Taylor Expansion of Eisenstein Series and Applications

点击翻译按钮获取中文摘要

The investigator will try to generalize the well-known Kronecker'sfirst limit formula to `holomorphic' Eisenstein series with charactersand study the tangent line of the Eisenstein series. He will then applythe results to study the central derivative of automorphic L-functions.In the second project, He, Kudla, and Rapoport will continue Kudla's fundamental work on the the central derivative of incoherent Eisenstein series. In particular, they will prove that the central derivative ofcertain incoherent Eisenstein series is the generating function of zerocycles on an arithmetic Shimura surface. He is also working with M. Stollon the arithmetic nature of a very nice genus two curve in relation withthe Birch and Swinnerton-Dyer conjecture. Finally, he is studying theexterior cube automorphic L-series of GU(3, 3) in relation with the associated Shimura variety. Number theory has been one of the most amazing subjects to study for more than two thousand years and will continue to be so. Even more so is its surprising and beautiful applications to communications andcryptography discovered during last 20 years. Automorphic forms and Eisenstein series the investigator is working on is at the heart ofthe modern number theory. Its advances would not only further ourunderstanding of the deep and beautiful subject of number theory but also find its way to improve our daily life in the e-world.