STRUCTURE OF FINE SELMER GROUPSIN ABELIAN p-ADIC LIE EXTENSIONSDEBANJANA KUNDU, FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO,AND SUJATHA RAMDORAI

Abstract. This paper studies fine Selmer groups of elliptic curves in abelian p-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer
group are trivial in the cyclotomic Zp -extension. The fine Selmer groups of elliptic curves withcomplex multiplication are shown to be pseudonull over the trivializing extension in some newcases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic
curves and the Generalized Greenberg’s Conjecture is clarified.

1. Introduction
The fine Selmer group (see §2.3) is a module over an Iwasawa algebra that is of interest in thearithmetic of elliptic curves. It plays a key role in the formulation of the main conjecture in Iwasawa
theory. Moreover, it enables us to propose analogues of important conjectures in classical Iwasawa
theory to elliptic curves over certain p-adic Lie extensions of their field of definition. J. Coates
and the third named author initiated a systematic study of the structure of fine Selmer groups andproposed two conjectures (see [CS05b, Conjectures A and B]). While Conjecture A is a generalization of the Iwasawa µ “ 0 Conjecture to the context of elliptic curves, Conjecture B is in thespirit of generalizing R. Greenberg’s pseudonullity conjecture to elliptic curves. Recently, there has
been a renewed interest in studying pseudonull modules over Iwasawa algebras, [BCG` 20, LP19].
It is thus natural to investigate Conjecture B, and this article makes progress in this direction.
These conjectures have been generalized to fine Selmer groups of ordinary Galois representations
associated to modular forms in [JS11], and their mod p-versions for supersingular elliptic curves
have been studied by the second and third author in [NS21]. This article restricts attention to
the fine Selmer groups of elliptic curves, with good reduction at a prime p, over abelian p-adic Lie
extensions of the base field.
We now outline the main results in the paper. Given a number field F and an odd prime
number p, let E{F be an elliptic curve, with good reduction at all the primes of F that lie above
p. Consider an admissible p-adic Lie extension L of F (see §2.2 for the precise definition) with
Galois group GalpL{F q “: GL{F . The dual fine Selmer group of E at a prime p over L is a finitely
generated module over the associated Iwasawa algebra (see §2.3). While Conjecture A assertsthat the dual fine Selmer group over the cyclotomic Zp -extension Fcyc {F is finitely generated asa Zp -module, Conjecture B is an assertion on the structure of the dual fine Selmer group overadmissible p-adic Lie extensions of dimension at least 2. This conjecture predicts that the dual
Date: December 23, 2022.
2020 Mathematics Subject Classification. Primary 11R23.
Key words and phrases. Fine Selmer groups, pseudonull, Conjecture A, Conjecture B, Generalized Greenberg’s
Conjecture.
1

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D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

fine Selmer group over any admissible p-adic Lie extension is pseudonull as a module over the
associated Iwasawa algebra. In this article, both conjectures are established in previously unknown
cases. Using a result of Greenberg, we prove a general theorem that gives sufficient conditions forthe dual fine Selmer group of E over the cyclotomic Zp -extension Fcyc to be trivial. More precisely,
we have the following theorem (we refer the reader to Corollary 3.5 for finer estimates):
Theorem 3.1. Let E{F be the base-change of a rational elliptic curve E{Q. Suppose that it has
rank 0 over F and that the Shafarevich–Tate group of E{F is finite. When E has CM by an orderof an imaginary quadratic field K, assume
that the Galois closure of F , denoted by F c ,` further˘
contains K. Then, the Selmer group Sel E{Fcyc is trivial for a set of prime numbers of density atleast rF c1:Qs . In particular, Conjecture A holds for E{F at all such primes.
Denote by F pEp8 q the field obtained by adjoining the coordinates of all p-power torsion points.
When p is a prime of good ordinary reduction, using a result of B. Perrin-Riou [PR81, Lemme 1.1(i)and Lemme 1.3] we prove that Conjecture B holds for special classes of admissible p-adic Lie
extensions whenever the dual fine Selmer group over the cyclotomic extension is finite for a CM
elliptic curve. We obtain the following result:
Theorem 4.6. Let E{F be an elliptic curve defined over a number field F . Suppose that F contains
the imaginary quadratic field K and that E has CM by OK . Assume further that p ě 3 is a primeof good ordinary reduction that splits in K and that GalpF pEp8 q{F q » Z2p . If the fine Selmer groupover the cyclotomic Zp -extension Fcyc {F is finite, then Conjecture B holds for pE, F pEp8 qq.
Over the cyclotomic Zp -extension Fcyc of F , there is a connection between the Galois group of
the maximal abelian unramified pro-p extension of Fcyc and the fine Selmer groups of elliptic curves
defined over F , see [CS05b, Theorem 3.4]. This phenomenon can be extended to (both abelian andnon-abelian) admissible p-adic Lie extensions of higher dimension. In fact, Conjecture B can beviewed as an elliptic curve analogue of an old conjecture of Greenberg on Galois modules associatedwith pro-p Hilbert class fields (see §2.4 for the precise statement). This has been explored in [CS05b,
p. 827]. It is therefore pertinent to investigate the precise connections between Conjecture B for
admissible, abelian p-adic Lie extensions, and Greenberg’s conjecture. For CM elliptic curves, theGeneralized Greenberg’s Conjecture is shown to be equivalent to Conjecture B for certain admissible
pro-p, p-adic Lie extensions in Theorem 4.10. This result provides a framework for proving newcases of the Generalized Greenberg’s Conjecture. In particular, we prove the following result1.
Theorem 5.4 and Corollary 5.5. Let K{Q be an imaginary quadratic field. If there exists
one CM elliptic curve E{K such that the dual fine Selmer group is pseudonull over the trivializing
extension KpEp8 q, then the Generalized Greenberg’s Conjecture holds for K and KpEp q.
Little is known about Conjecture B and the Generalized Greenberg’s Conjecture in full generality.
We recall some cases where Conjecture B is proven in the literature. When there is a unique prime
above p in the p-adic Lie extension of interest Conjecture B is proven in [Och09, Theorem 1.3] and[She18, § 4]. Also, when the p-adic Lie extension has large dimension there are explicit examples
where Conjecture B is known, detailed in [Bha07, Example 23]. Certain analogues of Conjecture B
have also been considered in [Jha12, LP19]. For evidence towards the Generalized Greenberg’s
Conjecture (both theoretical and computational) see [Tak21, Remark 1.3], as well as [Min86, McC01,
Oza01, NQDV05, Sha08, Fuj17]. As per the knowledge of the authors, most results in this latter
1The proof of Theorem 5.4 does not require E to be defined over K. This formulation is used in this introduction,
for simplicity.

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

3

direction require the crucial hypothesis that p does not divide the class number of the number
field. One exception is the result of R. Sharifi and W. McCallum, where the conjecture for Qpµp q
is proven under certain assumptions on a cup-product (see [MS03, Corollary 10.5]); another isof Sharifi [Sha08, Theorem 1.3], where computational evidence for the Generalized Greenberg’s
Conjecture is provided when F “ Qpµp q and p ă 1000 is an irregular prime. Our approach suggests
a new line of attack for the Generalized Greenberg’s Conjecture even in the case when p divides
the class number of the base field.
The paper consists of five sections. Section 2 is preliminary in nature; wherein we recall the
precise assertions of Conjecture A, Conjecture B, and the Generalized Greenberg’s Conjecture andwe introduce the main objects of study. In Section 3, new evidence for Conjecture A is provided
by proving the triviality of the fine Selmer group over the cyclotomic extension. Some simple casesof Conjecture B are proven in Section 4. In Section 5 the relation between Conjecture B for CM
elliptic curves and the Generalized Greenberg’s Conjecture is clarified.
2. Preliminaries
Throughout this article, p denotes an odd prime number. For an abelian group M and a positive
integer n, write Mpn for the subgroup of elements of M annihilated by pn . PutďnMp8 :“Mpn , Tp pM q :“ limÐÝ Mp .
ně1

and, when M is a discrete p-primary (resp. compact pro-p) abelian group M , its Pontryagin dual
is defined as
M _ “ Homcont pM, Qp {Zp q.
Given any p-adic analytic group G, its Iwasawa algebra is defined as
ΛpGq “ limÐÝ Zp rG{U s
U

for U running through all open, normal subgroups of G. When G is compact and p-valued in thesense of M. Lazard, ΛpGq is a noetherian Auslander regular ring (see [CSS03, Proposition 6.2]). In
the special case when G is abelian with no elements of order p, there is an isomorphismΛpGq » Zp JT1 , . . . , Td K.

where d is the dimension of G as a p-adic analytic manifold. If M is a compact (resp. discrete)ΛpGq-module then its Pontryagin dual is discrete (resp. compact). Given a finitely generated ΛpGqmodule M , its Krull dimension is defined as the Krull dimension of ΛpGq{ AnnpM q and it is denoted
dimpM q.
2.1. Suppose that G is an abelian p-analytic group without elements` of order˘ p. A finitely generated`ΛpGq-moduleMistorsion(resp.
pseudonull
)
if
dimpM
q
ď
dim
ΛpGq
´ 1 (resp. dimpM q ď˘
dim ΛpGq ´ 2). Equivalently (see [Ven02, p. 273]), M is pseudonull if there exists a prime ideal psuch thatAnnΛpGq pM q :“ ta P ΛpGq : aM “ 0u Ď p
and htppq ě 2 (see [NSW08, Definition 5.1.4]).
Let W (resp. M ) be a discrete (resp. compact) G-module. The profinite cohomology groups(resp. homology groups) of W (resp. M ) are denoted H i pG, W q (resp. Hi pG, M q). The subgroupof elements of W fixed by G is denoted W G , and MG denotes the largest quotient of M on which
G acts trivially.

