GROWTH OF p-FINE SELMER GROUPS AND p-FINE
SHAFAREVICH-TATE GROUPS IN Z/pZ EXTENSIONSDEBANJANA KUNDU

Abstract. In this paper we show that the p-ne Selmer Group can become
arbitrarily large as we vary over all Z/pZ extensions of a given number eldK and nd eective estimates on the conductor of such a Z/pZ-extension. In
fact, we show that the p-ne Shafarevich-Tate group can become arbitrarily
large on varying over all Z/pZ extensions of a given number eld. We explore
the close relationship in the size of p-ne Selmer groups and p-torsion of idealclass groups in quadratic extensions of number elds. Iwasawa Theory, Fine
Selmer groups, Class groups, p-rank

1. Introduction
Using genus theory, Gauss proved that the 2-torsion of the ideal class group of aquadratic number eld can be arbitrarily large. There is a known analogy between
the growth of ideal class groups and growth of Selmer groups of Abelian varieties.
For xed prime p, it is a folklore result that the p-torsion of the ideal class group can
become arbitrarily large in Z/pZ extensions of a xed number eld. Varying over
all Z/pZ-extensions of a global eld, the p-Selmer group is also known to become
arbitrarily large [1].
In [3], the study of the ne Selmer group was initiated. A key idea was toshow that the ne Selmer group approximates the ideal class group better than theclassical Selmer group. This was made more precise in [5]. Lim-Murty proved thatthe p∞ -ne Selmer group of an Abelian variety has unbounded growth on varying
over all Z/pZ-extensions of a xed number eld. Using this method of proof, we ndeective estimates on the conductor of such a Z/pZ-extension (see Theorem 3.4).
As per the knowledge of the author, such bounds can not be obtained by the methodof proof in [1].
It is known that the p-torsion of the classical Shafarevich-Tate group of an elliptic curve has unbounded growth in Z/pZ-extensions of a xed number eld [2].
Just like the ne Selmer group, one can dene a ne analogue of the classicalShafarevich-Tate group [11]. Lim-Murty asked the natural question whether thep-ne Shafarevich-Tate group has unbounded growth in Z/pZ-extensions. Using
the unboundedness result of Clark-Sharif, we provide an armative answer to theirquestion, for the case of elliptic curves in Theorem 4.5. It would be interesting togive an independent proof of this theorem.
For a xed number eld K , we don't know how to show that the p-ne Selmer
group has unbounded growth as one varies over all Z/nZ-extensions of K for
Date : March 5, 2020.
2010 Mathematics Subject Classication. Primary 11R23.
Key words and phrases. Iwasawa Theory, Fine Selmer groups, ne Shafarevich-Tate groups,
Class groups, p-rank.
1

2

DEBANJANA KUNDU

1 < n < p. Analogous results are conjectured to be true for the p-torsion of
the ideal class group. It is believed that such a result should be true for n = 2. In

Theorem 5.1, we prove these two conjectures are equivalent.
2. Preliminaries

Let K be a xed number eld and p be an odd rational prime. Let A be ad-dimensional Abelian variety dened over K . Let S be a nite set of primes of K
including the innite primes, the primes where A has bad reduction and the primes
above p. Fix an algebraic closure K/K and denote the absolute Galois group
Gal(K/K) by GK . Use the notation KS for the maximal subeld of K containing
K which is unramied outside S . Write GS (K) = Gal(KS /K).
The pk -Selmer group of an Abelian variety is dened as,




M
H 1 (Kv , A) [pk ] .
Selpk (A/K) = ker H 1 GS (K), A[pk ] →v∈SHere, H (Kv , M ) is the Galois cohomology of the decomposition group at v for any
G-module, M .
The pk -ne Selmer group is dened as
∗



(1)

RpSk (A/K) = ker H 1







M
GS (K), A[pk ] →H 1 Kv , A[pk ]  .
v∈SFor any number eld K , one has the following exact sequence
0 → RpSk (A/K) → Selpk (A/K) →M

