Theta functions, L-functions, and modular forms

Theta 函数、L 函数和模形式

• 批准号: RGPIN-2014-03847
• 负责人: Vatsal, Vinayak
• 金额: $ 1.31万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667658
• 关键词: Theta; functions; modular; forms

The principal subjects of my research are arithmetic properties of theta functions, especially applications of Mumford's algebraic theta functions to basic questions in arithmetic. While Mumford's theory has had spectacular applications to the construction of arithmetic moduli spaces associated to abelian varieties, it seems that nobody has looked at the applications of the theory to concrete problems in arithmetic. We propose one such project, coming from quadratic forms, and propose a method of solution (joint work with Jonathan Hanke). A second project, more theoretical, project along these lines is the construction of geometric modular forms of weight 3/2, and to construct a geometric version of the theta correspondence in this context. We also propose two entirely different lines of research. The first deals with Iwasawa theory and deformation theory of Galois representations associated to modular forms of weight one. This is joint work with R. Greenberg, and is closely related to recent research of Bellaiche and Dimitrov. The second project seeks to construct modular symbols associated to Eisenstein series over totally real fields. This generalizes earlier work of Stevens, Heumann, and the present author.

我的主要研究课题是theta函数的算术性质，特别是Mumford代数theta函数在基本算术问题中的应用。虽然Mumford的理论在构造与阿贝尔变相关的算术模空间方面有着惊人的应用，但似乎没有人把这个理论应用到具体的算术问题上。我们提出了一个这样的项目，来自二次型，并提出了一种解决方法（与Jonathan Hanke合作）。第二个项目，更理论化的，沿着这条线的项目是构造3/2的几何模形式，并在这种情况下构造对应的几何版本。我们还提出了两种完全不同的研究方向。第一部分涉及与权值1的模形式相关的伽罗瓦表示的Iwasawa理论和变形理论。这是与R. Greenberg的共同工作，与Bellaiche和Dimitrov最近的研究密切相关。第二个项目试图在全实域上构建与爱森斯坦级数相关的模块符号。这概括了史蒂文斯、休曼和本作者的早期工作。