Non-commutative Iwasawa theory of elliptic
curves at primes of multiplicative reduction.
Chern-Yang LEE
Emmanuel College
This dissertation is submitted for the degree of
Doctor of Philosophy at the
University of Cambridge
April 2010
Declaration
This dissertation is my own work and contains nothing which is the outcome
of work done in collaboration with others, except as specified in the text and
Acknowledgements.
i
Dedicated to Sek-Khuan Lee and Nga-Fong Toh
ii
Abstract
Let E be an elliptic curve defined over the rationals Q, and p be a prime
at least 5 where E has multiplicative reduction. This thesis studies the Iwa-
sawa theory of E over certain false Tate curve extensions F∞, with Galois group
G = Gal(F∞/Q). I show how the p∞-Selmer group of E over F∞ controls
the p∞-Selmer rank growth within the false Tate curve extension, and how it is
connected to the root numbers of E twisted by absolutely irreducible orthogo-
nal Artin representations of G, and investigate the parity conjecture for twisted
modules.
Title:
Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative
reduction.
Author:
Chern-Yang LEE
iii
Introduction
The Iwasawa main conjectures for elliptic curves provide a scope to study
Birch and Swinnerton-Dyer conjecture. For a fix prime p, on the algebraic side,
one studies the structure of the Iwasawa module Xp(E/F∞), which is the Pon-
tryagin dual of the p∞-Selmer group of the elliptic curve E over a p-adic Lie
extension F∞ of the base number field F over which E is defined. On the an-
alytic side, one expects the existence of a p-adic L-function, interpolating the
special values of complex L-functions. Main conjecture asserts certain coinci-
dence of both the algebraic module and the p-adic L-function.
There have been quite a number of cases studied by many experts, in either
the algebraic modules or the p-adic L-functions. There seems to have a common
level of difference when the prime p varies by its reduction type for E, (assum-
ing naively that E has the same reduction type at each place of F above p),
or when the p-adic Lie extension F∞ varies. Out of the reduction types, good
ordinary reduction has been most studied along with many different p-adic Lie
extensions, for instance when F∞ = Fcyc/F by Mazur [20], F∞ = F(Ep∞) by
Coates-Fukaya-Kato-Sujatha-Venjakob [4] and F∞ = ’False Tate curve exten-
sions’ by Coates-Fukaya-Kato-Sujatha [3].
Let G denote the Galois group of the p-adic Lie extension F∞/F and Λ
denote the corresponding Iwasawa algebra. Mazur conjectured that the Selmer
module Xp(E/Fcyc) isΛ-torsion when E has good ordinary reduction over places
of F above p. This is proved in some of the cases of F, or under other assump-
tions. It is further expected that Xp(E/F∞) is Λ-torsion for many other p-adic
Lie extensions. However, when the p-adic Lie extension is not abelian, for
instance when F∞ = F(Ep∞) with E an elliptic curve without complex multi-
plication, being Λ-torsion alone is not sufficient to formulate a main conjecture.
iv
In [4], the authors introduce a stronger assumption, calledMH(G)-Conjecture,
asserting that the algebraic module Xp(E/F∞) should belong to the category
MH(G). Equivalently, it means Yp(E/F∞), the quotient module of Xp(E/F∞)
by its p-primary part, is finitely generated over the Iwasawa algebra of H, where
H is the subgroup of G which fixes Fcyc. This MH(G)-Conjecture allows one
to attach a characteristic element to the module and go further to formulate the
main conjecture.
In [3], the alliance took the belief ofMH(G)-Conjecture over to study Iwa-
sawa Theory for E over certain False Tate curve extension F∞/F, which is a
non-commutative p-adic Lie group of dimension 2, where E is any elliptic curve
defined over any number field F that has good ordinary reduction over all places
of F above the odd prime p. The field F∞ is defined from an element m ∈ F×,
with certain constraints relating to the reductions of E. ByMH(G)-Conjecture,
they define an algebraic invariant τ being the rank of Yp(E/F∞) over the Iwa-
sawa algebra of HK , where HK is the subgroup of G = Gal(F∞/F) which fixes
F(µp∞). This τ seems to have a lot of control over the arithmetics of E over
the intermediate fields. When τ is odd, it guarantees and give lower bounds for
the Selmer rank growth in both the cyclotomic direction and radical direction.
When τ = 1, the corresponding bounds are hit and one can decide the Selmer
rank of the elliptic curve over infinitely many intermediate number fields. The
authors also consider the root number of the elliptic curve E twisted by all ir-
reducible orthogonal Artin representations of G and show their connection to τ
and prove a parity conjecture of the twisted modules.
In this thesis, I study the parallel results as [3] under the setting of a triple
(E, p,m), where E is an elliptic curve defined over F = Q, having semistable
reduction at all prime divisors of the integer m > 1. The main difference is
that I assume E has multiplicative reduction at the odd prime p ≥ 5, instead
of good ordinary reduction. These assumptions are made in Section 1.1, which
is mentioned as assumptions on (E, p,m) in several places throughout the thesis.
In chapter 2, I start by introducing the all-importantMH(G)-Conjecture and
some direct consequences from it and compute the Zp-coranks of several mod-
ules under this Conjecture. These computations will lead to a formula to obtain
v
the marvelous but ’conjectural’ value τ.
In chapter 3, I have a thorough investigation over the Q¯p-irreducible Artin
representations which factors through Q(µpn , p
n√m). I use V.Dokchitser’s for-
mula [8] to compute the root numbers of the elliptic curve twisted by these
Artin-representations. By Greenberg[9]-Guo[10], these root numbers are again
controlled by the parity of τ from the formula established in chapter 2.
In chapter 4, we once again relate the value τ to λn, the λ-invariant of the
finitely generated torsion module Xp(E/Lcycn ) where Ln = Q( p
n√m) for n ≥
1 via two approaches in computing the homological rank of Yp(E/F∞). By
Greenberg-Guo, this relation leads to the control of growth of p∞-Selmer ranks
within the False Tate curve tower by the value τ. I give an assertion on when the
parity conjecture of twisted modules holds.
vi
Acknowledgements
Above all, I would like to express my gratitude to my supervisor Prof. John
Coates, for always being my largest source of support, confidence and encour-
agements throughout the years that I have been in Cambridge. He never fails in
motivating me when it is most needed. His advice and guidance are very encour-
aging and beneficial to me. I appreciate his patience in correcting me numerous
times when I had not understood well enough. I am particularly grateful that
John gave extremely careful readings on my manuscripts and pointed out my
errors and room of improvements.
Apart from the general guidance of my supervisor, the work in this thesis was
greatly assisted by the the following advice from other mathematicians. First,
I would like to thank Christian Wuthrich for referring me to Jones’ paper [14]
which leads to computing the λ-invariant of the p∞-Selmer group of an elliptic
curve over Q(µp∞). Christian taught me SAGE by providing me some relevant
computations as examples, and has been very responsive to my questions by
email. I would like to thank Tim Dokchitser for providing me his Magma com-
mand in computing the Hasse-Weil L-value of an elliptic curve over a number
field and numerous troubleshooting and help with my calculations in Magma.
Also, I am grateful to Tom Fisher for introducing me a Pari-GP command by
Denis Simon which searches rational points of infinite order on an elliptic curve
over a number field, which successfully led me to the examples given at the end
of the thesis.
I would like to dedicate this thesis to my dearest parents, Sek-Khuan and
Nga-Fong, not only for being the main financial support of my PhD life, but
also for their confidence in me in pursuing my dream. I would like to add here
my appreciation of the love and patience by Choonpei, who from being my girl-
friend till becoming my wife, taking very good care of me on daily basis, and
sharing my joy during the process of the writing up.
I am grateful to have some moments free of financial burdens with the very
generous funds by Cambridge Commonwealth Trusts and Emmanuel College. I
would like to thank a very special person, Dato Seri Joseph Chong, for sponsor-
ing my whole undergraduate studies in Peking University and my Part III course
vii
in the University of Cambridge.
I would like to thank the Department of Pure Mathematics and Mathemat-
ical Statistics for providing such a wonderful academic environment and the
departmental administrator Sally Lowe for solving every administration prob-
lem. Finally, I wish to thank the Chinese Educational Body in Malaysia and my
secondary school, Chong Hwa High School Kluang for providing me the best
education in Malaysia!
viii
Contents
Abstract iii
Introduction iv
Acknowledgements vii
1 Setting and Background 1
1.1 The False Tate Curve Extension . . . . . . . . . . . . . . . . . 1
1.2 Iwasawa algebras and their modules . . . . . . . . . . . . . . . 2
1.3 Selmer Groups and Fundamental Diagram . . . . . . . . . . . . 8
1.4 Pontryagin Duality and Zp-coranks . . . . . . . . . . . . . . . . 13
2 The CategoryMH(G) 17
2.1 Mazur Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The categoryMH(G) . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Computation of the Zp-coranks of some modules . . . . . . . . 21
2.4 Deeply Ramified Theorem . . . . . . . . . . . . . . . . . . . . 31
2.5 Subconclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Root Number Computations 37
3.1 Root Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Computations of Root Numbers . . . . . . . . . . . . . . . . . 46
3.3 Parity Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 More on the representations ρχn . . . . . . . . . . . . . . . . . . 63
4 Homological Ranks and Rank Growth 72
4.1 Homological Ranks . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Computation of Homological Ranks of Yp(E/F∞) I . . . . . . . 77
4.3 Computation of Homological Ranks of Yp(E/F∞) II . . . . . . 79
ix
4.4 Rank Growth in the False Tate Curve Extension . . . . . . . . . 87
4.5 Λ(HK)-rank 1 case . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography 101
x
Chapter 1
Setting and Background
1.1 The False Tate Curve Extension
Let p be a fixed odd prime. We denote by µpn the group of pn-th roots of
unity, and µp∞ the group of p-power roots of unity.
Definition:. For any number field L, we denote by Lcyc the p-cyclotomic exten-
sion of L and ΓL the Galois group Gal(Lcyc/L). More precisely, it is the unique
Zp-extension of L which is contained in L(µp∞).
Definition:. For the fixed prime p, and any positive integer m def=
∏
qiri , where
qi are distinct primes with p-ri, we introduce the notations
Fn
def
= Q(µpn , p
n√
m) n ≥ 0, (1.1)
and we call the union
F∞
def
=
⋃
n≥0
Fn (1.2)
a false Tate curve extension over Q. It is clearly a Galois extension of Q.
Let E be an elliptic curve. In this paper, I shall study the Iwasawa Theory
within the false Tate curve extension F∞/Q with the following assumptions on
the triple (E, p,m), with prime decomposition m =
∏
qiri .
1
Assumption on (E, p,m) :
1. E is defined over Q,
2. p ≥ 5,
3. E has (split or non-split) multiplicative reduction at p,
4. E has semi-stable reduction at all primes qi dividing m.
Let us introduce more notations corresponding to the false Tate curve exten-
sion F∞. For each n ≥ 0, Fn is the composite of two subfields
Kn
def
= Q(µpn) and Ln
def
= Q( pn
√
m),
and we denote their respective unions by
K∞
def
=
⋃
n
Kn and L∞
def
=
⋃
n
Ln.
I shall use the simpler notation K def= K1, and denote the following composite
fields by
L′n
def
= KLn and L′∞
def
= KL∞.
We denote by G the Galois group Gal(F∞/Q). For any number field L con-
tained in F∞, we denote by GL ≤ G the open subgroup which fixes L and
HL ≤ G the closed subgroup which fixes Lcyc, and simply by H def= HQ. These
groups are compact p-adic Lie groups.
1.2 Iwasawa algebras and their modules
Definition:. For any compact p-adic Lie group G, we define the Iwasawa alge-
bra of G by
Λ(G) def= lim
←
U
Zp[G/U] (1.3)
where U runs through the open normal subgroups of G and the inverse limit
is taken with respect to the canonical projective maps. In particular, for any
2
number field L, we denote the Iwasawa algebra of ΓL as
Λ(ΓL)
def
= lim
←
n
Zp[ΓL/Γp
n
L ] (1.4)
where Γp
n
L runs through the open normal subgroups of ΓL and the inverse limit
is taken with respect to the canonical projective maps.
If the compact p-adic Lie group G is pro-p, the Iwasawa algebra Λ(G) is a
local ring, with the unique maximal ideal the kernel of the augmentation map
Λ(G) −→ Fp. If G in addition has no nontrivial element of order p, the Iwasawa
algebra Λ(G) is Noetherian and contains no nontrivial zero divisors.
The main purpose of this section is to introduce the structure theorem of
finitely generated Λ(Γ)-torsion modules, where Γ denotes any compact p-adic
Lie group isomorphic to Zp.
Let us denote by Λ def= Zp[[T ]], the ring of power series of one variable with
coefficients in Zp. It is endowed with a complete topology described below:
The group ring Zp[T ] has a group topology defined on it, induced by princi-
pal ideals
(
(T + 1)pn − 1), n ≥ 0 being a base of neighbourhoods of 0 ∈ Zp[T ].
Λ is simply the completion of this group ring under this topology, i.e:
Λ ∼= lim←
n
Zp[T ]/
(
(T + 1)p
n − 1).
Proposition 1.2.1. We have a non-canonical isomorphism of algebras
Λ(Γ) ∼= Λ
induced by γ 7→ 1 + T, where γ is any chosen topological generator of Γ.
Definition:. A Λ-module is said to be torsion if for each element of the module,
there is a non-zero element in Λ annihilating it.
Definition:. Let M, M′ be two finitely generated Λ-modules. We say M is
3
pseudo isomorphic to M′ and denote M
pseudo∼ M′ if there is a Λ-modules ho-
momorphism M −→ M′ with finite kernel and cokernel.
This relation is an equivalence relation among finitely generated Λ-torsion
modules.
Definition:. A non-constant polynomial
f (T ) = T n + an−1T n−1 + · · · + a0 ∈ Zp[T ]
is called distinguished if p | ai for all 0 ≤ i ≤ n− 1.
The ring (or algebra) Λ is a unique factorization domain, with p and irre-
ducible distinguished polynomials as the only irreducible elements, up to multi-
plication by a unit in Λ.
Theorem 1.2.1. Structure Theorem [29, Theorem 13.12]
Let M be a finitely generated Λ-module. Then
M
pseudo∼ Λr ⊕ (⊕si=1Λ/(pni))⊕ (⊕tj=1Λ/( f j(T )m j)) , (1.5)
where r, s, t, ni, m j are non-negative integers , and f j(T ) is distinguished and
irreducible.
Hence there’s the direct corollary
Corollary 1.2.1. Let M be a finitely generated Λ-torsion module. Then
M
pseudo∼ (⊕si=1Λ/(pni))⊕ (⊕tj=1Λ/( f j(T )m j)) , (1.6)
where s, t, ni, m j are non-negative integers, and f j(T ) is distinguished and irre-
ducible.
Definition:. For any finitely generated Λ-torsion module M, with a pseudoiso-
4
morphism given as in (1.6), we define an µ-invariant and a λ-invariant to it:
µΛ(Γ)(M)
def
=
s∑
i=1
ni,
λΛ(Γ)(M)
def
=
t∑
j=1
m j · deg( f j).
A finitely generated Λ(Γ)-torsion module M is also a finitely generated Zp-
module if and only if it has vanishing µ-invariant. In this case, its λ-invariant is
just the Zp-rank of M.
For any H which is a pro-p p-adic Lie group without non-trivial elements
of finite order, the Iwasawa algebra Λ(H) has no non-trivial zero divisors and
therefore admits a skew field of fractions, denote as QΛ(H). In this paper, we
will come across some abelian cases when H ∼= Zp.
Definition:. Let H be a pro-p p-adic Lie group without non-trivial elements of
finite order. For any finitely generated Λ(H)-module M, we define
rankΛ(H)M
def
= dimQΛ(H)(QΛ(H)⊗Λ(H) M), (1.7)
as the Λ(H)-rank of M.
In the case when H ∼= Zp, M is Λ(Zp)-torsion if and only if it has zero
Λ(Zp)-rank.
Definition:. For any compact p-adic Lie group G, for i ≥ 0, we define the i-th
cohomology group of G with coefficients in a leftΛ(G)-module M as the image of
the i-th right derived functor of HomΛ(G)(Zp, ∗) of M, and denote it by Hi(G,M).
Similarly, we define the i-th homology group of G with coefficients in a right
Λ(G)-module M as the image of the i-th left derived functor of
(∗ ⊗Λ(G) Zp) of
M, and denote it by Hi(G,M).
Definition:. If there exists a natural number r such that Hi(G,M) = 0 for all
integers i > r and every finitely generated Λ(G)-module M, we say that Λ(G)
5
has finite p-homological dimension and denote the smallest such r by hdp(G)
and call it the p-homological dimension of G. As we shall see later, ”dually”
we can define the same for cdp(G), the p-cohomological dimension of G as the
smallest r such that Hi(G,M) = 0 for all integer i > r and every discrete p-
primary Λ(G)-module M.
For a compact p-adic Lie group G having no nontrivial element of order p,
it has finite p-(co)homological dimension, equaling to the dimension of G as a
p-adic Lie group. See [18] and [25].
I shall prove the following two propositions together, they are both due to
Howson [13].
Proposition 1.2.2. Let M denote a finitely generated Λ(G)-module, where G is
a p-adic Lie group having no element of order p. Then
i For each i ≥ 0, Hi(G,M) is a finitely generated Zp-module;
ii
∑
i≥0(−1)irankZp
(
Hi(G,M)
)
is finite.
Proposition 1.2.3 (Howson). [13]
Let M denote a finitely generated Λ(H)-module, where H is a pro-p p-adic Lie
group having no element of order p. Then
rankΛ(H)M =
∑
i≥0
(−1)irankZp
(
Hi(H,M)
)
. (1.8)
Proof. For both the cases G and H, since they have no non-trivial p-torsion
elements, they both have finite p-homological dimensions. and hence the alter-
nating sums appeared in the proposition are just sums of finite terms. Hence, it
reduces to show Hi(G,M) and Hi(H,M) are finitely generated Zp-modules for
each i ≥ 0. We prove the latter situation first. Since M is a finitely generated
Λ(H)-module, it is a quotient module of a finitely generated free Λ(H)-module.
Hence we have an exact sequence of Λ(H)-modules
0 −→ M′ −→ Λ(H)n0 −→ M −→ 0. (1.9)
6
Since H is pro-p and have no elements of order p other than the trivial element,
Λ(H) is a Noetherian ring. M′ being a submodule of a finitely generated module
over a Noetherian ring, is finitely generated. Taking the H-homology of the
short exact sequence above, we obtain a long exact sequence of Zp-modules
· · · −→ Hi+1(H,Λ(H)n0) −→ Hi+1(H,M) −→ Hi(H,M′) −→ Hi(H,Λ(H)n0) −→ · · ·
(1.10)
· · · −→ H1(H,Λ(H)n0) −→ H1(H,M) −→ M′H −→
(
Λ(H)n0
)
H −→ MH −→ 0.
(1.11)
The right end of eq(1.11) shows that H0(H,M) = MH is a finitely generated
Zp-module since
(
Λ(H)n0
)
H
∼= Zn0p . Since M′ is a finitely generated Λ(H)-
module as well, the same argument deduces H0(H,M′) is a finitely generated
Zp-module. For Λ(H)n0 is a free Λ(H)-module, its higher homology groups are
vanishing, that is
Hi(H,Λ(H)n0) = 0
for i ≥ 1. Therefore the exact sequences eq(1.11) and eq(1.10) become
0 −→ Hi+1(H,M) −→ Hi(H,M′) −→ 0, f or i ≥ 1, (1.12)
0 −→ H1(H,M) −→ H0(H,M′) −→ Zn0p −→ H0(H,M) −→ 0. (1.13)
From the latter exact sequence, H1(H,M) must be a finitely generatedZp-module
as so are the rest of the terms. The same arguments, when applied to M′, deduce
that H1(H,M′) is a finitely generated Zp-module. The exact sequence eq(1.12)
provides an inductive way to ’climb’ up the homology degree by one to induc-
tively show that Hi(H,M) is finitely generated over Zp, via the parallel result for
both M and M′ in the preceding case.
Now for the case ofG, there exists an open normal subgroup which is pro-p. Let
us denote this pro-p subgroup by H. By the Hochschild-Serre spectral sequence,
we have
Hp(G/H,Hq(H,M))⇒ Hp+q(G,M) (1.14)
and hence Hi(G,M) are finitely generated over Zp by the fact that Hi(H,M) are
and the quotient group G/H is finite.
Since Λ(H) is a local ring, a Λ(H)-module is projective if and only if it is free.
On the other hand, Λ(H) has finite global dimension, hence there exists a pro-
7
jective (and hence free) resolution of M of finite length
0 −→ Λ(H)nl −→ · · · −→ Λ(H)n1 −→ Λ(H)n0 −→ M −→ 0. (1.15)
Left-tensor this exact sequence by ΛQ(H), the skew field of fractions of Λ(H),
which is a flat module, we get an exact sequence of finitely generated vector
spaces over ΛQ(H) and hence the alternating sum of their dimensions would be
zero along this exact sequence. This yields
rankΛ(H)M =
l∑
i=0
(−1)idimΛQ(H)(ΛQ(H)⊗Λ(H) Λ(H)ni)
=
l∑
i=0
(−1)ini.
(1.16)
On the other hand, the alternating sum of the values
∑
i≥0(−1)irankZpHi(H, ?)
is zero along any long exact sequence of finitely generated Λ(H)-modules. This
yields
∑
i≥0
(−1)irankZpHi(H,M) =
l∑
k=0
(−1)k
∑
i≥0
(−1)irankZpHi(H,Λ(H)nk)
=
l∑
k=0
(−1)krankZpH0(H,Λ(H)nk)
=
l∑
k=0
(−1)knk,
(1.17)
by the fact that the higher homology groups of a free module vanish as pointed
before. Hence the formula (1.8) is proved.
1.3 Selmer Groups and Fundamental Diagram
In this section, we want to obtain a fundamental diagram involving the p∞-
Selmer groups S elp(E/F∞) and S elp(E/K∞) where F∞ and K∞ are as given in
Section 1.1. Before this, I shall define the p∞-Selmer groups of E over arbitrary
algebraic extension of the defining field of E. I shall first define it over number
fields.
8
Throughout this section, k always denotes a number field, that is, a field con-
taining Q, with degree [k : Q] <∞. Let k denote any fixed separable closure of
k. We denote by kv the completion of k at the place v.
