and has semi-stablereduction at K. If E has complex multiplication, then G ∼= Z2p and the conclusion follows
from Proposition 4.3.
We will now consider the case of elliptic curves without complex multiplication. Supposethat E has split multiplicative reduction. By the theory of Tate curves, there exists q ∈ K×such that E(K) ∼= K /q Z as Gal(K/K)-modules. Since K is assumed to contain E[p], italso contains µp . Therefore, K(µp∞ ) is a Zp -extension of K. Write H = Gal(K∞ /K(µp∞ ))and Υ = Gal(K(µp∞ )/K). The theory of Tate curves tells us that K∞ is obtained fromK(µp∞ ) by adjoining all the p-power roots of q. Thus, K∞ is a false-Tate extension of K.
Furthermore, there is a short exact sequence
0 −→ µp∞ −→ E[p∞ ] −→ Qp /Zp −→ 0
of Gal(K/K)-modules. Taking H-cohomology, we obtain the following exact sequenceδ

0 → µp∞ → H 0 (H, E[p∞ ]) → Qp /Zp −→ H 1 (H, µp∞ ) → H 1 (H, E[p∞ ]) → H 1 (H, Qp /Zp ) → 0,

where the final zero comes from the fact that H ∼= Zp has p-cohomological dimension
1. By Kummer theory, the connecting homomorphism, denoted by δ, is an isomorphism.

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Debanjana Kundu, Meng Fai Lim

Thus, we obtain H 0 (H, E[p∞ ]) ∼= µp∞ and H 1 (H, E[p∞ ]) ∼= H 1 (H, Qp /Zp ). Writennpp
Hn = H and Υn = Υ ; then a similar argument also yields H 0 (Hn , E[p∞ ]) ∼= µp∞ and1 (H , Q /Z ) as Υ -modules. By the inflation-restriction sequence,
H 1 (Hn , E[p∞ ]) ∼
H
=
n
p
p
n
we obtain
0 −→ H 1 (Υn , E[p∞ ]Hn ) −→ H 1 (Gn , E[p∞ ]) −→ H 1 (Hn , E[p∞ ])Γn −→ 0.
Now H 1 (Υn , E[p∞ ]Hn ) ∼= H 1 (Υn , µp∞ ) which vanishes by Lemma 2.3. By Kummer∼
theory, we know that H = Tp µp∞ as Υ-modules. Therefore,H 1 (Hn , E[p∞ ])Υn ∼= Hom(Tp µp∞ , Qp /Zp )Υn ;= H 1 (Hn , Qp /Zp )Υn ∼the last of which is easily seen to be finite with p-power order O(n). This proves the
proposition when E has multiplicative reduction.
When E does not have complex multiplication and has good ordinary reduction, thedimension of G is 3 (cf. [4, Proposition 2.8]). We have a short exact sequence of G-modules(4.1)

b ∞ ] −→ E[p∞ ] −→ E[pe ∞ ] −→ 0,
0 −→ E[pb (resp. E)
e denotes the formal group (resp. reduced curve) of E. Taking Gn where E
invariance,





1
∞
1
∞
1
∞
b
e
H Gn , E[p ] −→ H Gn , E[p ] −→ H Gn , E[p ] .
e ∞ ]) is finite with p-power order
The discussion in [11, p. 274] implies that H 1 (Gn , E[pb ∞ ]). For this, we further analyse the structure
O(n). It remains to estimate H 1 (Gn , E[p∞eof G. The Gal(K̄/K)-action on E[p] induces a group homomorphism ρ : Gal(K̄/K) −→∞kere
Aut(E[p ]). We denote L∞ := K̄ ρ which is a Zp -extension of K contained in K∞(recall that we are assuming that E[p] is realized over K). By definition, Gal(K/K ur )e ∞ ], where K ur is the maximal unramified extension of K. Hence, theacts trivially on E[pextension L∞ is an unramified Zp -extension of K contained in K∞ . Set M∞ = L∞ (µp∞ );b ∞ ] is not rational over M∞ . Indeed,this is a Zp -extension of L∞ . We claim that E[pe ∞ ] is realized over L∞ , it is also realized over M∞ . If E[pb ∞ ] is also rational
since E[p∞over K∞ , then E[p ] must be rational over M∞ which in turn implies that K∞ = M∞ .
This contradicts the fact that G = Gal(K∞ /K) has dimension 3. By dimension counting,
Gal(K∞ /M∞ ) is a one-dimensional p-adic Lie group. By enlarging K, we may assume that
Gal(K∞ /M∞ ) ∼= Zp . For n  0, there exists a constant c independent of n such that

.


p n
b n+c ] .
Gal K∞ /M∞= Gal K∞ M∞ E[pnnNow, write Un = Gal(K∞ /M∞ )p and Vn = Gal(M∞ /K)p . For large enough n, theinflation-restriction sequence gives us the following exact sequence


Vn



b n+c ] −→ H 1 Gn , E[pb ∞ ] −→ H 1 Un , E[pb ∞]0 −→ H 1 Vn , E[p.

