GALOIS GROUPS ATTACHED TO POINTS OF FINITE ORDER ON ELLIPTIC CURVES OVER NUMBER FIELDS (D’APRÈS SERRE)

JACQUES VÉLU

1. Introduction

Let E be an elliptic curve defined over a number field K and equipped with a K-rational point O. It is always possible to give E a model of the following type:

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, (1)

where a_{i} ∈ K, the cubic in (1) is nonsingular, and where O is the “point at infinity” of the cubic.

The points of E defined over K form a group E(K), where A + B is calculated as follows:

[The diagram explaining the addition of points on elliptic curves is not repreoduced here.]

The theorem of Mordell-Weil states that if K is a number field, then E(K) is finitely generated. Thus we have

E(K) ≃ E(K)_{tors} ⊕ ℤ^{r_{K}},

where E(K)_{tors} is the finite subgroup of E(K) formed by points of finite order, and where r_{K} is called the rank of E over K. The computation of r_{K} is not always possible although it is known how to do that for certain curves. It is the subject of numerous conjectures.

The computation of E(K)_{tors} is easy since there is an algorithm allowing to find all its points and therefore its structure.

2. Points of finite order on an elliptic curve over a number field

These have the following properties:

a) The points E[N] of order dividing N on the curve E defined over K form a group isomorphic to (ℤ/Nℤ)^{2}.

b) The coordinates of points of E[N] are algebraic over K, and the algebraic extension K_{N} = K(E[N]) is Galois over K. We put G_{N} = Gal(K_{N}/K).

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c) The sum of two points A and B in E[N] is a point C whose coordinates are rational functions over K of the coordinates of A and B. Consequently, if σ ∈ G_{N}, then σ(A + B) = σ(A) + σ(B), and there is a homomorphism G_{N} ⟶ Aut(E[N]). Moreover, since E[N] generates K_{N} over K, this homomorphism is injective.

Thus we have

G_{N} is isomorphic to a subgroup of Aut(E[N]) ≃ GL(2, ℤ/Nℤ). Next we pass to the limit. We define

E_{ℓ^{∞}} = ⋃_{n} E[ℓ^{n}]  = lim_{⟶} E[ℓ^{n}]
E_{∞} = ⋃_{N} E[N]  = lim_{⟶} E[N]
K_{ℓ^{∞}} = ⋃_{n} K_{ℓ^{n}},  K_{∞} = ⋃_{N} K_{N}
G_{ℓ^{∞}} = Gal(K_{ℓ^{∞}}/K), G_{∞} = Gal(K_{∞}/K).

Then G_{ℓ^{∞}} is isomorphic to a subgroup of

lim_{⟵} GL(2, ℤ/ℓ^{n}ℤ) = GL(2, ℤ_{ℓ}),

and G_{∞} is isomorphic to a subgroup of

lim_{⟵} GL(2, ℤ/Nℤ) = GL(2, ℤ̂).

3. The analogue of the multiplicative group

An algebraic group that can be studied more easily than an elliptic curve E/K is the multiplicative group G_{m}.

The points of order N of G_{m} are the N^{th} roots of unity μ_{N} which form a group isomorphic to ℤ/Nℤ. Consider the field K(μ_{N}); this is a Galois extension of K, and the group Gal(K(μ_{N})/K) is a subgroup of Aut(μ_{N}) which in turn is isomorphic to (ℤ/Nℤ)^{×} ≃ GL(1, ℤ/Nℤ). By passing to the limit, we get

Gal(K(μ_{ℓ^{∞}}/K) ≃ subgroup of ℤ_{ℓ}^{×} ≃ GL(1, ℤ_{ℓ}),
Gal(K(μ_{∞}/K) ≃ subgroup of ℤ̂^{×} ≃ ∏_{ℓ} GL(1, ℤ_{ℓ}).

The following equivalent theorems are known to hold:

Theorem 1 The group Gal(K(μ_{∞}/K) is isomorphic to an open subgroup of ℤ̂^{×}.

Theorem 1’ For all primes ℓ, the group Gal(K(μ_{ℓ^{∞}}/K) is isomorphic to an open subgroup of ℤ_{ℓ}^{×}, with equality for almost all ℓ.

