Iwasawa Theory of Fine Selmer Groups
R. Sujatha
Mini Course-CMS Winter Meet 2019
TorontoDecember 6, 2019

R. Sujatha (UBC Vancouver)Iwasawa Theory

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I WASAWA T HEORY OF C LASS G ROUPS

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Iwasawa Theory

We will fix a number field F /Q and an odd prime p for simplicity.

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December 6, 20193 / 42

Iwasawa Theory

We will fix a number field F /Q and an odd prime p for simplicity.
A Zp -extension of F is a Galois extension F∞ /F such that
[
F∞ =
Fn
n

with each Fn /F a cyclic extension, Gal (Fn /F ) ' Z/pn Z.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20193 / 42

Example F = Q
Consider the tower
Q = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn ⊂ · · · ⊂[

Qn =: Qcycnwhere Qn is the unique subfield of Q(ζpn ) such that
Gal(Qn /Q) ' Z/pn Z.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20194 / 42

Example F = Q
Consider the tower
Q = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn ⊂ · · · ⊂[

Qn =: Qcycnwhere Qn is the unique subfield of Q(ζpn ) such that
Gal(Qn /Q) ' Z/pn Z.
So Γ := Gal(Qcyc /Q) ' lim Z/pn Z ' Zp .
←−R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20194 / 42

Example F = Q
Consider the tower
Q = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn ⊂ · · · ⊂[

Qn =: Qcycnwhere Qn is the unique subfield of Q(ζpn ) such that
Gal(Qn /Q) ' Z/pn Z.
So Γ := Gal(Qcyc /Q) ' lim Z/pn Z ' Zp .
←−Qcyc , called the cyclotomic Zp -extension is the unique Zp extension of
Q. It is contained inside Q(ζp∞ ).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20194 / 42

Example F = Q
Consider the tower
Q = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn ⊂ · · · ⊂[

Qn =: Qcycnwhere Qn is the unique subfield of Q(ζpn ) such that
Gal(Qn /Q) ' Z/pn Z.
So Γ := Gal(Qcyc /Q) ' lim Z/pn Z ' Zp .
←−Qcyc , called the cyclotomic Zp -extension is the unique Zp extension of
Q. It is contained inside Q(ζp∞ ).
Gal (Q (ζp∞ ) /Q) ' Z×p ' Zp × Z/(p − 1) ' (1 + pZp ) × Z/(p − 1)

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20194 / 42Field Diagram
Q(ζp∞ )
Z/(p − 1)
Qcyc Zp

Zp
Q(ζp )
Q

Z/(p − 1)

For a number field F , the cyclotomic Zp -extension always exists.
Fcyc = F · Qcyc .
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Leopoldt Conjecture

Let r1 denote the number of real places of F and r2 the number of(non-conjugate) complex places.

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Leopoldt Conjecture

Let r1 denote the number of real places of F and r2 the number of(non-conjugate) complex places.
Therefore,[F : Q] = r1 + 2r2 .

Conjecture (Leopoldt Conjecture)
Let F be a number field. Then F admits r2 + 1 independentZp -extensions.
In particular, if F is totally real, Fcyc is the unique Zp -extension of F .

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20196 / 42

Leopoldt Conjecture

Let r1 denote the number of real places of F and r2 the number of(non-conjugate) complex places.
Therefore,[F : Q] = r1 + 2r2 .

Conjecture (Leopoldt Conjecture)
Let F be a number field. Then F admits r2 + 1 independentZp -extensions.
In particular, if F is totally real, Fcyc is the unique Zp -extension of F .
Brumer proved the Leopoldt Conjecture for Abelian extensions F /Q.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20196 / 42

Anti-Cyclotomic Zp -Extension

If F /Q is an imaginary quadratic field, then F admits two linearly
independent Zp -extensions.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20197 / 42

Anti-Cyclotomic Zp -Extension

If F /Q is an imaginary quadratic field, then F admits two linearly
independent Zp -extensions.
One of them is the cyclotomic Zp -extension.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20197 / 42

