The Central Derivatives of Automorphic L-Functions

杨9700777自从格罗斯和扎吉尔十年前证明了他们著名的公式以来，已经发现了许多应用，包括伯奇和斯温纳顿-戴尔猜想和高斯的班数问题。Gross-Zagier公式已被推广到p元情形和高权情形。这个项目的主要目的之一是利用Kudla关于Eisenstein级数中心导数的最新结果来证明Gross-Zagier公式，而不受“虚二次域”的任何限制。这是与库德拉的联合项目。库德拉的想法是非常概念性的，应该能够适用于其他古典团体。从这个角度来看，似乎也有可能将Gross-Zagier公式推广到O(n，2)或U(n，1)型的Shimura变种。这将是首席调查员的长期目标。在这个项目的第二部分，首席研究员计划继续研究赫克·L的主导系数-CM数域的函数。相关的黑克字符有一个对称的中心，并具有根号1或-1。当根数为1时，PI证明了一个公式，它将中心Hecke L值表示为某一从U(1)到其自身的θ提升的内积，并应用该公式证明了一族椭圆曲线族的秩为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，该公式将有关的Hecke L函数的中心导数与与U(1)相关的某个Shimura变种上两个不同圈的高度配对联系起来。此外，PI计划应用该公式来证明在某些特殊情况下Hecke L-函数的中心导数不会消失。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将开始在Picard模曲面上处理类似的问题。这一建议是数学朗兰兹计划的一部分。朗兰兹计划是数论的一部分。数论是研究整体数的性质的学科，是数学中最古老的分支。从一开始，数论中的问题就为在该学科的其他不同部分创造新的数学提供了动力。朗兰德的计划是一种将数论与微积分相结合的一般哲学，它体现了研究整数的现代方法。现代数论具有很强的技术性和深刻性，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.