CHAPTER 21 Lecture 21

1. Local Behavior of Cohomology Classes

1.1. X_{0}(11) over Finite Fields. We continue with the analysis of the classes

c_{η}^{(l)} ∈ H^{1}(G_{F^{(1)}}, B)[p]
d_{η}^{(l)} ∈ H^{1}(𝔤^{(l)}, B(F^{(l)}))[p]
κ_{η}^{(l)} ∈ H_{s}^{1}(G_{F} , B[p])

for the modular curve B = X_{0}(11) = J_{0}(11). We begin by considering the action of Fr_{l} on B[p]. (Here by Fr_{l} we mean the Frobenius of l in the extension of F generated by the coordinates of B[p]. This makes sense since B has good reduction at l and l ≠ p.) Fr_{l} has characteristic polynomial x^{2}−a_{l}x+l on the two-dimensional 𝔽_{p}-vector space B[p]; since l is good for p, we have

x^{2} − a_{l}x + l ≡ x^{2} − 1 (mod p).

In particular, we see immediately that Fr_{l} is a non-scalar involution on B[p]. (It is non-scalar since it has distinct eigenvalues.)

Next consider the points of B over the residue fields k_{0} and k of ℚ_{l} and F_{l} respectively; of course, k_{0} = 𝔽_{l} and k = 𝔽_{l^{2}} since l is inert in F/ℚ. It is a standard fact that

|B(k_{0})| = 1 + l − a_{l}

and

|B(k)| = (1 + l)^{2} − a_{l}^{2}.

In particular, since l is good for p we see that B(k_{0}) has a point of order p, and all of B[p] is contained in B(k). Note that B(k_{0}) does not contain all of B[p], since Fr_{l} projects to the non-trivial element of Gal(k/k_{0}) and acts as a non-scalar, and thus in particular is non-trivial.

If we consider the eigenspaces of B(k) under the action of Fr_{l} (or equivalently under the action of the non-trivial element of Gal(F/ℚ) = Gal(k/k_{0})), we obtain a decomposition

B(k)_{p} = B(k)_{p}^{+} ⊕ B(k)_{p}^{−}
= B(k_{0})_{p} ⊕ B(k)_{p}^{−}

where B(k)_{p} is the p-primary component of B(k); we do not actually obtain such a decomposition of all of B(k) because of problems with 2. In any event, we do find that

B(k)[p] = B(k_{0})[p] ⊕ B(k)[p]^{−}

0Last modified December 17, 1999

and our above analysis shows that both direct summands are cyclic of order p. This implies that B(k_{0})_{p} and B(k)_{p}^{−} are themselves cyclic, and combining all of this yields the following lemma.

Lemma 1.1. B(k)_{p}^{+} = B(k_{0})_{p} is cyclic of order
p^{ord_{p}(1+l−a_{l})}

and B(k)_{p}^{−} is cyclic of order

p^{ord_{p}(1+l+a_{l})}.

1.2. The Kolyvagin-Flach Theorem. Recall that our eventual goal is to prove Lecture 20, Theorem 1.2 and to derive Lecture 20, Corollary 1.3 from it. For the moment let us discuss this second issue. The difficulty is that Theorem 1.2 only yields Kolyvagin-Flach classes for primes l such that the basic Heegner point P is not a p^{th}-power in B(F_{l}). Using the notation from earlier in the course, our Kolyvagin-Flach theorems to this point have required that we have such classes for all l ∈ L_{τ}, where L_{τ} is the set of places of F for which there exists a place w of F_{J} over v such that Fr_{F_{J} /F} (w) is the fixed non-scalar involution τ ∈ Gal(F_{J}/ℚ). In our case we can identify L_{τ} with the set of l which are good for p. (This essentially comes from the fact that for any l ≠ p, the action of Fr_{l} on B[p] has characteristic polynomial x^{2} − a_{l}x + l; which ones yield non-scalar involutions is then clear.) We, however, only have Kolyvagin-Flach classes for the subset L′_{τ} of L_{τ} for which P is not a p^{th}-power locally.

Let us define this set L′_{τ} purely field theoretically. Let F′ be a (probably non-Galois) extension of F generated by some p^{th} root of P. Then L′_{τ} is precisely the set of place of F which are good for p and which have no degree 1 places of F′ over them. (The existence of such a place means exactly that P is a p^{th}-power over the completion of F.)

We will sketch the proof of the Kolyvagin-Flach theorem in this new situation. Abstractly, we add a new field F′/F and we define F′_{J} = F′F_{J} and L′ = F′L. (Again we are using our earlier notation.) We make the hypothesis that F_{J} ≠ F′_{J}.

[figure: L′; F′_{J}; L; Γ; F′; F_{J}; Δ_{J}; F; F_{0}]

We claim that in this situation it is enough to have a Kolyvagin-Flach system for L′_{τ} (as defined above). Specifically, we want to show that if φ(Fr_{w}) = 0 for some w dividing each v ∈ L′_{τ}, then φ = 0. To see this, we first lift τ̃ to an element τ̃′ ∈ Gal(L′/F_{0}). We now consider φ((g′τ̃′)^{2}) only for those g′ ∈ Gal(L′/F_{J}) − Gal(L′/F′_{J}). The exact same proof works for these g′ to show that

φ(Gal(L′/F_{J}) − Gal(L′/F′_{J})) = 0.

