Derivatives of Eisenstein Series L - Functions and the Theta Correspondence

该奖项支持调查三个问题，涉及theta对应，Siegel-Weil公式，价值观和衍生物的爱森斯坦系列和L-功能，代数周期志村品种。问题一涉及局部θ对应的基本性质，特别是第一次出现的问题。 问题二是关于某些Shimura簇上代数圈的高度对与Siegel Eisenstein级数的中心导数的Fourier展开之间的可能关系。问题三 涉及应用的关系问题II的衍生物，在中心的对称性，某些朗兰兹L-功能，推广公式的格罗斯和Zagier。 研究福尔斯属于朗兰兹的一般数学范畴 程序。朗兰兹程序是数论的一部分，数论是研究整数的性质，是数学最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分创造新的数学提供了动力。朗兰兹纲领是一种将数论与微积分联系起来的一般哲学;它体现了研究整数的现代方法。 现代数论是非常技术性和深刻的，但它在理论计算机科学和编码理论等领域有着惊人的应用。

This award supports an investigation into three problems involving the theta correspondence, the Siegel-Weil formula, values and derivatives of Eisenstein series and L-functions, and algebraic cycles on Shimura varieties. Problem I concerns the basic properties of the local theta correspondence, especially the question of the first occurence. Problem II concerns a possible relation between the height pairings of algebraic cycles on certain Shimura varieties and the Fourier expansions of central derivatives of Siegel Eisenstein series. Problem III involves the application of the relation of Problem II to the derivative, at the center of symmetry, of certain Langlands L-functions, generalizing the formula of Gross and Zagier. The research falls in the general mathematical area of the Langlands program.The Langlands program is part of Number Theory, which is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.