Anti-Cyclotomic Zp -Extension

If F /Q is an imaginary quadratic field, then F admits two linearly
independent Zp -extensions.
One of them is the cyclotomic Zp -extension. The other is theanti-cyclotomic Zp -extension.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20197 / 42

Anti-Cyclotomic Zp -Extension

If F /Q is an imaginary quadratic field, then F admits two linearly
independent Zp -extensions.
One of them is the cyclotomic Zp -extension. The other is theanti-cyclotomic Zp -extension.
The anti-cyclotomic Zp -extension (denoted Fac /F ) is the uniqueZp -extension of F which is Galois over Q but not Abelian over Q.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20197 / 42Field Diagram
e to be the composite of all Zp -extensions of F .
Denote F

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20198 / 42Field Diagram
e to be the composite of all Zp -extensions of F .
Denote F
If F /Q is an imaginary quadratic extension, thenR. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 20198 / 42Field Diagram
e to be the composite of all Zp -extensions of F .
Denote F
If F /Q is an imaginary quadratic extension, then
e
F
Zp

Zp

Fcyc

Fac
Zp

R. Sujatha (UBC Vancouver)

F
Z/2
Q

Iwasawa Theory

Zp

December 6, 20198 / 42

Equivalent Formulation of the Leopoldt Conjecture

Let F{p} /F be the maximal extension of F unramified outside primes
above p.

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Equivalent Formulation of the Leopoldt Conjecture

Let F{p} /F be the maximal extension of F unramified outside primes
above p.

Theorem
The Leopoldt Conjecture is equivalent to the following assertion:

H 2 Gal F{p} /F , Qp /Zp = 0

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Some Properties of Zp -Extensions

Let F∞ /F be a Zp -extension. Then1

Any prime q - p is unramified in F∞ /F .

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Some Properties of Zp -Extensions

Let F∞ /F be a Zp -extension. Then1

Any prime q - p is unramified in F∞ /F . At least one prime p | pramifies in this extension.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201910 / 42

Some Properties of Zp -Extensions

Let F∞ /F be a Zp -extension. Then1

Any prime q - p is unramified in F∞ /F . At least one prime p | pramifies in this extension.

2

For F∞ = Fcyc , the extension ramifies at every prime p | p.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201910 / 42

Some Properties of Zp -Extensions

Let F∞ /F be a Zp -extension. Then1

Any prime q - p is unramified in F∞ /F . At least one prime p | pramifies in this extension.

2

For F∞ = Fcyc , the extension ramifies at every prime p | p.

3

Qcyc /Q is totally ramified at p.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201910 / 42

Iwasawa algebra
Let G be a profinite group.

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December 6, 201911 / 42

Iwasawa algebra
Let G be a profinite group.
Define the Iwasawa algebraΛ(G) := lim Zp [G/H]←−
where the inverse limit runs over all open normal subgroups H of G andis taken with respect to the natural surjection maps.

R. Sujatha (UBC Vancouver)Iwasawa Theory

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Iwasawa algebra
Let G be a profinite group.
Define the Iwasawa algebraΛ(G) := lim Zp [G/H]←−
where the inverse limit runs over all open normal subgroups H of G andis taken with respect to the natural surjection maps.
Let G = Γ = Gal (Fcyc /F ) ' Zp . Then

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201911 / 42

Iwasawa algebra
Let G be a profinite group.
Define the Iwasawa algebraΛ(G) := lim Zp [G/H]←−
where the inverse limit runs over all open normal subgroups H of G andis taken with respect to the natural surjection maps.
Let G = Γ = Gal (Fcyc /F ) ' Zp . Then∼Λ(Γ) = Zp JΓK −→ Zp JT K
γ 7→ 1 + T

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201911 / 42Pseudo-Isomorphism and Pseudo-Nullity

Let M and N be finitely generated Λ(Γ)-modules.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201912 / 42Pseudo-Isomorphism and Pseudo-Nullity

Let M and N be finitely generated Λ(Γ)-modules.
We say, M is pseudo-isomorphic to N (denoted M ∼ N) if there exists a
Λ(Γ)-homomorphism ϕ : M → N such that both ker(ϕ) and coker(ϕ) are
finite.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201912 / 42Pseudo-Isomorphism and Pseudo-Nullity

