CHAPTER 18 Lecture 18

1. Representations on Torsion Points

1.1. The Tree of a Modular Pair. For this section fix a modular pair (E, C_{N}) defined over the complex numbers ℂ. (The precise choice of C_{N} will only come into play later.) We associate to this pair a graph (actually a building) Γ(E, N) in the following way. The vertices of Γ(E, N) are in bijection with the cyclic subgroups of E(ℂ) of order relatively prime to N. Two vertices C_{1} and C_{2} are joined by an edge if one is contained in the other with index some prime l. We will consider this edge to have weight l.

For any prime l not dividing N, we also define a subgraph Γ_{l}(E) consisting of those vertices of Γ(E, N) corresponding to cyclic subgroups of l-power order (and all of the edges in Γ(E, N) connecting them). Γ_{l}(E) is in fact a tree, as one easily sees.

For any l not dividing N, we define a Hecke correspondence T_{l} on the vertices vert(Γ(E, N)) by sending a vertex v to all vertices which are joined to v by an edge

Figure 1. The Beginning of the Graph Γ for l = 2 of weight l. This induces a homomorphism on divisor groups

0Last modified December 17, 1999

T_{l} : ℤ[Γ(E, N)] → ℤ[Γ(E, N)];

here by ℤ[Γ(E, N)] we mean the free abelian group on the vertices of Γ(E, N).

Most Hecke correspondences do not restrict to Γ_{l}(E); the only one that does is T_{l}.

1.2. Galois Actions on Γ(E, N). Suppose now that the modular pair (E, C_{N}) is defined over a subfield K of ℂ. We let E_{tors}^{(N)} be the subgroup of the torsion points of E of order relatively prime to N. Define K(E, N) to be the splitting field of the natural G_{K}-action on E_{tors}^{(N)}; equivalently, K(E, N) is generated over K by the coordinates of all of the prime to N torsion points of E(ℂ). If K is a number field, then K(E, N) is certainly an infinite extension of K.

Note that Aut(E_{tors}^{(N)}) also acts on the graph Γ(E, N): if φ ∈ Aut(E_{tors}^{(N)}) and v ∈ vert(Γ(E, N)) corresponds to the cyclic group C ⊆ E_{tors}^{(N)}, then φv is the vertex corresponding to φ(C). One easily checks that this action preserves edges, so that it really is an automorphism of the graph Γ(E, N).

Now, any choice of isomorphism

E_{tors}^{(N)} ≅ ∏_{l∤N} ℚ_{l}/ℤ_{l} × ℚ_{l}/ℤ_{l}

yields an isomorphism

Aut(E_{tors}^{(N)}) ≅ ∏_{l∤N} GL_{2}(ℤ_{l}).

Note that the subgroup of scalar matrices is independent of the choice of “basis”of E_{tors}^{(N)} (it is just those elements of the automorphism group which act on each E[M] as a scalar), and furthermore that it acts trivially on Γ(E, N), since any cyclic subgroup is taken to itself by scalars. Thus the map

∏_{l∤N} GL_{2}(ℤ_{l}) ≅ Aut(E_{tors}^{(N)}) → Aut(Γ(E, N))

factors through ∏ PGL_{2}(ℤ_{l}):

\xymatrix{∏_{l∤N} GL_{2}(ℤ_{l}) \ar[rr] \ar@{-->}[dr] &  & Aut(Γ(E, N)) \\  & ∏_{l∤N} PGL_{2}(ℤ_{l}) \ar[ur]}

Of course, Gal(K(E, N)/K) acts in a natural way on E_{tors}^{(N)}, yielding a representation

ρ_{E,N} : Gal(K(E, N)/K) → Aut(E_{tors}^{(N)}) ≅ ∏_{l∤N} GL_{2}(ℤ_{l}).

The action of Aut(E_{tors}^{(N)}) on Γ(E, N) yields a second representation

pρ_{E,N} : Gal(K(E, N)/K) → Aut(Γ(E, N)) ≅ ∏_{l∤N} PGL_{2}(ℤ_{l});

of course, pρ_{E,N} is just the composition of ρ_{E,N} with the natural quotient map

∏_{l∤N} GL_{2}(ℤ_{l}) → ∏_{l∤N} PGL_{2}(ℤ_{l}).

Let us now connect all of this. Let Γ(E, N) be the graph associated to a modular pair (E, C_{N}) defined over a subfield K of ℂ. Then we can define a natural map

vert(Γ(E, N)) → X_{0}(N)(ℂ)

by sending a vertex corresponding to a cyclic group C of order prime to N to the modular pair

(E/C, C_{N} + C/C).

