1998/06/03 by Valery Gritsenko, V. Gritsenko, Klaus Hulek +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.math/9806011
20 pages, Latex2e RIMS Preprint 1203
arxiv created 1998/06/03 · openalex publication_date 1998/06/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 A3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group Γ3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is Δ13. The form Δ1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group Γ3. 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 Δ1 is an element of a short series of modular forms with this last property. Using the fact that Δ1 is a weight 3 cusp form with respect to the group Γ3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space \cal A3(2).