Height pairings on unitary and orthogonal Shimura varieties

单一和正交志村品种的高度配对

• 批准号: 1201480
• 负责人: Benjamin Howard
• 金额: $ 15.99万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201480&HistoricalAwards=false
• 关键词: Height; pairings; unitary; orthogonal; Shimura

The Principal Investigator will study the arithmetic of integral models of Shimura varieties, with particular emphasis on explicit Arakelov theory and the relations between intersection multiplicities of special cycles and the coefficients of Eisenstein series. The prototype of such a relation is the Gross-Zagier theorem, whose proof proceeds by computing the arithmetic intersection multiplicities of Heegner points on modular curves, and then comparing these multiplicities with the Fourier coefficients of the kernel function for the Rankin-Selberg convolution L-function. The PI will prove similar relations for cycles on unitary and orthogonal Shimura varieties, and use these relations to prove Gross-Zagier type theorems for modular forms of higher weight.Shimura varieties are particular kinds of higher-dimensional surfaces, and their rich geometry and arithmetic puts them among the central objects of study in modern mathematics. They play an essential role in the Langlands program, which is of increasing interest in theoretical physics, and are essential tools for understanding elliptic curves and abelian varieties, which have applications to cryptography. The Principal Investigator's research into Shimura varieties is motivated partly by their connections to the conjecture of Birch and Swinnerton-Dyer, one the the Clay Mathematics Institute's million-dollar Millenium Prize Problems. The strongest known results in the direction of this conjecture come from work of Gross and Zagier, whose methods relied on the detailed study of one-dimensional Shimura varieties called modular curves. The PI will study higher dimensional versions of these results in order to better understand the Birch and Swinnerton-Dyer conjecture, and its generalizations and variants.

主要研究者将研究志村簇的积分模型的算法，特别强调显式Arakelov理论和特殊圈的交叉多重性与Eisenstein级数系数之间的关系。 这种关系的原型是Gross-Zagier定理，其证明过程是计算模曲线上Heegner点的算术交叉重数，然后将这些重数与Rankin-Selberg卷积L函数的核函数的傅立叶系数进行比较。 PI将证明酉和正交Shimura簇上圈的相似关系，并利用这些关系证明更高权模形式的Gross-Zagier型定理。Shimura簇是高维曲面的特殊类型，其丰富的几何和算术使其成为现代数学的中心研究对象。 它们在朗兰兹计划中扮演着重要的角色，朗兰兹计划在理论物理学中越来越受到关注，并且是理解椭圆曲线和阿贝尔簇的重要工具，这些都应用于密码学。 主要研究者对志村变种的研究部分是出于它们与伯奇和斯温纳顿-戴尔猜想的联系，这是克莱数学研究所的百万美元千禧年奖问题之一。 在这个猜想的方向上，最强的已知结果来自格罗斯和扎吉尔的工作，他们的方法依赖于对称为模曲线的一维志村簇的详细研究。 PI将研究这些结果的高维版本，以便更好地理解Birch和Swinnerton-Dyer猜想及其推广和变体。