Let M and N be finitely generated Λ(Γ)-modules.
We say, M is pseudo-isomorphic to N (denoted M ∼ N) if there exists a
Λ(Γ)-homomorphism ϕ : M → N such that both ker(ϕ) and coker(ϕ) are
finite.
The (Krull) dimension of Λ(Γ) = 2.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201912 / 42Pseudo-Isomorphism and Pseudo-Nullity

Let M and N be finitely generated Λ(Γ)-modules.
We say, M is pseudo-isomorphic to N (denoted M ∼ N) if there exists a
Λ(Γ)-homomorphism ϕ : M → N such that both ker(ϕ) and coker(ϕ) are
finite.
The (Krull) dimension of Λ(Γ) = 2. A finitely generated Λ(Γ)-module M
is called pseudo-null if

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201912 / 42Pseudo-Isomorphism and Pseudo-Nullity

Let M and N be finitely generated Λ(Γ)-modules.
We say, M is pseudo-isomorphic to N (denoted M ∼ N) if there exists a
Λ(Γ)-homomorphism ϕ : M → N such that both ker(ϕ) and coker(ϕ) are
finite.
The (Krull) dimension of Λ(Γ) = 2. A finitely generated Λ(Γ)-module M
is called pseudo-null if
M is finite (equivalently has Krull dimension 0).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201912 / 42Structure Theorem: Iwasawa and Serre

Theorem
Let M be a finitely generated Λ(Γ)-module. Then

!  s
t
M
M
m
M ∼ Λr ⊕Λ/pni ⊕ Λ/fj j 
i=1

j=1

where fj are distinguished polynomials in Zp [T ].

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Invariants for M

The Λ(Γ)-rank of M is r .

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Invariants for M

The Λ(Γ)-rank of M is r .
The µ-invariant is definedµ(M) =

t
X

ni .

i=1

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201914 / 42

Invariants for M

The Λ(Γ)-rank of M is r .
The µ-invariant is definedµ(M) =

t
X

ni .

i=1

The λ-invariant is definedλ(M) =sX

mj deg(fj ).

j=1

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201914 / 42Classical Theorem

Theorem (Iwasawa)
Let F∞ /F be a Zp -extension and let en be the integer so that pen ||hnwhere hn is the order of the class group of Fn . There exist integers
λ, µ ≥ 0 and ν such that
en = λn + µpn + ν
for all n sufficiently large where λ, µ, ν are all independent of n.

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December 6, 201915 / 42

Iwasawa’s Conjecture: Setting up the Diagram
Let F∞ /F be any Zp -extension.

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December 6, 201916 / 42

Iwasawa’s Conjecture: Setting up the Diagram
Let F∞ /F be any Zp -extension.
Mn /Fn is the maximal unramified p-extension.

Hn = Gal Mnab /Fn = p − Hilbert class field of Fn
H = lim Hn←−n

L = lim Mnab←−nR. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201916 / 42

Iwasawa’s Conjecture: Setting up the Diagram
Let F∞ /F be any Zp -extension.
Mn /Fn is the maximal unramified p-extension.

Hn = Gal Mnab /Fn = p − Hilbert class field of Fn
H = lim Hn←−n

L = lim Mnab←−n

Mn0 /Fn is the maximal unramified p-extension completely decomposed
at all primes above p.
Hn0 = Gal

Mn0ab /Fn

H0 = lim Hn0
←
−
n

0

L = lim Mn0←−nR. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201916 / 42Field Diagram

L

F∞

Mn
Mnab
Fn

F

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December 6, 201917 / 42

Iwasawa’s Conjecture: Defining the Modules

L, L0 are Galois extensions over F∞ . Consider the Λ(Γ)-modules

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December 6, 201918 / 42

Iwasawa’s Conjecture: Defining the Modules

L, L0 are Galois extensions over F∞ . Consider the Λ(Γ)-modules
Xnr = Gal (L/F∞ )Xcs = Gal L0 /F∞

