GROWTH OF FINE SELMER GROUPS IN INFINITE TOWERSDEBANJANA KUNDU
In this paper, we study the growth of ne Selmer groups intwo cases. First, we study the growth of ne Selmer ranks in multiple Zp extensions. We show that the growth of the ne Selmer group is unboundedin such towers. We recover a sucient condition to prove the µ = 0 Conjecture for cyclotomic Zp -extensions. We show that in certain non-cyclotomic
Zp -towers, the µ-invariant of the ne Selmer group can be arbitrarily large.
Second, we show that in an unramied p-class eld tower, the growth of the
ne Selmer group is unbounded. This tower is non-Abelian and non p-adic
analytic.
Abstract.

1. Introduction
Let E be an elliptic curve dened over a number eld F . The Mordell-Weiltheorem says that the group of rational points of E , denoted E(F ), is nitely
generated. Selmer groups play a crucial role in studying the rational points on
elliptic curves. In [21], Mazur introduced the Iwasawa theory of Selmer groups. He
described the growth of the p-primary part of the Selmer group in Zp -towers andshowed that in the ordinary case the growth is controlled.
In [6], Coates and Sujatha introduced the study of a certain subgroup of theSelmer group, called the ne Selmer group. They showed that these subgroups
have stronger niteness properties than the classical Selmer group.
In the 1980's in a series of papers, Cuoco and Monsky studied the growth of classgroups in Zdp -towers when d > 1. In Section 3, we study the growth of the p-rankof ne Selmer group in multiple Zp -extensions. We prove this growth is unboundedin general. When d = 1, we recover a sucient condition to prove the Classicalµ = 0 Conjecture. In a specic Z2p -tower, it is possible to provide an asymptotic
formula. In Section 4 we study the growth of ne Selmer groups in non-cyclotomic
Zp -extensions. We prove an analogue of Iwasawa's theorem [12, Theorem 1] for neSelmer groups and show that there exist non-cyclotomic Zp -extensions where thene Selmer group can have arbitrarily large µ-invariant.
Greenberg had developed a general theory to study the growth of the p-primary
component of the Selmer group in any p-adic analytic Galois extension. In [24],Murty and Ouyang studied the growth of p-Selmer ranks for Abelian varieties with
CM, in p-class eld tower of F . By the work of Boston [3] and Hajir [9], we knowthat this tower denes an innite Galois extension of F whose Galois group is not
p-adic analytic. Hence, growth in these towers is not covered by the general theory
developed by Greenberg. Murty-Ouyang showed that the Selmer rank in such a
Date : March 1, 2020.
2010 Mathematics Subject Classication. Primary 11R23.
Key words and phrases. Iwasawa Theory, Fine Selmer groups, Class groups, p-rank.
1

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DEBANJANA KUNDU

tower is unbounded [24]. We prove that in fact the ne Selmer rank in such a tower
is unbounded.
2. Preliminaries
φ

Let M −
→ N be a homomorphism of Abelian groups (modules etc.), we denoteby M [φ] the kernel of φ.

Denition 2.1. For an Abelian group G, its p-rank rp (G) := dimZ/pZ G[p].
2.1. Review of Fine Selmer Group. Throughout this article p will denote an

odd prime. Let F be a number eld and A/F be an Abelian variety. Let S = S(F )be a nite set of primes in F , which contains the Archimedean primes (denoted S∞ ),the primes of bad reduction of E (denoted Sbad ) and the primes above p (denoted
Sp ). Denote by FS the maximal algebraic extension of F unramied outside S and
set GS (F ) = Gal(FS /F ). Dene the p-ne Selmer group RpS (A/F ) and p∞ -neSelmer group Rp∞ (A/F ) as in [31].
Let K be an imaginary quadratic eld and p be a rational prime that splits as
pp in K . Let F/K be any nite Galois extension and E be an elliptic curve denedover it.
Dene the p-ne Selmer group as


M


RpS (E/F ) := ker H 1 GS (F ), E[p] →H 1 Fv , E[p]  .
v∈Sdene RpSn (E/F ) for any n≥ 1. Taking direct limits, dene theAnalogously,
p∞ -ne Selmer group (also called the p-primary ne Selmer group),Rp∞ (E/F ) := lim RpSm (E/F )−→mwhere the limit is with respect to the maps induced by E[pm ] ,→ E[pm+1 ]. An
equivalent denition is the following,M
Rp∞ (E/F ) = ker H 1 (GS (F ), E[p∞ ]) →H 1 (Fv , E[p∞ ]) .
v∈SThough it appears a priori that the denition depends on S , one can check [17,Lemma 4.1] that the p∞ -ne Selmer group is independent of this choice. The sameis true for the p∞ -ne Selmer group. There are analogous denitions for every
nite extension, F 0 /F . Suppose F∞ /F is contained in FS with Gal(F∞ /F ) beinga p-adic Lie group. Set

