Iwasawa theory for elliptic curves over imaginary quadratic fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ASSIMO

B

ERTOLINI

Iwasawa theory for elliptic curves over imaginary

quadratic fields

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 1-25

<http://www.numdam.org/item?id=JTNB_2001__13_1_1_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Iwasawa

_theory

for

_elliptic

curves over

imaginary

quadratic

fields

par MASSIMO BERTOLINI

RÉSUMÉ. Soit E une courbe

_elliptique

sur

Q,

soit K un _corps

quadratique imaginaire,

et soit _K~ une

_Zp-extension

de K.

Étant

donné un ensemble 03A3 de

_places

de K contenant les

_places

au-dessus de _{p et} les

_places

de mauvaise réduction de

_E,

nous notons K03A3 l’extension maximale de K non ramifiée en-dehors de E. Cet

article est consacré à l’étude de la structure des groupes de

co-homologie

Hi (K03A3/K~,

Ep~)

pour i = _{1, 2, et} de la

composante

p-primaire du _groupe de Selmer

_{Selp~ (E/K~),}

considérés comme

modules discrets sur

_l’algèbre

d’Iwasawa de

K~/K.

ABSTRACT. Let E be an

_elliptic

curve over

Q,

let K be an

imag-inary

quadratic field,

and let K~ be a

_Zp-extension

of K. Given a set 03A3 of _primes of K,

_containing

the _primes above _p, and the

primes of bad reduction for _E, write K03A3 for the maximal

alge-braic extension of K which is unramified outside E. This _paper is devoted to the

_study

of the structure of the

_cohomology

_groups

Hi (K03A3/K~, Ep~)

for i = 1, 2, and of the p-primary Selmer

group

Selp~ (E/K~),

viewed as discrete modules over the Iwasawa

alge-bra of

_K~/K.

CONTENTS

Introduction 2

1. Selmer _groups 3

2.

_Special

values of L-series and Selmer groups 4

3. Descent Iwasawa modules 6

4. Finite submodules of

_{Selpoo(E/Koo)dual}

8

5. Growth numbers and

_Heegner

_points

10

6. A

_duality

theorem 14

7. On the structure of

_Selpoo

_{(E / Koo)dual}

20

References 24

Manuscrit regu le 28 octobre 1999.

Introduction

Let E be an

_elliptic

curve defined over a number field

_K,

and let

_K~

be

a

_Zp-extension

of

K,

where _p is a rational

_prime.

Given a set E of

_primes

of

_K,

_containing

the

_primes

above _p, the

_primes

of bad reduction for E and the Archimedean

_primes,

write

KE

for the maximal

_algebraic

extension of K which is unramified outside E.

The

_diophantine

_properties

of E with values in the

_layers

of

_I~~

can be

encoded in the Galois

_cohomology

_groups

_{HZ(K~/K~, Ep~ ), i}

=

_{1, 2}

and

in the

_p-primary

Selmer group

Selpoo(E/Koo)

of E over

Koo.

Denote

by

A

the Iwasawa

_algebra

attached to the extension

_K~/K,

_i.e.,

the

_completed

group

ring

7~p QrD ,

with r : =

Gal(K,,./K).

The above groups are

equipped

with a natural structure of discrete A-modules.

This _{paper is} devoted to the

_study

of the A-module structure of the above _groups, when E is an

elliptic

curve defined over the rationals and K is an

_imaginary

_quadratic

field. This

_setting

is

_particularly

_rich,

as

_elliptic

curves

over Q

_carry a modular

_structure,

and _moreover, when

Ago

is the

anticyclotomic

Zp-extension

of an

_imaginary

field

_K,

the

_points

of E over

the finite subextensions of

_~oo

have

_interesting

_growth

_properties,

due in certain cases to the _presence of

_systematic

families of

_points.

Let

Kn

denote the subfield of

~oo

having degree

over K

equal

to

_pn.

After

_reviewing

in section 1 the definition of Selmer _group, we state in section 2 two natural

_conjectures

on the ranks of the Mordell-Weil _groups

E(Kn).

These

_conjectures

are

_compatible

with the Birch and

Swinnerton-Dyer conjectures,

and with the

_expected

behaviour of the

_special

values of the

_complex

L-series of

_E/K

twisted

_by

finite order

_complex

characters of

Gal(Koo/ K).

Assuming

the above

_conjectures,

and

_using

_general

theorems

of

_Greenberg

_[8],

we show in section 3 a result on the structure of the A-module

_{Hi (K’E/ Koo, Epoo), i}

=

1, 2.

In section

5,

we assume that

Koo

is

the

_{anticyclotomic}

_Zp-extension

of

_K,

_p is a

good ordinary

_prime

for E and

there are

_Heegner

_points

defined over

_K,

_satisfying

a mild

non-vanishing

condition 1.

In this

_setting

the

_Pontryagin

dual

Xoo

of the Selmer _group

Selpoo(E/Koo)

has

_positive

_A-rank,

and we can _prove the above

conjec-tures. The

_proof

uses results of

(1),

which

provide

a

partial

proof

of a Main

Conjecture

of Iwasawa

_theory

formulated

_by

Perrin-Riou in

_[14].

We also consider the

_problem

of the existence of non-trivial finite A-submodules of

Xoo.

If

_Xoo

is a torsion

A-module,

this

problem

is studied

by

Greenberg

in

(8~ ;

we review some of his results in section 4. If the A-rank of

Xoo

is

pos-itive,

we

study

this

problem

in section

7,

working

in the

setting

of section

5,

and

_making

use of a

duality

theorem

proved

in section 6.

1 Note Added in Proof: This condition has recently been _{proved in many} cases _{by Christophe}

Acknowledgements.

We thank

_{Ralph Greenberg}

for

_interesting

conversa-tions on the

_topics

of this _paper.

1. Selmer groups

In this

_section,

let E be an

elliptic

curve defined over a number field K

and let _p be a rational

prime.

Define E and

Kr

as in the introduction.

The _group

_Epoo

of p-power torsion

points

of E is

naturally

a

Gal(K/K)-module. Given an extension L of K contained in

_Kr

and a

_{prime w}

of

L,

write

_Lw

for the

_compositum

of the

_completions

of the finite subextensions

L’ of

_L/K

at the

_prime

of L’ below w. If v is a

_prime

of

K,

let

Lv

:=

ITwlv

Lw,

the

_{product being}

taken over the

_primes

of L above v.