It is therefore natural to ask the same question for the groups G_{ℓ^{∞}} and G_{∞} associated to an elliptic curve.

4. Results

If E does not have complex multiplication, then the following three equivalent theorems are true:

Theorem 2 The group G_{∞} is isomorphic to an open subgroup of GL(2, ℤ̂).

Theorem 2’ a) For all primes ℓ, the group G_{ℓ^{∞}} is isomorphic to an open subgroup of GL(2, ℤ_{ℓ});

b) for almost all ℓ, we have G_{ℓ^{∞}} ≃ GL(2, ℤ_{ℓ}).

Theorem 2” For all primes ℓ, the group G_{ℓ^{∞}} is isomorphic to an open subgroup of GL(2, ℤ_{ℓ});

b) for almost all ℓ, we have G_{ℓ} ≃ GL(2, 𝔽_{ℓ}).

Property a) of Theorems 2’ and 2” is proved in [1, IV-11]. The equivalence of Theorems 2, 2’ and 2” is proved in [1, IV-19]. Property 2”.b) is proved in [2, p. 294].

Remark 1. Theorem 2 can be reformulated in another form:

Theorem 2”’ If the elliptic curve E defined over K does not have complex multiplication, then there exists a finite extension L/K such that Gal(L(Etors)/L) is isomorphic to an open subgroup of GL(2, ℤ̂).

It is clear that Theorem 2 implies Theorem 2”’ by taking L = K. Conversely, Theorem 2”’ implies Theorem 2; in fact, since Gal(L_{∞}/L) ≃ Gal(K_{∞}/K_{∞}∩L) which is a subgroup of Gal(K_{∞}/K), this implies that if Gal(L_{∞}/L) is isomorphic to an open subgroup of GL(2, ℤ̂), then so is Gal(K_{∞}/K).

L_{∞} = K_{∞}L
[figure]
K_{∞}  L
[figure]
K_{∞} ∩ L

K

Remark 2. In [1], Theorems 2, 2’ and 2” are proved under the assumption that the invariant j of E is not an integer in K, while in [2]

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it is proved assuming that E does not have CM. Since it is known that curves with CM have integral j-invariant (the converse is false), the result in [2] is stronger than that in [1]. Moreover, if E has CM, it is known how to describe the groups G_{ℓ^{∞}}. Let A denote the endomorphism ring of such an E. This is an order in the complex quadratic field A ⊗ ℚ. Then (A ⊗_{ℤ} ℤ_{ℓ})^{×} embeds into GL(2, ℤ_{ℓ}). We have the following theorem ([2, p. 302]):

Theorem. G_{∞} is isomorphic to an open subgroup of ∏_{ℓ}(A ⊗_{ℤ} ℤ_{ℓ})^{×}.

5. Sketch of the proof of property 2’.a)

The group G_{ℓ^{∞}} is isomorphic to a closed subgroup of GL(2, ℤ_{ℓ}). It is therefore a Lie group, and its Lie algebra G_{ℓ} is isomorphic to a subalgebra of M_{2}(ℚ_{ℓ}). The theory of Lie groups [3] shows that property a) is equivalent to the fact that G_{ℓ} ≃ ℚ_{ℓ}. Let G′_{ℓ} denote the commutator of G_{ℓ} in M_{2}(ℚ_{ℓ}). It is a field, hence either ℚ_{ℓ} or a quadratic extension of ℚ_{ℓ}. If G_{ℓ} ≃ M_{2}(ℚ_{ℓ}), then G′_{ℓ} ≃ ℚ_{ℓ}. It can be shown that the converse holds. Finally, one shows using the theory of locally algebraic representations developed by Serre that G′_{ℓ} cannot be a quadratic extension of ℚ_{ℓ}, and property a) follows.

6. Method of proof of property 2”.b)

This is what we need to prove:

Given an elliptic curve defined over a number field K and without complex multiplication, then for almost all primes ℓ we have G_{ℓ} ≃ GL(2, 𝔽_{ℓ}).

A. It is known that G_{ℓ} is isomorphic to a subgroup of GL(2, 𝔽_{ℓ}); let us therefore make a list of all subgroups of GL(2, 𝔽_{ℓ}).

