On the arithmetic of genus 2 curves with (4,4)-split Jacobians

In this thesis, we study genus 2 curves whose Jacobians allow a decomposition into two elliptic curves. More specifically, we are interested in genus 2 curves C whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We restrict to base fields of characteristic distinct from 2 or 3, but we do not require them to be algebraically closed. In the first half of the thesis, we obtain a full classification of principally polarized abelian surfaces that can arise from gluing two elliptic curves together along their 4-torsion and we derive the relation which their absolute invariants must satisfy. In the process, we derive a description of Richelot isogenies between Jacobians of genus 2 curves. Previous literature only considered Richelot isogenies whose kernels are pointwise defined over the base field. We also obtain a Galois theoretic characterization of genus 2 curves which admit multiple Richelot isogenies on their Jacobians. As a corollary to this classification, we obtain a model for the universal elliptic curve over the modular curve of elliptic curves with 4-torsion anti-isometric to E[4]. The final chapter of the thesis considers elements of order m of the Shafarevich-Tate group of an elliptic curve E, denoted Sha(E/k)[m]. For a given elliptic curve, E, we consider the question of making Sha(E/k)[4] visible in the sense of Mazur. We show that the visibility argument for m=4 is less tractable than the arguments in the m=2 and m=3 cases. In the m=4 case, we encounter a challenge of trying to find rational points on a K3 surface. We also show that finding the appropriate twist of this surface is a non-trivial problem. Nevertheless, in particular cases, one can proceed with this construction and we conclude the thesis by working through a couple of examples in detail.