AN INTRODUCTION TO
IWASAWA THEORY FOR ELLIPTIC CURVES
Abstract. We present the first few sections of Greenberg’s article “Introduction to Iwasawa
Theory for Elliptic Curves”. In the process, we review the construction of Zp -extensions of
the rational numbers, discuss Iwasawa’s result on the growth of the class group in towers of
cyclotomic extensions, and analyze the notion of a Selmer group of an elliptic curve defined
over a number field.
Part I
Zp -Extensions. Let p be a prime number. Denote Kn = Q(ζpn+1 ) as the splitting field of
n+1
the polynomial xp − 1 over Q. The Galois group is known explicitly from the isomorphism
×
∼
Z/pn+1 Z
−−−→
Gal Kn /Q
a mod pn+1 −−−→ σa : ζpn+1 7→ ζpan+1
We have two short exact sequences of interest: one coming from the Teichmüller Character
×
ω
{0} −−−→ Z/pn Z −−−→
Z/pn+1 Z
−−−→
F×
−−−→ {1}
p
a mod pn −−−→ 1 + p a mod pn+1
ω
n
−−−→ bp mod pn+1
b mod pn+1
and another coming from surjecting onto the Sylow p-Subgroup.
×
n+1
Z
−−−→
Z/pn Z
−−−→ {0}
{1} −−−→ F×
p −−−→ Z/p
b mod pn+1 −−−→ bp−1 mod pn+1
×
Here Z/pn Z is an additive subgroup of the multiplicative
group Z/pn+1 Z . Let Fn =
P
Q(αn ) be the field generated by the resolvent αn = a∈F×p ζpan+1 , so that we have the field
diagram
{1}
Kn
F×
p
Fn
Z/pn Z
K0 = Q(ζp )
×
Z/pn+1 Z
where the boldSlines denote the groups
Gal
F
/F
'
Gal
K
/K
' Z/pn Z. Upon den
0
n
0
S
noting K∞ = n≥0 Kn and F∞ = n≥0 Fn as the respective composite fields, we have the
Galois groups
Gal F∞ /Q ' Gal K∞ /Q(ζp ) ' Zp .
F0 = Q
1
We call both K∞ = Q(ζp∞ ) and F∞ cyclotomic Zp -extensions of Q. In general, given a field
F0 = F , we wish to create a tower F0 ⊆ F1 ⊆ · · · ⊆
F
⊆
·
·
·
such
that
Gal
F
/F
'
n
n
0
S
Z/pn Z. If we denote F∞ = n≥0 Fn , then Gal F∞ /F ' Zp .
Remark. One can create more exotic extensions than Zp . The multiplicative
group Gm
over the field K = Q yields the finite extension Kn = Q Gm [pn+1 ] = Q(ζpn+1 ), so that
×
Gal Kn /K ' Z/pn+1 Z . Hence Gal K∞ /K ' GL1 (Z). We say that the compositum
K∞ = Q(ζp∞ ) is a GL(1)-extension of Q. More generally, given an abelian variety
E of
n+1
dimension d over a number field
K, denote the finite extension Kn = K E[p ] . Then
Gal Kn /K ,→ GL2d Z/pn+1 Z coming from the action on the Tate module, so that the
compositum K∞ = K E[p∞ ] is contained in an GL(2d)-extension of K.
Relation with Fermat’s Last Theorem. The following was first conjectured in 1635 by
Pierre de Fermat (August 17, 1601 – January 12, 1665).
Conjecture (Fermat, 1635). For integral exponents n > 2, the integer
solutions a, b, and c to the equation an + bn = cn must satisfy a b c = 0.
As noted in Wikipedia, “[this] famous Last Theorem was first discovered by [Fermat’s] son
in the margin on his father’s copy of an edition of Diophantus, and included the statement
that the margin was too small to include the proof.” Indeed, to prove such a conjecture, one
need only consider the odd prime exponents p. If not n is not divisible by any odd prime,
then n = 2m+2 is a power of 2, and one can write

2m

x = a m
2m
y = b2
x4 + y 4 = z 4
where
=⇒
xyz = abc .


