Università di Pisa
Dipartimento di Matematica
Corso di Laurea in Matematica
Tesi di Laurea Magistrale
Mazur’s Control Theorem
and Applications
Candidata:
Jessica Alessandrì
Relatore: Prof. Andrea Bandini Controrelatore: Prof.ssa Ilaria Del Corso
Introduction ii
1 Preliminary Results 1
1.1 Zp-extensions . . . 1
1.1.1 The Iwasawa algebra . . . 3
1.2 Cohomology of profinite groups . . . 6
1.2.1 Cohomology of procyclic groups . . . 11
1.2.2 Cohomological dimension . . . 13
1.3 Elliptic Curves . . . 18
1.3.1 Weil Pairing . . . 21
1.3.2 Formal Group of an Elliptic Curve . . . 23
1.3.3 Elliptic Curves over local fields . . . 25
1.3.4 Good and Bad Reduction . . . 28
1.3.5 Elliptic Curves over global fiels . . . 30
1.3.6 The Selmer Groups . . . 35
1.4 Λ-modules . . . 37
2 Mazur’s Control Theorem 45 2.1 Study of the Kummer maps . . . 45
2.2 Control theorem . . . 51
2.3 Applications and Examples . . . 58
2.3.1 Mazur’s Theorem . . . 58
2.3.2 Triviality of SelE(F∞)p. . . 58
2.3.3 Order of the Tate-Shafarevich group . . . 59
2.3.4 Corank of the Selmer group . . . 60
2.3.5 Example with X0(11) . . . 61
2.3.6 Main Conjecture . . . 64
Bibliography 67
Introduction
Iwasawa theory started in the late 1950’s from the work of Kenkichi Iwa-sawa, who studied a particular type of infinite Galois extensions, called Zp
-extensions, namely extensions of a number field with Galois group isomorphic to Zp, the ring of p-adic integers, for some prime p. In particular, he studied
the behaviour of (the p-torsion part of) ideal class groups In in the sub-fields Kn of a Zp-extension K∞/K, in order to recover information about
the "class group" I = lim←−In, where the transition maps of the projective
limit are the norm maps. This group is a module over the Iwasawa Algebra Zp[[Gal(K∞/K)]], so the idea is to study this ring and the modules defined
over it.
In the early 1970s, Barry Mazur in his work Rational points of abelian varieties with values in towers of number fields [4] presented a generalization of Iwasawa theory for abelian varieties. Some of the motivations were the study of the growth of the rank of an abelian variety over the subfields in a Zp-extension and the Birch and Swinnerton-Dyer conjecture.
One of the main theorems proven by Mazur is the so called Control Theorem, concerning the behaviour of Selmer groups in a Zp-extension for
an abelian variety with good, ordinary reduction at a prime p. The main goal of this thesis is, following Greenberg’s idea [2], to prove Mazur’s Control Theorem using Galois cohomology and to provide some applications mainly referred to the case of elliptic curves with good, ordinary reduction at a prime p.
We now give some motivations and details of our work. Let E be an elliptic curve defined over a number field K. The Mordell-Weil theorem 1.3.38 states that the abelian group E(K) of the points in E with coordinates in K is finitely generated, so
E(K) ' Zr×T,
where T = E(K)tors is a finite abelian group and r = rankZE(K). If K
is an infinite algebraic extension of Q this may not be true in general, but there are extensions such that E(K) is finitely generated, and the following theorem gives a sufficient condition.
Theorem 1. Let K be a Galois extension of Q. Suppose that E(K)tors is
finite and that rankZE(L) is bounded as L varies over the finite extensions of Q contained in K. Then E(K) is finitely generated.
We look for examples of K such that the hypotheses of this theorem are verified, which is not easy, especially for the one on rankZE(L). The following theorem by Mazur gives us conditions to control the rank of an elliptic curve over the subfields of a Zp-extension.
Theorem 2. Let E be an elliptic curve defined over a number field F . Let p be a prime such that E has good, ordinary reduction at all primes in F lying over p. Assume that E(F ) and XE(F )(p) are finite. Let F∞ =S Fn be a
Zp-extension of F . Then the rank of E(Fn) is bounded as n varies.
The group XE(F ) is the Tate-Shafarevich group, and XE(F )(p) denotes
its p-primary subgroup. We will prove this theorem as a corollary of Mazur’s Control theorem (see Corollary 2.3.1), and we will need results about mod-ules defined over the Iwasawa algebra Zp[[Gal(F∞/F )]]. The idea is to study
the behaviour of certain groups, called Selmer groups, instead of studying directly E(F ) or the Tate-Shafarevich group. In fact, if we restrict to ellip-tic curves with good, ordinary reduction at a prime p, then the p-primary part of the Selmer group for E over a number field, or over certain infinite extensions of Q, can be easily described in terms of the Galois cohomology for the p-torsion part of E. Here is the statement of the Control Theorem (which is proved in Section 2.2):
Theorem 3 (Mazur’s Control Theorem). Let p be a prime such that E has good, ordinary reduction at all primes in F over p. Let F∞ = SnFn be a
Zp-extension of F . Then the natural maps
SelE(Fn)p−→ SelE(F∞)Gal(Fp ∞/Fn)
have finite kernels and cokernels, of bounded order as n → ∞.
To prove theorem 2 we will show, via the control theorem, that the dual of SelE(F∞)pis a torsion Zp[[Gal(F∞/F )]]-module and this, using properties
of finitely generated modules over the Iwasawa algebra, will give us a bound on rankZE(Fn) for all n. We will also use the structure of these modules
to get a bound on the Zp-corank of the (p-torsion part of) Selmer groups
SelE(Fn) and from that we derive an analogous bound on rankZE(Fn).
In Chapter 1 we recall some basic results on Iwasawa theory and el-liptic curves. In particular, we show some properties of Zp-extensions and
the Iwasawa algebra, proving that it is isomorphic to Λ = Zp[[T ]], the ring
of power series in one variable over Zp. We list some results on cohomology
of profinite groups which we will widely use for the main results of Chap-ter 2, we give the definition of cohomological dimension and Potryagin dual,
Introduction iv
and we prove the Corank Lemma, another fundamental tool for the proofs in Chapter 2. We recall some properties of elliptic curves, in particular we focus on elliptic curves defined over local fields, studying what happens when we reduce the curve modulo a uniformizer. We introduce the Weil pairing and the Kummer pairing, and give two equivalent definitions of Selmer groups. We end this introductory chapter by discussing some properties of Λ-modules and defining the Iwasawa invariants λ and µ.
In Chapter 2 we first study the p-primary subgroups of the Selmer groups and to do so we describe the images of the Kummer maps that define those groups. We then prove Mazur’s control theorem for elliptic curves with good ordinary reduction: we divide the proof in steps and we use the description of the Kummer maps for each step. In the last section we show some ap-plications of the control theorem: we use properties of Λ-modules to prove Mazur’s theorem 2 and some results concerning the growth of the order of the Selmer and the Tate-Shafarevich groups. We conclude by giving two examples: first we study an elliptic curve with positive µ invariant, then we briefly introduce L-functions and the Main Conjecture of Iwasawa theory for elliptic curves to show how Mazur’s theorem and the Main Conjecture can be combined to provide partial results on the Birch and Swinnerton-Dyer conjecture.
Preliminary Results
In this introductive chapter we will see the main tools that we shall use later to prove the main theorems. We shall state the main results on Iwasawa theory and elliptic curves we are going to use in the rest of this thesis. Most of them are well known and we shall provide references for them plus some sketches of proofs when needed.
1.1
Z
p-extensions
A Zp-extension of a number field K is a Galois extension K∞/K with
Gal(K∞/K) ' Zp, the ring of p-adic integers. Recalling that the only closed
subgroups of Zp are 0 and pnZp for some n, from the fundamental theorem
of infinite Galois theory we get:
Proposition 1.1.1. Let K∞/K be a Zp-extension. Then, for each n ≥ 0,
there exists a unique field Kn⊆ K∞ of degree pn over K, and these Kn are
the only fields between K and K∞.
From that it follows that we can regard a Zp-extension as a sequence of
fields
K = K0⊂ K1 ⊂ · · · ⊂ K∞=
[ Kn
with Gal(Kn/K) ' Z /pnZ.
It is possible to prove that every number field has at least one Zp
-extension, namely the cyclotomic Zp-extension. It is obtained by
let-ting K∞ be an appropriate subfield of K(ζp∞) =S
nK(ζpn), where ζpn is a
primitive pn-th root of unity.
Proposition 1.1.2. Let K∞/K be a Zp-extension and let l be a prime
(possi-bly archimedean) of K which does not lie above p. Then K∞/K is unramified
at l.
Proof. Let I ⊆ Gal(K∞/K) ' Zp be the inertia group for l. It is closed, so
I = 0 or I = pnZpfor some n ≥ 0. If I = 0 we are done. Suppose I = pnZp.
1.1 Zp-extensions 2
We have that I is infinite and we may assume that l is non-archimedean (if l is archimedean, I must have order 1 or 2). Set l0 = l and for each n, choose inductively ln ∈ Kn lying above ln−1. Let cKn be the completion of Kn at
ln, and let dK∞=SKcn. Then we have
I ⊆ Gal( dK∞/ bK).
