Isogenies

Today I want to talk a bit about isogenies. In particular, I’m going to be thinking about the category of Abelian varieties over some algebraically closed field K but some approximation of this discussion could probably be had in a number of other contexts as well. A good reference for what I’m talking about is Mumford’s Abelian Varieties.

If we take abelian varieties X, Y then Hom(X, Y) (i.e. the morphisms of varieties that preserve the group structures) is a a finitely generated free abelian group. In particular, End(X,X) has the natural structure of a \mathbb{Z}-algebra.

Of course by Yoneda, studying Hom( \_, Y) is essentially the same thing as studying Y and so understanding the Hom(X,Y) is crucial. Vector spaces are generally easier to understand than modules, and so it’s reasonable to consider Hom^0(X,Y)=Hom(X,Y) \otimes \mathbb{Q}, End^0(X,Y)=End(X,Y) \otimes \mathbb{Q}. Of course, we lose something by passing to \mathbb{Q}. To give a concrete example, the endomorphism ring of an elliptic curve with complex multiplication is given by an order in a quadratic imagery extension of \mathbb{Q}, but there are multiple non-isomorphic orders within a given field extension. However, we expect (and are correct in our expecations) that studying \mathbb{Q}-algebras and vector spaces is easier than working over \mathbb{Z}.

So now we are working in a new category where the objects are still abelian varieties over K but the morphisms are given by Hom^0(X,Y).  If \alpha \otimes r: X \to Y has an inverse \beta \otimes s then \alpha \circ \beta \otimes rs=1 \in End^0(X, X) and so \alpha \circ \beta=[n]_X, \beta \circ \alpha=[m]_Y (where [n]_X denotes multiplication by n in X). The converse is also clear.

Since each [n] is surjective and has finite kernel,  \alpha \circ \beta=[n]_X gives \alpha is surjective and \beta has finite kernel and \beta \circ \alpha=[m]_Y gives \beta is surjective and \alpha has finite kernel.

Conversely, if \alpha: X \to Y is surjective and has finite kernel then we see that \ker \alpha is a finite group scheme and so killed by some [n]_X. Then Y \simeq X / \ker \alpha and so there exists some \beta: Y \to X giving \beta \circ \alpha=[n]_X. Similarly, we get \alpha \circ \beta=[m]_Y.

So to recap, we’ve now shown that the invertible morphisms in this new category are precisely those that are surjective with finite kernel. But this is the ‘traditional’ definition of isogeny, and we have arrived at it through purely wholesome ideas! Another way to look at this is that we’ve ‘localized’ our original category to force the [n] morphisms to be invertible and have shown that in this new category, the isomorphisms are precisely the isogenies.

I already find this quite satisfying, but I should mention some benefits of working up to isogeny.  One can prove the Poincare reducibility theorem which says that if Y is a subvariety of X then we can find a complement of Y in X. i.e there exists a Z \subset X so that Z \cap Y is finite and Z \times Y is isogenous (isomorphic in our current category!) with X. Using the standard sorts of arguments one can make with any ‘simple’ object, we can actually show that any abelian variety X is isogenous to a product {X_1}^{n_1} \times ... \times {X_k}^{n_k} where X_i and X_j are not isogenous for i \neq j. Further, one shows that End^0({X_i}^{n_i}) \equiv M_{n_i}(D_i) (the matrix ring over some division algebra).  Thus, classifying abelian varieties up to isogeny becomes an issue of understanding which division algebras can appear.

Furthermore (and perhaps most importantly), we have for finite fields and number fields Tate’s Isogeny theorem which allows us to conclude that abelian varieties are isogenous if and only if their rational Tate modules (i.e. T_{\ell}(X) \otimes \mathbb{Q}_{\ell}) are isomorphic as Galois modules.

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3 thoughts on “Isogenies

  1. Nice post! I think this is good motivation for why one calls your localized category the ‘isogeny category’.

    One nitpick though. The statement “Thus classifying abelian varieties up to isogeny becomes an issue of understanding which division algebras appear” is a little misleading. Namely, take two elliptic curves E_1,E_2/\mathbb{Q}. Then, \mathrm{End}^0(E_1)=\mathbb{Q}=\mathrm{End}^0(E_2) regardless of whether E_1 and E_2 are isogenous.

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    1. Ha! Well I said that we’re only working over an algebraically closed field at the beginning for this reason. But actually, knowing how the theory works over other fields/rings is something I don’t feel great about.

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