CHAPTER 20 Lecture 20

1. The Euler System for X_{0}(11)

1.1. The Set-Up. For simplicity we now restrict our investigations to the case N = 11; it is in many ways typical of the general case, and it will help to cut down on the notation. Recall that X_{0}(11) is a curve of genus 1 with good reduction away from 11, and we make it into an elliptic curve by using the cusp at infinity as the base point. This yields a canonical isomorphism

X_{0}(11) ≅ J_{0}(11),

and we will also use B to denote this elliptic curve.

X_{0}(11) can actually be realized by the Weierstrass equation

y^{2} + y = x^{3} − x^{2} − 10x − 20,

and the group B(ℚ) of rational points is cyclic of order 5. The standard everywhere regular differential on X_{0}(11) has a Fourier expansion

ω(q) = η(q)^{2}η(11q)^{2}

= q ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{11n})^{2}

= ∑_{n=1}^{∞} a_{n}q^{n}

= q − 2q^{2} − q^{3} + 2q^{4} + q^{5} + 2q^{6} − 2q^{7} − 2q^{9} − 2q^{10} + q^{11} + · · · .

The image 𝕋 of the Hecke algebra in End_{ℚ}(J_{0}(11)) = ℤ is just ℤ, and the Hecke operator T_{l} (for l ≠ 11) on J_{0}(11) can be identified with its eigenvalue a_{l}. For l = 11, U_{11} is identified with a_{11} = 1. The Atkin-Lehner involution w_{11} is the negative of this, and thus is the involution −1.

For every p, we obtain a Galois representation

\overline{ρ}_{p} : G_{ℚ} → Aut(B[p]) ≅ GL_{2}(𝔽_{p}).

\overline{ρ}_{p} is surjective for all p ≠ 2, 5, 11. For p = 5 this lack of surjectivity is forced by the fact that B has a rational point of order 5; the image of \overline{ρ}_{5} in GL_{2}(𝔽_{5}) is simply the subgroup of diagonal matrices with 1 in the upper left coordinate.

We let F be any quadratic imaginary field ℚ(\sqrt{−d}) in which 11 is inert; by quadratic reciprocity, this condition says precisely that

d ≡ −1, 2, −3, −4, −5 (mod 11).

(Note that this automatically eliminates the fields ℚ(\sqrt{−1}) and ℚ(\sqrt{−3}).) We also fix a factorization 11 = N · N̄ in O_{F}.

0Last modified December 17, 1999

Now fix a prime p ≠ 2, 3, 5, 11. Our maximal ideal 𝔪 is just (p), and it is easily seen to be good. Our element η is a power p^{s} of p, and a prime l, inert in F/ℚ, is good for η if and only if

l + 1 ≡ a_{l} ≡ 0 (mod p^{s}).

Fix such an l.

For later use, we define the basic Heegner point P ∈ B(F) by

P = Tr_{F^{1}/F} x^{1}.

1.2. The Construction. We now briefly review our construction of Lecture 19 in this simplified notation. We begin with a point

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

We now “reduce P^{(l)} modulo p^{s}”; that is, we take P_{η}^{(l)} to be image of P^{(l)} in

B(F^{(l)}) ⊗_{ℤ} ℤ/p^{s}ℤ[𝔤^{(l)}].

As discussed in Lecture 19, Section 1.3, P_{η}^{(l)} actually lies in the submodule

B(F^{(l)}) ⊗_{ℤ} I^{(l)}.

We define P_{η}^{(l)} to be the image of P_{η}^{(l)} in

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

We also fix an isomorphism

𝔤^{(l)} ⊗_{ℤ} ℤ/p^{s}ℤ ≅ ℤ/p^{s}ℤ;

this allows us to consider P_{η}^{(l)} as an element of

B(F^{(l)}) ⊗_{ℤ} ℤ/p^{s}ℤ = B(F^{(l)})/p^{s}B(F^{(l)}).