Rp∞ (E/F∞ ) := lim Rp∞ (A/F 0 ).
−→where the inductive limit is over nite extensions of F contained in F∞ .
3. Growth of p-Fine Selmer Ranks in Zdp -Extensions
In this section, we study the growth of ne Selmer groups in Zdp -extensions of anumber eld. These are Abelian pro-p, p-adic analytic extensions. In Theorem 3.3,
we obtain an estimate for the rank growth of ne Selmer groups in a specic Z2p
extension. In the general setting, we can only prove that the p-rank growth isunbounded under certain assumptions (see Theorem 3.8). Our result is a renementof the result of Lim-Murty [16, Proposition 5.1].

3

3.1. Notation: Consider a Zdp -extension F∞ /F . Set Σ = Gal(F∞ /F ) ' Zdp . FurnΣnther, write Σn = Σp and Fn = F∞
. Here, Fn is a (Z/pn Z)d -extension of F inside
F∞ . The Iwasawa algebra Λd , is the completed group ring of Σ over Zp ; let Λd be
the completion (with respect to the powers of the augmentation ideal) of the Z/pZ
group ring of Σ. We have,Λd ' Zp [[T1 , T2 , . . . , Td ]];Λd ' Z/pZ[[T1 , T2 , . . . , Td ]].

Let M (F∞ ) (resp. MS (F∞ )) be the maximal Abelian unramied pro-p extension of
F∞ (resp. the maximal Abelian unramied pro-p extension of F∞ such that primesin S are totally split). The Iwasawa-Greenberg module (resp. S -Iwasawa-Greenberg
module) is denoted by X := Gal(M (F∞ )/F∞ ) (resp. XS := Gal(MS (F∞ )/F∞ )).
Let X (resp. XS ) be the reduction mod p of the Iwasawa-Greenberg module (resp.
S -Iwasawa-Greenberg module). Dene a (resp. a0 ) to be the height of the local
ring Λd / Ann X (resp. Λd / Ann XS ); so 1 ≤ a, a0 ≤ d.
3.2. Trivialising Extension of an Elliptic Curves with CM. Let K be an
imaginary quadratic eld. Suppose further p 6= 2, 3 splits as pp̄ in K . Let E/K be
an elliptic curve with good reduction at p and CM by OK . Set

F = K(E[p]), F∞ = K(E[p∞ ]), G = Gal(F∞ /F ), G∞ = Gal(F∞ /K).

Remark 3.1. The hypotheses, EndF (E) is the maximal order of K , involves no real
loss of generality since every E/F with Q ⊗Z EndF (E) isomorphic to K is isogenous
over F to one with this property.
We refer to F∞ as the trivializing extension of F . By the Weil pairing, F∞
contains the cyclotomic Zp -extension of F , which in turn is denoted by Fcyc .
The group G is pro-p and is isomorphic to Z2p . By the theory of complex multiplication, the Galois group Gal(F/K) is Abelian. Since p splits completely in K/Q,
the order of Gal(F/K) is coprime to p. It then follows that G∞ ' G × ∆ where∆ ' Gal(F/K).

Proposition 3.2. Let E be an elliptic curve over the imaginary quadratic eld K
with CM by OK . Set F = K(E[p]) and let F∞ be the trivialising extension. Then
rp Rp∞ (E/Fn ) − 2rp Cl(Fn ) = O(1)where Fn = K(E[pn ]) is a (Z/pn Z)2 -extension of F contained in F∞ and Cl(Fn )is the class group of Fn .
Proof. The proof is identical to that of [17, Theorem 5.1] provided the nite primes
Sf (F ) are nitely decomposed in F∞ /F . Recall that primes above p ramify, andall ramied primes in this extension are nitely decomposed.
By hypothesis, E attains good reduction at all the primes of F [28, Lemma 2] or[30]. The denition of Rp∞ (E/Fn ) is independent of S , so choose S to be Sp ∪ S∞ .
The condition of nite decomposition is met [7, Page 45], and the proposition
follows.
Using the main result in [23], there is the following interesting corollary.

Theorem 3.3. Let E be an elliptic curve over an imaginary quadratic eld K ,with CM by OK . Set F = K(E[p]) and F∞ be the trivialising extension. Then for

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DEBANJANA KUNDU

Fn and a as dened before,
rp Rp∞ E/Fn = 2cpan + O(p(a−1)n ).