Assume that E is an elliptic curve defined over the number field k.
Definition:. For any abelian group A, and n a positive integer, we denote by An
the subgroup consisting of all n-torsion elements, by Ators the torsion subgroup
of A, by A(p) or Ap∞ the p-primary subgroup of A for any prime p. Further-
more, when A is a G-module for any topological group G, let Hi(G, A) denote
the i-th cohomological group, defined with continuous cochains. For any field
F, we use the notation Hi(F, A) def= Hi(Gal(F/F), A) for F a separable closure of
F and A a Galois module.
The multiplication by n map of the elliptic curve induces the exact sequence
of Gal(k/k)-modules:
0 −→ En −→ E(k) [n]−→ E(k) −→ 0
where En denotes the kernel of the multiplication by nmap. Taking theGal(k/k)-
cohomology of this short exact sequence yields a long exact sequence:
0 −→ En(k) −→ E(k) [n]−→ E(k) −→ H1(k, En) −→ H1(k, E) [n]−→ H1(k, E)
which provides the short exact sequence:
0 −→ E(k)
[n]E(k)
−→ H1(k, En) −→ H1(k, E)n −→ 0
For any place v of k, we can view E as defined over the completion field kv.
Similarly, we can do exactly the same as before and obtain
0 −→ E(kv)
[n]E(kv)
−→ H1(kv, En) −→ H1(kv, E)n −→ 0
Taking the inductive limit over all n ≥ 1, which is an exact functor, to these
short exact sequences, we obtain the following commutative diagram with exact
9
rows:
0 −→ E(k)⊗Q/Z κ−→ H1(k, Etors) λ−→ H1(k, E) −→ 0
0 −→
∏
v
E(kv)⊗Q/Z
? Q
κv−→
∏
v
H1(kv, Etors)
? Q
λv−→
∏
v
H1(kv, E)
?
−→ 0
(1.18)
where direct product runs over all places v of k. The left vertical map is just the
canonical embedding into each component. The middle and right vertical maps
are restriction maps into each v-component.
Definition:. The Tate-Shafarevich group of E over k,X(E/k) is defined to be
the kernel of the right vertical map in diagram (1.18), which is
X(E/k) = ker
(
H1(k, E) −→
∏
v
H1(kv, E)
)
, (1.19)
of which v runs over all places of k in the product
∏
v
above.
It is conjectured that over number field k, the groupX(E/k) is finite.
Definition:. The Selmer group of E over k, S el(E/k) is defined to be the preim-
age ofX(E/k) under the surjection λ. By the exactness of the second row of the
commutative diagram (1.18), we can equivalently give its definition as:
S el(E/k) = ker
(
H1(k, Etors) −→
∏
v
H1(kv, E)
)
. (1.20)
where v runs over all places of k in the product
∏
v
.
Definition:. Let S elp(E/k) denote the p-primary subgroup of the Selmer group
for E over k, or briefly the p∞-Selmer group for E over k. This is by definition,
the p-primary subgroup of (1.20), which is
S elp(E/k) = ker
(
H1(k, Ep∞)→
∏
v
H1(kv, E)p∞
)
(1.21)
10
where v runs over all places of k. We endow these p-primary groups S elp(E/k) ⊆
H1(k, Ep∞) with discrete topology and hence they are equipped with discrete Zp-
module structures.
Combined with the commutative diagram (1.18), we obtain the following
exact sequence:
0 −→ E(k)⊗Q/Z −→ S el(E/k) −→X(E/k) −→ 0, (1.22)
and its p-primary analogue:
0 −→ E(k)⊗Qp/Zp −→ S elp(E/k) −→X(E/k)(p) −→ 0. (1.23)
Now I would want to extend these definitions from over a number field to
over any general algebraic extension.
Let us fix the following notations:
Definition:. Let L denote any algebraic extension of k, which could be of infinite
degree over k, and w denote a place of L. We write Lw to be the union of the
completion at w of all number fields contained in L.
With this notion of Lw, all of the above leading to the commutative diagram
eq(1.18) still holds for L and Lw in place of k and kv respectively. Thus, we can
defineX(E/L), S el(E/L) and S elp(E/L) in the same fashion as above with k
and kv replaced by L and Lw respectively. We endow too to S elp(E/L) a discrete
Zp-module structure. Equivalently,X(E/L), S el(E/L) and S elp(E/L) are just
the direct limits ofX(E/k′), S el(E/k′) and S elp(E/k′) respectively, where the
limits are taken with respect to restriction maps, with k′ running over all the sub-
fields of Lwhich are finite extensions over k. Thus, the exact sequences eq(1.22)
and eq(1.23) hold also for k being replaced by L.
Let S be a finite set of non-Archimedean places of k containing
i all places of k above the chosen prime p,
ii all places of k at which E has bad reduction.
11
Denote kS the maximal extension of k which is unramified outside S ∪ S∞,
where S∞ is the set of all Archimedean places of k. Denote GS (L)
def
= Gal(kS /L)
for any intermediate field L, k ⊂ L ⊂ kS and S (L) the set of places of L above S .
Given any non-Archimedean place v of k, such that E attains good reduction
E˜ modulo v, denote by kv a separable closure of the field of completion kv, by kv
a separable closure of the residue field kv, by Eˆ the formal group associated to
the elliptic curve E/kv, byM the maximal ideal of the ring of integers of kv. By
classical theorem of elliptic curve, we have short exact sequence
0 −→ Eˆ(M) −→ E(kv) −→ E˜(kv) −→ 0.
The formal group Eˆ(M) contains no non-trivial p-primary part for any prime p
coprime to the characteristic of kv. Hence, as long as the good reduction prime
v does not divide p, we have
E(kv)p∞ ∼= E˜(kv)p∞
and thus v is unramified over k(Ep∞) and hence k(Ep∞) ⊂ kS .
So. we get for any intermediate field L, k ⊂ L ⊂ kS , the isomorphism
S elp(E/L) ∼= Ker
(
H1(GS (L), Ep∞)→
⊕
v∈S∪S∞
Jv(L)
)
(1.24)
where Jv(L)
def
=
⊕
w|v
H1
(
Lw, E(Lw)
)
p∞ with w runs over all places (finitely many)
of L above v, when [L : k] <∞. When [L : k] =∞, we define
Jv(L)
def
= lim
→
[L′:k]<∞
⊕
w′|v
H1
(
L′w′ , E(L′w′)
)
p∞ ,
with w′ runs over all places (finitely many) of L′ where the direct limit are taken
over all subfields L′ of L which is of finite degree over k. Since we are assuming
p is an odd prime, Jv(L) is trivial for any archimedean place v of k, hence we
may omit S∞ in the direct sum in the isomorphism above.
Hence, in our settings made in Section 1.1, taking L = K∞ and F∞, we
12
obtain the following commutative diagram, with exact rows:-
0 - S elp(E/F∞)HK - H1(GS (F∞), Ep∞)HK - ⊕v∈S Jv(F∞)HK
0 - S elp(E/K∞)
rK∞
6
- H1(GS (K∞), Ep∞)
resK∞
6
- ⊕v∈S Jv(K∞)
⊕v∈S hv
6
(1.25)
where the right upward map is just the restriction map. In order that F∞ ⊂ KS
such that the notation GS (F∞) makes sense, we require S to contain all the
places of K that ramify over F∞. In particular, we can just let S = S (K) =
S p ∪ S bad ∪ S ram, where S p denotes the singleton of unique prime of K above p,
S bad denotes the set of primes of K where E has bad reduction, and S ram denotes
the set of primes of K dividing m, where m is the positive integer which defines
F∞ = Q(µp∞ , p
∞√m).
1.4 Pontryagin Duality and Zp-coranks
Definition:. Let M and T be two topological spaces. For each compact subset
U ⊂ M and open subset V ⊂ T, construct a subset
h(U,V) def= {h ∈ C(M,T ) | h(U) ⊂ V}
where C(M,T ) consists of all continuous map from M to T .
{h(U,V) | U ⊂ M is compact,V ⊂ T is open.}
forms a topological base for C(M,T ) and we call this the compact-open topol-
ogy.
Definition:. Give the p-primary group Qp/Zp the discrete topology and view
it as a discrete Zp-module. Hence, it is locally compact and Hausdorff. For
any topological Zp-module M, we denote the Zp-module of all continuous Zp-
homomorphisms by
Mˆ
def
= HomZp(M,Qp/Zp) (1.26)
and call it the Pontryagin dual of M. We shall always endow Mˆ with the induced
13
compact-open topology. When M is a profinite, or equivalently compact and to-
tally disconnected, (or discrete torsion respectively), Zp-module, Mˆ is discrete
torsion (or profinite respectively) as a topological Zp-module.
Pontryagin duality is a reflexive contravariant functor between the category
of discrete torsion Zp-modules and the category of profinite Zp-modules.
Definition:. We say a discrete torsion Zp-module, M, is cofinitely generated as
a Zp-module if its Pontryagin dual Mˆ is a finitely generated Zp-module. We can
then define its Zp-corank as the Zp-rank of its Pontryagin dual and denote the
value by corankZpM.
Let L denote an arbitrary algebraic extension of the base field k, of which
the elliptic curve E is defined over.
Definition:. Let S elp(E/L) be the p∞-Selmer group of E over L, denote by
Xp(E/L)
def
= HomZp
(
S elp(E/L),Qp/Zp
)
(1.27)
its Pontryagin dual. It is a compact Zp-module. Moreover, when L/k is Galois,
then the Galois group Gal(L/k) acts continuously from the left on H1(L, Ep∞),
and hence S elp(E/L) is equipped as a discrete left Zp[Gal(L/k)]-module. We
shall endow its Pontryagin dual Xp(E/L) a left Zp[Gal(L/k)]-module structure
induce by the following definition:
(σ f )(s) = σ( f (σ−1s))
for any σ ∈ Gal(L/k), f ∈ Xp(E/L) and s ∈ S elp(E/L). In addition, when
L/k is a p-adic Lie extension, then the same expression in (1.27) gives Xp(E/L)
a Λ
(
Gal(L/k)
)
-module structure, where Λ
(
Gal(L/k)
)
is the Iwasawa algebra
which is define in the same fashion as (1.3).
Proposition 1.4.1. Assume that L is a finite extension of the base field k, then
H1(GS (L), Ep∞) is a cofinitely generated Zp-module. In particular, S elp(E/L)
is a cofinitely generated Zp-module. Moreover, for any number fields extension
14
k ⊆ L ⊆ L′, we have
corankZpS elp(E/L) ≤ corankZpS elp(E/L′). (1.28)
Proof. By Nakayama lemma, it is sufficient to check the finiteness of
̂H1(GS (L), Ep∞)/p
( ̂H1(GS (L), Ep∞)) = Coker( ̂H1(GS (L), Ep∞) p−→ ̂H1(GS (L), Ep∞)).
This is the Pontryagin dual of
(
H1(GS (L), Ep∞)
)
p = Ker
(
H1(GS (L), Ep∞)
p−→ H1(GS (L), Ep∞)
)
.
Taking the GS (L)-invariants from the short exact sequence
0 −→ Ep −→ Ep∞ p−→ Ep∞ −→ 0
yields a long exact sequence which contains
H1(GS (L), Ep) −→ H1(GS (L), Ep∞) p−→ H1(GS (L), Ep∞).
By its exactness, we get that
(
H1(GS (L), Ep∞)
)
p is the homomorphic image of
the finite group H1(GS (L), Ep) and hence is finite.
For the second argument, it is sufficient to show that the kernel of the induced
restriction map S elp(E/L) −→ S elp(E/L′) is finite. Indeed, this kernel injects
into the kernel of the restriction map
H1(GS (L), Ep∞)
rL′/L−→ H1(GS (L′), Ep∞).
When L′ is Galois over L, ker(rL′/L) = H1(Gal(L′/L), E(L′)p∞) is clearly finite.
When L′ is not Galois over L, take its Galois closure L′′ ∈ kS , then ker(rL′/L) ⊂
ker(rL′′/L) where the latter is finite since L′′ is Galois over L.
Looking at the short exact sequence (1.23)
0 −→ E(L)⊗Qp/Zp −→ S elp(E/L) −→X(E/L)(p) −→ 0.
15
Hence, the Zp-corank of S elp(E/L) is an upper bound of the Mordell-Weil
rank of E over L. Since conjecturally X(E/L)(p) is finite, the Zp-corank of
S elp(E/L) should actually reflect the Mordell-Weil rank over L.
However, when L is an infinite extension of k, S elp(E/L) is not in general a
cofinitely generated Zp-module. For instance when L = kcyc is the p-cyclotomic
extension of k, we shall later see in Lemma 2.1.2 that S elp(E/kcyc) is a co-finitely
generated Λ(Γk)-module. Even if its Λ(Γk)-corank vanishes, we can observe
from (1.6) that the positivity of the µ-invariant of its Pontryagin dual would pre-
vent S elp(E/kcyc) from being a cofinitely generated Zp-module.
16
Chapter 2
The CategoryMH(G)
2.1 Mazur Conjecture
Within this section, k always denotes an arbitrary number field, i.e a finite
extension of Q.
In this section, I will mainly state some results and conjectures regarding
the Λ(Γk)-rank of the compact profinite group Xp(E/kcyc), where p is a rational
prime and E is an elliptic curve defined over number field k.
Lemma 2.1.1. Nakayama [1, p.2,Theorem]
LetΛ be a compact topological ring with 1 and let I be a (left) ideal with In → 0
in Λ. For M any compact profinite (left) Λ-module, IM = M implies M = 0.
As a consequence of Nakayama lemma, if M/IM is a finitely generated Zp-
module, then M is a finitely generated Λ-module.
Lemma 2.1.2. [19, Theorem 4.5 (a)]
For any prime p, Xp(E/kcyc) is a finitely generated Λ(Γk)-module.
We shall see later that the nullity of the Λ(Γk)-rank of Xp(E/kcyc) depends
crucially on the reduction type of the primes of k above the prime p.
Theorem 2.1.1. Mazur Control Theorem [20, c.f Proposition 6.4]
For a fixed odd prime p, and a number field k, denote by Γn
def
= Γ
pn
k the open
17
subgroup of Γk = Gal(kcyc/k) of index pn, for each integer n ≥ 0, and let kn
denote the subfield of kcyc, fixed by Γn. Suppose E is an elliptic curve defined
over k, with good ordinary reduction at all primes of k lying above p, then the
restriction map
S elp(E/kn)
r¯kn−→ S elp(E/kcyc)Γn (2.1)
has finite kernel and cokernel, for n ≥ 0 and moreover, the kernels and coker-
nels are of bounded order when n varies.
Mazur conjectured the following statement [20]
Conjecture 2.1.1. (Mazur)
If p is a prime such that E/k has potentially good ordinary reduction or poten-
tially multiplicative reduction at all primes of k above p, then S elp(E/kcyc) is
Λ(Γk)-cotorsion.
The same argument fails for any potentially good supersingular reduction
places above p, due at prior to Konovalov [15] and later to Schneider.
Theorem 2.1.2. [24, Corollary 5]
corankΛ(Γk)S elp(E/k
cyc) ≥ r(E, k)
where
r(E, k) def=
∑
pss
[kv : Qp]
which the sum
∑
pss runs over each place v of k above p at where E has poten-
tially supersingular reduction.
Schneider conjectured that
Conjecture 2.1.2. The equality always holds in the theorem above.
Under our assumption on (E, p,m) in Section 1.1, we expect the validity
of another stronger conjecture, namely Conjecture 2.2.1 introduced in the fol-
lowing section, which will implies the validity of Mazur’s Conjecture for every
18
intermediate number field Q ⊆ k ⊂ F∞.
2.2 The categoryMH(G)
Assume from now on that we are working over the False Tate curve ex-
tension introduced in Section 1.1 and recall the notations G def= Gal(F∞/Q),
H def= Gal(F∞/Qcyc). We recall that we are assuming in particular that E has
multiplicative reduction at p. For any Λ(G)-module M, M(p) denotes its sub-
module consisting of all elements of p-power order.
Definition:. Let MH(G) denote the category of all finitely generated Λ(G)-
modules M, such that M/M(p) is finitely generated over Λ(H).
Throughout this paper, I will assume the validity of the following conjecture
made in [4]. .
Conjecture 2.2.1. Under the assumption made on (E, p,m) in Section 1.1, Xp(E/F∞)
belongs to the categoryMH(G).
We first derive some direct consequences from this conjecture. Write L an
arbitrary number field contained in F∞, and denote
HL
def
= Gal(F∞/Lcyc),
ΓL
def
= Gal(Lcyc/L),
Yp(E/F∞)
def
= Xp(E/F∞)/Xp(E/F∞)(p),
Yp(E/Lcyc)
def
= Xp(E/Lcyc)/Xp(E/Lcyc)(p).
Now, Xp(E/F∞) belongs toMH(G) means that Yp(E/F∞) is a finitely gen-
erated Λ(H)-module. Since [H : HL] is finite, Λ(H) is a finitely generated
Λ(HL)-module. Hence Yp(E/F∞) is also a finitely generated Λ(HL)-module.
Therefore
Yp(E/F∞)HL is finitely generated over Zp
19
Observing from the following commutative diagram with exact rows,
Xp(E/F∞)HL−→Yp(E/F∞)HL−→ 0
Xp(E/Lcyc)
?
−→ Yp(E/Lcyc)
?
−→ 0
(2.2)
the cokernel of the first vertical map is Pontryagin dual to the kernel of the
restriction map
rLcyc : S elp(E/Lcyc) −→ S elp(E/F∞)HL
which is a subgroup of
ker(rLcyc) ⊆ H1
(
HL, Ep∞(F∞)
) ∼= Ep∞(F∞)HL
where the isomorphism is due to the Poincare duality, whenever HL is pro-p,
i.e. when µp ⊂ L. Finally, we see its finiteness since Ep∞(F∞) is cofinitely
generated over Zp and Ep∞(F∞)HL = Ep∞(Lcyc) is finite by Ribet’s Theorem
[22, Theorem 1.1]. Hence, for any intermediate field L containing µp, both the
vertical maps in the commutative diagram (2.2) above have finite cokernels, and
Consequence 1: Yp(E/Lcyc) is finitely generated over Zp.
Consequence 2: Xp(E/Lcyc) is Λ(ΓL)-torsion.
In fact, these two consequences hold for any intermediate number field L ⊂
F∞. Indeed, they hold for L′
def
= L(µp). Write ∆L
def
= Gal(L′cyc/Lcyc) which is
a finite group of order coprime to p. Therefore the restriction map induces an
isomorphism
S elp(E/Lcyc) ∼= S elp(E/L′cyc)∆L
Yp(E/L′cyc) is finitely generated overZp and so is Yp(E/L′cyc)∆L and hence Yp(E/Lcyc).
Similarly, Xp(E/L′cyc) is Λ(ΓL′)-torsion, and so is Xp(E/L′cyc)∆L ∼= Xp(E/Lcyc)
with the same group action by ΓL and ΓL′ , so we obtain Consequence 2 for L.
The validity of Conjecture 2.2.1 is essential to define the Λ(HK)-rank of
Yp(E/F∞), which will play a large part in the rest of this thesis.
20
2.3 Computation of the Zp-coranks of some mod-
ules
In the rest of this chapter, we suppose the triple (E, p,m) satisfies the as-
sumption made in Section 1.1 and shall always assume the validity of Conjec-
ture 2.2.1. Let L denote any subfield of F∞ with [L : Q] < ∞. From the
consequences in section 2.2, we deduce that Xp(E/Lcyc) is Λ(ΓL)-torsion and its
λ-invariant is
λΛ(ΓL)(Xp(E/L
cyc)) def= dimQp(Xp(E/L
cyc)⊗Zp Qp).
In the rest of this chapter, for simplicity, we shall assume further that the
subfield L contains K def= Q(µp) as this implies that HL ∼= Zp. For instance when
L = K and L = Fn for n ≥ 1. The other cases for instance when L = Ln for
n ≥ 0 will be treated separately in Section 4.3 and we shall see then the results
are very identical.
For simplicity, we shall assume that S = S (K) = S ram ∪ S bad ∪ S p as given
at the end of Section 1.3. Now, I would want to rewrite the fundamental dia-
gram (1.25) by replacing S by S (Lcyc) in the direct sum. Recall that there are
only finitely many primes of Kcyc above each rational prime and [L : K] < ∞,
S (Lcyc) is again a finite set.
Let us denote for any place u of Lcyc,
Ju(F∞)
def
= lim
−→
[L′:Lcyc]<∞
L′⊂F∞
⊕
w′|u
H1
(
L′w′ , E(L′w′)
)
p∞ ,
where the limit is taken via restriction map, and
Ju(Lcyc)
def
= H1
(
Lcycu , E(L
cyc
u )
)
p∞ ,
21
we have
H2(HL, Ep∞(F∞))
0 - S elp(E/F∞)HL - H1(GS (F∞), Ep∞)HL
6
λ
HL
F∞- ⊕u∈S (Lcyc)Ju(F∞)HL
0 - S elp(E/Lcyc)
rLcyc
6
- H1(GS (Lcyc), Ep∞)
resLcyc
6
λLcyc- ⊕u∈S (Lcyc)Ju(Lcyc)
⊕u∈S (Lcyc)hu
6
H1(HL, Ep∞(F∞))
6
(2.3)
where the vertical upward sequence is the inflation-restriction exact sequence.
This doesn’t make any difference since Jv(Lcyc) = ⊕u|vJu(Lcyc), Jv(F∞) = ⊕u|vJu(F∞),
and hv = ⊕u|vhu, where u always denotes a place of Lcyc and v denotes a place of
K
We shall be interested at the difference between the “Zp-coranks” of the two
terms on the left and for this, we need to apply snake lemma. We may check the
surjectivity of λLcyc by the theorem below, which does not depend on the settings
and assumptions made at the beginning of this section.
Theorem 2.3.1. (Hachimori and Venjakob) [12, Theorem 7.2]
For E an elliptic curve defined over number field k and any odd prime p, assume
G = Gal(F/k) is a pro-p, p-adic Lie group with no p-torsion and E(F)p∞ is
finite. If Xp(E/F) is Λ(G)-torsion, then we have
1. H2(GS (F), Ep∞) = 0
2. H1(GS (F), Ep∞)
λF- ⊕v∈S Jv(F) is surjective
here S is a set of primes of k containing all the primes above p, all places at
where E has bad reduction and primes which are ramified in F/k.
