Taylor Expansion of Eisenstein Series and Applications

研究者将把著名的Kronecker第一极限公式推广到带字符的“全纯”爱森斯坦级数，并研究爱森斯坦级数的切线。然后，他将应用这些结果来研究自同构l函数的中心导数。在第二个项目中，他、库德拉和拉波波特将继续库德拉在非相干爱森斯坦级数的中心导数方面的基础工作。特别地，他们将证明某些非相干爱森斯坦级数的中心导数是算术Shimura曲面上零环的生成函数。他还与M. Stollon一起研究了一条非常好的二格曲线的算术性质，这条曲线与Birch猜想和Swinnerton-Dyer猜想有关。最后，他正在研究GU（3,3）的外部立方体自同构l系列与相关的Shimura变种的关系。数论是两千多年来最令人惊奇的研究课题之一，并将继续如此。在过去的20年里，它在通信和密码学上的惊人和美丽的应用更是如此。研究者正在研究的自同构形式和爱森斯坦级数是现代数论的核心。它的进步不仅会进一步加深我们对数论这一深奥而美丽的学科的理解，而且还会找到改善我们在电子世界中的日常生活的方法。

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.