When a = 1, c is an integer and when a = 2, c = rankΛ2 (X).
Proof. Consider [23, Theorem 1.9] when d = 2. Monsky proves that
rp Cl(Fn ) = cpan + O p(a−1)n .
When a = 1, c ∈ Z; when a = 2, c = rankΛ2 (X). The theorem follows.



3.3. General Zdp -Extension. Using a variant of Monsky's result in [23], we showthat the ne Selmer rank growth is unbounded in a general Zdp -extension.

Lemma 3.4. [17, Lemma 3.2] Let G be any pro-p group and M be a discrete Gmodule that is conitely generated over Zp . Dene h1 (G) := rp H 1 (G, Z/pZ) .

If h1 (G) is nite then rp (H 1 (G, M )) is nite. Furthermoreh1 (G)rp (M G ) − rp (M/M G )G ≤ rp H 1 (G, M ).
≤ h1 (G) corankZp (M ) + logp M/Mdiv

Lemma 3.5. Let A be a d-dimensional Abelian variety dened over a number

eld F . Consider a nite set of primes S containing Sp ∪ Sbad ∪ S∞ . SupposeA(F )[p] 6= 0. Then




(1)
rp Rp∞ A/F ≥ rp ClS (F ) rp A (F ) [p] − 2d




(2)
rp RpS A/F ≥ rp ClS (F ) rp A (F ) [p] − 2d.
Proof. Inequality 1 is proven in [17, Lemma 4.3]. We therefore provide a proof of
Inequality 2. The proof is similar but we include it for the sake of completeness.
Let HS (F ) = H be the p-Hilbert S -class eld of F . Denote the Galois group,
Gal(HS (F )/F ) = ClS (F ) by G . We call ClS (F ) the S -class group of F . We have
rp ClS (F ) = rp ClS (F )/p ClS (F )= dimZ/pZ H 1 (G, Z/pZ)= h1 (G).
The rst equality follows from niteness of the S -class group. The second equality
is the denition of the S -class group. By Lemma 3.4, to prove Inequality 2, itsuces to prove




(3)
rp RpS A/F ≥ rp H 1 G, A(H)[p] .
To prove this inequality, consider the following commutative diagram with exact
rows.
L
1
0 −→ RpS (A/F
) −→ H 1 (GS (Fv∈S H (Fv , A[p]) ), A[p]) −→



yα
yβ
yγ
L
L
S
1
1
0 −→ Rp (A/H) −→ H (GS (H), A[p]) −→v∈S
w|v H (Hw , A[p])5

The vertical maps are given by restriction maps. Write γ = ⊕v γv whereMγv : H 1 Fv , A[p] →H 1 Hw , A[p] .
w|v

By the ination-restriction sequence, ker γv = H 1 (Gv , A[p]) where Gv is the decomposition group of G at v . By the denition of the p-Hilbert S -class eld, all theprimes in S(F ) split completely in HS ; hence Gv = 1. Thus, ker γ is trivial.
By ination-restriction, ker β = H 1 G, A(H)[p] . By a diagram chasing argument,
H 1 G, A (H) [p] ,→ RpS (A/F ).
This implies
rp RpS (A/F ) ≥ rp H 1 G, A (H) [p] .
The result follows.

Remark 3.6.

Lemma 3.5 holds at every layer Fn /F in the Zdp -tower.

Repeating the argument as in the original paper of Monsky, it is possible toprove the following variant of [23, Theorem 1.9].

Theorem 3.7. With notation introduced at the start of the section, there is apositive real constant c, such that

 0


0
rp ClS (Fn ) = cpa n + O p(a −1)n ,where ClS (Fn ) is the S -class group of Fn .
The fact that c > 0, follows from [22, Corollary to Theorem 1.8].

Theorem 3.8. Let A be an Abelian variety dened over F , and let p be an

odd prime such that A(F )[p] 6= 0. Let F∞ /F be a Zdp -extension such that XS
is innite. LetFn be a (Z/pn Z)d -extension of F inside F∞ . Then as n → ∞,rp Rp∞ (A/Fn ) → ∞.
Proof. By Theorem 3.7, if XS is innite then rp ClS (Fn ) approaches innity as
n approaches innity. The conclusion follows from Lemma 3.5.
Remark 3.9. Theorem 3.8 is a ne Selmer variant of [16, Proposition 5.1]. Unlike
the result of Lim-Murty, we do not need to impose any restrictions on the reduction
type at p. This comes at a cost that we need a smaller group XS to be innite.
3.4. Growth in Zp -Extensions. We focus on the case d = 1. Let F be a number
eld and F∞ /F be any Zp -extension with Γ = Gal(F∞ /F ) ' Zp . There is an
isomorphism Λ(Γ) ' Zp [[T ]] given by γ − 1 7→ T , where γ is a topological generatorof Γ. By the structure theorem for a nitely generated Λ(Γ)-module M , there is a
pseudo-isomorphism
s
t
.
.
M
M
Λ(Γ)
Λ(Γ)  lj M → Λ(Γ)r ⊕
mi ⊕
p
f
i=1

j=1

j

where s, t are nite, mPis, lj > 0, and each fj is a distinguished polynomial. If
r = 0, dene µ(M ) := i=1 mi .

