When is an abelian surface isomorphic or isogenous to a product of elliptic curves?
Let \(\Lambda\) be a lattice in \(\mathbb C^ 2\) defining an abelian surface \(A\cong\mathbb C^ 2/\Lambda\). One can define an alternating map \(\phi: \Lambda\times \Lambda \to \mathbb C\). The author proves that \(A\) is isomorphic to a product of elliptic curves if and only if \(\phi\) is hyperbolic, i.e. there exists a \(\mathbb Z\)-basis \(e_ 1,f_ 1,e_ 2,f_ 2\) of \(\Lambda\) such that \(\phi (e_ 1,f_ 1)=\phi (e_ 2,f_ 2)=0\). After that the author proves that if \(\phi\) is hyperbolic then the rank of \(\phi\) (e.g. the dimension of the \({\mathbb Z}\)-module spanned by the values of \(\phi\)) is 2, 3 or 4. If \(\text{rank}(\phi)=2\), then \(A\) is always a product of elliptic curves. In the other cases the author gives an algorithm in order to decide if \(A\) is a product or not.
scientific article; zbMATH DE number 4171103