What is an isogeny? (2019)

https://www.johndcook.com/blog/2019/04/21/what-is-an-isogeny/

by hasheddan • 4 years ago

22 10 4 years ago

sn41• 4 years ago

Hmm.. What's the difference between this and a homomorphism? Are there homomorphisms which are not isogenies? I'm still not very clear on this.

klyrs• 4 years ago

Great question. There's a subtle point the article glosses over "... with coefficients in K." If you have two elliptic curves that just happen to be cyclic groups over finite fields, you can construct homomorphisms between them with finite kernels. But, you won't have any hope of finding morphisms unless those finite fields are the same. (same characteristic? It's been a while)

In less fancy language: isogenies are homomorphisms you can describe with simple formulas involving (x, y) coordinates.

m00n2• 4 years ago

Good question, no there is no difference for elliptic curves, which you can think of as 1-dimensional geometric objects (curves) which posses group structure. A good map in this category should respect the geometry (be a so called rational map, ie defined by polynomials in a suitable coordinate system) and the group structure. Interestingly all these maps are either constant (map everything to 0) or surjective.

For higher dimensional geometric groups (abelian surfaces etc) one usually wants to make a distinction and calls the surjective homomorphisms with finite preimage isogenies.

gotoeleven• 4 years ago

Very glad to read the link and find out this isn't a highly potent form of intersectional bigotry that was created in a lab by critical studies theorists.

snotrockets• 4 years ago

I found this talk by Luca De Fao is a very good introduction to isogneys and SIDH (caveat emptor: I can grok some algebra, but am not cryptographer) https://m.youtube.com/watch?v=EAe5dqWcxh4

ur-whale2• 4 years ago

" Two curves E/K and E‘/K are isogenous over K if there exists a morphism φ : E → E‘ with coefficients in K mapping the neutral element of E to the neutral element of E‘."

What does it mean for a morphism to "have elements in K" ????

Not sure I understand this so-called definition.

m00n2• 4 years ago

A map between algebraic curves is defined by polynomials. That the map is defined over K means you can find a coordinate system such that the equations of the curves and the equations of the morphisms have coefficients in K and not some larger ring, eg the complex numbers or a large extension field of F_p(field of p elements)

ur-whale2• 4 years ago

A map between algebraic curves is defined by polynomials.

That's a statement that requires IMO justification.

Is that really always the case?

Does it somehow stem from the fact that the source and destination sets are - albeit large - finite?

Again: not obvious.

kup0• 4 years ago

This is one of those things where I think there are 2 or 3 layers of math concepts above the concept of isogeny that I have to understand before I can understand isogenies.

I tried following a trail upwards from isogeny and most of the related terms I didn't understand either. Nice.

What is an isogeny? (2019)

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