# isomorphism of extensions by abelian varieties

Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence

$0 \to G \to P \to A \to 0$,

here one can give $P$ the structure of an abelian variety. And $P$ can be viewed as principal $G$-bundle over $A$. In general, is there a way to determine if two given elements $P$ and $P'$ in $\text{Ext}(A, G)$ are isomorphic?

In particular, assume that $A$ is an elliptic curve. Given two extensions $0 \to G \to P \to A \to 0$ and $0 \to G \to P' \to A \to 0$ with morphisms $g : G \to G$, $f : P \to P'$, and $h : A \to A$ so that the resulting diagram commutes, $g$ is an isomorphism, and $h$ is an isogeny with kernel $(\mathbb{Z}/n\mathbb{Z})^2$. Can we show that $P$ and $P'$ are isomorphic or is there an example otherwise?

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In your example in the last paragraph, wouldn't you expect only an isogeny? Take for instance $P$ to be the trivial extension. If $h$ is an isogeny, so should be $f$. Right? –  Sándor Kovács Mar 9 '11 at 5:56
...I mean that if $h$ is not an isomorphism, then neither is $f$. –  Sándor Kovács Mar 9 '11 at 5:56
In the example, $f$ is an isogeny with kernel $(\mathbb{Z}/n\mathbb{Z})^2$. I just wonder if there is a way to construct an isomorphism between $P$ and $P'$ (this isomorphism does not necessarily fit into the given diagram with $g$ and $h$). –  Tuan Mar 9 '11 at 16:00