22
Proof. By Poitou-Tate global duality, we have the exact sequence
0 −→ S elp(E/F) −→ H1(GS (F), Ep∞) λF−→
⊕
v∈S
Jv(F) −→ ̂Rp(E/F)
−→ H2(GS (F), Ep∞) −→ 0 (2.4)
where ∗̂ denotes the Pontryagin dual of ∗, and
Rp(E/F)
def
= lim←−−
n,M
S el(Epn/M)
where the inverse limit is taken with respect to corestriction maps (c.f for in-
stance [21, page 232] for the definition), and the maps induced by ”multiplica-
tion by p” maps Epn+1 −→ Epn .
Here,
S el(Epn/M)
def
= ker
(
H1(GS (M), Epn) −→
⊕
v∈S
Jv(M)
)
.
Since G def= Gal(F/k) is a pro-p infinite group and E(F)p∞ is finite, we have
lim←−
M
E(M)p∞ = 0
with M runs over the finite extensions of k, contained in F. Therefore,
0 −→ Rp(E/F) −→ lim←−
M
HomZp( ̂S elp(E/M),Zp)
follows from taking inverse limit with respect to corestriction maps from the
short exact sequence
0 −→ E(M)p∞ −→ lim←−n S el(Epn/M) −→ HomZp(
̂S elp(E/M),Zp) −→ 0
Further, we see that from the restriction map
resM : S elp(E/M) −→ S elp(E/F)Gal(F/M)
23
we have exact sequence
0→ lim←−
M
HomZp( ̂(ker(resM)),Zp)
→ lim←−
M
HomZp( ̂(S elp(E/M)),Zp)
→ lim←−
M
HomZp( ̂(S elp(E/F))Gal(F/M),Zp) (2.5)
and E(F)p∞ is finite again implies ker(resM) ⊂ H1(Gal(F/M), E(F)p∞) is finite
and hence the vanishing of the first term of this exact sequence (2.5), and thus
Rp(E/F) is a submodule of lim←−
M
HomZp( ̂(S elp(E/F))Gal(F/M),Zp)
of which the latter is isomorphic to
lim←−
M
HomΛ(G) ̂(S elp(E/F),Zp[G/Gal(F/M)]) ∼= HomΛ(G) ̂(S elp(E/F),Λ(G))
via
f 7→
(
x ∈ ̂S elp(E/F) 7→
∑
σ∈G/Gal(F/M)
f (σ−1x)σ ∈ Zp[G/Gal(F/M)]
)
and this last term HomΛ(G) ̂(S elp(E/F),Λ(G)) = 0, and hence Rp(E/F) = 0,
when Xp(E/F) is Λ(G)-torsion. Hence two statements of this theorem follow
from the exact sequence (2.4).
There are two other facts which hold regardless of the settings and validity
of Conjecture 2.2.1, and will be very useful for further computations.
Theorem 2.3.2. (Ribet)
For A an abelian variety defined over a number field k. Let k∞ be the com-
positum of k(µp∞) for all rational primes p. The torsion subgroup of A(k∞) is
finite.
Proof. See [22, Theorem 1.1].
24
Lemma 2.3.1. (Hachimori-Matsuno)
Let Kq be a finite extension ofQq, containing µp, with p 6= q are distinct primes,
p is odd. For any elliptic curve E defined over Kq, we have
1. If E has good reduction over Kcycq
def
= Kq(µp∞), then
E(Kcycq )p∞ ∼=
Ep∞ if E(Kq)p∞ 6= 0,0 if E(Kq)p∞ = 0.
2. If E has split multiplicative reduction over Kcycq , then
E has split multiplicative reduction over Kq and
E(Kcycq ) ∼= Kcycq ∗/gZ
as Gal(Kcycq /Kq)-modules for some g ∈ Kq, and
E(Kcycq )p∞ ∼=< µp∞ , g1/p
n
> /gZ
as Gal(Kcycq /Kq)-submodules, for some integer n ≥ 0 such that g1/pn ∈
Kcycq but g1/p
n+1 6∈ Kcycq .
3. If E has non-split multiplicative reduction over Kcycq , then
E(Kcycq )p∞ = 0.
Proof. See [11, Proposition 5.1]
Let us resume the settings and assumptions from the beginning of this sec-
tion from now on.
Corollary 2.3.1. Assuming Conjecture 2.2.1, we have
1. H2(GS (Lcyc), Ep∞) = 0
2. H1(GS (Lcyc), Ep∞)
λLcyc−→ ⊕u∈S (Lcyc)Ju(Lcyc) is surjective
Proof. This is merely a check of the validity of the criterions of the theorem
of Hachimori and Venjakob in the settings as given in section 1.1. The elliptic
25
curve E is defined over Q and hence over L. Also, ΓL
def
= Gal(Lcyc/L) ∼= Zp is
a pro-p, p-adic Lie group without p-torsion and E(Lcyc)p∞ is finite by Ribet’s
theorem. Lastly, Xp(E/Lcyc) is Λ(ΓL)-torsion by the validity of Conjecture 2.2.1.
Corollary 2.3.2. Assuming Conjecture 2.2.1, the co-finitely generated Λ(ΓL)-
module S elp(E/F∞)HL is cotorsion over Λ(ΓL) and
λΛ(ΓL)
(
Xp(E/F∞)HL
)
= λΛ(ΓL)
(
Xp(E/Lcyc)
)
+
∑
u∈S (Lcyc)
corankZpker(hu). (2.6)
Proof. By snake lemma, we have an exact sequence
0→ ker(rLcyc)→ H1(HL, Ep∞(F∞))→ ⊕u∈S (Lcyc)ker(hu)→ coker(rLcyc)→ H2(HL, Ep∞(F∞)) = 0.
The nullity on the right is due to HL ∼= Zp having p-cohomological dimen-
sion 1. Since Ep∞(F∞) is co-finitely generated over Zp, by Poincare duality,
H1(HL, Ep∞(F∞)) ∼= (Ep∞(F∞))HL has vanishingZp-corank since (Ep∞(F∞))HL =
Ep∞(Lcyc) is finite again by Ribet’s theorem. Hence we proved this corollary
by assuming the next proposition and the deeply ramified theorem later, which
show each ker(hu) has finite Zp-corank.
Let S good, S ns, S s denote the set of good, non-split multiplicative, split multi-
plicative reduction primes of E over K respectively. The notation S ∗(Lcyc) means
the set of places of Lcyc above S ∗, for ∗ = p, bad, good, s, ns, and ram. It is
clear to see that S ∗(Lcyc) is the set of ∗-reduction primes of E over Lcyc for ∗ =
bad, good, s or ns, because Gal(Lcyc/K) is a pro-p group and p ≥ 5, hence the
reduction type doesn’t change within this pro-p extension Lcyc/K.
Proposition 2.3.1. For u ∈ S (Lcyc)− S p(Lcyc),
1. If u 6∈ S ram(Lcyc), then
ker(hu) = 0.
26
2. If u ∈ S ram(Lcyc) ∩ S ns(Lcyc), then
ker(hu) = 0.
3. If u ∈ S ram(Lcyc) ∩ S s(Lcyc), then
ker(hu) ∼= Qp/Zp ⊕ Au,
where Au is some abelian group of finite order.
4. If u ∈ S ram(Lcyc) ∩ S good(Lcyc), then
(a) if E(Lcycu )p∞ = 0, then
ker(hu) = 0,
(b) if E(Lcycu )p∞ 6= 0, then
ker(hu) ∼= (Qp/Zp)2.
Proof. Let HL,u be the decomposition subgroup of HL at w, a place of F∞ above
u. For u ∈ S (Lcyc),
Ju(Lcyc)
hu−→ Ju(F∞)HL
has kernel
ker(hu) = H1(HL,u, E(F∞,w))p∞ .
When u 6∈ S p(Lcyc), by Lutz’s Theorem E(Lcycu )⊗Zp Qp/Zp = 0, hence
ker(hu) ∼= H1(HL,u, E(F∞,w)p∞)
by Kummer Theory.
• When u ∈ S (Lcyc)− S ram(Lcyc)− S p(Lcyc),
u is unramified over F∞/Lcyc
⇒ u splits completely over F∞/Lcyc
⇒ HL,u = 0
27
⇒ ker(hu) = 0
• When u ∈ S ram(Lcyc)− S p(Lcyc),
then w is the unique prime of F∞ above u. Look at the tower of fields
below:-
F∞
Ln′′
def
= LnLcyc
Lcyc Ln′
def
= LnL
L Ln = Q(m1/p
n
)
Q
Write qi the rational prime lying below w, this is a prime divisor of m
which is distinct from p. For each n ≥ 0, write u′′, u′ and uL the prime of
L′′n , L
′
n and L respectively, which lies below w. Since the triple (E, p,m)
satisfies the assumption made in Section 1.1, according to the final state-
ment in the assumption, E does not have additive reduction at u, u′′n , u
′
n
and uL.
In order to understand E(F∞,w)p∞ , we apply the theorem of Hachimori-
Matsuno, quoted as Lemma 2.3.1 above, with Kq = L′n,u′n for all n ≥ 0.
Noticing that since µp ⊂ L by assumption, we have µp ⊂ L′n,u′n and L′′n,u′′n =
L′n,u′n(µp∞)
28
– When E has non-split multiplicative reduction at u, then
E has non-split multiplicative reduction at u′′n and u
′
n since L
′′
n/L
cyc
and L′′n/L
′
n are pro-p extension and p is odd for all n ≥ 0. By part(3)
of Lemma 2.3.1, we see
E(L′′n,u′′n )p∞ = 0
for all n ≥ 0, and hence
E(F∞,w)p∞ = 0
and finally
ker(hu) = 0.
– When E has good reduction at u, then
E has good reduction at qi by semistability and the assumption that
E has no additive reduction at qi. Furthermore,
∗ if E(Lcycu )p∞ = 0, then
E(LuL)p∞ = 0.
Since uL is totally ramified in L′n for all n ≥ 0, L′n,u′n and LuL
have the same residue field, denoted by l. Since l has finite
characteristic coprime to p, write E˜ the reduction of E modulo
uL, we have
E(LuL)p∞ ∼= E˜(l)p∞ ∼= E(L′n,u′n)p∞
and hence for all n ≥ 0,
E(L′n,u′n)p∞ = 0
and by part(1) of Lemma 2.3.1,
E(L′′n,u′′n )p∞ = 0.
29
Therefore, we have again
E(F∞,w)p∞ = 0
and
ker(hu) = 0.
∗ if E(Lcycu )p∞ 6= 0, then
by part(1) of Lemma 2.3.1, we have
E(F∞,w)p∞ = E(L′′n,u′′n )p∞ = E(L
cyc
u )p∞ = Ep∞ ∼= (Qp/Zp)2
as HL,u = Gal(F∞,w/Lcycu )-modules with trivial action. There-
fore,
ker(hu) ∼= H1(HL,u, E(F∞,w)p∞) = Hom(HL,u, E(F∞,w)p∞) ∼= (Qp/Zp)2.
– When E has split multiplicative reduction at u, then
for all n ≥ 0, E has split multiplicative reduction at u′′n . By
E(L′′n,u′′n )p∞
∼= Qp/Zp ⊕ finite group
and since by Poincare duality,
H1(HL,u, E(F∞,w)p∞) ∼= (E(F∞,w)p∞)HL,u
the statement follows since
(E(F∞,w)p∞)HL,u = E(Lcycu )p∞ ∼= Qp/Zp ⊕ finite group
by part(2) of Lemma 2.3.1 again.
30
2.4 Deeply Ramified Theorem
In this section, we continue assuming the same settings as the previous sec-
tion. We have given the description of ker(hu) for all u ∈ S (Lcyc) − S p(Lcyc).
The main purpose of this section is to complete the description of ker(hu) when
u ∈ S p(Lcyc). Let us denote by p a place of Lcyc above p, by p˜ a place of F∞
above p and denote by HL,p the corresponding decomposition group.
Theorem 2.4.1. For any p ∈ S p(Lcyc), we have
corankZpker(hp) =
1 if E has split multiplicative reduction at p,0 if E has non-split multiplicative reduction at p.
(2.7)
This result is just a simple consequence of Coates-Greenberg [5, Proposition
4.3] and Greenberg [9, Section 3]. We recall the proof.
Let hp be the map
Jp(Lcyc) = H1(Lcycp , E)p∞
hp−→ H1(F∞,p˜, E)HLp∞ = Jp(F∞)HL . (2.8)
Using the local Kummer exact sequence,
0 −→ E(F)⊗Qp/Zp κF−→ H1(F, Ep∞) −→ H1(F, E)p∞ −→ 0
for any local field F, we can rewrite the map hp in (2.8) as
H1(Lcycp , Ep∞)/Im(κLcycp )
hp−→ (H1(F∞, p˜, Ep∞)/Im(κF∞,p˜))HL (2.9)
Throughout this section, we shall be dealing with the case where F is an alge-
braic extension of Qp, for the particular fixed odd prime p.
We shall use a very important theorem by Coates-Greenberg about deeply
ramified extensions to eventually show the main theorem of this section.
For an elliptic curve E of semistable reduction over local field F, and let Ê
31
be the formal group over OF attached to the minimal model of E over F. We
have an GF-invariant submodule
C
def
= Ê(MF)p∞ ⊆ Ep∞
whereMF denotes the maximal ideal of the ring of integers of a separable clo-
sure of the local field F. Hence, we obtain a short exact sequence ofGF-modules
0 −→ C −→ Ep∞ −→ D −→ 0
where D denotes Ep∞/C. This deduces a long exact sequence
· · · −→ H1(F,C) λF−→ H1(F, Ep∞) piF−→ H1(F,D) −→ · · · (2.10)
Theorem 2.4.2. (Coates-Greenberg)
Assume that E is defined over local field F, a finite extension of Qp. Then for
any deeply ramified extension F/F, we have
Im(κF) = Im(λF).
Proof. See [5, Proposition 4.3].
Proof of Theorem 2.4.1. Since F = F∞, Lcyc are deeply ramified extensions
over Qp, by the theorem of Coates-Greenberg, we have
Im(piLcycp )
hp−→ Im(piF∞,p˜)HL
which is actually the restricted map of dp which lives in the commutative dia-
32
gram below:-
· · · - H1(F∞,p˜,C)
λF∞,p˜- H1(F∞,p˜, Ep∞)
piF∞,p˜- H1(F∞, p˜,D)
· · · - H1(Lcycp ,C)
6
λ
Lcycp- H1(Lcycp , Ep∞)
6
pi
Lcycp- H1(Lcycp ,D)
dp
6
Hence, we have identified the following
ker(hp) ∼= Im(piLcycp ) ∩ ker(dp)
= ker(dp)
(2.11)
as piLcycp is surjective since GLcycp has p-cohomological dimension 1.
From the knowledge about Tate curves,
• when E has split multiplicative reduction at p, we have
C ∼= µp∞
as GQp-modules and
D ∼= Qp/Zp
as GQp-modules with trivial action. So
ker(dp) ∼= Hom(HL,p,Qp/Zp) ∼= Qp/Zp
as groups, and the theorem follows in this case.
33
• when E has non-split multiplicative reduction at p, we have
C ∼= µp∞ ⊗ φ
as GQp-modules and
D ∼= Qp/Zp ⊗ φ
as GQp-modules with φ the unramified non-trivial quadratic character of
GQp . Since F∞, p˜/Qp is totally ramified, we have
DGF∞,p˜ = 0
and hence
ker(dp) = 0.
2.5 Subconclusion
We continue to assume that the triple (E, p,m) satisfies the assumptions
made in Section 1.1 and that Yp(E/F∞) is finitely generated over Λ(H). Again,
write L to denote any subfield of F∞ which is a finite extension over K = Q(µp).
Recall that in this case, HL is always isomorphic to Zp. We shall apply the re-
sults here in later chapters for L = K and Fn for n ≥ 1.
Proposition 2.5.1. We have
rankΛ(HL)Yp(E/F∞) = λΛ(ΓL)
(
Xp(E/F∞)HL
)
(2.12)
Proof. Considering cdp(HL) = 1, Howson’s formula (1.8) gives
rankΛ(HL)Yp(E/F∞) = rankZpYp(E/F∞)HL − rankZpH1
(
HL,Yp(E/F∞)
)
(2.13)
34
On the other hand, we have an exact sequence
0→ H1(HL, Xp(E/F∞)(p))→ H1(HL, Xp(E/F∞))→ H1(HL,Yp(E/F∞))→
H0(HL, Xp(E/F∞)(p))→ H0(HL, Xp(E/F∞))→ H0(HL,Yp(E/F∞))→ 0.
Since Hi(HL, Xp(E/F∞)(p)) being p-primary for i = 0, 1, we conclude that
rankZpHi(HL,Yp(E/F∞)) = λΛ(ΓL)Hi(HL, Xp(E/F∞))
for i = 0, 1 and the proof is complete once we prove the following lemma.
Lemma 2.5.1. We have
H1
(
HL, S elp(E/F∞)
)
= 0 (2.14)
Proof. By definition, we have
0 −→ S elp(E/F∞) −→ H1(GS (F∞), Ep∞) λF∞−→
⊕
v∈S
Jv(F∞)
and hence we have our target module lies in the long exact sequence
0 −→ S elp(E/F∞)HL −→ H1(GS (F∞), Ep∞)HL ρF∞−→
(
Im(λF∞)
)HL
−→ H1(HL, S elp(E/F∞)) −→ H1(HL,H1(GS (F∞), Ep∞)) (2.15)
So, it suffices to show
1. H1(GS (F∞), Ep∞)HL
ρF∞−→ (ImλF∞)HL is surjective
2. H1(HL,H1(GS (F∞), Ep∞)) = 0
For 1, clearly,
Im(λF∞)
HL ⊆
(⊕
v∈S
Jv(F∞)
)HL
Corollary 2.3.1 says that λLcyc is surjective, also,
coker(⊕u∈S (Lcyc)hu) ⊆ ⊕w∈S (F∞)H2(HL,u, E(F∞,w))p∞ = 0
35
since the p-cohomological dimension of HL,u
cdp(HL,u) 6 1
by the commutativity of the fundamental diagram eq(2.3),
H1(GS (F∞), Ep∞)HL
λ
HL
F∞−→ (⊕v∈S Jv(F∞))HL
is surjective, hence so is ρF∞ .
For 2, using the Hochschild-Serre spectral sequence,
H2(GS (Lcyc), Ep∞) −→ H1(HL,H1(GS (F∞), Ep∞)) −→ H3(HL, Ep∞(F∞))
is exact. The first term H2(GS (Lcyc), Ep∞) = 0 due to Corollary 2.3.1. The last
termH3(HL, Ep∞(F∞)) = 0 since cdp(HL) = 1. Hence, H1(HL,H1(GS (F∞), Ep∞)) =
0.
We have
Proposition 2.5.2.
rankΛ(HL)Yp(E/F∞) = λΛ(ΓL)
( ̂S elp(E/Lcyc)) + ∑
u∈S (Lcyc)
corankZpker(hu). (2.16)
Proof. This is just the association of eq(2.6) and eq(2.12).
36
Chapter 3
Root Number Computations
3.1 Root Numbers
The aim of this section is to give the definition of root numbers, more par-
ticularly the root numbers of the representations attached to elliptic curves and
their twists by self-dual representations. I shall outline these definitions with the
acknowledgement of various facts without repeating the technical proofs, which
can mostly be found in Rohrlich’s paper [23] and Deligne’s paper [6].
Definition:. Let E be an elliptic curve defined over a number field K. We define
the global root number as
w(E) =
∏
v
w(E/Kv),
where w(E/Kv) is the local root number and the product runs through all places
v of K. If ρ is a self-dual representation of Gal(K¯/K), we define the twisted
global root numbers similarly as
w(E, ρ) =
∏
v
w(E/Kv, ρv),
where ρv denote the restricted representation of ρ to a pre-fixed decomposition
subgroup Gal(K¯v/Kv) for each place v of K.
Hence, it is down to defining the local root numbersw(E/Kv) andw(E/Kv, ρv)
for all places v of K. Let F denote the local field Kv. By abuse of notation, we
still write the twisted local root number as w(E/F, ρ), without the subindex v
37
to ρ, with understanding that ρ means here its restriction to Gal(F¯/F), which
can be identified as a subgroup of Gal(K¯/K) by choosing a place v¯ of K¯ above
v. The choice of v¯ will determined the subgroup Gal(F¯/F) up to conjugation
and hence the isomorphic class of ρ is determined independent of the choice of
place. The definition of the local root numbers will be given separately in the
case where either F is Archimedean or non-Archimedean.
When F is non-Archimedean: Let F be a finite extension of Qp, and denote
by F¯ a algebraic closure of F. We denote by OF the ring of integers in F, and
piF a fixed local parameter of F. We shall also fix a Φ ∈ Gal(F¯/F) which acts
as the inverse of the Frobenius in the residue fields, and call Φ the geometric
Frobenius.
Definition: (Weil groups and Weil-Deligne groups). The Weil group of F, de-
noted by WF
def
= W(F¯/F) is the subgroup of Gal(F¯/F) generated by the inertia
subgroup I
def
= IF and the geometric Frobenius Φ. The Weil-Deligne group of F,
denoted by W ′F , is defined to be the group:
W ′F
def
= CoWF
where WF the Weil group of F acts on C by
gzg−1 = ωF(g)z, g ∈ WF , z ∈ C
where ωF : WF −→ C× is the unramified character, which sends Φ to the
reciprocal of the order of the residue field of F.
The Weil group WF is equipped with a topology satisfying:-
1. The subgroup IF is open in WF ,
2. The relative topology on IF coincides to the one from Gal(F¯/F),
3. Left multiplication by Φ is a homeomorphism.
The topology of the Weil-Deligne groupW ′F is regarded as the product topology
of the Cartesian product of WF and C.
38
Definition:. A representation ρ′ = (ρ,N) of W ′F is a continuous homomorphism
ρ′ : W ′F −→ GL(V)
where V is a finite dimensional vector space over C, such that ρ′ |C is analytic.
Here, the correspondent pair (ρ,N) is given by
(ρ,N) = (ρ′ |WF , (logρ′(z))/z)
for any z ∈ C×. Conversely, given any ρ a representation of WF on V, that is
a continuous homomorphism from WF to GL(V), and N any nilpotent endomor-
phism on V, satisfying
ρ(g)Nρ(g)−1 = ωF(g)N
for all g ∈ WF , then ρ′ can be recovered by
ρ′(gz) = ρ(g)exp(zN)
where z ∈ C.