H 1 (Kv , A[pk ]).

v∈SConsider the limit versions of the above dened objects. Dene

Selp∞ (A/K) := lim Selpk (A/K) = ker H 1 (GS (K), A[p∞ ]) →−→M

H 1 (Kv , A)[p∞ ] ,v∈Swhere the limit is w.r.t maps induced by inclusions A[pk ] ,→ A[pk+1 ]. It has asubgroup, the discrete ne Selmer group
Rp∞ (A/K) := lim RpSk (A/K).
−→

Note that Rp∞ (A/K) is independent of S . However, RpS (A/K) is not.
For the classical Selmer group, one has the short exact sequence
0 → A(K)/pk → Selpk (A/K) → X(A/K)[pk ] → 0,

where A(K) is the Mordell-Weil group. In [11], a ne subgroup of the Mordell-Weilgroup is dened; it is the following kernel
0 → Mpk (A/K) → A(K)/pk →M

A(Kv )/pk .

v|p

It is now natural to dene the ne Shafarevich-Tate group by the exact sequence,
0 → Mpk (A/K) → RpSk (A/K) → pk (A/K) → 0.

3

One can view pk (A/K) as a subgroup of X(A/K)[pk ]. To see this we repeat theargument in [11, Page 3]. Consider the following diagram
0

−→

0

−→

k
A(K)/p




y
L
k
v|p A(Kv )/p−→−→

Selpk
(A/K)



y
L
1
k
v|p H (Kv , A[p ])

−→

−→

k
X(A/K)[p
]
−→



0
y
L
1
k
v|p H (Kv , A)[p ] −→From the above diagram, by an application of the snake lemma, one obtains thefollowing exact sequence
0 → Mpk (A/K) → RpSk (A/K) → X(A/K)[pk ] → Cpk ,where Cpk is the cokernel of the left vertical map in the above diagram. Thus,pk (A/K) is a subgroup of X(A/K)[pk ] with quotient in Cpk .
2.1. p-rank. For an Abelian group G, dene its p-rank, denoted by rp (G), as
dimZ/pZ G[p]. We record some elementary estimates.
[5, Lemma 3.2] Consider the following short exact sequence of ofconitely generated Abelian groups

Lemma 2.1.

P → Q → R → S.

Then
rp (Q) − rp (R) ≤ 2rp (P ) + rp (S) .

Denote the p-Hilbert S -class eld of K by HS (K) or HS . It is the maximal
Abelian unramied p-extension of K where all primes in S split completely.
The following lemma is a variant of [5, Lemma 4.3]. The proof is identical. It
provides a lower bound for the p-rank of p-ne Selmer group in terms of the p-rankof the S -class group.
Lemma 2.2. Let A/K be a d-dimensional Abelian variety.

Let S be a nite setof primes of K including the innite primes, the primes where A has bad reduction
and the primes above p. Suppose A(K)[p] 6= 0. Thenrp (RpS (A/K)) ≥ rp (ClS (K))rp (A(K)[p]) − 2d.
Remark 2.3. Under the slightly stronger assumption that A[p] ⊆ A(K), we canget better estimates. This assumption forces A[p] ' (Z/pZ)2d as GS (K)-modules.
Now, GS (K) acts trivially on A[p] and hence we have

H 1 (GS (K), A[p]) = Hom(GS (K), A[p]).

We have similar equalities for the local cohomology groups as well. Thus,
RpS (A/K) = Hom(ClS (K), A[p]) ' ClS (K)[p]2d

as Abelian groups (see [5, Page 87] or [3, Lemma 3.8] or [9, 6.1]). Therefore
rp (RpS (A/K)) = 2drp (ClS (K)).

0

0

4

DEBANJANA KUNDU

3. Unboundedness of p-fine Selmer groups in Z/pZ-Extensions andEffective Estimates

Recall the Grunwald-Wang theorem [8, Theorem 9.2.8].
Theorem 3.1. Let S be a nite set of primes of a global eld K and let G be a

nite Abelian group. For all p ∈ S , let the nite Abelian extensions Kp | Kp be
given such that Gal(Kp | Kp ) may be embedded into G. Then there exists a global
Abelian extension K | K with Galois group G such that K has the given completions
Kp for all p ∈ S .