Let U be the units of bK. By local class field theory, there is a continuous surjective homomorphism U → I. But we know also that
U ' Za˜l ×T,
where T is a finite group, a ∈ N and ˜l is the prime in Q such that l|˜l. Now since I ' pnZp has no torsion, we must have a continuous surjective map
Z˜al → pnZp → pnZp/pn+1Zp.
But Za˜l has no closed subgroups of index p, so there is a contradiction. Hence
we have always I = 0 and the proof is complete.
In other words, we say that Zp-extensions are unramified outside p.
Proposition 1.1.3. Let K∞/K be a Zp-extension. At least one prime
ram-ifies in this extension, and there exists n ≥ 0 such that every prime which ramifies in K∞/K is totally ramified.
Proof. Since K has ideal class group Cl(K) of finite order, if H is the Hilbert class field of K (the maximal abelian unramified extension of K), we have that Gal(H/K) ' Cl(K) is finite. Thus there is at least a prime which ramifies in K∞/K. By the above proposition, only finitely many primes
ramify in K∞/K. Call them p1, . . . , psand let I1, . . . , Isbe the corresponding
inertia groups. Then T Ii is a closed subgroup of Zp, so it is equal to pnZp
for some n ≥ 0. The fixed field of pnZp is Kn and since Gal(K∞/Kn)
is contained in each Ij we have that all primes above each pj are totally
ramified in K∞/Kn.
We now introduce the Iwasawa algebra of a profinite group G = lim
←−iGi. Suppose that the homomorphisms ϕij : Gj → Gi are surjective
for i ≥ j ≥ 0. We then obtain Zp-algebra homomorphisms
ψij : Zp[Gj] → Zp[Gi],
where Zp[Gk] denotes the group ring for Gk over Zp. So we define the
Iwasawa algebra for G as the completed group ring for G over Zp, i.e.
where the inverse limit is defined by the surjective maps ψij. Since Zp[Gi]
is a compact topological ring, Zp[[G]] is a compact topological ring as well.
For each i there is a continuous, surjective Zp-algebra homomorphism ψi :
Zp[[G]] → Zp[Gi]. Moreover, the group algebra Zp[G] can be identified with
a subring of Zp[[G]] in a natural way and it is dense.
We are interested in the case G = Γ = Gal(K∞/K) ' Zp, taking Gn =
Γn= Γ/Γp
n
' Gal(Kn/K).
1.1.1 The Iwasawa algebra
The topological ring Zp[[Gal(K∞/K)]] is called the Iwasawa algebra
associ-ated with the Zp-extension K∞/K. We shall study modules over this algebra
in Section 1.4 (mainly Selmer groups, see Section 1.3.6). Their structure will be determined via a noncanonical isomorphism between the Iwasawa alge-bra and the ring of power series in one variable over Zp which we denote by
Λ := Zp[[T ]].
First we investigate some properties of Λ = Zp[[T ]]. It is a Noetherian
ring and if g(T ) = P∞
i=0biTi ∈ Λ, then g(T ) is invertible if and only if
b0 ∈ Z×p. Thus Λ is a local ring with maximal ideal m = (p, T ) and we
have Λ/m ' Z /p Z. Actually Λ/mk is finite for every k ≥ 1. Giving Λ the m-adic topology, it becomes a topological ring and, since Λ ' lim
←−Λ/mk, it is a compact topological ring.
Theorem 1.1.4. Let g(T ) = P∞
i=0biTi ∈ Λ, with g(T ) /∈ pΛ. Let d be
the smallest integer such that bd ∈ Z×p. Let f (T ) ∈ Λ, then there exists r(T ) ∈ Zp[T ] of degree < d and h(T ) ∈ Λ such that
f (T ) = g(T )h(T ) + r(T ).
Furthermore, h(T ) and r(T ) are uniquely determined by f (T ) and g(T ).
Proof. First we prove uniqueness. If g(T )h(T ) + r(T ) = g(T )h0(T ) + r0(T ), then s(T ) = r0(T ) − r(T ) is divisible by g(T ) in Λ. Let ˜Λ = Λ/pΛ. We observe that ˜Λ ' Fp[[T ]] is an integral domain, so p is a prime element
of Λ. Assume that s(T ) 6= 0, then we can write s(T ) = pms0(T ), where
s0(T ) /∈ pΛ and m ≥ 0. Now since g(T )|s(T ) and p is irreducible in Λ, we
have that g(T )|s0(T ). But in ˜Λ, ˜s0(T ) (that is the image of s0(T ) under the
projection Λ → ˜Λ) is a non zero polynomial of degree < d, divisible by ˜g(T ) and therefore by Tdin ˜Λ. That is impossible, so s(T ) = 0, thus r0(T ) = r(T ) and this implies that h0(T ) = h(T ).
For the existence we use the fact that ˜Λ is a discrete valuation ring and we lift back to Λ the quotient and the reminder of the division of ˜f (T ) by ˜
g(T ) in ˜Λ as in the Hensel Lemma. If ˜f (T ) =P∞
i=0a˜iTi, then we can write
˜ f (T ) = Td( ∞ X i=d ˜ aiTi−d) + d−1 X i=0 ˜ aiTi.
1.1 Zp-extensions 4
In ˜Λ, ˜g(T ) and Tddiffer by an invertible, so they generate the same ideal. We can write ˜f (T ) = ˜g(T )˜h(T )+ ˜r(T ), where ˜h(T ) ∈ ˜Λ and ˜r(T ) is a polynomial in ˜Λ of degree < d. Lifting back to Λ we get
f (T ) = g(T )h1(T ) + r1(T ) + pf1(T )
where h1(T ), f1(T ) ∈ Λ and r1(T ) is a polynomial of degree < d. Applying
the same argument to f1(T ) and substituting in the previous equation, we
have
f (T ) = g(T )h2(T ) + r2(T ) + p2f2(T )
with h2(T ), f2(T ) ∈ Λ, r2(T ) is a polynomial of degree < d and h2(T ) ≡
h1(T ) (mod pΛ), r2(T ) ≡ r1(T ) (mod p Zp[T ]). If we iterate this argument
we obtain, for each n ≥ 1
f (T ) = g(T )hn(T ) + rn(T ) + pnfn(T )
with hn(T ), fn(T ) ∈ Λ, rn(T ) is a polynomial of degree < d and hn(T ) ≡
hn−1(T ) (mod pn−1Λ), rn(T ) ≡ rn−1(T ) (mod pn−1Zp[T ]). Now if h(T ) =
limn→∞hn(T ) and r(T ) = limn→∞rn(T ), we can write f (T ) = g(T )h(T ) +
r(T ), where h(T ) ∈ Λ and r(T ) is a polynomial of degree < d.
Corollary 1.1.5. Let g(T ) be as in Theorem 1.1.4. Then Λ/(g(T )) is iso-morphic to Zdp as Zp-module.
Proof. Consider the map
Λ/(g(T )) → {r(T ) | r(T ) ∈ Zp[T ], deg(r(T )) < d} = X
f (T ) + (g(T )) 7→ r(T ),
where r(T ) is the reminder of the division of f (T ) by g(T ). It is an isomor-phism of Zp-modules. This concludes since X ' Zdp.
Definition 1.1.6. A polynomialPd
i=0ciTi ∈ Zp[T ] is called distinguished
if cd= 1 and ci∈ p Zp for all i < d. Equivalently, if it is monic and all of its
roots in Qp have positive valuation.
Corollary 1.1.7. Let g(T ) be as in Theorem 1.1.4. Then there exists a unique distinguished polynomial g0(T ) such that g(T ) = u(T )g0(T ), where
u(T ) ∈ Λ× and g0(T ) has degree d.
Proof. The multiplication by T induces an endomorphism ϕ of Λ/(g(T )). Since Λ/(g(T )) is free of rank d as Zp-module, there exists a monic
polyno-mial g0(T ) of degree d that annihilates Λ/(g(T )). In fact, g0(T ) is the
charac-teristic polynomial of the endomorphism ϕ. Thus (g0(T )) ⊆ (g(T )). Now if
we apply the previous corollary to g0(T ), we find that Λ/(g0(T )) ' Zdp0, with
d0 ≤ d. But, since (g0(T )) ⊆ (g(T )), there exists a surjective homomorphism
Ψ : Zd0
and so d0 = d and Ψ is an isomorphism. Thus (g(T )) = (g0(T )) and g0(T )
is a distinguished polynomial.
The uniqueness follows easily by observing that Λ/(g(T )) can be iden-tified with Zp[T ]/(g0(T )) and g0(T ) is the minimal polynomial of the
endo-morphism ϕ.
Corollary 1.1.8. Let g(T ) ∈ Λ be nonzero. Then we can write g(T ) as
g(T ) = pmu(T )g0(T )
with m ≥ 0, u(T ) ∈ Λ× and g0(T ) ∈ Zp[T ] distinguished, in a unique way.
Furthermore g(T ) is irreducible if and only if either m = 1 and g0(T ) = 1 or m = 0 and g0(T ) is irreducible in Qp[T ].
Proof. It is obvious that g(T ) = pmg1(T ) with g1(T ) ∈ Λ − pΛ and m ≥ 0
(in a unique way). So if we apply Corollary 1.1.7 to g1(T ), the first part is proved. Now p is clearly irreducible in Λ, so suppose m = 0. Since Qp is the
field of fractions of Zp, it follows immediately that g(T ) is irreducible in Λ
if and only if g0(T ) is irreducible in Qp[T ].
