Isogenies - Elliptic Curves: Number Theory and Cryptography - page 401

A similar situation holds over Q , as was proved by Faltings in 1983. Namely,
if E 1 ,E 2 are elliptic curves over Q , then the L -series of E 1 (see Section 14.2)
equals the L -series of E 2 if and only if E 1 and E 2 are isogenous over Q .This
theorem arose in his proof of Mordell's conjecture that an algebraic curve of
genus at least 2 has only finitely many rational points.

Exercises

12.1 Let L be the lattice Z + Z i .

(a) Show that [1 + i ]: C /L → C /L is an isogeny. List the elements of
the kernel and conclude that the isogeny has degree 2.

(b) Let 0 C /L is an

isogeny of degree a 2 + b 2 .( Hint: The proof of Lemma 12.1 shows
that the degree is the determinant of a + bi acting on the basis
{

= a + bi ∈ Z + Z i . Show that [ a + bi ]: C /L →

1 ,i } of L .)

12.2 Let E = C /L be an elliptic curve defined over C .Let n be a positive
integer. Let [ α ]: C /L → C /L 1 be an isogeny and assume that E [ n ] ⊆
Ker α . By multiplying by α − 1 , we may assume that the isogeny is given
by the map z → z and that L ⊆ L 1 ,so L 1 /L is the kernel of the isogeny.
For convenience, we continue to denote the isogeny by [ α ].

(a) Show that E [ n ]= n L/L .
(b) Let α 1 : C /L C / n L be the map given by z

→ → z . Show that

there is an isomorphism β : C / n L C /L such that β ◦ α 1 =[ n ]

(= multiplication by n on E ).

(c) Observe that α factors as α 2 ◦ α 1 ,where α 1 is as in (b), and where

α 2 : C / n L β − 1 . → C /L 1 is given by z → z .L t α 3 = α 2 ◦

Conclude that α factors as α 3 ◦ [ n ].
(d) Let γ : E → E 1 be an isogeny with Ker γ Z n 1 ⊕ Z n 2 with n 1 |n 2 .
Show that γ equals multiplication by n 1 on E composed with a
cyclic isogeny whose kernel has order n 2 /n 1 .

12.3 Let [ α ]: C /L 1 → C /L 2 be an isogeny, as in Section 12.1.

(a) Show that deg([ α ]) = deg([ α ]) ( Hint: multiplication by N/α cor-
responds to the matrix N ( a ij ) − 1 , in the notation of the proof of
Lemma 12.1).
(b) Show that [ α ]=[ α ].