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D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

2.2. For a number field F , denote by Fcyc its cyclotomic Zp -extension. Suppose that S “ SpF q isa finite set of primes of F containing the primes above p and the archimedean primes. Let FS be
the maximal extension of F unramified outside S and set GS pF q “ GalpFS {F q. For any (finite or
infinite) extension L{F contained in FS , denote by GS pLq the Galois group GalpFS {Lq. Throughout
the paper, the focus is on S-admissible p-adic Lie extensions L{F , in the following sense:
Definition 2.1. An S-admissible p-adic Lie extension is a Galois extension L{F satisfying thefollowing conditions:`˘‚ the group Gal L{F is a pro-p, p-adic Lie group with no elements of order p;‚ the field L contains the cyclotomic Zp -extension Fcyc ;‚ the field L is contained in FS .
Next, we record some conjectures pertaining to the modules associated with maximal abelianunramified pro-p extension of admissible p-adic Lie extensions. The first conjecture we mention wasformulated by K. Iwasawa in [Iwa73b, pp. 1–2] for the cyclotomic Zp -extension.
Iwasawa µ “ 0 Conjecture. Let LpFcyc q denote the maximal abelian unramified pro-p extensionof Fcyc and set
Fcyc
Xnr“ GalpLpFcyc q{Fcyc q.
FThen, the µ-invariant associated with Xnrcyc is trivial.
F

In [Iwa73a, Theorem 5], Iwasawa proved that Xnrcyc is a torsion ΛpΓq-module; in view of thisF
result, the Iwasawa µ “ 0 Conjecture is equivalent to saying that Xnrcyc is finitely generated over
Zp . When F {Q is an abelian extension, the Iwasawa µ “ 0 Conjecture is known to be true by thework [FW79] by B. Ferrero and L. Washington.
Next, we mention a conjecture of Greenberg (see [Gre01b, Conjecture 3.5]) which is formulated
for certain abelian p-adic Lie extensions.
Generalized Greenberg’s Conjecture. Let Fr denote the compositum of all Zp -extensions
of
´
¯F
r
r
r
r
and let LpF q denote the maximal abelian unramified pro-p extension of F . Then Gal LpF q{F isa pseudonull module over the Iwasawa algebra ΛpGalpFr{F qq “ Zp JGalpFr{F qK.

2.3. Fix a number field F and an admissible extension L{F . Write GL{F for the compact, pro-p,p-adic Lie group GalpL{F q and ΛpGL{F q for the associated Iwasawa algebra. The main objectsof study will be modules over ΛpGL{F q that arise in Iwasawa theory, such as the Selmer groupand the fine Selmer group. Let E be an elliptic curve defined over F . Choose a set S “ SpF q
containing the primes above p, the primes of bad reduction of E{F , and the archimedean primes.
Write S Ě Sp Y Sbad Y S8 , where the notation Sp , Sbad , and S8 are self-explanatory. For a finite
extension L{F and a prime v of F , define
à 1
à 1`
˘
(1)
Jv pLq “
H pLw , Eq ppq,
andKv pLq “
H Lw , Ep8
w|v

w|v

where the direct sum is taken over all primes w of L lying above v. Taking direct limits, define
Jv pLq “ limÝÑ Jv pLq,
LandKv pLq “ limÝÑ Kv pLq
L

where L varies over finite sub-extensionsof L{F . Given any finite extension `L{F ˘contained in L,`˘
the p-primary Selmer group Sel E{L and the p-primary fine Selmer group R E{L are defined by

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

the exactness of the following sequences:`˘
0 ÝÑ Sel E{L ÝÑ`˘0 ÝÑ R E{LÝÑ`˘
H 1 GS pF q, Ep8
ÝÑ
˘
`
ÝÑ
H 1 GS pF q , Ep8àvPSpLqà5

Jv pLq,Kv pLq.

vPSpLqMoreover, by [CS05b, Equation (58)] we can relate these groups as follows
à `
`
˘
`
˘
˘
(2)
0 ÝÑ R E{L ÝÑ Sel E{L ÝÑE pLw q b Qp {Zp .
wPSp pLq

Define SelpE{Lq “ limÝÑL SelpE{Lq and RpE{Lq “ limÝÑL RpE{Lq. It can then be shown (see [CS00,
pp. 14–15] and [CS05b, Equation (46)]) that
¸
˜
à
˘
`
`
˘
1
Jv pLqSel E{L – ker H GS pLq , Ep8 ÝÑ
vPS

and
˜
`

˘

R E{L – ker H

¸
1

`

˘

GS pLq , Ep8 ÝÑàKv pLq .

vPS

Taking direct limits of (2), we obtain that`˘`˘0 ÝÑ R E{L ÝÑ Sel E{L ÝÑ lim
ÝÑ

à

`

˘
E pLw q b Qp {Zp .

L wPSp pLq

Finally, we set a notation for the Pontryagin dual of these groups:
`
˘_
`
˘_
(3)
XpE{Lq :“ Sel E{Land YpE{Lq :“ R E{L .
These are compact ΛpGL{F q-modules and it follows from (2) that YpE{Lq is a quotient of XpE{Lq.
In this paper, we are interested in a certain class of S-admissible p-adic Lie extensions generated
by the p-primary torsion points of an elliptic curve. When the elliptic curve E{F is clear from thecontext, we writeďF8 :“F pEpn q.
ně1

It follows from the Weil pairing that F8 contains Fcyc and the choice of S ensures that F8 iscontained in FS . The Galois group GalpF8 {F q has no p-torsion if p ě 5 (see, for example, [How98,
Lemma 4.7]) and contains an open, normal, pro-p subgroup (see [DdSMS99, Corollary 8.34]). In
fact, the extension F8 {F pEp q is always pro-p and hence S-admissible. If E is an elliptic curve withCM, and F contains the field of complex multiplication, then GalpF8 {F q contains an open subgroupwhich is abelian and isomorphic to Z2p .
2.4. Fix a number field F . In this section, we record the two conjectures formulated by Coates
and the third named author in [CS05b] which will be studied in this paper.
Conjecture A. ([CS05b, Section 3]) Let E be an elliptic curve defined over F . Then YpE{Fcyc q isa finitely generated Zp -module.
This conjecture is closely related to the Iwasawa µ “ 0 Conjecture. Their connection can bemade precise:
Theorem 2.2 ([CS05b, Theorem 3.4]). Let E{F be an elliptic curve and suppose that GalpF8 {F qis pro-p. Then Conjecture A for E{F is equivalent to the Iwasawa µ “ 0 Conjecture for F .

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D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

In [Ari14, Theorems A,C], there are more examples for which Conjecture A holds.
The dimension theory for finitely generated modules over Iwasawa algebras allows framing ananalogue of the Generalized Greenberg’s Conjecture in a more general setting. This is Conjecture B
and concerns the dual fine Selmer group over admissible p-adic Lie extensions (not necessarily
abelian) of dimension ě 2. It asserts that this module is smaller than intuitively expected.
Conjecture B. ([CS05b, Section 4]) Let E{F be an elliptic curve and let L{F be an S-admissible
p-adic Lie extension such that GL{F` “˘ GalpL{F q has dimension strictly greater than 1. Then
Conjecture A holds for E{F and Y E{L is a pseudonull ΛpGL{F q-module.
2.5. Fix a number field F and let T denote a finitely generated Zp -module, endowed with acontinuous action of GS pF q, where S contains the primes above p, the archimedean primes, andthe primes v such that the inertia group of v does not act trivially on T . Note that if T is the Tate
module Tp E of an elliptic curve E{F , then the inertia group of v acts trivially on T for every prime vof good reduction. Fix an S-admissible extension L{F . Define the i-th Iwasawa cohomology groupas the inverse limit
`
˘
`
˘
i
(4)
ZSi T {L “ limÐÝ H GS pLq, T , for i “ 0, 1, 2,
L

where L ranges over all finite extensions of F contained in L and the limit is taken with respect
to the corestriction maps. It is well-known that ZS0 pT {Lq vanishes (see, for example, [CS05b,nn
Proposition 2.1]). In this article, we consider T “ Zp p1q “ limÐÝ µp or T “ Tp pEq “ limÐÝn Ep . HereZp p1q denotes the Tate twist of Zp . We remark that the dual fine Selmer group Zp p1q has also beenstudied under various guises in [Sch79, NQD92]. The weak Leopoldt conjecture is known to be true
for the cyclotomic Zp -extension, see [NSW08, Theorem 10.3.25]. In other words,`
˘
H 2 GS pFcyc q, Qp {Zp “ 0.
`˘Hence H 2 GS pLq, Qp {Zp vanishes (see [CS05b, p. 815 (20)]). An argument identical to [CS05b,
Lemma3.1] ˘but for the module Qp {Zp , shows that this vanishing is equivalent to`` the fact that˘
2
ZS2 Zp p1q{L is ΛpG`L{F q-torsion.
Analogously,[CS05b,
Lemma
3.1]
shows
that
H
GS pLq, Ep8 “˘
0 if and only if ZS2 Tp pEq{L is ΛpGL{F q-torsion, but the equivalent of the weak Leopoldt conjecture
is not known
in the case of elliptic curves. When GS pLq acts trivially on Ep8 , then`˘
H 2 GS pLq, Ep8 “ 0 (see for example [CS05b, Lemma 2.4]).
The following notions will be useful in the reformulation of Conjecture B in Section 4.2. Fori ě 0 and T “ Zp p1q, choose S to be a finite set of places of F containing the primes above p andthe archimedean primes. For a finite extension L{F , let OL r1{Ss be the subring of L consistingiof elements that are integral at every finite place of L not lying over S, and let Hétdenote étale
cohomology. An equivalent definition of the i-th Iwasawa cohomology group is the following (see[Kat06, § 2.2 p. 552])
`
˘
`
˘
i
(5)
ZSi Zp p1q{L “ limÐÝ Hét OL r1{Ss, Zp p1q
L

where L ranges over all finite extensions of F contained in L and the limit is taken with respect
to the corestriction maps. The dual fine Selmer group of Zp p1q was introduced in [CS05a] and isdefined as GalpM pLq{Lq where M pLq is the maximal abelian, pro-p unramified extension of L suchthat all primes above p split completely. An equivalent definition has been given in [Kat06, §2.4,
p. 554]. In particular,`
˘(6)
YpZp p1q{Lq “ limÐÝ Pic OL r1{Ss p8 .
L