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201918 / 42

Iwasawa’s Conjecture: Defining the Modules

L, L0 are Galois extensions over F∞ . Consider the Λ(Γ)-modules
Xnr = Gal (L/F∞ )Xcs = Gal L0 /F∞

Theorem (Iwasawa)
The modules Xnr and Xcs are finitely generated, torsion Λ(Γ)-modules.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201918 / 42

Iwasawa’s Conjecture: Defining the Modules

L, L0 are Galois extensions over F∞ . Consider the Λ(Γ)-modules
Xnr = Gal (L/F∞ )Xcs = Gal L0 /F∞

Theorem (Iwasawa)
The modules Xnr and Xcs are finitely generated, torsion Λ(Γ)-modules.
Therefore, by the Structure Theorem, one can define the µ, λ-invariants.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201918 / 42

Iwasawa’s Conjecture

Conjecture
For the cyclotomic Zp -extension,µ(Xnr ) = 0.

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December 6, 201919 / 42

Iwasawa’s Conjecture

Conjecture
For the cyclotomic Zp -extension,µ(Xnr ) = 0.
Iwasawa proved that when F = Q, µ = λ = ν = 0.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201919 / 42

Iwasawa’s Conjecture

Conjecture
For the cyclotomic Zp -extension,µ(Xnr ) = 0.
Iwasawa proved that when F = Q, µ = λ = ν = 0.
More generally, this holds when F /Q is an Abelian extension by thework of Ferrero-Washington (1979).
Another proof was given by Sinnott (1984).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201919 / 42

Iwasawa’s Conjecture: Equivalent Formulation

Let F be a number field that contains ζp .

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December 6, 201920 / 42

Iwasawa’s Conjecture: Equivalent Formulation

Let F be a number field that contains ζp .

TheoremIwasawa µ = 0 Conjecture is equivalent to the following two assertions
combined:1
H 2 Gal F{p} /F , Qp /Zp = 0 and
2
H 2 Gal F{p} /F , Z/p = 0.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201920 / 42

Vandiver’s Conjecture

A closely related conjecture is the following:

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201921 / 42

Vandiver’s Conjecture

A closely related conjecture is the following:

Conjecture (Vandiver’s Conjecture)
A prime p does not divide the class number of the maximal real sub-fieldof the p-th cyclotomic field Q(ζp ).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201921 / 42

Vandiver’s Conjecture

A closely related conjecture is the following:

Conjecture (Vandiver’s Conjecture)
A prime p does not divide the class number of the maximal real sub-fieldof the p-th cyclotomic field Q(ζp ).
It is known for primes less than 163 million (2008) and in particular is
known for all regular primes.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201921 / 42A generalization due to Greenberg

Conjecture (Greenberg(1971, 1976))
Let F be a totally real field and Fcyc /F be the cyclotomic Zp -extension.
Thenµ(Xnr ) = λ(Xnr ) = 0.
In particular, Xnr is finite.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201922 / 42A generalization due to Greenberg

Conjecture (Greenberg(1971, 1976))
Let F be a totally real field and Fcyc /F be the cyclotomic Zp -extension.
Thenµ(Xnr ) = λ(Xnr ) = 0.
In particular, Xnr is finite.
This conjecture was further generalized (2001). We will study thisconjecture in the next few slides.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201922 / 42Greenberg’s Pseudonullity Conjecture: Setting up thenotatione be the compositum of all Zp -extensions of F .
Let F

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201923 / 42Greenberg’s Pseudonullity Conjecture: Setting up thenotatione be the compositum of all Zp -extensions of F .
Let F
e . Write
Let L/F be a finite Galois extension contained in F
A(L) = p − Sylow subgroup of Cl(L).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201923 / 42Greenberg’s Pseudonullity Conjecture: Setting up thenotatione be the compositum of all Zp -extensions of F .
Let F
e . Write
Let L/F be a finite Galois extension contained in F
A(L) = p − Sylow subgroup of Cl(L).
ConsiderA := lim A(L).
←
−
norm