For 1 m _oo, the

_pm-Selmer

_group of E over L is defined

_by

the

exactness of the natural _sequence

where we extend

multiplicatively

our functors on local fields with values in

the

_category

of Abelian _groups. We find

_directly

from the definitions that

the limit

_{being compiled by}

means of the _maps induced

_by

the natural

inclusions

_{En -}

_Epn+1.

If

_L/K

is a finite

_extension,

define the _pro-p Selmer _group of E over L

to be

where the inverse limit is with

_respect

to the _maps induced

_by

_Ep,,+,

~

Rep- -

When the

_{p-torsion points}

_{Ep (L)}

of E over L are

_trivial,

then

Sp (EIL)

is

_equal

to the Tate module

_{Homzp (~p/7~p,}

_{Selpoo (E/L))}

of

_Selpoo

_(E/L);

in

particular,

it is a free

_Zp-module.

_Letting

E(L)p

:=

E(L)

0

_Zp

denote the

p-adic

completion

of

_E(L),

there is a natural inclusion

E(L)p - Sp(E/L).

Define

_{Koo, Kn,}

r and A as in the

introduction;

let

rn

:=

Gal(Koo/ Kn) =

rpn

and

_Gn

:=

Gal(Kn/K)

=

r/rn.

Given a discrete A-module

_M,

we _say

that M is

_{cofinitely generated,}

_resp.

_cotorsion,

or cofree over A if its

Pon-tryagin

dual

Mdual

=

Homzp

(lVl,

Qp/Zp)

is

_finitely

_generated,

_resp.

_torsion,

or free as a A-module.

The inflation-restriction _{exact sequence}

_gives

where the direct limit is taken with

_respect

to the natural restriction

map-pings.

One can show that the

Pontryagin

dual

Xoo

of

Selpoo(E/Koo)

is

Define the A-module

the limit

_being

taken with

_respect

to the _{corestriction maps.}

Lemma 1.1. Assume that E has

_{good ordinary}

reduction at the

_primes

_of

K

_(above

_p)

which are

ramified

in

_K~.

Then there is a canonical

identifi-cation

-Proof.

See

_[14,

lemme

_5,

_p.

_417].

D

It follows that

_6’p(E/~oo)

is a torsion free

A-module,

having

the same

rank over A as

Zoo.

2.

_Special

values of L-series and Selmer _groups

In this section and in the

_sequel

of the paper, let E be an

elliptic

curve

defined

_{over Q}

and let K be an

_{imaginary quadratic}

field. Write E for a

finite set of rational

_primes

_including

_p and the

_primes

of bad reduction for E.

_(Note

that in the definition of the Selmer _group for extensions of K no reference to the Archimedean

_primes

is

_needed.)

Let

~oo

stand for a

Zp-extension

of

_K,

and retain the notations of the

_previous

section.

Given a finite order

character X :

F -

C~,

write

L(E/K, x, s)

for the

L-series of

_E/K

twisted

_by

_x.

_By

the fundamental work initiated

_by

Wiles and

_Taylor,

E is known to be modular.

_Hence,

_{L(E/K, x, s)}

can be

con-tinued

_analytically

to the whole

_complex

_plane,

so that it is defined at s=1.

We review some

_conjectures

on the behaviour of

L ( E/ K, x,1 ) ,

which,

combined with the Birch and

_{Swinnerton-Dyer}

_conjectures,

lead to

predic-tions on the rank of the Mordell-Weil _groups

E(Kn)

and on the

_Zp-corank

of

_{Selpao (E/Kn ) .}

Our _purpose is that of

_motivating

certain

_assumptions

that will be made later in the _paper.

Recall that the

_{Zp-extensions}

of K which are Galois

over Q

are the

cyclotomic

Zp-extension,

which is Abelian over

Q,

and the

anticyclotomic

Zp-extension,

whose Galois group

over Q

is

pro-dihedral.

Their

composite

is the

_{Z2 -extension}

of

_K,

which contains all

_{Zp-extensions}

of K.

If

Ago

is different from the

_{anticyclotomic}

_Zp-extension

of

_K,

it is

ex-pected

that

_L(E/K,

_x,1)

is non-zero for almost

all X

as above. For the

cyclotomic

Zp-extension

of

_K,

this is

_proved

in Rohlrich’s _paper

_[17].

Suppose

now that

K~

is the

anticyclotomic

Zp-extension

of K. It is con-venient to

_distinguish

two cases. We _say,

following

Mazur’s

_terminology,

that E is in the

_generic

case if either E does not have

complex

multi-plications

or the field of

_{complex multiplication}

of E is different from K.

Let E be in the

_generic

case.

For X

an

anticyclotomic character,

then

L(E/K, X, s)

and

_{L(E/K, X, 2 - s)}

are related

_by

a functional

_equation,

whose

_sign

is the same for all ramified _X. We call this common

sign

the

_sign

of

_{(E, Koo/ K).}

_(It

can be different from the

_sign

correspond-ing

to unramified characters when _p divides the conductor of

_E.)

If the

sign

of

_{(E, Koo/ K)}

is

_+1,

_resp.

_-1,

it is

_expected

_{(cf. [11]}

and

_[12])

that

L(E/K, x,1),

resp. the first derivative

L’(E/K, x,1)

is non-zero for almost

all X.

We now turn our attention to

_elliptic

curves in the

_exceptional

case.

There is a factorization

where

_’l/JE

is the Grossencharacter attached to

_Fez

and e is the

quadratic

character associated with K. The Hecke L-series

_appearing

in the above factorization

_satisfy

a functional

_equation

_relating

their values at s and

2 - s.

Here,

we define the

sign

of

(E, K~/K)

to be the common

sign

of the functional

_equations

of the L-series

_L(’OE,

_X,

_s),

so that the

_sign

of

_(E,

_{Koo/ K)}

determines the

_parity

of one half the order of

_vanishing

of

L(E/K, X, s).