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

7

Moreover, there is an exact sequence (see for example, [CS05a, p. 330 (2.6)])
à
`
˘
`
˘
0 Ñ Y Zp p1q{L Ñ ZS2 Zp p1q{L Ñ
Zp Ñ Zp Ñ 0
vPSpLqand an isomorphism (see [Rub00, §I.6.1])
´
¯
`
˘
`
˘
(7)
R Qp {Zp {L » Hom Gal M pLq {L , Qp {Zp .
Since the dual fine Selmer group is independent of the choice of S, it is not included in the notation.
3. Fine Selmer Groups in the Cyclotomic Extension
The results in this section provide evidence for Conjecture A. First, we prove that for a set ofordinary primes of positive density, the Selmer group is trivial over the cyclotomic Zp -extension
for rank 0 elliptic curves. Next, we provide evidence for Conjecture A for a class of elliptic curves
defined over p-rational number fields.
3.1. Trivial Fine Selmer Groups in the Cyclotomic Tower. Throughout this section, assume
that E{Q is a rational elliptic curve. Fix a number field F and consider the base-change E{F ofthe curve to F . Given a prime number p,` by slight˘ abuse of notation, we denote by Fcyc {F thecyclotomic Zp -extension and by Γ “ Gal Fcyc {F » Zp its Galois group, without mention of theprime p, as it can be inferred by the context.
Ăv ; it is a curve over the residue field
At a prime v in F , the reduction of E modulo v is denoted E
rv pκv q .
κv . Following [Maz72, Section 1(b)], a prime v | p is called anomalous if p divides E
In the remaining part of this section, we extend results of Greenberg [Gre99, Proposition 5.1] andC. Wuthrich [Wut07, Section 9] to base fields other than Q. In Theorem 3.1 we provide evidence
for Conjecture A for elliptic curves over a general number field. We stress that the prime p is not
fixed in the remainder of this section and will vary over primes of good reduction.
In the statement of the next theorem we denote by F c the Galois closure of F {Q.
Theorem 3.1. Let E{F be the base-change of a rational elliptic curve E{Q. Suppose that it has
rank 0 over F and that the Shafarevich–Tate group of E{F is finite. When E has CM by anorder in an
quadratic field K, assume further that F c contains K. Then the Selmer` imaginary˘group Sel E{Fcyc is trivial for a set of prime numbers of density at least rF c1:Qs . In particular,
Conjecture A holds for E{F at all such primes.
Proof. By assumption, the Selmer group over F is finite since both the Mordell–Weil and the
Shafarevich–Tate groups are finite. If we further know that p is a prime of good ordinary reduction` for E, ˘it follows from Mazur’s Control Theorem that the cyclotomic p-primary Selmer group
Sel E{Fcyc is ΛpΓq-cotorsion (see [Gre01a, Corollary`
˘ 4.9]). In this setting, let fE pT q be a power
series generating the characteristic ideal of X E{Fcyc . Since SelpE{F q is finite, fE p0q ‰ 0. Denoteppq

by cv the local Tamagawa number at a prime v and by cv the highest power of p dividing it. Then,[Gre99, Theorem 4.1] asserts that
¨
˛¨
˛
N
ź
ź
2
2
ppq ‚˝r
˝‚(8)
fE p0q „cv
Ev pκv qp|SelpE{F q|
EpF qpv badv|pwhere a „ b for a, b P Qˆp indicates that a, b have the same p-adic valuation.
For a prime number p, consider the following five properties:

8

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA(i) p is a prime of good ordinary reduction for E;(ii) E has no non-trivial p-torsion points defined over F ;(iii) E{F has good ordinary reduction at all primes v | p and all these primes are non-anomalous;(iv) the p-primary part XpE{F qp8 of the Shafarevich–Tate group is trivial;
ppq(v) p does not divide the local Tamagawa number, i.e, cv “ 1 for every prime v of bad
reduction.
Since E{F is assumed to have rank 0, the condition EpF qp “ 0 implies that SelpE{F q “ XpE{F qp8 .
It follows from (8) that for a prime number satisfying (i)–(v) above, fE p0q is a unit.
`
˘
When fE p0q` is a unit,
˘ elementary properties of characteristic power series show that X `E{Fcyc ˘(and hence
Y ˘E{Fcyc ) is finite, (see notation introduced in (3)). Equivalently, both`Sel E{F`˘ cycand R E{Fcyc are finite. When EpF qp “ 0, [Gre99, Proposition
4.14]
implies
that
X
E{F
has
cyc
`
˘
no non-trivial
finiteΛpΓq-submodules.
In
otherwords,
X
E{Fistrivial,wheneveritisfinite.
cyc`
˘Thus, Y E{Fcyc is also trivial. Hence, Conjecture A holds for E{F when E{F is an elliptic curve
satisfying (i)–(v).
To complete the proof, we show that for E{F satisfying the assumptions of the theorem, properties (i)–(v) hold for a set of prime numbers of density at least rF c1:Qs .
When E{Q is an elliptic curve without CM, we know by [Ser81, Théorème 20] that all primesin Q outside a set of density 0 have good ordinary reduction. When E{F is an elliptic curve with
CM by an order in K, Deuring’s Criterion (see, for instance, [Lan87, Chapter 13, §4, Theorem 12])asserts that the primes of ordinary reduction are those lying above rational primes that split in K{Qand the density of such prime numbers equals 1{2 by the Chebotarev density theorem. Next, it
follows from the celebrated result [Mer96, Théorème] of L. Merel that for all but finitely many prime
numbers, we have EpF qp “ 0. Assuming the finiteness of the Shafarevich–Tate group, condition(iv) holds for all but finitely many prime numbers, and the same is true for (v) since the local
Tamagawa number cv is equal to 1 at the primes of good reduction.
The analysis of (iii) requires more care. By definition, a prime v | p is anomalous when av “
arv pκv q is congruent to 1 pmod pq. Observe that by the Hasse bound, |av | ď 2 |κv |.
1 ` |κv | ´ E
Therefore, if v | p is a prime in F that splits completely, so that κv “ Fp , then av ” 1 pmod pq
implies that av “ 1 for p ą 5. By the Chebotarev density theorem, the density of rational primes
that split completely in F c is rF c1:Qs . Therefore, at least rF c1:Qs of the primes in Q split in F , aswell. By the previous discussion, the density of rational primes which split completely in F andwhose divisors are primes of good ordinary reduction for E{F is at least rF c1:Qs . Finally, since E isdefined over Q, the Modularity Theorem guarantees that E is associated with an eigencuspform of
weight 2. This allows us to appeal to the work of V. K. Murty [Mur97]. We conclude from [Mur97,
pp. 288–289 or Theorem 5.1 and Remark 5.2] that for E{Q, the set of prime numbers with theproperty that ap “ 1 has density 0. Since for all prime numbers p that split completely and for allv | p, we have av pE{F q “ ap pE{Qq, we deduce that the set of prime numbers p such that av “ 1 forat least one v | p is a set of density 0. This completes the proof of the theorem.
□
Remark 3.2.
(1) It should be clear from the proof that one can insist that at all primes dividing the prime
numbers in the set of positive density whose existence is stated in the theorem, the reduction
type is good and ordinary.

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

9(2) The key difficulty in extending this result to elliptic curves defined over F is that we relyon [Mur97] to show that anomalous primes have density 0. Since these results are provenfor normalized weight 2 eigencuspforms, we need to invoke the Modularity Theorem.
An analogous statement can be proven in the supersingular case as well.
Theorem 3.3. Let E{Q be an elliptic curve, and suppose that SelpE{F q is finite. Then Conjecture A
holds for E{F for all but finitely many primes of supersingular reduction.
Proof. For an elliptic curve E{F it is known that the Selmer group is not ΛpΓq-cotorsion at a prime2
p of supersingular reduction, see` [CS00,
˘ p. 19]. However, there is a notion of ˘-Selmer groups
˘
when p ą 3, denoted by Sel E{Fcyc . In the setting of the theorem, and under the additional`˘hypothesis that p ą 3 is an unramified prime in F , it is known that Sel˘ E{Fcyc are ΛpΓq-cotorsion,see [Kim13, first line of the proof of Corollary 3.15]. Therefore, in this case, we can define` a pair
˘ ofsigned characteristic power series fE˘ pT q for the Pontryagin duals XpE{Fcyc q˘ of Sel˘ E{Fcyc . It
follows from the definitions that the fine Selmer group is a subgroup of the signed Selmer groups.
To prove the theorem it thus suffices to show that either of the signed Selmer groups is finite forall but finitely many primes of good supersingular reduction as this will ensure that the fine Selmer
group is also `finite˘ and its corresponding µ and λ invariants vanish.
When Sel E{F is finite and p ą 3 is an unramified prime in F , we know from [Kim13, Theorem 1.2] thatź(9)fE˘ p0q „ |SelpE{F q|cvppq .
v bad`
˘˘
If fE p0q „ 1, then it follows from the Structure Theorem that Sel˘ E{Fcyc are finite. To complete
the proof we show that fE˘ p0q „ 1 for all but finitely many primes of good supersingular reduction.

