Automorphic Forms and L-Functions

这个研究项目涉及数论。它关注自同构形式——在一个大的对称离散群下不变的函数。当应用对称性的顺序无关紧要时，可以用19世纪发现的傅立叶分析来研究这些函数；然而，对于更复杂的对称，需要新的想法，而我们的知识还远远不够完整。基本的朗兰兹泛函猜想预测，高度对称的函数是理解多项式方程解的关键，在连续（函数）和离散（解）之间架起一座桥梁。在怀尔斯证明费马猜想的过程中，朗兰兹猜想的一个特例起到了关键作用，但朗兰兹猜想的大多数特例仍未得到证明。该项目将提供关于自同构形式的新信息，包括功能，以及数论和其他与自同构形式相关的数学领域中的量。更详细地说，这个项目包括与功能、l -函数和覆盖组相关的问题。在一系列问题中，作者提出了一种新的构造——扭转重积分的应用和推广，使分析l函数的非一般自同构形式成为可能。第二组问题是关于一般复盖群上的自同构形式，即元复群。主要研究者建议给出广义的例子，说明涉及线性群（或Weil描述的双覆盖）上自同构形式的结构可以扩展到一般覆盖群。第三组问题涉及涉及自同构形式、表示论和数论的新结构，包括与Iwahori-Hecke代数和量子群有关的p进群上的泛函的研究。这项研究将促进我们对约化群及其覆盖的自同构形式的认识，并对数论、表示理论和弦理论产生影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns number theory. It focuses on automorphic forms---functions that are invariant under a large discrete group of symmetries. When the order of applying the symmetries does not matter, such functions may be studied by Fourier analysis, discovered in the nineteenth century; however, for more complicated symmetries new ideas are needed and our knowledge is far from complete. The fundamental Langlands Functoriality Conjectures predict that highly symmetric functions are the keys to understanding solutions to polynomial equations, making a bridge between the continuous (functions) and the discrete (solutions). A specific case of the conjectures played a key role in Wiles's proof of Fermat's Conjecture, but most cases of the Langlands Conjectures are still unproved. This project will provide new information about automorphic forms, including functoriality, and about quantities in number theory and in other areas of mathematics related to automorphic forms.In more detail, this project includes problems related to functoriality, L-functions, and covering groups. In one series of problems, the principal investigator proposes to give applications of, and to generalize, a new recent construction, the twisted doubling integral, which makes it possible to analyze L-functions for non-generic automorphic forms. A second series of problems concerns automorphic forms on general covering groups, the metaplectic groups. The principal investigator proposes to give broad examples of the principle that constructions involving automorphic forms on linear groups (or the double covers described by Weil) can be extended to general covering groups. A third set of problems concerns new constructions involving automorphic forms, representation theory, and number theory, including the study of functionals on p-adic groups related to Iwahori-Hecke algebras and quantum groups. This research will advance our knowledge of automorphic forms on reductive groups and their covers, with consequences for number theory, representation theory, and string theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.