GROWTH OF SELMER GROUPS AND FINE SELMER GROUPSIN UNIFORM PRO-p EXTENSIONSDEBANJANA KUNDU
In this article, we study the growth of (ne) Selmer groups ofelliptic curves in certain innite Galois extensions where the Galois group G,
is a uniform, pro-p, p-adic Lie group. By comparing the growth of (ne)Selmer groups with that of class groups, we show that it is possible for theµ-invariant of the (ne) Selmer group to become arbitrarily large in a certainclass of nilpotent, uniform, pro-p Lie extension. We also study the growth of
ne Selmer groups in false Tate curve extensions.
Dans cet article, nous étudions la croissance de groupes de Selmer (ns)de courbes elliptiques dans certaines extensions innies de Galois oú le groupede Galois G est un groupe de Lie uniforme, pro-p, p-adique. En comparant lacroissance des groupes (ns) de Selmer avec celle des groupes de classes, nous
montrons qu'il est possible que l'invariant µ du groupe (n) de Selmer devienne
arbitrairement grand dans une certaine classe de nilpotents, uniformes, pro-p
Lie extension. Nous étudions également la croissance de groupes de Selmerns dans de fausses extensions de courbe de Tate.
Abstract.

1. Introduction
Iwasawa theory began as the study of ideal class groups over innite towers of
number elds. Kenkichi Iwasawa introduced the notion of a µ-invariant to study thegrowth of (p-ranks) of ideal class groups in Zp -extensions. In [13], he constructed
Zp -extensions over number elds with arbitrarily large µ-invariants. This notion ofa µ-invariant was later generalized to all uniform pro-p groups [12], [30]. In [10],Hajir and Maire investigated uniform pro-p groups which are realisable as Galois
groups of extensions of number elds with arbitrarily large µ-invariant.
In the study of rational points on Abelian varieties, the Selmer group plays animportant role. In [21], exploiting the intimate connection between class groups
and Selmer groups, Mazur developed an analogous theory to study the growth of
Selmer groups of Abelian varieties in Zp -extensions. He showed that the Selmer
groups of Abelian varieties and ideal class groups have similar growth patterns in
Zp -extensions. When the Abelian variety has good ordinary reduction at p, it ispossible to associate a µ-invariant to the Selmer group. In [3], Coates and Sujatha
showed that the ne Selmer group has even stronger niteness properties than theclassical Selmer group. They showed that the growth of ne Selmer groups mimics
the growth of ideal class group in a general p-adic analytic extension containing
the cyclotomic Zp -extension. In [20], this was further investigated by Lim andMurty wherein they extended this analogy to some non p-adic analytic extensions
as well. In [16], the author showed that there exist non-cyclotomic Zp -extensions
Date : October 18, 2020.
2010 Mathematics Subject Classication. Primary 11R23.
Key words and phrases. Selmer groups, Fine Selmer groups, Class groups, p-rank.
1

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DEBANJANA KUNDU

over number elds where the µ-invariant associated to the ne Selmer group canbe made arbitrarily large. In this article, the goal is to extend these methods andinvestigate the growth of Selmer groups and ne Selmer groups of Abelian varieties(and their associated µ-invariants) in extensions of the kind studied by Hajir-Maire.
In Section 3, we develop a general strategy to show that the µ-invariant of (ne)Selmer groups can be arbitrarily large in extensions where it is known that the µinvariant associated to the class group is arbitrarily large. We give explicit examplesof nilpotent, uniform, pro-p, p-adic Lie extensions of number elds with arbitrarily
large µ-invariant of (ne) Selmer groups. In Section 4, we study the growth of ne
Selmer groups in metabelian extensions (in particular, false Tate curve extensions).
All the results we prove in this paper are for elliptic curves, but can be easily
generalized to Abelian varieties.
2. Preliminaries
Throughout this article, p is an odd prime.
2.1. Selmer Groups and Fine Selmer Groups [2], [3], [32]. Let E be an
elliptic curve dened over a xed number eld F . Let S be a nite set of primes
in F containing the Archimedean primes, the primes above p, and the primes ofbad reduction of E ; for short write S ⊇ S∞ ∪ Sp ∪ Sbad . For any (nite or innite)extension L/F , denote by LS the maximal extension of L unramied outside S ;for the Galois group Gal(LS /L), set the notation GS (L). For a GS (L)-module M ,its i-th Galois cohomology group is denoted by H i GS (L) , M . If w is a placeof L, write Lw for its completion at w; when L/F is innite, it is the union ofcompletions of all nite sub-extensions of L. For local elds, the cohomology groupH i (Lw , M ) is with respect to the absolute Galois group of Lw . For an Abeliangroup A, we use the notation A[p] to denote its p-torsion points and A(p) to denote
its p-primary part.
The p-primary Selmer group ts into an exact sequence

0 → E(F ) ⊗ Qp /Zp → Sel(E/F ) → X(E/F )(p) → 0
where E(F ) is the group of F -rational points called the Mordell-Weil group andX(E/F ) is the Shafarevich-Tate group.
The ne Selmer group is a subgroup of the classical Selmer group obtained by
imposing stronger conditions at primes above p. The p-primary ne Selmer
Group is dened by the following kernelM
0 → R E/F → Sel E/F →H 1 Fv , E[p∞ ] ,v|pwhere E[p∞ ] is the set of all the p-power division points of the elliptic curve.
For an innite Galois extension L/F , the p-primary Selmer group, Sel E/L ,and the p-primary ne Selmer group, R E/L , are dened as follows


