Question

Let $\varphi : A \to B$ be an isogeny between 2 abelian varieties of dimension $g$. Are there known conditions for the $\ker\varphi$ so that this induces an isomorphism between $A$ and $B$? For example, if $\ker\varphi \cong (\mathbb{Z}/n\mathbb{Z})^{2g}$, then $A \cong B$, because $\varphi$ factors through the multiplication map $A \xrightarrow{\times n} A$ followed by an isomoprhism $A \to B$. I wonder if there are other cases that induce isomorphisms.

Answer 1

Kevin's comment is right on the money, but here it is in more detail: I will give a general criterion for an isogeny $\varphi: A \rightarrow B$ of abelian varieties to induce an isomorphism upon passage to the kernel.

Let me work over an unnamed algebraically closed field. Suppose that $A = B$ and $\eta \in \operatorname{End}(A)$ is a surjective endomorphism of $A$. (N.B.: If $A$ is simple -- i.e., contains no proper nontrivial subvariety -- then any nonzero endomorphism is surjective. In particular this holds for all elliptic curves.) Then $\eta$ is also an isogeny: i.e., its kernel is a finite subgroup scheme, say $K$ and -- essentially, by the first isomorphism theorem for groups, as Kevin says -- it follows that there is an induced isomorphism

$A/K \stackrel{\sim}{\rightarrow} A$.

This condition is also necessary: if $\varphi: A \rightarrow B$ is an isogeny such that $B \cong A$, then composing with this isomorphism gives a surjective endomorphism of $A$ and the resulting map factors through an isomorphism $A/(\operatorname{ker}(\varphi)) \rightarrow B$. Thus all examples arise from a surjective endomorphism of $A$ as above, well-defined up to isomorphisms on the source and target.