Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 203

is therefore the tangent line at this point, and the slope is computed by implicit
differentiation of s = t 3 + Ats 2 + Bs 3 :

ds
dt

=3 t 2 + As 2 +2 Ast ds

dt +3 Bs 2 ds

dt .

Solving for ds/dt yields the expression in the statement of the lemma when
t 1 = t 2 = t and s 1 = s 2 = s .

Since s 1 ≡ s 2 ≡ 0(mod p ), we find that the denominator

B ( s 2 + s 1 s 2 + s 1 ) 1 − A ( s 1 + s 2 ) t 1 − ≡ 1(mod p ) .

Since p r

|t i ,wehave

t 2 + t 1 t 2 + t 1 + As 2 ≡ 0(mod p 2 r ) .

Therefore, α ≡ 0(mod p 2 r ). Since p 3 r |s i ,wehave

β = s i − αt i ≡ 0(mod p 3 r ) .

The point P 3 is the third point of intersection of the line s = αt + β with
s = t 3 + As 2 t + Bs 3 . Therefore, we need to solve for t :

αt + β = t 3 + A ( αt + β ) 2 t + B ( αt + β ) 3 .

This can be rearranged to obtain

0= t 3 + 2 Aαβ +3 Bα 2 β

1+ Bα 3 + Aα 2 t 2 + ··· .

The sum of the three roots is the negative of the coe cient of t 2 ,so

2 Aαβ +3 Bα 2 β
1+ Bα 3 + Aα 2

t 1 + t 2 + t 3 = −

0(mod p 5 r ) .

≡

The last congruence holds because p 2 r α and p 3 r 0

(mod p r ), we have t 3 ≡ 0(mod p r ). Therefore, s 3 = αt 3 + β ≡ 0(mod p 3 r ).
By Lemma 8.3, P 3 ∈ E r .Moreover,

| | β .Sin e t 1 ≡ t 2 ≡

λ r ( P 1 )+ λ r ( P 2 )+ λ r ( P 3 ) ≡ p −r ( t 1 + t 2 + t 3 ) ≡ 0(mod p 4 r ) .

Therefore, λ r is a homomorphism. This completes the proof of Theorem 8.1.

COROLLARY 8.6
Letthe notationsbeasin T heorem 8.1. If n> 1 and n is not a pow er of p ,
then E 1 con tains no points of exact order n .(Seealso T heorem 8.9.)