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