This can be identified with the maximal Abelian unramified p-extension
e.
of F

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201923 / 42Greenberg’s Pseudonullity Conjecture: Setting up thenotatione be the compositum of all Zp -extensions of F .
Let F
e . Write
Let L/F be a finite Galois extension contained in F
A(L) = p − Sylow subgroup of Cl(L).
ConsiderA := lim A(L).
←
−
norm

This can be identified with the maximal Abelian
 unramified p-extensioneeof F . It is a Λ(G)-module where G = Gal F /F ' Zdp . Here,d ≤ r1 + r2 − 1 (equality iff the Leopoldt Conjecture is true).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201923 / 42Greenberg’s Pseudonullity ConjectureWith notation as introduced in the last slide,Λ(G) ' Zp JT1 , . . . , Td K.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201924 / 42Greenberg’s Pseudonullity ConjectureWith notation as introduced in the last slide,Λ(G) ' Zp JT1 , . . . , Td K.

Theorem (Greenberg)
A is a finitely generated torsion Λ(G)-module.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201924 / 42Greenberg’s Pseudonullity ConjectureWith notation as introduced in the last slide,Λ(G) ' Zp JT1 , . . . , Td K.

Theorem (Greenberg)
A is a finitely generated torsion Λ(G)-module.

Conjecture (Pseudonullity Conjecture)
A is pseudonull, equivalently
dim A ≤ d − 1.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201924 / 42

I WASAWA T HEORY OF E LLIPTIC C URVES

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201925 / 42

Selmer Groups of Elliptic Curves
Consider an elliptic curve E/F and p be an odd prime.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201926 / 42

Selmer Groups of Elliptic Curves
Consider an elliptic curve E/F and p be an odd prime. Let S be a finite
set of primes containing the primes above p, the Archimedean primes
and the primes of bad reduction of E.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201926 / 42

Selmer Groups of Elliptic Curves
Consider an elliptic curve E/F and p be an odd prime. Let S be a finite
set of primes containing the primes above p, the Archimedean primes
and the primes of bad reduction of E.
Let FS /F be the maximal extension unramified outside S.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201926 / 42

Selmer Groups of Elliptic Curves
Consider an elliptic curve E/F and p be an odd prime. Let S be a finite
set of primes containing the primes above p, the Archimedean primes
and the primes of bad reduction of E.
Let FS /F be the maximal extension unramified outside S.
The Selmer group of E/L for a finite Galois extension L/F contained in
FS is given by the exact sequence
λL
0 → Sel(E/L) → H 1 (Gal (FS /L) , Ep∞ ) −→M

Jv (Ep∞ /L)

v ∈S

where
Jv (Ep∞ /L) =M

H 1 (Lw , E) (p).

w|v

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201926 / 42Comments

1

The Galois group Gal(L/F ) acts on H 1 (Gal (FS /L) , Ep∞ ) andJv (Ep∞ /L).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201927 / 42Comments

1

The Galois group Gal(L/F ) acts on H 1 (Gal (FS /L) , Ep∞ ) andJv (Ep∞ /L). Therefore, the Selmer group is endowed with a natural
Galois action.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201927 / 42Comments

1

2

The Galois group Gal(L/F ) acts on H 1 (Gal (FS /L) , Ep∞ ) andJv (Ep∞ /L). Therefore, the Selmer group is endowed with a natural
Galois action.
There is an analogous exact sequence by taking direct limits

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201927 / 42Comments

1

2

The Galois group Gal(L/F ) acts on H 1 (Gal (FS /L) , Ep∞ ) andJv (Ep∞ /L). Therefore, the Selmer group is endowed with a natural
Galois action.
There is an analogous exact sequence by taking direct limits
λ0 → Sel(E/F∞ ) → H 1 (Gal (FS /F∞ ) , Ep∞ ) −−∞→

M

Jv (Ep∞ /F∞ ) .

v ∈S

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201927 / 42F INE S ELMER G ROUPS

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201928 / 42Fine Selmer GroupWe define!
1

R(E/L) := ker H (Gal (FS /L) , Ep∞ ) →M

Kv1 (Ep∞ /L)

v ∈S

where
Kv1 (Ep∞ /L) =M

H 1 (Lw , Ep∞ ) .

w|v

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201929 / 42Fine Selmer GroupWe define!
1

R(E/L) := ker H (Gal (FS /L) , Ep∞ ) →M

Kv1 (Ep∞ /L)

v ∈S

where
Kv1 (Ep∞ /L) =M

H 1 (Lw , Ep∞ ) .

w|v

Taking direct limits as before, define
R(E/F∞ ) := lim R(E/L)−→
L

where L runs over all finite extensions of F contained in F∞ .