It follows that, up to multiplication by an invertibile element of Λ×, the irreducible elements of Λ are either p or distinguished polynomials that are irreducible in Zp[T ] (or, equivalently, in Qp[T ]). Now, since Zp[T ] is a UFD,
it is clear that Λ is a UFD too.
Remark 1.1.9. For n ≥ 1, let ωn= (1 + T )p
n
− 1. They are distinguished polynomials in Λ and we have that
Λ ' lim←−
n
Zp[[T ]]/(ωn).
It is not hard to prove this isomorphism, using that Λ is a UFD (for more details see [13], §7.1).
The next theorem shows, as we anticipated, that the rings Zp[[Γ]] and Λ
can be identified, with a non canonical identification, that depends on the choice of a topological generator γ0 of Γ.
Theorem 1.1.10. Let γ0 be a topological generator of Γ. There is a unique
isomorphism
ε : Λ → Zp[[Γ]]
of topological Zp-algebras such that ε(T ) = γ0− 1.
Proof. The ring Λ = Zp[[T ]] is a compact Zp-algebra. For each n ≥ 0,
consider ωn = (1 + T )p
n
1.2 Cohomology of profinite groups 6
distinguished polynomial of degree pn, Λ/(ωn) is isomorphic to Zppn as a Zp -module, with a Zp-basis given by {(1 + T )i+ (ωn) | 0 ≤ i < pn}. We can
then define an isomorphism
Λ/(ωn) −→ Zp[Γ/Γp n ] 1 + T + (ωn) 7−→ γ0Γp n .
Thus we obtain a continuous surjective Zp-algebra homomorphism Λ →
Zp[Γ/Γp
n
] for all n ≥ 0, that maps T to (γ0 − 1)Γp
n
. It induces a con-tinuous Zp-algebra homomorpshim ε : Λ → Zp[[Γ]]. Now the image of ε is
dense, and since Λ is compact, it is also closed, so ε is surjective. It is clear that ε(T ) = γ0 − 1. This implies that ε is uniquely determined, and since
ker(ε) =T
n(ωn) = 0, ε is also injective. To conclude that ε is a
homeomor-phism, we show that it is a closed mapping: let H ⊆ Λ be a closed subset, it is compact since Λ is compact. So ε(H) is compact in a Hausdorff space (since Zp[[Γ]] is profinite), hence closed.
1.2
Cohomology of profinite groups
Since our focus will be on Zp-extensions, we want to give some definitions
and results concerning the cohomology of profinite groups. From now on G (unless explicitly specified otherwise) will be a profinite group.
First we see how direct and projective limits behave when passing in cohomology. Suppose that (Gi)i∈I is a projective system of finite groups
and (Ai)i∈I a direct system of Gi-modules for all i ∈ I and the transition
maps ϕij : Gj → Gi and fij : Ai → Aj form compatible pairs, meaning that
fij(ϕij(σ)a) = σ(fij(a)), i.e. the following diagram commutes for all σ ∈ Gj
Ai Aj Ai Aj fij ϕij(σ) σ fij .
Then the Hn(Gi, Ai) form a direct system of abelian groups with induced
transition maps
Hn(Gi, Ai) −→ Hn(Gj, Aj)
ψ 7−→ ψ : σ 7→ fe ij(ψ(ϕij(σ))) and the following holds:
Proposition 1.2.1. If G = lim←−Gi and A = lim−→Ai, then
Example 1.2.2. Take G = Zp, the projective limit of the p-groups Z /pnZ. So if we let Gn= Zp/pn
Zp and Γn = pnZp, given A a discrete Zp-module,
we have that A =S
nAΓn = lim−→AΓn. Thus
Hn(Zp, A) ' lim−→Hn(Gn, AΓn) = lim−→Hn(Zp/pnZp, Ap
n
Zp).
We will study discrete abelian groups, but, in general, for a Hausdorff, abelian and locally compact group A we define:
Definition 1.2.3. The group
A∨ = Homcont(A, R/ Z)
is the Pontryagin dual of A (where R/ Z has the quotient topology). We endow it with the compact-open topology (viewing A∨ as a subspace of Hom(A, R/ Z)).
Theorem 1.2.4 (Pontryagin duality). If A is a Hausdorff abelian locally compact topological group, then the same is true for A∨. The canonical homomorphism
A → (A∨)∨
a 7→ τa
where τa: A∨ → R/ Z, φ 7→ φ(a), is an isomorphism of topological groups.
If A is a discrete torsion group (as the ones we are interested in), then A∨ ' Hom(A, Q/ Z). If A is a G-module, Q/ Z is also a G-module with a trivial action, so A∨ becomes a G-module, with the action given by:
(σ · f )(a) = σ(f (σ−1a)) = f (σ−1a) for all σ ∈ G, for all f ∈ A∨ and for all a ∈ A.
Note that Zp and Qp/ Zp, like Q/ Z, are always viewed as discrete
G-modules with trivial action.
The following theorem gives us a powerful tool, that will be used repeat-edly in our proofs. For a proof, see [8], Proposition 1.6.7.
Theorem 1.2.5 (Five Term Exact Sequence). Let H be a closed normal subgroup of G and let A be a G-module. We have the following exact sequence
0 −→ H1(G/H, AH)−→ Hinf 1(G, A)−→ Hres 1(H, A)G/H
tg
−→ H2(G/H, AH)−→inf H2(G, A).
(1.1)
The maps inf and res are the well-known inflation and restriction maps, the map between H1(H, A)G/H and H2(G/H, AH) is called transgression, which we will not define, since our focus will be on the restriction-inflation
1.2 Cohomology of profinite groups 8
part and since in most of our cases we will have H2(G/H, AH) = 0.
Recall that if G is a finite group, given a G-module A, the modified cohomology groups are
b Hn(G, A) = ( AG/ N GA for n = 0 Hn(G, A) for n ≥ 1, where NGA is the image of the norm map
NG: A → A, a 7→
X
σ∈G
σa.
For profinite groups G = lim
←−Gi, by passing to the direct limit we can define
the modified cohomology groups as well:
b
Hn(G, A) = lim−→Hb
n
(Gi, A).
Proposition 1.2.6. Let G be a profinite group and U an open subgroup of G. Then for every G-module A such that bHn(U, A) = 0, we have
[G : U ] · bHn(G, A) = 0.
In particular, if G is finite we have |G| · bHn(G, A) = 0. If, moreover, A is a finitely generated abelian group, then bHn(G, A) is also a finitely generated abelian group, hence finite, since it is annihilated by |G|.
Remark 1.2.7. If G is a profinite group, since Hn(G, A) = lim−→Hn(G/U, AU), with the limit taken on the normal open subgroups of G, then the Hn(G, A) are torsion groups for every n ≥ 1.
Definition 1.2.8. Let p be a prime. A profinite group G is called a pro-p-group if for every U normal open subpro-p-group of G the pro-p-group G/U is a finite p-group, i.e. G is the projective limit of finite p-groups.
Example 1.2.9. It is clear that Zp = lim←− Z/pnZ is a pro-p-group.
For a closed subgroup H of a profinite group G and a natural number n, we say that [G : H] is prime to n if the index [G : U ] is prime to n for every open subgroup U containing H.
Definition 1.2.10. A p-Sylow subgroup of a profinite group G is a closed subgroup Gp such that:
(1) Gp is a pro-p-group,
The Sylow theorems hold for profinite groups as well, for a proof see [8], Proposition 1.6.9.
Theorem 1.2.11. Let G be a profinite group and p a prime. Then
(1) there exists a p-Sylow subgroup Gp;
(2) every pro-p-subgroup is contained in a p-Sylow subgroup;
(3) the p-Sylow subgroups of G are conjugate.
Example 1.2.12. Let bZ be the profinite completion of Z, that is bZ = lim
←−
n∈N
Z /n Z. We can write bZ = Qq primeZq, so the p-Sylow subgroup of bZ is
Zp.
In studying Zp-modules, and in general G-modules, we are often reduced
to study the submodule formed by those elements that have order a power of p. If A is an abelian group, this subgroup is called the p-primary part A(p) of A. If A is a torsion group, then A =L
pA(p). For the p-primary
part of the torsion group Hn(G, A), we have:
Proposition 1.2.13. Let A be a G-module and Gp a p-Sylow subgroup of G. Then the homomorphism
res : Hn(G, A)(p) → Hn(Gp, A)
is injective. If Gp is open in G, the homomorphism
cor : Hn(Gp, A) → Hn(G, A)(p)
is surjective.
Proof. The proposition holds for any closed subgroup H with [G : H] prime to p: if H is such a subgroup, suppose that H is open, then the map
[G : H] = cor ◦ res : Hn(G, A)(p) → Hn(G, A)(p)
is an isomorphism, since [G : H] is prime to p, so res : Hn(G, A)(p) → Hn(H, A) is injective and cor : Hn(H, A) → Hn(G, A)(p) is surjective. If H is not open, since H is itself profinite, the injectivity of the map res follows from Proposition 1.2.1.
Corollary 1.2.14. If Hn(Gp, A) = 0 for every prime p, then Hn(G, A) = 0.
Two interesting properties of p-primary modules are the following:
Proposition 1.2.15. Let H be a finite p-group and let A be a p-primary H-module.