M
M


lim
0 → Sel E/L → H 1 GS (L), E[p∞ ] →H 1 (Lw , E) (p) ,
−→
L w|v
v∈S


M
M



lim
0 → R E/L → H 1 GS (L), E[p∞ ] →H 1 Lw , E[p∞ ]  .
−→
v∈S

L

w|v

3

The inductive limit is taken with respect to the restriction maps and L runs overall nite extensions of F contained in L. Also, note that
Sel E/L = lim Sel E/L ,−→LR E/L = lim R E/L .
−→L

2.1.1. Control Theorem. Let F be a number eld and L/F be a p-adic analytic
extension with Galois group Gal(L/F ) ' G. Let E be an elliptic curve dened overF . The study of the natural restriction map
GsL/F : Sel E/F → Sel E/L
is called the control problem. Mazur proved the following result.

Theorem 2.1 (Control Theorem [21]). Let L/F be a Zp -extension and let E be an

elliptic curve dened over F with good ordinary reduction at primes above p. Thenboth ker(sL/L ) and coker(sL/L ) are nite and bounded as L/F varies over all niteextensions inside L.
In [7], Greenberg formulated a general plan to attack this problem. He proved
generalizations of Mazur's Control Theorem stated below.
Set E(L)[p∞ ] to denote all the p-power torsion points in L, g be the Lie algebraof Gal(L/F ) ' G, and dp (resp. ip ) be the Lie algebra of the decomposition group(resp. inertia subgroup) at p. For any Lie algebra l, denote by l0 , the derived Lie
subalgebra.

Theorem 2.2 (Greenberg [7]). Assume E has potentially ordinary reduction at allprimes of F lying over p. Assume that L/F is a p-adic Lie extension satisfying theproperty that d0p = i0p for all primes p above p. Further suppose that g is reductiveor E(L)[p∞ ] is nite. Then both ker(sL/L ) and coker(sL/L ) are nite as L varies
over all nite extensions of F inside L.
Some examples of p-adic Lie extensions L/F , where the property d0p = i0p holds
for all primes p | p, include:(1) when G is Abelian.
(2) when the inertia subgroup has nite index in G for all p | p.
(3) when G admits a faithful, nite-dimensional p-adic representation of HodgeTate type at p | p.
With the same setting as above, for the ne Selmer group, there is an analogous
control problem. It involves studying the natural restriction map
GrL/F : R E/F → R E/L .
The following result is known.

Theorem 2.3 (Control theorem for ne Selmer groups [26, Chapter VII, Section4]). Let F be a number eld and E be an elliptic curve dened over F . Let L/F bea Zdp -extension where d ≥ 1, and suppose all primes in S are nitely decomposed.
Then both ker(rL/L ) and coker(rL/L ) are nite as L varies over all nite extensionsof F inside L.
Remark 2.4.
(1) The Control Theorem for ne Selmer groups is independentof the reduction type at p.

4

DEBANJANA KUNDU(2) When d = 1, the Control Theorem is proved for all Zp -extensions [32].
Moreover, the order of ker(rL/L ) and coker(rL/L ) are bounded independentof L.
2.2. Iwasawa Theory of Uniform pro-p Groups [4], [12], [30]. Let G be a
nitely generated pro-p group. For two elements x, y ∈ G, dene the commutator[x, y] := x−1 y −1 xy . For closed subgroups H1 , H2 of G, let [H1 , H2 ] be the closed
subgroup generated by all commutators [x1 , x2 ] with xi ∈ Hi .

Denition 2.5. A pronite group G, is uniform if it is topologically nitely gen-erated by d generators, and there exists a (unique) ltration by the p-descending
central series of G. In other words, we have
G = G0 ⊃ G1 ⊃ . . . Gn ⊃ . . .
dsuch that each Gn+1 is normal in Gn , and Gn /Gn+1 ' Z/pZ . In particular, a
uniform p-adic analytic group is always pro-p.
For a d-dimensional uniform pro-p group G, one has [G : Gn ] = pdn , for all n.
A well-known and important fact is the following.

Theorem 2.6 ([4, Theorem II.8.32]). Every p-adic analytic pro-p group is a closed
subgroup of GLm (Zp ) for some integer m and contains an open uniform subgroup.

Let Λ(G) = Zp JGK := limH Zp [G/H] be the completed Iwasawa algebra of G,←−
where H runs over all open normal subgroups of G. Set Ω(G) = Fp JGK = Zp JGK/p.
Both Ω(G) and Λ(G) are local, Noetherianrings without zero divisors [4, Chapter7] (see also [12]). Denote by Q Ω (G) the fraction skew eld of Ω(G). If M isa nitelygenerated Ω(G)-module, the rank of M , written as rankΩ(G) (M ), is theQ Ω (G) -dimension of M ⊗(Ω(G)) Q(Ω(G)).

Denition 2.7. Let M be a nitely generated Λ(G)-module. Set
r(M ) = rankΩ(G) M [p] ;µ(M ) =XrankΩ(G) M [pi+1 ]/M [pi ]i≥0where M [pi ] denotes the pi -torsion points of M for all i ∈ Z≥0 .
It follows that µ(M ) ≥ r(M ) and r(M ) = 0 if and only if µ(M ) = 0.
Recall that for any G-module M , the co-invariant MG is the largest quotient of M
on which G acts trivially. When M is a discrete p-primary Abelian group or a compact pro-p Abelian group, dene its Pontryagin dual M ∨ := Homcont (M, Qp /Zp ).
The following result of Perbet measures the growth of nitely generated Λ(G)modules where G is a uniform, pro-p, p-adic Lie group.

