Langlands Correspondence, L-functions and Automorphic Forms
朗兰兹对应、L 函数和自守形式
基本信息
- 批准号:1162299
- 负责人:
- 金额:$ 34.5万
- 依托单位:
- 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:
- 关键词:
项目摘要
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.
局部 Artin 根数的存在是由 Langlands 和 Deligne(在某些情况下由 Dwork 更早)建立的,是在局部域 F 上 Weil-Deligne 群的 n 维连续表示与不可约容许表示之间建立局部 Langlands 对应关系的关键对象GL(n,F)。事实上,只有当 Weil 群表示的张量积所附加的根数和 L 函数等于由两个 GL(.,F) 的对应表示的 Rankin 乘积因子定义的根数和 L 函数后,才能获得唯一的对应关系。还有一些其他实例,这些对象被定义为 GL(n,F) 的表示,例如 GL(n,C) 的外部正方形和对称正方形表示,以及当 n 小于或等于 8 时的外部立方体朗兰兹-沙希迪方法。作为本提案的第一个主题,研究者将研究一种稳健的技术,该技术可用于通过对应关系为 Weil 小组定义的因素证明这些因素的相等性。所涉及的技术包括变形论证以及 Jacquet 和 Ye 提出的贝塞尔函数的广义 Shalika 胚芽展开,这似乎适合推广到其他组。他还将在他即将出版的书中使用亚瑟的结果来解决有关附加到亚瑟包的朗兰兹包的某些问题,以及经典群的某些算术问题(韦尔定律),以及它们对一般旋转群的推广。计算内窥镜方面的经典群的交织算子的残数是他多年来一直合作追求的,也应该受益于亚瑟的角色身份,他将作为该项目的一部分进行探索。他还将研究函子性的某些表征理论后果。 接下来,他将继续共同致力于通过 Langlands-Shahidi 方法研究 p 进 L 函数,并追求 Langlands 关于超越内窥镜和互易性的新思想,以及该方法对循环群和覆盖群的可能推广。提案涉及对研究生和博士后的培训,并包括针对他们的具体问题。研究者希望有几名新学生加入他和普渡大学数论小组的其他成员,并参与教授高级课程(例如,p-adic L-函数、自同构形式、实李群的表示论)来训练他们。 Artin L-函数理论及其与朗兰兹互易律(对应)的联系是数论中最美丽的部分之一,研究者希望不同水平的学生能够在不同的研讨会上进行研究。在另一个层面上,他参与组织会议并担任多个著名期刊和小组的编辑委员会成员。此外,他仍然参与指导和少数族裔招聘工作,目前在该系的毕业生招聘委员会任职,重点负责招聘女性和少数族裔学生。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Freydoon Shahidi其他文献
Freydoon Shahidi的其他文献
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{{ truncateString('Freydoon Shahidi', 18)}}的其他基金
L-functions, Fourier Transforms, and Gamma Factors
L 函数、傅立叶变换和伽玛因子
- 批准号:
1801273 - 财政年份:2018
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Langlands Reciprocity and Automorphic Forms
朗兰兹互易和自守形式
- 批准号:
1500759 - 财政年份:2015
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Problems in The Theory of Automorphic Forms and L-functions
自守形式和L-函数理论中的问题
- 批准号:
0700280 - 财政年份:2007
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Conference on Automorphic Forms and the Trace Formula; October 13-16, 2004; Toronto, Canada
自守形式和迹公式会议;
- 批准号:
0405874 - 财政年份:2004
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
Automorphic L-Functions and Langlands Functoriality
自同构 L 函数和朗兰兹函数性
- 批准号:
0200325 - 财政年份:2002
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Special Semester Program on Automorphic Forms, Shimura Varieties and L-functions; January 1-May 31, 2003, Fields Institute, Toronto, Canada
自守形式、志村簇和 L 函数特别学期课程;
- 批准号:
0211133 - 财政年份:2002
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
Shimura Varieties, the Trace Formula, Congruences and Galois Representations
志村簇、迹公式、同余式和伽罗瓦表示法
- 批准号:
0071404 - 财政年份:2000
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
Automorphic L-Functions, Endoscopy, and Representation Theory
自同构 L 函数、内窥镜检查和表示理论
- 批准号:
9970156 - 财政年份:1999
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
Mathematical Sciences: Automorphic L-functions and Interwining Operators
数学科学:自守 L 函数和交织算子
- 批准号:
9622585 - 财政年份:1996
- 资助金额:
$ 34.5万 - 项目类别:
Continuing Grant
Mathematical Sciences: Automorphic L-Functions and the Theory of Endoscopy
数学科学:自同构 L 函数和内窥镜理论
- 批准号:
9301040 - 财政年份:1993
- 资助金额:
$ 34.5万 - 项目类别:
Standard Grant
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