Conjecture (Classical µ = 0 Conjecture [11]). Let Fcyc /F be the cyclotomic Zp extension and X(Fcyc ) be the associated Iwasawa module. Then µ(X(Fcyc )) = 0.

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DEBANJANA KUNDU





Remark 3.10. It is known that µ X Fcyc = µ XS Fcyc [25, Corollary 11.3.6].

Lemma 3.11. [16, Lemma 5.3] Let M be a nitely generated Λ(Γ)-module andnwn = (1 + T )p − 1. For n  0,
 .

 n
M
rp
(p, w ) M = r(M ) + s(M ) p + O(1).
n

The Pontryagin dual of the p∞ -ne Selmer group is denoted Y (A/F∞ ). Also,.
∨
Y (A/F∞ )pY (A/F ) ' Rp∞ (A/F∞ )[p] .
∞We prove a simple variant of Inequality 1 , which follows from the next lemma.

Lemma 3.12. [17, Lemma 5.2] Let F∞ be any Zp extension of F such that all

the primes above p are nitely decomposed. Let Fn be the subeld of F∞ such that
[Fn : F ] = pn . Let S = Sp ∪ S∞ . Then
rp Cl(Fn ) − rp ClS (Fn ) = O(1).

Proof. The only non-Archimedean primes in S are the primes above p, they arenitely decomposed in any F∞ /F .
Proposition 3.13. Let A/F be an Abelian variety and p be an odd prime. Suppose

A has good reduction everywhere over F and A(F )[p] 6= 0. Let F∞ be a Zp -extensionof F and Fn be the subeld of F∞ such that [Fn : F ] = pn . Then



(4)
rp Rp∞ (A/Fn ) ≥ rp Cl(Fn ) rp A(Fn )[p] + O(1)Proof. The denition of Rp∞ (A/Fn ) is independent of S , choose S = S(Fn ) to beprecisely the set of Archimedean primes and the primes above p. The proposition
follows from Lemma 3.5 upon noting that Lemma 3.12 applies in this case.
Using Proposition 3.13 we give the ne Selmer variant of a theorem of Lim-Murty[16, Theorem 5.6] and esnavi£ius [4, Proposition 7.1].

Theorem 3.14. Let p 6= 2. Let A be an Abelian variety dened over F with good
reduction everywhere over F and A(F )[p] 6= 0. Let F∞ be any Zp -extension of F .
Then




r Y (A/F∞ ) + s Y (A/F∞ ) ≥ s X(F∞ ) rp A (F∞ ) [p]where X(F∞ ) is the Iwasawa module over the Zp -extension F∞ /F .
Proof. Set Γn = Gal(F∞ /Fn ). Consider the following commutative diagram withthe vertical maps given by restriction
L
∞
1
∞
0 →
R(A/F
→
H 1 (GS (F→vn H (F n)n ), A[p ])n,vn , A[p ])






rn
fn
γn
y
y
y

Γn
L
0 → R(A/F∞ )Γn → H 1 (GS (F∞ ), A[p∞ ])Γn →limn vn H 1 Fn,vn , A[p∞ ]−→Note that rp ker(fn ) ≤ 2d. Thus ker(rn ) has bounded p-rank. The Control
Theorem for ne Selmer groups guarantee that coker(rn ) is also nite and bounded7

independent of n (see [29, chapter VII] and [31]). Further,





r Y (A/F∞ ) + s Y (A/F∞ ) pn = rp Rp∞ A/Fn + O(1)≥ rp Cl(Fn ) rp A(Fn )[p] + O(1).
The rst equality follows from Lemma 3.11 upon observing that the Pontryagin
dual of Rp∞ (A/F∞ )Γn [p] is equal to Y /(p, wn )Y . The second inequality follows
from Proposition 3.13. There exists n0 , such that for n ≥ n0.
.
Cl(F )X(F∞ )  w  → 0.
p Cl(F ) →
p, n
wn0

The kernel of this map is bounded independent of n [25, Lemma 11.1.5]. Since
X(F∞ ) is always a nitely generated torsion Λ(Γ)-module (see [25, Proposition
11.3.1]), by Lemma 3.11

rp Cl(Fn ) = s(X(F∞ ))pn + O(1).
The result follows since rp (A(F∞ )[p]) = rp (A(Fn )[p]) for suciently large n.