When the

_sign

of

_(E,

_{Koo/ K)}

is

_+1,

it is

_again

_expected

that

L(E/K, X,1)

is non-zero for almost all _x; when the

_sign

of

(E, K~/K)

is

-1,

then

_{L(E/K, X,}

_s)

should vanish to exact order 2 for almost all _X.

By combining

the above

_expectations

with the Birch and

Swinnerton-Dyer conjectures,

one is lead to formulate the

following

conjecture.

See

[11,

sec.

18]

and

[12].)

Given two functions

_{f, g :}

I~Y -&#x3E;

_C,

we write

f (n)

g(n)

+

_O(1)

if

_{f (n) -}

_g(n)1

is bounded

_by

a constant

_independent

of n.

Conjecture

2.1.

_{If Koo is a}

_{7GP-extension}

_{of the}

_{imaginary quadratic}

_field

K,

there exists an

_integer

r =

r(E,

_E

{O,

_1,

21

depending

_on E and

such that

Conjecture

2.1 is

_equivalent

to the statement that is

equal

to r

_pn

-~ O ( 1 ) .

If the

_p-primary

_part

of the

Shafarevich-Tate group of is finite for all n, it is also

equivalent

to the

equality

of

rankzE(Kn)

and r ~

_pn

+

_0(1).

We formulate

_conjecture

2.1 in the weakest form

_apt

to be assumed as a

working hypothesis

in the next sections.

_Following

_Mazur,

we call r

the

_growth

number of

_(E,

_{Koo/ K).}

The above discussion indicates that

r should be 0 or 1 if E is in the

generic

_case, and 0 or 2 if E is in the

exceptional

case.

Given a rational

prime l,

let

Kn,l

:= the sum

_being

denotes the _group of the local

_points

of E at l. Write

for the

_{p-adic completion}

of

_E(Kn,l).

By

definition of Selmer _group, there are natural _maps

m

the direct limit

_being

with

_respect

to the natural _maps. Note that is

_{always equal}

to

Conjecture

2.2. yVe have =

r(E,

pn

₊

0(1).

Assume that the "weak

_{Leopoldt conjecture"}

is satisfied. Then

_conjecture

2.1 and

_conjecture

2.2 are

equivalent.

In

particular,

this is the case when the

_growth

number r is

_equal

to 0. In section 5 we

provide

evidence for these

_conjectures,

when is the

anticyclotomic

Zp-extension

and there are

_Heegner

_points

defined over

We can show that

they

follow from a mild

non-vanishing

_assumption

on the

Heegner

points.

3. Descent Iwasawa modules

Using

results of

_Greenberg

_[8],

and

_assuming

the

_validity

of the

conjec-tures 2.1 and

_2.2,

in this section we _prove a result on the structure of the

descent A-modules

_{H’(Kr /Koo, Epao ),}

i =

_{1, 2.}

This result holds for all odd

_primes

_p,

_irrespective

of whether

_they

are

_ordinary

or

_{supersingular}

primes

for

_E,

or

_primes

of bad reduction. It is an

analogue

in the current context of the theorems 1 and 2 of

_[7],

which

_study

the case of a modular

elliptic

curve over the

cyclotomic

Zp-extension

of Q

having analytic

rank

over Q

at most 1.

Theorem 3.1. Assume that the

_conjectures

2.1 and 2.2

_hold,

and that _p

is an odd

_prime.

Then:

1. the discrete A-module has corank over A

equal

to 2.

Moreover,

its

_Pontryagin

dual has no non-zero

finite submodule;

2.

_{E P_)}

= 0.

Proof. Greenberg

([8,

prop.

3,

4 and

5])

has shown the

following:

a. is a cofree

A-module;

b. if

_Epoo)

=

0,

then the

Pontryagin

dual of has no non-zero finite

submodule;

c.

corankAH1(K’E/ Koo, Epoo) -

= 2.

It follows that

_part

1 and

_part

2 of theorem 3.1 are

equivalent.

Moreover,

if s =

s(E,

Koo/ K)

denotes the A-corank of

Ep.),

then we

have

_5~2.

Recall that

_Selpao

_(E/Kn)

is defined

_by

the exact _sequence

For _every rational

_{prime l,}

the local Tate

_duality

_gives

a

perfect

pairing

By

the

_theory

of the formal groups attached to

_elliptic

curves, we find that

and,

for 1

_~

_p, and

_{H1 (Kn,l,}

are finite. It follows that

The inflation-restriction sequence

gives

readily

From the

_theory

of A-modules and the fact that

_Epoo(Koo))

is

_finite,

we deduce

By combining conjecture

2.1 with

₍₁₎

and

_(4),

we obtain

Together

with

_(3),

this shows that s - r 2. When r =

0,

this concludes the

_proof.

In

_general,

consider the natural _map

By

conjecture

2.2 and the above remarks

The

_global

_reciprocity

law of class field

_theory

_implies

that

_Im(pn)

and

Im(6n)

are

isotropic

under the local Tate

pairing

Hence,

by

(3), (5)

and

_(6),

we

_get

This

_implies

that s

_{2, which,}

_together

with the

_{inequality s &#x3E;}

2 observed

before,

gives s

=

2,

_{as was} to be shown. D

4. Finite submodules of

_Selpoo

It is of interest to formulate conditions under which the

_quotient

of has no non-trivial finite A-submodule. See in

particular

the

_applications

to the structure of

_Xoo

of section 8. Let T be the order of the torsion

_subgroup

of

Lemma 4.1.

Proof.

It is

_enough

to show that = 0 for all the

primes A

of above 1 E

_{E 2013 {?}.}

_By

_proposition

2 of

_~8~,

this holds if

_Ep.

is finite. But is zero under our

_assumption

on _p. 13 To

_simplify

the

_arguments,

we assume from now on that _{p is} a

_prime

of

good

reduction for E. The case where _p is an

_ordinary

_prime

is

_radically

different from the case where

_{p is}

supersingular.

We first consider the latter case.

Lemma 4.2.

_{If p is a supersingular}

_prime

_for

_E,

then

_H1(Koo,p,

_E)poo

is

equal

to zero.

Proof. By

local Tate

_duality,

_H1(Koo,p,

_E)poo

is the dual of

_limn

_E(Kn,p)p,

the inverse limit

_being

taken with

_respect

to the _{corestriction maps.} The claim follows from well-known results on the norm

_mapping

for Lubin-Tate

formal _groups of

_height

2. See

_[20].