Theorem 2.8 (Perbet [24]). Let G be a uniform pro-p group of dimension d and
M be a Λ(G)-module of rank ρ(M ). Then for suciently large n,
dimFp MGn /p = ρ(M ) + r(M ) pdn + O pn(d−1) ,ρ(M )+µ(M ))pdn +O npn(d−1)# MGn /pn = p(.

52.3. Growth of Class Groups in Uniform pro-p Lie Extensions. Let p be an
odd prime and let F be a number eld. Let L/F be a uniform pro-p Lie extension,i.e. Gal(L/F ) ' G where G is a uniform, pro-p, p-adic Lie group. Let L/F be a
nite subextension of L/F . Denote by Cl(F )p the p-Sylow subgroup of the classgroup of F . Dene the Iwasawa module

X L/F := lim Cl(L)p←−L

where the inverse limit is taken over all number elds L in L/F with respect tothe norm map. By a Nakayama's Lemma argument, it is known that
X(L/F ) is
and rL/F =a torsion Λ(G)-module [1](see also [11]). Set µL/F = µ X L/F
r X L/F . Perbet proved the following result by classical descent.

Theorem 2.9 (Perbet [24]). Let X(L/F ) be as dened above. Then for suciently

large n,
rp Cl(Fn ) = rL/F pdn + O pn(d−1) ,log Cl(Fn )p /pn = µL/F pdn log p + O npd(n−1) ,where for any Abelian group A, the p-rank rp (A) := dimFp (A[p]).
Remark 2.10. In [18], Lei obtained a more precise version of the above result when
G is a metabelian extension.
2.4. p-Rational Fields. Let p be an odd prime. Let F be a number eld and let
FSp denote the maximal pro-p extension of F unramied outside Sp , i.e. the primes
above p. The number eld F is called p-rational if Gal(FSp /F ) is pro-p free.

Theorem 2.11 ([6, Theorem IV.3.5]). Let F be a number eld that contains aprimitive p-th root of unity. Then, F is p-rational if and only if there exists a
unique prime p above p and the p-part of the p-class group of F is trivial.

Remark 2.12.
(1) It is believed that given any number eld, it should be prational for all primes p outside a set of density zero [8, Page 99].
(2) Let F be a p-rational number eld which is Galois over Q. Suppose p is
the unique prime above p in F , p - Cl(F ) , and p − 1 | [F : Q]. Then, p is
totally ramied in FSp /F [8, Remark 6.4].
2.5. False Tate Curve Extensions. Let p be a xed odd prime and let F be anumber eld containing the group of p-th roots of unity, denoted by µp . The false
Tate curve extension, denoted by F∞ , is obtained by adjoining the p-powerroots of a xed integer m > 1 to the cyclotomic Zp -extension of F , which in turnis written as Fcyc . Therefore,1
F∞ = F µp∞ , m pn : n = 1, 2, . . . .

The Galois group, G = Gal F∞ /F ' Zp o Zp , is a solvable group with no elementof nite order. This is a non-Abelian pro p p-adic Lie extension of cohomologicaldimension 2 [28]. Set H = Gal F∞ /Fcyc ' Zp . The extension F∞ /F is containedin GS (F ) where S is a nite set of primes in F containing the Archimedean primes,
the primes above p, the primes of bad reduction of E and primes dividing m. For
short we write, S ⊇ S∞ ∪ Sp ∪ Sbad ∪ Sm .

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DEBANJANA KUNDU

3. Growth of Selmer Groups in Uniform pro-p Extensions
3.1. Brief Review of the Result of Hajir-Maire [10, Section 4]. Given a
uniform pro-p group G, Hajir-Maire developed a general strategy to construct Gextensions of number elds with arbitrarily large µ-invariant.

Proposition 3.1 ([10, Proposition 4.1]). Suppose G is a uniform pro-p group andL/M is a Galois extension of a number eld such that
(1) Gal(L/M) ' G;(2) there are nitely many primes that are ramied in L/M;
(3) there are innitely many primes of M that split completely in L/M.
Then, there exist G-extension of number elds with arbitrarily large associated µinvariant.
The main theorem they prove is the following.

