Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$ to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain such an elliptic curve then $A$ is isogenous to $E \times E'$, where $E'$ is the complementary elliptic curve contained in $A$. This is by the Poincare reducibility theorem. Is there some condition on $H^0\left(\Omega^1_A\right)$, or for that matter any sort of Hodge theoretic condition, to determine the rank of this isogeny?