1.2 Cohomology of profinite groups 10
(1) If H0(H, A) = 0, then A = 0.
(2) If pA = 0 and there exists q such that bHq(H, A) = 0, then A is an induced H-module and, in particular, cohomologically trivial.
For a proof of this proposition, see [8], Proposition 1.6.12 for (1) and Proposition 1.8.3 for (2).
Remark 1.2.16. If a p-primary G-module A 6= 0 is simple, since for every a ∈ A the G-orbit of a is finite (this follows from the fact that the stabilizer of an element is an open subgroup of G), A is finitely generated as Z-module: just consider the submodule hσ(a)|σ ∈ GiZand conclude by simplicity. Then A is finite and there exists a unique prime p such that pA = 0.
Corollary 1.2.17. Let G be a pro-p-group. Then every simple p-primary module A is isomorphic to Z /p Z. In particular, if A is a p-primary G-module
A = 0 ⇔ AG = 0
Proof. If A is a (nonzero) discrete simple p-primary G-module, by the pre-vious remark, then pA = 0. Now since A = S AH, where the H are open
subgroups of G, there exists U an open subgroup of G such that AU 6= 0 and so by Proposition 1.2.15 applied to G/U and AU we have
H0(G/U, AU) = AG 6= 0.
So by simplicity AG = A, meaning that A is a Fp-vector space with trivial
action of G, hence it has dimension 1.
It follows immediately that every finite p-primary G-module admits a descending chain A = A0 ⊇ A1 ⊇ A2 ⊇ · · · ⊇ Ak = {id} such that Ai/Ai+1' Z /p Z for all i = 0, . . . , k − 1.
Let A be a p-primary abelian group. We can regard it as a Zp-module
and endow it with the discrete topology. In this case its Pontryagin dual
A∨ = Homcont(A, Qp/ Zp)
is a compact Zp-module.
Definition 1.2.18. We say that A is a confinitely generated Zp-module
if A∨ is a finitely generated Zp-module, and we call Zp-corank of A the rank of A∨ as Zp-module. We then write corankZpA = rankZpA∨.
Example 1.2.19. • The Pontryagin dual of Qp/ Zp is Zp. • A = Qp/ Zp
r
Therefore if A is a cofinitely generated Zp-module, since Zp is a PID, we
can write
A∨ ' Zrp×T,
for some r ≥ 0 and some finite group T (r = corankZpA). But then A ' (A∨)∨'Qp/ Zpr× T∨.
Example 1.2.20. If A is Zp-cofinitely generated of corank r, its divisible
part is Adiv' (Qp/ Zp)r and [A : Adiv] < ∞.
We will show that, if Kv is a finite extension of Qp, and if A is a cofree Zp-module, then H1(Gal(Kv/Kv), A) is a cofinitely generated Zp-module (it
is the corank lemma, that we will see at the end of this section).
1.2.1 Cohomology of procyclic groups
We know that, if G is a finite cyclic group, for every G-module A we have
b
H2n(G, A) = AG/ NGA and Hb
2n−1
(G, A) = ker NG/(σ − 1)A
where σ is a generator of G. If G is infinite, we want to give an explicit expression of the cohomology groups, at least for n = 0, 1, similar to the latter.
Definition 1.2.21. A procyclic group G is a profinite group that is topo-logically generated by an element σ, i.e. G is the closure of the subgroup generated by σ, hσi = {σn|n ∈ Z}.
For example, Zp = lim←− Z/pnZ and bZ = lim←− Z/n Z are additive procyclic groups, both generated by the element (1, 1, . . . ). Actually, every procyclic group is a quotient of bZ: if G is a procyclic group topologically generated by σ, then for every n we have the surjective homomorphism
Z /n Z → G/Gn [1] 7→ [σ],
so passing to the limit we have a continuous surjective homomorphism bZ → G. But now, since bZ =QpZp, every procyclic group can be written as
G ' Z /n Z ×Y
p∈S
Zp
where n is a natural number and S ⊆ {p prime | p - n}. Now assume that G is torsion-free, i.e. G =Q
p∈SZp and let N(S) be the set of natural numbers
not divisible by prime numbers q /∈ S. Then for every n ∈ N(S) the groups Gn(n-th powers) are open subgroups of G and
G = lim←− n∈N(S) G/Gn' lim ←− n∈N(S) Z /n Z .
1.2 Cohomology of profinite groups 12
Definition 1.2.22. An abelian group X is called S-divisible if X = nX for all n ∈ N(S), and is called S-torsion if X =S
n∈N(S)X[n], where X[n] =
{x ∈ X | nx = 0}.
Proposition 1.2.23. Let G = Q
p∈SZp be a torsion-free procyclic group,
with topological generator σ, and let A be a G-module.
(1) If A is S-torsion, then H1(G, A) ' A/(σ − 1)A.
(2) If A is torsion or S-divisible, then Hn(G, A) = 0 for n ≥ 2. For a proof, see [8], Proposition 1.7.7.
One easily obtains the following proposition, that will help us compute the cohomology groups for p-primary groups (we will use it, in fact, applied to the p-primary part of an elliptic curve).
Proposition 1.2.24. Let Γ ' Zp and let γ be a topological generator of Γ.
(1) Suppose that A is a finite abelian p-group, on which Γ acts continuously. Then
H1(Γ, A) = A (γ − 1)A.
(2) Let A be a p-primary abelian group on which Γ acts continuously. Then
H1(Γ, A) = A (γ − 1)A.
(3) If A ' (Qp/ Zp)rthen H0(Γ, A) and H1(Γ, A) have the same Zp-corank
and, if H0(Γ, A) is finite, then H1(Γ, A) = 0.
Proof. The first two points are immediate consequences of the previous proposition. For (3), consider the exact sequences
1 → AΓ→ A−→ (γ − 1)A → 1γ−1 and
1 → (γ − 1)A → A → A
(γ − 1)A → 1 . (1.2)
From the exactness of the first sequence, it follows that
corank A = corank AΓ+ corank(γ − 1)A. From the second we get instead:
corank A = corank A/(γ − 1)A + corank(γ − 1)A,
so corank AΓ = corank A/(γ − 1)A. Now suppose that H0(Γ, A) is finite,
then 0 = corank H0(Γ, A) = corank H1(Γ, A), and also H1(Γ, A) is finite. But since H1(Γ, A) = (γ−1)AA , this means that (γ − 1)A is an infinite open submodule of A = (Qp/ Zp)r, which has only finite proper submodules. Thus
1.2.2 Cohomological dimension
A fundamental numerical invariant of a profinite group G is its cohomological dimension.
Definition 1.2.25. The cohomological dimension cd G of G is the smal-lest integer n such that
Hq(G, A) = 0 ∀q > n
for all A torsion G-modules, and it is ∞ if no such integer exists.
Let p be a prime. The p-cohomological dimension cdpG of G is the
smallest integer n such that the p-primary part
Hq(G, A)(p) = 0 ∀q > n
for all A torsion G-modules, and it is ∞ is no such integer exists.
Remark 1.2.26. To show that the p-cohomological dimension of G is n, it is enough to show that n is the smallest integer such that Hq(G, A) = 0 for all q > n, for all A p-primary G-modules. Indeed, every torsion G-module can be written as A =L
pA(p) and we have that Hq(G, A)(p) = Hq(G, A(p)).
Since every abelian torsion group decomposes into the direct sum of its p-primary parts, we have that
cd G = sup
p
cdpG.
To compute the p-cohomological dimension of G, we can actually compute only the dimension of its p-Sylow subgroup.
Proposition 1.2.27. If H is a closed subgroup of G, then
cdpH ≤ cdpG
and we have an equality in each of the following cases
(1) [G : H] is prime to p,
(2) H is open and cdpG < ∞.
For a proof, see [8], Proposition 3.3.5. From the proposition we have immediately the following
Corollary 1.2.28. If Gp is a p-Sylow subgroup of G, then
cdpG = cdpGp = cd Gp.
1.2 Cohomology of profinite groups 14
Proposition 1.2.29. cdpG = 0 if and only if the order of G is prime to p. Proof. It is clear that if the order of G is prime to p, then G has p-cohomological dimension 0. Suppose now that the order of G is not prime to p, then by the previous corollary, we can suppose that G is a pro-p-group. If G = lim←−
nGn6=
{1}, then for each n ≥ 0, since Gnis a finite p-group, there exists a continuous surjective homomorphism Gn→ Z /p Z. This yields a continuous surjective
homomorphism G → Z /p Z. Thus H1(G, Z /p Z) ' Hom(G, Z /p Z) 6= 1, meaning that cdpG ≥ 1.
We are interested in the cohomological dimension of Zp: since cd(Zp) =
cdp(Zp), from Remark 1.2.26 we want to compute Hn(Zp, A) for all p-primary
Zp-modules A. Thanks to Proposition 1.2.23 we have that those cohomology
groups are trivial for n ≥ 2. Now by the previous proposition we have that cd(Zp) ≥ 1, thus we conclude cd(Zp) = 1.
We can use this information to simplify the five term exact sequence for Zp-extensions.
