Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs $(m,n)$ the locus of polarised abelian surfaces of type $(1,d)$ that contain two complementary elliptic curve of exponents $m,n$, denoted $\mathcal{E}_d(m,n)$ is non-empty. We show that if $d$ is square-free, the locus $\mathcal{E}_d(m,n)$ is an irreducible surface (if non-empty). We also show that the loci $\mathcal{E}_d(d,d)$ can have many components if $d$ is an odd square. As an application, we show that for a genus $3$ curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings $f_i:C\rightarrow E_i,\ i=1,2,3$ satisfy $lcm(deg(f_1),deg(f_2))=lcm(deg(f_1),deg(f_3))=lcm(deg(f_2),deg(f_3))$.