Lecture 24, Wednesday 05.05.04

0 −−−−→ {O, (0, 0)} −−−−→ E(Q) φ −−−−→ E(Q) α −−−−→ Q×/Q× 2 0 −−−−→ {O, (0, 0)} −−−−→ E(Q) ψ −−−−→ E(Q) β −−−−→ Q×/Q× . Finally, we have ψ ◦ φ = [2]E (multiplication by 2 on E) and φ ◦ ψ = [2]E. Thus although the situation is more involved than for the 2-descent on curves with three rational points of order 2, we have managed to break up the multiplicationby-2 map into two homomorphisms φ and ψ between E and the associated curve E. Multiplication by 2 has a kernel of order 4 (three points of order 2 and the point at infinity), whereas the isogenies φ and ψ have kernels of order 2 (which is why they are called 2-isogenies).