D

Proposition

4.3. Let p be a

_prime

of supersingular

reduction

for

E such

that p ( 2T.

Assume that the

_conjectures

2.1 and 2.2 hold.

Then,

A-module

_of

rank 2 with no non-trivial

_finite

A-submodule.

Proof.

By

combining

lemma 4.1 with lemma

_4.2,

we see that

is zero.

By definition,

_{Selpao (E / Koo)}

is

equal

to The

result follows from theorem 3.1. D

Remark. Lemma 4.2 shows that if _{p is}

_{supersingular,}

the A-coranks of

Selpoo

and

_Epoo)

are

_equal.

_Thus,

the results of

Green-berg

recalled in the

_proof

of theorem 3.1

_imply

that for

_{supersingular}

_primes

the A-rank of is

_always

at least 2.

We now turn to consider

_primes

_p of

_{good ordinary}

reduction for E. In the remainder of this

_section,

we review an

_argument

of

_Greenberg

([8,

sec.

7])

which shows that if is

_A-torsion,

it has no non-trivial finite A-submodule

for almost all

_ordinary

p

(see

theorem 4.5

below).

This

argument

does not

will be treated in the sections

_5,

6 and

_7,

from a different

_viewpoint.

For a

strengthening

of theorem

4.5,

see

_{Greenberg’s forthcoming publications.}

The

_following

lemma clarifies the relation between the

_growth

number defined in section 2 and the A-rank of

Lemma 4.4. Let _p be a

_prime

_{of good}

_ordinary

reduction

_for

_E,

and let

r =

rankAXoo .

Then we have

Proof.

Mazur’s "control theorem"

_([10,

ch.

_4])

shows that the natural re-striction _maps

-have finite kernel and

_cokernel,

of order bounded

_{independently}

of n. The

lemma follows from the

_theory

of A-modules. 0

Theorem 4.5

_(Greenberg).

Assume that:

(i)

p is a

_prime

of good ordinary

reduction

for

E;

(ii)

p

f

6T;

(iii)

is a torsion A-module.

Then has no non-trivial

finite

A-submodule.

Proof.

( ~8,

sec.

7])

Step

1. Since _p is

_ordinary

for

_E,

reduction mod _p

_gives

rise to an exact sequence

of modules over

Gxp

:=

Gal(Kp/Kp),

such that the inertia group

_Ip

C

_{G Kp}

at p acts

trivially

on

Epoo / F1 Epoo

and via the

cyclotomic

character xp on

F1 Epoo.

By

[8,

sec.

2]

and

by

lemma

4.1,

the Selmer _group of E over

K~

can be defined

by

the exact _sequence

where the last map is induced

by

(7).

Let

denote the Cartier dual of

_By

our

_assumptions

on _p, we

have that is _zero, and hence is

also zero. Then

[8,

_{prop. 1} and its

corollary

2]

shows that the A-module

H1(Koo,p,

Epoo

/ F1 Epoo)

is cofree of corank 2.

Step

2. Since

_by

our

assumptions

is

A-torsion,

lemma 4.4

implies

directly

that the

_conjectures

2.1 and 2.2 hold with r = 0.

By

theorem

_3.1,

Epoo )dual

has A-rank

_equal

to 2 and has no non-trivial finite

Step

3.

_By

_dualizing

the _sequence

_(8),

we find

where the

_injectivity

of the first _map follows from the fact that is

A-torsion and is free of the same rank as

Let C be the

_preimage

in of a finite A-submodule

C of

_{By Step}

1 and

_Step

_2,

C

is A-torsion free. Hence the

_theory

of A-modules shows that

13

must

_inject

in a free A-module of rank

_2,

with

finite cokernel. But this is

_{possible only}

if C = 0. D Remarks.

1. If the A-rank of is

_positive,

then the first _map of the sequence

(9)

is no

_longer

_injective,

and the

_argument

above cannot be

_performed

_(as

one sees from _easy

examples).

A natural situation where the rank of is

positive

arises because of the _presence of

_Heegner

_points

defined over the

layers

of the

_{anticyclotomic}

_Zp-extension

_Ago

of K. In the sections

_5,

6

and 8 we

_study

the structure of the Iwasawa modules and

in this _case, and in

_particular

the existence of non-trivial finite A-submodules of

2. Assume that E is in the

_generic

case and that _p is a

_prime

of

_good

ordinary

reduction for E.

_{Conjecturally,}

the rank of is

_positive

if and

_only

if is the

_{anticyclotomic}

_Zp-extension

of K and there exist

Heegner

points

defined over

Koo.

For,

the

conjectures

of section 2 combined

with lemma 4.4

_predict

that the the rank of is

_positive

when

Ago

is

anticyclotomic

and the

_sign

of

_(E,Koo/K)

is -1. If E is

_modular,

this is

precisely

the

_assumption

that

_guarantees

the existence of

_Heegner

_points

(see

the next

_section).

5. Growth numbers and

_Heegner

_points

We have observed that if is a torsion

A-module,

then the

conjectures

2.1 and 2.2 are

_verified,

with r = 0. In this

section,

let

Koo

be the

anticyclo-tomic

_Zp-extension

of K and _p be a

_prime

of

_{good ordinary}

reduction for E

(subject

to certain technical

_conditions).

_Assuming

the existence of

Heeg-ner

_points

defined over

satisfying

a mild

_{non-triviality}

_assumption,

we

show that the

_conjectures

2.1 and 2.2

_hold,

with r = 1. In

particular,

_we

may

apply

theorem 3.1 in this

setting.

Let K be such that

_disc(K)

is

_prime

to the conductor N of E. In order

to ensure the existence of

_{Heegner points}

on E defined over the

layers

of

Assumption

A.

The L-series

_{L(E/K, s)}

vanishes to odd order at s = 1.

