Automorphic Forms and L-Functions

这个研究项目是关于数论的。它集中在自守形式-函数是不变的下一个大的离散群的对称性。 当应用对称性的顺序无关紧要时，这种函数可以用世纪发现的傅立叶分析来研究;然而，对于更复杂的对称性，需要新的思想，我们的知识还远远不够。基本的朗兰兹功能性猜想预测，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）之间架起了一座桥梁。在怀尔斯证明费马猜想的过程中，一个特殊的朗兰兹猜想起了关键作用，但朗兰兹猜想的大多数情况仍然没有得到证明。这个项目将提供关于自守形式的新信息，包括函性，数论中的量以及与自守形式相关的其他数学领域。更详细地说，这个项目包括与函性，L-函数和覆盖群相关的问题。在一系列的问题，主要研究者提出的应用程序，并推广，一个新的最近建设，扭曲倍积分，这使得有可能分析L-函数的非一般自守形式。第二个系列的问题涉及一般覆盖群的自守形式，即元群。主要研究者建议给出广泛的例子的原则，建设涉及自守形式的线性群（或双覆盖所描述的韦尔）可以扩展到一般覆盖群。第三组问题涉及新的建设，涉及自守形式，表示论和数论，包括研究泛函的p-adic群有关岩堀-赫克代数和量子群。这项研究将推进我们对约化群及其覆盖上的自守形式的认识，并对数论、表示论和弦理论产生影响。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.