Theorem 3.2 ([10, Theorem 4.8]). Let G be a uniform pro-p group having an

automorphism τ of order m with xed-point-free action, where m is coprime to p.
Suppose F0 is a totally imaginary eld admitting a cyclic extension F/F0 of degree
m such that F is p-rational. Then there exists a nite p-extension K/F unramied
outside p and a G-extension L/K such that for any given integer N , there exists a
cyclic degree p extension K 0 over K(µp ) and a G-extension L0 /K 0 where L0 = LK 0
whose associated µ-invariant is greater than N .
3.1.1. Construction/Discussion: Let G be a uniform pro-p group having an automorphism τ of order m with xed-point-free action, where m is coprime to p. If
m = 2, then G ' Zdp for some d ≥ 1 [25, Corollary 4.6.10].
Suppose F0 is a totally imaginary eld admitting a cyclic extension F/F0 of
degree m such that F is p-rational. For n suciently large, set K0 (resp. K ) to be
the n-th layer of the cyclotomic Zp -extension of F0 (resp. F ). It follows that K is
p-rational. By construction, K/K0 is a cyclic extension of degree m.
Let KSp be the maximal, pro-p, unramied outside Sp extension of K . Then,
there exists an intermediate eld K ⊂ L ⊂ KSp with Gal(L/K0 ) ' G o hτ i [10,Proposition 4.6]. The conjugation action of τ is xed-point-free by assumption;equivalently the action of hτ i is xed-point-free, if m is a prime (not equal top). Under this additional assumption, G o hτ i is a Frobenius group [25, Theorem
4.6.1(d)]. Hence G is a nilpotent uniform pro-p group [25, Corollary 4.6.10].
Every place q which is totally inert in K/K0 and is not ramied in L/K splits
completely in this extension [10, Proposition 4.7]. By the Chebotarev density theorem, there are innitely many primes that remain totally inert in the Galois extension K/K0 .
Without loss of generality assume K contains µp (otherwise replace K by K(µp )in this paragraph). Choose an integer t ≥ 1, and primes q1 , . . . , qt in OK which
split completely in L/K . There exists a Z/pZ-extension, K 0 /K in which each ofthese qi ramify. Indeed, let q0 be a prime ideal coprime to q1 · · · qt which is in the
inverse of q1 · · · qt , i.e. q0 q1 · · · qt is a principal ideal generated by α (say). Then
K 0 := K(α1/p ) is a cyclic degree p extension of K where q1 , . . . , qt ramify.
3.2. Strategy to Extend the Above Result. To extend Theorem 3.2 to (ne)Selmer groups of elliptic curves, we adopt the following strategy. We write it down
for Selmer groups. For ne Selmer groups, the strategy is identical.

7

Step 1: Let G be a d-dimensional uniform pro-p p-adic Lie group. Let K be anumber eld that admits an innite extension L/K with Gal(L/K) ' G suchthat the associated µ-invariant in this extension is arbitrarily large. Recall that
[G : Gn ] ' (Z/pZ)nd . Let Kn /K be the eld xed by Gn . Let E/K be an elliptic
curve such that E(K)[p] 6= 0. Then, the rst task is to show




(1)
rp Sel E/Kn ≥ rp Cl(Kn ) rp E(Kn )[p] − 2.
This is a consequence of the following lemma.

Lemma 3.3 ([19, Proposition 4.1(i)]). Let M be a number eld and E be an elliptic
curve with good reduction everywhere over M and E(M)[p] 6= 0. Then,




(2)
rp Sel E/M ≥ rp Cl(M) rp E(M)[p] − 2.
0
Step 2: With notation
 as in Step 1, we construct a cyclic extension Kn /Kn suchthat rp Sel E/Kn0
can be made arbitrarily large.
A key ingredient in this step is the following result from genus theory.

Theorem 3.4 ([27]). Let M be a number eld and L/M be a Z/pZ-extension. Lett be the number of primes that ramify in L/M. Then,×rp Cl(L) ≥ t − 1 − rp OM.

Let K be the number eld considered in Step 1. Suppose there exists a Z/pZextension K 0 /K such that the number of primes t that ramify in K 0 /K can be madearbitrarily large. Further, suppose these primes split completely in the G-extension
L/K . Set L0 = LK 0 ; then Kn0 = Kn K 0 is the eld xed by Gn in L0 /K 0 .
0of) Inequality 1 and Theorem 3.4, both rp Cl(Kn0 ) andBy (the Kn-version
rp Sel E/Kn0
can be made arbitrarily large. In fact [10, Page 609],
(3)
rp Sel(E/Kn0 ) ≥ [Kn0 : K] t − r2 (K 0 ) − 1 − 2
where r2 denotes the number of pairs of complex embeddings of the number eld.
In obtaining the above inequality we have used that rp E(Kn0 )[p] ≥ 1.

Remark 3.5. When G is a uniform pro-p group having an automorphism τ of orderm with xed-point-free action, and m is coprime to p, the number elds K and
K 0 exist by the work of Hajir-Maire. They are constructed as in Theorem 3.2 (seeSection 3.1.1 for the details).
Step 3: We apply Greenberg's Control Theorem (Theorem 2.2). This allows thecomparison of the growth estimate obtained from genus theory (in Inequality 3)
and that obtained from the theorem of Perbet (Theorem 2.8).
Recall that the Pontryagin dual of the Selmer group (and hence the Pontryagin
dual of the ne Selmer group) is always a nitely generated Λ(G)-module where G
is a uniform pro-p group. This is a consequence of the Nakayama's Lemma in thissetting [1].
In our setting, Perbet's theorem can be applied. It gives the following equality.
G n 
(4)
rp Sel E/L0
= ρ(X0 ) + r(X0 ) pdn + O(pn(d−1) )where ρ(X0 ) (resp. r(X0 )) is the Λ(G)-rank (resp. Ω(G)-rank) of the dual Selmer
group X(E/L0 ) = X0 . By construction, Gal(L0 /K 0 ) ' G. If L0 /K 0 is such that

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DEBANJANA KUNDU

Greenberg's Control Theorem is applicable and the p-ranks of the kernel and cokernel of the restriction map are nite and bounded (see for example [7, Proposition3.4]), then for an elliptic curve E with (potentially) ordinary reduction at K 0 ,G n 
+ O(1).
rp Sel E/L0
= rp Sel E/Kn0This allows the comparison of the leading terms of Inequality 3 and Equation 4.
We getρ(X0 ) + r(X0 ) ≥ pt − pr2 (K 0 ) − p.
In Step 2, it was guaranteed by construction that t can be made arbitrarily large. It
follows that ρ(X0 ) + r(X0 ) (and hence ρ(X0 ) + µ(X0 )) can be made arbitrarily large.
If X0 is Λ(G)-torsion then ρ(X0 ) = 0. Hence, µ(X0 ) can be made arbitrarily large.
3.3. Growth of Selmer groups when G is Abelian. When G is an Abelian
pro-p group, i.e. G ' Zdp (for d ≥ 1), the result of Hajir-Maire can be applied (withm = 2) and Greenberg's Control Theorem holds.
With notation as before, invoking the strategy described in Section 3.2, thefollowing result is immediate.