The above

_assumption

_implies

that E is in the

_generic

case. The condi-tion on

L(E/K, s)

is

equivalent

to

_E(N)

=

_1,

where recall that _E denotes the Dirichlet character attached to K. It follows that for all finite order

chaxacters x

of r the L-series

_{L(E/K, X,}

_s)

vanishes to odd order at s = 1. Under

_assumption

_A,

we can construct

_Heegner

_points

defined over the

layers

of

_coming

from a Shimura curve

parametrization

X ~ E of

E,

the definition of the curve X

_depending

on which

_prime

factors of N are

split

or inert in K. In the case when all the

_primes

_dividing

N are

split

in

K,

X is the modular curve

Xp(N).

See

[5]

for more details.

The extension

Kn

is contained in a

_ring

class field of conductor

pkn.

Assume that

_kn

is minimal.

_(For

_example,

if _p does not divide the class _group of

_{K, kn}

=

n ~-1. )

For all n _we

may define a

_Heegner

_point

_{an E}

E(Kn)

by tracing

from

_K~pkn~

down to

_Kn

a

_{Heegner point}

in

By

viewing

an as a

point

of

E(Kn)p

=

E(Kn) 0

Zp,

define the submodule

of

_generated

_by

the

_Gn-orbit

of an. The module

E(EIK,,,)p

does

not

_depend

on the choice of the

_{Heegner point}

_an, but does

_depend

on the

choice of modular

_{parametrization.}

_{Let ap denote the Fourier coefficient}

1-~ p -

#E(Fp).

In order to be able to

_{apply directly}

the results of

_~1~,

we

impose

the same

assumptions

as in

~1~,

with the

exception

that we do not

require

that the

_Heegner

_points

an come

necessarily

from a

_{parametrization}

of E

_by

the modular curve

Xo(N).

Since the formal

properties

of the an are

the same in all cases

(see [5]),

the

_arguments

of

[1]

_go

through unchanged

and the results contained there hold in our more

general

_setting.

(Some

of

the other

_assumptions

could also be weakened

_somewhat.)

Assumption

B.

(1)

The

_{imaginary quadratic}

field K is such that

_disc(K)

is

_prime

to N

and

_C7K

=

~~1}.

(2)

p f

where

_EIEO

denotes the _group of

connected

_components

of the N6ron model of E over

Spec(OK) .

(3)

The Galois

_{representation}

_{pp :} is

_surjective.

(4)

0,1, 2

(mod p)

if _p

_splits

in

_K,

and

_{ap 0}

_0,1,

-1

_{(mod p)}

if _p is inert in K.

Fixing

E and

_{K, assumption}

B holds for a set of

_primes

_p of

_density

1: see

[1, §2.2]

for more details. In the

sequel

of this

section,

we will

_tacitly

work

Lemma 5.1. For all _n, corestriction induces

_surjective

_maps

In

_particular,

we have natural inclusions C

_{£(E / Kn+1)p.}

Proof.

See

_[1,

_§2.1,

prop.

4] (based

on

_computations

contained in

_[14]).

0

Define the Iwasawa module of the

_Heegner

_points

to be

I., fI

where the inverse limit is taken

_{with respect}

to the _{corestriction maps.} The module is

_cyclic

over A.

_Moreover,

it

_injects

_naturally

in

(the

latter module

_being

defined in section

_1).

Since the A-module is torsion-free

_by

lemma

1.1,

we conclude that either is

_isomorphic

to A or it is zero. One checks that is non-zero if and

_only

if _an has infinite order for some n.

(See

[1, §2.1]

for

details.)

Remark.

_By

_modifying

the definition of the as in

[14],

it is

possible

to define an Iwasawa module of

_Heegner

_{points enjoying}

the same

properties

of under weaker

_assumptions

than the ones above.

For

_simplicity,

we do not _pursue this here. Define the A-module

the inverse limit

_being

taken with

_respect

to the the _{corectriction maps.}

Proposition

5.2. 1. The A-module

_Ê(Koo,p)p

is

_{free of}

rank 2.

2. The natural _maps

are

_{isomorphisms.}

_(Thus,

_by

_{part 1,}

_E(Kn,p)p

is

_{isomorphic to}

Proof.

Part l.

_By

the local Tate

_duality,

is identified with the

Pontryagin

dual of Consider the local descent exact se-quence

Let

_FK,,

denote the residue

_algebra

of

_Assumption

B

_implies

that

Ep(FK..)

is zero. Then the _sequence

(1)

of section 4 induces the exact

This _sequence ends with a zero since has

p-cohomological

dimen-sion 1

_(see

_[21]).

The results of

_[6]

show that the

_image

of

HI

_(Koo,p,

F1

_{Ep~ )}

in

_H1(Koo,p,

_{Ep~ )}

_equals

the

_image

of

_{0 Qp/Zp}

in

_H1(Koo,p,

_Ep.)

-It follows that is identified with

Since is zero under our

assumptions,

we also have that is zero. Then

[8,

_proposition

1 and its

_corollary

_2]

implies

that

HI

_(K~,p,

is A-cofree of rank 2. This concludes the

_proof

of

_part

1.

Part 2. Under our

assumptions,

the results of

[10,

ch.

4]

show that the

corestriction _maps

"""’/7"’ " """’/7"’" ,

are

_{surjective. Therefore,}

the natural _maps -

E(Kn,p)p

are

also

_surjective.

_By

_{part 1,}

is

_isomorphic

to _in

par-ticular,

it has

_Zp-rank

_equal

to

_2pn.

Since the

_Zp-rank

of

_{E(Kn~p) p}

is

_equal

to

_{2pn by}

the

_theory

of formal _groups, we conclude that

E(Kn,p)p

is an

isomorphism.

0

Theorem 5.3. Assume that is non-zero. Then the

_conjectures

~.1 and ~.,~

_hold,

with

_r(E,

_{Koo/ K)}

= 1.

Proof.

Part 1

Under our

_assumptions,

the results of

[1, §3.1]

show that the A-rank of is

_equal

to 1. Lemma 4.4

_implies

that

_conjecture

2.1 holds with

r (E, Koo/ K)

= 1. Part 2

Step

l. Consider the natural localization _maps

By taking

the inverse limit with

_respect

to the corestriction

_mappings,

we

obtain a _map

By

our is a free A-module of rank 1.

_By

proposi-tion

_5.2,

is torsion-free. Hence _o~~~p is

_injective

if and

_only

if it is non-zero. This is

equivalent

to

_saying

_{that an,p} is non-zero for some n.