In [23, section 4], Rohrlich explains a recipe to obtain C representations of
Weil-Deligne group from l-adic Galois representations. Let l be a rational prime
different from p and choose a nontrivial character tl : I −→ Ql. The recipe is
from the following:-
Proposition 3.1.1. [23, p.131] Let τl : Gal(F¯/F) −→ GL(Vl) be an l-adic
representation, where Vl is a finite dimensional vector space over Ql.
1. There is a unique nilpotent endomorphism Nl of Vl such that
τl(i) = exp(tl(i)Nl)
for i in some open subgroup of I and furthermore, τl(g)Nlτl(g)−1 = ωF(g)Nl
for all g ∈ WF .
2. The map σl : WF −→ GL(Vl) defined by sending g = Φmi with m ∈ Z
and i ∈ I to τl(g)exp(−tl(i)Nl), is a homomorphism which is trivial on an
open subgroup of I.
3. σl(g)Nlσl(g)−1 = ωF(g)Nl for all g ∈ WF .
39
4. Pick an embedding ι : Ql ↪→ C and consider the extension of scalars
of the vector space Vl via ι, we can associate Nl and σl from 1 and 2
above canonically to Nl,ι and σl,ι as endomorphism and representation
on Vl ⊗ι C respectively, and the isomorphic class of the representation
σ′l,ι = (σl,ι,Nl,ι) is independent on the choice of Φ and tl.
Definition:. Suppose that E is an elliptic curve defined over F, pick l a prime
different from p and an embedding ι : Ql ↪→ C. Let τE/F,l : Gal(F¯/F) −→
GL((Tl(E)⊗Ql)∗) be the contragredient of the natural Galois representation on
the Tate-module Tl(E)⊗Ql. Denote by σ′E/F,l,ι = (σE/F,l,ι,NE/K,l,ι) the associated
C-representation of W ′F defined by the recipe above. The isomorphic class of
σ′E/F,l,ι is independent of the choice of l and ι and hence we shall just denote by
σ′E/F as we will only be interested in the isomorphic classes of representations
from now on.
We shall define the -factor of any representation ρ′ = (ρ,N) of the Weil-
Deligne group W ′F , which at last will lead to the definition of root number. This
definition goes very deep into a theorem of Langlands [17] regarding the exis-
tence of a function .
For any non-trivial additive character ψ : F −→ C×, let nF(ψ) denote the
largest integer n such that ψ(pi−nF OF) = 1. For any quasicharacter χ : F
× −→
C×, denote aF(χ) as 0 if χ(O×F ) = 1 or otherwise the smallest integer m such that
χ(UF,m) = 1, where UF,m is the subgroup of units in F that are congruent to 1
modulo pimF . By fixing a Haar measure dxF on F, we can define
Definition: (Local -factors for quasicharacter χ). Let c denote any element of
F× with valuation equals to nF(ψ) + aF(χ). Then we define
(χ, ψ, dxF)
def
=

χ(c)
ωF (c)
× ∫
OF
dxF i f χ is unrami f ied∫
c−1O×F
χ−1(x)ψ(x)dxF i f χ is rami f ied.
(3.1)
We can identify any quasicharacter χ : F× −→ C× with a one dimensional
representation of the Weil group WF , since any of the latter must factor through
the abelian quotient WabF which is isomorphic to F
× via the Artin reciprocity
map.
40
We shall need a corollary of Brauer Induction Theorem and an existence
theorem due to Langlands and Deligne [6] to generalize the definition above to
arbitrary virtual C-representations of Weil group.
Proposition 3.1.2. [23, Corollary 2 of Section 2] Let ρ be a representation of WF
and denote by [ρ] its class in the Grothendieck group of virtual representations.
Then we have
[ρ] = (dimρ) · [1F] +
∑
(L,χ,χ′)
cL,χ,χ′[IndLF(χ− χ′)] (3.2)
where the sum runs over triples with L all finite extensions of F in F¯, χ and χ′ all
quasicharacters of L×, with almost all cL,χ,χ′ ∈ Z are zero. Here, the notation
[IndLF(χ − χ′)] denotes the class [IndLFχ] − [IndLFχ′] and 1F denotes the trivial
quasicharacter of WF .
Theorem 3.1.1. There exists a unique function (?,?, ?) satisfying the follow-
ing
i
(ρ2, ψ, dxF) = (ρ1, ψ, dxF) · (ρ3, ψ, dxF)
for any short exact sequence 0 → ρ1 → ρ2 → ρ3 → 0 of representations
of the Weil group WF .
ii For any finite extension L of F in F¯, and any Haar measure dxL of L,
(IndLFρ, ψ, dxF)
(ρ, ψ ◦ trL/F , dxL) =
(
(IndLF1L, ψ, dxF)
(1L, ψ ◦ trL/F , dxL)
)dimρ
for any virtual representation ρ of Weil group WL = WF∩Gal(F¯/L), where
1L denotes the trivial quasicharacter of WL and trL/F is the trace map from
L to F.
iii For any finite extension L of F in F¯, the -factor of any quasicharacter χ
of L×, (χ, ψL, dxL) is given by the formula in (3.1) with all the factors
considered as defined with respect to field L.
41
This theorem of Langlands in fact holds for all local fields, including the
Archimedean fieldsR andC. The only difference when dealing with Archimedean
fields is on the criterion iii, which appoints the values of the  for quasicharacter
differently. See below (3.7) and (3.9).
We can now explicitly give a formula for (ρ, ψ, dxF) as a result of this the-
orem.
Definition: (Local -factors for representations of Weil groups). Let ρ be as
given in the proposition above (3.2), we define its -factor as the one which
exists as described in the Theorem above. Namely,
(ρ, ψ, dxF)
def
= (1F , ψ, dxF)dimρ ·
∏
(L,χ,χ′)
(
(χ, ψ ◦ trL/F , dxL)
(χ′, ψ ◦ trL/F , dxL)
)cL,χ,χ′
(3.3)
where each term on the right hand side is given in (3.1).
Definition: (Local -factors for representations of Weil-Deligne group). The
local -factor of ρ′ = (ρ,N) is defined by
(ρ′, ψ, dx) def= (ρ, ψ, dx)δ(ρ′)
where
δ(ρ′) := det
(−Φ | V I/V IN)
with V IN = V
I ∩ ker(N).
A representation ρ′ of W ′F is called essentially symplectic if there exists a
ρ′ ⊗ ωkF-invariant bilinear non-degenerate symplectic form on V for certain real
number k.
The representation σ′E/F that defined earlier is essentially symplectic. For
the dual of the Weil-pairing, which is symplectic, isW ′F-invariant under the rep-
resentation σ′E/F ⊗ ω1/2F . [23, c.f section 16] .
42
Definition: (Local root numbers). The root number of ρ′ = (ρ,N) is defined as
wF(ρ′, ψ) =
(ρ′, ψ, dxF)
| (ρ′, ψ, dxF) | .
This value is independent on the choice of the Haar measure dxF on F, and
furthermore independent on the choice of the additive character ψ, if ρ′ is es-
sentially symplectic, and we shall simplify the notation by just wF(ρ′) and in this
case, it takes value ±1.
Definition: (Local root numbers for elliptic curves). The (local) root number
for an elliptic curve E defined over a non-Archimedean local field F is
w(E/F) def= wF(σ′E/F).
For any orthogonal C-representation ρ of Gal(F¯/F), we can also define the
twisted (local) root number, which is again independent of the choice of the
additive character ψ by
w(E/F, ρ) def= wF(σ′E/F ⊗ ρ).
When F is Archimedean: Let F be R or C here.
Definition: (Weil groups and Weil-Deligne groups). When F = C, we define its
Weil-Deligne group W ′C and its Weil group WC as
W ′C = WC
def
= C×;
when F = R, we define its Weil-Deligne group W ′R and its Weil group WR as
W ′R = WR
def
= C× ∪ JC×,
where J2 = −1, and JzJ−1 = z¯ for all z ∈ C×.
Definition:. When E is an elliptic curve defined over C, we define a represen-
tation
43
σ′E/C = σE/C : W
′
C −→ GL(C2)
z 7→
(
1/z 0
0 1/z¯
)
;
when E is an elliptic curve defined over R, we define a representation
σ′E/R = σE/R : W
′
R −→ GL(C2)
z 7→
(
1/z 0
0 1/z¯
)
Jz 7→
(
0 1/z
−1/z¯ 0
)
for z ∈ C×.
Definition:. Let
ψR,y(x) = e2piiyx, (3.4)
ψC,y(z) = e2pii·trC/R(yz) (3.5)
be additive characters onR andC respectively, with y ∈ R× and y ∈ C× respec-
tively, and let dR denote the Lebesgue measure on R and dC the twice Lebesgue
measure on C.
In contrast to the non-Archimedean case (3.1), we can define the -factor for
quasicharacter of WF where F is R or C. Firstly, it is easily seen that in both
cases, we have canonical isomorphisms F× ∼= WabF . Hence we can identify any
quasi-character of F× with an isomorphic class of one-dimensional representa-
tions of the Weil-group WF .
Definition: (Local -factors of Archimedean fields). When F = R, all the quasi-
characters χm,r of R× are parameterized by a pair (m, r) ∈ {0, 1} × C, given
as
χm,r(x)
def
= (sgnx)m|x|r (3.6)
and we define
(χm,r, ψR,y, dR)
def
= (i · sgny)m|y|r (3.7)
where y ∈ R× as appeared in (3.4).
44
When F = C, all the quasi-characters χι,m,r of C× are parameterized by a triple
(ι,m, r) ∈ {identity, complex conjugation}×N× C, given as
χι,m,r(z)
def
= ι(z)m|zz¯|r (3.8)
and we define
(χι,m,r, ψC,y, dC)
def
= (i · ι(y))m|yy¯|r (3.9)
where y ∈ C× as appeared in (3.5).
According to the remark after Theorem 3.1.1, when we replace criterion iii
by (3.7) and (3.9), we can define a unique function (?, ψF,y, dF) for every vir-
tual representation of the Weil-group (or equivalently the Weil-Deligne group)
WF , where F is R and C, in view of the corollary of the Brauer Theorem stated
earlier. Finally, we can define the local root numbers w(?, ψF,y) for every virtual
representation of W ′F exactly the same as the non-Archimedean case. Similarly,
when ρ′ is essentially symplectic, w(ρ′, ψF,y) takes value ±1 and is independent
of the choice of y ∈ F×. The Weil group WF is of no difference to the Weil-
Deligne group W ′F when F is Archimedean, and hence the notation ρ
′ = (ρ,N)
will always mean N = 0 and ρ′ = ρ in this case.
Proposition 3.1.3. We have
(i) w(E/C) def= w(σE/C) = −1
(ii) w(E/R) def= w(σE/R) = −1
(iii) w(E/C, ρ) def= w(σE/C ⊗ ρ) = (−1)dimρ
(iv) w(E/R, ρ) def= w(σE/R ⊗ ρ) = (−1)dimρ
for any elliptic curve E defined over C and any self-dual representation ρ of WC
and WR in (iii) and (iv) respectively.
We conclude this section by quoting a formula by V.Dokchitser, of which I
shall apply to compute certain root numbers in next chapter.
45
Theorem 3.1.2. (V.Dokchitser)
Let E be an elliptic curve defined over Q, and ρ an Artin representation which
is self-dual. Let S add and S multi be the set of rational primes at where E has
additive reduction and multiplicative reduction respectively. If ρ is unramified
at all places of S add, then
w(E, ρ) = w(E)dimρ·(−1)dimρ− ·
∏
p∈Smulti
sdimρ−dimρ
Ip
p ·det(Φp|ρIp)·
∏
p∈S add
det(Φp|ρ)Np(E)
where ρ− denotes the eigenspace of ρ(τ) of eigenvalue -1, where τ is the complex
conjugation, and the conductor of E has prime factorization
∏
p p
Np(E). HereΦp
is an geometric Frobenius element, Ip is the corresponding inertia subgroup and
sp =
−1 if E has split multiplicative reduction at p,1 if E has non-split multiplicative reduction at p.
Proof. See [8].
3.2 Computations of Root Numbers
Recall the tower of fields:-
Fn = Q(µpn ,m1/p
n
)
Kn = Q(µpn)
K0 = Q
and we denote Nn
def
= Gal(Fn/Kn), Gn
def
= Gal(Fn/Q) and Hn
def
= Gal(Kn/Q).
46
Using external semi-direct product
Gn = Nn oϕn Hn
where
ϕn : Hn → Aut(Nn)
h 7→(n 7→ nh · n · n−1h )
where nh ∈ Gn is a lifting of h.
Definition:. Let
ρχn
def
= IndKnQ χn,
where χn denotes any character of exact order pn of the Galois groupGal(Fn/Kn).
We will have deeper discussions on these Artin representations ρχn in Sec-
tion 3.4. We shall see, in Proposition 3.4.1 that ρχn are self dual and we shall
assume this fact in this section.
Throughout this section, we shall assume again that (E, p,m) satisfies the
assumptions made in Section 1.1. It is clear to see that the criteria in Theorem
3.1.2 are met for ρ = ρK
def
= IndKQ1 and ρχn , since they are unramified outside
p ·m and by the assumption, E has semistable reduction at all the prime divisors
of p ·m. We shall apply the formula in Theorem 3.1.2 in computing the quotient
between root numbers
w(E/K) = w(E, IndKQ1) and w(E, ρχn).
Let us first compute the terms appearing in the formula in Theorem 3.1.2 for
ρ = ρK and ρχn respectively.
Proposition 3.2.1. We have
1. dimρ−K =
1
2dimρK =
1
2 (p− 1)
2.
{
dimρIpK = 1,
dimρIqK = dimρK = p− 1, when q 6= p
47
3.
{
det(Φp|ρIpK ) = 1,
det(Φq|ρIqK ) = ( qp ), when q 6= p
where Ip and Iq denotes the inertia subgroups at primes p and q respectively of
the abelian extension K/Q.
Proof. Fix an embedding K ↪→ C and pick ξp, a fixed primitive pth root of unity.
Let gi ∈ Gal(K/Q) such that gi(ξp) = ξip with i a primitive root modulo p. Then
gi generates Gal(K/Q) and we denote gil
def
= (gi)l for any l modulo p− 1.
ρK = IndKQ1K acts on the vector space spanned by {gi, . . . . . . , gip−1}. Since
gil · gi j = gil+ j , the matrix of ρK(gil) under this basis is(
0 Ip−1−l
Il 0
)
where Ik denotes the k × k identity matrix.
In particular, since the image of complex conjugation in Gal(K/Q) takes ξp 7→
ξ−1p = ξ
p−1
p = (ξp)
i
p−1
2 hence it is g
i
p−1
2
, and the corresponding matrix is
(
0 I p−1
2
I p−1
2
0
)
which has zero trace. Therefore
dimρ+K = dimρ
−
K =
1
2
dimρK =
1
2
(p− 1)
which justifies part(1).
Since ρK factors through K, it can be viewed as a representation of Gal(K/Q).
Since p is totally ramified over K/Q, the inertia subgroup Ip = Gal(K/Q). Sup-
pose
p−1∑
k=1
ck · gik ∈ ρIpK
i.e
ρK(gil) · (
p−1∑
k=1
ck · gik) =
p−1∑
k=1
ck · gik
48
for every 1 ≤ l ≤ p− 1, i.e
(cp−l, . . . , cp−1, c1, c2, . . . , cp−1−l) = (c1, . . . , cp−1)
When l runs over 1, 2, 3, . . . , p− 1, we see this criterion is equivalent to
c1 = c2 = . . . = cp−1
i.e dimρIpK = 1. Since prime q 6= p is unramified over K/Q, the inertia subgroup
Iq = {id} ⊂ Gal(K/Q). Hence,
dimρIqK = dimρK = p− 1
that shows part(2).
For prime q 6= p, denote q−1 as the inverse of the residue of q modulo p. Since
Φq(ξp) = ξq
−1
p ≡ ξi−dp where q ≡ id mod p for some d modulo p− 1. The matrix
of ρK(Φq) is (
0 Id
Ip−1−d 0
)
by picking the least positive residue of d modulo p− 1.
This matrix has
determinant = (−1)parity of permutation of p−d,p−d+1,...,p−1,1,2,...,p−d−1
= (−1)(p−1−d)×d
= (−1)d
= (
q
p
).
The image of Φp in Gal(K/Q) is trivial, since p is totally ramified over K/Q
hence
det(Φp|ρIpK ) = 1
.
Recall that the notion m =
∏
qrii , p - ri.
49
Proposition 3.2.2. We have
1.
dimρ−χn =
1
2
dimρχn =
1
2
pn−1(p− 1)
2. • For prime p,
– if p | m, then
dimρIpχn = 0.
– if p - m, then
dimρIpχn =
{
1, for r ≥ n,
0, for n > r.
where r ≥ 0 be the integer such that pr+1 ‖ mp−1 − 1.
• For prime qi | m but qi 6= p,
dimρIqiχn = 0.
• For prime q - mp,
dimρIqχn = dimρχn = p
n−1(p− 1).
3.
det(Φq|ρIqχn) =
{
1, for prime q | mp,
( qp ), for prime q - mp.
where Ip and Iq denotes the inertia subgroups at primes p and q respectively of
the extension Fn/Q.
Proof. Fix an embedding Fn ↪→ C, and pick ξpn , a fixed primitive pn-th root
of unity. Let σi ∈ Hn, such that σi(ξpn) = ξipn with i a primitive root modulo
pn, then σi generates Hn, a cyclic group of order pn−1(p − 1), and we denote
σil
def
= (σi)l for any l modulo pn−1(p− 1).
There is a canonical isomorphism betweenGal(Fn/Kn) andZ/pnZ, nk ∈ Gal(Fn/Kn)
is correspondent to k modulo pn, via the relation nk( p
n√m) = ξkpn · pn
√
m.
50
The multiplication in Gn is carried out in the following fashion:-
(nk, σil)× (nk′ , σil′ ) = (nk × ϕn(σil)(nk′), σil × σil′ )
Suppose χn is a character of Gal(Fn/Kn) ∼= Z/pnZ of exact order pn, i.e χn is
fully determined by eχn
def
= χn(n1), a primitive element in µpn .
Denote gil
def
= (1, σil) ∈ Gn, ρχn acts on the vector space spanned by {gi, gi2 , . . . , gipn−1(p−1)}.
Since
(nk, σil)× (1, σi j) = (nk × ϕn(σil)(1), σil+ j)
= (nk, σil+ j)
(1, σil+ j)× (ϕn(σil+ j)−1(nk), 1) = (1× ϕn(σil+ j)(σil+ j)−1(nk)), σil+ j)
= (nk, σil+ j)
hence ρχn((nk, σil)) acts on the vector space C-linearly via sending
gi j 7→ χn(ϕn(σil+ j)−1(nk))× gil+ j
Computation of the dimension of subspaces of ρχn
• dimρ−χn:-
Since complex conjugation takes ξpn 7→ ξ−1pn = ξi
pn−1(p−1)
2
pn and fixes p
n√m ∈
R, hence the image of complex conjugation in Gn is correspondent to
(1, σ−1) = (1, σ
i
pn−1(p−1)
2
) and the correspondent matrix is
 0 I pn−1(p−1)2
I pn−1(p−1)
2
0

which has zero trace. Hence,
dimρ+χn = dimρ
−
χn
=
1
2
dimρχn =
1
2
pn−1(p− 1)
• dimρInertiaχn :-
51
– For prime q - mp,
q is unramified over Fn/Q, hence the inertia subgroup Iq = {id} and
dimρIqχn = dimρχn = p
n−1(p− 1)
– For prime qi | m but qi 6= p,
qi is unramified over Kn/Q, but totally ramified over Fn/Kn hence
Iqi ∼= Nn ⊂ Gn.
Let
pn−1(p−1)∑
j=1
c j · gi j ∈ ρIqiχn
i.e
ρχn((nk, 1))(
pn−1(p−1)∑
j=1
c j · gi j) = (
pn−1(p−1)∑
j=1
c j · gi j)
for all k modulo pn, i.e
c j × χn(σ˜i j−1 · nk · σ˜i j) = c j
for each j and all k modulo pn, where σ˜i j is any lifting in Gn of σi j .
For each j, since χn is of exact order pn, χn(σ˜i j
−1 · nk · σ˜i j) = 1 if and
only if σ˜i j
−1 · nk · σ˜i j = id which holds if and only if nk = id. Taking
k 6= 0 modulo pn implies that c j = 0 for each j, therefore
dimρIqiχn = 0.
– For prime p,
∗ if p | m,
then p is totally ramified over Fn/Q, and the inertia subgroup
Ip ∼= Gn.
Hence ρIpχn ⊆ ρIqiχn = 0, so
dimρIpχn = 0.
52
∗ if p - m, let r ≥ 0 be the largest integer such that pr+1 | mp−1−1,
then it is shown in [28, Theorem 5.2 and Theorem 5.5] that
· for n > r ≥ 0, there are pr distinct primes of Fn above p.
Each of these has residue degree 1 and ramification degree
pn−r × pn−1(p− 1) and the inertia subgroup Ip contains the
subgroup N prn .
· for r ≥ n ≥ 1, there are pn distinct primes of Fn above
p. Each of these has residue degree 1 and ramification de-
gree pn−1(p − 1). Let % be one of these primes, the corre-
sponding inertia subgroup Ip is just the decomposition sub-
group, which consists of pairs (nkl , σil), indexed by lmodulo
pn−1(p− 1), where nkl is the unique element in Nn such that
nkl · (σ˜il(%)) = %, where σ˜il = (1, σil) ∈ Gn.
Again let
pn−1(p−1)∑
j=1
c j · gi j ∈ ρIpχn
i.e
ρχn((nk, σil))(
pn−1(p−1)∑
j=1
c j · gi j) = (
pn−1(p−1)∑
j=1
c j · gi j)
for all (nk, σil) ∈ Ip.
In the case when n > r ≥ 0, since there exists an non-trivial
element nk ∈ N prn ⊂ Ip. For the same reason as before, we have
dimρIpχn = 0.
In the case when r ≥ n ≥ 1, (nk, σil) ∈ Ip if and only if nk = nkl .
Hence, for any l modulo pn−1(p− 1) we have
pn−1(p−1)∑
j=1
c j · χn(ϕn(σi j+l)−1nkl) · gi j+l =
pn−1(p−1)∑
j=1
c j · gi j .
