The Arithmetic of Genus Two Curves with (4,4)-split Jacobians

In this paper we study genus 2 curves whose Jacobians are (4, 4)-isogenous to a product of elliptic curves. Such Jacobians are called (4, 4)-split. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We give a generic model such that any genus 2 curve with optimally (4, 4)-split Jacobian can be obtained as a specialization. We also describe the locus of (4, 4)-split Jacobians in the moduli space of genus 2 curves. Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies. We also give a general description of Richelot isogenies between Jacobians of genus 2 curves. Previously, only Richelot isogenies with kernels that are pointwise defined over the base field were considered.