Isogenies and Finite-Field Group Structure

Elliptic curves can be defined over finite fields. Their rational points form a finite group, including the point at infinity. An isogeny is a non-constant morphism of elliptic curves that preserves the identity and group structure. One can study it algebraically without relying on a real-number picture of a smooth curve.

Finite-field algebraic geometry still has notions of topology; “there is no topology” is not the reason these maps exist. The useful distinction for an implementation is between algebraic group operations and visual intuition about real coordinates.

For nonzero integer m, the multiplication map [m] sends P to mP and has isogeny degree m^2. Its group property is:

\[ [m](P + Q) = [m]P + [m]Q. \]

Testing this equality on a small set of points illustrates the group law. It is not a proof that an arbitrary implementation is correct or that the construction is cryptographically secure.

A structure-preserving map may be useful within a specified representation. It does not establish that the points encode true statements, that their addition performs semantic reasoning, or that an isogeny retrieves missing source records. Those claims need an explicit model and separate evaluation.

The earlier toy code is not retained as an implementation reference. Use the documented relation rules for current platform behaviour, encoding and proof boundaries for cryptographic scope, and research notes for open questions.