ON REMARKS OF LIFTING PROBLEMS FOR ELLIPTIC CURVES 1 ...

ON REMARKS OF LIFTING PROBLEMS FOR ELLIPTIC CURVES 7
Theorem 2.7. Suppose E is an elliptic curve over F 2 [T ] defined by
y 2 + xy = x 3 + a 2 x 2 + a 6 , a 2 and a 6 ∈ F 2 [T ]
and the points R, R ′ of E(F 2 (T )) satisfy that
|h(R) − h(R ′ )| > 5 6 deg a 6.
Then
ĥ(R) > ĥ(R′ ) if and only if h(R) > h(R ′ ).
Proof. First, we need the following lemma.
Lemma 2.8. Suppose E is an elliptic curve over F 2 [T ] defined by
Then
y 2 + xy = x 3 + a 2 x 2 + a 6 , a 2 , a 6 ∈ F 2 [T ].
− 1 4 deg a 6 ≤ ĥ(P ) − 1 2 h(P ) ≤ 1 6 deg a 6.
Proof. First, we write x(P ) = f/g where f and g are relative prime polynomials of
F 2 [T ]. Then
x(2P ) = f 4 + a 6 g 4
f 2 g 2 .
Let k = gcd(f 4 + a 6 g 4 , f 2 g 2 ) = gcd(a 6 , f 2 ), then
(3)
h(2P ) = max{deg(f 4 + a 6 g 4 ), deg(f 2 g 2 )} − deg k
≤ max{4 deg f, 4 deg g + deg a 6 , 2 deg f + 2 deg g}
≤ 4h(P ) + deg a 6
since deg k ≤ deg a 6 .
If 4 deg f ≠ 4 deg g + deg a 6 , then
Hence,
(4)
deg(f 4 + a 6 g 4 ) = max{4 deg f, 4 deg g + deg a 6 }.
h(2P ) ≥ 4h(P ) − deg a 6 .
Now, suppose that 4 deg f = 4 deg g + deg a 6 . Then
h(2P ) ≥ 2 deg f + 2 deg g − deg k
(5)
≥ 4 deg f − 3 2 deg a 6 ≥ 4h(P ) − 3 2 deg a 6.
With (3),(4) and (5), we have
− 3 2 deg a 6 ≤ h(2P ) − 4h(P ) ≤ deg a 6 .
By applying 2 n P instead of P in the above,
−(1 + · · · + 4 n−1 ) 3 2 deg a 6 ≤ h(2 n P ) − 4 n h(P ) ≤ (1 + · · · + 4 n−1 ) deg a 6 .
Since
lim
n∑
n→∞
i=1
1
4 n = 1 3 and ĥ(P ) = 1 2 lim
n→∞
1
4 n h(2n P ) for any point P,