Example 1.2.30. Let K∞/K be a Zp-extension and for every n ≥ 0 consider
K ⊂ Kn ⊂ K∞. For our purposes, we consider A a p-primary Zp-module
and put GK∞ = Gal(K/K∞) and GKn = Gal(K/Kn), where K is a fixed
algebraic closure of K. Since GKn/GK∞ ' Gal(K∞/Kn) ' pnZp we get the
five-term exact sequence:
0 → H1(GKn/GK∞, AGK∞) inf −→ H1(GKn, A) res −→ H1(GK∞, A)GKn/GK∞ tg −→ H2(GKn/GK∞, A GK∞)−→inf H2(G Kn, A).
Since GKn/GK∞ ' pnZp' Zp one has cd(pnZp) = 1 and so
H2(GKn/GK∞, A
GK∞) = 0.
The sequence above becomes then a short exact sequence:
0 → H1(GKn/GK∞, A GK∞)→ Hinf 1(G Kn, A) res → H1(GK∞, A) GKn/GK∞ → 0
We now give the definition of cohomological dimension of a field, and we investigate some of its properties.
Definition 1.2.31. The cohomological dimensions cdp(K), cd(K) of a
field K are defined as the cohomological dimension cdpGK and cd GK of its
absolute Galois group GK = Gal(K/K), where K is a fixed algebraic closure of K.
The p-cohomological dimension for a field with char(k) = p > 0 plays an exceptional role (for a proof, see [9], §2.2, Proposition 3):
We now consider a prime number p different from the characteristic of the field considered. We write Hn(K, A) for Hn(GK, A), where A is any
GK-module.
Proposition 1.2.33. If F/K is a finite extension and if cdp(K) < ∞ or
p - [F : K], then cdp(F ) = cdp(K).
This is a consequence of Proposition 1.2.27, since GF is isomorphic to an
open subgroup of GK.
Theorem 1.2.34. Let K be a field, complete with respect to a discrete val-uation with perfect residue field k. If p 6= char(k) is a prime number and cdp(k) < ∞, then
cdp(K) = cdp(k) + 1.
For a proof, see [8], Theorem 6.5.15.
From this theorem we find that, for any prime p, the p-cohomological dimension of Qp is 2: we have that the p-cohomological dimension of its
residue field is cdp(Fp) = cdp(GFp) = cdpZ and since the p-Sylow of bb Z
is Zp, and we showed that cdp(Zp) = 1, we conclude that cdp(Fp) = 1.
Furthermore, thanks to the previous proposition, the same is true for every finite extension of Qp.
But what if the extension is infinite? In particular, what if the extension is a Zp-extension?
Definition 1.2.35. Let K/F be an infinite extension. We define the profi-nite degree of K/F as
lcm{[L : F ]| F ⊆ L ⊆ K L/F finite} We can regard it as a supernatural numberQ
l primelal, with 0 ≤ al ≤ ∞.
We say that l∞ divides the profinite degree of K/F if for every n ≥ 0 there exists a finite extension with ln | [L : F ] (i.e. the power of l that divides [L : F ] is unbounded as L/F varies).
Lemma 1.2.36. Let l be a prime and Kv be an extension of Ql such that
p∞ divides the profinite degree of Kv/Ql. Then GKv has p-cohomological dimension 1.
For the proof we will need the following (for a proof see [9], §5.6, Lemma 3)
Lemma 1.2.37. In the same hypotheses of the previous lemma, the p-primary subgroup of the Brauer group of Kv is 0, i.e.
H2(Kv, Kv ×
1.2 Cohomology of profinite groups 16
Proof. (of Lemma 1.2.36) We have H2(Kv, Kv
×
)(p) = 0 and applying the same argument to any exten-sion of Kv we get H2(H, Kv
×
)(p) = 0 for every H closed subgroup of GKv.
Consider the following exact sequence
1 −→ µp −→ Kv × p −→ Kv×−→ 1. It induces in cohomology: H1(H, Kv × ) −→ H2(H, µp) −→ H2(H, Kv × ), and since by Hilbert 90 theorem H1(H, Kv
×
) = 0 and, as we have just seen, H2(H, Kv
×
)(p) = 0, we deduce H2(H, µp) = 0. Note that Kv(µp)/Kv is an
extension (possibly trivial) of degree prime to p: since we are only interested in p-cohomological dimension we have cdpGKv = cdpGKv(µp) and we can
always assume µp ⊂ Kv so that µp ' Z /p Z as GKv-modules. Thus in
particular if Gp is a p-Sylow subgroup of GKv H2(Gp, Z /p Z) = 0. Now
if M is a p-primary Gp-module H2(Gp, M ) = 0. In fact, using Corollary
1.2.17, we can construct M0 = {id} ⊂ M1 ⊂ · · · ⊂ Mk−1 ⊂ Mk = M with
Mi/Mi−1' Z /p Z and we have the short exact sequence
0 → Mn−1→ Mn→ Z /p Z → 0
that induces
H2(Gp, Mn−1) → H2(Gp, Mn) → H2(Gp, Z /p Z) = 0
and so H2(Gp, Mn−1) H2(Gp, Mn). Proceeding in this way we arrive to
0 = H2(Gp, M1) = H2(Gp, Z /p Z) H2(Gp, M2), i.e. H2(Gp, M2) = 0, and
so by iteration we must have H2(Gp, M ) = 0.
Now we know that the restriction H2(Kv, M )(p) → H2(Gp, M ) is
injec-tive and thus H2(Kv, M )(p) = 0, as we wanted.
We end this section on cohomology with a proof of the Corank Lemma. We use two theorems of Tate.
Theorem 1.2.38 (Tate Duality). Let K be a p-adic local field. Let A be a finite GK module and let bA(1) = Hom(A, µp∞). Then the cup-product
Hi(K, bA(1)) × H2−i(K, A)−→ H∪ 2(K, µp∞) ' Q/ Z
induces for 0 ≤ i ≤ 2 an isomorphism of finite abelian groups
This is proved in [8], Theorem 7.2.6.
We write bA(1) to emphasize the fact that the action of GK is given by
the action on µp∞, i.e. by the cyclotomic charater χcyc.
For the second theorem, let K be a local field and p the characteristic of its residue field. If A is a finite GK-module of order prime to char K (if char K > 0), we define the Euler-Poincarè characteristic of A:
χ(K, A) =
2
Y
j=0
| Hj(K, A)|(−1)j.
Theorem 1.2.39 (Tate). If |A| = a, then
χ(K, A) = kakK
where k · kK is the normalized valuation on K.
For a proof, see [8], Theorem 7.3.1.
Now for the Corank Lemma, let Kv be a finite extension of Qp, where p
is a prime, and suppose A is a discrete GKv-module.
Theorem 1.2.40 (Corank Lemma). Suppose that A ' (Qp/ Zp)r as a Zp
-module.
(1) Let Kv be a finite extension of Qp. Then H1(Kv, A) is a cofinitely
generated Zp-module of corank equal to
r[Kv : Qp] + corankZpH
0(K
v, A) + rankZpH
0(K
v, bA(1)).
(2) Let Kv be a finite extension of Ql, where l 6= p. Then H1(Kv, A) is a
cofinitely generated Zp-module of corank equal to
corankZpH
0(K
v, A) + rankZpH
0(K
v, bA(1)).
Proof. We first show that H1(Kv, A) is a cofinitely generated Zp-module. To
do so, consider the short exact sequence
0 → A[p] → A→ A → 0p that induces in cohomology
H1(Kv, A[p])→ Hi 1(Kv, A) p
→ H1(Kv, A).
We have im(i) = ker(p) = H1(Kv, A)[p], thus we have a surjective map
H1(Kv, A[p]) → H1(Kv, A)[p]. Now let Lv be a finite Galois extension of
Kv such that GLv acts trivially on A[p] (so H
1(L
v, A[p]) = Hom(GLv, A[p])).
From the inflaction-restriction sequence (1.1) with G = GKv and H = GLv
0 −→ H1(Gal(Lv/Kv), A[p]GLv) inf
−→ H1(Kv, A[p]) res
1.3 Elliptic Curves 18
we have that
ker(res) = H1(Gal(Lv/Kv), A[p]GLv) = H1(Gal(Lv/Kv), A[p])
and it is finite, since Gal(Lv/Kv) is finite. Moreover we have H1(Lv, A[p]) =
Hom(Gal(Mv/Lv), A[p]), where Mv is the maximal extension of Lv such
that Gal(Mv/Lv) is an elementary abelian p-group (since the elements of
A[p] have order p). But Lv has only a finite number of extensions of degree
p, so Mv/Lv is finite. Hence H1(Lv, A[p]) is finite, so H1(Kv, A[p]) is finite
too. It follows that H1(Kv, A)[p] is finite since H1(Kv, A[p]) H1(Kv, A)[p].
This implies that H1(Kv, A) is a Zp-cofinitely generated module.
The same statement is true for H0(Kv, A), thanks to Proposition 1.2.24.
We now see how to use the two theorems of Tate to show that H2(Kv, A)
is also cofinitely generated as a Zp-module and the formulas for the rank of
H1(Kv, A) hold.