In fact, the modular pair is defined over K(E, N), since C is, so the image of the map is contained in X_{0}(N)(K(E, N)). Furthermore, this mapping is compatible with the Hecke correspondences on Γ(E, N) and X_{0}(N)(ℂ), as is clear from the definitions. Composing with the natural map

X_{0}(N)(K(E, N)) → J_{0}(N)(K(E, N)),

(this map is an embedding if X_{0}(N) has genus greater than zero, and is trivial if X_{0}(N) has genus 0) we obtain a Hecke equivariant map

vert(Γ(E, N)) → J_{0}(N)(K(E, N)).

1.3. The Image of ρ_{E,1}. We now consider the case N = 1 above. Thus we have an elliptic curve E, defined over some field K, and we are considering the respresentation of G_{K} on E_{tors}, which we will just write as ρ_{E}. The image of this representation turns out to depend very much on whether or not E has complex multiplication. Recall that E is said to have complex multiplication (or CM for short) if End_{ℂ}(E) is strictly larger than ℤ. It follows that End_{ℂ}(E) is an order in an imaginary quadratic field. (See [Si-AEC, Chapter 6, Theorem 5.5]. For a nice introduction to the theory of complex multiplication, see [Si-2, Chapter 2]. Unfortunately, it does not cover complex multiplication in the generality in which we will need it. We will comment more on that later.) In the non-CM case, we have the following theorem of Serre.

Theorem 1.1 (Serre). Let K be a number field and let E be an elliptic curve over K without complex multiplication. Then the representation

ρ_{E} : G_{K} → ∏ GL_{2}(ℤ_{l}) ≅ GL_{2}(ℤ̂)

has open image. In particular, the image has finite index, and the composition

G_{K} → GL_{2}(ℤ̂) → GL_{2}(ℤ_{l})

is surjective for almost all primes l.

Proof. See [Se-EC] or [La-EF, Chapter 17].

In the CM case, ρ_{E} is very far from surjective; to see this, note that the image of ρ_{E} must commute with all elements of End_{K}(E). Thus

ρ_{E}(G_{K}) ⊆ Aut_{End_{K}(E)}(E_{tors}) ⊆ Aut(E_{tors)}.

If End_{K}(E) is larger than ℤ, then this subgroup is much smaller than the whole of Aut(E_{tors}). Furthemore, although End_{K}(E) need not be larger than ℤ even though E has complex multiplication, there exists a finite extension L/K such that End_{L}(E) is larger than ℤ (see [Si-2, Chapter 2, Theorem 2.2]), which is enough to force ρ_{E}(GL) to be small.

One can actually characterize ρ_{E}(G_{K}) fairly precisely in the CM case. Let F be the quadratic imaginary field by which E has complex multiplication, with Galois group 𝔤 = Gal(F/ℚ) = {1, τ}. Consider the group

(O_{F} ⊗_{ℤ} ℤ_{l})^{∗}/ℤ_{l}^{∗};

here O_{F} is the ring of integers in F, and

ℤ_{l}^{∗} ↪ (O_{F} ⊗ ℤ_{l})^{∗}

is induced by the natural inclusion

ℤ ↪ O_{F} .

We define D_{l} to be the semi-direct product

D_{l} = 𝔤 ⋉ (O_{F} ⊗ ℤ_{l})^{∗}/ℤ_{l}^{∗},

where τ ∈ 𝔤 acts on (O_{F} ⊗ ℤ_{l})^{∗}/ℤ_{l}^{∗} by inversion.

D_{l} can actually be realized as a subgroup of PGL_{2}(ℤ_{l}). This is done by first choosing a ℤ-basis of O_{F}, which is a free rank 2 ℤ-module. The action of (O_{F} ⊗ℤ_{l})^{∗} on O_{F} ⊗ ℤ_{l} now yields an embedding

(O_{F} ⊗ ℤ_{l})^{∗} ↪ GL_{2}(ℤ_{l})

which in turn yields

(O_{F} ⊗ ℤ_{l})^{∗}/ℤ_{l}^{∗} ↪ PGL_{2}(ℤ_{l}).

Next, one observes that this group is a Cartan subgroup of PGL_{2}(ℤ_{l}), meaning that it becomes a maximal torus over the algebraic closure. (Recall that a torus is a group isomorphic to a product of multiplicative groups.) In this situation, it is a general fact that the group has index 2 in its normalizer, and that non-trivial elements of the normalizer act by inversion. Thus we have realized

D_{l} ↪ PGL_{2}(ℤ_{l})

as the normalizer of

(O_{F} ⊗ ℤ_{l})^{∗}/ℤ_{l}^{∗} ↪ PGL_{2}(ℤ_{l}).