But this implies that φ vanishes on all of Gal(L′/F_{J}), since we already know that it vanishes on the group minus a proper subgroup (since we assumed F′_{J} ≠ F_{J}) and φ is a homomorphism. This completes the proof of the Kolyvagin-Flach theorem in this restricted situation.

1.3. Finiteness of the Kolyvagin-Flach Classes. The first step in proving Theorem 1.2 is the following lemma. For the remainder of the lecture we assume that F^{(1)} = F, for purely notational reasons; the general proofs are the same except that we can avoid a few corestrictions.

Lemma 1.2. If v ≠ l, then res_{v}(κ_{η}^{(l)}) ∈ H_{s}^{1}(G_{v}, B[p]) is trivial.

Proof. Recall that we had a class d_{η}^{(l)} ∈ H^{1}(𝔤^{(l)}, B(F^{(l)})) which mapped to κ_{η}^{(l)} = c_{η}^{(l)} ∈ H_{s}^{1}(G_{F} , B[p]). The commutative diagram

\xymatrix{H^{1}(𝔤^{(l)}, B(F^{(l)})) \ar[r] \ar[d] & H^{1}(D_{w}, B(F^{(l)})) \ar[d] \\ H^{1}(G_{F} , B) \ar[r] & H^{1}(G_{v}, B)}

where w is a place of F^{(l)} lying over v and D_{w} is the decomposition group of w in 𝔤^{(l)}, shows that it will suffice to show that res_{w}(d_{η}^{(l)}) = 0. We will do this by showing that all of H^{1}(D_{w}, B(F^{(l)})) vanishes.

More generally, let Φ/Φ_{0} be an unramified, Galois extension of local fields. (In the case above we take Φ = F_{w}^{(l)} and Φ_{0} = F_{v}.) Let B be an abelian variety over Φ_{0} with good reduction over O_{Φ_{0}}. We claim that

H^{1}(Gal(Φ/Φ_{0}), B(Φ)) = 0.

This is essentially a consequence of Lang’s theorem; we begin by noting that by properness

B(Φ) = B(O_{Φ}) = \underleftarrow{lim} B(O_{Φ}/𝔪^{n})

where 𝔪 is the maximal ideal of O_{Φ}. Thus we must show that

H^{1}(Gal(Φ/Φ_{0}), B(O_{Φ}/𝔪^{n})) = 0

for all n. For n = 1 this is precisely Lang’s theorem; for n > 1, one uses the Greenberg functor to express B(O_{Φ}/𝔪^{n}) as the k-valued points of an n-dimensional smooth, connected algebraic group. Lang’s theorem again completes the proof.

The above argument works for v ≠ 11, l. For v = 11, B has bad reduction, so one must use the Neron model of B and a more refined analysis in order to apply Lang’s theorem. The key is that the Neron model is just the product of the multiplicative group and a cyclic group of order 5; since 5 is prime to 11, this group does not enter in to the calculations, and Lang’s theorem applies directly to 𝔾_{m}; we omit the details.

162 21. LECTURE 21

1.4. Singular Depth of the Kolyvagin-Flach Classes. We next must compute the singular depth of κ_{η}^{(l)} at v = l. Recall the defining diagram

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

(Recall that we are assuming F^{(1)} = F.) We have

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

and

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

Let d̃_{η}^{(l)} be the image of d_{η}^{(l)} under the map

H^{1}(𝔤^{(l)}, B(F^{(l)})) → H^{1}(𝔤^{(l)}, B(F_{l}^{(l)})).

(Note that F^{(l)}/F is totally ramified at F, so 𝔤^{(l)} is the decomposition group of l.)

The residue field of F_{0,l} is k_{0} = 𝔽_{l}, while those of F_{l} and F_{l}^{(l)} are k = 𝔽_{l^{2}}. Consider the map

B(F_{l}^{(l)}) → B(k).

The kernel is pro-l, so, since d̃_{η}^{(l)} maps to an element of order p in the cohomology, we find that d̃_{η}^{(l)} = 0 if and only if ˜ d̃_{η}^{(l)} = 0, where ˜ d̃_{η}^{(l)} is the image of d̃_{η}^{(l)} under

H^{1}(𝔤^{(l)}, B(F_{l}^{(l)})) → H^{1}(𝔤^{(l)}, B(k)).

Since 𝔤^{(l)} acts trivially on k, we have that

H^{1}(𝔤^{(l)}, B(k)) = Hom(𝔤^{(l)}, B(k)).

Now, however, our earlier notational simplifications come back to haunt us. Really we must consider

Hom(𝔤^{(l)}, B(k)) ⊗ I^{(l)}/(I^{(l)})^{2} = Hom(𝔤^{(l)}, B(k) ⊗ I^{(l)}/(I^{(l)})^{2})

and ask what

˜d̃_{η}^{(l)} : 𝔤^{(l)} → B(k) ⊗ I^{(l)}/(I^{(l)})^{2}

is.

Lemma 1.3. For all g ∈ 𝔤^{(l)},

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