48 2. Elliptic curves

part of E(Q), then the determinant of H({Pi}i=1) r is called the elliptic

regulator of E/Q.

Theorem 2.8.5. Let E/Q be an elliptic curve. Then the Néron-Tate

pairing ·, · associated to E is a non-degenerate symmetric bilinear

form on E(Q)/E(Q)torsion, i.e.,

(1) For all P, Q ∈ E(Q), P, Q = Q, P .

(2) For all P, Q, R ∈ E(Q) and all m, n ∈ Z,

P, mQ + nR = m P, Q + n P, R .

(3) Suppose P ∈ E(Q) and P, Q = 0 for all Q ∈ E(Q). Then

P ∈ E(Q)torsion. In particular, P is a torsion point if and

only if P, P = 0.

The properties of the Néron-Tate pairing follow from those of the

canonical height in Proposition 2.7.3 (see Exercise 2.12.12). Theorem

2.8.5 has the following important corollary:

Corollary 2.8.6. Let E/Q be an elliptic curve and let P1,P2,...,Pr ∈

E(Q) be rational points. Let H be the elliptic height matrix associated

to {Pi}i=1.

r

Then:

(1) Suppose det(H) = 0 and u = (n1,...,nr) ∈ Ker(H), with

ni ∈ Z. Then the points {Pi}i=1 r are linearly dependent and

∑r

k=1

nkPk = T , where T is a torsion point on E(Q).

(2) If det(H) = 0, then the points {Pi}i=1 r are linearly indepen-

dent and the rank of E(Q) is ≥ r.

Here is an example of how the Néron-Tate pairing is used in

practice:

Example 2.8.7. Let E/Q be the elliptic curve y2 = x3 + 2308x2 +

665858x. Put

P = (−1681,25543), Q = (−338,26), and

R =

332929

36

, −

215405063

216

.