A Question in Number Theory and Arithmetic Geometry: The Colmez Conjecture

这个项目是现代数论的核心，有着深刻的历史背景和现代理论意义。该奖项的重点是证明L函数的公式。在数论中，L-函数是黎曼ζ-函数的推广。相关的公式通常是连接分析和代数的桥梁，这是两个不同的数学领域。这样一个公式的最基本的例子是关于黎曼zeta函数在2处的值的欧拉公式。在简单的术语中，它断言所有正整数平方的倒数之和正好等于（pi/2）的平方。这里π =3.14159通常是圆的周长与直径之比。这类公式除了在数学上有重大影响外，也是可以用来向世界推广数学的完美话题。PI提出证明数论中关于L-函数的两个公式。第一个是CM交换簇的Colmez猜想，第二个是四元数Shimura曲线的模高公式。PI提出的方法是他以前的联合工作的显着扩展的格罗斯-Zagier公式和平均科尔梅兹猜想。它们涉及自守形式理论、志村簇和阿拉克洛夫几何，对公式的研究将使我们能够更好地理解这些领域之间深刻的相互作用。

This project lies at the heart of modern number theory, which has a deep historical background as well as modern theoretical implications. The focus of this award is to prove formulae for L-functions. In number theory, L-functions are generalizations of the Riemann zeta-function. The relevant formulae are usually bridges relating analysis to algebra, two different areas of mathematics. The most elementary example of such a formula is Euler's formula about the value of the Riemann zeta-function at 2. In plain terminology, it asserts that the summation of the reciprocals of the squares of all positive integers, is exactly equal to the square of (pi/2). Here pi=3.14159... is the usual ratio of a circle's circumference to its diameter. Besides their significant impact in mathematics, such formulae are also perfect topics that can be used to popularize mathematics to the world.The PI proposes to prove two formulae about L-functions in number theory. The first one is the Colmez conjecture for CM abelian varieties, and the second one is a formula for the modular heights of quaternionic Shimura curves. The methods proposed by the PI are significant extensions of his previous joint works on the Gross-Zagier formula and the averaged Colmez conjecture. They involve the theories of automorphic forms, Shimura varieties and Arakelov geometry, and the study of the formulae will enable us to better understand the profound interactions of these areas.