CHAPTER 19 Lecture 19

1. Hecke Operators on Heegner Points

1.1. Galois Actions on Cyclic Subgroups. Let us review briefly our notation and hypotheses. N is a fixed positive integer and F is a quadratic imaginary extension of ℚ in which each prime dividing N splits completely. We further require that F is not ℚ(i) or ℚ(\sqrt{−3}). We fix an embedding F ↪ ℂ and an ideal N of F such that O_{F} /N is cyclic of order N. For each n relatively prime to N we let O_{F,n} be the order ℤ + nO_{F} of conductor n in O_{F}, and we let F^{(n)} be the ring class field of F of conductor n. Thus

Gal(F^{(n)}/F) ≅ Pic(O_{F,n})

and

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

We now further assume that X_{0}(N) has positive genus and we fix an embedding X_{0}(N) ↪ J_{0}(N). x^{(n)} is the point of X_{0}(N) corresponding to the modular pair (E^{(n)}, C_{N}^{(n)}), where E^{(n)} = ℂ/O_{F,n}, N_{n} = N ∩ O_{F,n} and C_{N}^{(n)} = N_{n}^{−1}/O_{F,n}. x^{(n)} actually lies in X_{0}(N)(F^{(n)}), and we will also regard it as an element of J_{0}(N)(F^{(n)}) via our fixed embedding.

Fix a prime l not dividing N. We claim that for any n relatively prime to lN,

Gal(F^{(nl)}/F^{(n)}) = Gal(F^{(l)}/F^{(1)}).

This follows from the fact that F^{(n)} ∩ F^{(l)} = F^{(1)}, which is true since the two fields have relatively prime conductor (in the sense of class field theory; see [Cox, Exercise 9.20]). We denote this Galois group by 𝔤^{(l)}.

[figure: F^{(nl)}; 𝔤^{(l)}; F^{(n)}; F^{(l)}; 𝔤^{(l)}; F^{(1)}]

Fix such an n. As we have said, the elliptic curve E^{(n)} has a model over F^{(n)}; in fact, it is defined over an index 2 subfield of F^{(n)} (see [Shi, Theorem 5.7]) and thus the model over E^{(n)} is only well-defined up to twisting by a quadratic character. In any event, fixing such a model yields a Galois representation

G_{F^{(n)}} → Aut(E^{(n)}[l]).

0Last modified December 17, 1999

The image of G_{F^{(n)}} must commute with the endomorphisms of E^{(n)} defined over F^{(n)}, which is all of O_{F,n} (see [Si-2, Theorem 2.2.6]). Thus the image of the Galois representation actually lies in

Aut_{O_{F,n}/lO_{F,n}}(E^{(n)}[l]).

E^{(n)}[l] is free of rank 1 over O_{F,n}/lO_{F,n} ≅ O_{F} /lO_{F}, so we have a Galois representation

G_{F^{(n)}} → Aut_{O_{F,n}/lO_{F,n}} ≅ (O_{F,n}/lO_{F,n})^{∗} ≅ (O_{F} /lO_{F})^{∗}.

We are actually only interested in the action of G_{F^{(n)}} on the cyclic subgroups of E^{(n)} of order l, so we need only consider the induced map

G_{F^{(n)}} → (O_{F} /lO_{F})^{∗}/𝔽_{l}^{∗}

since scalars act trivially on the set of cyclic subgroups. This map is surjective and factors through Gal(F^{(nl)}/F^{(n)}) ([Shi, Theorem 5.7]), yielding a map

𝔤^{(l)} → (O_{F} /lO_{F})^{∗}/𝔽_{l}^{∗}

which is now an isomorphism, since each group has the same order.

Let us now assume that l is inert in F/ℚ. Then O_{F} /lO_{F} ≅ 𝔽_{l^{2}}. Now, E^{(n)}[l] is an 𝔽_{l^{2}} vector space of dimension 1, and thus an 𝔽_{l} vector space of dimension 2. One thus sees immediately that 𝔽_{l^{2}}^{∗}/𝔽_{l}^{∗} acts simply transitively on the collection ℙ_{𝔽_{l}}(E^{(n)}[l]) of one dimensional 𝔽_{l} subspaces of E^{(n)}[l], which are precisely the cyclic subgroups of E^{(n)} of order l. Combining all of this, we have seen that 𝔤^{(l)} acts simply transitively on the collection of cyclic subgroups of E^{(n)} of order l.

In order to restate our above analysis in a more useful way we first must define a trace map

Tr_{(nl,n)} : J_{0}(N)(F^{(nl)}) → J_{0}(N)(F^{(n)}).

This is nothing more than the natural map

x ↦ ∑_{σ∈𝔤^{(l)}} σx.

Proposition 1.1. With the above notation and hypothesis (in particular, l is inert in F/ℚ), we have

T_{l}x^{(n)} = Tr_{(nl,n)} x^{(nl)},

where T_{l} is the usual Hecke operator on J_{0}(N).

