Langlands Correspondence, L-functions and Automorphic Forms

朗兰兹对应、L 函数和自守形式

• 批准号: 1162299
• 负责人: Freydoon Shahidi
• 金额: $ 34.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1162299&HistoricalAwards=false
• 关键词: Langlands; Correspondence; functions; Automorphic; Forms

Local Artin root numbers whose existence were established by Langlands and Deligne (in some cases earlier by Dwork) are crucial objects in establishing the local Langlands correspondence between n-dimensional continuous representations of the Weil-Deligne group over a local field F and irreducible admissible representations of GL(n,F). In fact, a unique correspondence is obtained only after the root numbers and L-functions attached to tensor products of representations of Weil group are shown to equal to those defined by Rankin product factors for corresponding representations of two GL(.,F). There are some other instances where these objects are defined for representaions of GL(n,F) such as exterior square and symmetric square representations of GL(n,C), as well exterior cube when n is less than or equal to 8 by means of the Langlands-Shahidi method. As the first topic in this proposal, the investigator will study a robust technique which can be used to prove the equality of these factors by those defined for Weil group through the correspondence. Techniques involved are a deformation argument as well as a generalized Shalika germ expansion for Bessel functions by Jacquet and Ye which seems to be amenable to generalization to other groups. He will also use Arthur's results in his upcoming book to resolve certain questions concerning the Langlands packet attached to an Arthur packet, as well as certain arithmetic questions (Weyl laws) for classical groups, and their generalizations to general spin groups. Computing the residues of intertwining operators for classical groups in terms of endoscopy which he has been pursuing in collaboration for many years, should also benefit from Arthur's character indentities which he will explore as part of this project. He will also study certain representation theoretic consequences of functoriality. Next he will continue his joint work on studying p-adic L-functions through the Langlands-Shahidi method, and pursue Langlands new ideas on Beyond Endoscopy and Reciprocity, as well as the possible generalization of the method to loop groups and covering groups.The proposal involves training of graduate students and postdocs and includes specific problems for them. The investigator expects several new students to join him and other members of the Number Theory group at Purdue and is involved in teaching high level courses (e.g.,p-adic L-functions, automorphic forms, representation theory of real Lie groups) to train them. Theory of Artin L-functions and its connection with reciprocity law (correspondence) of Langlands is one of the most beautiful parts of number theory which the investigator hopes can be studied by students of different level in different seminars. On another level, he is involved in organizing conferences as well as serving in editorial boards of several prominent journals as well as panels. Moreover, he remains involved in mentoring and minority hiring and currently serves on the Department's Graduate Recruitment Committee with emphasis on recruiting women and minority students.

Langlands和Deligne建立的本地Artin根数（在某些情况下是由DWORW）建立了在Weil-Deligne组的N维连续表示之间建立局部langlands对应的至关重要的对象，而局部领域F和GL（N，F）的不可替代表示。实际上，仅在张量的张量产物的根数和l功能上，才能获得唯一的对应关系，显示了Weil组表示的张量产品等于由Rankin产品因子定义的，对于两个GL（。，F）的相应表示形式。在其他一些情况下，当n langlands-shahidi方法n小于或等于8时，在其他情况下定义了这些对象为GL（N，F）的代表（N，F）的代表，例如GL（N，C）的外部正方形和对称的正方形表示。作为本提案中的第一个主题，研究人员将研究一种可靠的技术，该技术可用于通过通信为Weil组定义的因素证明这些因素的平等性。所涉及的技术是雅克和Ye的贝塞尔功能的变形论证以及对贝塞尔功能的广义shalika胚芽扩展，这似乎可以概括对其他群体的概括。他还将在即将出版的书中使用亚瑟的结果来解决有关亚瑟包装附加的兰兰兹包的某些问题，以及针对古典群体的某些算术问题（Weyl Laws），以及它们对一般旋转群体的概括。在他从事的内窥镜检查中，计算经典群体相互交织的操作员的残留物已合作多年，也应该从亚瑟的性格契约中受益，他将作为该项目的一部分探索。他还将研究功能性的某些代表理论后果。 接下来，他将继续他的共同努力，通过Langlands-Shahidi方法研究P-ADIC L功能，并在内窥镜检查和互惠方面追求Langlands的新想法，以及该方法可能概括循环组和涵盖小组的概括。该建议涉及研究生和博客的培训，并包括对他们的具体问题。调查人员希望几名新学生和普渡大学的数字理论小组的其他成员一起，并参与教授高级课程（例如，P-Adic L功能，自动形式，真实谎言组的代表理论）来培训他们。 Artin l功能理论及其与兰兰兹互惠定律（对应）的联系是数字理论中最美丽的部分之一，研究者希望可以在不同级别的不同级别的学生中对其进行研究。在另一个层面上，他参与组织会议，并在几个著名期刊和小组的编辑委员会中任职。此外，他仍在参与指导和少数族裔招聘，目前是该部门的研究生招聘委员会，重点是招募妇女和少数民族学生。