Let M be a finite GKv-module. Assume that |M | = p
a. Then cM (1) is
also a GKv-module of order pa. From Theorem 1.2.39, taking the canonical valuation on Kv we get 2 Y j=0 | Hj(Kv, M )|(−1) j = ( p−a[Kv:Qp] if v|p 1 if v - p
and by Theorem 1.2.38 H2(Kv, M ) is the Pontryagin dual of H0(Kv, cM (1))
(and so they have the same order). We extend these results to infinite GKv
-modules A =S
nA[pn] applying the above results to M = A[pn] for all n ≥ 0
(which are finite). We get:
2 X j=0 (−1)jcorank Hj(Kv, A) = ( −[Kv : Qp] corank A if v|p 0 if v - p
and H2(Kv, A) is the Pontryagin dual of H0(Kv, bA(1)) and hence
corank H2(Kv, A) = rankZpH
0(K
v, bA(1)).
These results conclude the proof in both cases.
Given an elliptic curve over a finite extension of Qp, with the Corank
Lemma we will able to compute the rank of the image of the map induced by the inclusion ker(π) ,→ E(p), where π is the reduction modulo v and E(p) is the p-primary subgroup of Etors, that is the torsion subgroup of E.
1.3
Elliptic Curves
Elliptic curves are non-singular curves of genus 1 with a specific base point. Every such curve can be written (thanks to the Riemann-Roch theorem) as
the locus in P2 of a cubic equation with only one point (the base point) on the line at ∞, i.e. as an equation of the form (using non-homogeneous coordinate to ease notation)
E : y2+ a1xy + a3y = x3+ a2x2+ a4x + a6, (1.3)
with the basepoint O = [0, 1, 0], called the Weierstrass equation of the curve. If a1, . . . , a6 ∈ K, we say that E is defined over K.
If char(K) 6= 2, 3, we can change coordinates to get a simpler equation:
E : y2 = x3+ Ax + B . (1.4)
This equation has associated quantities
∆(E) := −16(4A3+ 27B2), j(E) := 1728(4A3)/∆(E).
Definition 1.3.1. The quantity ∆ above is called the discriminant of the Weierstrass equation, j is called the j-invariant of the elliptic curve E.
Proposition 1.3.2. (1) The curve given by a Weierstrass equation is non singular if and only if ∆ 6= 0.
(2) Two elliptic curves are isomorphic over K if and only if they have the same j-invariant.
(3) Let j0 ∈ K. Then there exists an elliptic curve (defined over K(j0)) with j-invariant j0.
For a proof, see [11], §III, Proposition 1.4.
Let E be an elliptic curve given by a Weierstrass equation and let O be the point at infinity. We define an operation on E by the following rule:
Let P, Q ∈ E, L the line connecting P and Q (tangent line if P = Q), and R the third point of intersection of L with E. Let L0 be the line connecting R and O. Then P + Q is the point such that L0 intersects E at R, O and P + Q.
It is not hard to prove the following (in [11], Ch.III.2 there is a proof with explicit formulas for the coordinates of the point P + Q):
Theorem 1.3.3. The composition law defined above makes E into an abelian group with identity element O.
Now suppose that a Weierstrass equation has discriminant ∆ = 0, so it has a singular point. If we discard the singular point we still have a group law.
1.3 Elliptic Curves 20
Definition 1.3.4. Let E be a curve given by a Weierstrass equation. The non-singular part of E, denoted Ens, is the set of non-singular points of E. Similarly, if E is defined over K, then Ens(K) is the set of non-singular
points of E(K).
Proposition 1.3.5. Let E be a curve given by a Weierstrass equation with discriminant ∆ = 0, so E has a singular point. Then the composition law defined above makes Ens into an abelian group. Furthermore:
(1) suppose that E has a node, then Ens is isomorphic to K×; (2) suppose that E has a cusp, then Ens is isomorphic to K
+
.
For a proof, see [11], Ch.III, Proposition 2.5.
For each m ∈ Z we can define the multiplication by m [m] : E → E
in the following way:
[m]P = P + · · · + P (m terms) for m > 0, [0]P = O, and [m]P = [−m](−P ) for m < 0.
This map will be one of the most powerful tools for the study of elliptic curves.
Definition 1.3.6. Let E be an elliptic curve and m ∈ Z, m 6= 0. The m-torsion subgroup of E is the set of points of E of order m:
E[m] = {P ∈ E | [m]P = O}.
The torsion subgroup of E is the set of points of finite order
Etors= ∞
[
m=1
E[m].
The structure of the torsion points is given by the following theorem (see [11], Ch.III, Corollary 6.4).
Theorem 1.3.7. (1) If char(K) = 0 or if m is prime to char(K), then
E[m] ' (Z /m Z) × (Z /m Z). (2) If char(K) = p, then one of the following holds:
E[pn] ' Z /pnZ for all n = 1, 2, 3, . . . in this case we say that E/K is ordinary, or
E[pn] = 0 for all n = 1, 2, 3, . . . and in this case we say that E/K issupersingular.
From that, by taking the direct limit we get that if char(K) = 0 then
Etors' Q/ Z ×Q/ Z .
Actually, we will be interested in the p-primary part of an elliptic curve defined over a field with char(K) = 0. In this case, we take E[pk] ' Z /pkZ × Z /pkZ for every k ∈ N and by passing to the direct limit, our study will be focused on
E(p) = [
k∈N
E[pk] ' Qp/ Zp×Qp/ Zp.
If char(K) = p and E is ordinary, then
E(p) ' Qp/ Zp,
while if E is supersingular E(p) = 0, i.e. it has no p-torsion points.
Given an elliptic curve over a field K we have an obvious action of the absolute Galois group GK on E, given by the action on the coordinates. Of course, it acts on the group E[m] and for all P ∈ E[m] we have
[m](Pσ) = ([m]P )σ = O.
Thus, when E[m] ' Z /m Z × Z /m Z, choosing a basis for E[m], we obtain a representation
GK → Aut(E[m]) ' GL2(Z /m Z).
1.3.1 Weil Pairing
Consider now E an elliptic curve over K. We fix an integer m ≥ 2, prime to p = char(K), if p > 0. Let S, T ∈ E[m] and choose a function F ∈K(E) (the function field of E over the algebraic closure of K) with divisor
div(F ) = m(T ) − m(O).
Letting T0∈ E with [m]T0= T , there is a function G ∈ K(E) such that
div(G) = [m]∗(T ) − [m]∗(O) = X
R∈E[m]
(T0+ R) − (R),
where [m]∗is the pullback of the multiplication by m. The functions F ◦ [m] and Gm have the same divisor, so we may assume that
F ◦ [m] = Gm. Then for every point X ∈ E we have
1.3 Elliptic Curves 22
Therefore we can define a pairing1, called Weil em-pairing
em : E[m] × E[m] −→ µm
(S, T ) 7−→ G(X + S) G(X) ,
(1.5)
where X ∈ E is any point in which G(X + S) and G(X) are defined and non-zero. We give some of its basic properties (for a proof see [11], Ch.III, Proposition 8.1).
Proposition 1.3.8. The Weil em-pairing is:
(1) bilinear:
em(S1+ S2, T ) = em(S1, T )em(S2, T ),
em(S, T1+ T2) = em(S, T1)em(S, T2);
(2) alternating:
em(S, T ) = em(T, S)−1;
(3) non-degenerate: if em(S, T ) = 1 for all S ∈ E[m], then T = O;
(4) compatible with Galois action: for all σ ∈ GK,
em(S, T )σ = em(Sσ, Tσ);
(5) compatible with multiplication: if S ∈ E[mm0] and T ∈ E[m], then emm0(S, T ) = em([m0]S, T ).
This basic properties of the Weil paring imply its surjectivity.
Corollary 1.3.9. The pairing em is surjective. In particular, if E[m] ⊂
E(K), then µm ⊂ K∗.
Proof. The image of em is a subgroup of µm, say equal to µd. So for all
S, T ∈ E[m],
1 = em(S, T )d= em([d]S, T ).
Because the pairing is non-degenerate, we have that [d]S = O and since S is arbitrary, we must have d = m. Now suppose that E[m] ⊂ E(K). Then from the Galois invariance of the em-pairing, em(S, T ) ∈ K∗ for all S, T ∈ E[m]. Therefore µm ⊂ K∗.
1
We can regard a pairing between R-modules M × N → L as a R-linear map ϕ : M → HomR(N, L). We say that the pairing is perfect if ϕ is an isomorphism of R-modules.
1.3.2 Formal Group of an Elliptic Curve
Let R be a ring.
Definition 1.3.10. A (one-parameter commutative) formal group F defined over R is a power series F (X, Y ) ∈ R[[X, Y ]] satisfying:
(1) F (X, Y ) = X + Y + terms of degree ≥ 2;
(2) F (X, F (Y, Z)) = F (F (X, Y ), Z) (associativity);
(3) F (X, Y ) = F (Y, X) (commutativity);
(4) there is a unique power series i(T ) ∈ R[[T ]] such that F (T, i(T )) = 0 (inverse);
(5) F (X, 0) = X and F (0, Y ) = Y .
We call F (X, Y ) the formal group law of F .
In general a formal group is merely a group operation, but if the ring R is local and complete, and if the variables are specialized to values in the maximal ideal m of R, then the power series giving the formal group will converge.
Definition 1.3.11. Suppose that R is a local ring. The group associated to F /R, denoted F (m), is the set m with the group operations
x ⊕F y = F (x, y) for x, y ∈ m (addition),
Fx = i(x) for x ∈ m (inverse).
Proposition 1.3.12. Let R be a local ring and let F be a formal group defined over R.