11b ∞ ]) = E[pb n+c ] is finite, it follows from an application of Lemma 2.3 that
Since H 0 (Un , E[pb ∞ ]) = 0. Now, by appealing to Lemma 2.2, we obtain the required equality,
H 1 (Un , E[pb ∞ ] = ordp H 1 Vn , E[pb n+c ] ≤ 2(n + c) = O(n).
ordp H 1 Gn , E[pWe now come to the situation when E does not have complex multiplication and has
good supersingular reduction. Under this assumption, the dimension of G is 4 (cf. [41, IV
A.2.2]). In particular, it is an open subgroup of GL2 (Zp ). By the discussion in [47, p. 302],
and upon enlarging K if necessary, we may assume that G = Z × H, where Z ∼= Zp and
n
cyc
∞
Z
p
n
H = Gal(K∞ /K ). We claim that E[p ] is finite, where Zn = Z . Suppose for now
that the claim holds. Then, by Lemma 2.3, we have H 1 (Zn , E[p∞ ]) = 0. Therefore, thespectral sequence
H i (Hn , H j (Zn , E[p∞ ])) =⇒ H i+j (Gn , E[p∞ ])degenerates to yield
H i Hn , E[p∞ ]Zn ∼= H i Gn , E[p∞ ] .
Z , we see that for sufficiently large n, Z pn = Gal
Writing L∞ = K∞E[pn+c ]K∞ /L∞
where c is a constant independent of n. By Lemma 2.1, we see that
ordp H 1 Gn , E[p∞ ] = ordp H 1 Hn , E[p∞ ]Zn ≤ 3 ordp E[p∞ ]Zn = O(n).

It therefore remains to verify that E[p∞ ]Zn is finite. Without loss of generality, it suffices
to show that E[p∞ ]Z is finite. Let T be the Tate module of E[p∞ ]Z . Then T ⊗ Qp
is a Gal(K/K)-submodule of Tp E ⊗ Qp . Since E is an elliptic curve without complex
multiplication, the latter is an irreducible Gal(K/K)-module by [41, IV 2.1]. So, T ⊗ Qp iseither zero or the whole of Tp E ⊗Qp . As E[p∞ ] is not realized over L∞ , we have T ⊗Qp = 0;equivalently, E[p∞ ]Z is finite.
Finally, we record an estimate on the Zp -coranks of the first cohomology groups inintermediate extensions of a general p-adic Lie extension. This result is independent of the
reduction type of E.
Proposition 4.8. Let K∞ be a d-dimensional uniform p-adic Lie extension of K. Then, 
dimZ/pZ H 1 Gn , E (K∞ ) [p∞ ] [p] = O(1).
In particular,corankZp H 1 Gn , E (K∞ ) [p∞ ] = O(1).
Proof. This follows from Lemma 2.1 and noting that h1 (Gn ) = d for all n.

Debanjana Kundu, Meng Fai Lim

12

5. A General Control Theorem
We fix an algebraic closure Q of Q. Any algebraic (possibly infinite) extension of Q isthen a subfield of this fixed algebraic closure, Q. When the subfield is a finite extensionof Q, we call it a number field. Let E be an elliptic curve defined over a number field,
F , and let S be a finite set of primes containing the primes above p, the primes of bad
reduction of E, and the Archimedean primes. Let FS be the maximal algebraic extensionof F unramified outside S. For every (possibly infinite) extension L of F contained in FS ,write GS (L) = Gal(FS /L). Let S(L) be the set of primes of L above S. If L is a finite
extension of F and w is a place of L, we write Lw for its completion at w; when L/F isinfinite, it is the union of completions of all finite sub-extensions of L.
Over each L, the p-primary Selmer group and p-primary fine Selmer group are defined
as follows
M


0 −→
Sel E/L −→H 1 GS (L), E[p∞ ] −→H 1 (Lw , E) [p∞ ],w∈S(L)0 −→
R E/L −→M


H 1 GS (L) , E[p∞ ] −→


H 1 Lw , E[p∞ ] .

w∈S(L)

Using the definition, it is a simple observation that R(E/L) is independent of the choiceof S (see [30, Lemma 4.1]). Indeed, this is because we have the following exact sequenceM0 −→ R E/L −→ Sel E/L −→E(Lw ) ⊗ Qp /Zp .
w|p

A uniform p-adic Lie extension F∞ of F is one where Gal(F∞ /F ) is a uniform group. For agiven uniform p-adic Lie extension F∞ contained in FS , define R(E/F∞ ) = limL R(E/L),−→
where L runs through the finite sub-extensions of F∞ /F . In other words, we have


M



R E/F∞ ∼Kv1 E/F∞  ,= ker H 1 GS (F∞ ) , E[p∞ ] −→v∈S
where Kv1 E/F∞ = limL w∈S(L) H 1 Lw , E[p∞ ] . For a finite extension, L/F , we shall−→L
sometimes write Kv1 (E/L) = w|v H 1 Lw , E[p∞ ] .
For a finite extension L of F contained in F∞ , we have the following commutative
diagram


0

−→

0

−→

R(E/L)

yrL
Gal(F∞ /L)
R E/F∞

L

−→

−→


H 1 GS (L) , E[p∞ ]yhL
Gal(F∞ /L)
1
H GS (F∞ ) , E[p∞ ]

−→

−→

M

Kv1 (E/L)
v∈S 

ygL
M 1
Gal(F∞ /L)
Kv E/F∞v∈S

with exact rows, where the maps hL and gL are the restriction maps on cohomology, and rL
is the map induced by these. The above diagram will be the main tool for our discussion.
We first prove a general result independent of any hypothesis on the reduction type.

13

For the remainder of the section, the elliptic curve E will be defined over the number
field, F , and F∞ will be a uniform p-adic Lie extension over F contained in FS with Galoisngroup, G = Gal(F∞ /F ). We will write Gn = Gp and denote by Fn the fixed field of Gn .
Theorem 5.1. Let E be an elliptic curve defined over F , and let F∞ be a d-dimensional
uniform p-adic Lie extension of F . Then the kernel and cokernel of the restriction map
G
rn : R E/Fn → R E/F∞ n
are cofinitely generated over Zp . Furthermore,corankZp (ker rn ) = O(1)

and



corankZp (coker rn ) = O p(d−1)n .