I claim the equality
χn(ϕn(σi j+l)−1nkl) =
χn(ϕn(σi j+l)−1nk j+l)
χn(ϕn(σi j)−1nk j)
,
53
from which we deduce that
∑pn−1(p−1)
j=1 c j · gi j ∈ ρIpχn if and only
if
c j
χn(ϕn(σi j)−1nk j)
=
cl
χn(ϕn(σil)−1nkl)
for all j, l modulo pn−1(p− 1), and thus
dimρIpχn = 1.
The claim follows from the equality
ϕn(σi j+l)−1nkl · ϕn(σi j)−1nk j = ϕn(σi j+l)−1nk j+l
⇔ nklσ˜il · nk jσ˜i j · σ˜i j+l−1 = nklσ˜il · nk jσ˜il−1 = nk j+l
which clearly holds since nklσ˜il · nk jσ˜i j · σ˜i j+l−1 is an element in
Nn which maps σ˜i j+l(%) to %.
When q = p, since the residue degree over Fn is trivial, Φp is trivial and
det(Φp|ρIpχn) = 1.
When q = qi | m but qi 6= p, we have shown that ρIqiχn = 0 and hence
det(Φqi|ρIqiχn ) = 1.
How does Φq act on Fn, q - mp.
The aim here is to compute det(Φq|ρχn) which takes value ±1. Hence this is
equal to computing det(φq|ρχn) where φq is an arithmetic Frobenius element. As
φq : ξpn 7→ ξqpn = ξi
d
pn
for some d modulo pn−1(p− 1) such that q ≡ id mod pn, hence
φq = (nk, σid) ∈ Nn oϕn Hn = Gn
for some k modulo pn. Since
(nk, σid) · (1, σi j) = (nk, σid+ j) = (1, σid+ j) · (ϕn(σid+ j)−1nk, 1)
54
the matrix of ρχn(φq) under basis {gi, gi2 , . . . , gipn−1(p−1)} is
0BBBBBBBBBBBBBBBBBBBBBBB@
0 . . . . . . 0 χn(ϕn(σ
−1
id+1
)nk) 0 . . . . . . 0
0 . . . . . . . . . 0 χn(ϕn(σ
−1
id+2
)nk) 0 . . . 0
. . . . . . . . . . . . . . . . . .
. . . . . . . . .
0 . . . . . . . . . . . . . . . 0 χn(ϕn(σ
−1
ip
n−1(p−1)−1 )nk) 0
0 . . . . . . . . . . . . . . . . . . 0 χn(ϕn(σ
−1
ip
n−1(p−1) )nk)
χn(ϕn(σ
−1
i1
)nk) 0 . . . . . . . . . . . . . . . . . . 0
0 χn(ϕn(σ
−1
i2
)nk) 0 . . . . . . . . . . . . . . . 0
. . . . . .
. . . . . . . . . . . . . . . . . . . . .
0 . . . 0 χn(ϕn(σ
−1
id
)nk) 0 . . . . . . . . . 0
1CCCCCCCCCCCCCCCCCCCCCCCA
This matrix has
determinant = (−1)parity of permutation of d+1,d+2,...,pn−1(p−1),1,2,...,d ·
pn−1(p−1)∏
j=1
χn(ϕn(σ−1id+ j)nk)
= (−1)(pn−1(p−1)−d)×d · 1 (Applying the lemma below)
= (−1)d
= (
q
p
)
Lemma 3.2.1. Choose gil
def
= (1, σil) as a lifting of σil , we have g−1il · nk · gil =
ni−l·k mod pn . and further
pn−1(p−1)∏
j=1
χn(ϕn(σ−1id+ j)nk) = 1
.
Proof.
g−1il · nk · gil : p
n√
m
gil7−→ pn√m nk7−→ ξkpn · pn
√
m
g−1
il7−→ ξi−lkpn · pn
√
m
55
hence g−1il · nk · gil = ni−l·k mod pn . Therefore
pn−1(p−1)∏
j=1
χn(ϕn(σ−1id+ j)nk) = χn
pn−1(p−1)∏
j=1
g−1il · nk · gil

= χn
pn−1(p−1)∏
j=1
ni−l·k mod pn

= χn
(
nP(r,p)=1
r mod pn r·k
)
= χn(n0)
= 1
Theorem 3.2.1. Suppose the triple (E, p,m) satisfies the assumptions made in
Section 1.1. We have
w(E, ρχn) = w(E, ρK) ·
∏
qi 6=p∈Smulti
(
qi
p
)
· s(1−dimρ
Ip
χn )
p (3.10)
for n ≥ 1. Moreover, if we assume further that p ‖ mp−1 − 1 if (p,m) = 1, then
w(E, ρχn) = w(E, ρK) ·
∏
qi 6=p∈Smulti
(
qi
p
)
· sp (3.11)
for n ≥ 1.
Proof. Applying the formula of Theorem 3.1.2, by Proposition 3.2.1 and Propo-
56
sition 3.2.2, we have
w(E, ρχn)
w(E, ρK)
=
w(E)dimρχn
w(E)dimρK
· (−1)
dimρ−χn
(−1)dimρ−K ·
∏
q∈Smulti
sdimρχn−dimρ
Iq
χn
q
sdimρK−dimρ
Iq
K
q
· det(Φq|ρ
Iq
χn)
det(Φq|ρIqK )
·
∏
q∈S add
(
det(Φq|ρχn)
det(Φq|ρK)
)Nq(E)
=
∏
q 6=p∈Smulti
det(Φq|ρIqχn)
det(Φq|ρK) ·
sdimρ
Ip
χn
p
sp
· 1
det(Φp|ρIpK )
·
∏
q∈S add
(
det(Φq|ρχn)
det(Φq|ρK)
)Nq(E)
=
∏
qi 6=p∈Smulti
1
det(Φqi|ρK)
·
∏
q-mp∈Smulti
det(Φq|ρχn)
det(Φq|ρK) ·
s(dimρ
Ip
χn−1)
p
det(Φp|ρIpK )
·
∏
q∈S add
(
det(Φq|ρχn)
det(Φq|ρK)
)Nq(E)
=
∏
qi 6=p∈Smulti
(
qi
p
)
·
∏
q-mp∈Smulti
(
q
p
)
(
q
p
) · s(dimρIpχn−1)p · ∏
q∈S add

(
q
p
)
(
q
p
)
Nq(E)
=
∏
qi 6=p∈Smulti
(
qi
p
)
· s(1−dimρ
Ip
χn )
p .
3.3 Parity Conjecture
In this section, we continue from the previous section in assuming that the
triple (E, p,m) satisfies the assumption in Section 1.1.
Lemma 3.3.1. Let p be an odd prime and K def= Q(µp), for any rational prime
q 6= p, we have
(−1)#S q(K) =
(
q
p
)
where S q(K) is the set of primes of K above q.
Proof. Since K/Q is a cyclic extension of even order, namely p − 1, there is
a unique quadratic intermediate extension R = Q(
√
p∗), where p∗ = ±p s.t.
57
p∗ ≡ 1 mod 4.
q 6= p splits into even many primes over K/Q
⇔ The decomposition subgroup of q is of even index in Gal(K/Q)
⇔ The decomposition subgroup of q is a subgroup of Gal(K/R)
⇔ q splits in R
⇔
(
p∗
q
)
= 1
hence the lemma follows since
(
p∗
q
)
=
(
q
p
)
by Quadratic Reciprocity Law.
Proposition 3.3.1. Suppose the triple (E, p,m) satisfies the assumptions made
in Section 1.1. Write K∞
def
= Q(µp∞), then we have
(−1)#{S ram(K∞)∩S s(K∞)−S p(K∞)} =
∏
qi 6=p∈Smulti
(
qi
p
)
. (3.12)
where the sets S ram(K∞), S s(K∞) and S p(K∞) are as defined in the paragraph
preceding Proposition 2.3.1, in the case Lcyc = K∞.
Proof. Since K∞/K is a pro-p extension with p ≥ 5 an odd prime, the reduction
type is invariant over this extension and there are only odd many primes of K∞
above each prime of K. Hence
(−1)#{S ram(K∞)∩S s(K∞)−S p(K∞)} = (−1)#{S ram(K)∩S s(K)−S p(K)}
By assumption (E, p,m), none of the prime divisors qi of m is of additive reduc-
tion type for E, hence any qi ∈ S ram(K) ∩ S s(K) − S p(K) is lying above some
rational prime qi | m, qi 6= p, at where E/Q has split or non-split multiplicative
reduction. Therefore we have plainly the partition
S ram(K) ∩ S s(K)− S p(K) =
∐
qi 6=p∈Smulti
S qi(K) ∩ S s(K)
For each qi( 6= p) | m, qi ∈ S multi,
• If E has split multiplicative reduction at qi,
then E has split multiplicative reduction at all primes qi’s of K above qi
58
and hence S qi(K) ⊂ S s(K) and by the lemma above,
(−1)#S qi (K)∩S s(K) = (−1)#S qi (K) =
(
qi
p
)
• If E has non-split multiplicative reduction at qi, then
– if qi is inert over R = Q(
√
p∗), we have(
p∗
qi
)
=
(
qi
p
)
= −1
On the other hand, E attains split multiplicative reduction at the
prime of R above qi and hence S qi(K) ⊂ S s(K). Thus,
(−1)#S qi (K)∩S s(K) = (−1)#S qi (K) =
(
qi
p
)
= −1
– if qi splits over R = Q(
√
p∗),
we have (
p∗
qi
)
=
(
qi
p
)
= 1
and also, #S qi(K) is even and so is S qi(K) ∩ S s(K) and hence
(−1)#S qi (K)∩S s(K) = 1
We see the proposition follows by multiplying through all such qi’s.
Theorem 3.3.1. Under the assumptions of the triple (E, p,m) made in Section
1.1, assume further that p ‖ mp−1 − 1 if E has split multiplicative reduction at
p and (p,m) = 1. Assuming the validity of Conjecture 2.2.1, we have
(−1)rankΛ(HK )Yp(E/F∞) = w(E, ρχn)
for all ρχn as defined in Section 3.2.
We shall apply two theorems below to k = K = Q(µp) to finally filling up
the gaps to the proof of Theorem 3.3.1.
59
Theorem 3.3.2. (Greenberg-Guo)[9, Proposition 3.10][10, Section 5]
Assume that E is an elliptic curve defined over a number field k and that S elp(E/kcyc)
is Λ(Γk)-cotorsion, where p is any odd prime, then
λΛ(Γk)
(
Xp(E/kcyc)
) ≡ corankZpS elp(E/k) mod 2 (3.13)
Proof. Let us use the notations introduced in Theorem 2.1.1. For brevity, we
denote λ def= λΛ(Γk)
(
Xp(E/kcyc)
)
; for each n ≥ 0, we denote S n def= S elp(E/kn),
its maximal divisible subgroup by Dn
def
= S elp(E/kn)div and Qn
def
= S n/Dn, which
is a finite abelian p-group. Since Cassels-Tate pairing on Qn is non-degenerate
and skew-symmetric, we can write Qn ∼= Mn ⊕ Mn for Mn a maximal isotropic
subgroup of Qn.
Since the restriction map
H1(kn, Ep∞)
rkn−→ H1(kcyc, Ep∞)
has kernel ker(rkn) = H
1(Γn, E(kcyc)p∞) ∼= E(kcyc)p∞/(γpn − 1)E(kcyc)p∞ which
is clearly of finite order, bounded by the order of E(kcyc)p∞ which is finite by
Ribet’s Theorem. Therefore we have an upper bound corankZpS n ≤ λ for all
n ≥ 1. Let λ′ def= max
n≥0
{corankZpS n}, then there exists some integer n0 ≥ 0
such that for all n ≥ n0, we have corankZpS n = λ′. Let Q∞ = S∞/D∞ =
lim
−→
Qn, where S∞
def
= S elp(E/kcyc) = lim−→ S n and D∞
def
= lim
−→
Dn. We have D∞ ∼=(
Qp/Zp
)λ′ and hence
corankZpQ∞ = λ− λ′.
We shall go on and prove that corankZpQ∞ is even and λ′ has the same parity as
corankZpS 0.
For any finite abelian p-group Q, write uniquely Q ∼= Z/pa(1)Z ⊕ Z/pa(2)Z ⊕
· · · ⊕ Z/pa(l)Z with a(1) ≥ a(2) ≥ · · · ≥ a(l) ≥ 0. We call this sequence
(a(1), a(2), · · · , a(l)) the index of Q and the largest r such that a(r) 6= 0 the number
of generators of Q. Since Q∞ has finite Zp-corank, all the finite subgroup in(Qn)
has bounded number of generators, say l. Using the canonical map Qn
in−→ Q∞,
we have in(Qn) ⊂ in+1(Qn+1) and hence for each 1 ≤ i ≤ l, b(i)n is non-decreasing
when n increases and hence either converges to infinity or to a finite bound,
where the sequence (b(1)n , b
(2)
n , · · · , b(l)n ) denotes the index of in(Qn). The Zp-
60
corank of Q∞ is just the number of
#{1 ≤ i ≤ l | b(i)n →∞}.
For n ≥ n0, since Dn −→ D∞ is surjective, we can lift each element in the
kernel of Qn −→ Q∞ to an element in the kernel of S n −→ S∞, hence we have
#
(
ker(Qn
in−→ Q∞)
)
≤ # (ker(S n −→ S∞)) ≤ #
(
ker(rkn)
)
which is finite, bounded independent on n ≥ n0 by |E(kcyc)p∞ |, and therefore
ker(in), and hence Qn have bounded numbers of generators. Without loss of
generality, by increasing the value l above, let us assume l is a bound for the
number of generators of Qn for all n ≥ n0. Suppose pN is an exponent of
ker(in) for all n ≥ n0 and N is an integer, and denote by (a(1)n , a(2)n , · · · , a(l)n ) and
(c(1)n , c
(2)
n , · · · , c(l)n ) the index of Qn and Qn/(Qn)pN respectively, then obviously,
c(i)n = max{a(i)n − N, 0} for all 1 ≤ i ≤ l. We have surjections Qn −→ in(Qn) and
in(Qn) −→ Qn/(Qn)pN since ker(in) ⊆ (Qn)pN for n ≥ n0, hence we have
a(i)n ≥ b(i)n ≥ c(i)n ≥ a(i)n − N
for all 1 ≤ i ≤ l and n ≥ n0. This implies the equality
#{1 ≤ i ≤ l | a(i)n →∞} = #{1 ≤ i ≤ l | b(i)n →∞},
since we have 0 ≤ a(i)n − b(i)n ≤ N for all the i’s and n’s above. Thus, we showed
that corankZpQ∞ is even since a(1)n = a
(2)
n ≥ a(3)n = a(4)n ≥ · · · ≥ a(2 j−1)n = a(2 j)n =
· · · ≥ a(l)n by the decomposition Qn ∼= Mn ⊕ Mn above.
On the other hand, the restriction map S 0 −→ SGal(kn/k0)n has finite kernel and
cokernel and hence so is the kernel and cokernel of D0 −→ DGal(kn/k0)n . Since
the degree of any non-trivial irreducible representation over Qp ofGal(kn/k0) ∼=
Z/pnZ is divisible by p− 1, we have
corankZpS 0 = corankZpD0
mod(p−1)≡ corankZpDn = corankZpS n
since p is odd, this implies that the Zp-corank of S elp(E/k) has the same parity
as that of S elp(E/kn) which the latter equals λ′ when n ≥ n0.
61
Theorem 3.3.3 (T,V. Dokchitser). [7, Theorem 1.2]
Let E be an elliptic curve defined over Q. For every abelian extension k/Q and
every prime p,
w(E/k) = (−1)corankZpS elp(E/k) (3.14)
Proof. See the proof of [7, Theorem 2.8].
Proof of Theorem 3.3.1
Proof. From Proposition 2.5.2, eq(2.16) for L = K, we have
(−1)rankΛ(HK )Yp(E/F∞) = (−1)λΛ(ΓK )(Xp(E/K∞)) · (−1)
P
u∈S (K∞) corankZpker(hu)
The former term of the right hand side, by the two theorems eq(3.13) and
eq(3.14) above, for k = K, is
(−1)λΛ(ΓK )(Xp(E/K∞)) = w(E/K) (3.15)
The latter term of the right hand side, by Theorem 2.4.1 eq(2.7), Proposition
2.3.1 and Proposition 3.3.1 eq(3.12), is
(−1)
P
u∈S (K∞) corankZpker(hu) = sp · (−1)#{S ram(K∞)∩S s(K∞)−S p(K∞)}
= sp ·
∏
qi 6=p∈Smulti
(
qi
p
)
.
(3.16)
Lastly, the product of eq(3.15) and eq(3.16) is given in Theorem 3.2.1 eq(3.11)
and hence proved
(−1)rankΛ(HK )Yp(E/F∞) = w(E, ρχn).
62
3.4 More on the representations ρχn
In [26, Proposition 25], Serre provides a recipe to constructively list out all
of the isomorphic classes of representations of any finite group of the form
G = N o H
with N an abelian normal subgroup ofG and H a subgroup ofG. The arguments
are subject to representations over C but in fact, they hold in general over any
algebraically closed field K, of characteristic 0, for instance Q¯p. I shall rephrase
the recipe here.
Firstly, any irreducible representation of N is one dimensional since N is abelian.
Hence, we can identify them as elements in Hom(N, Q¯×p ). Since N is normal in
G, there’s a G-action on it, namely
(gχ)(n) = χ(g−1ng) (3.17)
for g ∈ G, n ∈ N, χ ∈ Hom(N, Q¯×p ).
Choose a system of representatives for the H-orbits in Hom(N, Q¯×p ), denoted by
{χα}. Let Hα ≤ H denote the stabilizer subgroup of χα, then we can extend χα
to a representation χ˜α of Gα
def
= N o Hα by
χ˜α(nhα) = χα(n)
for n ∈ N, hα ∈ Hα. Indeed, plainly we have
χ˜α(nhαn′h′α) = χ˜α(nhαn
′hα−1hαh′α)
= χα(nhαn′hα−1)
= χα(n)χα(hαn′hα−1)
= χ˜α(nhα)(hα−1χα)(n′)
= χ˜α(nhα)χ˜α(n′h′α)
On the other hand, composing the canonical projection Gα → Hα with any
irreducible representation ηα of Hα, we obtain another irreducible representation
η˜α of Gα. Eventually, we obtain a representation ρ(χα,ηα) of G by induction:
ρ(χα,ηα)
def
= IndGGα(χ˜α ⊗ η˜α)
63
Lemma 3.4.1 (J-P. Serre). [26, Proposition 25]
ρ(χα,ηα) runs over all isomorphic classes of irreducible Q¯p-representations of
G when χα runs over the representatives of the H-orbits of Hom(N, Q¯×p ) and
ηα runs over all the isomorphic classes of irreducible representations of the
stabilizer subgroup Hα.
Proof. Since both χ˜α and η˜α are irreducible, their tensor product χ˜α ⊗ η˜α is
irreducible. Using Frobenius reciprocity, we have
< IndGGα(χ˜α ⊗ η˜α), IndGGα(χ˜α ⊗ η˜α) >G=< χ˜α ⊗ η˜α,ResGα IndGGα(χ˜α ⊗ η˜α) >Gα
where < V,W >G denotes the inner product of the characters of the representa-
tions V and W of G. Let Tα = {ti ∈ G} be a left transversal for Gα, G has a left
action on Tα, induced by left multiplication. ThisG-action inscribes the induced
module
IndGGα(χ˜α ⊗ η˜α) ∼= ⊕ti∈Tαti(χ˜α ⊗ η˜α). (3.18)
The direct summands ti(χ˜α ⊗ η˜α) on the right are not G-stable, but permuted
under the left action of G. Restricted to the action by Gα, we may rewrite the
above decomposition courser with regard to their Gα-orbits, we have
ResGα Ind
G
Gα(χ˜α ⊗ η˜α) ∼=
⊕
Gαti
(⊕t∈Gαtit(χ˜α ⊗ η˜α)).
whereGαti ⊂ Tα runs over allGα-orbits in Tα. Each of the summand⊕t∈Gαtit(χ˜α ⊗ η˜α)
is nowGα-stable and hence this is a decomposition of representations ofGα. For
each ti ∈ Tα, it is easily seen that its stabilizer subgroup in Gα is Gα ∩ tiGαti−1.
Hence it is obvious that ⊕t∈Gαtit(χ˜α ⊗ η˜α) is an induced representation from the
subgroup Gα ∩ tiGαti−1. More precisely,⊕
t∈Gαti
t(χ˜α ⊗ η˜α) =
⊕
s∈S α,ti
s(ti(χ˜α ⊗ η˜α))
= IndGαGα∩tiGαti−1(ti(χ˜α ⊗ η˜α))
where S α,ti is a left transversal of Gα ∩ tiGαti−1 in Gα. Applying Frobenius
reciprocity again, we obtain from the above that
< IndGGα(χ˜α ⊗ η˜α), IndGGα(χ˜α ⊗ η˜α) >G=
∑
Gαti
< ResGα∩tiGαti−1(χ˜α⊗η˜α), ti(χ˜α⊗η˜α) >Gα∩tiGαti−1
64
The Gα ∩ tiGαti−1-action on ti(χ˜α ⊗ η˜α) is by the following:
For gα ∈ Gα such that ti−1gαti ∈ Gα, gα = titi−1gαtiti−1 maps ti(χ˜α ⊗ η˜α) to
ti(ti−1gαtiχ˜α ⊗ ti−1gαtiη˜α). Hence if we let (χ˜α ⊗ η˜α)ti denote the vector space
(χ˜α⊗ η˜α), with gα above sending (χ˜α⊗ η˜α) to (ti−1gαtiχ˜α⊗ ti−1gαtiη˜α), then (χ˜α⊗
η˜α)ti is equipped as a Gα ∩ tiGαti−1-module which is Gα ∩ tiGαti−1-isomorphic
to ti(χ˜α ⊗ η˜α), given by left multiplication by ti .
Let t0 be the unique element in Tα that lies inGα. Clearly, t0Gαt0−1 = Gα and so
the summand with regard to the orbit Gαt0 is
< ResGα∩t0Gαt0−1(χ˜α ⊗ η˜α), t0(χ˜α ⊗ η˜α) >Gα∩t0Gαt0−1 =< χ˜α ⊗ η˜α, (χ˜α ⊗ η˜α)t0 >Gα
=< χ˜α ⊗ η˜α, χ˜α ⊗ η˜α >Gα= 1.