m
z = c2
Fermat himself showed (albeit posthumously) that x y z = 0 using the Method of Infinite
Descent, so that a b c = 0. On the other hand, if n is divisible by an odd prime p, then one
can write


x = an/p


n/p
xp + y p = z p
where
=⇒
xyz = abc
.
y = bn/p


 z = cn/p
In 1770, Leonhard Euler (April 15, 1707 – September 18, 1783) proved the conjecture for
p = 3; in 1825, Peter Dirichlet (February 13, 1805 – May 5, 1859) proved it for p = 5; and
in 1839, Gabriel Lamé (July 22, 1795 – May 1, 1870) proved in for p = 7. In fact, in 1847,
Lamé attempted to prove
for all odd primes p by factoring over the ring Z[ζp ].
Qp the conjecture
j
The idea was to write j=1 (x + ζp y) = z p , then conclude that each factor x + ζpj y = αjp for
some element αj ∈ Z[ζp ]. Unfortunately, Ernst Kummer (January 29, 1810 – May 14, 1893)
had shown some years before that unique factorization fails in this ring – and therefore such
a conclusion was not valid. In 1850, Kummer showed the following.
Theorem (Kummer, 1850). Assume that p is an odd prime that does not
divide the order of the class group of Z[ζp ]. Then the conjecture is valid.
2
Using his theory of ideal class groups for the cyclotomic field K = Q(ζp ), he was able to
show that each ideal x + ζpj y OK = apj for some ideal aj in OK = Z[ζp ]. The assumption
allowed him to conclude aj = αj OK for some element αj ∈ Z[ζp ] because apj = z p OK is a
principal ideal, so that each factor x + ζpj y = αjp . He was able to formulate a contradiction
from this point.
Remark. Any odd prime p satisfying the assumption in the theorem is called a regular
prime; any prime that divides the order of the class group Z[ζp ] is called an irregular prime.
Kummer calculated that the only irregular primes p < 165 are p = 37, 59, 67, 101, 103, 131,
or 149. It is known that there are infinitely many irregular primes; it is not known whether
there are infinitely many regular primes, but it appears that approximately e−1/2 ≈ 61% of
primes are regular. Hence Kummer settled many cases of the conjecture – but unfortunately
not infinitely many cases.
Iwasawa’s Results. Let me explain more about the class group mentioned above. Let K
be a number field with ring of integers OK ; in practice, we will choose K = Q(ζpn+1 ) and
OK = Z[ζpn+1 ]. Recall that we say a is a (fractional) ideal of K if a is a nonzero, finitely
generated OK -submodule of K. Equivalently, it is a nonzero subset of K such that x a ⊆ OK
for some nonzero x ∈ OK . A (fractional) ideal a of K is said to be principal if a = α OK for
some nonzero α ∈ K. The following is a standard result from Algebraic Number Theory.
Proposition.
• The collection IK of (fractional) ideals a of K forms an abelian group
under multiplication. Explicitly,
( n
)
X
−1
a·b=
αi βi αi ∈ a, βi ∈ b
and a = x ∈ K x a ⊆ OK .
i=1
• The collection PK of principal ideals α OK forms an abelian subgroup of
IK .
• The (ideal) class group C`(OK ) = IK /PK of K is a is finite group.
Moreover, we have an exact sequence
×
{1} −−−→ OK
−−−→ K × −−−→
IK
−−−→ C`(OK ) −−−→ {1}
α −−−→ α OK
We call the order hK = C`(OK ) = IK : PK of the (ideal) class group the class number.
The following explains the usefulness of considering regular primes.
Corollary. Assume that a ∈ IK satisfies ad ∈ PK for some d relatively prime
to hK . Then a ∈ PK .