Proof. By definition,

T_{l}x^{(n)} = ∑_{C_{l}⊆E^{(n)}[l]}(E^{(n)}/C_{l}, C_{N}^{(n)} + C_{l}/C_{l}),

where the sum is over all cyclic subgroups of E^{(n)} of order l. On the other hand,

Tr_{(nl,n)} x^{(nl)} = ∑_{σ∈𝔤^{(l)}} σx^{(nl)}

= ∑_{σ∈𝔤^{(l)}} σ(E^{(nl)}, C_{N}^{(nl)})

= ∑_{σ∈𝔤^{(l)}} σ(ℂ/O_{F,nl}, N_{nl}^{−1}/O_{F,nl})

Note that

(ℂ/O_{F,nl}, N_{nl}^{−1}/O_{F,nl}) = ((ℂ/O_{F,n})/C_{l}, (N_{n}^{−1}/O_{F,n})/C_{l})

for some unique cyclic group of order l C_{l} ∈ E^{(n)}, since O_{F,n}/O_{F,nl} is cyclic of order l. Furthermore, by definition

σ((ℂ/O_{F,n})/C_{l}, (N_{n}^{−1}/O_{F,n})/C_{l}) = (ℂ/O_{F,n})/σC_{l}, (N_{n}^{−1}/O_{F,n})/σC_{l});

combined with the two expressions above and the transitivity of the action of 𝔤^{(l)} on the cyclic subgroups of E^{(n)} of order l, this proves the proposition.

Although we will not use it, we sketch the corresponding result in the case that l splits in F/ℚ. Let λ and λ̄ be the primes of F lying over l. In this case,

𝔤^{(l)} = 𝔽_{l}^{∗} × 𝔽_{l}^{∗}/𝔽_{l}^{∗},

the last 𝔽_{l}^{∗} being the diagonal copy of 𝔽_{l}^{∗}. So 𝔤^{(l)} is cyclic of order l − 1, and it acts on the ℙ_{𝔽_{l}}(E^{(n)}[l]) of cyclic subgroups of order l in E^{(n)}. Of course, 𝔤^{(l)} can not act on this projective space principally homogeneously. In fact, it fixes the two distinguished subgroups E^{(n)}[λ] and E^{(n)}[λ′], and then acts principally homogeneously on ℙ_{𝔽_{l}}(E^{(n)}[l]) − {E^{(n)}[λ], E^{(n)}[λ′]}. In this case, then, the formula of Proposition 1.1 must be modified slightly. Specifically, if one denotes by σ_{λ} and σλ̄ the elements of Gal(F^{(n)}/F) produced by λ ∩ O_{F,n} and λ̄ ∩ O_{F,n} under the isomorphism

Pic(O_{F,n}) ≅ Gal(F^{(n)}/F),

then one has the formula

Tr_{(nl,n)} x^{(nl)} = (T_{l} − σ_{λ} − σλ̄)x^{(n)}
= (T_{l} − σ_{λ} − σ_{λ}^{−1})x^{(n)}.

With a little care, then, one can derive the general formula

Tr_{(nl,n)}(−σ_{λ}x^{(nl)} + σ_{λ}^{2}x^{(n)}) = (1 − T_{l}σ_{λ} + lσ_{λ}^{2})x^{(n)}

of Mazur and Rubin, which works independently of the behaivor of l in F/ℚ.

1.2. Hecke Algebras. We will now relate all of our data to our earlier notation. For simplicity we now require that N is squarefree. Let 𝕋 ⊂ End_{ℚ}(J_{0}(N)) be the subalgebra generated by 1, T_{l} for l prime not dividing N, and U_{q} for q dividing N. 𝕋 is a commutative, finite, free ℤ-algebra. (See [DI, Sections 3.3-3.4].)

Let 𝔪 be a maximal ideal of 𝕋 such that 𝕋/𝔪 has characteristic p ≠ 2. Let B be the abelian variety J_{0}(N), and let

W = ⋃_{r} B(\overline{ℚ})[𝔪^{r}]

be the 𝔪-divisible subgroup of B(\overline{ℚ}). W is of course also p-divisible. Both 𝕋 and G_{ℚ} act on W, and the actions commute since 𝕋 is defined over ℚ.

W is actually a module over

𝕋_{𝔪} = \underleftarrow{lim} 𝕋/𝔪^{r}.

We will write A for this ring; it satisfies all of our usual hypotheses. Let H be the Cartier dual of W; it is a free A module of finite rank. We have a Galois representation

ρ : G_{ℚ} → Aut_{A}(H)

and a residual representation

\overline{ρ} : G_{ℚ} → Aut_{k}(H ⊗_{A} k),

where k = 𝕋/𝔪 = 𝕋_{𝔪}/𝔪𝕋_{𝔪}.

