Isogenies

Motivation

Prerequisites: in this section we assume the reader is somewhat familiar with elliptic curves and begin by considering morphisms (maps) between elliptic curves.

Humans are fascinated with symmetries. The guiding star of theoretical physics is the study of dualities. How much one thing can be said to be another leads to strange and beautiful links between areas of mathematics that appear to be totally distinct.

A cruical piece of building this understanding is how one can map between objects which share structure. When we learn about topology, we are given the fun: "A doughnut is the same as a mug"; a statement which says within topology, we identify objects related by continuous functions.

Elliptic curves are beautiful mathematical objects. The fact that a geometric comes hand-in-hand with a algebraic group law is, to me, incredible. The study of isogenies is the study of maps (morphisms) between elliptic curves which preserves the origin. This condition is enough to preserve the group scheme of the elliptic curve.

In short, isogenies allow us to map between curves preserving their geometric properties (as projective varieties) and algebraic properties (the group associated with point addition).

Isogenies of Elliptic Curves

Definition: We say an isogeny $\phi : E_1 \to E_2$ between elliptic curves defined over a field $k$is a surjective morphism of curves which includes a group homomorphism $E_1(\bar{k}) \to E_1(\bar{k})$

References

• https://arxiv.org/pdf/1711.04062.pdf

• https://math.mit.edu/classes/18.783/2019/LectureNotes5.pdf

• https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/ell_curve_isogeny.html