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
.