The relevance of D_{l} to us rests on the fact that ρ_{E}(G_{K}) is conjugate to a subgroup of D_{l}.

2. Heegner Points

2.1. Ring class fields. For an excellent introduction to orders and ring class fields and proofs of the results in this section, see [Cox, Chapter 2, Sections 7 and 9].

Let F be an imaginary quadratic field. A subring R of the ring of integers O_{F} is called an order if it is finitely generated and if it generates all of F over ℚ. It follows easily that R is a lattice in ℂ. One can also show that all orders are of the form

O_{F,n} = ℤ + nO_{F}

for some integer n.

The ideal theory of O_{F,n} is quite similar to that of O_{F}; in particular, if 𝔞 is an ideal of O_{F} relatively prime to n, then 𝔞 ∩ O_{F,n} is an invertible ideal in O_{F,n}. Furthermore, the ideal class group Pic(O_{F,n}) of O_{F,n} (this is not defined exactly as one might expect; see [Cox, p. 136]) is an extension of the ideal class group of O_{F} by the abelian group

(O_{F} /nO_{F})^{∗}/(ℤ/nℤ)^{∗}.

Orders have important connections to the class field theory of F. Specifically, for any positive integer n we define the ring class field F^{(n)} of conductor n to be the unique abelian extension of F, unramified away n, such that a prime 𝔭 of O_{F} splits completely in F^{(n)} if and only if 𝔭 is principal with a generator α such that α is congruent to a rational integer modulo n. (That F^{(n)} exists follows from the usual statements of class field theory; see [Cox, p. 145 and pp. 179–180].)

Note that F^{(1)} is just the Hilbert class field of F. Furthermore, it follows from the definition that Gal(F^{(n)}/F) is isomorphic to the ideal class group of O_{F,n}; from this and the computation of the ideal class group of O_{F,n}, we see that

Gal(F^{(n)}/F^{(1)}) ≅ (O_{F} /nO_{F})^{∗}/(ℤ/nℤ)^{∗}.

F^{(n)} is also Galois over ℚ; here one finds that

Gal(F^{(n)}/ℚ) ≅ 𝔤 ⋊ Pic(O_{F,n})

where τ ∈ 𝔤 operates on Pic(O_{F,n}) in the natural way, which works out to just be by inversion. In fact, F^{(n)} can also be characterized as the maximal generalized dihedral extension of ℚ of conductor n; see [Cox, Chapter 2, Section 9, Part D].

2.2. Heegner Points. We will now need some of the theory of complex multiplication over non-integrally closed fields. Most of what we need is contained in [Cox, Chapter 3, Section 11, esp. Theorem 11.1]; alternately, [La-EF] and [Shi] have extensive treatments of the theory of complex multiplication.

Let N be a positive integer and let F be a quadratic imaginary field in which every prime dividing N splits completely. (The existence of such an F follows easily from the quadratic reciprocity law.) For simplicity, we assume that F is not ℚ(i) or ℚ(\sqrt{−3}); the existence of non-trivial units complicates things somewhat. We also must fix an embedding F ↪ ℂ. Our hypothesis on F implies that there is an ideal N of O_{F} such that O_{F} /N is cyclic of order N. Fix such an ideal N. We also have that N^{−1}/O_{F} is cyclic of order N.

Now, let n be an integer relatively prime to N. We consider the order O_{F,n} and the ideal N_{n} = N ∩ O_{F,n}. N_{n} is invertible in O_{F,n} (see the previous section), and we still have that N_{n}^{−1}/O_{F,n} is cyclic of order N.

We associate to each pair (O_{F,n}, N_{n}^{−1}) the modular pair

(ℂ/O_{F,n}, N_{n}^{−1}/O_{F,n})

defined over ℂ; note that O_{F,n} is a lattice, so ℂ/O_{F,n} really is an elliptic curve, and that N_{n}^{−1}/O_{F,n} is a cyclic subgroup of order N. This modular pair then defines a Heegner point

x^{(n)} ∈ X_{0}(N)(ℂ).

For our applications, we will need the following theorem, which is essentially [Cox, Theorem 11.1].

Theorem 2.1. Let n be relatively prime to N. Let E be the elliptic curve over ℂ defined by ℂ/O_{F,n} and let C be the cyclic subgroup of order N of E defined by N_{n}^{−1}/O_{F,n}. Then the modular pair (E, C) is defined over the ring class field F^{(n)}. In particular,

x^{(n)} ∈ X_{0}(N)(F^{(n)}).