We explain why. Since the class group has finite order hK , we see that ahk ∈ PK . Write
r
s
r hK + s d = 1 for some integers r and s. But then a = ahK · ad ∈ PK . In the specific
case where
d = p: if p is a regular prime, then it is relatively prime
K = Q(ζ
p ), wep choose
p
to hK = C`(Z[ζp ]) . If a = z OK then ap ∈ PK , so a ∈ PK , showing that a = α OK is
principal.
The conclusion here is we can prove the conjecture when we know about the elements of ppower order in the (ideal) class group C`(Z[ζp ]). In the 1950’s, Kenkichi Iwasawa (September
11, 1917 – October 26, 1998) studied precisely this group. Allow me to state the precise
3
result. Denote Kn = Q(ζpn+1 ) so that OKn = Z[ζpn+1 ] is its ring of integers. We can compute
the group IKn of (fractional) ideals, the subgroup PKn of principal ideals, and the quotient
group C`(Z[ζpn+1 ]) = IKn /PKn as the (ideal) class group. Let en denote the exponent
p
of
∞ en
which exactly divides the order of the (ideal) class group, that is, p = C` Z[ζpn+1 ] [p ]
in terms of the subgroup
pn
∞
C`(OK )[p ] = a ∈ IK /PK a ∈ PK for some n ≥ 1 ⊆ C`(OK ).
Iwasawa’s main result is that the exponent en does not grow too “quickly.”
Theorem (Iwasawa, 1950’s). There exist integers µ, λ, and ν, only dependent on p, such that en = µ pn + λ n + ν for “sufficiently large” n.
In fact, Iwasawa showed that µ = ν = 0.
Remark. One can attempt to replace Kn = Q(ζpn+1 ) with an arbitrary Zp -extension of number fields. Indeed, we have seen that this example can be constructed via the multiplicative
group Gm over K = Q. Given a number field F0 = F , we can always create a tower of number fields F0 ⊆ F1 ⊆ · · · ⊆ Fn ⊆ · · · such that we have the group Gal Fn /F0 ' Z/pn Z. We
can define the group IFn of (fractional) ideals, the subgroup PFn of principal ideals, and the
quotient group C`(OFn ) = IFn /PFn as the (ideal) class group. Write pen = C`(OFn )[p∞ ].
The main question is whether there exist integers µ, λ, and ν, only dependent on the extension F∞ /F , such that en = µ pn + λ n + ν for “sufficiently large” n. We’ll discuss this
throughout the seminar!
Part II
Class Field Theory. In order to generalize Iwasawa’s ideas for elliptic curves, we discuss
a different way to view the (ideal) class group. We recall a result shown by Johann Peter
Gustav Lejeune Dirichlet (February 13, 1805 – May 5, 1859).
Theorem (Dirichlet, 1846). Let L be a number field with ring of integers
OL . Then U (L) = OL× is a finitely generated abelian group under multiplicar
tion. More precisely, U (L) ' U (L)tors
× Z where U (L)tors is a finite group of
roots of unity and r = rankZ U (L) = r1 + r2 − 1 is a nonnegative integer in
terms of
L ⊗Q R = R
· · × R} × C
· · × C} .
| × ·{z
| × ·{z
r1 copies
r2 copies
This is known as Dirichlet’s Unit Theorem. The set U (L) = OL× is called the unit group of
L; this is a subgroup of the multiplicative group Gm (L) = L× . For each positive integer n,
the map α 7→ αn induces a short exact sequence
n
{1} −−−→ µn −−−→ U (L) −−−→ U (L) −−−→ {1}.
Each of these groups can be viewed as a Z-module with action by GL = Gal(L/L), and
hence action by the decomposition group GLv = Gal(Lv /Lv ) corresponding to a choice of
embedding L ,→ Lv for each place v. This induces the following exact diagram coming from
4
group cohomology:
{1} −−−→
OL×
κ
−−−→
× n
OL