But we know that an has infinite order for some _n, and hence

_on,p(an)

is non-zero. In

conclusion,

_o,,,.,p is

injective.

Step

2.

_By

_{fixing isomorphisms}

A _0’,,,,,p

By

proposition 5.2,

the

_image

_{of an,p is identified with}

Hence,

by

the

_theory

of

_A-modules,

we find

Step

3. _{Since O’n,p} is the restriction of the map

the above

_equality

_gives

The claim follows from

_part

_1,

since

D Remark. As we have observed in section

_2,

in our

setting

L’(E/K, x,1)

is

conjectured

to be non-zero for all but

finitely

_many finite order characters x of r. A natural

generalization

of the

Gross-Zagier

formula states that

L’(E/K,

x,1)

is

_equal

to the canonical

_height

of the

_x-component

of the

Heegner

point

an, up to a non-zero constant. This would

_imply

that for n

large enough

an has infinite order.

Thus,

it is natural to

_conjecture

that

is

_always

non-zero: see

[11,

sec.

19].

By

combining

theorem 3.1 with theorem

_5.3,

we obtain the

_following.

Theorem 5.4. Assume that non-zero.

1. The discrete A-module has corank over A

equal

to 2.

Moreover,

its

_Pontryagin

dual has no non-zero

finite

submodule.

2.

_{H2(K’E/ Koo,}

= 0.

Remark.

_Proposition

5.2 and theorem 5.3 _may be viewed as

_analogues

in our

_setting

of the results of Iwasawa on the structure of the modules of the local units and the

_cyclotomic

units over the

cyclotomic

_Zp-extension

of

Q.

(See

for

_example

_[9].)

6. A

_duality

theorem

In this

_section,

we _prove a

general duality

theorem which will be used in

the next section to

_study

the existence of finite submodules of in the

case of the

_{anticyclotomic}

_Zp-extension

of K.

_Here,

K will be a number

field and an

_arbitrary

_Zp-extension

of K. We make the

following

assumptions

on

(E, p, Koo/ K).

1.

_{p t}

_2#(E/E°),

where denotes the group of connected

components

of the N6ron model of E over the

ring

of

integers

of K. 2. E has

_good

reduction above _p.

3. The Galois

_{representation}

_{pp :}

_Aut(Ep)

contains a Cartan

subgroup

of

_{Aut(Ep) - GL2(Fp).}

4. The local norm

mappings Nv :

E(Kn,v) ~

E(Kv)

are

surjective

for all

primes v

of K.

Remark. In the case when E is an

elliptic

curve defined over the rationals

and K is an

imaginary quadratic

field,

one checks

_by

_using

the results of

[10,

ch.

_4]

that these

_assumptions

follow from the

_assumption

B of the

previous

section.

_By

condition

_3,

is _zero, and hence is also

zero. Condition 4

implies

that E has

ordinary

reduction at all the

_primes

of K which are ramified in

(~10,

ch.

4]).

Define the universal norm submodule of

Sp(E/K)

to be

Theorem 6.l. Under the above

_assumptions,

there exists a

perfect

canon-ical

_pairing

Remark. The existence of a

_pairing

like the one

_above,

_having

finite

right

radical,

is often used to define the

_{p-adic height}

_pairing

on

Sp(E/K)

attached to

_Koo.

See

_{[18], [19]}

and

_[15].

For our _purposes, we need that

the

_pairing

_{(( , ))}

is

_perfect,

not

_just

up to

quasi-isomorphisms.

The

proof

uses theorem 3.2 of

[4].

We

_study

the existence of finite submodules of

_by

means of the

_following

corollary.

Corollary

6.2. has no non-trivial

_finite

A-submodule

_if

and

_{only if}

Sp(EIK)IUSp(EIK)

has no torsion.

Proof.

Theorem 6.1

_gives

an identification of

Sp(E/K)/USp(E/K)

with

X~.

If M is a non-trivial finite A-submodule of then

Mr

is a

non-trivial finite submodule of

_X~.

_Conversely,

if the torsion

_subgroup

of

Sp(E/K)/USp(E/K)

is

_non-trivial,

the

_{corresponding}

finite submodule of

X~

is a non-trivial finite A-submodule of

(such

that the action of A

factors

_through

_{A/(7 - 1)11}

=

Zp) .

D

If is a torsion

_A-module,

so that

USp(E/K)

is _zero,

corollary

6.2

_gives

under the current more restrictive

assumptions

a new

_proof

of the result of

Greenberg

recalled in section 4. Our

_goal

is to

_apply

the above

_corollary

to

an

elliptic

curve defined over the rationals with values in the

anticyclotomic

positive

A-rank due to the _presence of

_Heegner

_points

over the

_layers

of

K~.

Proof of theorem 6.1. It follows

_by

_combining

the

_following

results. Fix

_positive

_integers

m and n. To ease notation in the next

_lemma,

let

H :=

Kn

and G :=

Gn.

Denote

by

I the

augmentation

ideal of the _group

ring

Thus,

we have a canonical identification

G/Gpm

=

Proposition

6.3. There is

_perfect

canonical

_pairing

_of

Tate

_cohomology

groups

Proof.

Under our

assumptions,

the Hochschild-Serre

spectral

_sequence can

be used to show that the restriction _map induces an

_isomorphism

between

Selpm (E/K)

and

_Selpm(E/H)G

_(see

for

_example

_{[1, §2.3]).}

In

_particular,

we _may

identify

Selpm(E/K)

with a submodule of

Selpm(E/H).

Write resp.

Selpm(E/H)O

for the

image,

resp. the

kernel of the corestriction _map

_{coresH~K :}

_{Selpm (E / K).}

Thus,

the group

ft’(G,

Selpm (E / H) ),

resp.

ÎI-1 ( G,

Selpm(E/H))

is

equal

by

definition to

resp.

Let,

be a

_generator

for G.

_By

[4,

theorem

3.2],

_given

_y E

there exists z E

H1(H,Epm)

such

_{that y}

=

(~y - 1)z.

If v is a

_prime

of

_K,

we

_{write zv}

for the natural

_image

of z in

H1

(Hv,

E)Pm .

