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 .

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