An Introduction to Automorphic Representations: With a View Toward Trace Formulae
Getz, Jayce R., Hahn, Heekyoung
- 出版商: Springer
- 出版日期: 2024-03-02
- 售價: $2,580
- 貴賓價: 9.5 折 $2,451
- 語言: 英文
- 頁數: 609
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 303141151X
- ISBN-13: 9783031411519
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商品描述
The goal of this textbook is to introduce and study automorphic representations, objects at the very core of the Langlands Program. It is designed for use as a primary text for either a semester or a year-long course, for the independent study of advanced topics, or as a reference for researchers. The reader is taken from the beginnings of the subject to the forefront of contemporary research. The journey provides an accessible gateway to one of the most fundamental areas of modern mathematics, with deep connections to arithmetic geometry, representation theory, harmonic analysis, and mathematical physics.
The first part of the text is dedicated to developing the notion of automorphic representations. Next, it states a rough version of the Langlands functoriality conjecture, motivated by the description of unramified admissible representations of reductive groups over nonarchimedean local fields. The next chapters develop the theory necessary to make the Langlands functoriality conjecture precise. Thus supercuspidal representations are defined locally, cuspidal representations and Eisenstein series are defined globally, and Rankin-Selberg L-functions are defined to give a link between the global and local settings. This preparation complete, the global Langlands functoriality conjectures are stated and known cases are discussed.
This is followed by a treatment of distinguished representations in global and local settings. The link between distinguished representations and geometry is explained in a chapter on the cohomology of locally symmetric spaces (in particular, Shimura varieties). The trace formula, an immensely powerful tool in the Langlands Program, is discussed in the final chapters of the book. Simple versions of the general relative trace formulae are treated for the first time in a textbook, and a wealth of related material on algebraic group actions is included. Outlines for several possible courses are provided in the Preface.
商品描述(中文翻譯)
本教科書的目標是介紹和研究自守表示,這是Langlands計劃的核心。它旨在作為一學期或一年課程的主要教材,用於獨立學習高級主題,或作為研究人員的參考資料。讀者將從這個主題的起點一直引領到當代研究的前沿。這個旅程提供了一個通往現代數學最基礎領域的可靠入口,與算術幾何、表示論、調和分析和數學物理有著深刻的聯繫。
本書的第一部分致力於發展自守表示的概念。接下來,它提出了Langlands函子性猜想的粗略版本,該猜想是由對非阿基米德局部域上的可分類可接受表示的描述所啟發的。接下來的章節發展了使Langlands函子性猜想具體化所需的理論。因此,局部地定義了超分類表示,全局地定義了尖峰表示和Eisenstein級數,並定義了Rankin-Selberg L-函數,以建立全局和局部環境之間的聯繫。完成這些準備後,將陳述全局Langlands函子性猜想並討論已知的情況。
接下來是對全局和局部環境中區別表示的研究。在一章中解釋了區別表示和幾何之間的聯繫,特別是局部對稱空間(尤其是Shimura變體)的上同調。Langlands計劃中一個極其強大的工具——追蹤公式,在本書的最後幾章中進行了討論。這是第一次在教科書中對一般相對追蹤公式進行了簡單的處理,並包含了大量關於代數群作用的相關材料。在前言中提供了幾個可能課程的大綱。
作者簡介
Jayce R. Getz is Associate Professor of Mathematics at Duke University. His research in number theory spans several subjects --- algebraic geometry, automorphic representations, harmonic analysis, and trace formulae. His monograph with M. Goresky (Institute for Advanced Studies), Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, was awarded the 2011 Ferran Sunyer i Balaguer Prize.
Heekyoung Hahn is Associate Research Professor of Mathematics at Duke University. Her research covers additive combinatorics, Langlands functoriality, and related issues in the representation theory of algebraic groups.作者簡介(中文翻譯)
Jayce R. Getz 是杜克大學的數學副教授。他在數論方面的研究涵蓋了多個主題,包括代數幾何、自守表示、調和分析和追蹤公式。他與M. Goresky(高等研究院)合著的專著《帶有交集同調和二次基底變換的希爾伯特模形式》獲得了2011年的Ferran Sunyer i Balaguer獎。
Heekyoung Hahn 是杜克大學的數學副研究員。她的研究涵蓋了加法組合學、Langlands函子性以及代數群表示論中的相關問題。