Proof. Consider the following diagram
0
0

−→
−→

R(E/F
 n)

yrn
G
R E/F∞ n

−→
−→


∞
H 1 GS (Fn ) , E[p ]yhnG
H 1 GS (F∞ ) , E[p∞ ] n−→

1
vn ∈S(Fn ) HLFn,vn , E[p∞ ]


ygn

−→

1
v∈S Kv

L

E/F∞



Gn

with exact rows. By the Hochschild-Serre spectral sequence, we have
ker hn = H 1 Gn , E(F∞ )[p∞ ]and coker hn ⊆ H 2 Gn , E(F∞ )[p∞ ] .
Lemma 2.1 asserts that dimZ/pZ (ker hn )[p] = O(1) and dimZ/pZ (coker hn )[p] = O(1).
In particular, corankZp (ker hn ) (hence, also corankZp (ker rn )) is finite and bounded independent of n.
For the estimate of coker rn , we now study the growth of corankZp (ker gn ). By Shapiro’s
Lemma,
M
ker gn =H 1 Gn,vn , E(F∞,vn )[p∞ ] .
vn ∈S(Fn )
If v is a prime which splits completely in F∞ /F , then H 1 Gn,vn , E(F∞,vn )[p∞ ] = 0 for
every vn above v. Thus, we consider primes which do not split completely
in F∞ /F .

By Proposition 4.1, for v - p, the group H 1 Gn,vn , E(F∞,vn )[p∞ ] is finite and has
no Zp -corank contribution. We are reduced to studying v|p which do not split completely in F∞ /F . In this case, the dimension of the decompositiongroup
 of G at v is(d−1)nat least 1, so the number of primes of Fn above each v is O p. By Proposi∞tion 4.8, each H 1 Gn,vn , E(F∞,vn )[p ] has bounded Zp -corank growth. It follows thatcorankZp (ker gn ) = O p(d−1)n . This completes the proof of the theorem.

Corollary 5.2. Retain the setting of Theorem 5.1. Then,





corankZp R(E/Fn ) − corankZp R(E/F∞ )Gn = O p(d−1)n .
Inparticular,
 R(E/F∞ ) is cotorsion over Zp JGK if and only if corankZp R(E/Fn ) =O p(d−1)n .

14Debanjana Kundu, Meng Fai Lim

Proof. The first assertion follows from Theorem 5.1. By Harris’ result (see Proposition 3.2),





corankZp R(E/F∞ )Gn = corankZp JGK R(E/F∞ ) pdn + O p(d−1)n .
The final assertion is now immediate from this and the above estimate.



We now consider an analogue of Theorem 5.1 for the p-rank.
Theorem 5.3. Retain the setting of Theorem 5.1. Then,





dimZ/pZ R(E/Fn )[p] − dimZ/pZ R(E/F∞ )Gn [p] = O p(d−1)n .
In particular, the Pontryagin dual of R(E/F∞ ) is a torsion Zp JGK-module with trivial
µG -invariant if and only if dimZ/pZ R(E/Fn )[p] = O p(d−1)n .
Proof. The proof of this estimate is similar to (and easier than) that of Theorem 5.1. Thus,







dimZ/pZ R(E/Fn )[p] = O p(d−1)n ⇔ dimZ/pZ R(E/F∞ )Gn [p] = O p(d−1)n .



 
By Pontryagin duality, the latter is equivalent to dimZ/pZ R(E/F∞ )∨ /p Gn = O p(d−1)n .
In view of Proposition 3.4, this is the same as µG R(E/F∞ )∨ /p = 0. Equivalently (seeLemma 3.3), R(E/F∞ )∨ is a torsion Zp JGK-module with trivial µG -invariant.
When G = Zp , the conclusion of the theorem was mentioned in [30, Proof of Theorem
5.5]. Thus, Theorem 5.3 is a generalization of the discussion there. This theorem is related
to a conjecture of Coates-Sujatha (see [6]) which we now describe.
Conjecture (Conjecture A). Let F cyc denote the cyclotomic Zp -extension of F . The fineSelmer group R(E/F cyc ) is cofinitely generated over Zp , i.e., R(E/F cyc )∨ is a torsion
Zp JΓK-module with trivial µΓ -invariant, where Γ = Gal(F cyc /F ).

By Theorem 5.3, Conjecture A holds if and only if dimZ/pZ R(E/Fn )[p] = O(1) (see [30,discussion in the proof of Theorem 5.5]). For a general p-adic Lie extension, the following
result holds.
Corollary 5.4. Let E be an elliptic curve defined over F , and let F∞ be a d-dimensional
uniform p-adic Lie extension of F contained in FS . Suppose that F∞contains F cyc . IfConjecture A of Coates-Sujatha holds, then dimZ/pZ R(E/Fn )[p] = O p(d−1)n .
Proof. If Conjecture A holds, R(E/F cyc )∨ is finitely generated over Zp . By [6, Lemma3.2], this implies R(E/F∞ )∨ is finitely generated over Zp JHK where H = Gal(F∞ /F cyc ).
Applying an observation of Howson [14, Lemma 2.7], we see that R(E/F∞ )∨ is torsion
over Zp JGK with trivial µG -invariant. The result follows from combining these observations
with Theorem 5.3.
Next, we prove an H-analogue of Theorem 5.1.

15

Theorem 5.5. Let E be an elliptic curve defined over F . Let F∞ be a uniform p-adic Lie
extension of F contained in FS with Galois group, G = Gal(F∞ /F ) of dimension d ≥ 2.
Suppose that F∞ contains F cyc . Write H = Gal(F∞ /F cyc ) and let Fn be the fixed field ofn
Hn := H p . Then the kernel and cokernel of the restriction mapH
sn : R E/Fn −→ R E/F∞ n
are cofinitely generated over Zp . Furthermore,corankZp (ker sn ) = O(1)

and



corankZp (coker sn ) = O p(d−2)n .