y
{1}
n
H 1 GL , U (L) [n] −−−→ {0}


y
Y
Y
∼
H 1 GLv , U (Lv )
−−−→
H 1 GLv , U (Lv ) −−−→
H 1 GL , µn


y
−−−→
v
v
√ √
Here, κ is the Kummer map which sends α to that cohomology class ξ : σ 7→ σ n α / n α.
We review a fundamental result which culminates in the study of Local Class Field Theory.
Theorem. Let U (L) = OL× denote the unit group
of a number field L. There
−1
1
exists a map λ : C`(OL ) → H GL , U (L) which induces an isomorphism
"
#
Y
1
1
C`(OL ) ' ker H GL , U (L) →
H GLv , U (Lv ) .
v
This requires some explanation because it is not a standard result in the literature. Let H
denote the Hilbert class field of L, and denote GH/L = Gal(H/L) as its Galois group. The
map α 7→ α OH induces the short exact sequence
{1} −−−→ U (H) −−−→ Gm (H) −−−→ PH −−−→ {1}
of GH/L -modules, which yields the exact sequence
Gm (L) −−−→ H 0 GH/L , PH
λ−1
−−−→ H 1 GH/L , U (H) −−−→ H 1 GH/L , Gm (H) .
Every fractional ideal a of L becomes
principal in H, so that a OH = α OH ; hence the GH/L
0
1
invariant ideals H GH/L , PH = IL . Hilbert’s Theorem 90 asserts H GH/L , Gm (H) =
{0}, so that H 1 GH/L , U (H) ' IL /PL = C`(OL ). Since H/L is an unramified extension,
and the short exact sequence
ord
v
{1} −−−→ U (Lv ) −−−→ Gm (Lv ) −−−→
Q −−−→ {0}
comes from those ramified extensions, the composition
Inf
λ−1
C`(OL ) −−−→ H 1 GH/L , U (H) −−−→ H 1 GL , U (L) −−−→ H 1 GLv , U (Lv )
must be trivial for each place v. (I am thankful to Jiu-Kang Yu for assisting with this
argument.)
Selmer Groups for Number Fields. This result motivates a definition. We define the
n-Selmer group of a number field L as the abelian group
#
"
Y
(n)
Sel (L) = ker H 1 GL , µn →
H 1 GLv , U (Lv )
v
5
so that we have the following exact diagram:
{1} −−−→
{1} −−−→
OL×
κ
−−−→
× n
OL


y
Sel(n) (L)


y
OL×
κ
−−−→
× n
OL


y
H 1 GL , µn


y
{1}
λ
−−−→
C`(OL )[n]

