Theta functions, L-functions, and modular forms

我研究的主要主题是 theta 函数的算术性质，特别是 Mumford 的代数 theta 函数在算术基本问题中的应用。虽然芒福德的理论在与阿贝尔簇相关的算术模空间的构造方面有着令人瞩目的应用，但似乎没有人研究过该理论在算术中具体问题中的应用。我们提出了一个这样的项目，来自二次形式，并提出了一种解决方法（与乔纳森·汉克共同工作）。第二个项目，更理论化，沿着这些思路的项目是构造权重 3/2 的几何模形式，并在这种情况下构造 theta 对应的几何版本。我们还提出了两种完全不同的研究方向。第一个涉及与重量一模形式相关的岩泽理论和伽罗瓦表示的变形理论。这是与 R. Greenberg 的合作，与 Bellaiche 和 Dimitrov 的最新研究密切相关。第二个项目试图在完全真实的场上构建与爱森斯坦系列相关的模块化符号。这概括了史蒂文斯、休曼和本文作者的早期工作。

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.