The following proposition is proved in [5]. We repeat the proof here because it
plays a crucial role in proving our main result.
Proposition 3.2. [5, Proposition 6.1] Let S be a nite set of primes of K contain-

ing the Archimedean primes. Then there exists a sequence {Ln } of distinct number
elds such that each Ln is a Z/pZ extension of K and such that for every n ≥ 1,
rp (ClS (Ln )) ≥ n.

Proof. Set r1 and r2 to denote the number of real and the number of pairs ofcomplex places of K . Let S1 be a set of primes of K containing S such that
|S1 | = |S| + r1 + r2 + δ + 1,

where δ = 1 if K contains a primitive p-root of unity, and is 0 otherwise.
By the Grunwald-Wang theorem, there exists a Z/pZ extension L1 /K such that
it is ramied at all nite places of S1 and is unramied outside of it. Using [8,Proposition 10.10.3],
rp (ClS (L1 )) ≥ |S1 | − |S| − r1 − r2 − δ = 1.

Repeat the above process; choose a set S2 containing S1 with the property|S2 | = |S1 | + 1 = |S| + r1 + r2 + δ + 2.

By Grunwald-Wang theorem, there exists a Z/pZ-extension L2 /K ramied at allthe nite places of S2 and unramied outside of it. L2 is distinct from L1 by
construction. For this eld,rp (ClS (L2 )) ≥ 2.

Since K has innitely many primes, we can continue this process indenitely. Eachof the Li 's are distinct by construction. This proves the proposition.
Remark 3.3. Proposition 3.2 implies the following:With the same setting as Lemma 2.2,
sup{rp RpS (A/L) | L/K is a cyclic extension of degree p} = ∞.

Here sup is over the conductor of L/K .

The above remark shows that the p-ne Selmer group of an Abelian variety A/K
becomes arbitrarily large on varying over all Z/pZ-extensions of K . The proof of
Proposition 3.2 suggests that it should be possible to nd an eective estimate on
the conductor. Indeed, we can prove the following theorem.

5
Theorem 3.4. Let A be an Abelian variety of dimension d, dened over a numbereld, K . Let S be a nite set of primes as dened above. Suppose A(K)[p] 6= 0.
Given a non-negative integer N , there exists a Z/pZ extension L/K with norm of
the conductor, NK/Q (f(L/K)) ∼ κN where κ is a constant depending on S , A andK , such that rp (RpS (A/L)) ≥ N .

When K = Q, the notation simplies considerably. We prove the above theorem
in detail for the special case K = Q.
Theorem 3.5. Let A/Q be an Abelian variety of dimension d. Let S be a nite setof primes containing the Archimedean primes, the primes above p, and the primesof bad reduction of A. Suppose A(Q)[p] 6= 0. Given a non-negative integer N , there
exists a Z/pZ extension L/Q of conductor f(L/Q) ∼ κN where κ is a constantdepending on S and A, such that rp (RpS (A/L)) ≥ N .

Proof. Let L/Q be a Z/pZ-extension and let P be the set of rational primes which
ramify in L. Since L/Q is a Galois extension,
there is a unique p above p, if p isQramied in L. The conductor, f(L/Q) = q∈P fq where(fq =q p−1 ,
when (q, p) = 1
pp−1+sp|p , otherwise.

