CHAPTER 22 Lecture 22

1. Completion of the Proofs

1.1. Bounding of the Shafarevich-Tate Group. We continue with the notation of the previous lectures. We assume the following lemmas for this section; we will prove Lemma 1.1 later in the lecture. (Recall that the cohomology class

˜d̃_{η}^{(l)} lies in H^{1}(𝔤^{(l)}, B(k)), where k is the residue field of F_{l}^{(1)} (and of F_{l}^{(l)} and F).

Lemma 1.1. For all g ∈ 𝔤^{(l)},

˜d̃_{η}^{(l)}(g) = ± (\frac{(l + 1) Fr_{l} −a_{l}}{p} ) P ⊗ (1 − g).

Lemma 1.2.

κ_{η}^{(l)} ∈ H_{s}^{1}(G_{F} , B[p])^{+}.

We now prove the main theorem of the course. Recall that P ∈ B(F) is the basic Heegner point.

Theorem 1.3. If P ∈ pB(F_{l}), then Supp κ_{η}^{(l)} = ∅. If P ∉ pB(F_{l}), then Supp κ_{η}^{(l)} = {l}, and κ_{η}^{(l)} has singular depth 1 at l.

Proof. Recall from Lecture 21 that we had reduced this to determining the behavior of any representative crossed homomorphism of

˜d̃_{η}^{(l)} ∈ H^{1}(𝔤^{(l)}, B(k)).

In fact, since 𝔤^{(l)} acts trivially on B(k), the homomorphism

˜d̃_{η}^{(l)} : 𝔤^{(l)} → B(k)

is well-defined, and we must determine when it vanishes.

This can be done easily from Lemma 1.1 and Lemma 1.1 of Lecture 21. Specifically, Fr_{l} acts on P by −1, so

˜d̃_{η}^{(l)}(g) = ± (\frac{l + 1 + a_{l}}{p} ) P ⊗ 1 − g.

But since P ∈ B(k)_{p}^{−} and B(k)_{p}^{−} is cyclic of order p^{ord_{p}(l+1+a_{l})}, ˜ d̃_{η}^{(l)} vanishes if and only if P̃ is divisible by p in B(k)_{p}^{−}. Our analysis at the end of Lecture 21 shows that this happens if and only if P̃ is divisible by p in B(F_{l}^{(1)}), which completes the proof.

0Last modified December 17, 1999

Corollary 1.4. If P is of infinite order in B(F), then

� (J_{0}(11)/ℚ)_{p} = 0

for all but finitely many p.

Proof. If P is of infinite order in B(F), then P lies in pB(F) for only finitely many p. For any p outside of this set and distinct from 2, 5 and 11, Theorem 1.3 and our Kolyagin-Flach theorems apply to yield the vanishing of � (J_{0}(11)/ℚ)_{p}.

In fact, P is of infinite order and is a generator of B(F), as can be shown by various (non-elementary) methods. In particular, this means that Corollary 1.4 applies for all p ≠ 2, 5, 11. A more refined analysis can show that � (J_{0}(11)/ℚ)_{p} = 0 for these p as well, so that

� (J_{0}(11)/ℚ) = 0.

1.2. Explicit Determination of Cocycles 1. In preparation for the proof of Lemma 3 we consider the following situation: let H be a subgroup of a group G with quotient G/H = Γ. Let M be a G-module such that M^{H} = 0. Then the inflation-restriction sequence yields an isomorphism

H^{1}(G, M) ⟶^{≅} H^{1}(H, M)^{Γ}.

We wish to give an explicit inverse map, on the level of cocycles.

So let c : H → M be a crossed homomorphism representing a class in H^{1}(H, M). Suppose further that the cohomology class of c is invariant under the action of G, which can be given explicitly by

c^{g}(h) = g^{−1}c(ghg^{−1}).

Thus for each g ∈ G, there is an element b(g) ∈ M such that

(c^{g} − c)(h) = (h − 1)b(g)

for all h ∈ H. In fact, b(g) is unique since M^{H} = 0. We now define a cocycle

C : G → M

by C(g) = b(g^{−1}). It is not difficult to check that C(g) is a crossed homomorphism and its class in H^{1}(G, M) is independent of the choice of crossed homomorphism c. Furthermore, C restricts to c on H, so that C is the desired extension of c.

1.3. Explicit Determination of Cocycles 2. We abstract the situation of Lemma 1.1 somewhat before giving the proof. Let p be a fixed prime number and let B be an abelian variety defined over a number field K. Let L/K be a finite extension such that B(L) has no non-trivial p-torsion. Set G = G_{K}, H = G_{L} and Γ = G/H. Suppose that we are given Q ∈ B(L) such that the image of Q lies in (B(L)/pB(L))^{Γ}. We wish to determine a crossed homomorphism

κ_{Q} : Γ → B(L)

to which Q maps in the following commutative diagram.

\xymatrix{ &  &  & 0 \ar[d] \\  &  &  & H^{1}(Γ, B(L)) \ar[d] \\  &  & H^{1}(G, B[p]) \ar[r] \ar[d] & H^{1}(G, B) \ar[d] \\ 0 \ar[r] & (B(L)/pB(L))^{Γ} \ar[r] & H^{1}(H, B[p])^{Γ} \ar[r] & H^{1}(H, B)}

Now, since Q is Γ-invariant in B(L)/pB(L), for every γ ∈ Γ there exists R(γ) ∈ B(L) such that

γQ − Q = pR(γ).