Proof. Consider the following diagram
0 −→

R(E/F
 n)

ysn
H
0 −→ R E/F∞ n∞
H 1 GS (Fn ) , E[p ]yβnH−→ H 1 GS (F∞ ) , E[p∞ ] n
−→

−→

1
v∈S K
v

L

E/Fn


yγn

−→

1
v∈S Kv

L

E/F∞



H n

with exact rows. The kernels and cokernels of the vertical maps are cofinitely generated
over Zp (see [6, Proof of Lemma 3.2]). An argument similar to Theorem 5.1 shows thatboth corankZp (ker βn ) and corank Zp (coker βn ) are bounded independent of n, and thatcorankZp (ker γn ) = O p(d−2)n .

Remark 5.6. In some special cases more can be said:(1) When E(F∞ )[p∞ ] is finite, both ker βn and coker βn are finite and bounded independent of n. In particular, |ker sn | = O(1).
 Indeed, by applying the inflation-restrictionsequence ker βn = H 1 Hn , E(F∞ )[p∞ ] and coker βn ⊆ H 2 Hn , E(F∞ )[p∞ ] . SinceE(F∞ )[p∞ ] is finite, it follows that ker βn and coker βn are finite. Upon noting thath1 (Hn ) = d − 1 for all n, it follows from Lemma 2.2 that the bound is independentof n.
(2) When F∞ = F (E[p∞ ]), both ker βn and coker βn are finite [5, Corollary 6]. Thus,
ker sn is finite. In fact, dimZ/pZ (ker sn )[p] = O(1), but we are unable to estimate
ordp |ker sn |.
(3) If F∞ /F is a p-adic Lie extension such that the primes in S are finitely decomposed,
then ker γn has finite Zp -corank bounded independent of n.
Before continuing further, we recall another conjecture of Coates-Sujatha [6, Conjecture B].
Conjecture (Conjecture B). Let F∞ be a p-adic Lie extension of F containing F cyc , whichis unramified outside a finite set of primes, and whose Galois group G = Gal(F∞ /F ) has
dimension ≥ 2. Then R(E/F∞ )∨ is a pseudo-null Zp JGK-module.
This conjecture is very much open. Some examples verifying Conjecture B are given
in [3, 17, 23, 25, 36, 43]. As a corollary of Theorem 5.5, we show that Conjecture B
can be characterized in terms of the growth of the fine Selmer groups in the intermediate
cyclotomic extensions.

Debanjana Kundu, Meng Fai Lim

16

Corollary 5.7. Retain the setting of Theorem 5.5. Suppose that R(E/F cyc ) is a cofinitely
generated Zp -module. Then every R(E/Fn ) is cofinitely generated over Zp and R(E/F∞ )is cofinitely generated over Zp JHK. Furthermore,





corankZp R(E/Fn ) − corankZp R(E/F∞ )Hn = O p(d−2)n .
In particular, R(E/F∞ )∨ is pseudo-null over Zp JGK if and only if
corankZp R(E/Fn ) = O p(d−2)n .
Proof. Since R(E/F cyc ) is a cofinitely generated Zp -module, we may apply [6, Lemma
3.2] to conclude that R(E/F∞ ) is cofinitely generated over Zp JHK. Once again applying [6,Lemma 3.2] yields that every R(E/Fn ) is also cofinitely generated over Zp . This establishes
the first assertion. The second assertion on the estimate is an immediate consequence ofTheorem 5.5. For the final assertion, we remind the readers of an equivalent definition ofpseudo-nullity (due to Venjakob) when R(E/F∞ ) is cofinitely generated over Zp JHK. By[48, Example 2.3 and Proposition 5.4], we know that R(E/F∞ )∨ is pseudo-null over Zp JGK if
and only if R(E/F∞ ) is cotorsion over Zp JHK. By the result of Harris (cf. Proposition 3.2),





corankZp R(E/F∞ )Hn = corankZp JHK R(E/F∞ ) p(d−1)n + O p(d−2)n .
The final assertion follows from combining these observations with the estimate of thecorollary.

6. Control Theorem over Certain p-adic Lie Extensions
In this section, we consider specific p-adic Lie extensions and prove sharper Control
Theorems for these extensions. These results are then applied to give asymptotic estimateson the growth of non-divisible part of fine Selmer groups in the intermediate subfield of
the p-adic Lie extension.
Throughout this section, F is a number field and F∞ is a uniform p-adic Lie extension
nof F with Galois group, G. We write Gn = Gp and denote by Fn the fixed field of Gn .
As a start, we record a finiteness result which will be useful for our discussion.
Lemma 6.1. Let E be an elliptic curve defined over F , and F∞ be a p-adic Lie extensionof F at which E[p∞ ] is not realized over F∞ . Suppose that at least one of the following
statements is valid.
(a) The elliptic curve, E, has no complex multiplication.
(b) The p-adic Lie extension, F∞ , contains the cyclotomic Zp -extension F cyc .
Then E(F∞ )[p∞ ] is finite.
Proof. If statement (a) holds, the conclusion follows from [28, Lemma 6.2]. When statement(b) is valid, the conclusion follows from [52, Proposition 10].


