The Central Derivatives of Automorphic L-Functions

杨9700777 自从Gross和Zagier在十年前证明了他们著名的公式以来，已经发现了许多应用，包括Birch和Swinnerton-Dyer猜想以及Gauss的类数问题。 Gross-Zagier公式被推广到p-adic情形和更高权的情形。 本项目的主要目的之一是利用Kudla最近关于Eisenstein级数中心导数的结果证明Gross-Zagier公式，而不对“虚二次域”进行任何限制。 这是一个与Kudla合作的项目。 Kudla的想法是相当概念化的，应该能够对其他经典群起作用。 从这一点来看，似乎也可以将Gross-Zagier公式推广到O（n，2）或U（n，1）型的Shimura簇。 这将是主要研究者的长期目标。 在本项目的第二部分，主要研究者计划继续他的CM数域的Hecke L-函数的主导系数的研究。 所涉及的Hecke特征标具有对称中心并且具有根数1或-1。 当根数为1时，PI证明了一个公式，将中心Hecke L值表示为从U（1）提升到自身的某个θ的内积，并应用该公式证明了一族椭圆曲线的秩为0。 当根数为-1时，中心L值自动为0。 PI计划证明一个Gross-Zagier公式，通过调整Kudla的想法，将相关Hecke L-函数的中心导数与与U（1）相关的某个Shimura簇上的两个不同循环的高度配对联系起来。 此外，PI计划应用该公式来证明Hecke L-函数的中心导数在某些特殊情况下不为零。 特别是，他想证明某些CM椭圆Q-曲线具有有理秩1。 当前两部分取得重大进展时，PI将开始在Picard模块化表面上解决类似的问题。 这个建议是数学的一部分，被称为朗兰兹纲领。 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。 从一开始的问题在数论提供了一个驱动力，创造新的数学在其他不同的部分纪律。 朗格兰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。

Yang 9700777 Since Gross and Zagier proved their celebrated formula ten years ago, a lot of applications have been found, including to the Birch and Swinnerton-Dyer conjecture and to Gauss's class number problem. The Gross-Zagier formula has been extended to the p-adic case and to higher weight case. One of the main purposes of this project is to prove the Gross-Zagier formula without any restriction on the ``imaginary quadratic field'' using Kudla's recent results on the central derivative of Eisenstein series. This is a joint project with Kudla. Kudla's idea is quite conceptional and should be able to work on other classical groups. From that point of view, it seems also possible to extend the Gross-Zagier formula to Shimura varieties of type O(n,2) or U(n, 1). This will be a long term goal of the Principal Investigator. In the second part of this project, the Principal Investigator plans to continue his study of the leading coefficients of Hecke L-functions of CM number fields. The concerned Hecke characters have a center of symmetry and have root number 1 or -1. When the root number is 1, the PI has proved a formula to express the central Hecke L-value as the inner product of some theta lifting from U(1) to itself, and applied the formula to prove that a family of elliptic curves have rank 0. When the root number is -1, the central L-value is 0 automatically. The PI plans to prove a Gross-Zagier formula to relate the central derivative of the concerned Hecke L-function to height pairing of two distinct cycles on some Shimura variety associated to U(1) by adapting Kudla's idea. Furthermore, the PI plans to apply the formula to prove that the central derivatives of the Hecke L-functions do not vanish in some special cases. In particular, he wants to prove that certain CM elliptic Q-curves have rational rank 1. When significant progress is made in the first two parts, the PI will start o tackle similar questions on the Picard modular surfaces. This proposal is an th e part of mathematics known as the Langlands program. The Langlands program is part of Number Theory. Number Theory 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 part of the discipline. The Langland's program is a general philosophy that connect 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 application in areas like theoretical computer science and coding theory.