The modularity of the Barth-Nieto quintic and its relatives
The modularity of the Barth-Nieto quintic and its relatives
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DOI：
10.15488/216
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发表时间：
2000-10
影响因子：
0.5
通讯作者：
K. Hulek;J. Spandaw;B. Geemen;D. Straten
中科院分区：
文献类型：
--
作者：
K. Hulek;J. Spandaw;B. Geemen;D. Straten
具有完全二级结构的O_1；3U极化阿贝尔曲面的模空间与Barth-Nieto五次曲面的双层覆盖是二次的。Barth和Nieto已经证明，这些品种分别具有Calabi-Yau模型Z和Y。本文利用Weil猜想证明了Y和Z是刚性的，并证明了它们共同的第三类上同调群的L函数是模函数，正如Fontaine和Mazur的猜想所预言的那样。对应的模形式是G106U组的权重为4的唯一归一化尖点形式。根据泰特的猜想，这应该意味着万能椭圆曲线S106U的纤维平方Y和Verrill的刚性Calabi-Yau ZA3都具有相同的L函数，它们在Q上是对应的。我们通过给出显式映射来证明确实是这样。
The moduli space of O1; 3U-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third etale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group G1O6U. By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S1O6U, and Verrill's rigid Calabi-Yau ZA3 , which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.