The Central Derivatives of Automorphic L-Functions

自从十年前Gross和Zagier证明了他们著名的公式以来，人们发现了许多应用，包括Birch猜想和Swinnerton-Dyer猜想以及高斯的类数问题。Gross-Zagier公式已推广到p进情况和更高权重情况。本课题的主要目的之一是利用Kudla最近关于爱森斯坦级数中心导数的结果，证明不受“虚二次域”限制的Gross-Zagier公式。这是和Kudla合作的项目。库德拉的想法是非常概念化的，应该可以在其他古典团体中发挥作用。从这个角度来看，似乎也可以将Gross-Zagier公式推广到O（n,2）或U（n, 1）型的Shimura变量。这将是首席研究员的长期目标。在项目的第二部分，项目负责人计划继续研究CM数域的Hecke l -函数的领先系数。有关的Hecke字符有一个对称中心，根数为1或-1。当根数为1时，PI证明了一个将中心Hecke l值表示为从U(1)到自身的某个θ的内积的公式，并应用该公式证明了一组椭圆曲线的秩为0。当根数为-1时，中心l值自动为0。PI计划通过采用Kudla的思想，证明一个Gross-Zagier公式，将有关Hecke l函数的中心导数与与U(1)相关的Shimura变量上的两个不同循环的高度对联系起来。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会消失。特别地，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将开始解决皮卡德模块表面上的类似问题。这个建议是数学中被称为朗兰兹纲领的一部分。朗兰兹程序是数论的一部分。数论是对整数性质的研究，是数学中最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分中创造新的数学提供了动力。朗兰程序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奥的，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.