(i) Since F is fixed, there are only finitely many primes which can ramify in F . In other words,(9) holds for all but finitely many primes.
(ii) By assumption, SelpE{F q is finite. There are only finitely many primes which can divide
its order.
(iii) The local Tamagawa number cv is equal to 1 at the primesof good reduction. Therefore,ś
there are only finitely many primes which can divide v bad cv .
`˘
Therefore, as p varies over all supersingular primes of E, `both signed
Selmer groups Sel˘ E{Fcyc˘
are finite for all but finitely many such primes. Hence, R E{Fcyc is also finite for such p.
□
Remark 3.4. In fact, more is true. [Kim13, Theorem 1.1 (or Theorem 3.14)] applies in the setting´
of Theorem 3.3 and ensures that` the ˘X pE{Fcyc q does not contain any non-trivial`
˘ finite index´
submodules. Therefore, if Sel E{Fcyc is finite, it must be trivial. Since R E{Fcyc is a subgroup`˘of Sel´ E{Fcyc , it must be trivial as well. For the assertion that XpE{Fcyc q` has no non-trivial
finite index submodules, the additional hypothesis that p is completely split in F is required.
Combining Theorems 3.1 and 3.3, the next result is immediate.
Corollary 3.5. Let E be CM rational elliptic curve and let E{F be its base-change to F . Supposethat E{F has rank 0, that the Shafarevich–Tate group of E{F is finite, and that the Galois closure
F c of F contains K. Then Conjecture A holds for E{F for a set of prime numbers of density
1
1
2 ` rF c :Qs .
2We avoid giving the precise definition of these Selmer groups because their definition is intricate and also not
relevant for the remainder of this paper. For a precise definition, we refer the reader to [Kob03] or [Kim13].

10

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

Proof. By Deuring’s Criterion we know that 1{2 of the primes are supersingular and Theorem 3.3
asserts that there is a contribution of density 1{2. But, there is also a contribution from the primesof good ordinary reduction by Theorem 3.1. The corollary follows.
□
Let us now turn to a special class of number fields, called p-rational number fields.
3.2. Conjecture A over p-Rational Number Fields. For the number field F and a fixed primep, choose S to be a finite set of primes of F containing the primes above p and the archimedean
primes. The weak Leopoldt conjecture for L{F is the following assertion (see for example [NSW08,
Theorem 10.3.22])
´
¯
`
˘
(10)
H 2 Gal FS {L , Qp {Zp “ 0.
It is known to hold for the cyclotomic Zp -extension Fcyc {F (see [NSW08, Theorem 10.3.25]). If (10)holds for a finite set S as above, it also holds for the set Σ “ Sp YS8 (see [NSW08, Theorem 11.3.2]).
Therefore, the weak Leopoldt Conjecture is independent of the choice of S, when S contains Σ.
Henceforth, fix S `“ Σ.˘ An equivalent
formulationof the Iwasawa µ “ 0 Conjecture for F is the`˘assertion that GΣ Fcyc “ Gal FΣ ppq{Fcyc is a free pro-p group (see [NSW08, Theorem 11.3.7]).
Moreover, a pro-p group G is free if and only if its p-cohomological dimension cdp pGq is less or equal
to 1 (see [NSW08, Corollary 3.5.17]). Combining these results with [Ser97, Chapter I, Section 4,
Proposition 21], one obtains the following equivalent formulation:
´ `
¯
˘
(11)
the Iwasawa µ “ 0 Conjecture for F is true ðñ H 2 GΣ Fcyc , Z{pZ “ 0.
To state the results in this section, we recall the notion of a special class of number fields, called
p-rational, which were introduced in [MN90]. We refer the reader to [Gra03, Theorem IV.3.5 andDefinition IV.3.4.4] for a detailed discussion.
Definition 3.6. Denote by FSp the maximal extension `of F unramified
outside Sp and let FSp ppq{F˘
be its maximal pro-p sub-extension. Set GSp pF q “ Gal FSp ppq{F . If GSp pF q is free pro-p, then F
is called p-rational.
Some examples of p-rational fields include:(i) the field Q of rational numbers;(ii) imaginary quadratic fields such that p does not divide the class number (see [Gre16, Proposition 4.1.1]);(iii) cyclotomic fields Qpµpn q, where p is a regular prime and n ě 1 (combine [Gra03, Example II.7.8.1.1] with [Was97, Proposition 13.22]);(iv) more generally, number fields F containing µp with the property that #Sp pF q “ 1 and suchthat p does not divide the class number of F (see [Gra03, Theorem 3.5-(iii)]).
p-rational number fields have been studied by Greenberg in [Gre16], where he explains heuristic
reasons to believe that a number field F should be p-rational for all primes outside a set of density 0(see [Gre16, §7.4.4]). In [BR20, Table 4.1], R. Barbulescu and J. Ray provide examples of nonabelian p-rational number fields.
The following result is easily deduced from the aforementioned results in Galois cohomology. A
proof is included for the sake of completeness.
Theorem 3.7. Let F be a p-rational number field. Then the following assertions hold.
(i) The Iwasawa µ “ 0 Conjecture holds for F .

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

11(ii) Suppose that F contains µp and that E{F is an elliptic curve such that EpF qp ‰ 0. Then
Conjecture A holds for E{F .
Proof.
(i) Since p ‰ 2, we can replace Sp by Σ in the definition of p-rational fields. This is because
the archimedean primes `are unramified
in FSp ppq{F when p is odd. By definition, if F is
˘
p-rational, GΣ pF q “ Gal FΣ ppq{F has p-cohomological dimension at most 1. Hence`˘
H 2 GΣ pF q, Z{pZ “ 0.
Since GΣ pFcyc q “ GalpFΣ ppq{Fcyc q is a closed normal subgroup of GΣ pF q, it follows from
[NSW08, Proposition 3.3.5] that`˘`˘
cdp GΣ pFcyc q ď cdp GΣ pF q ď 1.
Thus H 2 pGΣ pFcyc q, Z{pZq “ 0, and the result follows from (11).
(ii) Since F Ě µp and EpF qp ‰ 0 by assumption, the Weil pairing ensures that F pEp q{F iseither trivial or of degree p. Thus, F pEp8 q{F is pro-p. The theorem follows from the firstpoint together with Theorem 2.2.
□
4. Conjecture B for Elliptic Curves with CM: Special Cases
In this section, we provide evidence for Conjecture B. First, in Section 4.1 we provide sufficientconditions for Conjecture B to hold when p is a prime of good ordinary reduction, see Theorem 4.6.
In Section 4.2 we give a different formulation of Conjecture B for CM elliptic curves and provecases of the conjecture when p is a prime of good supersingular reduction. We start with a lemma
about good reduction of CM elliptic curves that can be found extracted from [Rub99, proof of
Theorem 5.7-(i)].
Lemma 4.1. Let F be a number field and let E{F be an elliptic curve with CM by an order inside
the ring of integers OK of an imaginary quadratic field K. Let p be an odd prime number andsuppose that the following hypotheses hold:(i ) E has good reduction at all primes above p.
(ii ) The Galois group G “ GalpF8 {F q is isomorphic to Z2p , where F8 denotes F pEp8 q.
Then E has good reduction everywhere over F .
Proof. It follows from the theory of complex multiplication that F contains the Hilbert class fieldK 1 of K. Since the extension F8 {F is a p-extension and rF pEp q : F s is prime-to-p, it follows thatF “ F pEp q. Therefore, K 1 pEp q Ď F .
Since all primes above p are of good reduction, we only need to check that at primes away from p,
the curve E has good reduction. This follows from the criterion of Néron–Ogg–Shafarevich, because
every such prime is unramified in the Z2p -extension F8 {F .
□4.1.

Fix a number field F . We will work in the following setting.

Ass 1(i) p ‰ 2, 3 is a fixed prime which splits in an imaginary quadratic field K;(ii) E is an elliptic curve defined over F with CM by OK , and K is contained in F ;(iii) E has good reduction at primes above p;(iv) the Galois group G “ GalpF8 {F q is isomorphic to Z2p , where F8 denotes F pEp8 q.

In the setting of Ass 1, write H “ GalpF8 {Fcyc q, and fix a finite set S containing Sp Y S8 .

12

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

Note that Ass 1 ensures that E has good ordinary reduction at p, see [Lan87, Chapter 13
Theorem 12 (Deuring’s Criterion)]. Observe that given any p-adic Lie group G and a finitely
generated ΛpGq-module M , the group MG :“ H0 pG, M q is finitely generated as a Zp -module.
Lemma 4.2. Suppose that Ass 1 holds. Then, the following map of ΛpHq-modules is a pseudoisomorphism, i. e. it has a finite kernel and cokernel,`
˘`˘
Y E{F8 H Ñ Y E{Fcyc .
Proof. Let L be a finite extension of F contained in FS . For each v P S, write Wv pLq “À
w|v EpLw q b Qp {Zp . We have the maps
àrcyc : SelpE{Fcyc q ÝÑ
Wv pFcyc qv|pandr8 : SelpE{F8 q ÝÑà

Wv pF8 qv|pwhere Wv pFcyc q (resp. Wv pF8 )) is the direct limit of Wv pLq with respect to the restriction map as
L ranges over all finite extensions of F contained in Fcyc (resp. F8 ). Write CpFcyc q (resp. CpF8 q)
for the image of rcyc (resp. r8 ). Consider the following diagram0

RpE{Fcyc qp

SelpE{Fcyc qp

CpFcyc qpβγα

RpE{F8 qH
p0

SelpE{F8 qH
p0

CpF8 qHp

Note that β is an isomorphism (see [PR81, Lemme 1.1(i) and Lemme 1.3]). Therefore kerpβq andcokerpβq are trivial; hence kerpαq “ 0. Further, observe that there is an inclusion
¨
˛
à
`
˘ δv
H
ker γ Ď ker ˝ Kv Fcyc ÝÑ Kv pF8 q ‚.
v|pNow, observe thatàv|pkerpδv q “à`˘
H 1 Hv , EpF8,v qp8 .

v|pThis latter object is known to be finite by using an argument identical to [CS05b, proof of
Lemma 4.2]. Therefore, by the snake lemma, cokerpαq must be finite.
□
Remark 4.3. Another way to prove this lemma was pointed out to us by the referee. Consider thefundamental diagram
`
˘
`
˘
À
0
RpE{Fcyc qp
H 1 GS pFcyc q, Ep80vPS Kv Fcyc
α0

RpE{F8 qH
p

γ

β

`
˘H
H 1 GS pF8 q, Ep8ÀvPS Kv pF8 q

`
1

H

.

