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Graduate Texts in Mathematics 288 Graduate Texts in Mathematics Series Editors Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board Alejandro Adem, University of British Columbia David Eisenbud, University of California, Berkeley & MSRI Brian C. Hall, University of Notre Dame Patricia Hersh, University of Oregon J. F. Jardine, University of Western Ontario Jeffrey C. Lagarias, University of Michigan Eugenia Malinnikova, Stanford University Ken Ono, University of Virginia Jeremy Quastel, University of Toronto Barry Simon, California Institute of Technology Ravi Vakil, Stanford University Steven H. Weintraub, Lehigh University Melanie Matchett Wood, Harvard University Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. More information about this series at http://www.springer.com/series/136 John Voight Quaternion Algebras 123 John Voight Department of Mathematics Dartmouth College Hanover, NH, USA This book is an open access publication. The Open Access publication of this book was made possible in part by generous support from Dartmouth College via Faculty Research and Professional Development funds, a John M. Manley Huntington Award, and Dartmouth Library Open Access Publishing. ISSN 0072-5285 ISSN 2197-5612 (electronic) Graduate Texts in Mathematics ISBN 978-3-030-56692-0 ISBN 978-3-030-56694-4 (eBook) https://doi.org/10.1007/978-3-030-56694-4 Mathematics Subject Classification: 11E12, 11F06, 11R52, 11S45, 16H05, 16U60, 20H10 © The Editor(s) (if applicable) and The Author(s) 2021. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution- NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which per- mits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. 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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Goal Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ring theory, number theory, K-theory, group theory, geometric topology, Lie theory, functions of a complex variable, spectral theory of Riemannian manifolds, arithmetic geometry, representation theory, the Langlands program—and the list goes on. Quaternion algebras are especially fruitful to study because they often reflect some of the general aspects of these subjects, while at the same time they remain amenable to concrete argumentation. Moreover, quaternions often encapsulate unique features that are absent from the general theory (even as they provide motivation for it). With this in mind, the main goal in writing this text is to introduce a large subset of the above topics to graduate students interested in algebra, geometry, and number theory. To get the most out of reading this text, readers will likely want to have been exposed to some algebraic number theory, commutative algebra (e.g., module theory, localization, and tensor products), as well as the fundamentals of linear algebra, topology, and complex analysis. For certain sections, further experience with objects in differential geometry or arithmetic geometry (e.g., Riemannian manifolds and elliptic curves), may be useful. With these prerequisites in mind, I have endeavored to present the material in the simplest, motivated version—full of rich interconnections and illustrative examples—so even if the reader is missing a piece of background, it can be quickly filled in. Unfortunately, this text only scratches the surface of most of the topics covered in the book! In particular, some appearances of quaternion algebras in arithmetic geometry that are dear to me are absent, as they would substantially extend the length and scope of this already long book. I hope that the presentation herein will serve as a foundation upon which a detailed and more specialized treatment of these topics will be possible. I have tried to maximize exposition of ideas and minimize technicality: sometimes I allow a quick and dirty proof, but sometimes the “right level of generality” (where things can be seen most clearly) is pretty abstract. So my efforts have resulted in a level of exposition that is occasionally uneven jumping between sections. I consider this a feature of the book, and I hope that the reader will agree v vi Preface and feel free to skip around (see How to use this book below). I tried to “reboot” at the beginning of each part and again at the beginning of each chapter, to refresh our motivation. For researchers working with quaternion algebras, I have tried to collect results otherwise scattered in the literature and to provide some clarifications, corrections, and complete proofs in the hope that this text will provide a convenient reference. In order to provide these features, to the extent possible I have opted for an organizational pattern that is “horizontal” rather than “vertical”: the text has many chapters, each representing a different slice of the theory. I tried to compactify the text as much as possible, without sacrificing com- pleteness. There were a few occasions when I thought a topic could use further elaboration or has evolved from the existing literature, but did not want to over- burden the text; I collected these in a supplementary text Quaternion algebras companion, available at the website for the text at http://quatalg.org. As usual, each chapter also contains a number of exercises at the end, ranging from checking basic facts used in a proof to more difficult problems that stretch the reader. Exercises that are used in the text are marked by .. For a subset of exercises (including many of those marked with .), there are hints, comments, or a complete solution available online. How to Use This Book With apologies to Whitman, this book is large, it contains multitudes—and hope- fully, it does not contradict itself! There is no obligation to read the book linearly cover to cover, and the reader is encouraged to find their own path, such as the following. 1. For an introductory survey course on quaternion algebras, read just the introductory sections in each chapter, those labeled with ., and supplement with sections from the text when interested. These introductions usually contain motivation and a summary of the results in the rest of the chapter, and I often restrict the level of generality or make simplifying hypotheses so that the main ideas are made plain. The reader who wants to quickly and gently grab hold of the basic concepts may digest the book in this way. The instructor may desire to fill in some further statements or proofs to make for a one semester course: Chapters 1, 2, 11, 25, and 35 could be fruitfully read in their entirety. 2. For a minicourse in noncommutative algebra with emphasis on quaternion algebras, read just Part I. Such an early graduate course would have minimal prerequisites and in a semester could be executed at a considered pace; it would provide the foundation for further study in many possible directions. 3. For quaternion algebras and algebraic number theory, read Part I and II. This course would be a nice second-semester addition following a standard first-semester course in algebraic number theory, suitable for graduate stu- dents in algebra and number theory who are motivated to study quaternion Preface vii algebras as “noncommutative quadratic fields”. For a lighter course, Chapters 6, 20,and21 could be skipped, and the instructor may opt to cover only the introductory section of a chapter for reasons of time and interest. To reinforce concepts from algebraic number theory, special emphasis could be placed on Chapter 13 (where local division algebras are treated like local fields) and 18 (where maximal orders are treated like noncommutative Dedekind domains).

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