Here, 1 ≤ sp|p ≤ valp (p) = p. The rst case is called tame ramication, and thesecond is the case of wild ramication (see [7, Chapter VII] or [6]).
Taking natural log,
(2)

log(f(L/Q)) = (p − 1)X

log q + sp|p log p.

q∈P

The goal is to nd the minimal conductor of L for which rp (RpS (A/L)) is unbounded, i.e. rp (RpS (A/L)) ≥ N for any given non-negative integer, N . From
Lemma 2.2, it is enough to nd a Z/pZ-extension Ln(N ) /Q such thatrp (ClS (Ln )) ≥2d + N=: n(N ) = n.
rp (A(Ln )[p])Note that rp (A(Ln )[p]) is a positive constant, less than or equal to 2d.
Let S = {v1 , . . . , vk }∪S∞ be the nite set of primes containing the Archimedean
primes, the primes above p, and the primes of bad reduction of A. We construct
Sn as in the proof of Proposition 3.2. Here, r1 = 1, r2 = 0 and δ = 0. Therefore
we must choose Sn such that |Sn | = |S| + 1 + n.
Q
Dene M = ki=1 vi . To construct Sn from the given set S , we need to add n + 1
many primes. Choose the rst prime p1 - M . By the Prime Number Theorem weknow that we can nd p1 ∼ log M . Now choose p2 - M p1 ; here p2 ∼ log(M log M ).
We have S ∪{p1 , p2 } = S1 . We continue to choose, in the same way, as many primes
as required to form Sn . Using Equation 2, as n → ∞,log(f(Ln /K)) ∼ (p − 1)n log log M.

Equivalently, f(Ln /Q) ∼ cn with c a constant that depends on the given set S . Bydenition of n(N ), f(Ln(N ) /Q) ∼ κN for a constant κ that depends on the set S
and the Abelian variety A.
6

DEBANJANA KUNDU

The computation for proving the general case is similar. We point out somesimilarities and dierences. Consider the tower of number elds L ⊃ K ⊃ Q where[L : K] = p. By hypothesis, L/K is Galois. If q | q is a prime in K that ramiesin L, there will be a unique prime Q|q. The denition of the conductor carries
through. But now, we are
Q interested in the NK/Q (f(L/K)) so as to be able to do
estimates. Dene M = i N (vi ) and construct Sn from S by adding r1 + r2 + δ + n
many primes. Choose p1 - M as before and the required element of Sn is p1 | p1 .
From here, the proof follows as before.


By Equation 1, rp Selp (A/K) ≥ rp RpS (A/K) . Thus, Theorem 3.4 holds on replacing rp (RpS (A/K)) ≥ N by rp (Selp (A/K)) ≥ N .


Remark 3.6.

4. Unboundedness of the fine Shafarevich-Tate group in
Z/pZ-extensions

In this section, we provide answers to the following question asked in [5].
Question 4.1. Let A be an Abelian variety dened over a number eld K . Suppose

A(K)[p] 6= 0. Issup{rp p∞ (A/L) | L/K is a cyclic extension of degree p} = ∞?

For elliptic curves, the answer to this question is precise. It is a corollary of
results proved in [2] and [11]. We record these previously known results.
Lemma 4.2. [11, Lemma 3.1] Let v | p and Kv /Qp be a nite extension of degree

nv . Then# E(Kv )/pk = pk·nv · # E(Kv )[pk ] .

b av ) has nite index in E(Kv )The lemma follows from the observation that E(mbwhere, E stands for the formal group associated to E and mav is any power of the
maximal ideal in the ring of integers in Kv . Therefore,b a )/pk#E(m#E(Kv )/pkv=.
b av )[pk ]#E(Kv )[pk ]#E(mb av ) ' mav where the isomorphism is given by the formal
For suciently large a, E(mb av )[pk ] = 0 and
logarithm [10, Theorem IV.6.4b]. The lemma follows since E(m
a
k
k·nv
b
E(mv )/p = p.
Recall that the quotient of X(E/K)[pk ] and pk (E/K) is contained in thecokernel of the map E(K)/p → ⊕v|p E(Kv )/pk , denoted by Cpk . Lemma 4.2 showsQ
that the codomain of this map has size bounded by pk[K:Q] v|p #E(Kv )[pk ]. Thus,
Proposition 4.3. [11, Proposition 3.2] The index of pk (E/K) inside X(E/K)[pk ]is bounded by(3)[X(E/K)[pk ] : pk (E/K)] ≤ pk[K:Q]

Y#E(Kv )[pk ]v|pThe focus is on the case k = 1, ie the p-ne Shafarevich-Tate group. When E isan elliptic curve dened over a number eld, K , and L/K is a degree p-extension,
there are only nitely many w|p in L. For
Qeach w|p, #E(Lw )[p] is nite and bounded[10, Corollary III.6.4b]. Therefore, # w|p E(Lw )[p] is nite and bounded as wevary over all Z/pZ-extensions, L/K .