Recall that in the cyclotomic Zp -extension, all the primes are nitely decomposed. Hence, Lemma 3.12 holds for all nite sets S ⊇ Sp ∪ Sbad ∪ S∞ . In this case,
we may drop the hypothesis of good reduction everywhere from Theorem 3.14. toobtainr Y (A/F∞ ) + s Y (A/F∞ ) ≥ s XS (F∞ ) rp A(F∞ )[p] .
An immediate corollary is the following.

Corollary 3.15. Let p 6= 2. Ifthere exists one
 elliptic curve E/F with E(F )[p] 6= 0such that r Y E/Fcyc = s Y E/Fcyc = 0, then the Classical Iwasawa µ = 0
Conjecture holds for Fcyc /F .




We remind the readers that r Y E/Fcyc = s Y E/Fcyc = 0 is precisely
Coates-Sujatha Conjecture A [6]. The above corollary is true if we consider Abelianvarieties in general, but we formulate it in this way since the conjecture of CoatesSujatha is for elliptic curves.

Remark 3.16. Corollary 3.15 is an eective variant of known results in the literature(see [6, Theorem 3.4], [17, Theorem 5.4]). In [1], Aribam had used [6, Theorem 3.4]
to give new evidence of the Iwasawa µ = 0 Conjecture.
4. Arbitrarily Large µ-Invariants of Fine Selmer Groups in
Non-Cyclotomic Zp -Extensions
We begin by recalling a theorem proven by Iwasawa [12, Theorem 1].

Theorem 4.1 (Iwasawa's Theorem for Non-Cyclotomic Extensions). Let F be thecyclotomic eld of p-th or 4-th roots of unity according as p > 2 or p = 2. For
any given integer N ≥ 1, there exists a cyclic extension L/F of degree p and a
Zp -extension L∞ /L such thatµ X (L∞ ) ≥ N.

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DEBANJANA KUNDU

The close relationship between ne Selmer groups and class groups in Zp -towers
raises the following natural question: can the µ-invariant associated with a ne
Selmer group be arbitrarily large in a non-cyclotomic Zp -extension. We answer thisquestion in the armative under the following hypothesis which we will call the(universal) analogue of the Weak Leopoldt Conjecture.
Hypothesis: Let K be any number eld and A be an Abelian variety dened over
K . Let p be an odd prime and K∞ /K be any Zp -extension of K . Then
H 2 GS (K∞ ) , A[p∞ ] = 0.

Theorem 4.2. Let F be the cyclotomic eld of p-th roots of unity for p > 2.

Let A/F be an Abelian variety of dimension d such that A(F )[p] 6= 0. Supposethe (universal) analogue of the Weak Leopoldt Conjecture holds. Given an integerN ≥ 1, there exists a cyclic Galois extension L/F of degree p and a Zp -extension
L∞ /L such thatµ Y A/L∞ ≥ N.
Remark 4.3.
(1) It is believed that for all number elds and all Zp -extensions
F∞ /F , the cohomology group H 2 GS (F∞ ), A[p∞ ] = 0. This is often
called the Abelian variety analogue of the Weak Leopoldt Conjecture overF . This implies Y (A/F∞ ) is Λ(Γ)-torsion where Γ = Gal(F∞ /F ) ' Zp
(see [20, Theorem 2.2] or [6, Lemma 3.1] for the case when primes in S
are nitely decomposed). Thus, conjecturally r Y (A/F∞ ) should be 0.
When F∞ = Fcyc , this is known in a large number of cases by a deep resultof Kato [13] For the Zp -extension unramied outside p, it was proved in [27](see also [15]). For non-cyclotomic extensions, there is little unconditional
evidence towards this conjecture (see [2]).
(2) In view of recent results of Hajir-Maire [10], we believe it might be possible
to extend the above theorem to more general situations. It would be interesting to consider analogues of results of Ozaki [26, Theorem 2] for p∞ -neSelmer groups.
4.1. Review of Iwasawa's Result. In proving Theorem 4.1, Iwasawa cruciallyused a result of Chevalley [5] on ambiguous class number formula.

Denition 4.4. Let F be a number eld and L/F be a cyclic Z/pZ-extension. Let
G = Gal(L/F ) = hσi. An ideal class [a] ∈ Cl(L) is called• ambiguous if [a]σ = [a], i.e. there exists an element α ∈ L× such thataσ−1 = (α).
• strongly ambiguous if aσ−1 = (1).

The subgroup of the class group Cl(L) consisting of ambiguous ideal classes (resp.
strongly ambiguous ideal classes) is denoted by Am(L/F ) (resp. Amst (L/F )).

