2.8. Linear independence of rational points 47 where T is a torsion point. Otherwise, if no such relation exists, we say that the points are linearly independent over Z. Example 2.8.2. Let E/Q : y2 = x3 + x2 − 25x + 39 and let P1 = 61 4 , − 469 8 , P2 = − 335 81 , − 6868 729 , P3 = (21,96). The points P1, P2 and P3 are rational points on E and linearly de- pendent over Z because −3P1 − 2P2 + 6P3 = O. Example 2.8.3. Let E/Q : y2 +y = x3 −x2 −26790x+1696662 and put P1 = 59584 625 , 71573 15625 , P2 = 101307506181 210337009 , 30548385002405573 3050517641527 . The points P1 and P2 are rational points on E, and they are linearly dependent over Z because −3P1 + 2P2 = (133, −685), and (133, −685) is a torsion point of order 5. Now that we have defined linear independence over Z, we need a method to prove that a number of points are linearly independent. The existence of the Néron-Tate pairing provides a way to prove in- dependence. Definition 2.8.4. The Néron-Tate pairing attached to an elliptic curve is defined by ·, · : E(Q) × E(Q) → R, P, Q = h(P + Q) − h(P ) − h(Q), where h is the canonical height on E. Let P1,P2,...,Pr be r rational points on E(Q). The elliptic height matrix associated to {Pi}i=1 r is H = H({Pi}i=1) r := ( Pi,Pj )1≤i≤r, 1≤j≤r . The determinant of H is called the elliptic regulator of the set of points {Pi}i=1. r If {Pi}i=1 r is a complete set of generators of the free
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