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