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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.