Automorphic Representations and L-Functions for the General Linear Group: Volume 1 (Cambridge Studies in Advanced Mathematics, Series Number 129) 1st Edition by Dorian Goldfeld | (PDF) Free Download

Description

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. The authors keep definitions to a minimum and repeat them when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. They also include concrete examples of both global and local representations of GL(n), and present their associated L-functions. The theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several of the proofs are here presented for the first time, including Jacquet’s simple and elegant proof of the tensor product theorem. Finally, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises, this book will motivate students and researchers to begin working in this fertile field of research.

User’s Reviews

Editorial Reviews: Review “This book treats the subject without imposing many prerequisites, and the pace is leisurely throughout. The authors also make an effort to explain every term. Accordingly, students are likely to find it quite accessible.” Solomon Friedberg for Mathematical Reviews Book Description This modern, graduate-level textbook does not assume prior knowledge of representation theory. Includes numerous concrete examples and over 250 exercises. About the Author Dorian Goldfeld is a Professor in the Department of Mathematics at Columbia University, New York.Joseph Hundley is an Assistant Professor in the Department of Mathematics at Southern Illinois University, Carbondale. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a terrific introduction to automorphic representation theory.If you’d like to learn the basics of automorphic representations but have very little background then this is definitely the book for you. The authors make almost no assumptions about what you know (they define a representation!) yet they cover quite a bit of material. I enjoyed the stark style of the writing: brief and to the point. The proofs are very short and clean and many of them are previously unpublished.I’m so grateful the authors for their wonderful book!

⭐This is such an amazing, valuable introductory textbook on the theory of automorphic representations. I enjoyed reading it a lot and I highly recommend it to beginners in this subject. It is very nicely written, clear and down-to-earth.Even though the end-goal of this book is to learn the representation-theoretic language, the classical picture remains indispensable as it provides concrete examples. The author has provided a summary in a chapter or two in order to make the book self-contained. For readers that are not familiar with p-adic, adelic languages and Haar measures, the authors developed them from scratch.The authors often provide context and comment before discussing a proposition. They also break up longer proofs into steps. For each step, they state the goal first before diving into the argument. Having a glance at the goals first, the reader will have a good road map in mind and will know what to expect. The argument is always clear and complete. The authors have tried hard to make the material motivated. From time to time, in order to assist readers to grasp new concepts, the authors include details of verifying some concrete examples really meet the definitions.Let me move onto content. (1). This book discusses the Godement-Jacquet theory extensively with remarkable clarity. This is not found in most literature. They have carried out both the local and global theory, calculations at both archimedean and non-archimedean places, and for each type of irreducible admissible representations, with full detail!(2). The part about the asymptotics of matrix coefficients (Chp. 8), especially the archimedean version, is amazingly informative and once again it can hardly be located in literature.(3). It includes the new, simple proofs of many deep but fundamental results in the setting of GL(2). Just to name a few, Bernstein-Zelevinsky’s theorem (irreducible smooth representation is admissible), Tensor Product Theorem, rapid decay of cusp forms, subquotient = subrepresentation for cuspidal automorphic representations.(4). Vol. 2 of the book moves on to discuss the theory for the high-rank case (i.e., GL(n) for n>=3). Although it is much shorter than vol. 1, it does much more than simply generalizing the argument for GL(2). The theory of the high-rank case is indeed much deeper and there is no way to include every single detail like vol 1 (say the Bernstein-Zelevinsky classification of irreducible smooth reps). However, the authors did a wonderful job of summarizing the results (mostly related to classification of reps), which are scattered throughout literature, in one place and stating them in an explicit and easy-to-understand manner. They also provide useful comment and pinpoint the references in which the interested readers can find the proofs of the results.There is one small thing the authors could have done better. It is better to use only one set of convention for principal series representations throughout the book. (The version in Chp 6 is unnormalized).

⭐good

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