A discrete extension of Γ_{1,p}^{\circ}(2) in SP(4,R) and the modular form of the Barth-Nieto quintic

Abstract

We construct a maximal discrete extension of the paramodular group with a full level-2 structure. The corresponding Siegel variety parametrizes (birationally) the space of Kummer surfaces associated to (1,p)-polarized abelian surfaces with a level-2 structure. In the case p=3 this is related to the Barth-Nieto quintic and in this case we also determine the space of cusp forms of weight 3.