Jacobians of genus 2 curves with full level 3 structure and the related elliptic fibrations

It is known that the moduli space of Jacobians of genus 2 curves with full level 3 structure is birational to the Burkhardt quartic threefold. Bruin and Nasserden showed that the Burkhardt quartic is rational over Q, and determined a model for the universal genus 2 curve over it with certain description of its level 3 structure. We give a different construction of the same moduli space and the universal genus 2 curve over it, and give an explicit description of the 3-torsion group J(C)[3] with explicit Sp(4,3) action. Our construction is based on a simple observation that finding 3-torsion points of the Jacobian J(C) of the genus 2 curve s²=f(t) naturally leads us to the equation of the form Y² = X³ + f(t), which can be regarded as a rational elliptic surface whose Mordell-Weil lattice is of type E₈. We then use R. Kloosterman's family of rational elliptic threefolds whose Mordell-Weil rank equals 6. As an application, we also discuss certain elliptic K3 surfaces associated with the Jacobian J(C). This is joint work with Abhinav Kumar.