(1) For each n ≥ 1, the map
F (mn)/F (mn+1) → mn/mn+1 (1.6)
induced by the identity map on sets is an isomorphism of groups.
(2) Let p be the characteristic of the residue field k (we admit the case p = 0). Then every torsion element of F (m) has order a power of p. Proof. (1) Since the underlying sets are the same, it suffices to show that
the map is a homomorphism: for x, y ∈ mn,
x ⊕Fy = F (x, y)
= x + y + . . .
1.3 Elliptic Curves 24
(2) For this proof we assume that R is Noetherian (for a general proof, see [11], Ch.IV, Proposition 3.2). We show that there are no non-zero torsion elements with order prime to p. Let m ≥ 1 be prime to p (if p = 0, take any arbitrary m) and x ∈ F (m) with [m](x) = 0. We show inductively that x ∈ mnfor all n ≥ 1 (this implies that x = 0). Suppose x ∈ mnand let ¯x be the image of x in F (mn)/F (mn+1). We have that ¯
x has order dividing m, but from (1) we have that F (mn)/F (mn+1) is isomorphic to the k-vector space mn/mn+1, so it has only p-torsion. Hence ¯x = 0, meaning that x ∈ mn+1.
Let E be an elliptic curve given by a Weierstrass equation with coeffi-cients in R
E : y2+ a1xy + a3y = x3+ a2x2+ a4x + a6
with the point at infinity O = [0, 1, 0]. We change variables
z = −x
y w = − 1 y
so that the point O is now the point (z, w) = (0, 0) and z is a local uniformizer at O (i.e. z has a zero of order 1 at O). The Weierstrass equation becomes
w = z3+ a1zw + a2z2w + a3w2+ a4zw2+ a6w3= f (z, w). (1.7)
Now if we substitute this equation into itself recursively, we get an expression of w as a power series in z:
w = z3(1 + A1z + A2z2+ . . . )
with Ai ∈ Z[a1, . . . , a6] for all i. Thanks to Hensel’s Lemma, this
proce-dure converges and moreover we get that w(z) is the unique power series in Z[a1, . . . , a6][[z]] satisfying w(z) = f (z, w(z)). Since x = wz and y = −w1,
using the power series for w(z) we can obtain a Laurent series for x and y
x(z) = z w(z) = 1 z2 − a1 z − a2− a3z − (a4+ a1a3)z 2− . . . y(z) = −1 w(z) = − 1 z3 = a1 z2 + a2 z = a3+ (a4+ a1a3)z + . . .
also with coefficients in Z[a1, . . . , a6]. The pair (x(z), y(z)) is a formal
solu-tion of the Weierstrass equasolu-tion of E, i.e. a solusolu-tion in the quotient field of formal power series.
We want to construct a power series that will formally give the addition law on E. Let z1 and z2 be two indeterminates and let w1 = w(z1) and
w2 = w(z2). The line connecting (z1, w1) and (z2, w2) has equation w =
λz − ν, where the slope λ = w2−w1
z2−z1 ∈ Z[a1, . . . , a6][[z1, z2]] and ν = w1 −
obtaining a cubic expression in z that has z1 and z2 as roots. We look for the third root z3 and we find that it can be expressed as a power series in z1 and z2, again with coefficients in Z[a1, . . . , a6]:
z3= −z1− z2+
a1λ + a3λ2− a2y − 2a4λν − 3a6λ2ν
1 + a2λ + a4λ2+ a6λ3
.
We need now the formula for the inverse. The inverse for (x, y) is (x, −y − a1x − a3), so since z = −x/y we find that the inverse of (z, w) has
z-coordinate i(z) = x(z) y(z) + a1x(z) + a3 = z −2− a 1z−1− . . . −z−3+ 2a 1z−2+ . . . ∈ Z[a1 , . . . , a6][[z]]
and w-coordinate w(i(z)). So the formal addition law is given by
F (z1, z2) = i(z3(z1, z2))
= z1+ z2− a1z1z2− a2(z12z2+ z1z22) + . . .
that is a power series in z1 and z2 with coefficients in Z[a1, . . . , a6]. From
the properties of the addition law on E we find that F (z1, z2) has the
corre-sponding properties of commutativity, associativity and inverse. The formal group associated to E, denoted bE, is given by this formal series F (z1, z2).
Suppose that the elliptic curve E is defined over K, the quotient field of a local and complete ring R with maximal ideal m. The power series x(z) and y(z) give a map
m −→ E(K)
z 7−→ (x(z), y(z)). This map gives a homomorphism from bE(m) to E(K).
1.3.3 Elliptic Curves over local fields
Let K be a local field, complete with respect to a discrete valuation v. We denote by R the ring of integers of K and by m the maximal ideal of R. Consider π a uniformizer (we assume that v is normalized so that v(π) = 1) for R and denote by k the residue field of R. We have the natural reduction map, that we denote by a tilde:
R −→ k
t 7−→ ˜t. Now let E/K be an elliptic curve and let
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6
be a Weierstrass equation of E/K. We can change coordinates in order to find a Weierstrass equation with all coefficients ai ∈ R. Then the discrimi-nant satisfies v(∆) ≥ 0 and, since v is discrete, we can look for an equation with v(∆) as small as possible.
1.3 Elliptic Curves 26
Definition 1.3.13. A Weierstrass equation as above is called minimal Weierstrass equation for E at v if v(∆) is minimized subject to the condition a1, a2, a3, a4, a6 ∈ R.
If we choose a minimal Weierstrass equation for E/K, we can reduce the coefficients modulo π to obtain a curve over k:
e
E : y2+ ˜a1xy + ˜a3y = x3+ ˜a2x2+ ˜a4x + ˜a6.
Furthermore, if P ∈ E(K), we can find homogeneous coordinates P = [x0, y0, z0] with x0, y0, z0 ∈ R and at least one of x0, y0, z0 in R×. Then
the reduced point eP = [ ˜x0, ˜y0, ˜z0] is in eE(k). So we have a reduction map
E(K) −→ E(k)e P 7−→ P .e Now eE may be singular. We define
E0(K) = {P ∈ E(K) | eP ∈ eEns},
the set of points with non-singular reduction, and
E1(K) = {P ∈ E(K) | eP = eO},
the kernel of reduction.
Remark 1.3.14. Obviously ∆( eE) = ^∆(E), so if v(∆) = 0, then e∆ 6= 0, e
E = eEnsand E0(K) = E(K).
Proposition 1.3.15. The sequence of abelian groups
0 → E1(K) → E0(K) → eEns(k) → 0,
where the right-hand map is the reduction modulo π, is exact.
Proof. We give a sketch of the proof. The surjectivity on the right is obtained by Hensel’s Lemma (a point in eEns(k) satisfies the reduced Weierstrass
equa-tion ef (x, y) = 0 and the partial derivatives of ef at the point are nonzero). To complete the proof one has to show that the reduction is a homomor-phism and that E0(K) is a subgroup of E(K), the exactness in the middle follows from the definition of E1(K) and E0(K). These two facts can be
proved by noticing that the reduction takes lines to lines (see [11], Ch.VII, Proposition 2.1).
Proposition 1.3.16. Let E/K be given by a minimal Weierstrass equation, let bE/R be the formal group associated to E and let w(z) ∈ R be the power series obtained by substituting recursively (1.7). Then the map
b E(m) −→ E1(K) z 7−→ z w(z), − 1 w(z) is an isomorphism.
For a proof, see [11], Ch.VII, Proposition 2.2.
Proposition 1.3.17. Let E/K be an elliptic curve and m ≥ 1 an integer prime to char(k). Then
(1) E1(K) has no non-trivial points of order m;
(2) if the reduced curve eE is non-singular, then the reduction map E(K)[m] → eE(k)
is injective.
Proof. Thanks to the previous proposition the above exact sequence becomes
0 → bE(m) → E0(K) → eEns(k) → 0.
From this and using (2) of Proposition 1.3.12, we get (1). Now if eE is non-singular, then E0(K) = E(K) and eEns(k) = eE(k), so the m-torsion of E(K)
injects into eE(k) and this proves (2).
The above proposition gives us a method to find the torsion subgroup of an elliptic curve defined over a number field: given a number field K and Kv its completion with respect to some discrete valuation v, clearly E(K)
injects into E(Kv). So by applying the proposition with different v’s, we can
obtain information about the torsion in E(K) by computing the points of eE with coordinates in the residue fields kv.
Example 1.3.18. Let E/Q be the elliptic curve E : y2+ y = x3− x + 1.
Its discriminant is ∆ = −611 = −13 · 47, so its reduction modulo 2 (de-noted eE) is non-singular. One easily checks that eE(F2) = {O}, so from the
proposition we get that E(Q) has no non-zero torsion points.
Now what can we say about the action of Galois on the torsion points? Let Knr be the maximal unramified extension of K and Iv the inertia sub-group of GK. We have the following exact sequence:
1 → GK/Knr → GK → GKnr/K → 1,
where we use the notation GL/F = Gal(L/F ). But we also know that the unramified extensions of K correspond to the extensions of the residue field k, so the sequence becomes
1 → Iv → GK → Gk → 1.
We can then say that Iv is the set of elements of GK which act trivially on
1.3 Elliptic Curves 28
Definition 1.3.19. Let Σ be a set on which GK acts. We say that Σ is unramified at v if the action of Iv on Σ is trivial.