Theorem 3. Let H ⊆ GL(2, 𝔽_{ℓ}) be a subgroup with ℓ | # H. Then either H contains SL(2, 𝔽_{ℓ}), or H is contained in some Borel subgroup, i.e. a subgroup consisting of elements of the form ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ).

Proof. a) If ℓ | # H, then H contains an element of order ℓ.

b). Lemma Every element s of order ℓ in GL(2, 𝔽_{ℓ}) fixes a unique line D_{s}.

In fact, s^{ℓ} = 1, so if λ is an eigenvalue of s, then λ^{s} = 1, and s fixes the line generated by an eigenvector associated to the eigenvalue 1. This is unique since if s fixes two lines, then s has the form ( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) with respect to the basis consisting of the vectors spanning the lines. But such an s does not have order ℓ.

c). If t ∈ H, then D_{tst^{−1}} = tD_{s}.

Thus there are two possibilities for H:

1. All lines D_{s} associated to all elements s of order ℓ in H are equal to one line D. By c), each t ∈ H fixes D. Taking a vector spanning D as our first basis vector, we see that H must be a Borel subgroup.

2. There exist s, s′ ∈ H of order ℓ such that D_{s} ≠ D_{s′}. Taking the vectors spanning these lines as our basis, then s = ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) and s′ = ( \begin{matrix}1 & 0 \\ a′ & 1 \end{matrix} ), and since s^{k} = ( \begin{matrix}1 & ka \\ 0 & 1 \end{matrix} ), H contains the matrices ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) which generate SL(2, 𝔽_{ℓ}) by [4, p. 104]. This proves Theorem 3.

For classifying the subgroups H of GL(2, 𝔽_{ℓ}) whose order is not divisible by ℓ, we look at the subgroups of PGL(2, 𝔽_{ℓ}) with order prime to ℓ.

Theorem 4. If H̃ is a subgroup of PGL(2, 𝔽_{ℓ}) with order prime to ℓ, then H̃ is isomorphic to a subgroup of one of the following groups:

i) a dihedral group;

ii) the alternating group A_{4}; iii) the symmetric group S_{4}; iv) the alternating group A_{5}.

This result is proved using the Lefschetz principle:

a) Since the order of H̃ is prime to ℓ, a Hensel-type argument shows that H̃ is isomorphic to a subgroup of PGL(2, ℚ_{ℓ}).

b) Since ℂ is isomorphic as a field to ℚ_{ℓ}, H is isomorphic to a subgroup of PGL(2, ℂ).

c) PGL(2, ℂ) is the automorphism group of the Riemann sphere ℙ_{1}(ℂ). Take a point on ℙ_{1}(ℂ) and consider its orbit under H̃. One obtains a regular polyhedron whose automorphism group contains H̃ as a subgroup. There are the following possibilities:

polyhedron  automorphism group
polygon  dihedral group
tetrahedron  A_{4}
cube or octahedron  S_{4}
dodecahedron or icosahedron  A_{5}

Theorem 4 has the following consequence:

Theorem 4’. If H is a subgroup of GL(2, 𝔽_{ℓ}) of order prime to ℓ, then H is one of the following:

i) H is contained in a Cartan subgroup; ii) H is contained in the normalizer of a Cartan subgroup; iii) H is isomorphic to A_{4}, S_{4} or A_{5}.

Proof i) If H̃ is cyclic

B) Now that we have these theorems classifying the subgroups of GL(2, 𝔽_{ℓ}), we use the information given by the fact that E is an elliptic curve to eliminate the different cases.

We show that:

a) The extension K_{ℓ}/K is unramified outside the places of bad reduction or the places of K dividing ℓ.

Almost all the places of K are unramified over ℚ and at almost all places of K, the curve E has good reduction. Therefore, for almost all ℓ the places of K dividing ℓ are unramified over ℚ, and E has good reduction there.

b) At such a place v, one can study the inertia subgroup of places w in K_{ℓ} above v. A local argument proves

Theorem 5. The inertia subgroup is

i) either a cyclic subgroup of order ℓ − 1 isomorphic to a 1/2-Cartan subgroup ( \begin{matrix}∗ & 0 \\ 0 & 1 \end{matrix} ),

ii) or a cyclic subgroup of order ℓ^{2} − 1 isomorphic to 𝔽_{ℓ^{2}}^{×}.