Theorem (Ambiguous Class Number Formula). The number of ambiguous idealclasses is given by
(5)
(6)

pT −1
[EF : EF ∩ N L× ]pT −1
# Amst (L/F ) = h(F ) ×[EF : N EL ]# Am(L/F ) = h(F ) ×

9where h(F ) is the class number of the base eld F , T is the number of ramied
primes, EF is the unit group of F , and EF ∩ N L× is the subgroup of units that arenorms of elements of L. Moreover, the above two formulas are equivalent.
Proof. See [5] or [14, Theorem 1].



The following variant of the ambiguous class number formula can be proved very
similar to [14, Theorem 1].

Proposition 4.5. Let F be a number eld and L/F be a cyclic Z/pZ-extension
with σ a generator of the Galois group G = Gal(L/F ). Let D be the degree of theextension L/Q and T be the number of primes of F that ramify in L. Then,
rp Amst L/F ≥ T − D.
4.2. Proof of Theorem 4.2. The proof can be divided into four main steps. The
rst three are the same as that of Theorem 4.1.
Step 1: Let F+ be the maximal totally real subeld of F . Let Fnc /F be a noncyclotomic Zp -extension which satises a special property: if Fnnc is the n-th layerof the Zp -tower, then Fnnc is Galois over F+ . Furthermore, Gn = Gal(Fnnc /F+ )is the dihedral group of order 2pn . Such a non-cyclotomic Zp -extension exists by
Iwasawa's construction [12, Section 2].
Step 2: Let l+ - p be a prime ideal of F+ which is inert in F , and be the uniqueprime ideal of F above l+ . Note l+ is unramied in Fnnc . Using group theoretic
properties of the dihedral group and class eld theory, it can be shown that l is
totally split in Fnnc [12]. This holds for every n, therefore l is totally split in Fnc /F .
By Chebotarev density theorem, there are innitely many prime ideals l+ in F+
which are inert in F . Thus, there are innitely many prime ideals in F which split
completely in Fnc /F .
Step 3: Choose prime ideals l1 , . . . , lt , t ≥ 1, in F which are prime to p and aretotally split in Fnc /F . We know from Step 2 that there are innitely many suchprimes. Let η be a non-zero element of F which is divisible exactly by the rstpower of li for 1 ≤ i ≤ t. Set√ 
L=F pη ;L∞ = LFnc .
Note Fnc ∩ L = F and L∞ /L is a Zp -extension. Let Ln be the n-th layer of the Zp
tower L∞ /L, then Ln /Fn is a cyclic extension of degree p. More precisely,√ (7)
Ln = Fnnc p η ,n ≥ 0.
We need to prove a technical lemma. Recall that the denition of the p-primary
ne Selmer group is independent of the choice of S . Denote the subset of niteprimes of S be Sf . Further, set Sf = s0 .

Lemma 4.6. Let F be the cyclotomic eld of p-th roots of unity for p > 2. Let
A/F be an Abelian variety of dimension d such that A(F )[p] 6= 0. Suppose theanalogue of the weak Leopoldt conjecture holds. Let L/F be a Z/pZ-extension as
constructed in Theorem 4.1. Then, rp Rp∞ (A/Ln ) ≥ rp A (Ln ) [p] s X(L∞ ) pn − 4ds0 pn + c
where Ln is as dened in (7) and c is a constant.

10

DEBANJANA KUNDU

Proof. The proof follows the same steps as Proposition 3.13. But primes in the Zp extension L∞ /L need not be nitely decomposed, so the analysis is more intricate.
Step a: Consider the following short exact sequence for all n [25, Lemma 10.3.12],
αn
ClS (Ln ) → 0.
Z|Sf (Ln )| → Cl(Ln ) −−→

Comparing p-ranks in this short exact sequence [17, Lemma 3.2], we obtain
rp Cl (Ln ) − rp ClS (Ln ) ≤ 2 Sf (Ln ) ≤ 2s0 pn .

Step b: Lemma 3.5 applied to the number eld Ln yieldsrp Rp∞ A/Ln ≥ rp ClS (Ln ) rp A (Ln ) [p] − 2d.
Therefore, we obtainrp Rp∞ A/Ln ≥ rp Cl (Ln ) − 2s0 pn rp A (Ln ) [p] − 2d≥ rp Cl (Ln ) rp A (Ln ) [p] − 4ds0 pn − 2d.
By the Structure Theorem (in particular Lemma 3.11),
rp Cl (Ln ) = s X(L∞ ) pn + O(1).
Plugging this back into the above inequality, proves the lemma.