The last equality holds because χ˜α ⊗ η˜α is an irreducible representation of Gα,
and the second equality holds because (χ˜α⊗ η˜α)t0 isGα-isomorphic to (χ˜α⊗ η˜α)
since they share the same character.
For the summand with regard to other orbits, say Gαti with ti 6= t0, when re-
stricted further to N ≤ Gα ∩ tiGαti−1,
ResN(χ˜α ⊗ η˜α) ∼= χα⊕degηα
and since N is normal in G, ti−1Nti = N
ResN(χ˜α ⊗ η˜α)ti ∼= tiχα⊕degηα .
SinceGα is the stabilizer subgroup of χα inG, tiχα 6= χα for all ti 6= t0. Therefore,
their restrictions to N are disjoint implying they are disjoint, and
< ResGα∩tiGαti−1(χ˜α ⊗ η˜α), ti(χ˜α ⊗ η˜α) >Gα∩tiGαti−1= 0
for ti ≤ t0. As a result, < ρ(χα,ηα), ρ(χα,ηα) >G= 1, hence ρ(χα,ηα) is irreducible.
Look at the restriction of IndGGα(χ˜α ⊗ η˜α) ∼= ⊕ti∈Tαti(χ˜α ⊗ η˜α) to N, each of the
summands ti(χ˜α ⊗ η˜α) is isomorphic to (χ˜α ⊗ η˜α)ti as N-modules, since N CG.
Hence, ResNρ(χα,ηα) ∼= ⊕ti∈Tα(χ˜α⊗ η˜α)ti ∼= (⊕ti∈Tαtiχα)⊕degηα which only involves
the G-orbit of χα in Hom(N, Q¯×p ) and so it decides the representative χα. Once
this is decided, the subspace (χ˜α⊗ η˜α)t0 is identify as the N-submodule of ρ(χα,ηα)
which N acts via the character χα. As obtained earlier, (χ˜α ⊗ η˜α)t0 ∼= χ˜α ⊗ η˜α as
Gα = NHα-modules. Restricted to the Hα, this is simply ηα and hence distinct
65
pair of (χα, ηα) give non-isomorphic ρ(χα,ηα).
Lastly, to prove that these {ρ(χα,ηα)} give a complete set of irreducible Q¯p-representations
of G up to isomorphism, it is sufficient to check if the sum of the squares of the
degrees of these representations equals to |G|. Easily, we have∑
α
∑
ηα
(deg(ρ(χα,ηα)))
2 =
∑
α
∑
ηα
(|G/Gα|degηα)2
=
∑
α
|G/Gα|2
∑
ηα
(degηα)2
=
∑
α
|H/Hα|2|Hα|
= |H|
∑
α
|H/Hα|
= |H||Hom(N, Q¯×p )|
= |H||N|
= |G|
Now I would like to apply this argument to give a collective description of
the representations ρχn of the Galois group Gn = Gal(Fn/Q), defined in Section
3.2.
Allowing the abuse of notation, we denote again by ρχn the composition of
itself to the canonical projection G = Gal(F∞/Q) −→ Gn = Gal(Fn/Q).
Proposition 3.4.1. The irreducible self-dual Artin representations of G of de-
gree greater than 1 are all orthogonal. Namely, up to G-isomorphism, these
representations are precisely given by ρχn where χn is any character of exact
order pn of the Galois group Nn = Gal(Fn/Kn), with n runs over all positive
integers.
Lemma 3.4.2. Up to Gn-isomorphism, ρχn is the unique Q¯p-representation of
Gn which does not factor through Fn−1(µpn).
66
Proof of lemma. Identifying the galois groups Gal(Fn/Ln) and Gal(Kn/Q) by
their actions on µpn and still refer the former by Hn. From the argument above,
more precisely eq(3.17), we know that Gn = NnHn acts on Hom(Nn, Q¯×p ) from
the left, where Nn acts trivially and so most of the time, we just focus on the
Hn-action.
For 0 ≤ k ≤ n,
Orbk
def
= {characters of exact order pn−k} ⊆ Hom(Nn, Q¯×p )
forms a Hn-orbit. Since Hn is a finite cyclic group of order pn−1(p − 1), all
characters in Orbk share the same stabilizer subgroup Hp
n−k−1(p−1)
n , when k < n;
whereas Orbn = {trivial character} has stabilizer subgroup Hn
For k = 0, from the recipe in the lemma above, the stabilizer subgroup is trivial
1 ≤ Hn, and this leaves no other choice for the corresponding η0 but η0 = id, the
trivial representation of degree 1. Picking any χn ∈ Orb0, we have
ρχn = ρ(χn,η0).
This obviously does not factor through Fn−1(µpn). In fact, not even the Nn-
submodule χn of ρχn factors through Fn−1(µpn), because χn is of exact order pn
hence the it cannot be trivial over subgroup N pn−1n .
For 0 < k < n, we shall see that for any irreducible Q¯p-representation ηk of
the stabilizer subgroup Hpn−k−1(p−1)n ≤ Hn and any χ ∈ Orbk, the corresponding
induced representation ρ(χ,ηk) is factored through Fn−1(µpn). To see this, we can
choose Tk = {ti ∈ Hn} be a left transversal for Hpn−k−1(p−1)n . This could serve too
as a left transversal for subgroup NnHp
n−k−1(p−1)
n ≤ NnHn. In view of the decom-
position (3.18), each of the direct summand ti(χ˜ ⊗ η˜k) of ρ(χ,ηk) is Nn-invariant
and hence a subspace of representation of Nn. As seen before, the representation
ti(χ˜ ⊗ η˜k) is isomorphic to (χ˜ ⊗ η˜k)ti . Since Nn acts trivially on η˜k, this direct
summand, when viewed as a representation of Nn, is isomorphic to (tiχ)⊕deg ηk
by the notation in (3.17). With tiχ ∈ Orbk, (tiχ)(N pn−1n ) = (tiχ)pn−1(Nn) = 1 since
tiχ has exact order pn−k dividing pn−1
Lastly for k = n, χ = id ∈ Orbn. The corresponding induced representation
ρ(χ,ηn) is just η˜n which is trivial when restricted to the whole Nn, for any irre-
ducible representation ηn of Hn.
Lemma 3.4.3. Let charφ denote the character of a representation φ of subgroup
67
H ≤ G, then the character of IndGHφ is given by
charIndGHφ(g) =
t−1gt∈H∑
t∈T
charφ(t−1gt)
for any g ∈ G and any given set T of coset representatives of H in G.
Lemma 3.4.4. [26, Proposition 39]
Let charρ denote the character of an irreducible representation ρ of a finite
group G, then the value
1
|G|
∑
g∈G
charρ(g2) =

0, if ρ is not self-dual,
1, if ρ is self-dual and orthogonal,
−1, if ρ is self-dual and symplectic.
(3.19)
Proof. Proof omitted.
Proof of Proposition 3.4.1. Since F∞ is the direct limit of Fn, any Artin repre-
sentation of Gmust factor throughGn for certain n ≥ 1. Therefore, it is sufficient
to search among the irreducible representations of Gn for a fixed n ≥ 1.
As seen in the proof of Lemma 3.4.2, for k=n, ρ(id,ηn) = η˜n is of dimension 1.
For 0 ≤ k < n, I want to compute the summation
1
|Gn|
∑
g=nlhil′∈NnHn
charρ(θk ,ηk )(g
2) (3.20)
As before, nl ∈ Nn denotes the element which sends pn
√
m to ξlpn p
n√m and leaves
ξpn unchanged; hil′ ∈ Hn denotes the element which sends ξpn to ξil
′
pn and leaves
pn
√
m unchanged, where i ∈ Z is a chosen primitive root modulo pα for all α ≥ 1.
By the formula in Lemma 3.4.3, we can rewrite the sum in (3.20) as
1
p2n−1(p− 1)
∑
g=nlhil′
t−1g2t∈NnHp
n−k−1(p−1)
n∑
t∈Tk
θ˜k(t−1g2t)η˜k(t−1g2t)
68
where θk ∈ Orbk, ηk is an irreducible representation of Hpn−k−1(p−1)n , Tk is a left
transversal of Hpn−k−1(p−1)n in Hn, which can hence be treated as a left transversal
of NnHp
n−k−1(p−1)
n in NnHn. Since NnH
pn−k−1(p−1)
n is normal in NnHn,
{t ∈ Tk | t−1(nlhil′ )2t ∈ NnHp
n−k−1(p−1)
n } = {t ∈ Tk | (nlhil′ )2 ∈ NnHp
n−k−1(p−1)
n }
=
Tk when (hil′ )2 ∈ Hp
n−k−1(p−1)
n
∅ when (hil′ )2 6∈ Hpn−k−1(p−1)n
(3.21)
Simple computation gives hil′nlh
−1
il′ = nl·il′ . Hence, the summation above can
be further rewritten as
1
p2n−1(p− 1)
pn−k−1(p−1)|2l′∑
l′∈Z/pn−1(p−1)Z
∑
t∈Tk
∑
l∈Z/pnZ
tθk(nl+l·il′ )ηk(hi2l′ )
=
1
p2n−1(p− 1)
pn−k−1(p−1)|2l′∑
l′∈Z/pn−1(p−1)Z
ηk(hi2l′ )
∑
θ∈Orbk
∑
l∈Z/pnZ
θ(nl(1+il′ ))
=
1
p2n−1(p− 1)
∑
l′
ηk(hi2l′ )|Orbk||Z/pnZ|
=
1
pk
∑
l′
ηk(hi2l′ )
(3.22)
where the summation
∑
l′ runs over l
′ in the set
Lk
def
= {l′mod pn−1(p− 1) : pn−k | 1 + il′ and pn−k−1(p− 1) | 2l′}.
Since
pn−k | 1 + il′ ⇔ il′ ≡ −1mod pn−k
⇔ l′ ≡ pn−k−1(p− 1)/2mod pn−k−1(p− 1)
⇔ l′ ≡ pn−k−1(p− 1)/2 + b · pn−k−1(p− 1)mod pn−1(p− 1)
⇒ 2l′ ≡ (2b + 1)pn−k−1(p− 1)mod pn−1(p− 1)
⇒ pn−k−1(p− 1) | 2l′,
(3.23)
69
where b runs over integers 0 ≤ b ≤ pk − 1, the set Lk can be rephrased by
Lk = {l′ ≡ pn−k−1(p−1)/2+b·pn−k−1(p−1)mod pn−1(p−1) : 0 ≤ b ≤ pk−1}.
Noticing that when l′ runs over residues in Lk, hi2l′ runs over all the elements of
the cyclic subgroup Hpn−k−1(p−1)n of order p
k, the summation in the last line of
(3.22) is hence equal to 1 when ηk is the trivial representation of degree 1; and
equal to 0 otherwise.
In conclusion, the self-dual irreducible representations of Gn are ρχn−k ,id, where
χn−k is any character of Nn of exact order pn−k, for all 0 ≤ k < n. These are all
orthogonal in view of the Lemma 3.4.4.
I shall conclude this section with yet another proposition which will be
needed in a proof in next chapter.
Proposition 3.4.2. The fixed subspace
(
ρχk
)Gal(Fn/Ln) is one dimensional for n ≥
k ≥ 1.
Proof. Since Gal(Fn/Ln) is a cyclic group generated by σi as given in the proof
of Proposition 3.2.2, it is clear that
(
ρχk
)Gal(Fn/Ln) is just the fixed subspace of
ρχk by the generator σi. Recall again the notation σil from proof of the same
theorem, Gal(Fn/Ln) = {σil | l ∈ Z/(pn−1(p− 1))Z}.
Let Tk denote a left transversal of Gal(Fk/Q) for Gal(Fk/Kk), we can actually
pick a nice choice here with
Tk = Gal(Fk/Lk).
We can then write
ρχk
∼=
⊕
τ j∈Tk
τ j(χk).
When k = n, we have Tk = {τ j} = {σil}, so we can shuffle the order of {τ j} of
the decomposition above by {σil} and denote by an (pk−1(p − 1))-tuple {al} a
general vector in the vector space underlying ρχk , with each al a general vector
(and in this case a field element) of the underlying space σil(χk) (which is one-
dimensional, hence in this case it is just Q¯p). Notice that for each l modulo
70
pk−1(p− 1),
σi · σil = σil+1 ,
this shows that the action of σi via the representation ρχk is taking
(a1, a2, a3, · · · , apk−1(p−1)−1, apk−1(p−1)) 7→ (apk−1(p−1), a1, a2, · · · apk−1(p−1)−2, apk−1(p−1)−1).
Therefore the fixed vector space consists of the diagonal tuples
(a, a, a, · · · , a, a), ∀a ∈ Q¯p
hence
(
ρχk
)Gal(Fk/Lk) ≡ Q¯p.
When n > k, since the representation ρχk factors through Gal(Fk/Q), the action
of σi ∈ Gal(Fn/Ln) < Gal(Fn/Q) via this representation is the corresponding
action of its image in the canonical surjection
Gal(Fn/Q) −→ Gal(Fk/Q).
This shows that the image of σi can be determined by its Galois action on Fk,
or more precisely, on pk
√
m and on any primitive pk-th root of unity ξpk , and
in particular a particular choice ξp
n−k
pn . Since σi fixes p
n√m, it must fix pk√m =
( pn
√
m)pn−k ; and it takes ξpk to ξipk with i modulo p
k. Since i is a primitive root
modulo pn, it is also a primitive root modulo pk and hence the image of σi in
Gal(Fk/Q) is a generator of the cyclic subgroup Gal(Fk/Lk) which reduces to
the case when n = k.
71
Chapter 4
Homological Ranks and Rank
Growth
4.1 Homological Ranks
In this section, I am to establish some knowledge of the growth of the
Mordell-Weil ranks of E over the fields within the False Tate curve extension
tower. The methods are similar to those used in [3], but we deal with the case of
multiplicative reduction at p.
We assume throughout this chapter again the validity of Conjecture 2.2.1,
which claims that Xp(E/F∞) belongs to the category MH(G) for any triple
(E, p,m) satisfying the assumptions made in Section 1.1. This implies that
Yp(E/F∞) is a finitely generated Λ(H)-module, and hence is finitely generated
Λ(H∗)-module for any H∗ a subgroup of H of finite index. Moreover, for each
number field L ⊂ F∞, we have seen that the validity of Conjecture 2.2.1 implies
the validity of Mazur’s Conjecture over L.
Definition:. Put
τ
def
= rankΛ(HK )Yp(E/F∞)
sE/L
def
= corankZpS elp(E/L) = dimQ¯p
(
Xp(E/L)⊗Zp Q¯p
)
for any number field L ⊂ F∞.
72
Recall the notations Ln
def
= Q( pn
√
m) and L′n
def
= Q(µp, p
n√m).
Definition:. For n ≥ 0, let HLn def= Gal(F∞/Lcycn ), HL′n
def
= Gal(F∞/L′cycn ) its
maximal pro-p subgroup, and denote the λ-invariant by
λn
def
= λΛ(ΓLn )
(
Xp(E/Lcycn )
)
= rankZpYp(E/L
cyc
n ).
Let us once and for all, introduce the same notation ∆ for all of the cyclic
Galois groups of order p − 1 appearing in the tower, they can be identified by
their action on the group µp. For instance:
∆
def
= Gal(F∞/Lcyc∞ ) ∼= Gal(K∞/Qcyc) ∼= Gal(K/Q).
Meanwhile, denote by ∆ˆ def= Hom(∆,Z×p ) the dual of ∆, and by 1 the trivial ele-
ment of ∆ˆ.
Definition:. For a ring R, we can define K0(R), the Grothendieck group of R
being the abelian group with generators [P], one for each isomorphism class in
the category of finitely generated projective R-modules with relations
[P2] = [P1] + [P3]
if there is a short exact sequence of finitely generated projective R-modules
0 −→ P1 −→ P2 −→ P3 −→ 0.
Definition:. Let M be a finitely generated Λ(H)-module and
· · · −→ P j+1 −→ P j −→ P j−1 −→ · · · −→ P0 −→ M −→ 0 (4.1)
be a finite projective resolution of M, that is P j = 0 for j 0. We denote by
[M] def=
∑
i≥0
(−1)i[Pi] (4.2)
73
the formal alternating series, which is a finite sum.
Proposition 4.1.1. This well-defines an element [M] ∈ K0(Λ(H)).
Proof. Clearly, since Λ(H) has finite global homological dimension, such an
finite projective resolution eq(4.1) exists for the finitely generated module M.
We need to show that the right side of eq(4.2), as an element of K0(Λ(H)), is
independent on the choice of finite projective resolution of M. Take any finite
projective resolution of M:
· · · −→ P′j+1 −→ P′j −→ P′j−1 −→ · · · −→ P′0 −→ M −→ 0. (4.3)
We may assume P j = P′j = 0 for j  0. By long Schanuel’s lemma, [16,
Corollary (5.5)] since Pi, P′i are all projective, we have
P0 ⊕ P′1 ⊕ P2 ⊕ P′3 ⊕ P4 ⊕ · · · ∼= P′0 ⊕ P1 ⊕ P′2 ⊕ P3 ⊕ P′4 ⊕ · · ·
which yields the equality∑
i≥0
(−1)i[Pi] =
∑
i≥0
(−1)i[P′i]
in K0(Λ(H)).
Lemma 4.1.1. There is a canonical isomorphism
l∆ : K0
(
Λ(H)
) −→⊕
χ∈∆ˆ
Z (4.4)
where [Λ(H) ⊗Zp[∆] Zp(χ)] is correspondent to (0, · · · , 0, 1, 0, · · · , 0) where 1
appears in the χ-component.
Proof. This isomorphism is in fact the composition of the following isomor-
phisms:
Firstly, we have H ∼= HKo∆. Let J denote the kernel of the canonical surjection
Λ(H) −→ Zp[∆]. This surjection induces a group homomorphism
K0
(
Λ(H)
) −→ K0(Zp[∆])
74
which sends the isomorphic class of any finitely generated projective Λ(H)-
module M to the isomorphic class of the module M/JM. The injectivity of
this homomorphism is proved in [4, Lemma 3.5], using Nakayama’s lemma
by the fact that HK ∼= Zp is a pro-p open normal subgroup of H and hence
J is an ideal of Λ(H) with Jn → 0. The surjectivity of this group homo-
morphism is proved in [2, Chapter III Proposition 2.12] that we can write any
finitely generated projective Zp[∆]-module as Im(e∆) for some idempotent e∆ in
Mn(Zp[∆]), the endomorphism ring of Zp[∆]n for some integer n ≥ 1. In [2,
Chapter III Proposition 2.10], since Λ(H) is J-adically complete, e∆ has a idem-
potent lifting eH in Mn(Λ(H)), the endomorphism ring of Λ(H)n. So, we have
Im(eH)/J(Im(eH)) ∼= Im(e∆).
Secondly since the ring homomorphism
Zp[∆] −→
∏
χ∈∆ˆ
Zp
induced by sending
δ 7→ (χ(δ))χ∈∆ˆ
is an isomorphism too, (indeed, this Zp-modules homomorphism is given by a
Vandermonde matrix of determinant ±∏1≤<i< j≤p−1(α j − αi) ∈ Z×p for {αi} the
set of (p − 1)−th roots of unity in Z×p .) K0(Zp[∆]) is hence further isomorphic
to
⊕
χ∈∆ˆ K0(Zp) ∼=
⊕
χ∈∆ˆ Z. From this construction, we see that this homo-
morphism l∆ sends [M], for any finitely generated projective Λ(H) module, to
(
∑
i≥0(−1)irankZpHi(HK ,M)(χ))χ∈∆ˆ. The second argument is trivial by the con-
struction of the homomorphism.
Definition:. Let hLn be the group homomorphism
hLn : K0
(
Λ(H)
) −→ Z
defined by
[M] 7→
∑
i≥0
(−1)irankZpHi(HLn ,M)
for M any finitely generatedΛ(H)-module, and being extended to the Grothendieck
group linearly.
This term
∑
i≥0(−1)irankZpHi(HLn ,M) appears in [13, Section 2] where it is
75
denoted as χ∗(HLn ,M) or hmrankΛ(HLn )(M) which is called the homological rank
of M if this value is finite.
Let us check that this homomorphism hLn is well defined. Since HLn (and
H) has no non-trivial element of order p, Λ(HLn) (and Λ(H)) has finite p-
homological dimension. So the sum
∑
i≥0(−1)irankZpHi(HLn ,M) is a finite sum
for any finitely generated Λ(H)-module M. When M is a finitely generated pro-
jective Λ(H)-module, suppose we have the relation [M] = [M′] + [M′′], which
by the definition of K0(Λ(H)), is a short exact sequence of finitely generated
projective Λ(H)-modules
0 −→ M′ −→ M −→ M′′ −→ 0.
From this, we get a long exact sequence
· · · −→ Hm(HLn ,M′) −→ Hm(HLn ,M) −→ Hm(HLn ,M′′) −→
· · · −→ H1(HLn ,M′) −→ H1(HLn ,M) −→ H1(HLn ,M′′)
−→ H0(HLn ,M′) −→ H0(HLn ,M) −→ H0(HLn ,M′′) −→ 0
and Hm(HLn ,M′) = Hm(HLn ,M) = Hm(HLn ,M′′) = 0 for m 0. The alternating
sum of the Zp-ranks along this long exact sequence is zero, and this yields∑
i≥0
(−1)irankZpHi(HLn ,M) =
∑
i≥0
(−1)irankZpHi(HLn ,M′)+
∑
i≥0
(−1)irankZpHi(HLn ,M′′)
hence this hLn is well-defined in K0(Λ(H)). By Proposition 4.1.1, when M is
a finitely generated Λ(H)-module, having a finite projective resolution as in
eq(4.1), we can split up this resolution into short exact sequences and hence
obtain long exact sequences of the corresponding HLn homology groups. Gath-
ering these together, we have∑
i≥0
(−1)irankZpHi(HLn ,M) =
∑
j≥0
(−1) j
∑
i≥0
(−1)irankZpHi(HLn , P j).
On the other hand, since [M] =
∑
j≥0(−1) j[P j] we have also
hLn([M]) =
∑
j≥0
(−1) j
∑
i≥0
(−1)irankZpHi(HLn , P j).
76
Hence hLn is well-defined by in the definition above.
Now, although Yp(E/F∞) is not a projectiveΛ(H)-module, we can still com-
pute the value hLn([Yp(E/F∞)]) since Yp(E/F∞) is a finitely generated Λ(H)-
module by assuming Conjecture 2.2.1. In the next two sections, we shall carry
out this computation in two different approaches, leading to its relation to τ and
λn respectively.
