Isogenies - Elliptic Curves: Number Theory and Cryptography - page 382

We have the fundamental relation:

[ α ] ◦ [ α ]=deg([ α ]) ,

where the integer deg([ α ]) denotes integer multiplication on C /L 1 . It is easy
to show (see Exercise 12.3) that

[ α ]=[ α ]

and that

◦ [ α ]=deg([ α ]) = deg([ α ]) , [ α ]

which is integer multiplication on C /L 2 .
A situation that arises frequently is when α = 1. This means that we have
L 1 ⊆ L 2 and the isogeny is simply the map

z mod L 1 −→ z mod L 2 .

The kernel is L 2 /L 1 . An arbitrary isogeny [ α ] can be reduced to this situation
by composing with the isomorphism C /L 2 →

C /α − 1 L 2 given by multiplica-

tion by α − 1 .

PROPOSITION 12.4
Let C ⊂ E 1 = C /L be a finite subgroup. T hen there exist an ellipticcurve
E 2 = C /L 2 and an isogeny fro m E 1 to E 2 w hose kernelis C .

PROOF C can be written as L 2 /L 1 for some subgroup L 2 of C containing
L 1 .If N is the order of C ,then NL 2 ⊆ L 1 ,so L 1 ⊆ L 2 ⊆ (1 /N ) L 1 .Bythe
discussion following Theorem B.5 in Appendix B, L 2 is a lattice. Therefore,
C /L 1 → C /L 2 is the desired isogeny.

Given two elliptic curves and an integer N , there is a way to decide if
they are N -isogenous. Recall the modular polynomial Φ N ( X, Y ) (see Theo-
rem 10.15 and page 324), which satisfies
Φ N ( j ( τ 1 ) ,j ( τ 2 )) =
S

( j ( τ 1 ) − j ( S ( τ 2 ))) ,

∈ S N

where S N is the set of matrices ab

with a, b, d positive integers satisfying

0 d

ad = N and 0 ≤ b<d .

THEOREM 12.5
Let N be a positive integer and let Φ N ( X, Y ) be the N thmodu ar polynom ial,
as in T heorem 10.15. Let E i = C /L i have j -invariant j i for i =1 , 2 .Then
E 1 is N -isogenous to E 2 ifand onlyif Φ N ( j 1 ,j 2 )=0 .