Taylor Expansion of Eisenstein Series and Applications

爱森斯坦级数的泰勒展开及其应用

• 批准号: 0070476
• 负责人: Tonghai Yang
• 金额: $ 8.7万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070476&HistoricalAwards=false
• 关键词: Taylor; Expansion; Eisenstein; Series; Applications

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.

将著名的Kronecker第一极限公式推广到具有特征的全纯Eisenstein级数，并研究Eisenstein级数的切线。然后，他将applythe结果，以研究中心衍生的自守L-functions.In第二个项目，他，库德拉，和Rapoport将继续库德拉的基本工作的中心衍生的不连贯爱森斯坦系列。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曲面上零圈的生成函数。他还与M。Stollon的算术性质的一个非常好的亏格两曲线的关系与伯奇和Swinnerton-Dyer猜想。最后，他正在研究GU（3，3）的外立方自守L-级数与相关的志村簇的关系。 两千多年来，数论一直是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，它在通信和密码学方面的应用更是令人惊讶和美丽。自守形式和爱森斯坦级数的研究是现代数论的核心。它的进步不仅将进一步加深我们对数论这一深刻而美丽的主题的理解，而且还将改善我们在电子世界中的日常生活。