We make two hypothesis on the maximal ideal 𝔪. First, we assume that H is free of rank 2 over 𝕋_{𝔪}. Second, we assume that

\overline{ρ} : G_{ℚ} → GL_{2}(k)

is surjective. We call any maximal ideal with these properties good. We will see later that these hypotheses imply all of the hypotheses of Kolyvagin’s theorem.

We also assume that there is an η ∈ 𝕋 such that η𝕋 ∈ 𝔪, η : B → B is an isogeny and

ℤ ∩ η𝕋 = p^{s}ℤ

for some s ≥ 1. We will say that a prime l (inert in F/ℚ and not dividing N) is good for η if

T_{l} ∈ ηA

and

l + 1 ∈ ηA.

Note that in particular this implies that p^{s} divides l + 1, and thus that p^{s} divides the order of 𝔤^{(l)}.

1.3. Computations. We continue with the above notation, and we suppose that we have a good maximal ideal 𝔪, an η as above, and a prime l which is good for η. We define

P^{(l)} = ∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ g ∈ B(F^{(l)}) ⊗_{ℤ} ℤ[𝔤^{(l)}].

Let P_{η}^{(l)} be the “reduction of P^{(l)} modulo η”; that is, P_{η}^{(l)} is the image of P^{(l)} in

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ} ℤ[𝔤^{(l)}].

Note that

𝕋/η ⊗_{𝕋} B(F^{(l)})

is killed by p^{s} since 𝕋/η is, so

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ} ℤ[𝔤^{(l)}] = 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} ℤ/p^{s}ℤ[𝔤^{(l)}].

Next, let I^{(l)} be the augmentation ideal of ℤ/p^{s}ℤ[𝔤^{(l)}], so that there is an exact sequence

0 → I^{(l)} → ℤ/p^{s}ℤ[𝔤^{(l)}] → ℤ/p^{s}ℤ → 0,

the last map being given by evaluating each g ∈ 𝔤^{(l)} at 1. Each of these groups is free over ℤ/p^{s}ℤ, so the exact sequence splits over ℤ/p^{s}ℤ. In particular, the sequence

\begin{aligned}0 & ⟶ & 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} I^{(l)} & ⟶ & 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} ℤ/p^{s}ℤ[𝔤^{(l)}] \\ & ⟶ & 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} ℤ/p^{s}ℤ & ⟶ & 0 \end{aligned}

is still exact.

We claim that

P_{η}^{(l)} ∈ 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} I^{(l)}.

1. HECKE OPERATORS ON HEEGNER POINTS 153

To prove this, it suffices to show that P_{η}^{(l)} is in the kernel of the summation map

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} ℤ/p^{s}ℤ[𝔤^{(l)}] → 𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} ℤ/p^{s}ℤ. Equivalently, we must show that the image of P^{(l)} under

B(F^{(l)}) ⊗_{ℤ} ℤ[𝔤^{(l)}] → B(F^{(l)}) ⊗_{ℤ} ℤ = B(F^{(l)})

is contained in ηB(F^{(l)}). This image is

∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ 1 = ∑_{g∈𝔤^{(l)}} gx^{(l)}
= Tr_{(l,1)} x^{(l)}
= T_{l}x^{(l)}

by Proposition 1.1. But by hypothesis,

T_{l} ∈ η𝕋,

so this establishes the claim.

Next, let P_{η}^{(l)} be the image of P_{η}^{(l)} in

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} I^{(l)}/(I^{(l)})^{2}.

Note that the map

𝔤^{(l)} → I^{(l)}

sending g to g − 1 induces an isomorphism

𝔤^{(l)} ⊗_{ℤ} ℤ/p^{s}ℤ → I^{(l)}/(I^{(l)})^{2}.

Thus we can (and do) regard P_{η}^{(l)} as an element of

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} (𝔤^{(l)} ⊗_{ℤ} ℤ/p^{s}ℤ) .

Note also that 𝔤^{(l)} ⊗ ℤ/p^{s}ℤ is cyclic of order p^{s}, so

𝕋/η ⊗_{𝕋} B(F^{(l)}) ⊗_{ℤ/p^{s}ℤ} (𝔤^{(l)} ⊗_{ℤ} ℤ/p^{s}ℤ) ≅ 𝕋/η ⊗_{𝕋} B(F^{(l)}),

at least as abelian groups.

We claim that P_{η}^{(l)} is fixed under the action of 𝔤^{(l)} on the B(F^{(l)}) coordinate. To see this, we compute for σ ∈ 𝔤^{(l)}

σP^{(l)} − P^{(l)} = ∑_{g∈𝔤^{(l)}} σgx^{(l)} ⊗ g − ∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ g

= ∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ gσ^{−1} − ∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ g

= ∑_{g∈𝔤^{(l)}} gx^{(l)} ⊗ g(σ^{−1} − 1)