18.782 Arithmetic Geometry Lecture Note 24
نویسنده
چکیده
̄ As an example, the negation map that send P ∈ E(k) to its additive inverse is an isogeny from E to itself; as noted in Lecture 23, it is an automorphism, hence a surjective morphism, and it clearly fixes the identity element (the distinguished rational point O). Recall that a morphism of projective curves is either constant or surjective, so any nonconstant morphism that maps O1 to O2 is automatically an isogeny. The composition of two isogenies is an isogeny, and the set of elliptic curves over a field k and the isogenies between them form a category; the identity morphism in this category is simply the identity map from an elliptic curve to itself, which is is clearly an isogeny. Given that the set of rational points on an elliptic curve form a group, it would seem natural to insist that, as morphisms in the category of elliptic curves, isogenies should preserve this group structure. But there is no need to put this requirement into the definition, it is necessarily satisfied.
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