˘
The map β is the restriction
map;˘ it is surjective and kerpβq “ H H, EpF8 qp8 . Similarly,`À
1kerpγq “
Hv , EpF8,v qp8 . To show that kerpγq is finite, we use an argument similarvPSp H
to [CS05b, proof of Lemma 4.2]. First, recall a result of H. Imai [Ima75, Theorem] which asserts

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

13

is a torsion ΛpHv q-module (sincethat EpFcyc,v qp8 is finite. Note that Hv » Zp and EpF8,v q_` p8
˘
it is` in fact finitely˘generated over Zp ). It follows that H 1 Hv , EpF8,v qp8 is also finite. In fact,
H 1 Hv , EpF8,v qp8 “ 0 which can be proven in the same way as [CSW01, Lemma 5.4] using thefact that Hv has p-cohomological dimension 1. Furthermore, since EpFcyc qp8 is finite by a result ofK. Ribet [Rib81, Theorem 1], the global version of the above argument ensures that kerpβq is also
finite, see also [CS05b, pp. 834–835]. Applying the snake lemma, the lemma follows.
Since E is an elliptic curve with CM, both G and H are abelian. Under the assumption thatG » Z2p , we further know that ΛpHq » Zp JT K. We now state an equivalent condition for a ΛpGqmodule to be pseudonull.
Proposition 4.4. Let M be a finitely generated ΛpGq-module which is also finitely generated as aΛpHq-module. Then the module M is ΛpGq-torsion. Further, M is ΛpHq-torsion if and only if it
is ΛpGq-pseudonull.
Proof. Note that G » H ˆΓ where Γ » Zp . The first assertion follows from the fact that ΛpGq is not
finitely generated over ΛpHq. The second assertion is a special case of [Ven03a, Proposition 5.4]. □Lemma 4.5. Let M be a finitely generated ΛpGq-module which is also finitely generated over ΛpHq.
If MH is finite, then M is a pseudonull ΛpGq-module.
Proof. We are grateful to the referee for suggesting the following proof, which is simpler than the
one we had in a first version of our manuscript. Since H – Zp , it follows from the structure theoryof ΛpHq-modules that whenever MH is finite, then M is torsion over ΛpHq. The conclusion of thelemma is now immediate from Proposition 4.4.
□
The main theorem of this section is the following.
Theorem 4.6. Suppose that Ass 1 holds. If YpE{Fcyc q is finite, then Conjecture B holds for
pE, F8 q.
`
˘
Proof. By Lemma 4.2, if YpE{Fcyc q is finite, then so is Y E{F8 H . The theorem follows from
Lemma 4.5.
□
Remark 4.7. We point out that for a given prime p, we cannot conclude that Conjecture B holds
for pE, F8 q for a rank 0 elliptic curve E{F with CM by combining Theorems 3.1 and 4.6. This isbecause, in the proof of Theorem 3.1 it was required that the elliptic curve does not admit any
non-trivial p-torsion point over F . However, in proving Theorem 4.6, we assume that F8 {F is a
pro-p extension; hence F must contain non-trivial p-torsion points.
Another case where we can show Conjecture B is the following.
Proposition 4.8. Suppose that Ass 1 holds. Further assume that XpE{Fcyc q is a finitely generated
Zp -module of Zp -rank 2 and that EpF8 q has a point of infinite order. Then Conjecture B holds for
pE, F8 q.
Proof. By Ass 1, we know that E{F has good reduction everywhere. Next, it follows from [How02,
Theorem 2.8] thatrankΛpHq XpE{F8 q “ rankZp XpE{Fcyc q.
We explain this briefly. To apply [How02, Theorem 2.8] one must assume that Conjecture 2.5 ibid.
holds. As mentioned on p. 649 ibid., this conjecture is equivalent to Conjecture 2.6 ibid. when all

14

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

primes above p have good ordinary reduction. This conjecture predicts that XpE{Fcyc q is ΛpΓqtorsion and our hypothesis that XpE{Fcyc q is a finitely generated Zp -module accounts for it. For
the final assumption in Theorem 2.8 ibid, the inclusion µp Ď F is ensured by the Weil pairing.
It is shown in [CS05b, Theorem 4.5-(ii)] thatrankΛpHq YpE{F8 q ď rankΛpHq XpE{F8 q ´ 2.
and our assumption, combined with (4.1), implies that rankΛpHq YpE{F8 q “ 0, showing thatYpE{F8 q is ΛpGF8 {F q-pseudonull.
□
Remark 4.9. We are grateful to the referee for the following observation. In[CS05b, Theorem 4.5-(i)]it is shown that if rankΛpHq XpE{F8 q is odd, thenrankΛpHq YpE{F8 q ď rankΛpHq XpE{F8 q ´ 1.
Since XpE{Fcyc q is a finitely generated Zp -module, if rankZp XpE{Fcyc q “ 1, it would follow thatYpE{F8 q is ΛpGF8 {F q-pseudonull. Unfortunately, though, one can show that under our assumptions
rankΛpHq XpE{F8 q is always even, and therefore the above argument cannot be used to show Conjecture B in more cases. The argument forcing rankΛpHq XpE{F8 q to be even comes from CM theory:
indeed, by functoriality, the ΛpHq-module XpE{F8 q comes endowed with a structure of an OK Č
module, and therefore it is ultimately a ΛpHqbOK -module. For brevity, denote ΛpHqbOK by ΛpHq,
Č “ FracpΛpHqq.
Č Clearly, ΛpHqČ is a finite extension of ΛpHq ofand set QpHq “ FracpΛpHqq, QpHqČČ
Č “ 2.
rank 2: in particular, it is integral so that QpHq “ ΛpHq bΛpHq QpHq and dimQpHq QpHq
Now, by definition,´¯Č
2 ¨ rankΛpHq
Č XpE{F8 q “ 2 ¨ dimQpHq
Č XpE{F8 q bΛpHq
Č QpHq´Č“ 2 ¨ dimQpHq
Č XpE{F8 q bΛpHqČ ΛpHq bΛpHq QpHqq´“ 2 ¨ dimQpHq
Č XpE{F8 q bΛpHq QpHqq´“ dimQpHq XpE{F8 q bΛpHq QpHqq“ rankΛpHq XpE{F8 q,
showing that rankΛpHq XpE{F8 q is indeed even.
4.2. Reformulation of Conjecture B. Let E{F be an elliptic curve, and let L be an S-admissible
p-adic Lie extension containing the trivializing extension F8 . Throughout this section we supposethat Conjecture A holds for E{F . Since GS pLq acts trivially on Ep8 , Conjecture B for pE, Lq has
an equivalent formulation in terms of the pseudonullity of the Galois group GalpM pLq{Lq, where
M pLq is the maximal unramified abelian pro-p extension of L such that all primes above p in L
split completely.
Reformulation (see [CS05b, p. 827]). Let E{F be an elliptic curve, and let` L be an ˘S-admissible,
p-adic Lie extension over F such that GS pLq acts trivially on Ep8 . Then Y Zp p1q{L is ΛpGL{F qpseudonull.
The next result asserts that for an S-admissible p-adic Lie extension L{F containing F8 , theL
ΛpGL{F q-pseudonullity of the Iwasawa module Xnr
is equivalent to the pseudonullity of a certainL
quotient module. (The notation Xnr was introduced at the beginning of this section). This result is well-known to experts and follows from results available in the literature, see for example

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

15[Ven03b, Theorem 4.9]. For the convenience of the reader, a proof is provided here purely relying
on techniques that are more germane to our paper.
Theorem 4.10. Let E{F be an elliptic curve with CM by an order in an imaginary quadratic field
K such that K Ď F and suppose that GalpF8 {F q » Z2p . Let L{F be an abelian S-admissible p-adic
Lie extension containing F8 . Then, the following statements are equivalentL(a) The Iwasawa µ “ 0 Conjecture is true for F and Xnr
is ΛpGL{F q-pseudonull.
(b) Conjecture B holds for pE, Lq.
(c) The Iwasawa µ “ 0 Conjecture is true for F and YpZp p1q{Lq is ΛpGL{F q-pseudonull.
Proof. Since E{F has CM by the imaginary quadratic field K contained in F and F8 {F is a Z2p extension, it follows that F contains K 1 pEp q where K 1 is the Hilbert class field of K (see Lemma 4.1).
Moreover, since L{F is an abelian extension containing F8 and, by definition of being admissible,
it contains no element of order p, it must be a Zdp -extension for some d ě 2. It follows that the only
primes that can ramify in this extension are the primes above p and therefore we can assume thatS “ Sp Y S8 .
Equivalence of (a) and of (c) We need to show that`˘L
Y Zp p1q{L is ΛpGL{F q-pseudonull ðñ Xnr
is ΛpGL{F q-pseudonull.
`˘
L
Write Xcs
to denote the Galois group Gal M pLq{L . It is known by the work of U. Jannsen (seefor example [Ven03b, Proposition 4.7-(ii)]) that there is an exact sequence
à
` ˘
G
L
L
IndGL{F,v
Zp ÝÑ XnrÝÑ XcsÝÑ 0.
L{FvPScs YSram

