Problems in The Theory of Automorphic Forms and L-functions

从Langlands计划的最新进展中，研究人员提出了一些项目，既要在功能性方面取得新的进展，也要从现有的项目中受益。第一个包括研究朗兰兹“超越内窥镜”的手段，定期和相对的轨迹公式，以及使用其他庞加莱系列除了爱森斯坦系列的无限维群的可能性，希望捕捉新的伴随行动，发生在对偶设置，因为现在看来，爱森斯坦系列在这些群体不导致任何新的L-功能。第二组计划包括建立从一般自旋群到GL（n）的强转移以及从拟分裂特殊正交群到GL（n）的转移，通过函子性利用Harder-Mahnkopf周期得到L-函数的特殊值结果，以及尝试利用Langlands-Shahidi方法通过Harder的某些思想得到这样的结果;一个一般理论的贝塞尔函数所规定的调查工作的地方系数与眼睛上证明稳定的根数对称和外部平方L-函数的GL（n），除其他外，以及平等的根数从不同的方法。最后，研究者将研究某些局部缠绕算子的奇异剩余，希望将它们解释为某些加权轨道积分，以及局部群和L函数的表示论中的其他问题。朗兰兹纲领是一个将算术或几何性质的对象与分析性质的对象联系起来的问题和命题的庞大集合。这种相互作用通常被称为“功能性”。怀尔斯著名的费马大定理的证明就是这样一个基本的例子。在他的职业生涯中，研究人员开发了一种理论通常被称为朗兰兹-沙希迪方法，通过与一些数学家的合作，最近导致了一些新的和令人惊讶的情况下函的后果，如新的界限特征值的拉普拉斯对某些双曲黎曼曲面。目前的建议提出了一些问题，试图扩大函性的情况下，以及使用它们来建立数论和群表示的新成果。其中包括理解L-函数值的超越性，黎曼zeta函数的推广，在某些整数和半整数上，与后者的积分值一致，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。

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.