Theorem 3.6. Let G ' Zdp where d ≥ 1. Suppose F0 is a totally imaginaryeld. Let E/F0 be an elliptic curve with good reduction everywhere over F0 andordinary reduction at primes above p. Further suppose E(F0 )[p] 6= 0. Let F/F0
be a cyclic extension of degree coprime to p such that F is p-rational. Given any
integer N > 0, there exists a number eld K 0 /F and a Zdp -extension L0 /K 0 suchthat ρ(X0 ) + µ(X0 ) ≥ N . If further, X0 is Λ(G)-torsion, then µ(X0 ) ≥ N .
Remark 3.7. If E/F0 is an elliptic curve with complex multiplication (CM) by animaginary quadratic eld contained which is a subeld of F0 , then E acquires good
reduction everywhere over F0 (E[p]) [29]. If p ≥ 5, [F0 (E[p]) : F0 ] is coprime to p
and by the Weil pairing F0 (E[p]) ⊇ F0 (µp ).
3.4. Growth of Selmer groups when G is a non-Abelian nilpotent uniform
pro-p group. Throughout this section, let p ≥ 5.

The nilpotent uniform pro-p groups of dimension ≤ 2 are always Abelian. This
follows from the fact that every 2-dimensional nilpotent Lie algebra is Abelian. In
this section, we work with a specic example of a non-Abelian nilpotent uniform
pro-p group of dimension 3 considered in [10].
In [5, Theorem 7.4], González-Sánchez and Klopsch proved that up to isomorphism, every non-Abelian, nilpotent, uniform, pro-p group of dimension 3 isparametrized by s ∈ N and represented byDEs
G(s) = x, y, z | [x, z] = [y, z] = 1, [x, y] = z p .
The center of G(s) is pro-cyclic and there exists a short exact sequence

1 → Zp → G(s) → Z2p → 1.
These groups were studied by Hajir-Maire. They proved that G(s) is a uniform
pro-p group having an automorphism τ of order 3 with xed-point-free action whenp ≡ 1 (mod 3) [10, Proposition 5.1]. The following result is known about growth ofclass groups (and the associated µ-invariant) in G(s)-extensions of number elds.

9

Theorem 3.8 ([10, Corollary 5.3]). Suppose p is a regular prime, i.e. p - Cl(Q(µp ))and satisfy the additional property p ≡ 1 (mod 3). For each s ∈ N, there exist G(s)extensions of number elds with arbitrarily large µ-invariants.
In this section, we prove the following result.

Theorem 3.9. Let s ∈ N and G ' G(s). Suppose p is a xed regular prime withp ≡ 1 (mod 3). Suppose F0 is a totally imaginary eld containing µp and E/F0 is
an elliptic curve without CM with good reduction everywhere, ordinary reduction atp, and E(F0 )[p] 6= 0. Let F/F0 be a cyclic Z/3Z such that F is p-rational, Galois
over Q, and p - Cl(F ) . Given any integer N > 0, there exists a number eldK 0 /F and a G(s)-extension L0 /K 0 such that ρ(X0 ) + µ(X0 ) ≥ N . If further, X0 isΛ(G)-torsion, then µ(X0 ) ≥ N .

Remark 3.10. Since G(s) is a non-Abelian, nilpotent, uniform pro-p group, thecorresponding Lie algebra can not be reductive.
To prove the theorem, we adopt the general strategy developed in Section 3.2.
We need to verify that Greenberg's Control Theorem is in fact applicable to thisextension. For this, we need the following lemmas.

Lemma 3.11. Let the number eld K 0 and a G(s)-extension L0 /K 0 with arbitrarily
large µ-invariant be constructed as in Theorem 3.2. Let E/K 0 be an elliptic curve
without CM. Then, E(L0 )[p∞ ] is nite.
To prove the lemma, recall the following result of Zarhin.

Theorem 3.12 ([33, Main Theorem]). Let X be a g-dimensional Abelian variety
over a number eld M. Assume that the Hodge group of X is semi-simple. If L/M
is an innite extension such that its intersection with Mcyc /M is of nite degree
over M, then X(L)tors is nite.
Proof of Lemma 3.11. When E is an elliptic curve without CM, it is known thatits Hodge group is SL2 , and therefore semi-simple [22, Section 2]. Further, the0/K 0 is necessarily of nite degreeintersection of the G(s)-extension L0 /K 0 and Kcyc
000
over K . This is because by construction of L /K , there are innitely many primes0/K 0 all primes arethat are not nitely decomposed in L0 /K 0 . However, in Kcyc
0
nitely decomposed. Thus, for a non-CM elliptic curve E/K , Theorem 3.12 implies
that E(L0 )[p∞ ] is nite.
Lemma 3.13. Keep the notation as in Theorem 3.9. Let the number eld K 0