Step 4: We are assuming that the analogue of the weak Leopoldt Conjectureholds for the Zp -extension L∞ /L where L∞ = LFnc . Now,s Y (A/L∞ ) pn = rp Rp∞ A/Ln + O(1) ≥ rp A (Ln ) [p] s X(L∞ ) pn − 4ds0 pn + O(1)≥ rp A (Ln ) [p] t − p(p − 1) − 4ds0 pn + O(1).
The rst equality follows from Lemma 3.11. The last line follows from Proposition 4.5 by observing that at least tpn primes of Fn ramify in Ln (by construction)and [Ln : Q] = p(p−1)pn . By Step 2 we know that t can be chosen to be arbitrarily
large. Therefore, given N ≥ 1 there exists L/F such thats Y (A/L∞ ) ≥ N.

Since µ Y (A/L∞ ) ≥ s Y (A/L∞ ) , the theorem follows.

Remark 4.7. The condition A/F is an Abelian variety such that A(F )[p] = 0
is a mild condition. We can base change to a nite extension F 0 /F such thatA(F 0 )[p] 6= 0. Theorem 4.2 can then be stated (and proved) in terms of F 0 provided
the (universal) analogue of the Weak Leopoldt Conjecture holds.
5. Growth of Fine Selmer Groups in Infinite Class Field Towers
In this section, we consider growth of ne Selmer groups in non-Abelian andnon-p adic analytic (innite) extensions. Let K be an imaginary quadratic eldand p be an unramied prime in K . Further suppose p ≥ 5. Let F/K be anynite Galois extension where p is unramied and E/F be an elliptic curve with CM
by OK . Let S be a nite set of primes in F containing the Archimedean primes,
primes above p, and the primes of bad reduction of E .

11Denition 5.1. Set HS (F ) to be the p-Hilbert S -class eld of F . It is the maximal

Abelian unramied p-extension of F such that all the primes in S split completely.

By class eld theory, Gal HS (F )/F ' ClS (F ), the S -ideal class group of F .
S
Set F∞
to be the maximal unramied p-extension of F such that all the primes inS
S split completely. Set Γ = ΓF = Gal(F∞/F ) and write {Γn }n≥0 for its derived
series. For each n ≥ 0, the xed eld Fn corresponding to Γn is the p-Hilbert
S -class eld of Fn−1 .
Set r1 (F ) (resp. r2 (F )) to mean the number of real (resp. number of pairs ofcomplex) places of F . In [8], Golod and Shafarevich showed that ifq
rp ClS (F ) ≥ 2 + 2 r1 (F ) + r2 (F ) + δ + S \ S∞then ΓF is innite. Stark posed the following natural question.
Question. Does rp ClS (Fn ) tend to innity as n tends to innity?
For any pro-p group G, Z/pZ is a G-module with trivial G-action. In particular,
take G = Γn , the n-th term of the derived series. By class eld theory and nitenessof Cl(Fn ) (and hence of ClS (Fn )) we know
rp ClS (Fn ) = rp ClS (Fn )/p= dimZ/pZ H 1 Gal(HS (Fn )/Fn ), Z/pZ
= dimZ/pZ H 1 Γn , Z/pZ= dimZ/pZ Hom Γn , Z/pZ .
This is the number of minimal generatorsof Γn . By a result of Lubotzky andMann [18, Theorem A], rp ClS (Fn ) tends to innity if and only if ΓF is not padic analytic. It follows that the question posed by Stark is equivalent to asking
whether ΓF is p-adic analytic. The following result was shown independently by
Boston and Hajir [3], [9]. A dierent proof was also provided by Matar [19].

Theorem 5.2. Let F be a number eld. If either of the following inequalities holdp
rp Cl(F ) ≥ 2 + 2 r1 (F ) + r2 (F ),qor rp ClS (F ) ≥ 2 + 2 r1 (F ) + r2 (F ) + δ + S \ S∞ ,then ΓF is not p-adic analytic.
The main result we prove in this section is the following.

Theorem 5.3. Let F be a number eld containing the imaginary quadratic eld

K . Let p 6= 2, 3 be a prime that splits in K as pp̄ such that p is unramiedin F/K . Suppose E(F )[p] 6= 0. Let S be a nite set of primes in F containing
the Archimedean primes, primes above p and primes where E has bad reduction.
Assume that F satises the Golod-Shafarevich inequalityq
rp ClS (F ) ≥ 2 + 2 r1 (F ) + r2 (F ) + δ + S \ S∞ .

SLet F∞
be the maximal unramied (non-constant) pro-p extension of F whereprimes in S split completely and let Fn be the n-th layer of the p-Hilbert S -classeld tower. Then the p-rank of the p-ne Selmer group of E/Fn becomes arbitrarily
large as n → ∞.

