Barth-Nieto五次曲线的模形式
The Modular Form of the Barth-Nieto Quintic
摘要：
Barth和Nieto发现了一个显着的五次三重参数化Heisenberg不变的Kummer曲面，它属于具有（1，3）-极化和lecel 2结构的阿贝尔曲面。这个五次曲面的双覆盖也是一个Calabi-Yau簇，它与具有（1，3）-极化和2级结构的阿贝尔曲面的模空间{ \cal A}_3（2）是双有理等价的.因此，相应的仿模群Γ_3（2）具有唯一权为3的尖点形式.在本文中，我们找到了这个尖点形式，它是Δ_1^3。形式Δ_1是关于仿模群Γ_3的一个特征的显著权1尖点形式。它有几个有趣的特性。一个是它允许一个无穷乘积表示，另一个是它在Siegel空间中沿对角线1阶为零。实际上Δ_1是具有这最后一个性质的一小系列模形式的元素。利用Δ_1是关于群Γ_3（2）的权为3的尖点形式，给出了模空间{ \cal A}_3（2）的光滑投射Calabi-Yau模型的独立构造.
英文摘要：
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 Γ_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is Δ_1^3. The form Δ_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group Γ_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 Δ_1 is an element of a short series of modular forms with this last property. Using the fact that Δ_1 is a weight 3 cusp form with respect to the group Γ_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).
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