Here, Scs denotes the set of non-archimedean primes in S which are completely split in L{F andSram denotes the set of non-archimedean primes in S which are ramified in L{F . Note that in our
setting Scs “ H because every prime above p is finitely decomposed in Fcyc {F , and Sram “ Sp .
Since the base field F contains µp it follows from (7) that
`
˘
`
˘
L
Xcs
“ Gal M pLq{L » Y Zp p1q{L .
L
Lareand Xcs
Therefore, to complete the proof of the equivalence it is enough to show that Xnr
pseudo-isomorphic. In other words, it suffices to prove that
à
à
` ˘
G
Zp JGL{F K bZp JGL{F,v K Zp “
IndGL{F,v
Zp
L{F
vPSp

vPSp

is a ΛpGL{F q-pseudonull module. We know from [LNQD00, Théorème 3.2] (observe that since
F contains K 1 pEp q, condition piq ibid. is satisfied, by the Weil pairing) that for all v | p, the` ˘ÀGdecomposition group at v inside GL{F has dimension at least 2. It follows that vPSp IndGL{F,v
Zp
L{Fis ΛpGL{F,v q-pseudonull. This completes the proof of the equivalence.
Equivalence of (b) and (c) It follows from the discussion in [CS05b, p. 825] that
¯
`
˘ ´
(12)
Y E{L » YpZp p1q{Lq b E_.
8p
Here GL{F acts diagonally on the tensor product and E_
p8 is a Zp -module with a GL{F -action inducedby the GS pF q-action. This latter action makes sense because F8 is the trivializing extension of
Ep8 . In this setting, Conjecture A for E{F is equivalent to the Iwasawa µ “ 0 Conjecture for
F (see Theorem 2.2). Therefore, using [Ven03b, Proposition 2.12] and [OV02, Proposition 3.4],

16

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA`˘`˘
the isomorphism in (12) yields that Y E{L is ΛpGL{F q-pseudonull if and only if Y Zp p1q{L isΛpGL{F q-pseudonull.
□
Remark 4.11. In the above proof, the assumption that E has CM is not really used. We detail in§4.3 the proof in the general case.
We now prove a special case of Conjecture B in the supersingular reduction setting and provide
applications pertaining to universal norms. For the remainder of this section, we work in thefollowing setting:Ass 2(i) K is an imaginary quadratic field of class number 1;(ii) E is an elliptic curve defined over K, and with CM by OK ;(iii) p is an odd prime of good supersingular reduction for E;(iv) p does not divide the order of the Sp -class group of F “ KpEp q;

Remark 4.12. It follows from Ass 2-(ii) that the Galois group G “ GalpF8 {F q is isomorphic to Z2p
and we write H “ GalpF8 {Fcyc q.
Recall that the i-th Iwasawa cohomology group over F8 is defined, when S “ Sp , as
`
˘
`
˘
i
ZSi p Zp p1q{F8 “ limÐÝ Hét OL r1{ps, Zp p1q ,L

where L ranges over all finite extensions of F contained in F8 .
`˘
Proposition 4.13. Suppose that Ass 2 holds. Then ZS2p Zp p1q{F8 “ 0. In particular, Conjecture B holds for pE, F8 q and µp8 pF q is a universal norm from F8 .
Proof. We are grateful to the referee for suggesting the following proof, which is simpler than the
one we had in a first version of our manuscript. Using the Poitou–Tate sequence over F as in [Kat06,
p. 553 §2.4-(1)], we have that
à
`
˘
(13)
0 ÝÑ ClSp pF qp8 ÝÑ ZS2p Zp p1q{F ÝÑZp ÝÑ Zp ÝÑ 0.
vPSp pF q

Since p is a prime of supersingular reduction for E{K, we know that there exists a unique prime
above p in K. Moreover, p is totally ramified in the extension GalpF8 {Kq, see for example [PR04,
Section 1]. In particular, there is a unique prime above p in F , i. e. Sp “ 1. Combining this withthe assumption`
˘ that the p-Sylow subgroup of the Sp -class group is trivial yields, through (13), that
ZS2p Zp p1q{F “ 0. By Nekovar’s spectral sequence (see [Nek06, Corollary 8.4.8.4-(ii)]), we obtain
that
`
˘
`
˘
ZS2p Zp p1q{F8 G » ZS2p Zp p1q{F “ 0.
`˘Now, employing Nakayama’s Lemma we conclude that ZS2p Zp p1q{F8 “ 0.
Next, consider the exact sequence (see, for example, [CS05a, p. 330 (2.6)])
à
`
˘
`
˘
0 ÝÑ Y Zp p1q{F8 ÝÑ ZS2p Zp p1q{F8 ÝÑ
Zp ÝÑ Zp ÝÑ 0.
vPSp pF8 q`

˘
It follows from the first part of the proof that Y Zp p1q{F8 “ 0. Moreover, the same descent`˘Fargument as above using Nekovar’s spectral sequence shows that Y Zp p1q{Fcyc “ Xcscyc “ 0: inFFF
particular, µpXcscyc q “ 0. By [NSW08, Corollary 11.3.16], we know that µpXcscyc q “ µpXnrcyc q “ 0.
Therefore, we obtain that Iwasawa µ “ 0 Conjecture holds for F , and we can apply Theorem 4.10,
showing that Conjecture B holds.

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

17

To prove the final assertion, consider the exact sequence (see [CS05a, p. 335 (3.26)])´`˘`˘ τF8 {F`˘¯ÝÑ ZS1p Zp p1q{F0 ÝÑ H2 G, ZS2p Zp p1q{F8 ÝÑ ZS1p Zp p1q{F8 G ÝÝÝ´`˘¯ÝÑ H1 G, ZS2p Zp p1q{F8 ÝÑ 0.
`˘
We have shown above that ZS2p Zp p1q{F8 “ 0; hence, τF8 {F is an isomorphism. It follows that
µp8 pF q is a universal norm from F8 (see [CS05a, Corollary 3.27] for details).
□
The following corollary provides asymptotics for the growth of the p-primary torsion of the fineSelmer group at each layer of the Z2p -extension.
Corollary 4.14. Suppose that Ass 1 holds and that either(i) YpE{Fcyc q is finite; or(ii) XpE{Fcyc q is a finitely generated Zp -module of Zp -rank equal to 2 and EpF8 q has a point of
infinite order.
Then
´ `
¯
˘_
ordp R E{F pEpn q rp8 s “ O ppn q .
´ `¯˘_If, moreover, Ass 2 holds, then ordp R E{F pEpn q rp8 s “ 0.
Proof. Conjecture A holds by assumption in each case and Conjecture B holds by Theorem 4.6 incase (i) and by Proposition 4.8 in case (ii) The first claim follows from [KL22, Corollary 6.14].
When Ass 2 holds, a better estimate can be obtained, and we thank the referee for this observation. Since F8 is the trivializing extension, we have
ZS2p pTp E{F8 q » ZS2p pZp p1q{F8 q b Tp E
and thus Proposition 4.13 implies that ZS2p pTp E{F8 q “ 0. By Nekovar’s spectral sequence, we also
know that`
˘
ZS2p Tp E{F pEpn q » ZS2p pTp E{F8 qGn “ 0.
`
˘
`
˘
□
Since Y E{F pEpn q is contained in ZS2p Tp E{F pEpn q , the result follows.
4.3. The noncommutative setting. Even though this paper largely treats the commutative case,an analogue of Theorem 4.10 is valid even in the noncommutative setting. We would like to thankthe referee for pointing this out, and insisting that the general case be included. Let S be a finite
set of primes of F containing the primes above p , the archimedean primes and the primes of bad
reduction for E.
Theorem 4.15. Let E{F be an elliptic curve without complex multiplication and p ě 5 be a rational
prime. Assume that F contains the p-torsion points of E. Suppose that the elliptic curve has either
potential ordinary or potential multiplicative reduction at all the primes v | p, that F contains the ptorsion points of E, and let F8 “ F pEp8 q{F be the trivializing extension. Write S “ Sp Y Sbad Y S8
and set G “ GalpF8 {F q, H “ GalpF8 {Fcyc q. Then the following assertions are equivalent:
F8(a) The Iwasawa µ “ 0 Conjecture is true for F and Xnr
is ΛpGq-pseudonull.
(b) Conjecture B holds for pE, F8 q.
(c) The Iwasawa µ “ 0 Conjecture is true for F and YpZp p1q{F8 q is ΛpGq-pseudonull.

18

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

Proof. As in the proof of Theorem 4.10, the equivalence between (a) and of (c) will follow once we
F8
F8prove that Xnr
and Xcs
are pseudo-isomorphic. We know from [Coa99, Lemma 2.8] that GF8 {F,v
has dimension at least 2 at primes v P S; here we have used the fact that when v does not divide p,
it is not possible in our setting that E has bad but potential good reduction (see for example, [OV02,` ˘
ÀGZp is ΛpGF8 {F,v q-pseudonull.
the paragraph after Theorem 5.2]). It follows that vPS IndGFF8 {F,v
{F
8
We next prove the implication (c) ñ (b): assume that assertion (c) holds. Then, by [CS05b,
Lemma 3.8, p. 825], there is an isomorphism`
˘
YpE{F8 q » Y Zp p1q{F8 b E_
p8 ,where the tensor product is over Zp , and the action of GalpF pEp8 q{F q on the right hand-side is the
diagonal action. The hypothesis`
˘that the Iwasawa µ “ 0 Conjecture holds for F assures us of thefinite generation of Y Zp p1q{F8 as a ΛpHq-module. Hence, to show that assertion (b) holds, itsuffices to prove that YpE{F8 q is a torsion ΛpHq-module.
The pseudonullity of YpE{F8 q as a ΛpGq-moduleis ˘equivalent to YpE{F8 q being torsion as`a ΛpHq-module. By our hypothesis that Y Zp p1q{F8 is pseudonull, it follows that it is also
finitely
generated
and torsion as a ΛpHq-module. Thus there exists a finite, free ΛpHq-resolution of`˘
Y Zp p1q{F8 such that the alternating sum of the ΛpHq-ranks of the free modules in the resolution
is zero. Tensoring such a resolution over Zp with E_
p8 preserves the exactness and gives a finite,`˘.
Further,
the alternating sum of the ΛpHq-ranks of thefree ΛpHq-resolution of Y Zp p1q{F8 b E_
p8`
˘_
free modules is still zero, whence Y Zp p1q{F8 b Ep8 is ΛpHq-torsion. This proves assertion (b).
Note that this argument also proves the equality of the ΛpHq-ranks of YpZp p1q{F8 q and YpE{F8 q.
It remains to prove the implication (b) ñ (c). Suppose that Conjecture B is true for pE, F8 q.
Then Conjecture A is true and the dual fine Selmer group YpE{Fcyc q of E over the cyclotomic Zp extension is a finitely generated Zp -module. The vanishing of the Iwasawa µ-invariant for Fcyc is
a consequence of [CS05b, Theorem 3.4]. Now suppose that YpZp p1q{F8 q is not ΛpHq-torsion, andhence has positive rank as a ΛpHq-module. By the remark above on the equality of ΛpHq-ranks, this
implies that YpE{F8 q also has positive ΛpHq-rank, contradicting the hypothesis. This completes
the proof of the equivalence.
□
5. Conjecture B and the Generalized Greenberg’s Conjecture
The aim of this section is to clarify the connection between the Generalized Greenberg’s Conjecture and Conjecture B for CM elliptic curves. For the sake of brevity, we henceforth refer to theGeneralized Greenberg’s Conjecture as GGC.
Both conjectures pertain to the pseudonullity of certain Iwasawa modules. Even though Conjecture B was proposed as a generalization of GGC, the precise formulation of this connection israther intricate. Using Theorem 4.10, we make precise in which sense Conjecture B for CM elliptic
curves is a generalization of GGC (see Theorem 5.4).
Fix an imaginary quadratic field K and denote its Hilbert class field by K 1 . Given an elliptic
curve E{K 1 with CM by an order in K, set
F “ K 1 pEp q,