12

DEBANJANA KUNDU

Proof. In Lemma 3.5, we proved that if E(F )[p] 6= 0,rp RpS E/F ≥ rp ClS (F ) rp E (F ) [p] − 2.
In the exact same way, it is possible to prove the following inequality for every
extension Fn /F ,




(8)
rp RpS E/Fn ≥ rp ClS (Fn ) rp E (Fn ) [p] − 2.
Since we assume that F satises the Golod-Shafarevich inequality, by Theorem 5.2
we know ΓF is not p-adic analytic. By the theorem of Lubotzky-Mann, rp ClS (Fn )tends to innity as n → ∞. The theorem now follows from Inequality 8.
Remark 5.4.
(1) Theorem 5.3 is true for any d-dimensional CM Abelian variety.
(2) Theorem 5.3 is the ne Selmer variant of [24, Theorem 4] and implies [17,Theorem 7.2].
The author thanks the referee for pointing out that the following theorem canbe proved. It is the class group variant of the above theorem.
Let p be a xed odd prime and let F be a number eld. Let F∞ be the maximal
unramied p-extension of F . Now, set Σ = ΣF = Gal(F∞ /F ). Let {Σn }n≥0 be thederived series of Σ. For every n ≥, the xed eld Fn+1 corresponding to Σn+1 isthe p-Hilbert class eld of Fn .
By class eld theory and niteness of Cl(Fn ), we know
rp Cl(Fn ) = dimZ/pZ Hom Σn , Z/pZ .
This is the number of minimal generators of Σn . By a result of Lubotzky and Mann
[18, Theorem A], the p-rank rp Cl (Fn ) tends to innity if and only if ΣF is not
p-adic analytic.

Theorem 5.5. Let K be an imaginary quadratic eld and let F be a nite Galois

extension of K . Let p 6= 2, 3 be a prime that splits in K as pp̄. Let E/F be an
elliptic curve with CM by OK and suppose E(F )[p] 6= 0. Assume that F satises
the inequalityp(9)
rp Cl(F ) ≥ 2 + 2 r1 (F ) + r2 (F ).

Let F∞ be the maximal unramied p-extension of F and let Fn be the n-th layerof the p-Hilbert class eld tower. Then the p-rank of the p∞ -ne Selmer group of
E/Fn becomes arbitrarily large as n → ∞.
Proof. Let L = F (E[p]). Since E is an elliptic curve with CM, it has good reduction
at all the primes in L. Denote by HF , the p-Hilbert class eld of F and set
L0 = LHF . Notice that by hypothesis, the group G = Gal(L0 /L) ' Gal(HF /F )has order coprime to p.
To prove the theorem, it suces to show the following injection



(10)
H 1 Gal HF /F , E(HF )[p∞ ] ,→ Rp∞ (E/F ).
Indeed, once this injection is established, one has






(11)
rp Rp∞ E/F ≥ rp H 1 Gal HF /F , E(HF )[p∞ ](12)≥ rp Cl(F ) rp E (F ) [p] − 2

13where the second inequality can be proven in the same way as Lemma 3.5 uponobserving that rp E (F ) [p] = rp E (F ) [p∞ ] . Since F satises Inequality 9, by
Theorem 5.2 we know that ΣF is not p-adic analytic. By the theorem of LubotzkyMann, the p-rank rp Cl (Fn ) tends to innity as n → ∞. The theorem now follows
From Inequality 12.
It remains to show that Injection 10 holds. By an argument used in proving
Lemma 3.5, we have an injection
H 1 G, E L0 [p∞ ] ,→ Rp∞ E/L .
Taking Gal(L/F )-invariant, it follows that
Gal(L/F )Gal(L/F )H 1 G, E L0 [p∞ ],→ Rp∞ E/L.
Injection 10 follows from the above injection by the same argument as in [16, Theorem 4.1(ii)]. This nishes the proof of the theorem.
Remark 5.6.
(1) In general, this theorem can be proven for d-dimensional CM
Abelian varieties. For the case√ of CM elliptic curves, the theorem holds for
p = 3, provided K is not Q( −3) [16, Remark 4.5].
(2) Using analogous arguments, it should be possible to prove under the same
hypothesis that the p-rank of the p-ne Selmer group of E/Fn becomes
arbitrarily large as n → ∞.
Acknowledgement

We thank Kumar Murty for his constant support and encouragement. We are
grateful to R. Sujatha, Antonio Lei, Ahmed Matar, Meng Fai Lim, and Sudhanshu
Shekhar for helpful discussions. We thank all past and present members of the
GANITA Lab. We thank the referee for a careful reading of the manuscript, andfor suggesting the proof of Theorem 5.5.
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15
Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George
St., Room 6290, Toronto, Ontario, Canada, M5S 2E4Email address : dkundu@math.utoronto.ca