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.

以朗兰兹计划最近取得的进展为引领，研究者提出了一系列项目，既要在功能性方面取得新的进展，又要从现有的项目中受益。第一个包括通过正则和相对迹公式对朗兰兹“超越内窥镜”的研究，以及在无限维群上使用除爱森斯坦级数之外的其他庞加莱级数的可能性，以期捕获新的伴随动作发生在对偶设置中，因为现在看来这些群的爱森斯坦级数不会导致任何新的 L 函数。第二组项目包括建立从一般自旋群的强转移以及从准分裂特殊正交群到GL(n)的转移；通过函子性的 Harder-Mahhnkopf 周期得出 L 函数的特殊值结果，以及尝试使用 Langlands-Shahidi 方法通过 Harder 的某些想法获得此类结果；贝塞尔函数的一般理论，由研究者对局部系数的研究决定，着眼于证明 GL(n) 的对称和外部平方 L 函数的根数的稳定性等，以及从不同的方法获得的根数的相等性方法。最后，研究者将研究某些局部交织算子的奇异留数，希望将它们解释为某些加权轨道积分，以及局部群和 L 函数表示论中的其他问题。这些项目大多数是与其他数学家联合进行的。朗兰兹纲领是大量问题和猜想的集合，它将算术或几何性质的对象与分析特征的对象联系起来。这种互惠性通常称为“函数性”。怀尔斯著名的费马大定理证明中就以根本性的方式出现了这样的一个例子。在他的整个职业生涯中，研究者开发了一种通常称为 Langlands-Shahidi 方法的理论，通过与许多数学家的合作，该理论最近导致了许多新的和令人惊讶的函子性案例，其后果包括拉普拉斯算子特征值的新界限某些双曲黎曼曲面。目前的提案提出了一些问题，试图将函子性扩展到更大的案例类别，并使用它们在数论和群表示中建立新的结果。其中包括理解 L 函数值的超越性质、黎曼 zeta 函数在某些整数和半整数上的推广，与后者的整数值一致，以及分析其他具有算术意义的分析对象。该提案涉及研究生和博士后的培训以及与年轻研究人员的合作。