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201929 / 42

Some important sequences

1

κ

λ

0 → E(L) ⊗ Qp /Zp −→ Sel(E/L) −→ X(E/L)(p) → 0

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201930 / 42

Some important sequences

1

κ

λ

0 → E(L) ⊗ Qp /Zp −→ Sel(E/L) −→ X(E/L)(p) → 0
2

0 → R(E/L) → Sel(E/L) →M

E(Lw )p∞ ⊗ Qp /Zp

w|p

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201930 / 42

Pontryagin dual

Let G = Gal (F∞ /F ) be any pro-p, p-adic Lie group and Λ(G) be thecorresponding Iwasawa algebra.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201931 / 42

Pontryagin dual

Let G = Gal (F∞ /F ) be any pro-p, p-adic Lie group and Λ(G) be thecorresponding Iwasawa algebra. For example, G = Γ as before.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201931 / 42

Pontryagin dual

Let G = Gal (F∞ /F ) be any pro-p, p-adic Lie group and Λ(G) be thecorresponding Iwasawa algebra. For example, G = Γ as before.
Sel(E/F∞ ) and R(E/F∞ ) are finitely generated discrete Λ(G)-modules.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201931 / 42

Pontryagin dual

Let G = Gal (F∞ /F ) be any pro-p, p-adic Lie group and Λ(G) be thecorresponding Iwasawa algebra. For example, G = Γ as before.
Sel(E/F∞ ) and R(E/F∞ ) are finitely generated discrete Λ(G)-modules.
We work with the Pontryagin duals which makes them compact. These
are denoted X (E/F∞ ) and Y (E/F∞ ), respectively.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201931 / 42

Pontryagin dual

Let G = Gal (F∞ /F ) be any pro-p, p-adic Lie group and Λ(G) be thecorresponding Iwasawa algebra. For example, G = Γ as before.
Sel(E/F∞ ) and R(E/F∞ ) are finitely generated discrete Λ(G)-modules.
We work with the Pontryagin duals which makes them compact. These
are denoted X (E/F∞ ) and Y (E/F∞ ), respectively.
The Pontryagin dual of a p-primary module M is defined as
M ∨ = Hom (M, Qp /Zp ) .

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201931 / 42

C ONJECTURES

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201932 / 42

Conjecture of Mazur

Conjecture
Suppose E/F is an elliptic curve with good ordinary reduction at allprimes above p. Then X (E/Fcyc ) is a finitely generated torsion
Λ(Γ)-module.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201933 / 42

Conjecture of Mazur

Conjecture
Suppose E/F is an elliptic curve with good ordinary reduction at allprimes above p. Then X (E/Fcyc ) is a finitely generated torsion
Λ(Γ)-module.
By a deep result of Kato, this is now known in some cases such as
when F = Q or an Abelian extension.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201933 / 42

Conjecture of Mazur

Conjecture
Suppose E/F is an elliptic curve with good ordinary reduction at allprimes above p. Then X (E/Fcyc ) is a finitely generated torsion
Λ(Γ)-module.
By a deep result of Kato, this is now known in some cases such as
when F = Q or an Abelian extension.
For dual Selmer groups of elliptic curves over Q, we therefore have thestructure theorem.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201933 / 42

Conjecture of Mazur

Conjecture
Suppose E/F is an elliptic curve with good ordinary reduction at allprimes above p. Then X (E/Fcyc ) is a finitely generated torsion
Λ(Γ)-module.
By a deep result of Kato, this is now known in some cases such as
when F = Q or an Abelian extension.
For dual Selmer groups of elliptic curves over Q, we therefore have thestructure theorem. But there are lots of examples of elliptic curves withpositive µ-invariant.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201933 / 42Analogue of the Weak Leopoldt Conjecture