Furthermore, R(γ) is unique, since B(L) has no p-torsion. We claim that the crossed homomorphism

γ ↦ −R(γ)

represents the class κ_{Q} in H^{1}(Γ, B(L)) corresponding to Q.

To see this, recall first that the Kummer map

B(L)/pB(L) → H^{1}(H, B[p])

sends Q to the cocycle

h ↦ (h − 1)S ∈ B[p]

where S ∈ B(\overline{ℚ}) is such that pS = Q. The analysis of the previous section tells us how to pull this cocycle back to H^{1}(G, B[p]); in our new notation, we find the cocycle

g ↦ (g − 1)S − R(g).

This maps next to H^{1}(G, B). Here, however, (g − 1)S is trivial, so we are left with the cocycle

g ↦ −R(g).

This restricts to

γ ↦ −R(γ)

in H^{1}(Γ, B(L)), as claimed.

1.4. Proof of Lemma 1.1. Let us now apply the above analysis to the specific situation of Lemma 1.1. We have a diagram

\xymatrix{ &  &  & 0 \ar[d] \\  &  &  & H^{1}(𝔤^{(l)}, B(F^{(l)})) \ar[r] \ar[d] & H^{1}(𝔤^{(l)}, B(k)) \\  &  & H^{1}(F^{(1)}, B[p]) \ar[r] \ar[d] & H^{1}(F^{(1)}, B) \\ 0 & (B(F^{(l)})/pB(F^{(l)}))^{𝔤^{(l)}} \ar[r] & H^{1}(F^{(l)}, B[p])^{𝔤^{(l)}}}

(which one should really view as tensored with I^{(l)}/(I^{(l)})^{2}) and we wish to determine the cocycle of

H^{1}(𝔤^{(l)}, B(k))

corresponding to

P_{η}^{(l)} = ∑_{γ∈𝔤^{(l)}} γx^{(l)} ⊗ γ ∈ (B(F^{(l)})/pB(F^{(l)}))^{𝔤^{(l)}} ⊗ I^{(l)}/(I^{(l)})^{2}.

Note that since

∑_{γ∈𝔤^{(l)}} γx^{(l)} ⊗ 1 ∈ (B(F^{(l)})/pB(F^{(l)}))^{𝔤^{(l)}} ⊗ (I^{(l)})^{2},

we can rewrite P_{η}^{(l)} as

P_{η}^{(l)} = ∑_{γ∈𝔤^{(l)}} γx^{(l)} ⊗ (γ − 1).

Let us begin by rewriting P_{η}^{(l)}. We fix a generator γ_{0} of 𝔤^{(l)}. We write any γ ∈ 𝔤^{(l)} as γ_{0}^{log γ}, where 0 ≤ log γ < l + 1 = |𝔤^{(l)}|. Then an easy calculation shows that

γ − 1 = (γ_{0} − 1) ∑_{j=0}^{log γ−1}(γ_{0}^{j} − 1)

in the group ring ℤ[𝔤^{(l)}]. Furthermore, since γ_{0} −1 ∈ I^{(l)}, one easily computes from this that

γ − 1 ≡ log γ(γ_{0} − 1) (mod (I^{(l)})^{2}).

Thus we can rewrite P_{η}^{(l)} as

P_{η}^{(l)} = ∑_{γ∈𝔤^{(l)}} log γ · γx^{(l)} ⊗ (γ_{0} − 1).

Now, note that in ℤ[𝔤^{(l)}], we have the identity

(γ_{0} − 1) ∑_{γ∈𝔤^{(l)}} log γ · γ = l + 1 − ∑_{γ∈𝔤^{(l)}} γ.

We now compute

(γ_{0} − 1)P_{η}^{(l)} = ((γ_{0} − 1) ∑_{γ∈𝔤^{(l)}} log γ · γ) x^{(l)} ⊗ (γ_{0} − 1)

= (1 + l − ∑_{γ∈𝔤^{(l)}} γ) x^{(l)} ⊗ (γ_{0} − 1).

Our expression for γ − 1 in terms of γ_{0} − 1 shows that this extends to

(γ − 1)P_{η}^{(l)} = (1 + l − ∑_{γ∈𝔤^{(l)}} γ) x^{(l)} ⊗ (γ − 1)

for all γ ∈ 𝔤^{(l)}.

We can use this expression to determine R(γ), using the notation of the previous section. Using the fact that l is good for p and that

∑_{γ∈𝔤^{(l)}} γx^{(l)} = a_{l}x^{(1)},

we find that

R(γ) = (\frac{1 + l}{p} x^{(l)} − \frac{a_{l}}{p} x^{(1)}) ⊗ (γ − 1).

The analysis of the previous section shows that the cocycle

d_{η}^{(l)} ∈ H^{1}(𝔤^{(l)}, B(F^{(l)}))

is given by

d_{η}^{(l)}(γ) = −R(γ).

To complete the proof of Lemma 1.1 we must consider this formula over the residue field k. Here we have the reductions x̃^{(l)} and x̃^{(1)} in B(k), since k is the residue field of both F_{l}^{(l)} and F_{l}^{(1)}. The following lemma, which comes from a careful analysis of complex multiplication over finite fields, completes the proof of Lemma 1.1.

Lemma 1.5.

x̃^{(l)} = Fr x̃^{(1)}

where Fr is the generator of Gal(k/𝔽_{l}).