Clearly,

zv

belongs

to

_{H1(Hv, E);m.}

_By

the

_{assumption 4,}

an

_argument

based on the Hochschild-Serre

spectral

_sequence shows that restriction

gives

an

isomorphism

between

H1(Kv,

E)pm

and

H1(Hv,

E pm

It follows that we _may

identify

_zv with an element of

Hl(Kv,E)pm.

Write

for the local Tate

_pairing

_(~13,

ch.

_1]).

We define a

_pairing

by

letting

v

where x,

denotes the natural

_image

of x in and the sum runs over all

_primes

of K. Observe that the

pairing

[ , ~H

is well defined.

For,

if

_{(r -1)z}

=

(y -1)u

for elements u and z in then z - u

belongs

to

_H1(H,

_Epm

)G.

Since

_Ep(H)

is _zero, the restriction _map induces

an

isomorphism

of

Epm)

onto

Epm ) G.

Thus,

we _may

_regard

field

_theory

_([13,

lemma

_6.15,

_p.

_105])

_gives

= 0.

Moreover,

[ , ~H

does not

_depend

on the choice of the

_generator

_q.

Observe that _y

_belongs

to the

_right

kernel of

_[

,

~H

if and

only

if

there exists u E such that z - u

_belongs

to

_Selpm

_(E/H)

([13,

lemma

_6.15,

_p.

_105]).

This is

_equivalent

to

_saying

that _y

_belongs

to

_{(, -1)Selpm(E/H).}

Now,

assume that is

equal

to with * E

Selpm(E/ H).

Then

_{[X, Y] H}

= 0 for

all y

in

Selpm(E/H).

For,

the

_{global duality}

theorem

gives

where the second sum is over all the

_{primes w}

of

_H,

we

let [ , ~,

stand for

the local Tate

_pairing

relative to

Hw

and we write _zw for the

image

of z in

H

Thus far we have

proved

induces a

right

non-degenerate

pairing

But the _groups and have the

same

order,

since G is

cyclic

and

Selpm (E/H)

is finite.

Hence,

( , ~ H,m

is

_{non-degenerate.}

This concludes the

_proof

of

_proposition

6.3. D

We shall define the

_duality

of theorem 1

_{by compiling}

a limit of the

_pairings

defined in

_proposition

6.3. We

_begin

with a

compatibility

_property.

For

m ~ 77/, ?~ : 7/,

let

be defined

_by

_composing

the restriction _map with the _map induced

_by

the inclusion

_Epn

C Under our

assumptions,

the _map res

_gives

an

isomorphism

([1, §2.3])

Sel

By

abusing

notation

_somewhat,

we also write res for the induced _map

Let

be the _map induced

_{by .}

Since i

is contained in

_{coresKn/KSelpm(E/Kn)}

we obtain a _{map, still denoted}

_by

Write

_Im,n

for the

_augmentation

ideal of Let i =

in,n’ :

Gn ~

Gn,

be the canonical

_injection

defined

_by

_i

_{where §}

_{is any} lift

of g

to

_Gn,

with

_respect

to the natural

_{projection Gn 2013 Gn.}

The _{map i}

I --, -

-induces a canonical _map

Lemma 6.4. We have

for

any

given

Proof.

It follows

_directly

from the definition of our

pairings.

Lemma 6.5. We have

where the Lirrtit is taken with

_respect

to the _maps res.

Proof.

Note that

_{Selpm (E/Kn)}

_injects

via restriction into

_Selpm

Thus,

Let I

be a

topological

_generator

of r. The claim follows

by taking

the

direct limit of the short exact _sequences

Lemma 6.6. We have

fV fIV

the limit

_being

taken with

_respect

to the _{maps mp.}

Proof.

We start

_{by showing}

that

First,

we see that

This follows from

_taking

the inverse limit of the _maps 1BT

where

_NKn/K

denotes

_{multiplication by}

_EGEG.

_g E

_{Zp[Gn] .}

(Note

that we are

_working

with finite

_modules,

hence the inverse limit is an exact

functor.)

Moreover,

we have the

equality

Now take Urn of the short exact _sequence

m

We find

Although

the modules in the above _sequence are not

_finite,

a

simple

topo-logical

argument

(see

the sublemma 6.7

_below),

based on the

completeness

of in the

_{p-adic topology,}

shows that

_{lim n}

is exact on the above sequences,

thereby

proving

lemma 6.6. D

Sublemma 6.7. Let X be a

_{fcnitelg generated}

_Zp-module.

Let

_{Xn, n &#x3E;}

1

be a _sequence

of

Zp-submodules

of

X such that

Xn D

Xn+1 for

all n. Then

the inverse limit

_being

taken with

_respect

to the natural

_projection

_maps.

Proof.

Write

_Xoo

and let

_Y,

resp.

Yn

stand for resp.

Xn/ Xoo.

Hence,

_{nn Yn}

=

(0).

Our claim is

equivalent

to

showing

that the

inverse limit of the

_Yn

with

_respect

to the natural

_projections

is

_equal

to Y. Note that the

_Zp-rank

of

_Yn

is constant for all n

_greater

than a

_positive

integer

no .

By

the

theory

of

elementary

divisors,

we _{may write Y} = where Z contains with finite index

Yn

if n _{&#x3E; no.} We have

Since Z is

_complete

in the

_p-adic

_topology

and the

_Yn

are a basis of

neigh-bourhoods of the

_identity

of

_Z,

we conclude that

_lim~

Z/Yn =

Z. This

proves the sublemma. D

Theorem 1 is

_{proved by combining}

the results

_6.3-6.6,

since the direct limit of with

_respect

to the

_{maps j}

is

_naturally

identified with r

_0zp

7. On the structure of

_Selpao

We return to the

_study

of

_elliptic

curves defined over the

_rationals,

with

values in a

Zp-extension

of an

_imaginary

quadratic

field. In section

_4,

we

reviewed results of

_Greenberg

which show that has no non-trivial finite

A-submodule,

if _p is a

_prime

of

good ordinary

reduction for E and is a torsion A-module. In this

_section,

we consider the case where the A-module

has

_positive

rank and _{p is} an

_ordinary

_prime.

