Problems in The Theory of Automorphic Forms and L-functions

通过朗兰兹计划最近的进展，研究人员提出了一些项目，既要在功能上取得新的进展，又要受益于现有的东西。第一部分包括用正则迹公式和相对迹公式研究朗兰兹的“超越内窥镜”，以及在无限维群上使用除艾森斯坦级数以外的其他Poincar‘e级数的可能性，以期捕捉在对偶背景下发生的新的伴随作用，因为现在看来，这些群上的艾森斯坦级数不会产生任何新的L函数。第二组工作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；利用Hard-Mahnkopf周期通过函数论得到L-函数的特殊值结果，并尝试使用朗兰兹-沙希迪方法通过Hard的某些思想得到这样的结果；由研究者关于局部系数的工作所支配的贝塞尔函数的一般理论，着眼于证明GL(N)的对称和外方L函数的根数的稳定性，以及从不同方法得到的根数的相等性。最后，作者将研究某些局部交织算子的奇异剩余，希望将它们解释为某些加权轨道积分，以及局部群和L函数表示理论中的其他问题。这些项目中的大多数都是与其他数学家联合进行的。朗兰兹计划是一个庞大的问题和猜想的集合，它将算术或几何性质的对象与具有解析性质的对象联系起来。这种重复性通常被称为“功能”。这方面的一个例子出现在威尔斯著名的费马大定理的证明中。在他的整个职业生涯中，这位研究人员发展了一种通常被称为朗兰兹-沙希迪方法的理论，通过与许多数学家的合作，最近导致了一些新的和令人惊讶的函数性情况，结果如某些双曲黎曼曲面上拉普拉斯的特征值的新的界。本提案提出了一些问题，以试图将函数性扩展到更大类别的情况，并利用它们来建立数论和群表示的新结果。其中包括了解L函数的值的超越性，Riemann Zeta函数在某些整数和半整数处的推广，以及与后者的整数值一致的分析对象，以及分析其他具有算术意义的分析对象。该提案涉及对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.