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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635464
• 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，特别是应用芒福德的代数theta函数在算术的基本问题。虽然芒福德的理论有壮观的应用建设的算术模空间相关的阿贝尔品种，似乎没有人看的应用理论的具体问题的算术。我们提出了一个这样的项目，来自二次型，并提出了解决方案的方法（与Jonathan Hanke联合工作）。第二个项目，更理论化的，项目沿着这些线是构造权为3/2的几何模形式，并在此上下文中构造theta对应的几何版本。我们还提出了两条完全不同的研究路线。第一个涉及岩泽理论和变形理论的伽罗瓦表示与模形式的重量一。这是与R的合作。格林伯格，并密切相关的Bellaiche和Dimitrov最近的研究。第二个项目旨在构建与完全真实的域上的爱森斯坦级数相关的模符号。这概括了史蒂文斯，休曼和本作者的早期工作。