Talk:Isogeny
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Definition of "isogeny"
[edit]In its current version, this article defines "isogeny" by "an isogeny is a morphism of varieties between two abelian varieties (e.g. elliptic curves) that is surjective and has a finite kernel". I'm not even sure whether this definition is correct but, anyway, wouldn't it be simpler and clearer to use the standard definition of "isogeny" one can find in standard books on abelian varieties: "A homomorphism f: A -> B of abelian varieties A and B is called an isogeny if f is surjective and has a finite kernel"? See Lang's Abelian Varieties, [II, §1], remark after Theorem 6; Mumford's Abelian Varieties, II.6, Application 3; Milne's Abelian Varieties course notes, text before Proposition 7.1 in chapter I, http://www.jmilne.org/math/ Chrgue (talk) 21:14, 15 December 2010 (UTC)
- Some comments:
- The current definition is "a morphism of algebraic groups that is surjective and has a finite kernel," which seems to be couched in an algebraic geometry context.
- Encyclopedia of Mathematics (a handy example) defines an isogeny as an "epimorphism of group schemes with a finite kernel," which sounds like category theory.
- Our article epimorphism currently states "Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories."
- The definition of "isogeny" thus seems to require a clarification that depends on context.— Pingkudimmi 08:54, 19 July 2018 (UTC)
self isogenies
[edit]What structure do the set of self isogenies have? Do some or all of them have inverses so that those with inverse form a group? If so does the group of invertible selfisogenies transitively map every point on a curve, say, to every other point on the curve or perhaps points that are associated to monic polynomials are mapped just among themselves, analogous to a Galois group mapping roots to roots? (These questions could contribute to expansion of the article, although for my curiousity also. So I'm sure they meet talk page guidelines.) Thanks, Rich Peterson24.7.28.186 (talk) 16:31, 28 October 2011 (UTC)
The figure is no good
[edit]What are omega_1 and omega_2? Where does the lattice come from? — Preceding unsigned comment added by 2A02:1206:4553:25C0:BD20:90F:2CD3:BFB9 (talk) 21:29, 2 July 2019 (UTC)