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.
