Theta functions, L-functions, and modular forms

我研究的主要课题是theta函数的算术性质，特别是Mumford代数theta函数在算术基本问题中的应用。虽然芒福德的理论在构造与交换变种相关的算术模空间方面有着惊人的应用，但似乎还没有人研究过该理论在算术中的具体问题上的应用。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(与乔纳森·汉克联合工作)。第二个更具理论性的项目是构造权重为3/2的几何模形式，并在此背景下构造一个几何形式的theta对应。我们还提出了两条完全不同的研究路线。第一部分讨论与权1的模形式有关的岩泽理论和伽罗瓦表示的形变理论。这是与R·格林伯格的联合工作，与贝莱切和迪米特罗夫最近的研究密切相关。第二个项目寻求在全实域上构造与Eisenstein级数相关的模符号。这概括了史蒂文斯、海曼和本作者的早期工作。

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.