17

6.1. Zdp -extensions. When d = 1, the Control Theorem for fine Selmer groups has beenstudied in [27, 49]. Therefore, we concentrate on the case d ≥ 2. Pertaining to this,Rubin proved that the kernel and cokernel of the natural restriction map are finite (see[39, Chapter VII, §3]). Here, we refine this result by giving an estimate on the orders of
the kernels and cokernels.
Theorem 6.2. Let E be an elliptic curve defined over F , and F∞ be a Zdp -extension of F ,with d ≥ 2. Then the kernel and cokernel of the restriction map
G
rn : R E/Fn −→ R E/F∞ n
are finite. Furthermore,
ordp |ker rn | = O(n)

and



ordp |coker rn | = O p(d−1)n .

Proof. Consider the following diagram
0
0

−→
−→

R(E/F
 n)

yrn
G
R E/F∞ n

−→
−→


∞
H 1 GS (Fn ) , E[p ]

yhn
G
1
H GS (F∞ ) , E[p∞ ] n−→

1
vn ∈S(Fn ) HLFn,vn , E[p∞ ]


ygn

−→

1
v∈S Kv

L

E/F∞



Gn

with exact rows. By the Hochschild-Serre spectral sequence, we have
ker hn = H 1 (Gn , E(F∞ )[p∞ ])andcoker hn ⊆ H 2 (Gn , E(F∞ )[p∞ ]).

Using an argument similar to that in Proposition 4.3, we see that both H 1 (Gn , E(F∞ )[p∞ ])and H 2 (Gn , E(F∞ )[p∞ ]) are finite withorder O(n). In particular,
ordp |ker rn | =Lp-power1∞
O(n). It remains to estimate ker gn = vn H Gn,vn , E(F∞,vn )[p ] . As in the proof of
Theorem 5.1, we only need to consider the primes which do not split completely. Now, if
the decomposition group of G at the prime v is at least 2, it follows from Proposition 4.3
that H 1 Gn,vn , E(F∞,vn )[p∞ ] is finite with p-power order O(n). Since the decompositiongroup has dimension at least 2, the number of primes above such v grows like O p(d−2)n .
Hence, for such a prime v, we have
ordp

M






H 1 Gn,vn , E F∞,vn [p∞ ] = O np(d−2)n .

vn |v

If the decompositiongroup
at v has dimension 1, then the number of primes above such(d−1)na v grows like O p. But for these primes, we know that H 1 Gn,vn , E(F∞,vn )[p∞ ]is finite and the growth is bounded. This is Proposition 4.1(a) when v - p and Remark 4.4
when v|p. Hence,
ordp

M






H 1 Gn,vn , E F∞,vn [p∞ ] = O p(d−1)n .

vn |v

Combining these estimates yields the required estimate for ker gn ; hence, also for coker rn .
18Debanjana Kundu, Meng Fai Lim

Using a results of Liang-Lim, we estimate the growth of R(E/Fn )∨ [p∞ ].
Corollary 6.3. Let E/F be an elliptic curve, and F∞ be a Zdp -extension of F , with d ≥ 2.
Then





∨
∨  dnordp R E/Fn [p∞ ] = µG R E/F∞
p + O np(d−1)n .
If F∞ contains F cyc and Conjecture A of Coates-Sujatha holds, then




∨
ordp R E/Fn [p∞ ] = O np(d−1)n .
Finally, if F contains F cyc and Conjecture B of Coates-Sujatha is also valid, then




∨
ordp R E/Fn [p∞ ] = O p(d−1)n .
∨Proof. Since R E/F∞ is a finitely generated Zp JGK-module, by a result of Liang-Lim(cf [22, Theorem 2.4.1]) we know that





∨ 
∨  dn∞(d−1)nordp
R E/F∞[p ] = µG R E/F∞.
p + O np
Gn

The estimate in the corollary is now immediate from this and Theorem 6.2. For the second
assertion, it can be seen from the proof of Corollary 5.4 that if Conjecture A holds, then∨ 
µG R E/F∞= 0.
If both Conjecture A and Conjecture B are valid, applying [22, Proposition 2.2.1] we obtain




∨ 
ordp
R E/F∞[p∞ ] = O p(d−1)n .
GnCombining this with Theorem 6.2, we obtain the required estimate.



We mention another corollary of Theorem 6.2.
Corollary 6.4. Retain the settings of Theorem 6.2. Suppose that R(E/F ) is finite. Then
R(E/F∞ )∨ is torsion over Zp JGK.
Proof. By Theorem 6.2 and hypothesis of the corollary, R(E/F∞ )G is finite. The conclusionof the corollary now follows from this and the main theorem of [2, pp. 5-6].

In view of the above result, we make the following conjecture.
Conjecture (Conjecture Yd ). Let E be an elliptic curve defined over F , and F∞ be a
Zdp -extension of F . Then, R(E/F∞ )∨ is torsion over Zp JGK.
When d = 1, the above conjecture is made in [27, Conjecture Y].

19

6.2. Multi-False-Tate extensions. In this subsection, we suppose that the number field,
F , contains
p-th root of unity. For d ≥ 2, we will consider the extension
 a primitive
√
√
∞
∞
p
p
α1 , . . . ,αd−1 , where α1 , . . . , αd−1 ∈ F × whose image in F × /(F × )p
F∞ = F µp∞ ,∼ Zd−1 o Zp and we call F∞ /F a
are linearly independent over Fp . Then Gal(F∞ /F ) =pmulti-false-Tate curve extension of dimension d. Throughout this subsection, we assume
that E has (potential) good reduction at primes above p.
Theorem 6.5. Let E be an elliptic curve defined over a number field F with potential good
reduction at every prime of F above p. Let F∞ /F be a multi-false-Tate curve extension ofdimension d ≥ 2 which is contained in FS . Then the kernel and cokernel of the natural
restriction map
G
rn : R E/Fn −→ R E/F∞ nare finite with ker(rn ) = O(1) and ordp coker(rn ) = O p(d−1)n . If the dimension of the
decomposition group of G at every v ∈ S is at least 2, then ordp coker(rn ) = O np(d−2)n .
Proof. As before, we start by considering the following commutative diagram
0
0