(If

E has

supersingular

reduction at _p, we have seen in section 4

(proposition 4.3)

that is

essentially equal

to the

_Pontryagin

dual of

_{H1(K’E/ Koo,}

_Ep.),

and we _may

apply

theorem 3.1 to understand its

_structure.)

In view of the

_conjectures

of section

_2,

if we exclude curves in the

ex-ceptional

case can have

_positive

A-rank

only

when is the

anticy-clotomic

_Zp-extension

of K and there are

_Heegner

_points

defined over the

layers

of

_Therefore,

it is natural to

_place

ourselves in the

_setting

of

section

_5,

under the

_assumptions

A and B made there.

The main result of this section

_(theorem

_7.1)

_gives

a sufficient condition

ensuring

that does not have non-trivial finite A-submodules. As an

application

(theorem 7.2),

we obtain a

strengthening

of the main result of

111-Let

_£n

denote the natural

_image

of the module of

_Heegner

_points

E(E/Kn)p

in

_{E(Kn)/pE(Kn),}

and let be the Shafarevich-Tate

group of

E/Kn.

Define

m(E,Kn+1/Kn)

to be the kernel of the restriction map

Theorem 7.1. Assume:

1. 0

_for

some _n;

2.

_III(E,

= 0

for

all n &#x3E; 0.

Then has no non-trivial

_finite

A-submodule.

Remark. There is theoretical as well as

experimental

evidence in

_support

of the

_expectation

that

_Heegner

_points

tend to be non-trivial modulo _p. See also the discussion in

_[3,

section

_2~.

We recall the results of

_[1].

Assume that the Iwasawa module of the

_Heegner

points

8(E/Koo)p

is non-zero.

Then,

theorem 1 of

[1, §3.1]

shows that is a free A-module of rank 1. Write E A for a characteristic

power series of the

cyclic

torsion A-module and

let

_(X,,,,)tors

be the A-torsion submodule of

_Let,

be a

topological

generator

of r. Theorem 1 of

_{[1, §3.2]}

states the

_following:

1.

_{(-y -}

is a finite

A-module;

Thus,

in any case,

p2 -

is finite.

Note that the condition

_{~n ~ 0}

_implies

that 0.

_By

impos-ing

on the

_stronger

_assumptions

of theorem

_7.1,

we obtain the

following strengthening

of the above results.

Theorem 7.2. Under the

_assumptions

_of

theorem 7.1 we have:

Thus,

we have

always

poo 2

= 0.

Proof of theorem ?.1. The rest of this section is devoted to the

_proof

of theorem 7.1.

Let

_Rn

be the group

ring

Z/pZ[Gn] .

With q as

above,

denote

by

_’Yn

the natural

_image

_{of 7}

in

_Gn

and

_by

cvn the element qn - 1 of

Rn.

The

ring

.Rn

is a local

principal

ideal

ring,

whose maximal ideal is

generated by

wn. Since

R,~

may be identified with the

quotient

11/(~y -

of the PID

A

:=

_{A © Fp,}

we can

apply

to

Rn-modules

the structure

theory

for modules

over PID’S.

Therefore,

if M is a

finitely generated

Rn-module,

there is an

isomorphism

where

_{0 ik}

_pn.

Note also that the

lln-module

for 0 i

_pn,

is

_isomorphic

to via the _map induced

_by

_{multiplication}

_by

Moreover,

the dimension of the

_Fp-vector

_space is

_equal

to i .

_(See

[3,

section

_3]

for more

details.)

We will

_apply

the above remarks to the structure of the

_Rn-modules

Selp(E/Kn)

and,6n.

Let

_~n

:=

Gal(Kn+l/Kn).

Note that the restriction _map induces an

isomorphism

(See,

for

_example,

_{[1, §2.2].)}

_Thus,

we _may

regard,

and we will in the

sequel,

_Selp(E/Kn)

as a submodule of

Selp(E/Kn+1).

By

combining

(*)

with the

_surjectivity

of the corestriction maps on

Heegner

_points

(lemma

5.1),

we obtain the

equality

where denotes the norm

_operator

1: g

E In

_particular,

we may view

~n

as a submodule of

En+1.

We are

assuming

that there exists no such that the

Rno-module End is

norm-compatibility

of

_Heegner

_points,

we deduce that there is an

isomor-phism

for all n.

For n 2:

_no, define

Un

to be One checks that

Un

is

equal

to Observe that

_Un

is an

_Rn-module

in the natural _way,

and it is free of rank 1. If n _no, define

_Un

to be

_Again,

Un

is a free

Rn-module

of rank 1. It follows from the

isomorphisms

(*)

that

Un

is a submodule of

Selp ( E/ Kn ) .

Proposition

’1.3. For all n &#x3E;

_1~

we have an

equality

where the

Tn is

annihilated

_by

Proof.

See

_[2,

theorem

_12].

D

Remarks.

1. Theorem 12 of

_[2]

is

_{proved by building}

_upon

_{Kolyvagin’s}

methods

applied

to the Euler

_System

of

_Heegner

_points.

2. Since

to

pno ,

we have the natural identification

Thus

_by

_{proposition 7.3,}

for n &#x3E; _no,

Tn

is fixed

by

By

the

isomorphisms

( * ) ,

we _may assume that

_Tn

is

_equal

to

_{Tno ,}

for all n &#x3E; no .

In

_particular,

the

_growth

with n of the

Rn-module

Selp(E/Kn)

is

fully

accounted for

_by

the "universal norm" submodule

_Un.

The free rank 1

_{Rn-module Un,}

constructed from

_Heegner

_points,

is a

submodule of the

_p-Selmer

_group

_Selp(E/Kn).

Under our

_assumptions,

a

_stronger

statement holds.

Proposition

7.4. For all n &#x3E;

_{0, Un}

is a submodules

of

E(Kn)/pE(Kn).

Proof. By taking

the

_{Cn-cohomology}

of the _{exact sequence}

we obtain the exact _sequence

For all

_{places v}

of

_Kn,

the local

_cohomology

group

E((Kn+1)v))

is zero.

_For,

this _{group is} the dual of

E((Kn+1)v)),

by

local

Tate

_duality.

But this last group is zero,

by

assumption B,

(4)

of section 5.

_Therefore,

the _group

_{Hl(Qn,E(Kn+1))}

is

_equal

to