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- W1673711274 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)." @default.
- W1673711274 created "2016-06-24" @default.
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- W1673711274 date "1998-06-03" @default.
- W1673711274 modified "2023-10-05" @default.
- W1673711274 title "The Modular Form of the Barth-Nieto Quintic" @default.
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