Conjecture
Let E/F be an elliptic curve and p be an odd prime. For any
Zp -extension F∞ /F ,H 2 (FS /F∞ , Ep∞ ) = 0.
Equivalently, the (dual) fine Selmer group is Λ(Γ)-torsion.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201934 / 42Analogue of the Weak Leopoldt Conjecture

Conjecture
Let E/F be an elliptic curve and p be an odd prime. For any
Zp -extension F∞ /F ,H 2 (FS /F∞ , Ep∞ ) = 0.
Equivalently, the (dual) fine Selmer group is Λ(Γ)-torsion.
The equivalence of the two statements was shown by Perrin-Riou.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201934 / 42

Conjecture A

Conjecture (Coates-S.)
Let E be an elliptic curve over F and p be an odd prime. Y (E/Fcyc ) is
finitely generated as a Zp -module.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201935 / 42

Conjecture A

Conjecture (Coates-S.)
Let E be an elliptic curve over F and p be an odd prime. Y (E/Fcyc ) is
finitely generated as a Zp -module.
Equivalently, the elliptic curve analogue of the weak Leopoldt
Conjecture holds and µ(Y (E/Fcyc )) = 0.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201935 / 42

Conjecture A

Conjecture (Coates-S.)
Let E be an elliptic curve over F and p be an odd prime. Y (E/Fcyc ) is
finitely generated as a Zp -module.
Equivalently, the elliptic curve analogue of the weak Leopoldt
Conjecture holds and µ(Y (E/Fcyc )) = 0.
This conjecture is to be viewed as an analogue of Iwasawa’s µ = 0
Conjecture for the case of elliptic curves.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201935 / 42

Conjecture B

Let F∞ /F be an admissible p-adic Lie extension, i.e.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201936 / 42

Conjecture B

Let F∞ /F be an admissible p-adic Lie extension, i.e.
1

Fcyc ⊂ F∞ ⊂ FS .

2

Gal(FS /F ) = G is pro-p with no elements of order p.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201936 / 42

Conjecture B

Let F∞ /F be an admissible p-adic Lie extension, i.e.
1

Fcyc ⊂ F∞ ⊂ FS .

2

Gal(FS /F ) = G is pro-p with no elements of order p.

Conjecture (Coates-S.)
Suppose Conjecture A holds for E/Fcyc and G has dimension strictly
larger than 1 as a p-adic Lie group, then Y (E/F∞ ) is a pseudonull
Λ(G)-module.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201936 / 42

Conjecture B

Let F∞ /F be an admissible p-adic Lie extension, i.e.
1

Fcyc ⊂ F∞ ⊂ FS .

2

Gal(FS /F ) = G is pro-p with no elements of order p.

Conjecture (Coates-S.)
Suppose Conjecture A holds for E/Fcyc and G has dimension strictly
larger than 1 as a p-adic Lie group, then Y (E/F∞ ) is a pseudonull
Λ(G)-module.
This echoes Greenberg’s pseudonullity conjecture in the context of
elliptic curves.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201936 / 42

R ECENT R ESULTS

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201937 / 42Recent Evidence towards Conjecture A

Theorem (K.-S.)
Let F be a number field and E be an elliptic curve of rank 0 over F .
Assume that the Shafarevich-Tate group of E/F is finite. Varying overprimes of good ordinary reduction, Sel(E/Fcyc )(p) is trivial for all primes
outside a set of density 0.
In particular, Conjecture A holds for Y (E/Fcyc ).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201938 / 42Recent Evidence towards Conjecture A

Theorem (K.-S.)
Let F be a number field and E be an elliptic curve of rank 0 over F .
Assume that the Shafarevich-Tate group of E/F is finite. Varying overprimes of good ordinary reduction, Sel(E/Fcyc )(p) is trivial for all primes
outside a set of density 0.
In particular, Conjecture A holds for Y (E/Fcyc ).
This result was first proven for F = Q by Greenberg. To extend this to
the general number fields case it was necessary to use an effective
Chebotarev density result of Kumar Murty.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201938 / 42Evidence for Conjecture B: CM case