 −1
yλ
−−−→ {1}
n
H 1 GL , U (L) [n] −−−→ {0}


y
Y
Y
∼
−−−→
H 1 GLv , U (Lv ) −−−→
H 1 GLv , U (Lv )
v
−−−→
v
× n
As the quotient OL× / OL and class group C`(OL ) are finite groups, the n-Selmer group
Sel(n) (L) is a finite group as well. In what follows, we would like to find a similar short exact
sequence which does not assume L/Q is finite nor depend on a choice of positive integer n.
Remark. If L is a number
the nth roots of unity µn , then Kummer Theory
field containing
n
asserts that H 1 GL , µn ' L× / L× . In this case, the n-Selmer group can be realized as
)
(
×
L
n α OL = an for some a ∈ IL .
Sel(n) (L) ' α ∈
×
L
This definition appears in Franz Lemmermeyer’s paper Selmer Groups and Quadratic Reciprocity, which appeared online at http://arxiv.org/abs/1108.5674v1.
Elliptic Curves. Continue to let L be a number field, and consider an elliptic curve E
defined over L. That is, we consider the abelian variety
E:
y 2 + a1 x y + a3 y = x3 + a2 x2 + a4 x + a6
of dimension d = 1, where each coefficient ai ∈ L. Then the set
)
(
x2 x0 + a1 x1 x2 x0 + a3 x2 x2
0
2
E(L) = (x1 : x2 : x0 ) ∈ P2 (L) 3
2
= x1 + a2 x1 x0 + a4 x1 x20 + a6 x30
is an abelian group. In 1908, Jules Henri Poincaré (April, 29 1854 – July 17, 1912) conjectured bit more:
Theorem (Mordell, 1922; Weil, 1928). E(L) is a finitely generated
abelian group under addition. That is, E(L) ' E(L)tors × Zr where E(L)tors
is a finite group, and r = rankZ E(L) is a nonnegative integer.
This was shown in 1922 by Louis Joel Mordell (January 28, 1888 – March 12, 1972) for
L = Q, then generalized in 1928 to arbitrary number fields L (and abelian varieties of
arbitrary dimension d) by André Weil (May 6, 1906 – August 6, 1998). The proof relies on
the fact that the quotient group E(L)/n E(L) is finite.
We’ve seen that the unit group U (L) = OL× and the Mordell-Weil group E(L) are both
finitely generated abelian groups, so we make a general observation. Say that Γ ' Γtors × Zr .
6
For any positive integer n, we have the quotient
r
1 Z
Γtors
Z
Z
Γ
'
×
' Γ ⊗Z
' Γ ⊗Z n
nΓ
n Γtors
nZ
nZ
Z
as a finite group. Note that we have different structures depending on whether we take the
inverse limit or the direct limit:
Γ
Γ
lim n ' Γ ⊗Z Zp
yet
lim n ' Γ ⊗Z Qp /Zp .
←− p Γ
−→ p Γ
n
n
Both are Zp -modules, but only the one on the left is finitely generated. We’ll return to the
modules
OL×
E(L)
×
lim
' E(L) ⊗Z Zp .
and
lim
n ' OL ⊗Z Zp
p
n
←− O×
←− p E(L)
n
n
L
Towers of Number Fields. We would like to know if we can generalize the Mordell-Weil
Theorem to extensions larger than number fields – such as those extensions K/Q which are
algebraic. To this end, we have the following deep result.
Theorem. Denote K = Qab
Σ as the compositum of those finite, abelian extensions L of Q which are unramified outside a finite set Σ = {p1 , p2 , . . . , pn }
of primes pi . Let E be an elliptic curve defined over Q.
• E(K)tors is a finite group.
• rankZ E(L) is uniformly bounded.
• E(K) is a finitely generated abelian group.
The first statement was shown in 1981 by Kenneth Alan Ribet (June 28, 1948 – ). The
second was shown in a series of papers and preprints in the 1980’s by Kazuya Kato (January
17, 1952 – ) and David Ephraim Rohrlich (?? – ). (We also need the result that elliptic
curves E over Q are modular.) We show the third statement following a proof due to Barry
Charles Mazur (December 19, 1937 – ). Let L be a finite, abelian extension of Q, unramified
outside of Σ, such that rankZ E(L) is maximal. Then the quotient group E(K)/E(L) is
torsion, so that for each P ∈ E(K) we can find a positive integer m such that [m]P ∈ E(L).
Hence we have a well-defined perfect pairing
Gal(K/L) ×
E(K)
−→ E(K)tors
E(L)
which sends
σ, P
7→ σ(P ) − P.
But this induces
7 [t]P where
a group homomorphism E(K)/E(K)tors → E(L) defined by P →
t = E(K)tors . Now the Mordell-Weil Theorem asserts that E(L) is finitely generated, so
that E(K) must be as well.
Remark. Here is a special case of this result. Let K∞ = Q(ζp∞ ) be the compositum of the
cyclotomic fields Kn = Q(ζpn+1 ). Since each Kn is a finite, abelian extension of Q which is
unramified outside of Σ = {p}, we see that K∞ = Qab
Σ in this case. The result above implies
that, for any given elliptic curve E over Q, there exists some nonnegative integer n0 such
that
rankZ E(Kn ) = rankZ E(Kn0 )
for all n ≥ n0 .
7
Selmer Groups for Elliptic Curves. Let E be an elliptic curve over Q. Recall that
the Mordell-Weil Theorem for number fields L used the statement that, for any positive
integer n, the quotient E(L)/n E(L) is finite. We discuss how to generalize this to algebraic
extensions K over Q.
With K as the algebraic closure of K, we have the following short-exact sequence of abelian
groups:
[n]
{OE } −−−→ E[n] −−−→ E(K) −−−→ E(K) −−−→ {OE }.
Each of these groups can be viewed as a Z-module with action by GK = Gal(K/K), and
hence action by the decomposition group GKv = Gal(K v /Kv ) corresponding to a choice of
embedding K ,→ K v for each place v. This induces the following exact diagram coming from
Galois cohomology:
{OE } −−−→
{OE } −−−→
E(K)
κ
−−−→
n E(K)


y
SelE (K)


y
E(K)
κ
−−−→
n E(K)


y
H 1 GK , E[n]


y
{OE }
λ
(n)
−−−→
XE (K)[n]