−→
−→

R(E/F
 n)

yrn
G
R E/F∞ n

−→
−→


∞
H 1 GS (Fn ) , E[p ]

yhn
G
1
H GS (F∞ ) , E[p∞ ] n−→

1
vn ∈S(Fn ) HLFn,vn , E[p∞ ]


ygn

−→

1
v∈S Kv

L

E/F∞



Gn

By Lemma 6.1, E (F∞ ) [p∞ ] is finite. Now by applying the inflation-restriction sequence
and using Lemma 2.2, we see that ker(hn ) and coker(hn ) are finite and bounded independent of n.
We will now estimate ker gn . For the primes not above p, using Proposition 4.1 weconclude that
ordp

M




H 1 Gn,vn , E F∞,vn [p∞ ]

vn |v

 


O p(d−1)n ,=O np(d−2)n ,if the decomposition group at v has dimension 1,
if the decomposition group at v has dimension 2.

Since we assume that the elliptic curve E has potential
good reduction
at all primes
 ∞
1above p, it follows from Proposition 4.5 that H Gn,vn , E F∞,vn [p ] is finite withbounded growth. Thus,
ordp

M
vn |v




H 1 Gn,vn , E F∞,vn [p∞ ]Debanjana Kundu, Meng Fai Lim

20

 


O p(d−1)n ,=O p(d−2)n ,if the decomposition group at v has dimension 1,
if the decomposition group at v has dimension ≥ 2.

The assertions of the theorem are now immediate from these estimates.

Using a result of Perbet (see [37, Théorème 2.1]) we estimate the growth of R(E/Fn )[pn ].
Corollary 6.6. Let E be an elliptic curve defined over a number field F with potential good
reduction at every prime of F above p. Let F∞ /F be a multi-false-Tate curve extension ofdimension d ≥ 2 which is contained in FS . Then





 

∨  dn∨ 
p + O np(d−1)n .
n + µG R E/F∞
ordp R E/Fn [pn ] = rankZp JGK R E/F∞

If the Conjecture A of Coates-Sujatha holds, then





ordp R E/Fn [pn ] = O np(d−1)n .
∨Proof. Since R E/F∞ is a finitely generated Zp JGK-module, the result of Perbet implies
that
!
ordp




∨ 
R E/F∞

pn



=

Gn






∨ 
∨  dn
rankZp JGK R E/F∞
n + µG R E/F∞
p + O np(d−1)n .

The estimate is now immediate from Theorem 6.5. For the final remark, one sees thatfrom the proof of Corollary 5.4 that if Conjecture A holds, then


∨ 
∨ 
rankZp JGK R E/F∞
= µG R E/F∞
= 0.
Under the validity of Conjecture B of Coates-Sujatha, we have an even better upperbound.
Corollary 6.7. Retain the settings of Theorem 6.5. If both Conjecture A and Conjecture
B of Coates-Sujatha hold, then





ordp R E/Fn [pn ] = O p(d−1)n .
∨Proof. Since R E/F∞ is a finitely generated Zp JHK-module by the validity of Conjecture
A, we may apply [26, Proposition 2.4] to obtain
 !




∨ 
∨  (d−1)nordp
R E/F∞pn ≤ rankZp JHK R E/F∞
np
+ O p(d−1)n .
Gn


∨ 
If Conjecture B holds, then as in the proof of Corollary 5.7, rankZp JHK R E/F∞
= 0.
Hence,
 !



∨ 
ordp
R E/F∞
pn = O p(d−1)n .
GnCombining this with Theorem 6.5, we obtain the required estimate.

21

6.3. Trivializing Extension. We now consider the case of the trivializing extension. The
case of elliptic curve with and without complex multiplication will be treated separately.
When E has complex multiplication, it is well known that the Galois group Gal(F∞ /F )with F∞ = F (E[p∞ ]) contains an open subgroup which is Abelian and isomorphic to
Zp × Zp . In this situation, Theorem 6.2 (with d = 2) says that ordp (ker rn ) = O(n) andordp (coker rn ) = O (pn ). In the next theorem, we show that a sharper estimate for thecokernel is possible in this case.
Theorem 6.8. Let E/F be an elliptic curve with complex multiplication. Suppose thatF∞ = F E[p∞ ] and G = Gal(F∞ /F ) is uniform. Then the kernel and cokernel of therestriction mapsG
rn : R E/Fn −→ R E/F∞ n
are finite. Furthermore, ordp |ker rn | = O(n) and ordp |coker rn | = O(n).
Proof. In view of the proof of Theorem 6.2, it suffices to show that ordp |coker rn | = O(n).
This in turn reduces us to showing that ker gn has the same growth. Let K be the imaginary
quadratic field which gives E the complex multiplication. By base-changing, we may
assume that E[p] is realized over F , that K ⊆ F , that E has good reduction at every primeof F , and that Gal(F∞ /F ) ∼= Z2p . Choose S to be the set of primes above p and the infinite
primes. From the theory of complex multiplication, we see that F∞ is the compositum of
F and K∞ , where K∞ is the unique Z2p -extension of K. By [21, Théorème 3.2], there are
finitely many primes of K∞ above p. So, there are finitely many primes of F∞ above p.
Combining this latter observation with Proposition 4.3, shows that ordp |ker gn | = O(n). 
In this situation, Corollary 6.3 (or Theorem 6.8) yields∨∨  2nordp R E/Fn [p∞ ] = µG R E/F∞
p + O (npn ) .
In particular, if Conjecture A of Coates-Sujatha holds, then∨
ordp R E/Fn [p∞ ] = O (npn ) .
There is one case where we have the above estimate without assuming Conjecture A holds.
Proposition 6.9. Let K be an imaginary quadratic field at which the prime p splits completely in K, say p = pp. Let F0 be a finite extension of K which is unramified at p. Let
E be an elliptic curve defined over F0 which satisfies all the following properties.
(a) E has complex multiplication given by the ring of integers of K.
(b) E has good ordinary reduction at all primes of F0 above p.
(c) F0 (Etor ) is an Abelian extension of K.
Let F = F0 (E[p]) and F∞ = F0 (E[p∞ ]) = F (E[p∞ ]). Then∨
ordp R E/Fn [p∞ ] = O (npn ) .
∨ Proof. Under the hypothesis of the proposition, one can show that µG R E/F∞
=0
by appealing to the results of Gillard and Schneps. (For the details of this argument, weDebanjana Kundu, Meng Fai Lim

