Square-integrable automorphic forms, local Langlands correspondence and Gross-Prasad conjecture

平方可积自守形式、局部朗兰兹对应和格罗斯-普拉萨德猜想

• 批准号: 0801071
• 负责人: Wee Teck Gan
• 金额: $ 21.11万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801071&HistoricalAwards=false
• 关键词: Square; integrable; automorphic; forms; local

The PI proposes to study various basic questions in representation theory and the theory of automorphic forms arising from the Langlands program, and their applications to questions of arithmetic interest. For a number of years, the PI has been pursuing the construction and classification of square-integrable automorphic forms as predicted by Arthur's conjecture, especially in the context of the exceptional groups. The PI hopes to complete this study in the next 3-year period.He also intends to study the local Langlands correspondence for certain classical and exceptional groups. In another direction, the PI hopes to establish certain cases of the Gross-Prasad conjecture regarding the restriction of representations of an orthogonal or unitary group to a smaller one, which has applications to special values of L-functions. Finally, the PI proposes to establish the second term identity for the regularized Siegel-Weil formula in the context of the theory of theta correspondences.The Langlands program is an integral part of modern number theoretic research. Its initial goal is to understand certain groups which arise naturally in number theory and representation theory. In recent years, it has expanded beyond its traditional boundaries to connect with areas such as algebraic geometry and mathematical physics. It has already found unexpected applications in real life. Indeed, the Jacquet-Langlands correspondence, which is one of the first major results in the Langlands program, has been exploited to give a construction of the so-called Ramanujan graphs.These graphs are highly connected and serve as efficient networks.They have turned out to be very useful in communications theory. It is hoped that the questions investigated in this proposal can help to unearth more applications of this type.

PI建议研究表示论中的各种基本问题和朗兰兹纲领所产生的自守形式理论，以及它们在算术问题上的应用。多年来，PI一直在追求平方可积自守形式的构造和分类，正如亚瑟猜想所预测的那样，特别是在例外群的背景下。PI希望在未来3年内完成这项研究，他还打算研究某些经典和特殊群体的局部朗兰兹对应。 在另一个方向上，PI希望建立Gross-Prasad猜想的某些情况，关于正交或酉群的表示限制为较小的表示，这对L函数的特殊值有应用。最后，PI建议在theta对应理论的背景下建立正则化Siegel-Weil公式的第二项恒等式。Langlands程序是现代数论研究的一个组成部分。它的最初目标是理解在数论和表示论中自然出现的某些群。近年来，它已经超越了传统的界限，与代数几何和数学物理等领域联系起来。它已经在真实的生活中找到了意想不到的应用。事实上，雅凯-朗兰兹对应（Jacquet-Langlands correspondence）是朗兰兹纲领的第一个重要结果，它被用来构造所谓的拉马努金图（Ramanujan graphs），这些图是高度连通的，可以作为有效的网络，在通信理论中非常有用。希望本文所探讨的问题能有助于挖掘更多的此类应用。