Consequently, for almost all ℓ the group G_{ℓ} cannot be too small because of the following

Theorem 6. Let H ⊆ GL(2, 𝔽_{ℓ}) be a subgroup such that H contains a subgroup as in Theorem 5.i). Then exactly one of the following assertions holds:

• H = GL(2, 𝔽_{ℓ});

• H is contained in a Borel subgroup ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ),

• H is contained in the normalizer of a Cartan subgroup,

• ℓ = 5, and the image of H̃ of H in PGL(2, 𝔽_{ℓ}) is isomorphic to S_{4}.

Theorem 6’. Let H ⊆ GL(2, 𝔽_{ℓ}) be a subgroup such that H contains a subgroup as in Theorem 5.ii). Then

• either H = GL(2, 𝔽_{ℓ});

• or H is contained in the normalizer of a Cartan subgroup

Proof of Theorem 6.

If ℓ | #H then H ⊆ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix}) or SL(2, 𝔽_{ℓ}) ⊆ H but the map det : H ⟶ 𝔽_{ℓ}^{×} is onto: thus if SL(2, 𝔽_{ℓ}) ⊆ H then in fact H = GL(2, 𝔽_{ℓ}).

If (ℓ, #H) = 1, let H̃ denote the image of H in PGL(2, 𝔽_{ℓ}). H̃ contains a cyclic subgroup of order ℓ − 1, so if ℓ ≥ 7, then H̃ ≠ A_{4}, S_{4}, A_{5}; if ℓ = 5, then the only possibility is H̃ = S_{4}; if ℓ = 2, 3, since A_{4}, S_{4}, A_{5} contain elements of order 2 and 3 and since (ℓ, #H) = 1, we deduce that H̃ ≠ A_{4}, S_{4}, A_{5}. Finally, if H̃ is cyclic or dihedral, then H is contained in a normalizer of a Cartan subgroup.

Proof of Theorem 6’.

If ℓ | #H then we cannot have H ⊆ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix}) since (ℓ^{2} − 1) | #H and (ℓ^{2} − 1) ∤ ℓ(ℓ − 1)^{2}, and again SL(2, 𝔽_{ℓ}) ⊆ H implies H = GL(2, 𝔽_{ℓ}).

If (ℓ, #H) = 1, then H̃ contains a cyclic subgroup of order ℓ+1. For ℓ ≥ 5, this shows that H̃ ≠ A_{4}, S_{4}, A_{5}, and if ℓ = 2, 3, the argument in the proof of Theorem 6 applies.

Theorems 5, 6 and 6’ show

Theorem 7. For almost all ℓ, the group G_{ℓ} is

(i) isomorphic to GL(2, 𝔽_{ℓ}); (ii) contained in a Borel subgroup or a Cartan subgroup; (iii) or contained in the normalizer of a Cartan subgroup without being

contained in a Cartan subgroup.

It remains to eliminate the cases (ii) and (iii). For (iii), we proceed as follows: G_{ℓ} is contained in the normalizer of a Cartan subgroup N_{ℓ} but not contained in the Cartan subgroup C_{ℓ}. Thus the group G_{ℓ}/C_{ℓ} has order 2 (since (N_{ℓ} : C_{ℓ}) = 2), hence there exists a field K′_{ℓ} ⊂ K_{ℓ} which is a quadratic extension of ℚ with Galois group G_{ℓ}/C_{ℓ}. By considering all possibilities it can be shown that the places of K_{ℓ} (places dividing ℓ, places not dividing ℓ with good reduction, places not dividing ℓ with bad reduction) that K′_{ℓ} is unramified over ℚ_{ℓ}, and since there are only finitely many unramified quadratic extensions of K_{ℓ}, the case (iii) can occur only finitely often.

The case (ii) is much more difficult. Since G_{ℓ} is isomorphic to a Cartan or Borel subgroup, there are two characters on G_{ℓ} with values in 𝔽_{ℓ}^{×}, and using class field theory it can be shown that E has complex multiplication: this proves the theorem.

References

[1] J.P. Serre, Abelian l-adic representations and elliptic curves, McGill University lecture notes 1968

[2] J.P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331

[3] J.P. Serre, Lie algebras and Lie groups, Benjamin 1964

[4] N. Bourbaki, Algèbre Chap. III: Algèbre multilineaire Paris 1948