Problems in The Theory of Automorphic Forms and L-functions

自守形式和L-函数理论中的问题

• 批准号: 0700280
• 负责人: Freydoon Shahidi
• 金额: $ 41.25万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700280&HistoricalAwards=false
• 关键词: Problems; Theory; Automorphic; Forms; functions

Taking the lead from the recent progress in Langlands program, the investigator proposes a number of projects, both towards making new progress on functoriality as well as benefiting from what is available. The first includes a study of Langlands "Beyond Endoscopy" by means of both regular and relative trace formulas as well as the possibility of using other Poincar\'e series besides Eisenstein series on infinite dimensional groups with the hope of capturing the new adjoint actions that happen in the dual setting, since it now appears that Eisenstein series on these groups do not lead to any new L-functions. The second set of projects includes establishing the strong transfer from general spin groups as well as the transfer from quasisplit special orthogonal groups to GL(n); special value results for L-functions by means of Harder-Mahnkopf periods through functoriality as well as an attempt in using the Langlands-Shahidi method to obtain such results via certain ideas of Harder; a general theory of Bessel functions dictated by the investigator's work on local coefficients with an eye on proving stability for root numbers of symmetric and exterior square L-functions of GL(n), among others, as well as equality of root numbers obtained from different methods. Finally the investigator will study the singular residues of certain local intertwining operators hoping to interpret them as certain weighted orbital integrals, as well as other problems in representation theory of local groups and Lfunctions. Most of these projects are joint with other mathematicians.Langlands Program is a vast collection of problems and conjectures which connects objects of arithmetic or geometric nature to those of analytic character. Such reciprocities are usually called "Functoriality". One example of this appeared in a fundamental way in the celebrated proof of Fermat's Last Theorem by Wiles. Throughout his career the investigator has developed a theory usually called the Langlands-Shahidi method, which through collaboration with a number of mathematicians, has recently led to a number of new and surprising cases of functoriality with consequences such as new bounds on eigenvalues of Laplacian on certain hyperbolic Riemann surfaces. The present proposal suggests a number of problems to try to extend functoriality to a larger class of cases as well as using them to establish new results in number theory and group representations. Among them is understanding the transcendental nature of values of L-functions, generalizations of Riemann zeta functions, at certain integers and half-integers, in line with integral values of the latter, as well as analyzing other analytic objects of arithmetic significance. The proposal involves training of graduate students and postdocs and collaboration with younger investigators.

研究人员提出了许多项目，既可以从兰兰兹计划的最新进展中获得领先，既要在功能上取得新的进展，又要从可用的项目中受益。首先包括通过常规和相对痕量公式对Langlands进行“超越内窥镜检查”的研究，以及除了Eisenstein系列无限尺寸群体外，还可以使用其他Poincar \'E系列，希望捕获双重设置中发生的新邻接动作，因为现在似乎在这些团体上对这些新组都没有任何新的l-function。第二组项目包括从一般自旋组以及从Quasisplit特殊正交组转移到GL（n）的强大转移；通过功能性的较难时期，通过较难的mahnkopf时期以及尝试使用Langlands-Shahidi方法来通过某些更难的某些想法来获得此类结果的特殊价值结果；贝塞尔功能的一般理论由研究者在局部系数上的工作决定，目的是证明gl（n）等对称和外部平方l功能的根数，以及从不同方法获得的根数等等。最终，研究者将研究某些局部相互交织的操作员的奇异残基，希望将其解释为某些加权轨道积分，以及当地群体和LFunctions的表示理论中的其他问题。这些项目中的大多数是与其他数学家的联合。Langlands计划是大量问题和猜想，将算术或几何性质与分析性质的对象联系起来。这种互惠通常称为“功能性”。一个例子以一种基本的方式出现在著名的fermat撰写的威尔斯定理的证据中。在他的整个职业生涯中，调查员开发了一种通常称为Langlands-Shahidi方法的理论，该理论通过与许多数学家的合作，最近导致了许多新的和令人惊讶的功能性案例，以及诸如Laplacian在某些多重物质Riemann表面上的新界限。本提案提出了许多问题，以尝试将功能性扩展到较大类别的案例，并使用它们来建立数字理论和小组表示的新结果。其中包括了解L功能值的先验性质，在某些整数和半智商处的Riemann Zeta函数的概括，符合后者的积分值，并分析其他具有算术意义的分析对象。该提案涉及培训研究生和博士后，以及与年轻调查员的合作。