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Modular forms from Noether–Lefschetz theory
Algebra & Number Theory ( IF 1.3 ) Pub Date : 2020-10-13 , DOI: 10.2140/ant.2020.14.2335
François Greer

We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree 1, but heuristics for higher base degree match predictions from the topological string partition function.

中文翻译:

Noether-Lefschetz 理论的模形式

我们在投影空间中的超曲面上列举了非常普遍的 Weierstrass fibrations 的平滑有理曲线。这些数字的生成函数位于经典模形式的环中。证明方法使用周期堆栈上的拓扑交积和 Kudla 和 Millson 的上同调 theta 对应,用于正交类型的局部对称空间上的特殊循环。此处的结果仅适用于基度 1,但适用于来自拓扑字符串分区函数的更高基度匹配预测的启发式方法。
更新日期:2020-10-13
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