Mock modular forms and quantum modular forms
Mock modular forms and quantum modular forms
复制标题
模拟模块化形式和量子模块化形式
DOI：
10.1090/proc/12907
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发表时间：
2015
影响因子：
0.8
通讯作者：
Robert C. Rhoades
中科院分区：
文献类型：
--
作者：
D. Choi;Subong Lim;Robert C. Rhoades
在他给哈代的最后一封信中，Ramanujan介绍了mock theta函数。对于他的每个例子，Ramanujan声称有一个模形式的集合，使得对于每个单位根，都有这样的集合。此外，Ramanujan声称这个集合的大小一定大于。在他2001年的博士论文中，Zwegers证明了模拟theta函数是调和弱Maass形式的全纯部分。本文通过建立调和Maass形式的所有全纯部分的一个更一般的结果，证明了一定存在这样一个集合。这补充了格里芬、小野和罗伦的结果，即这样的集合不可能有大小。这些结果是在Zagier的量子模形式理论的背景下产生的。给出了从模形式空间到量子模形式的线性内射映射。此外，我们还给出了“Ramanujan径向极限”的表达式。参考文献
In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples, Ramanujan claimed that there is a collectionof modular forms such that for each root of unity, there is asuch thatMoreover, Ramanujan claimed that this collection must have size larger than. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size. These results arise within the context of Zagier’s theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for “Ramanujan’s radial limits” as-values. References