7
Theorem 4.4. [2] Let E/K be an elliptic curve. For any positive integer r , there

exists Z/pZ eld extensions L/K such that X(E/L) contains at least r elementsof order p i.e. there exists a Z/pZ eld extension L/K such that X(E/L)[p] isunbounded.

The above two results provides a positive answer to Question 4.1.
Theorem 4.5. Let E be an elliptic curve dened over the number eld K . Varying
over all Z/pZ-extensions, L/K , p (E/L) is unbounded.
Remark 4.6. In the case of elliptic curves, the question asked in [5] does not need

the assumption E(K)[p] 6= 0.

In general, we know that p (A/K) is a subgroup of X(A/K)[p] with quotientin Cp . We have
#Cp ≤

Y
v|p

#A(Kv )/pA(Kv ) ≤Y#H 1 (Kv , A[p]).

v|pThe right hand side of the inequality is nite and bounded [8, Theorem 7.1.8(iii)].
The next result now follows immediately.
Let A be an Abelian variety dened over the number eld K .
Varying over all Z/pZ-extensions L/K , p (A/L) is unbounded if and only if
X(A/L)[p] is unbounded.
Proposition 4.7.

(1) Theorem 4.5 is also seen to follow from Proposition 4.7 andthe theorem of Clark-Sharif without using the results of [11].
(2) In [4], Creutz has proven results on the unboundedness of X(A/L)[p].

Remark 4.8.

5. Growth of p-Fine Selmer Groups in Quadratic Extensions
We are unable to prove that the p-(ne) Selmer group can be arbitrarily large
in quadratic extensions of Q, but there are reasons to believe it should be true. In
this section, we prove that this question is equivalent to a well-known conjecture
about class groups of quadratic extensions.
Fix an odd prime p. Let E/K be an elliptic curve such that
E(K)[p] 6= 0. Let S be a nite set of primes in K containing the primes abovep, the primes of bad reduction of E and the Archimedean primes. As we vary over
all Z/2Z-extensions L/K ,
sup{rp RpS (E/L) | L/K is a quadratic extension} = ∞Theorem 5.1.

if and only if
sup{rp Cl(L) | L/K is a quadratic extension} = ∞.

To prove the theorem, we need to rst prove some lemmas.
Lemma 5.2. With the same setting as Theorem 5.1,

rp RpS (E/L) ≥ rp Cl(L) rp E(L)[p] + O(1).

Proof. It follows immediately from Lemma 2.2 upon observing that

(4)



rp Cl(L) − rp ClS (L) = O(1).

8DEBANJANA KUNDU

Indeed, this dierence depends only on S(L) , where S(L) is the set of nite primesof L above the primes of S in K . Note that S(L) is nite and bounded, in fact
less than |S|2 .

Set B = E(L)[p]. Dene RpS (B/L) by replacing E[p] with E(L)[p] in the denition of the p-ne Selmer group (see (1)).
Lemma 5.3. With the setting as Theorem 5.1,






rp RpS (B/L) − rp RpS (E/L) ≤ rp ClS (L) + O(1).

Proof. If B = E(L)[p] = E[p], there is nothing to prove. So, assume B 6= E[p].
Consider the following commutative diagram→ RpS (B/L)
→


s
y

0

→

0

RpS (E/L)→

H 1 (GS(L), B)

f
y

→

H 1 (GS (L), E[p]) →L

L

1v H (Lv ,

vH

1

B)


g
y

(Lv , E[p])where v runs over all the primes in the nite set S(L).
By hypothesis, E has an L-rational p-torsion point. This gives the short exact
sequence(5)
0 → B → E[p] → µp → 0.
This is because, if E has an L-rational p-torsion point, this point gives an injection
Z/pZ ,→ E[p]. Therefore,
0 → Z/pZ → E[p] → M → 0.