and a G(s)-extension L0 /K 0 with arbitrarily large µ-invariant be constructed as in
Theorem 3.2. Let E/K 0 be an elliptic curve. The inertia subgroup has nite index
in G(s) for all primes above p in K 0 .
Proof. Recall that by construction, K is the n-th layer of the cyclotomic Zp extension of F . Therefore, K/F is p-rational number eld containing µp suchthat KSp ⊆ FSp . There is a unique prime p above p in K (see Theorem 2.11), i.e.
Sp = {p}. Recall that the p-adic Lie extension L/K with Gal(L/K) ' G(s) is
contained in KSp . Since K/F is a p-power extension and p - Cl(F ) , it follows thatp - Cl(K) [31, Theorem 10.4(1)]. By Remark 2.12(2), the unique prime p | p is
totally ramied in FSp /F . Hence, the unique prime above p in K is totally ramiedin KSp /K ; the inertia subgroup of G(s) ' Gal(L/K) is maximal (in this case, ofdimension 3). Upon performing a base change to K 0 /K , for every prime above p

10

DEBANJANA KUNDU

in K 0 , the dimension of the inertia subgroup is still 3, i.e. the inertia subgroup has
nite index in the G(s)-extension over K 0 .
Proof of Theorem 3.9. Given F0 as in the statement of the theorem, construct
K 0 /F0 as in Theorem 3.2 and let L0 /K 0 be a G(s)-extension with arbitrarily largeµ-invariant. Such a G(s)-extension L0 /K 0 exists by the proof of Theorem 3.8.
By hypothesis, E/K 0 has good reduction everywhere and good ordinary reduction at primes above p. To apply Greenberg's Control Theorem, the following two
properties need to be veried.
(i) E(L0 )[p∞ ] is nite.
(ii) The inertia subgroup has nite index in G(s) for all primes p | p.
The rst property is veried in Lemma 3.11. This guarantees that the kernel ofthe restriction map is nite and bounded [7, Proposition 3.1]. The second property
is veried in Lemma 3.13. This guarantees that the cokernel is also nite andbounded [7, Proposition 4.4]. The general strategy developed in Section 3.2 proves
the theorem.
Remark 3.14. If p is a regular prime and m is an odd divisor of p − 1, then for any
k ≥ 1, the cyclotomic eld Q(µpk ) admits a Z/mZ extension K/Q(µpk ) which is
p-rational (see discussion following [10, Theorem 1.1]).
3.5. Growth of ne Selmer groups when G is Abelian. When G ' Zdp (ford ≥ 1), it is possible to prove an analogue of Theorem 3.6 for the ne Selmer groups.
The diculty lies in the fact that the control problem for ne Selmer groups is lessunderstood. Set the notation Y = Y(E/L) (resp. Y0 = Y(E/L0 )) to denote thePontryagin dual of the ne Selmer group over L (resp. L0 ).

Theorem 3.15. Let G ' Zdp where d ≥ 1. Suppose F0 is a totally imaginary eldcontaining µp . Let E/F0 be an elliptic curve with good reduction everywhere andE(F0 )[p] 6= 0. Let F/F0 be a cyclic extension of degree coprime to p such that F
is p-rational and p - Cl(F ). Given any integer N > 0, there exists a number eldK 0 /F and a Zdp -extension L0 /K 0 such that ρ(Y0 ) + µ(Y0 ) ≥ N . If further, Y0 isΛ(G)-torsion, then µ(Y0 ) ≥ N .
Remark 3.16.
(1) The denition of the ne Selmer group is independent of
the choice of the set S . By hypothesis, E has good reduction everywhere.
Hence, choose S = Sp ∪ S∞ . In our setting, the unique prime p | p is totallyramied in L/K . Observe that the prime(s) above p in K 0 are nitely
decomposed in the Zdp -extension L0 /K 0 .
(2) When d = 1, Wuthrich proved the Control Theorem for all Zp -extensions.
As will be clear from the proof, in this case there is no hypothesis on the
reduction type [16].
(3) It is always possible to choose S = Sp ∪ Sbad ∪ S∞ . If all the bad primesin S are nitely decomposed in L0 /K 0 , we need not assume E/F0 has good
reduction everywhere.

Denition 3.17. Let HS (M) be the maximal Abelian unramied extension of M

such that all the primes in S split completely in HS (M). By class eld theory, theGalois group Gal HS (M)/M ' ClS (M) where ClS (M) is the S -class group of M.
We begin by recording two elementary estimates.

11

Lemma 3.18 ([19, Lemma 3.2]). Let G be a pro-pgroup and let M be a discrete
G-module that is conitely generated over Zp . If rp H 1 G, Z/pZ is nite then







1
1
.
corankZp M + logp M/Mdiv
rp H (G, M ) ≤ rp H G, Z/pZ

Lemma 3.19 ([20, Lemma 3.2]). Consider the following short exact sequence ofconitely generated Abelian groups

P → Q → R → S.

Then
rp (Q) − rp (R) ≤ 2rp (P ) + rp (S) .
To prove Theorem 3.15, we will apply the general strategy. To complete Step 1of the general strategy, we need the following lemmas.

Lemma 3.20. Let L be any Zdp extension of a number eld M. Let S(M) be a nite

set of primes in M containing the primes above p and the Archimedean primes. Let
s0 be the number of non-Archimedean primes in S(M). Let Mn be the subeld of Lsuch that [M : Mn ] = pdn . Then
rp Cl (Mn ) − rp ClS (Mn ) ≤ 2s0 pdn .