22

refer readers to [31, Proposition 4.1].) Hence, it follows from this that we have the estimate
as asserted.

We now come to the case of an elliptic curve without complex multiplication.
Theorem 6.10. Let E be an elliptic
curve defined over F without complex multiplication.
Suppose that F∞ = F E[p∞ ] and G = Gal(F∞ /F ) is uniform. Then the kernel andcokernel of the restriction mapsG
rn : R E/Fn −→ R E/F∞ n
are finite.
 Furthermore, the power of p in ker rn is O(n) and the power of p in coker rn is
O np2n .
In [40], Serre proved that H i Gn , E[p∞ ] is finite for every i ≥ 0. For our purpose, weneed to go one step further by analyzing its growth (for i = 1, 2), which is the content of
the next lemma.
Lemma 6.11. Retain the setting of Theorem 6.10. Then for i = 1, 2, the groups
H i (Gn , E[p∞ ]) are finite and ordp H i Gn , E[p∞ ] = O(n).
Proof. Since E is an elliptic curve without complex multiplication, the group G has dimension 4. By the discussion in [47, p. 302], and enlarging F if necessary, we may assume that
G = Z × H, where Z ∼= Zp and H = Gal(F∞ /F cyc ). By Lemma 6.1(a), E[p∞ ]Zn is finite;
it follows from Lemma 2.3 that H 1 (Zn , E[p∞ ]) = 0. Therefore, the spectral sequence
H i (Hn , H j (Zn , E[p∞ ])) =⇒ H i+j (Gn , E[p∞ ])degenerates to yield
H i Hn , E[p∞ ]Zn ∼= H i Gn , E[p∞ ] .
Now, applying Lemma 2.1, we see that
ordp H 1 Gn , E[p∞ ] = ordp H 1 Hn , E[p∞ ]Zn ≤ 3 ordp E[p∞ ]Zn = O(n) and
 



3
2
∞
2
∞ Zn
≤
ordp E[p∞ ]Zn = O(n).
ordp H Gn , E[p ] = ordp H Hn , E[p ]2This completes the proof of the lemma.

Proof of Theorem 6.10. In view of Lemma 6.11, it remains to study the growth of ker gn .
By base-changing, we may assume that E has no additive reduction outside p. Let S be theset of primes of F consisting of the primes above p, the multiplicative primes of E outsidep, and the Archimedean primes. For all v ∈ S, the decomposition group has dimension ≥ 2(see [4, Lemma 2.8]). The rest of the argument proceeds as before building on Propositions
4.1 and 4.7.


23

Remark 6.12. When the elliptic curve has potential good ordinary reduction at all primes
above p, more can be said. Indeed, under this assumption, Greenberg showed (cf. [11,Proposition 5.3]) that the kernel of the restriction map on the classical Selmer groups
sn : Sel(E/Fn ) −→ Sel(E/F∞ )Gn is finite and bounded. Hence, ker rn is also bounded inthis case.
Corollary 6.13. Let E be an elliptic curve defined over a number field F without complex
multiplication and F∞ /F be the trivializing extension. Then






∨  4nordp R E/Fn [pn ] = µG R E/F∞
p + O np3n .
If Conjecture A of Coates-Sujatha holds, then





ordp R E/Fn [pn ] = O np3n .
Proof. Since F∞ /F is the trivializing extension, it follows from [6, Lemmas 2.4 and 3.1]∨
that R E/F∞ is a torsion Zp JGK-module. By [37, Théorème 2.1], we have
 !




∨ 
∨  4nordp
R E/F∞
pn = µG R E/F∞
p + O np3n .
Gn

The estimate follows from Theorem 6.10. The final assertion follows from Corollary 5.4. 
Corollary 6.14. Let E be an elliptic curve defined over a number field F and F∞ /F be
the trivializing extension. If both Conjecture A and Conjecture B of Coates-Sujatha hold,
then


∨
(a) ordp R E/Fn [p∞ ] = O (pn ) if E is an elliptic curve with complex multiplication.
(b) ordp R E/Fn [pn ] = O p3n if E does not have complex multiplication.
Proof. In the case when E is an elliptic curve with complex multiplication, the assertion
follows from Corollary 6.3 with d = 2. When E does not have complex multiplication, theproof of the statement is similar to the argument in Corollary 6.7.