Letting 𝔤^{(l)} act on B(F^{(l)})/p^{s}B(F^{(l)}) in the natural way, we have as before

P_{η}^{(l)} ∈ (B(F^{(l)})/p^{s}B(F^{(l)}))^{𝔤^{(l)}}.

We now relate P_{η}^{(l)} to Galois cohomology and in particular to our abelian variety finite/singular structure on B[p^{s}]. We begin with the exact sequence

0 → B[p^{s}] → B ⟶^{p^{s}} B → 0.

Taking the long exact sequence for G_{F^{(1)}} and G_{F^{(l)}} cohomology yields a commutative diagram with exact rows

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

Note that the two horizontal sequences are simply our finite/singular sequences for the Galois module B[p^{s}] over F^{(1)} and F^{(l)} respectively. In particular, we have an identification

H^{1}(G_{F^{(1)}}, B)[p^{s}] = H_{s}^{1}(G_{F^{(1)}}, B[p^{s}]).

1. THE EULER SYSTEM FOR X_{0}(11) 157

In fact, the vertical maps all have image in the 𝔤^{(l)}-invariant subgroups, yielding a diagram

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

Consider the map

φ : H^{1}(G_{F^{(1)}}, B[p^{s}]) → H^{1}(G_{F^{(l)}}, B[p^{s}])^{𝔤^{(l)}}.

By the inflation-restriction sequence, this has kernel H^{1}(𝔤^{(l)}, B(F^{(l)})[p^{s}]) and cokernel H^{2}(𝔤^{(l)}, B(F^{(l)})[p^{s}]). However, we must have B(F^{(l)})[p^{s}] = 0, since if B had rational p^{s}-torsion in a dihedral extension of ℚ, the image of the representation \overline{ρ}_{p} would factor through this dihedral group and an abelian subgroup of GL_{2}(𝔽_{p}), and thus be solvable. Since GL_{2}(𝔽_{p}) is not solvable for p ≠ 2, 3, our assumptions imply that this is not possible. In particular, this implies that φ is an isomorphism. The same is not true of ψ, but at the very least we can identify its kernel as H^{1}(𝔤^{(l)}, B(F^{(l)}))[p^{s}].

Let us now push our element P_{η}^{(l)} of (B(F^{(l)})/p^{s}B(F^{(l)}))^{𝔤^{(l)}} around the diagram. It yields an element of of H^{1}(G_{F^{(l)}}, B[p^{s}])^{𝔤^{(l)}}, and thus via φ^{−1} an element of H^{1}(G_{F^{(1)}}, B[p^{s}]). Pushing this forward we obtain an element

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

which maps to 0 under ψ, and thus also an element

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

We are now in a position to define our long awaited Kolyvagin-Flach system. We define

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

to be the image of c_{η}^{(l)} under the corestriction map

cor : H_{s}^{1}(G_{F^{(1)}}, B[p^{s}]) → H_{s}^{1}(G_{F} , B[p^{s}]).

We will now summarize the main remaining results; proofs will be given (hopefully) in later lectures.

Theorem 1.1. Let ε be the eigenvalue −a_{11} of W_{11}. Then κ_{η}^{(l)} is in the −ε eigenspace of H_{s}^{1}(G_{F} , B[p^{s}]).

Recall that P is the basic Heegner point and lies in B(F). We let F_{l} be the completion of F at l; since l is inert this makes sense. We also restrict now to the case η = p.

Theorem 1.2. If P ∈ pB(F_{l}), then Supp κ_{η}^{(l)} is empty. If P ∉ pB(F_{l}), then Supp κ_{η}^{(l)} = {l} and the depth of κ_{η}^{(l)} is 1.

Corollary 1.3. Let F = ℚ(\sqrt{−d}) where d ≡ −1, 2, −3, −4, −5 (mod 11). Let p be a prime distinct from 2, 3, 5, 11. Then if P is not divisible by p in J_{0}(11)(F), we have

� (J_{0}(11)/ℚ) = 0.

158 20. LECTURE 20

Corollary 1.4. If there exists a field F as above for which P is of infinite order, then the p-primary component of � (J_{0}(11)/ℚ) vanishes for almost all p.