Proof. Let S(Mn ) denote the primes in Mn above the primes in S(M). Set Sf (Mn )to denote all the non-Archimedean primes of S(Mn ). Consider the following short
exact sequence for all n [23, Lemma 10.3.12],
αnZ|Sf (Mn )| → Cl(Mn ) −−→
ClS (Mn ) → 0.

Observe that
 ker(αn ) is nite because the class group is always nite. Note thatrp ker (αn ) ≤ Sf (Mn ) and by niteness it follows that rp ker (αn ) /p ≤ Sf (Mn ) .
Using Lemma 3.19, compare the p-ranks in this short exact sequence; this givesrp Cl (Mn ) − rp ClS (Mn ) ≤ 2 Sf (Mn ) ≤ 2s0 pdn .
The last inequality follows from the fact that if all the non-Archimedean primes inS(M) undergo complete splitting in Mn , there are s0 pdn nite primes in S(Mn ). 

Lemma 3.21 ([20, Lemma 4.3]). Let E be an elliptic curve dened over the number
eld M with E(M)[p] 6= 0. Let S be a nite set of primes in M containing the primes
above p, the primes of bad reduction, and the Archimedean primes. Then,rp R E/M ≥ rp ClS (M) rp E(M)[p] − 2.
We can now prove Theorem 3.15. The proof involves invoking the strategy in
Section 3.2. We briey explain some of the dierences.

Proof of Theorem 3.15. Let F0 be as in the statement. Construct K and K 0 as in
Theorem 3.2. Let t be the number of primes that are inert in K/K0 and totally
split completely in L/K . By construction, these primes ramify in K 0 /K . Since
E/F0 has good reduction everywhere, set S = Sp ∪ S∞ . Recall that the prime(s)above p in K 0 are nitely decomposed in L0 . It follows that for n suciently large,

12

DEBANJANA KUNDU


rp Cl (Mn ) − rp ClS (Mn ) is nite and bounded. Over the n-th layer of the Zdp
extension L0 /K 0 , we have



 
 
≥ rp ClS Kn0 rp E Kn0 [p] − 2(5)
rp R E/Kn0
 
 
 

(6)
+ O(1) rp E Kn0 [p] − 2= rp Cl Kn0≥ pt − r2 K 0 − p pdn + O(1).
(7)
The rst inequality is an application of Lemma 3.21. The equality in the secondline follows from Lemma 3.20. The
is obtained by an application of last inequality 
0Theorem 3.4 and noting that rp E Kn [p] ≥ 1. By Theorem 2.8,
G n 

(8)
rp R E/L0= ρ(Y0 ) + r(Y0 ) pdn + O(pn(d−1) ).

Step 3 of the general strategy can now be carried out. The Control Theorem forne Selmer groups holds when G is Abelian and primes in S are nitely decomposed(Theorem 2.3). This is independent of the reduction type at primes above p. The
p-rank of the kernel and cokernel of the (global) restriction map
G n
hL0 /Kn0 : H 1 Kn0 , E[p∞ ] → H 1 L0 , E[p∞ ]is known to be nitebounded independent of n [7, Proposition 3.4]. Therefore, and the p-rank of ker rL0 /Kn0 enjoys the same properties. The above mentioned result
of Greenberg also ensures that coker rL0 /Kn0 is nite and bounded independentof n, if the same holds for the p-rank of the kernel of the local
 restriction map.

But the latter follows from Lemma 3.18 upon observing that rp H 1 Gn,v , Z/pZ
is bounded independent of n. Now by Lemma 3.19, it is possible to compare theleading terms of Inequality 7 and Equation 8. By construction, t can be madearbitrarily large. This proves the theorem.
Remark 3.22. At this point, we are unable to prove a Control Theorem for neSelmer groups in G(s)-extensions. This prevents us from completing Step 3 of thegeneral strategy and proving an analogue of Theorem 3.9 for ne Selmer groups.
Remark 3.23. It was only after this paper was published, while discussing with
Meng Fai Lim on another problem, we realized that the strategy developed in
Subsection 3.2 can be extended to all p-adic Lie extensions. Indeed, the strategy
works whenever the p-rank of ker(rL0 /K 0 ) has bounded p-rank which is trivially true
for p-adic Lie extensions.
4. Growth of Fine Selmer Groups in False Tate Curve Extensions
We will use the notation introduced in Section 2.5. In this section, the goal isto prove a non-commutative version of results in [20, Section 5] and [16, Section
3.3]. We prove our results for the false Tate curve extension F∞ /F with Galois
group G = Gal(F∞ /F ) ' H o Γ where both H and Γ are isomorphic to Zp . The
action of Γ on H is non-trivial; hence G is non-Abelian. Note that G is a solvableuniform pro-p group which is not nilpotent. This is because all nilpotent uniform
pro-p groups of dimension ≤ 2 are Abelian. Thus, we do not expect that the results
proved in Section 3 will extended easily to the false Tate curve extension.

13

Consider the setting described in Section 2.5. Let p be a xed odd prime and forsimplicity, set F = Q(µp ). Throughout this section, we make either of the following
hypothesis on m:(1) let m be an integer which is not a p-th power or(2) let p | m.
In either of the cases, it is guaranteed that the unique prime above p is totallyramied in F∞ /F (see [9, Lemma 3.9(ii)] or [17]).
By a deep result of Kato, for a modular elliptic curve, an analogue of the
Weak Leopoldt
Conjecture (WLC) holds over the cyclotomic extension Fcyc /F ,

2i.e. H GS Fcyc , Ep∞ is trivial [14]. By a Hochschild-Serre spectral seqeunceargument it follows that H 2 GS (F∞ ) , Ep∞ = 0. Thus, the elliptic curve analogue of the WLC is true over this false Tate curve extension [9, Remark 2.2].
In the main theorem of this section, we relate the growth of ne Selmer groups
and class groups in the false Tate curve extension.

