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 \({{\,\textrm{lcm}\,}}(\deg (f_1),\deg (f_2))={{\,\textrm{lcm}\,}}(\deg (f_1),\deg (f_3))={{\,\textrm{lcm}\,}}(\deg (f_2),\deg (f_3))\).