Today I want to talk about morphisms between curves and the special properties they have. This is sort of a broad-based topic, given that the study of morphisms and their properties is essentially the whole of algebraic geometry. I decided that perhaps the best way to narrow my focus is to push towards proving that if is a non-constant isogeny of elliptic curves over a field with a kernel of order then there exists a dual isogeny such that , the map sending each point to . The end result is that the path I present towards the dual isogeny is brisk but perhaps not optimal in that there are lots of facts I use which deserve a more unified treatment.
First things first: We recall from last time that an elliptic curves is an abstract curve with a distinguished rational point. An isogeny between elliptic curves is a morphism of curves which sends the distinguished point of to the distinguished point of .
Implicit in this business of a dual isogeny is that of a finite kernel, which implies that an isogeny must be finite and a homomorphism on the groups of points. We recall some facts about morphisms of curves:
Fact 1: If is a morphism of irreducible curves, is either all of or just a point.
The proof of this fact hinges entirely on the fact that is continuous and is irreducible. Being a morphism implies that it is continuous and the image of an irreducible set under a continuous map is irreducible. Then we are reduced to dimension-counting. Since has dimension 1, has dimension 0 or 1. The only 0-dimensional irreducible set is a point and the only 1-dimensional irreducible subset of is itself.
In the latter case, we have an injection of function fields . To see this, let be a rational function on defined on the open set . Then $latex f^*r = r\circ f$ is a rational function on defined on the open set . This map is a ring homomorphism, as and on the intersection of the domains of and . Since it is a ring homomorphism between fields, it is injective, so is isomorphic to a subfield of which we call .
Fact 2: is a finite extension of up to isomorphism.
We already know that . However, recall that we define the dimension of an irreducible variety to be the transcendence degree of its function field over . Since and are extensions of transcendence degree 1 over , is an algebraic extension of . Since both are finitely generated over , it is a finite algebraic extension.
We define the degree (likewise the separable and inseparable degrees) of a morphism to be the degree of the extension .
Inseparable morphisms are frequently encountered in arithmetic geometry and cannot be ignored, but it’s very easy to get bogged down in details when one studies them. We will revisit them in a future post and for now restrict to the case when is a separable morphism ( is a finite separable extension) and are elliptic curves.
Fact 3: Isogenies of elliptic curves induce group homomorphisms on and thus they ARE homomorphisms in the chord and tangent group law.
Recall first that if is an isogeny with a single point as its image, that single point must be , so .
So consider to be a nonconstant isogeny. Since it maps points to points, we can extend it linearly to divisors (). And if is the norm from down to then . Hence we have an induced map .
The kernel of this map is then , a finite subgroup of of size (in the algebraic closure).
Fact 4: There is an injective map(really a group isomorphism, but we don’t need that) , thus is a Galois extension since is separable.
Let be a point in and be the automorphism of be induced by translation. This is a nonconstant morphism and thus induces an automorphism of (This identification is injective by checking at ). For each , fixes every element of because for all points , and .
Fact 5: If is separable and has then there is a unique isogeny such that .
We know is Galois over . It may or may not be Galois over but we know the automorphism group of over contains the Galois group above. Hence every element of fixes and we have . This inclusion gives us an injection , which gives a unique morphism such that and thus . Thus .
Fact 6: For any natural number , described above is a nonconstant isogeny
By the group law, the function , and addition is continuous so we have a morphism. If we can show is nonconstant, then because , we need only prove that for is nonconstant. To do this, we find a nontrivial 2-torsion point , so that .
Thus it suffices to find nontrivial 2-torsion. If we work over a field of characteristic not 2, we can use coordinates and the transformation in a previous post on elliptic curves to write our elliptic curve as for that we can calculate in terms of . The nontrivial 2-torsion elements of our elliptic curve correspond to the places where the above equation has a vertical tangent line, thus the roots of the polynomial . Since there are exactly 3 such points in the algebraic closure of , the morphism is nonconstant. If is a field of characteristic 2, we have to get our hands dirty and compute a duplication formula(or else just look it up in Silverman’s book, under theorem II.2.3).
If then the -coordinate of is a rational function with denominator described above. Thus once more, the 2-torsion occurs when , which only occurs at finitely many points. The trouble is that in characteristic 2, , which is not a separable polynomial so there’s reason to suspect that is inseparable and there is no nontrivial 2-torsion(which is indeed the case as we’ll see next time). In this case we calculate a triplication formula, which will yield a similar polynomial formula for the denominator. This tells us that is a nonconstant morphism and there are nontrivial 3-torsion points, i.e. points such that . A similar argument shows that the integers coprime to 6(they must be 1 or 2 mod 3) give nonconstant morphisms, completing the proof.
Now we can state the following:
If is a nonconstant separable isogeny with kernel of size , $ there is a unique isogeny such that .
The proof follows from each of the facts given. In fact if , then would serve as a suitable dual. Next time, The Dual Isogeny 2: Inseparable Isogenies.