# 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.

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.