Isogenies - Elliptic Curves: Number Theory and Cryptography - page 403

(b) Show that if 2 Q = ∞ ,then

v Q u Q

x P + Q − x Q + x P−Q − x −Q =

( x P − x Q ) 2 + ( x P − x Q ) 3 ,

y P + Q − y Q + y P−Q − y −Q

u Q 2 y P + a 1 x P + a 3
( x P − x Q ) 3

v Q a 1 ( x P − x Q )+ y P − y Q
( x P − x Q ) 2

= − −

g Q g Q a 1 u Q −

− .

( x P − x Q ) 2

(c) Show that, in the notation of Theorem 12.16,
X ( P )= x ( P )+
∞

[ x ( P + Q ) − x ( Q )]

= Q ∈ C

Y ( P )= y ( P )+
∞

[ y ( P + Q ) − y ( Q )] .

= Q ∈ C

12.7 Let p ( T ) ,q ( T ) be polynomials with coe cients in a field K with no
common factor. Let X be another variable. Show that the polynomial
F ( T )= p ( T ) Xq ( T ), regarded as a polynomial with coe cients in

K ( X ), is irreducible. ( Hint: By Gauss's Lemma (see, for example,
[71]), if F ( T ) factors, it factors with coe cients that are polynomials
in X (that is, we do not need to consider polynomials with rational
functions as coecients).)

−

12.8 Recall that in Velu's formulas,
X = x +
Q

v Q

( x − x Q ) 2 .

u Q

x − x Q +

∈ S

(a) Show that g Q =0ifandonlyif2 Q = ∞ . Show that if 2 Q = ∞ ,
then g Q =0( Hint: the curve is nonsingular). Conclude that if

.
(b) Write the rational function defining X as p ( x ) /q ( x ), where p, q are
polynomials with no common factor. Show that q ( x )containsthe
product of ( x − x Q ) 2 for all points Q ∈ S with 2 Q = ∞ and that
it contains ( x − x Q )foreachpoint Q ∈ S with 2 Q = ∞ . Conclude
that deg q =# C − 1.
(c) Show that X

2 Q = ∞ then v Q =0,andthat u Q =0ifandonlyif2 Q = ∞

x has the form r ( x ) /q ( x )withdeg r< deg q .
(d) Use the fact that

−

p ( x )

q ( x ) = x + r ( x ) q ( x )

to prove that deg p =# C . This shows that the isogeny constructed
in Theorem 12.16 is separable.