Remark 6.15. It is natural to ask if the above growth formula can allow us to estimate thegrowth of fine Shafarevich-Tate groups (see [50] for the definition) as done for the classicalShafarevich-Tate groups in [10]. A key ingredient used in this proof is the p-divisibilityof E (Fn ) ⊗Zp Qp /Zp . However, as the fine Mordell-Weil group need not be p-divisible(see [50, §7]), the torsion part of the Pontryagin dual of the fine Shafarevich-Tate group
and R(E/Fn )[p∞ ]∨ need not agree. Therefore, the above asymptotic formula for the fineSelmer group does not carry over to the fine Shafarevich-Tate group automatically. For
Zp -extensions, asymptotic growth estimates for the fine Shafarevich-Tate group has beenobtained by the second named author in [27] under certain hypotheses. Unfortunately, theresults rely on finer aspects of the structure theory of Zp JΓK-modules which are not available
for more general Iwasawa algebras, and so the techniques of [27] do not carry over.

24

Debanjana Kundu, Meng Fai Lim

7. Examples
Example 7.1. Let E be the elliptic
curve defined byy 2 + y = x3 − x. Let p = 5. By [12, ∞
Example 4.13], we know that Sel E Q(µ5∞ , α−5 ) = 0 for every α = (±5)m . From this,∞

it is then easy to verify that for every finite extension F of Q contained in Q(µ5∞ , α−5 ),one has Sel(E/F ) = 0 which in turn implies that R(E/F ) = 0. By Corollary 6.4, it follows
that R(E/F∞ ) is cotorsion over Z5 JGal(F∞ /F )K for every multiple Z5 -extension F∞ of F .
Example 7.2. Let E be the elliptic curve 150a1 of Cremona’s table which is given by
y 2 + xy = x3 − 3x − 3.
Let p = 5 and F = Q(µ5 ). Let S be the set of primes of F lying above 2, 3, 5, and ∞. Then,
E has good ordinary reduction at the unique prime of F above 5 and split multiplicative
reduction at the unique primes of F above 2 and 3. By [3, Example 23] and [25, §6],
we know that R(E/F∞ )∨ is a pseudo-null Z5 JGal(F∞ /F )K-module when F∞ is one of thefollowing 5-adic extension








−∞
−∞
−∞
−∞
−∞
−∞
−∞
−∞
Q E[5∞ ], 35, Q E[5∞ ], 25 , 35
, Q E[5∞ ], 35 , 55
, Q E[5∞ ], 25 , 35 , 55
,






−∞
−∞
−∞
−∞
−∞
−∞
−∞
,
, L∞ E[5∞ ], 25 , 35 , 55
, L∞ E[5∞ ], 35 , 55
L∞ E[5∞ ], 25 , 35

where L∞ is any Zr5 -extension of F for 1 ≤ r ≤ 3. Therefore, Corollary 5.7 applies to yieldn
Gal(F /F cyc )5
corankZ5 R(E/Fn ) = O 5(d−2)n , where Fn = F∞ ∞.
Example 7.3. Let E be the elliptic curve 79a1 of Cremona’s tables given by
y 2 + xy + y = x3 + x2 − 2x.
Let p = 3 and F = Q(µ3 ). Let S be the set of primes of F lying above 3, 79 and ∞. By[17, p. 362] and [25, §6], we see that R(E/F∞ )∨ is a pseudo-null Z3 JGal(F∞ /F )K-module
when F∞ is one of the following 3-adic extensions




−∞
−∞
−∞
Q µ3∞ , 33
, Q µ3∞ , 33 , 793
.




Corollary 6.7 then tells us that ord3 R E/Fn [3n ] is O (3n ) and O 32n for the above
two extensions, respectively.
Example 7.4. Let E be one of the following elliptic curves(with Cremona label): 256a1,√
256a2, 256d1, 256d2 withcomplexmultiplication
by
Q(
−2)
or 121b1, 121b2 with complex√
multiplication by Q( −11). These elliptic curves have good reduction at p = 3. In [1,Theorem 3.11], it was shown that these satisfy Conjecture A for p = 3 over the base field
F = Q(E[3]) which is non-Abelian over Q. However, it does not seem that the validityof Conjecture B is known for any p-adic Lie extension over F . Nevertheless, we can still
apply Corollary 6.6 to obtain





ord3 R E/Fn [3n ] = O n3(d−1)n .

25for every multi-false-Tate extension, F∞ of F of dimension d ≥ 2, and apply Theorem 6.8
to yield
ord3 R E/Fn [3n ] = O (n3n ) ,when F∞ = F E[3∞ ] is the trivializing extension over F .
Example 7.5. Let E√be the elliptic curve 3136u1 from Cremona’s table with complex
multiplication by Q( −1). This is given by the equationy 2 = x3 − 343x.
This elliptic curve has good ordinary reduction at all primes satisfying p ≡ 1 (mod 4).
For p large, the validity of Conjecture
A does not appear to be verified in literature.

Nevertheless, when F∞ = F E[p∞ ] is the trivializing extension over F , we can still apply
Proposition 6.9 to obtain∨
ordp R E/Fn [p∞ ] = O (npn ) .
Acknowledgements
The authors thank Prof. Kumar Murty, Prof. John Coates, and Antonio Lei for
their interest and comments on an earlier draft. D. Kundu thanks Centre de Recherches
Mathématiques (CRM) for their hospitality and support where parts of this work were carried out during the thematic semester “Number Theory - Cohomology in Arithmetic” inFall 2020. She acknowledges the support of the PIMS Postdoctoral Fellowship. M. F. Lim
acknowledges support by the National Natural Science Foundation of China under Grant
Nos. 11550110172 and 11771164, and the Fundamental Research Funds for the Central
Universities of CCNU under grant CCNU20TD002. We thank the anonymous referee for
their comments which helped improve the exposition of the article.
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(Kundu) Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada,
V6T1Z2E-mail : dkundu@math.ubc.edu(Lim) School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences
Central China Normal University, Wuhan, 430079, P.R.China.
E-mail : limmf@mail.ccnu.edu.cn