F8 “ K 1 pEp8 q “ F pEp8 q,

G “ GalpF8 {F q,

G8 “ GalpF8 {Kq,1
G8
“ GalpF8 {K 1 q.

Ă1 , Fr) to be the compositum of all Zp -extensions of K (resp. ofr (resp. K
Note that G » Z2p . Set K
1
r is the unique
K , of F ). Since the Leopoldt conjecture is true for imaginary quadratic fields, K

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

19

Z2p -extension of K. For the rest of this section, we make the following assumption.
Ass 3(i) p is an odd prime that is unramified in K;
r “ K.
(ii) the prime p is such that K 1 X K

1By the theory of complex multiplication, G8 “ G ˆ ∆ and G8“ G ˆ ∆1 where ∆ » GalpF {Kq11(resp. ∆ » GalpF {K q) is a finite abelian group. Recall from [Ser68, Remark on p. IV-13] that
∆1 is a Cartan subgroup of GL2 pFp q and hence it either has order p2 ´ 1 or pp ´ 1q2 : in any case,p ∤ ∆1 .

Remark 5.1.
(1) In fact, it is forced by Ass 3-(i) that µp Ć K. This?can be seen as follows: the only pairpp, Kq for which µp Ă K is when p “ 3 and K “ Qp ´3q; but this contradicts Ass 3-(i).
(2) We now discuss Ass 3-(ii) in a little more detail. This assumption is trivially satisfied whenr is containedp does not divide the class number of K. But observe that, in general, K 1 X K
in the anti-cyclotomic Zp -extension of K, denoted by Kac . For a proof of this fact, see[Fuj13, Lemma 2.2]. Therefore, Ass 3-(ii) is equivalent to the following condition:(ii1 ) The prime p is such that K 1 X Kac “ K.
To know more about non-trivial examples where this condition is satisfied, we refer thereader to [Bri07]. For a specific example, see Example 4 ibid. Moreover, Ass 3-(ii) is
closely related to the notion of p-rationality (see [Bri07, p. 2133]) but we will not discuss
this point any further.
1
r The theory of complex
Set the notation K8
to denote the composite of the fields K 1 and K.
1r
multiplication guarantees that F8 “ F K “ F K8 . Recall that F8 is the trivializing extension for
the Galois representation associated to Tp E and it is an S-admissible p-adic Lie extension. We notethat F8 Ď Fr.
Denote by LpFrq (resp. LpF8 q) the maximal abelian unramified pro-p-extension of Fr (resp. of
F8 ). Denote by FS the maximal abelian pro-p extension of Fr unramified outside S. Set thenotation
´
¯
´
¯
`
˘
r
r
(14)
X F “ Gal LpFrq{Fr , X F8 “ Gal LpF8 q{F8 , X F “ Gal FS {Fr .
nr

nr

S

As in the previous sections, given any extension L{F , we denote by M pLq the maximal unramified
abelian p-extension of L where all primes above p in L split completely; this group is related to thefine Selmer group (see (7)). For most of the discussion, L will either be F8 or Fr. For convenience,the diagram of fields is drawn in Figure 1.
Recall the statement of GGC for F (the statement for K is analogous, by replacing F, Fr , ΛpGFr {F qr ΛpG Ă ), respectively).
by K, K,
K{KF
GGC. With notation as above, Xnr
is a pseudonull ΛpGFr {F q-module.
r

The following results are required to relate GGC to the pseudonullity of the fine Selmer group.
The first lemma assures pseudonullity over a larger tower, once it holds for a proper subextension.
Lemma 5.2 (Pseudonullity Lifting Lemma). Let n ě 3, let F{Q be a finite Galois extensionrcontaining µp , and denote by Fr the compositum of all Zp -extensions of F. Suppose that GalpF{Fq
»
pdq
n
pdq
pdq
d
F
Zp and let F Ĺ Fr be such that GalpF {Fq » Zp for some 2 ď d ă n. If Xnr is ΛpGF pdq {F qrpseudonull then GGC holds for F{F.

20

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

FS
LpFrq
Ă
F
XS

LpF8 q

M pFrqFXnr
ĂM pF8 q

Fr

F8
Xnr

F8
Zp

1
K8

Fcyc
Zp
G

r
K

F
Z2
p

∆1

K1
∆

ClpKq

K
Q

Figure 1. The diagram of fields occurring in Theorem 5.4
Proof. This lemma is a special case of [Ban07, Theorem 12]. Since F contains µp , the technicalconditions in the mentioned theorem are satisfied by [LNQD00, Theorem 3.2] or [Ban07, Remark 15].
□
The next result studies pseudonullity of Galois modules under base change.
Lemma 5.3 (Pseudonullity Shifting Down Lemma). Let F be a number field and let F pdq {F be a
K
is a ΛpGK{F1 qZdp -extension. Suppose that F1 {F is a finite extension and set K “ F1 ¨ F pdq . If Xnr
pdq

F
pseudonull module, then Xnr

is a ΛpGF pdq {F q-pseudonull module.
□

Proof. For a proof, see [Kle16, Theorem 3.1-(i)].

The purpose of the next result is to show that Conjecture B is indeed a generalization of GGC.
We resume the notation introduced at the beginning of this section.
Theorem 5.4. In the setting of Ass 3, suppose that there exists an elliptic curve E{K 1 with CM
by an order in K such that Conjecture B holds for pE, F8 q. Then GGC holds for K.
Proof. Let E{K 1 be an elliptic curve with CM by an order in K such that Conjecture B holds
F8
for pE, F8 q. Regarding it as being defined over F “ K 1 pEp q, Theorem 4.10 shows that XnrisΛpGF8 {F q-pseudonull.
1
1
Applying Lemma 5.3 with F “ K 1 , F1 “ F , F p2q “ K8
, and K “ F8 “ F K8
, the ΛpGF8 {F qK1F8
8
1 {K 1 q-pseudonullity of Xnr . Therefore, we havepseudonullity of Xnr
can be shifted down to ΛpGK8

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

shown that21

K1

1 {K 1 q-pseudonull.
Conjecture B for pE, F8 q ùñ Xnr8 is ΛpGK81
p2qr and K “ K 1 “ K 1 K,
r shows
Another application of Lemma 5.3 with F “ K, F1 “ K , F “ K,
8
1
K8
Ă
K
1 {K 1 q-pseudonullity of Xnr
that ΛpGK8
can be shifted down to ΛpGK{K
q-pseudonullity of Xnr. ThisĂis GGC for K.
□

Corollary 5.5. With the same hypotheses of Theorem 5.4, GGC holds also for any number field
L such that K 1 pµp q Ď L Ď F .
1
Proof. Applying Lemma 5.3 with F “ L, F1 “ F , F p2q “ LK8, and K “ F8 “ F F p2q , theF8
ΛpGF8 {F q-pseudonullity of Xnr obtained in Theorem 4.10 can be shifted down to ΛpGF p2q {F qp2qF
pseudonullity of Xnr. Therefore, we have shown that
p2q

F
Conjecture B for pE, F8 q ùñ Xnr

is ΛpGF p2q {L q-pseudonull.

