Isogenies - Elliptic Curves: Number Theory and Cryptography - page 392

Theisogeny isgiven by

v Q

( x − x Q ) 2

X = x +
Q

u Q

x − x Q +

∈ S

u Q 2 y + a 1 x + a 3 .

+ a 1 u Q − g Q q Q
( x − x Q ) 2

+ v Q a 1 ( x − x Q )+ y − y Q

Y = y −

( x − x Q ) 3 ( x − x Q ) 2

Q ∈ S

PROOF As in Section 8.1, let t = x/y and s =1 /y .Then t has a simple
zero and s has a third order zero at ∞ (see Example 11.3). Dividing the
relation y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 by y 3 and rearranging yields

s = t 3 a 1 st + a 2 st 2 a 3 s 2 + a 4 s 2 t + a 6 s 3 .

− − (12.1)

If we substitute this value for s into the right hand side of (12.1), we obtain

s = t 3 − a 1 ( t 3 − a 1 st + a 2 st 2 − a 3 s 2 + a 4 s 2 t + a 6 s 3 ) t

+ a 2 ( t 3 a 1 st + a 2 st 2 a 3 s 2 + a 4 s 2 t + a 6 s 3 ) t 2 +

− − ··· .

Continuing this process, we eventually obtain

= s = t 3 1 ···

1
y

a 1 t +( a 1 + a 2 ) t 2 ( a 1 +2 a 1 a 2 + a 3 ) t 3 +

− −

and

y = t − 3 + α 1 t − 2 + α 2 t − 1 + α 3 + α 4 t + α 5 t 2 + α 6 t 3 + O ( t 4 ) ,

where

α 1 = a 1 , α 2 = −a 2 , α 3 = a 3 , α 4 = − ( a 1 a 3 + a 4 ) ,
α 5 = a 2 a 3 + a 1 a 3 + a 1 a 4 ,
α 6 = − ( a 1 a 4 + a 1 a 3 + a 2 a 4 +2 a 1 a 2 a 3 + a 3 + a 6 ) ,

and where O ( t 4 ) denotes a function that vanishes to order at least 4 at ∞ .
Since x = ty ,wealsoobtain

x = t − 2 + α 1 t − 1 + α 2 + α 3 t + α 4 t 2 + α 5 t 3 + α 6 t 4 + O ( t 5 ) .

Substituting these expressions for x, y into the formulas given for X, Y yields
expressions for X, Y in terms of t . A calculation shows that

Y 2 + A 1 XY + A 3 Y = X 3 + A 2 X 2 + A 4 X + A 6 + O ( t ) ,

where the A i are as given in the statement of the theorem. Since X and Y
are rational functions of x, y , they are functions on E . The only poles of X
and Y are at the points in C , as can be seen from the explicit formulas for