Theorem (K.-S.)
Let E be a CM elliptic curve defined over a number field F and p be an
odd prime of good ordinary reduction.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201939 / 42Evidence for Conjecture B: CM case

Theorem (K.-S.)
Let E be a CM elliptic curve defined over a number field F and p be an
odd prime of good ordinary reduction.
Let F∞ = F (Ep∞ ). In this case, Gal(F∞ /F ) contains an open subgroupwhich is Abelian and isomorphic to Zp × Zp . Further, assume
G = Gal(F∞ /F ) is pro-p.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201939 / 42Evidence for Conjecture B: CM case

Theorem (K.-S.)
Let E be a CM elliptic curve defined over a number field F and p be an
odd prime of good ordinary reduction.
Let F∞ = F (Ep∞ ). In this case, Gal(F∞ /F ) contains an open subgroupwhich is Abelian and isomorphic to Zp × Zp . Further, assume
G = Gal(F∞ /F ) is pro-p.
If Y (E/Fcyc ) is finite, Y (E/F∞ ) is a pseudonull Λ(G)-module.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201939 / 42Evidence for Conjecture B: non-CM case for regularprimes

Theorem (K.-S.)
Let E be an elliptic curve defined over Q. Set F = Q(µp ) such that p isa regular prime. Then Conjecture B is true for Y (E/Q(Ep∞ )).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201940 / 42

Specific Example

Consider the elliptic curve E/Q defined by
E : y 2 + xy = x 3 − 3x − 3.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201941 / 42

Specific Example

Consider the elliptic curve E/Q defined by
E : y 2 + xy = x 3 − 3x − 3.
This is an elliptic curve of conductor 150 without CM.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201941 / 42

Specific Example

Consider the elliptic curve E/Q defined by
E : y 2 + xy = x 3 − 3x − 3.
This is an elliptic curve of conductor 150 without CM. With p = 5,
F∞ = Q(E5∞ ) is a pro-5 extension of F = Q(µ5 ).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201941 / 42

Specific Example

Consider the elliptic curve E/Q defined by
E : y 2 + xy = x 3 − 3x − 3.
This is an elliptic curve of conductor 150 without CM. With p = 5,
F∞ = Q(E5∞ ) is a pro-5 extension of F = Q(µ5 ).
It was shown in Coates-S. (2005) that either Conjecture B holds or there
is no point of infinite order over F∞ . The latter possibility was not ruledout in the intervening years, despite advances in computational
methods.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201941 / 42

Specific Example

Consider the elliptic curve E/Q defined by
E : y 2 + xy = x 3 − 3x − 3.
This is an elliptic curve of conductor 150 without CM. With p = 5,
F∞ = Q(E5∞ ) is a pro-5 extension of F = Q(µ5 ).
It was shown in Coates-S. (2005) that either Conjecture B holds or there
is no point of infinite order over F∞ . The latter possibility was not ruledout in the intervening years, despite advances in computational
methods.
Our theorem settles this example theoretically.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201941 / 42Relating Greenberg’s Conjecture with Conjecture B

Theorem (K.-S.)
Let E/F be an elliptic curve and p be a fixed odd prime.

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201942 / 42Relating Greenberg’s Conjecture with Conjecture B

Theorem (K.-S.)
Let E/F be an elliptic curve and p be a fixed odd prime. Lete be an admissible extension of F .
L = F∞ = F (Ep∞ ) or F

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201942 / 42Relating Greenberg’s Conjecture with Conjecture B

Theorem (K.-S.)
Let E/F be an elliptic curve and p be a fixed odd prime. Lete be an admissible extension of F .
L = F∞ = F (Ep∞ ) or F
L
Then Xnr is pseudonull (i.e. Greenberg’s Conjecture holds) if and only if
Conjecture B holds for Y (E/L).

R. Sujatha (UBC Vancouver)Iwasawa Theory

December 6, 201942 / 42