4.2 Computation of Homological Ranks of Yp(E/F∞)
I
Proposition 4.2.1. For n ≥ 0, we have
hLn([Λ(H)⊗Zp[∆] Zp(χ)]) =

pn − 1
p− 1 , χ 6= 1, (4.5)
pn − 1
p− 1 + 1, χ = 1. (4.6)
Proof. Since Λ(H)⊗Zp[∆] Zp(χ) is a projective Λ(HLn)-module,
Hi(HLn ,Λ(H)⊗Zp[∆] Zp(χ)) = 0
for i ≥ 1. Hence, by the definition, it remains to compute the Zp-rank of
H0(HLn ,Λ(H)⊗Zp[∆] Zp(χ)) = Zp ⊗Λ(HLn ) Λ(H)⊗Zp[∆] Zp(χ) (4.7)
where Zp⊗Λ(HLn )Λ(H) clearly is a free Zp-module on generators the right cosets
of HLn in H, hence is endowed naturally with an action of ∆ from the right. To
compute the Zp-rank of the module in eq(4.7), we first partition Zp⊗Λ(HLn )Λ(H)
by its ∆-orbits.
Zp ⊗Λ(HLn ) Λ(H) ∼=
⊕
∆−orbits Z
RZ (4.8)
where RZ is a free Zp-module of rank the length of the orbit Z. In fact, apart
from the singleton orbit {HLn}, the rest of the ∆-orbits are faithful. Hence we
77
have
RZ ∼=
{
Zp(1), when Z = singleton {HLn}, (4.9)
Zp[∆], when Z 6= singleton {HLn}, (4.10)
and there are p
n−1
p−1 copies of Zp[∆] and one copy of Zp(1). Hence, the result
follows from the trivial facts that
Zp(1)⊗Zp[∆] Zp(χ) ∼=
{ Zp(1), when χ = 1, (4.11)
0, when χ 6= 1. (4.12)
and
Zp[∆]⊗Zp[∆] Zp(χ) ∼= Zp(χ) (4.13)
as Zp[∆]-modules.
Proposition 4.2.2. For n ≥ 0, we have
hLn([Yp(E/F∞)]) =
∑
i≥0
(−1)irankZp
(
Hi
(
HK ,Yp(E/F∞)
)(1))
+τ· p
n − 1
p− 1 . (4.14)
Proof. Let Y denote Yp(E/F∞). From the construction of the group isomor-
phism l∆ in the proof of Lemma 4.1.1, by linearity we have
l∆([Y]) =
(∑
i≥0
(−1)irankZpHi(HK ,Y)(χ)
)
χ∈∆ˆ
or equivalently,
[Y] =
∑
χ∈∆ˆ
(∑
i≥0
(−1)irankZpHi(HK ,Y)(χ)
)
[Λ(H)⊗Zp[∆] Zp(χ)] ∈ K0
(
Λ(H)
)
78
Applying the formula in Proposition 4.2.1, we get
hLn([Y]) =(
pn − 1
p− 1 ) ·
∑
χ6=1
(∑
i≥0
(−1)irankZpHi(HK ,Y)(χ)
)
+ (
pn − 1
p− 1 + 1) ·
(∑
i≥0
(−1)irankZpHi(HK ,Y)(1)
)
=(
pn − 1
p− 1 ) ·
∑
χ∈∆ˆ
(∑
i≥0
(−1)irankZpHi(HK ,Y)(χ)
)
+ 1 ·
(∑
i≥0
(−1)irankZpHi(HK ,Y)(1)
)
and the summation
∑
χ∈∆ˆ
(∑
i≥0
(−1)irankZp
(
Hi(HK ,Y)
)(χ))
=
∑
i≥0
(−1)i
∑
χ∈∆ˆ
rankZp
(
Hi(HK ,Y)
)(χ)
=
∑
i≥0
(−1)irankZp
⊕
χ∈∆ˆ
(
Hi(HK ,Y)
)(χ)
=
∑
i≥0
(−1)irankZp
(
Hi(HK ,Y)
)
and the right hand side is just theΛ(HK)-rank of Y by Howson’s formula eq(1.8).
Hence eq(4.14) follows.
4.3 Computation of Homological Ranks of Yp(E/F∞)
II
On the other hand, we can relate hLn([Yp(E/F∞)]) with the invariant λn. Let
S p denote the singleton {p}, S ram denote the set of primes {qi : qi | m}. Let
S ∗(F) denote the set of primes of F lying above S ∗, for any algebraic extension
F overQ and S ∗ = S p or S ram. For each integer n ≥ 0, we denote by wun a fixed
place of F∞ lying above un, which denote a place of Lcycn . We shall denote the
corresponding decomposition subgroup by HLn,un .
Let us introduce the following notation.
Definition:. For any odd prime p, positive integer m > 1 and integer n ≥ 0,
79
define an integer
βp,m,n
def
=
{
0, if p | m,
min{n, r}, if p - m and pr+1 ‖ mp−1 − 1.
We saw before that the prime p decomposes into pβp,m,n totally ramified primes
over Q(µpn , p
n√m). We say the pair (p,m) satisfies assumption ”β = 0” if either
p | m or p ‖ mp−1− 1. Clearly, p totally ramifies to a unique prime of F∞ when
assuming ”β = 0”.
Theorem 4.3.1. Suppose the triple (E, p,m) satisfies the assumption made in
Section 1.1 and assuming the validity of Conjecture 2.2.1. When E has split
multiplicative reduction at p, we further assume ”β = 0”. Then for n ≥ 0, we
have
hLn([Yp(E/F∞)]) = λn +
∑
un
rankZp
(
Tp(E)Jun
)
+ δp (4.15)
where the un runs over all places of S ram(Lcycn )−S p(Lcycn ) in the sum, Jun denotes
the absolute Galois group of Lcycn,un , and
δp
def
=
{
1, if E has split multiplicative reduction at p;
0, if E has non-split multiplicative reduction at p.
(4.16)
Applying the fundamental diagram (2.3) with L replaced by number field Ln
80
for n ≥ 0, we have
H2
(
HLn , Ep∞(F∞)
)
0 - S elp(E/F∞)HLn - H1
(
GS (F∞), Ep∞
)HLn
6
λ
HLn
F∞- ⊕un∈S (Lcycn )Jun(F∞)HLn
0 - S elp(E/Lcycn )
r
Lcycn
6
- H1
(
GS (Lcycn ), Ep∞
)
res
Lcycn
6
λ
Lcycn - ⊕un∈S (Lcycn )Jun(Lcycn )
⊕
un∈S (Lcycn )
hun
6
H1
(
HLn , Ep∞(F∞)
)
6
(4.17)
where S = S f (Q) = S p ∪ S bad ∪ S ram denotes finite set of rational primes con-
sist of p and all the other primes where the elliptic curve E has bad reduction
and all the prime divisors of m; and S (F) denotes the set of primes of F lying
above S , for any algebraic extension F over Q. Applying Theorem 2.3.1 again
to F/k = Lcycn /Ln, since Ribet’s theorem implies the finiteness of E(L
cyc
n )p∞ and
the validity ofMH(G) conjecture ensures that Xp(E/Lcycn ) is Λ(ΓLn)-torsion, we
deduce that H2
(
GS (Lcycn ), Ep∞
)
= 0 and λLcycn is surjective.
Lemma 4.3.1. For each n ≥ 0, H1(HLn , Ep∞(F∞)) has vanishing Zp-corank.
Proof. Let ∆ denote the Galois group Gal(L′cycn /L
cyc
n ), we have the following
inflation-restriction exact sequence
0 −→ H1(∆, Ep∞(L′cycn )) −→ H1(HLn , Ep∞(F∞)) −→ H1(HL′n , Ep∞(F∞))∆
(4.18)
By Ribet’s theorem again, Ep∞(L′cycn ) is finite, and thus H
1
(
∆, Ep∞(L′cycn )
)
is fi-
nite. On the other hand, by Poincare duality, H1
(
HL′n , Ep∞(F∞)
) ∼= Ep∞(F∞)HL′n ,
which has vanishing Zp-corank since Ep∞(F∞)HL′n = Ep∞(L′cycn ) is again finite.
Consequently, corankZpH1
(
HLn , Ep∞(F∞)
)
= 0.
We will later see in Lemma 4.3.3 that ker(hun) is a co-finitely generated Zp-
81
module, for each un ∈ S (Lcycn ).
Proposition 4.3.1. For each n ≥ 0, S elp(E/F∞)HLn is a finitely generated
Λ(ΓLn)-cotorsion module and the λ-invariant of its Pontryagin dual is
λΛ(ΓLn )
̂(S elp(E/F∞)HLn ) = λΛ(ΓLn ) ̂S elp(E/L
cyc
n ) +
∑
un∈S (Lcycn )
corankZpker(hun)
(4.19)
Proof. Since
HLn = HL′n o ∆
with ∆ a cyclic group of order p−1, which is coprime to p, HLn and HL′n have the
same p-cohomological dimension, which is equal to 1 since HL′n
∼= Zp. Hence,
H2
(
HLn , Ep∞(F∞)
)
vanishes. The rest of the proof is identical to the proof of
Corollary 2.3.2.
Similarly to Lemma 2.5.1, we have
Lemma 4.3.2.
H1
(
HLn , Xp(E/F∞)
)
= 0 (4.20)
Proof. Since the validity of Conjecture 2.2.1 and Ribet’s theorem enable one to
checks the surjectivity of λLcycn by Theorem 2.3.1. Together with cdp(HLn) = 1,
this is essentially the same proof of Lemma 2.5.1.
Proposition 4.3.2. For any n ≥ 0, we have
Hi
(
HLn ,Yp(E/F∞)
)
= 0, f or i ≥ 1; (4.21)
rankZpH0
(
HLn ,Yp(E/F∞)
)
= λΛ(ΓLn )H0
(
HLn , Xp(E/F∞)
)
. (4.22)
Proof. By assumption, Yp(E/F∞) is a finitely generatedΛ(H)-module and hence
a finitely generated Λ(HLn)-module. By Proposition 1.2.2, Hi
(
HLn ,Yp(E/F∞)
)
is a finitely generatedZp-module for each i ≥ 0. Since HLn has p-cohomological
dimension 1, plainly Hi
(
HLn ,Yp(E/F∞)
)
= 0 for i ≥ 2. To observe the case
82
when i = 1, take the HLn-homology of the canonical short exact sequence of
Λ(HLn)-modules
0→ Xp(E/F∞)(p)→ Xp(E/F∞)→ Yp(E/F∞)→ 0. (4.23)
It yields an exact sequence of Λ(ΓLn)-modules
0 = H1
(
HLn , Xp(E/F∞)
)→ H1(HLn ,Yp(E/F∞))→ H0(HLn , Xp(E/F∞)(p))
→ H0
(
HLn , Xp(E/F∞)
)→ H0(HLn ,Yp(E/F∞))→ 0. (4.24)
In fact, each term in this exact sequence is Λ(ΓLn)-torsion. Indeed, the valid-
ity of Conjecture 2.2.1 implies that S elp(E/Lcycn ) is a finitely generated Λ(ΓLn)-
cotorsion module. Together with the remark preceding of Proposition 4.3.1,
H0
(
HLn , Xp(E/F∞)
)
is Λ(ΓLn)-torsion, hence so is H0
(
HLn ,Yp(E/F∞)
)
. On the
other hand, Xp(E/F∞)(p) is annihilated by some power of p and hence the ho-
mology group Hi
(
HLn , Xp(E/F∞)(p)
)
will be annihilated by this power of p, for
each i ≥ 0. In particular, they are Λ(ΓLn)-torsion, with trivial λΛ(ΓLn )-invariants
and so is the submodule H1
(
HLn ,Yp(E/F∞)
)
. Moreover, since multiplying by
p, (and hence by any power of p) is injective in Yp(E/F∞), the induced multi-
plying by p in H1
(
HLn ,Yp(E/F∞)
)
is again injective, (so is the multiplying by a
power of p map). Hence, H1
(
HLn ,Yp(E/F∞)
)
= 0 since it injects into a module
which is annihilated by some power of p.
Since the λΛ(ΓLn )-invariant is additive in exact sequences and it coincides the Zp-
rank upon finitely generated Zp-modules, eq(4.22) follows from taking λΛ(ΓLn )-
invariant along the long exact sequence above.
Lemma 4.3.3. For n ≥ 0 and any non-Archimedean place un of S (Lcycn ), we
have
1. For un 6∈ S ram(Lcycn ) ∪ S p(Lcycn ),
ker(hun) = 0
2. For un ∈ S p(Lcycn ),
ker(hun) ∼=
0 un ∈ S ns(Lcycn )Qp/Zp un ∈ S s(Lcycn ) (4.25)
83
3. For un ∈ S ram(Lcycn )− S p(Lcycn ),
say un = uqi,n being a place lying above some rational prime qi 6= p
dividing m, we have
̂ker(huqi ,n)
∼= Tp(E)Juqi ,n (4.26)
and this latter module remain the same regardless of the choice of the
place uqi,n above qi and n ≥ 0. In particular, its Zp-rank is dependent
only on qi but not n nor the choice of uqi,n. Here, Juqi ,n denotes the absolute
Galois group of Lcycn,uqi ,n .
Proof. Trivially,
ker(hun) = H
1(HLn,un , E(F∞,wun ))p∞ .
As a result of Lutz’s theorem, this is isomorphic to H1(HLn,un , Ep∞(F∞,wun )) in
case 1 and 3. Since the extension F∞/Lcycn is unramified outside S ram(L
cyc
n ) ∪
S p(Lcycn ), the corresponding inertia subgroup for un 6∈ S ram(Lcycn ) ∪ S p(Lcycn ) is
trivial and hence the corresponding decomposition subgroup HLn,un is a subgroup
of ∆. Therefore
ker(hun) ∼= H1(HLn,un , Ep∞(F∞,wun )) = 0.
So proved 1.
For 3, we first notice that
H1(HLn,uqi ,n , Ep∞(F∞,wuqi ,n ))
∼= H1(Juqi ,n , Ep∞).
Indeed, the subgroup of Juqi ,n which fixes F∞,wuqi ,n has no quotient of order di-
visible by p since F∞,wuqi ,n contains the maximal tamely ramified p-extension of
Qqi as uqi,n is infinitely ramified over F∞,wuqi ,n . On the other hand, H
1(Juqi ,n , Ep∞)
is Pontryagin dual to
H0(Juqi ,n ,Hom(Ep∞ , µp∞))
∼= H0(Juqi ,n ,Tp(E))
since Hom(Ep∞ , µp∞) ∼= Tp(E) by Weil pairing. This proves eq(4.26) and hence
shows that ker(huqi ,n) has finite Zp-corank. Moreover, by the assumption of the
triple (E, p,m), E has semistable reduction over each u ∈ S ram − S p and hence
has semistable reduction over un ∈ S ram(Lcycn ) − S p(Lcycn ). When E has good
reduction at qi, E has good reduction at uqi,n for all n 6= 0. Since qi 6= p, the
84
action of Juqi ,n on Tp(E) is unramified for every n ≥ 0 and thus this action factors
through its quotient by the inertia, which are all isomorphic among the integers
n ≥ 0 since {Lcycn }n≥0 is a totally ramified tower of extensions for uqi,0. Hence
Tp(E)Juqi ,n = Tp(E)
Juqi ,0
for all n ≥ 0. When E has multiplicative reduction at qi, E has multiplicative
reduction at uqi,n for all n 6= 0, as a result of Tate curve, we see that Tp(E)Iuqi ,n is
independent on n ≥ 0 and
rankZpTp(E)
Iuqi ,n = 1,
where Iuqi ,n denotes the inertia subgroup of Juqi ,n for each n ≥ 0. Now, since Juqi ,n
acts unramifiedly on the common Zp-ranked one module Tp(E)Iuqi ,n = Tp(E)Iuqi ,0
for each n ≥ 0, by the same argument before, {Lcycn }n≥0 being a totally ramified
tower of extensions for uqi,0 implies that Tp(E)
Juqi ,n has Zp-rank at most 1 which
is independent on n, and Case 3 is proved.
For Case 2, we need to apply the argument of theorem of deeply ramified again.
Let un = up,n denote a place in S p(Lcycn ). Obviously, both L
cyc
n,up,n and F∞,p˜ are
deeply ramified over Qp. Using the notations and arguments in Section 2.4,
Im(κF) = Im(λF) for both F = Lcycn,up,n and F∞,p˜, and hence the restriction map
hup,n can be rewritten as
Im(piLcycn,up,n )
hup,n−→ Im(piF∞,p˜)HK .
Since the absolute Galois groupGLcycn,up,n has p-cohomological dimension 1, piLcycn,up,n
is surjective and hence
ker(hup,n) ∼= H1(HLn,up,n ,DGF∞,p˜ )
with D as defined in Section 2.4.
• When up,n ∈ S s(Lcycn ),
since p is totally ramified over Lcycn , E has split multiplicative reduction at
p. Hence,
D ∼= Qp/Zp
85
as GQp-modules with trivial action. We have then
H1(HLn,up,n ,D
GF∞,p˜ ) ∼= Hom(HLn,up,n ,Qp/Zp) ∼= Qp/Zp
as groups, and hence proved the statement in this case.
• when up,n ∈ S ns(Lcycn ),
E has non-split multiplicative reduction at p. Hence,
D ∼= Qp/Zp ⊗ φ
as GQp-modules with φ the unramified non-trivial quadratic character of
GQp . Since F∞,p˜/Qp is totally ramified, we have
DGF∞,p˜ = 0
and hence
ker(hup,n) = 0.
Hence proved Case 2.
Proof of Theorem 4.3.1
Proof. Combining Proposition 4.3.1 and Proposition 4.3.2, immediately we get
hLn([Yp(E/F∞)]) = λΛ(ΓLn )
̂S elp(E/Lcycn ) +
∑
un∈S (Lcycn )
corankZpker(hun) (4.27)
By Lemma 4.3.3, we can rewrite the term
∑
un∈S (Lcycn ) corankZpker(hun) as
δp × #S p(Lcycn ) +
∑
un∈S ram(Lcycn )−S p(Lcycn )
rankZp
(
Tp(E)Jun
)
(4.28)
since each of the prime in S p(F∞) is totally ramified over p, each of the prime
in S p(Lcycn ) is totally ramified over p too, hence E has the same type of multi-
plicative reduction at all primes in S p(Lcycn ) as at p. Thus when δp = 0, eq(4.15)
follows. When δp = 1, by assumption ”β = 0”, we have #S p(Lcycn ) = 1 and
hence eq(4.15) follows.
86
From Proposition 4.2.2, noticing that there is a term∑
i≥0
(−1)irankZp
(
Hi
(
HK ,Yp(E/F∞)
))(1)
which is independent of n in the formula eq(4.14). We can easily get rid of this
term by taking the difference of the same equation eq(4.14) for for different n’s.
A similar phenomena actually happens to Theorem 4.3.1.
Lemma 4.3.4. For each rational prime qi 6= p dividing m, and any non-negative
integral pair (n, n′), we have∑
un|qi
rankZp
(
Tp(E)Jun
)
=
∑
un′ |qi
rankZp
(
Tp(E)Jun′
)
.
Proof. From the proof of Case 3 of Lemma 4.3.3, we have seen that rankZp
(
Tp(E)Jun
)
only depends on qi, not on n nor the choice of un. Hence, it suffices to show that
the number of primes over Lcycn above qi is again independent on n. Indeed, this
follows from the fact that qi is totally ramified within the tower {Ln}n≥0 of fields
and hence there is a unique prime above qi over each number field Ln, and each
of these splits into a same (finite) number of primes in its Zp-cyclotomic exten-
sion Lcycn .
4.4 Rank Growth in the False Tate Curve Exten-
sion
The property of having these highly complicated but n-independent terms
in both the expressions of hLn([Yp(E/F∞)]) essentially allows us to ’extinguish’
them and give a neat connection to link up the arithmetic invariants which at
prior seem unrelated.
From this point onward, we always assume the pair (p,m) satisfies hypothe-
sis ”β = 0” when E has split multiplicative reduction at p.
87
Proposition 4.4.1. For n ≥ 0, we have
λn+1 − λn = τpn (4.29)
Proof. From the discussion above, both sides of eq(4.29) equal to
hLn+1([Yp(E/F∞)])− hLn([Yp(E/F∞)])
by eq(4.15) and eq(4.14) respectively.
Corollary 4.4.1. We have
sE/L0 ≡ sE/L2 ≡ · · · ≡ sE/L2k ≡ · · · mod 2, (4.30)
and
sE/L1 ≡ sE/L3 ≡ · · · ≡ sE/L2k+1 ≡ · · · mod 2. (4.31)
Moreover, the values in eq(4.30) and eq(4.31) have the same parity if and only
if τ is even.
Proof. In view of Theorem 3.3.2 for k = Ln, we have
λn ≡ sE/Ln mod 2.
Hence, by taking mod 2 of eq(4.29), since p is odd, we obtain
sE/Ln+1 ≡ sE/Ln + τ mod 2,
and consequently
sE/Ln+2 ≡ sE/Ln mod 2.
Finally the statements follow from these.
Corollary 4.4.2. When τ is odd, we have
sE/L0 < sE/L1 < sE/L2 < · · · < sE/Ln−1 < sE/Ln < · · · (4.32)
88
and in particular, sE/Ln has a lower bound
sE/Ln ≥ sE/Q + n (4.33)
for n ≥ 1.
Proof. By Proposition 1.4.1 eq(1.28), the p-Selmer rank never decreases over
any field extensions. From the last statement of the corollary above, we see that
when τ is odd, the p-Selmer rank can never be unraised over the tower {Ln}n≥0
since
sE/Ln+1 6≡ sE/Ln mod 2
for all n ≥ 0. Therefore,
sE/Ln+1 ≥ sE/Ln + 1 (4.34)
for all n ≥ 0 and eq(4.32) follows. In particular, the lower bound given in
eq(4.33) is just the inductive lower bound of eq(4.34).