As discussed in Remark 5.1-(1), µp Ć K, hence L ‰ K and L admits at least two complex
r denote the compositum of all Zp -extensions of L, [Was97, Theorem 13.4]
embeddings. Letting L
F p2q
rimplies that GalpL{Lq– Znp for some n ě 3. Using Lemma 5.2 with F “ L, pseudonullity of Xnr
as a ΛpGF p2q {L q-module implies GGC for L.
□
Acknowledgements
D. K. is supported by a PIMS Postdoctoral fellowship. S. R. is supported by the NSERC Discovery Grant 2019-03987. We thank the referee for their timely and thorough reading of the manuscript
which helped strengthen some of the results and also contributed to a clearer exposition.
References
C. S. Aribam, On the µ-invariant of fine Selmer groups, J. Number Theory 135 (2014), 284–300.
A. Bandini, Greenberg’s conjecture and capitulation in Zdp -extensions, J. Number Theory 122 (2007),
no. 1, 121–134.
[BCG` 20] F. M. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi, and M. J. Taylor, Higher
Chern classes in Iwasawa theory, Amer. J. Math. 142 (2020), no. 2, 627–682.
[Bha07]
A. Bhave, Analogue of Kida’s formula for certain strongly admissible extensions, J. Number Theory
122 (2007), no. 1, 100–120.
[BR20]
R. Barbulescu and J. Ray, Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p-rationality conjecture, J. Théor. Nombres Bordeaux 32 (2020), no. 1, 159–177.
[Bri07]
D. Brink, Prime decomposition in the anti-cyclotomic extension, Math. Comp. 76 (2007), no. 260,
2127–2138.
[Coa99]J. Coates, Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication,
Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Math., vol. 1716, Springer, Berlin,
1999, pp. 1–50.
[CS00]
J. Coates and R. Sujatha, Galois cohomology of elliptic curves, Tata Institute of Fundamental Research
Lectures on Mathematics, vol. 88, Published by Narosa Publishing House, New Delhi; for the Tata
Institute of Fundamental Research, Mumbai, 2000.
, Fine Selmer groups for elliptic curves with complex multiplication, Algebra and number theory,[CS05a]Hindustan Book Agency, Delhi, 2005, pp. 327–337.
[CS05b], Fine Selmer groups of elliptic curves over p-adic Lie extensions, Math. Ann. 331 (2005), no. 4,
809–839.
[CSS03]
J. Coates, P. Schneider, and R. Sujatha, Modules over Iwasawa algebras, J. Inst. Math. Jussieu 2 (2003),
no. 1, 73–108.
[CSW01]
J. Coates, R. Sujatha, and J.-P. Wintenberger, On the Euler-Poincaré characteristics of finite dimensional p-adic Galois representations, Publ. Math. Inst. Hautes Études Sci. (2001), no. 93, 107–143.
[Ari14][Ban07]

22

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA[DdSMS99] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, second ed., Cambridge
Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999.
[Fuj13]
S. Fujii, On a bound of λ and the vanishing of µ of Zp -extensions of an imaginary quadratic field, J.
Math. Soc. Japan 65 (2013), no. 1, 277–298.
[Fuj17], On Greenberg’s generalized conjecture for CM-fields, J. Reine Angew. Math. 731 (2017), 259–
278.
[FW79]
B. Ferrero and L. C. Washington, The Iwasawa invariant µp vanishes for abelian number fields, Ann.
of Math. (2) 109 (1979), no. 2, 377–395.
[Gra03]
G. Gras, Class field theory: From theory to practice, Springer Monographs in Mathematics, SpringerVerlag, Berlin, 2003, Translated from the French manuscript by Henri Cohen.
[Gre99]R. Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997),
Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51–144.
[Gre01a], Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry 9 (2001),
407–464.
[Gre01b], Iwasawa theory—past and present, Class field theory—its centenary and prospect (Tokyo, 1998),
Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 335–385.
[Gre16], Galois representations with open image, Ann. Math. Qué. 40 (2016), no. 1, 83–119.
[How98]
S. Howson, Iwasawa theory of elliptic curves for p-adic Lie extensions, Ph.D. thesis, University ofCambridge, 1998.
[How02], Euler characteristics as invariants of Iwasawa modules, Proc. London Math. Soc. (3) 85 (2002),
no. 3, 634–658.
[Ima75]
H. Imai, A remark on the rational points of abelian varieties with values in cyclotomic Zp -extensions,
Proc. Japan Acad. 51 (1975), 12–16.
[Iwa73a]K. Iwasawa, On Zl -extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326.
, On the µ-invariants of Zℓ -extensions, Number theory, algebraic geometry and commutative[Iwa73b]
algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 1–11.
[Jha12]
S. Jha, Fine Selmer group of Hida deformations over non-commutative p-adic Lie extensions, Asian J.
Math. 16 (2012), no. 2, 353–365.
[JS11]
S. Jha and R. Sujatha, On the Hida deformations of fine Selmer groups, J. Algebra 338 (2011), 180–196.
[Kat06]
K. Kato, Universal norms of p-units in some non-commutative Galois extensions, Doc. Math. (2006),
no. Extra Vol., 551–565.
[Kim13]
B.-D. Kim, The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95 (2013),
no. 2, 189–200.
[KL22]
D. Kundu and M. F. Lim, Control theorems for fine Selmer groups, J. théor. Nombres Bordeaux (2022),
accepted for publication.
[Kle16]
S. Kleine, Relative extensions of number fields and Greenberg’s generalised conjecture, Acta Arith. 174(2016), no. 4, 367–392.
[Kob03]
S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003),
no. 1, 1–36.
[Lan87]
S. Lang, Elliptic functions, second ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New
York, 1987, With an appendix by J. Tate.
[LNQD00] A. Lannuzel and T. Nguyen Quang Do, Conjectures de Greenberg et extensions pro-p-libres d’un corps
de nombres, Manuscripta Math. 102 (2000), no. 2, 187–209.
[LP19]
A. Lei and B. Palvannan, Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction, Forum Math. Sigma 7 (2019), Paper No. e25, 81.
[Maz72]B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18(1972), 183–266.
[McC01]W. G. McCallum, Greenberg’s conjecture and units in multiple Zp -extensions, Amer. J. Math. 123(2001), no. 5, 909–930.
[Mer96]
L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124(1996), no. 1-3, 437–449.
[Min86]
J. V. Minardi, Iwasawa modules for Zdp -extensions of algebraic number fields, ProQuest LLC, Ann Arbor,
MI, 1986, Thesis (Ph.D.)–University of Washington.

FINE SELMER GROUPS IN ABELIAN p-ADIC LIE EXTENSIONS

23

A. Movahhedi and T. Nguyen Quang Do, Sur l’arithmétique des corps de nombres p-rationnels, Séminairede Théorie des Nombres, Paris 1987–88, Progr. Math., vol. 81, Birkhäuser Boston, Boston, MA, 1990,
pp. 155–200.
[MS03]
W. G. McCallum and R. T. Sharifi, A cup product in the Galois cohomology of number fields, Duke
Math. J. 120 (2003), no. 2, 269–310.
[Mur97]V. K. Murty, Modular forms and the Chebotarev density theorem. II, Analytic number theory (Kyoto,
1996), London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 287–
308.
[Nek06]
J. Nekovář, Selmer complexes, Astérisque (2006), no. 310, viii+559.
[NQD92]
T. Nguyen Quang Do, Analogues supérieurs du noyau sauvage, Sém. Théor. Nombres Bordeaux (2) 4(1992), no. 2, 263–271.
[NQDV05] T. Nguyen Quang Do and D. Vauclair, K2 et conjecture de Greenberg dans les Zp -extensions multiples,
J. Théor. Nombres Bordeaux 17 (2005), no. 2, 669–688.
[NS21]F. A. E. Nuccio Mortarino Majno di Capriglio and R. Sujatha, Residual supersingular Iwasawa theory
and signed Iwasawa invariants, Rendiconti del Seminario Mathematico di Padova (2021), accepted for
publication, DOI 10.4171/RSMUP/111.
[NSW08]
J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, second ed., Grundlehren dermathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, SpringerVerlag, Berlin, 2008.
[Och09]Y. Ochi, A remark on the pseudo-nullity conjecture for fine Selmer groups of elliptic curves, Comment.
Math. Univ. St. Pauli 58 (2009), no. 1, 1–7.
[OV02]Y. Ochi and O. Venjakob, On the structure of Selmer groups over p-adic Lie extensions, J. Algebraic
Geom. 11 (2002), no. 3, 547–580.
[Oza01]
M. Ozaki, Iwasawa invariants of Zp -extensions over an imaginary quadratic field, Class field theory—its
centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001,
pp. 387–399.
[PR81]
B. Perrin-Riou, Groupe de Selmer d’une courbe elliptique à multiplication complexe, Compositio Math.
43 (1981), no. 3, 387–417.
[PR04]
R. Pollack and K. Rubin, The main conjecture for CM elliptic curves at supersingular primes, Ann. of
Math. (2) 159 (2004), no. 1, 447–464.
[Rib81]
K. Ribet, Torsion points of abelian varieties in cyclotomic extensions, Enseign. Math 27 (1981), 315–
319.
[Rub99]K. Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer,
1999, AWS notes.
, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton,[Rub00]NJ, 2000, Hermann Weyl Lectures. The Institute for Advanced Study.
[Sch79]P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181–205.
[Ser68]J.-P. Serre, Abelian l-adic representations and elliptic curves, W. A. Benjamin, Inc., New YorkAmsterdam, 1968, McGill University lecture notes written with the collaboration of Willem Kuyk andJohn Labute.
[Ser81], Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ.
Math. (1981), no. 54, 323–401.
, Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion[Ser97]and revised by the author.
[Sha08]R. T. Sharifi, On Galois groups of unramified pro-p extensions, Math. Ann. 342 (2008), no. 2, 297–308.
[She18]
S. Shekhar, Comparing the corank of fine Selmer group and Selmer group of elliptic curves, J. Ramanujan
Math. Soc. 33 (2018), no. 2, 205–217.
[Tak21]
N. Takahashi, On Greenberg’s generalized conjecture for imaginary quartic fields, Int. J. Number Theory
17 (2021), no. 5, 1163–1173.
[Ven02]
O. Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc.
(JEMS) 4 (2002), no. 3, 271–311.
[Ven03a], A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory, J.
Reine Angew. Math. 559 (2003), 153–191, With an appendix by Denis Vogel.
[Ven03b], On the Iwasawa theory of p-adic Lie extensions, Compositio Math. 138 (2003), no. 1, 1–54.
[MN90]

24[Was97][Wut07]

D. KUNDU, F. A. E. NUCCIO, AND R. SUJATHA

L. C. Washington, Introduction to cyclotomic fields, 2 ed., Graduate Texts in Mathematics, vol. 83,
Springer-Verlag, New York, 1997.
C. Wuthrich, Iwasawa theory of the fine Selmer group, J. Algebraic Geom. 16 (2007), no. 1, 83–108.

(Kundu) Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
Email address: dkundu@math.toronto.edu(Nuccio) Université Jean Monnet Saint-Étienne, CNRS, Institut Camille Jordan UMR 5208, F-42023,
Saint-Étienne, France
Email address: filippo.nuccio@univ-st-etienne.fr(Sujatha) Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
Email address: sujatha@math.ubc.ca