Theorem 4.1. Let E be an elliptic curve dened over a number eld F . Let F∞

be the false Tate curve extension such that primes of bad reduction of E divide m.
Further assume E[p] ⊆ E(F ). Then,
rp R E/Fn − 2rp Cl (Fn ) = O(1).
Remark 4.2.
(1) It should be possible to weaken the hypothesis E[p] ⊆ E(F )slightly, i.e. the above theorem should hold under the weaker hypothesis
E(F )[p] 6= 0 (see [15]).
(2) It should be possible to generalize this result to any metabelian extension
considered in [18].
To prove the theorem, we need a series of lemmas. In the rst lemma we provethat the class group and S -class group have the same order of growth in the false
Tate curve extension. Recall that S ⊇ Sp ∪ Sbad ∪ Sm ∪ S∞ .

Lemma 4.3. Let F∞ /F be the false Tate curve extension of F . Let Fn be the n-thlayer of this false Tate curve extension, i.e.

Fn = Q µpn ,

√ 
m ,

pn

where either p | m or m is an integer that is not a p-th power. Then for suciently
large n,
rp Cl(Fn ) − rp ClS (Fn ) = O(1).
The proof is similar to Lemma 3.20 (see also [20, Lemma 5.2]).

Proof. As in the proof of Lemma 3.20, we obtain
rp Cl (Fn ) − rp ClS (Fn ) ≤ 2 Sf (Fn ) = O(1).
By the hypothesis on m, the last equality follows from the fact that primes in S arenitely decomposed in the false Tate curve extension F∞ /F [9, Lemmas 3.9]. 
We now dene the p-ne Selmer group of an elliptic curve. Let S be a nite
set of primes containing the primes above p, the primes of bad reduction of E , the14

DEBANJANA KUNDU

primes above m, and the Archimedean primes. Dene


M



RS E[p]/F = ker H 1 GS (F ) , E[p] →H 1 Fv , E[p]  .
v∈SThe next lemma is the ne Selmer group analogue of Lemma 4.3. As will be shownin the proof of Theorem 4.1, the p-ne Selmer group indeed depends on the set S .

Lemma 4.4. Let F∞ /F be the false Tate curve extension of F . Let Fn be the n-thlayer of this false Tate curve extension, i.e.

Fn = Q µpn ,

√ 
m .

pn

Let
YE be an elliptic curve dened over F satisfying the additional property thatv divides m. Then for suciently large n,v∈Sbadrp R E/Fn − rp RS E[p]/Fn= O(1).

Proof. Consider the commutative diagram below.

L
1
0 → RS (E[p]/F
H 1 GS (F→n) →v∈S(Fn ) H Fn,vn , E[p] n ) , E[p]






sn
fn
γn
y
y
y



L
1
∞
1
0 → R E/Fn [p] → H GS (Fn ) , E[p ] [p] →Fn,vn , E[p∞ ] [p]vn ∈S(Fn ) H
Both fn and γn are surjective. The kernel of these maps areker(fn ) = E(Fn )[p∞ ]/pM
ker(γn ) =E(Fn,vn )[p∞ ]/p.
vn ∈S(Fn )Observe that rp ker (sn ) ≤ rp ker (fn ) ≤ 2. Also, rp ker (γn ) ≤ 2 Sf (Fn ) .
Y
By assuming thatv divides m, it is guaranteed that all primes are nitely
v∈Sbaddecomposed in the false Tate curve extension [9, Lemma 3.11], i.e. for n suciently
large, Sf (Fn ) = O(1). By an application of the Snake Lemma, it follows thatrp coker (sn ) is nite and bounded. Applying Lemma 3.19 to the map sn gives
the desired result.

We are now in a position to prove the theorem.

Proof of Theorem 4.1. Let S be a nite set of primes containing the primes above p,
the primes above m and the Archimedean primes. By hypothesis, it is not needed toassume that S also contains the primes of bad reduction of E . Since E[p] ⊆ E(F ),we have the following isomorphism (as GS (Fn )-modules)
E[p] ' Z/pZ × Z/pZ.
Since the action of GS (Fn ) is trivial, the following equality holds

H 1 GS (Fn ) , E[p] = Hom GS (Fn ) , E[p] .

15There are similar identications for the local cohomology groups. It follows that(see [3, Lemma 3.8] or [26, Chapter I 6.1])
2
RS E[p]/Fn = Hom ClS (Fn ) , E[p] ' ClS (Fn ) [p] .
Thus, rp RS E[p]/Fn= 2rp ClS (Fn ) [p] . Combined with Lemmas 4.3 and
4.4, the proof of the theorem is complete.
Acknowledgements

We thank Kumar Murty for his continued support and all members of the
GANITA Lab for listening to the details. We extend our gratitude to Prof. Sujatha
Ramdorai, Prof. Christian Maire, Prof. Karl Rubin, Meng Fai Lim and Jishnu Ray
for answering many questions along the way. We thank Prof. Antonio Lei for hiscomments on an earlier draft of this article. Finally, we thank the referee for acareful reading of the article which helped improve the exposition.
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Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George
St., Room 6290, Toronto, Ontario, Canada, M5S 2E4Email address : dkundu@math.utoronto.ca