The Modular Form of the Barth-Nieto Quintic

Abstract

Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).