y
[n]
−−−→ {0}
[n]
H 1 GK , E(K) [n] −−−→ {0}


y
Y
Y
∼
H 1 GKv , E(K v )
H 1 GKv , E(K v ) −−−→
−−−→
λ
−−−→
v
v
Here we replace E(K)/n E(K) ' E(K) ⊗Z
E(K)
lim
' E(K) ⊗Z Q/Z
−→ n E(K)
1
Z
/Z
to find the direct limits
n
and
n
E(K)
lim n
' E(K) ⊗Z Qp /Zp .
−→ p E(K)
n
This can be summarized as the following.
Proposition. We have the short exact sequences
κ
λ
{OE } −−−→ E(K) ⊗Z Qp /Zp −−−→ SelE (K)p −−−→ XE (K)[p∞ ] −−−→ {0}






y
y
y
κ
λ
{OE } −−−→ E(K) ⊗Z Q/Z −−−→ SelE (K) −−−→ XE (K) −−−→ {0}
in terms of the p-primary Selmer group
"
SelE (K)p =
(pn )
lim Sel (K)
−→ E
n
#
= ker H 1 GK , E[p∞ ] →
Y
H 1 GKv , E(K)
v
and the Tate-Shafarevich group
"
#
Y
XE (K) = ker H 1 GK , E(K) →
H 1 GKv , E(K) .
v
8
×
Remark. A similar result holds for the class group. Recall that if we denote U (K) = OK
n
as the unit group of K, then the map α 7→ α induces a short exact sequence
n
{1} −−−→ µn −−−→ U (K) −−−→ U (K) −−−→ {1}.
The following becomes clear.
Proposition. We have the short exact sequences
κ
λ
×
{1} −−−→ OK
⊗Z Qp /Zp −−−→ Sel(K)p −−−→ C`(OK )[p∞ ] −−−→ {0}






y
y
y
κ
λ
×
{1} −−−→ OK
⊗Z Q/Z −−−→ Sel(K) −−−→ C`(OK ) −−−→ {0}
in terms of the p-primary Selmer group
"
(pn )
Sel(K)p = lim Sel
−→
#
(K) = ker H 1 GK , µp∞ →
n
Y
H 1 GKv , U (K)
v
Generalized Selmer Groups. We conclude the presentation by offering a general definition for Selmer groups and Tate-Shafarevich groups. Let K/Q be an algebraic extension,
and Γ be a divisible abelian group, that is, a group for which the sequence
n
{OΓ } −−−→ Γ[n] −−−→ Γ −−−→ Γ −−−→ {OΓ }
is exact for each positive integer n. We will assume that σ(P ⊕ Q) = σ(P ) ⊕ σ(Q) for all
P, Q ∈ Γ and σ ∈ Gal(K/K), so that the sequence above holds as Gal(K/K)-modules.
Using the arguments above, we have the short exact sequences
κ
λ
{OΓ } −−−→ Γ(K) ⊗Z Qp /Zp −−−→ SelΓ (K)p −−−→ XΓ (K)[p∞ ] −−−→ {0}






y
y
y
κ
λ
{OΓ } −−−→ Γ(K) ⊗Z Q/Z −−−→ SelΓ (K) −−−→ XΓ (K) −−−→ {0}
in terms of the K-rational points Γ(K) = H 0 Gal(K/K), Γ , the p-primary Selmer group
"
#
Y
SelΓ (K)p = ker H 1 GK , Γ[p∞ ] →
H 1 GKv , Γ
v
and the Tate-Shafarevich group
"
#
Y
XΓ (K) = ker H 1 GK , Γ →
H 1 GKv , Γ .
v
Clearly, these assumptions are true for Γ = U (Q), the units in Z, and for Γ = E(Q) relative
to an elliptic curve E defined over Q.
Questions. Denote Kn = Q(ζpn+1 ).
• Is XΓ (Kn ) a finite group? Is pen = XΓ (Kn )[p∞ ] finite integer?
• Do there exist integers µ, λ, and ν, only dependent on p, such that
en = µ pn + λ n + ν for “sufficiently large” n?
Kenkichi Iwasawa solved these questions for C`(OK ) = XU (Kn ). We will find that Barry
Mazur’s Control Theorem will help in answering these questions for XE (Kn ).
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