Important Notion: __Isogeny Class__

•An *isogeny* is a nontrivial algebraic map between two elliptic
curves. It is a group homomorphism.

__Examples:
__

1.Map any E to itself by z ! 2z
(called an *endomorphism*)

2.map C/Z[i] ! C/Z[2i] by z ! 2z

3.map C/Z[i] ! C/Z[i] by z ! iz (called complex
multiplication “CM”)

•__Tate’s Isogeny Theorem:__ two elliptic curves over Fq with the same number of points are *isogenous* over Fq (isogenies
exist between them in both directions).

•Related to commensurability.

•Isogenies give *an explicit reduction between DLOG* on different curves if they each have the same number of prime points. (Identical cyclic groups.)

•So because of Tate’s theorem, the selection problem can
be reinterpreted: is isogeny class a fine enough invariant for curve selection? Or is more needed?