For any finite Galois extension L over Q, since by assumption, E is defined
overQ, Xp(E/L)⊗ZpQ¯p is a Q¯p-representation ofGal(L/Q), and more generally,
a Q¯p-representation ofGal(L/L′) for any intermediate number field L′ ⊂ L. This
Q¯p-representation is of finite dimension sE/L. Hence by Maschke’s theorem, this
representation is semisimple and thus we have
(
Xp(E/L)⊗Zp Q¯p
)Gal(L/L′)
= Xp(E/L′)⊗Zp Q¯p
and decomposition of representation
Xp(E/L)⊗Zp Q¯p ∼= Xp(E/L′)⊗Zp Q¯p ⊕ AL/L′
where AL/L′ is the complement of Xp(E/L′)⊗Zp Q¯p in Xp(E/L)⊗Zp Q¯p which can
also be characterized by being the largestGal(L/Q)-invariant sub-representation
of Xp(E/L) ⊗Zp Q¯p, that contains no non-zero sub-representation on which
Gal(L/L′) acts trivially.
Using these properties of semisimple Galois representations, one can de-
duce the growth of the Zp-Selmer coranks over certain tower of finite Galois ex-
tensions, according to the respective growth over a tower of non-Galois tower.
89
More specifically, given the strict growth of p-Selmer rank of E over Ln, we
shall obtain certain strict growth over the tower of the respective Galois closures
Fn which inductively provides a lower bound of the growth of p-Selmer rank
of E over Fn. Here Fn is just the composite field KnLn, and more generally, the
composite field KiL j is Galois over Q when i ≥ j.
Proposition 4.4.2. When τ is odd, we have
sE/Fn ≥ sE/KnLn−1 + pn−1(p− 1) ≥ sE/Fn−1 + pn−1(p− 1) (4.35)
for all n ≥ 1. In particular, sE/Fn has a lower bound
sE/Fn ≥ sE/K + pn − 1 (4.36)
for n ≥ 1.
Proof. For n ≥ 1, from the preceding discussion, we have a decomposition of
Gal(Fn/Q)-representation
Xp(E/Fn)⊗Zp Q¯p =
(
Xp(E/KnLn−1)⊗Zp Q¯p
)⊕ AFn/KnLn−1 .
We shall first show that when τ is odd,
AFn/KnLn−1 6= 0.
Assume the contrary, we have
Xp(E/Fn)⊗ZpQ¯p =
(
Xp(E/KnLn−1)⊗Zp Q¯p
) ∼= (Xp(E/Fn)⊗Zp Q¯p)Gal(Fn/KnLn−1) .
Note that Gal(Fn/Ln) ∼= Gal(KnLn−1/Ln−1), taking the Gal(Fn/Ln)-invariant
from the above, we get
Xp(E/Ln)⊗Zp Q¯p =
(
Xp(E/Fn)⊗Zp Q¯p
)Gal(Fn/Ln)
∼=
((
Xp(E/Fn)⊗Zp Q¯p
)Gal(Fn/KnLn−1))Gal(KnLn−1/Ln−1) .
=
(
Xp(E/Fn)⊗Zp Q¯p
)Gal(Fn/Ln−1) .
= Xp(E/Ln−1)⊗Zp Q¯p,
90
hence sE/Ln = sE/Ln−1 , which contradicts Corollary 4.4.2 eq(4.32) as τ is odd.
Secondly, we shall try to determine this non-zero Gal(Fn/Q)-representation
AFn/KnLn−1 . By definition, AFn/KnLn−1 contains no non-zeroGal(Fn/Q)-sub-representation
on whichGal(Fn/KnLn−1) acts trivially. By semisimplicity, AFn/KnLn−1 is a direct
sum of irreducibleGal(Fn/Q)-representations on which the subgroupGal(Fn/KnLn−1)
does not act trivially. We have established in Lemma 3.4.2 that there is a unique
irreducible Q¯p-representation ofGal(Fn/Q) which does not factor throughGal(KnLn−1/Q),
denoted by ρχn . Therefore, we have
AFn/KnLn−1 ∼= (ρχn)⊕kn
for some integer kn ≥ 1, as AFn/KnLn−1 6= 0. Thus, we have
sE/Fn = sE/KnLn−1 + kn · pn−1(p− 1)
≥ sE/KnLn−1 + pn−1(p− 1)
as ρχn is of pn−1(p− 1) dimensional. This proves the left inequality of eq(4.35).
The right inequality of eq(4.35) is just a trivial fact since Xp(E/Fn−1)⊗Zp Q¯p is
a Gal(KnLn−1/Q)-subrepresentation of Xp(E/KnLn−1)⊗Zp Q¯p.
Lastly, eq(4.36) is just the inductive consequence of eq(4.35), as
sE/Fn ≥ pn−1(p− 1) + sE/Fn−1
≥ pn−1(p− 1) + pn−2(p− 1) + sE/Fn−2
≥ · · · · · · · · · · · ·
≥ pn−1(p− 1) + pn−2(p− 1) + · · · + p1(p− 1) + sE/F1
≥ pn−1(p− 1) + pn−2(p− 1) + · · · + p1(p− 1) + (p− 1) + sE/K1
= pn − 1 + sE/K
(4.37)
Definition:. Let ρ be an irreducible Q¯p-Artin representation which factors through
Gal(k/Q) for k a finite Galois extension of Q. Let sE,ρ denote the number of
copies of ρ occurring in the representation Xp(E/k)⊗Zp Q¯p.
91
Theorem 4.4.1. Suppose the triple (E, p,m) satisfies the assumption made in
Section 1.1 and assuming the validity of Conjecture 2.2.1. When E has split
multiplicative reduction at p, we further assume ”β = 0”, then for all absolutely
irreducible self-dual Artin representations ρ of G = Gal(F∞/Q) with dimension
greater than 1, we have
w(E, ρ) = (−1)sE,ρ . (4.38)
Proof. By Lemma 3.4.1 and Proposition 3.4.1, we have ρ ∼= ρχn for some χn
given in Section 3.2. From the proof of Proposition 4.4.2, since up to isomor-
phism, ρχn is the only irreducible representation of Gal(Fn/Q) which does not
factor through Gal(KnLn−1/Q), we have
Xp(E/Fn)⊗Zp Q¯p =
(
Xp(E/KnLn−1)⊗Zp Q¯p
)⊕ (ρχn)sE,ρχn .
Taking theGal(Fn/Ln)-invariants, and compute their Q¯p-dimensions, using Propo-
sition 3.4.2, and Greenberg-Guo Theorem 3.3.2, we obtain
sE,ρχn = sE/Ln − sE/Ln−1
mod 2≡ λn − λn−1.
Since p is odd, by Proposition 4.4.1, we have
τ
mod 2≡ λn − λn−1.
Therefore from the proof of Theorem 3.3.1 eq(3.15) and eq(3.16), together with
Theorem 3.2.1 eq(3.10), we conclude that
(−1)sE,ρχn = (−1)τ = w(E, ρχn) · sdimρ
Ip
χn
p .
The statement follows immediately in the case when sp = 1, that is when E has
non-split multiplicative reduction at p. In the case when E has split multiplica-
tive reduction at p, the statement follows by the description of dimρIpχn in Case
2 of Proposition 3.2.2, since under the assumption ”β = 0”, dimρIpχn = 0 for all
n ≥ 1.
92
4.5 Λ(HK)-rank 1 case
In this section, we carry on assuming ”β = 0” when E has split multiplica-
tive reduction at p.
In this very special case when rankΛ(HK )(Yp(E/F∞)) = 1, we shall prove that
the lower bounds in eq(4.33) and eq(4.36) are precisely the respective p-Selmer
ranks.
Theorem 4.5.1. When τ = 1, we have
sE/Ln = corankZpS elp(E/Q) + n (4.39)
and
sE/Fn = corankZpS elp(E/K) + p
n − 1 (4.40)
for n ≥ 1.
Proof. As the restriction homomorphism
S elp(E/K) −→ S elp(E/K∞)
has finite kernel, we have
sE/K = corankZpS elp(E/K) ≤ λΛ(ΓK )
(
̂S elp(E/K∞)
)
.
For the same reason,
sE/Fn = corankZpS elp(E/Fn) ≤ λΛ(ΓFn )
(
̂S elp(E/Fcycn )
)
.
By the theorem of Greenberg-Guo,
sE/K ≡ λΛ(ΓK )
(
̂S elp(E/K∞)
)
mod 2.
By Proposition 2.5.2 (for L = K and L = Fn),
τ = rankΛ(HK )(Yp(E/F∞)) = λΛ(ΓK )
( ̂S elp(E/K∞)) + ∑
u∈S (K∞)
corankZpker(hu),
93
rankΛ(HFn )(Yp(E/F∞)) = λΛ(ΓFn )
( ̂S elp(E/Fcycn )) + ∑
vn∈S (Fcycn )
corankZpker(hvn).
The criterion τ = 1 forces λΛ(ΓK )
( ̂S elp(E/K∞)) = 0 or 1. Hence, we have
equality
sE/K = λΛ(ΓK )
(
̂S elp(E/K∞)
)
= 0 or 1.
This again forces ∑
u∈S (K∞)
corankZpker(hu) = 1− sE/K = 1 or 0.
I now claim that∑
u∈S (K∞)
corankZpker(hu) ≡
∑
vn∈S (Fcycn )
corankZpker(hvn) mod 2.
Indeed, since Fcycn /K∞ is a Galois extension of degree p
n where p is an odd
prime, each u ∈ S (K∞) will only split into an odd number of primes in S (Fcycn ).
Moreover, the corresponding ramification degree and residue degree are both a
power of p. With p ≥ 5, the reduction type should remain unchanged in this
extension. By Proposition 2.3.1 and Theorem 2.4.1, both sums coincide in their
parity, and moreover∑
u∈S (K∞)
corankZpker(hu) ≤
∑
vn∈S (Fcycn )
corankZpker(hvn).
On the other hand, since [Λ(HK) : Λ(HFn)] = pn,
rankΛ(HFn )
(
Yp(E/F∞)
)
= pn · rankΛ(HK )
(
Yp(E/F∞)
)
= pn.
(4.41)
Hence,
sE/Fn ≤ λΛ(ΓFn )
(
̂S elp(E/Fcycn )
)
= pn −
∑
vn∈S (Fcycn )
corankZpker(hvn)
≤ pn −
∑
u∈S (K∞)
corankZpker(hu)
= pn − 1 + sE/K .
(4.42)
94
By eq(4.36), the right hand side is also the lower bound and hence we get the
equality
sE/Fn = λΛ(ΓFn )
(
̂S elp(E/Fcycn )
)
= pn − 1 + sE/K .
From the proof of Proposition 4.4.2, we see that when the lower bound eq(4.36)
is reached, every intermediate inequality in eq(4.37) is in fact equality. This
implies that the value kn = 1, that is AFn/KnLn−1 = ρχn for all n ≥ 1, and the
inequalities in eq(4.35) are equalities, except for the right inequality in n = 1
case. Hence, we have
Xp(E/Fn)⊗Zp Q¯p =
(
Xp(E/Fn−1)⊗Zp Q¯p
)⊕ ρχn (4.43)
for n ≥ 2 and
Xp(E/F1)⊗Zp Q¯p =
(
Xp(E/K)⊗Zp Q¯p
)⊕ ρχ1 . (4.44)
Thus,
Xp(E/Fn)⊗Zp Q¯p =
(
Xp(E/K)⊗Zp Q¯p
)⊕ ρχ1 ⊕ ρχ2 ⊕ · · · ⊕ ρχn
as a decomposition of Gal(Fn/Q)-representation. Taking the subspace fixed by
subgroup Gal(Fn/Ln), we get
Xp(E/Ln)⊗ZpQ¯p =
(
Xp(E/K)⊗Zp Q¯p
)Gal(Fn/Ln)⊕ρGal(Fn/Ln)χ1 ⊕ρGal(Fn/Ln)χ2 ⊕· · ·⊕ρGal(Fn/Ln)χn .
Since the canonical surjectionGal(Fn/Q) −→ Gal(K/Q) is still surjective when
restricted to subgroup Gal(Fn/Ln), we get(
Xp(E/K)⊗Zp Q¯p
)Gal(Fn/Ln)
=
(
Xp(E/K)⊗Zp Q¯p
)Gal(K/Q)
= Xp(E/Q)⊗Zp Q¯p.
Counting dimensions, by Proposition 3.4.2, we obtain eq(4.39).
Corollary 4.5.1. When τ = 1, we have
sE/Kn′Ln = sE/Fn = p
n − 1 + sE/K (4.45)
for all n′ ≥ n ≥ 1, and
sE/Kn′ = sE/K (4.46)
95
for n′ ≥ 1. In particular, in this case, we have the following refinements of
Greenberg-Guo:
λΛ(ΓK )
(
̂S elp(E/K∞)
)
= sE/K = 0 or 1 (4.47)
and
λΛ(ΓFn )
(
̂S elp(E/Fcycn )
)
= sE/Fn (4.48)
for n ≥ 1.
Proof. In the proof of the theorem above, we showed that the inequalities in
eq(4.35) are equalities, namely we have
Xp(E/K j+1L j)⊗ZpQ¯p = Xp(E/F j)⊗ZpQ¯p =
(
Xp(E/K j+1L j)⊗Zp Q¯p
)Gal(K j+1L j/K jL j)
(4.49)
for all j ≥ 1.
Since
Gal(K j+1L j/K j+1Ln) ∼= Gal(K jL j/K jLn)
for all j ≥ n, taking the Gal(K j+1L j/K j+1Ln)-invariant of eq(4.49), we get
Xp(E/K j+1Ln)⊗Zp Q¯p = Xp(E/K jLn)⊗Zp Q¯p. (4.50)
When j runs over n, n + 1, · · · , n′ − 1 we get
Xp(E/Kn′Ln)⊗Zp Q¯p = Xp(E/KnLn)⊗Zp Q¯p, (4.51)
hence proved eq(4.45).
SinceGal(Kn′L1/Kn′) ∼= Gal(KL1/K), eq(4.46) is just theGal(Kn′L1/Kn′)-invariant
of eq(4.51) in n = 1 case.
The refinements of Greenberg-Guo are true because both K and Fn contain µp
and hence
Fcycn =
⋃
n′≥n
Kn′Ln
and
K∞ = Kcyc =
⋃
n′≥1
Kn′ .
96
It is never easy to find the generators of the Mordell-Weil group of an elliptic
curve over a number field. However, in each of the following two examples of
our kind (E, p,m), I managed to obtain a rational point of infinite order over
the number field Q( p
√
m), by using the Pari-GP package ell.gp, by Denis Simon
[27]. The level of computation becomes much more complicated and massive
in the process to go on to number field Q( p2
√
m), over which we expect to reach
extra rank.
Example 1 The elliptic curve 70A1 in Cremona’s Table has Weierstrass equa-
tion
E : y2 + xy + y = x3 − x2 + 2x− 3 (4.52)
It has split multiplicative reduction at 2 and non-split multiplicative reduction at
5 and 7. This elliptic curve E has Mordell-Weil rank 0 over Q. Taking triple
(E, p,m) = (70A1, 5, 2) in our setting, then K := Q(µ5) and L1 := Q(θ) with
θ = 5
√
2.
P := (4θ4 − 2θ3 + θ2 + 2θ − 5,−3θ4 − 6θ3 + 17θ2 − 20θ + 17) (4.53)
is a rational point on E over L1, with height' 1.5505 hence is a point of infinite
order in E(L1). This indeed matches the prediction by our theorems, which tell
the Mordell-Weil rank of E over L1 is 1.
In fact L(E/K, 1) 6= 0 hence assuming Birch and Swinnerton-Dyer Conjecture,
E has Mordell-Weil rank over K, rE(K) = 0. Applying J.Jones’ formula [14,
Theorem 1], we know that (since 5 is the only prime ramified (totally) over
Kcyc/Q , hence e def= the number of split multiplicative reduction primes of K
which ramifies in Kcyc/K is equal to 0.) RE/K , the order of vanishing at T = 0 of
the characteristic polynomial of S el5(E/Kcyc) is ≥ rE(K). Moreover, up to mul-
tiplication by a 5-adic unit, the coefficient of the T rE(K) term in the characteristic
polynomial of S el5(E/Kcyc) is equal to
∏
v-∞
mv · |X(E/K)(p)||E(K)(p)|2 (4.54)
The product of the Tamagawa numbers
∏
v-∞mv = 8, the torsion part of
E(K) is isomorphic to Z/4Z and by assuming Birch and Swinnerton-Dyer Con-
jecture of the leading coefficient of the complex L-series of E over K at s = 1
97
in terms of the product involving the order ofX(E/K), we obtain conjecturally
thatX(E/K) is a trivial group. Hence, the product in (4.54) should be itself a
5-adic unit. Consequently, the coefficient of the T rE(K) term in the characteristic
polynomial of S el5(E/Kcyc) should be a 5-adic unit and this forces
µ(X5(E/Kcyc)) = 0 and λ(X5(E/Kcyc)) = rE(K) = 0.
Since the prime m = 2 is inert over Kcyc and E has split multiplicative reduction
at 2, by the third case of Proposition 2.3.1, together with eq(2.7) and Proposition
2.5.2 for L = K, we have
rankΛ(HK )Y5(E/F∞) = 1. (4.55)
So the prediction of the 5-Selmer rank of E over L1 by first assertion of Theorem
4.5.1 coincides the Mordell-Weil rank of E over L1, i.e sE/L1 = rE(L1) = 1. Hence
proved in this case that the 5-part of theX(E/L1) is finite.
Example 2 The elliptic curve 30A1 in Cremona’s Table has Weierstrass equa-
tion
E : y2 + xy + y = x3 + x + 23 (4.56)
It has split multiplicative reduction at 3 and non-split multiplicative reduction at
2 and 5. This elliptic curve E has Mordell-Weil rank 0 over Q. Taking triple
(E, p,m) = (30A1, 5, 3) in our setting, then K := Q(µ5) and L1 := Q(θ) with
θ = 5
√
3.
P := (2θ4 − 2θ3 + 2θ2 − 3, 4θ4 − 6θ2 + 12θ − 14) (4.57)
is a rational point on E over L1, with height' 0.5749 hence is a point of infinite
order in E(L1). This indeed matches the prediction by our theorems, which tell
the Mordell-Weil rank of E over L1 is 1.
In fact L(E/K, 1) 6= 0 hence assuming Birch and Swinnerton-Dyer Conjecture,
E has Mordell-Weil rank over K, rE(K) = 0. Applying J.Jones’ formula [14,
Theorem 1], we know that (since 5 is the only prime ramified (totally) over
Kcyc/Q , hence e def= the number of split multiplicative reduction primes of K
which ramifies in Kcyc/K is equal to 0.) RE/K , the order of vanishing at T = 0 of
the characteristic polynomial of S el5(E/Kcyc) is ≥ rE(K). Moreover, up to mul-
98
tiplication by a 5-adic unit, the coefficient of the T rE(K) term in the characteristic
polynomial of S el5(E/Kcyc) is equal to
∏
v-∞
mv · |X(E/K)(p)||E(K)(p)|2 (4.58)
The product of the Tamagawa numbers
∏
v-∞mv = 24, the torsion part of
E(K) is isomorphic toZ/12Z and by assuming Birch and Swinnerton-Dyer Con-
jecture of the leading coefficient of the complex L-series of E over K at s = 1
in terms of the product involving the order ofX(E/K), we obtain conjecturally
thatX(E/K) is a trivial group. Hence, the product in (4.58) should be itself a
5-adic unit. Consequently, the coefficient of the T rE(K) term in the characteristic
polynomial of S el5(E/Kcyc) should be a 5-adic unit and this forces
µ(X5(E/Kcyc)) = 0 and λ(X5(E/Kcyc)) = rE(K) = 0.
Since the prime m = 3 is inert over Kcyc and E has split multiplicative reduction
at 3, by the third case of Proposition 2.3.1, together with (2.7) and Proposition
2.5.2 for L = K, we have
rankΛ(HK )Y5(E/F∞) = 1. (4.59)
So the prediction of the 5-Selmer rank of E over L1 by first assertion of Theorem
4.5.1 coincides the Mordell-Weil rank of E over L1, i.e sE/L1 = rE(L1) = 1. Hence
proved in this case that the 5-part of theX(E/L1) is finite.
Remark:. I apply [14, Theorem 1] in the case L/K replaced by Kcyc/K. Hence,
e takes value 1 when E has a split multiplicative reduction at p and takes
value 0 when E has a non-split multiplicative reduction at p. Let r denote
the Mordell-Weil rank of E over K. Jones defines for each discrete Λ(ΓK)-
module with its Pontryagin dual a compact finitely generated Λ(ΓK)-torsion
module, an Iwasawa L-function L(G; s) def= FG(ω1−s − 1), for some ω ∈ Z×p ,
where FG(T ) denotes the characteristic element of the Pontryagin dual of G. Let
G2
def
= H1(spec(OKcyc), E0p∞) and G1
def
= H1(spec(OKcyc), Ep∞) be the f pq f coho-
mology groups, where we let E denote its Neron model over OKcyc , and by E0 its
connected component. This G1 is the flat Selmer group defined by Jones, which
99
he also proves to be quasi-isomorphic to Greenberg Selmer group and therefore
FG1(T ) = T
e × FG3(T ), where G3 def= S elp(E/Kcyc) is the classical Selmer group.
Jones shows that FG2(T ) = FG1(T ) and hence L(G2; s) = L(G1; s). This theorem
[14, Theorem 1] asserts that the Iwasawa L-function L(G2; s) has zero of order
at least r + e at s = 1, and the corresponding (r + e)-th coefficient is up to mul-
tiplication by a p-adic unit, given by a product involving some local invariants,
Schneider’s p-adic height regulator and the p-primary part of X(E/K) and
E(K). From the construction of the Iwasawa L-function, it is clear by simple
calculus that L(G3; s) has zero of order at least r at s = 1, and the correspond-
ing r-th coefficient is up to multiplication by a p-adic unit, given by the same
product given above. The computations in the two examples above are due to
this final statement.
Remark:. There are a lot more similar examples (E, p,m) which conjecturally
have rankΛ(HK )Yp(E/F∞) = 1. However, the two examples above are the only
ones among these of which ’ell.gp’ returns a rational point of infinite order over
Q( p
√
m).
In a forthcoming paper, I will further discuss the case left out by the re-
striction ”β = 0”, which is when E has split multiplicative reduction at p but
pr+1 ‖ mp−1−1 with a positive integer r. I will describe the growth of the Selmer
ranks within the False Tate curve tower in this case which would be slightly dif-
ferent from the relevant results in Chapter 4. However, I will prove the validity
of Theorem 4.4.1 with the assumption ”β = 0” dropped.
100
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