By Cartier duality and the Weil pairing, the short exact sequence turns into
0 → M ∨ → E[p] → µp → 0,

where µp is viewed as a quotient of E[p]. Since the Weil pairing is alternating, theorthogonal complement of Z/pZ is Z/pZ, thus M ∨ = Z/pZ as a subgroup of E[p].
0Taking the GS (L)-cohomologyof (5),
 ker(f ) = H (GS (L), µp ). As µp is niteand bounded, rp ker(s) ≤ rp ker(f ) = O(1). A similar argument for the local
cohomology yields rp ker(g) = O(1).
Therefore, to prove the lemma, it suces to prove
rp coker(s) ≤ rp coker(f ) + O(1) ≤ rp ClS (L) + O(1).

Indeed, by Lemma 2.1 applied to the map s,






rp RpS (B/L) − rp RpS (E/L) ≤ 2rp ker(s) + rp coker(s)= rp (coker(s)) + O(1).

Let
be the set of S -units of LS , the maximal unramied outside S extensionof L. We know µp ⊆ OS× and there exists a short exact sequence [8, Theorem 8.3.18]
OS×p

0 → µp → OS× −→ OS× → 0.

This yields a long exact sequence which can be rewritten as
(6)


p

×
×
0 → OL,S/ OL,S→ H 1 GS (L), µp → ClS (L)[p] → 0,

9×where OL,S
is the notation
for theS -units of L. We remark, (6) follows from




×
×
0
standard results H GS (L), OS ' OL,Sand H 1 GS (L), OS× ' ClS (L) [8,Theorem 8.3.11]. Therefore, (p-rank of) coker(f ) = H 1 (GS (L), µp ) is nite. Furthermore,
rp coker(f ) − rp ClS (L) ≤ rp

× .
OL,S

!
×
OL,S

p

.

Since S(L) is bounded by an absolute constant, the S -units analogue of Dirichlet'sUnit Theorem yields

rp coker(f ) − rp ClS (L) = O(1).

Equivalently,
rp coker(f ) = rp ClS (L) + O(1).

Therefore,





rp RpS (B/L) − rp RpS (E/L) ≤ rp ClS (L) + O(1).

This nishes the proof.

Proof of Theorem 5.1. In Lemma 5.2, we showedrp RpS (E/L) ≥ rp Cl(L) rp E(L)[p] + O(1).

This proves: if rp (Cl(L)) is arbitrarily large then so is the rp (RpS (E/L)). Equivalently, if rp (RpS (E/L)) is bounded then so is rp (Cl(L)).
We now prove the other direction.
Claim: If rp (Cl(L)) is bounded, the same is true for the rp (RpS (E/L)).

Justication: Supposerp Cl(L) = O(1). By Equation 4, rp Cl(L) is bounded if
and only if rp ClS (L) is bounded.
By hypothesis, the Galois action of GS (L) on E(L)[p] is trivial; the argument inRemark 2.3 yields
rp RpS (B/L) ≤ 2rp ClS (L) = O(1).

By Lemma 5.3, if rp ClS (L) is bounded,rp RpS (B/L) − rp RpS (E/L) ≤ O(1).

From the above two inequalities, the claim follows. This nishes the proof.

Acknowledgement

We thank Kumar Murty for helpful discussions and his encouragement. We
thank all past and present members of the GANITA Lab for listening to the details
over and over again. We extend our gratitude to K¦stutis esnavi£ius for hiscomments on an earlier draft, and to Arul Shankar for answering innumerable
questions on weekends when no one else was in the department. We are grateful tothe referee for a careful reading of this paper and for their helpful comments andcorrections.

10

DEBANJANA KUNDU
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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