Non-simple polarised abelian surfaces

We study the space of non-simple polarised abelian surfaces. Specifically, we describe the locus of polarised abelian surfaces of type $(1,d)$ that contain an elliptic curve of exponent $n$ and characterise its irreducible components. We find a condition on the complementary exponent $m$ and show that the locus of surfaces containing complementary elliptic curves of exponents $m,n$ is irreducible. We also find equations for this locus in the Siegel space.