Proposition 1.3.20. Let E/K be an elliptic curve and suppose that eE/k is non-singular. Let m ≥ 1 be an integer prime to char(k). Then E[m] is unramified at v.
Proof. Let F/K be a finite extension such that E[m] ⊂ E(F ) and let R0 be the ring of integers of F , m0 the maximal ideal of R0, k0 the residue field of R0 and v0 the valuation of F . Since E has non-singular reduction, we can take a minimal Weierstrass equation for E at v and have v(∆) = 0. Then also v0(∆) = 0, so the reduced curve eE/k0 is non-singular. Now from Proposition 1.3.17, the reduction map E[m] → eE(k0) is injective.
Let σ ∈ Iv and P ∈ E[m]. Since the inertia subgroup acts trivially onk,
σ acts trivially on eE(k0), so ^
Pσ− P = ePσ− eP = eO.
Now since Pσ− P ∈ E[m], the injectivity of the map E[m] → eE(k0) implies that Pσ− P = O.
We can actually give a kind of converse to this, known as the Néron-Ogg-Shafarevich criterion (see Theorem 1.3.25).
1.3.4 Good and Bad Reduction
Let E/K be an elliptic curve. We know that its reduction in general is not an elliptic curve, in particular, there are three possibilities:
Definition 1.3.21. Let E/K be an elliptic curve, and let eE be the reduced curve for a minimal Weierstrass equation.
(1) E has good reduction over K if eE is non-singular. (2) E has multiplicative reduction over K if eE has a node. (3) E has additive reduction over K if eE has a cusp.
In the last two cases, we say that E has bad reduction. Furthermore, if E has multiplicative reduction, the reduction is said to be split (respectively non-split) if the slopes of the tangent lines at the node are in k (respectively not in k).
We will work only with E having good, ordinary reduction, so that the reduced curve is still an elliptic curve. It is easy to characterize the type of reduction of an elliptic curve by its Weierstrass equation.
Proposition 1.3.22. Let E/K be an elliptic curve with minimal Weierstrass equation
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6.
Let ∆ be the discriminant of this equation. Then E has good reduction if and only if v(∆) = 0 (i.e. ∆ ∈ R×).
This and the characterization of multiplicative and additive reduction can be found in [11], Ch.VII, Proposition 5.1, the terminology originates form the isomorphisms of Proposition 1.3.5.
Example 1.3.23. Let p ≥ 5 be a prime. Then the elliptic curve
y2 = x3+ px + 1
has good reduction over Qp, since ∆ = −16(4p3+ 27).
We want to know how reduction types behave if we have a field exten-sion. The next theorem answers to our question (for a proof, see [11], §VII, Proposition 5.4).
Theorem 1.3.24. Let E/K be an elliptic curve.
(1) Let F/K be an unramified extension. Then the reduction type of E over K is the same as the reduction type over F .
(2) Let F/K be a finite extension. If E has good or multiplicative reduction over K, it has the same type of reduction over F .
(3) There exists a finite extension F/K such that E has good or (split) multiplicative reduction over F .
Now if an elliptic curve E/K has good reduction, and m ≥ 1 is an integer prime to char(k), then we have seen that the torsion group E[m] is unramified. As announced earlier, here is a partial converse.
Theorem 1.3.25 (Néron-Ogg-Shafarevich). Let E/K be an elliptic curve. The following are equivalent:
(1) E has good reduction over K;
(2) E[m] is unramified at v for all integers m ≥ 1 prime with char(k);
(3) E[m] is unramified at v for infinitely many integers m ≥ 1 prime with char(k).
We will not see the proof (that can be found in [11], Ch.VII, Theorem 7.1) of this important theorem, but we mention that it is based on a deep theorem about the structure of the group E(K)/E0(K).
1.3 Elliptic Curves 30
Theorem 1.3.26 (Kodaira, Néron). Let E/K be an elliptic curve. If E has split multiplicative reduction over K, then E(K)/E0(K) is a cyclic group of order v(∆). In all other cases, E(K)/E0(K) is a finite group of order at
most 4.
Corollary 1.3.27. The subgroup E0(K) is of finite index in E(K).
Another application of this theorem is the following proposition, that will be very useful for our later work. This and the previous corollary are proved in [11], Ch.VII.6.
Proposition 1.3.28. Let K be a finite extension of Qp. Then E(K) contains
a subgroup of finite index which is isomorphic to R+(i.e. R taken additively).
Proof (Sketch). The main steps are:
(1) from the above corollary we know that E(K)/E0(K) is finite and by Proposition 1.3.15 E0(K)/E1(K) is isomorphic to eEns(k), which is
finite by hypothesis;
(2) we know from Proposition 1.3.16 that E1(K) ' bE(m) and by
Proposi-tion 1.3.12 bE(m) has a filtration such that each quotient is isomorphic to mi/mi+1;
(3) the logarithm map (see [11], Ch.IV, §5) provides an isomorphism bE(mr) '
R+ if r is sufficiently large (see [11], Ch.IV, Theorem 6.4);
(4) we then have that bE(m) ' E1(K) has a subgroup of finite index
iso-morphic to R+.
So if K is a finite extension of Qp, since R+ is a free Zp-module with
rank [K : Qp], we have obtained
E(K) ' Z[K:Qp]
p ×T,
where T is a finite group. This is usually referred to as Lutz Theorem, a local version of the classical Mordell-Weil Theorem we shall see later (see Theorem 1.3.38).
1.3.5 Elliptic Curves over global fiels
Let K be a number field and E an elliptic curve over K. We denote by R the ring of integers of K, with Kv the completion of K at v, for a valuation v on K and with Rv, mv, kv the ring of integers, the maximal ideal and the
residue field associated to Kv, respectively. Moreover we write MK0 for the set of non-archimedean valuations on K and MK∞for the set of archimedean valuations on K. We suppose E[m] ⊂ E(K).
Definition 1.3.29. Let m ≥ 2 be an integer. The Kummer pairing
κ : E(K) × GK → E[m]
is defined as follows. Let P ∈ E(K), and choose Q ∈ E(K) such that [m]Q = P . We then define
κ(P, σ) = Qσ− Q.
Proposition 1.3.30. The Kummer pairing has the following properties.
(1) The Kummer pairing is well-defined.
(2) The Kummer pairing is bilinear.
(3) The kernel of the Kummer pairing on the left is mE(K), i.e.
{P ∈ E(K) | κ(P, σ) = O ∀σ ∈ GK} = mE(K) .
(4) The kernel of the Kummer pairing on the right is GL, where
L = K([m]−1E(K))
is the composite of all the fields K(Q), where Q ranges over the points of E(K) such that [m]Q ∈ E(K).
Therefore the Kummer pairing induces a perfect bilinear pairing
E(K)/mE(K) × Gal(L/K) → E[m].
Most of this proposition follows immediately if we reinterpret the Kum-mer pairing in terms of group cohomology. Recall the KumKum-mer exact se-quence of GK-modules:
1 −→ µm−→ K × m
−→ K×−→ 1
where the map m is the rising to the m-th power. By taking the GK -cohomology we get a long exact sequence from which we extract
1 → K×/(K×)m−→ Hδ 1(K, µm) → H1(K, K ×
), (1.8)
since by Hilbert 90 H1(K, K×) = 1, δ is an isomorphism. We follow the same procedure for elliptic curves.
Proof (of Proposition 1.3.30). We start with the following exact sequence of GK-modules
1.3 Elliptic Curves 32
As above, we take the GK-cohomology, obtaining a long exact sequence
0 −→ E(K)[m] −→ E(K) −→[m] E(K) →
δ
−→ H1(K, E[m]) −→ H1(K, E(K)) −→ H[m] 1(K, E(K)) → . . .
(1.10) We can then extract the following short exact sequence, which we call the Kummer sequence for E/K:
0 → E(K) mE(K) → H
1(K, E[m]) → H1(K, E(K))[m] → 0, (1.11)
where H1(K, E(K))[m] is the m-torsion subgroup of H1(K, E(K)).
Now recall how the boundary map δ is defined: let P ∈ E(K) and choose some Q ∈ E(K) such that [m]Q = P . Then a representative for δ(P ) is the cocycle
c : GK → E[m]
σ 7→ Qσ− Q,
which corresponds to the Kummer pairing κ(P, σ) defined earlier. Since E[m] ⊂ E(K), we have
H1(K, E[m]) = Hom(GK, E[m])
and so we have an injective homomorphism
˜
δ : E(K)
mE(K) −→ Hom(GK, E[m])
P 7−→ κ(P, ·)
This proves (1), (2), (3) of Proposition 1.3.30. To prove (4), take σ ∈ GL,
then we have
κ(P, σ) = Qσ− Q = O,
since Q ∈ E(L), by definition of L. Conversely, suppose that σ ∈ GK is such that κ(P, σ) = O for all P ∈ E(K). Then for every Q ∈ E(K) satisfying [m]Q ∈ E(K) we have
O = κ([m]Q, σ) = Qσ− Q. By the definition of L we have that σ fixes L, so σ ∈ GL.
Using the inflaction-restriction sequence we can also prove
Proposition 1.3.31. Let F/K be a finite Galois extension. If E(F )/mE(F ) is finite, then E(K)/mE(K) is also finite.
