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

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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.