The arithmetic of quaternion algebras John Voight [email protected] June 18, 2014 Preface Goal In the response to receiving the 1996 Steele Prize for Lifetime Achievement [Ste96], Shimura describes a lecture given by Eichler: [T]he fact that Eichler started with quaternion algebras determined his course thereafter, which was vastly successful. In a lecture he gave in Tokyo he drew a hexagon on the blackboard and called its vertices clock- wise as follows: automorphic forms, modular forms, quadratic forms, quaternion algebras, Riemann surfaces, and algebraic functions. This book is an attempt to fill in the hexagon sketched by Eichler and to augment it further with the vertices and edges that represent the work of many algebraists, geometers, and number theorists in the last 50 years. Quaternion algebras sit prominently at the intersection of many mathematical subjects. They capture essential features of noncommutative ring theory, number the- ory, K-theory, group theory, geometric topology, Lie theory, functions of a complex variable, spectral theory of Riemannian manifolds, arithmetic geometry, representa- tion 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 argumenta- tion. With this in mind, we have two goals in writing this text. First, we hope to in- troduce a large subset of the above topics to graduate students interested in algebra, geometry, and number theory. We assume that students have been exposed to some algebraic number theory (e.g., quadratic fields), commutative algebra (e.g., module theory, localization, and tensor products), as well as the basics of linear algebra, topology, and complex analysis. For certain sections, further experience with objects in arithmetic geometry, such as elliptic curves, is useful; however, we have endeav- ored to present the material in a way that is motivated and full of rich interconnections i ii and examples, so that the reader will be encouraged to review any prerequisites with these examples in mind and solidify their understanding in this way. At the moment, one can find introductions for aspects of quaternion algebras taken individually, but there is no text that brings them together in one place and that draws the connections between them; we have tried to fill this gap. Second, we have written this text for researchers in these areas: we have collected results otherwise scattered in the liter- ature, provide some clarifications and corrections and complete proofs in the hopes that this text will provide a convenient reference in the future. In order to combine these features, we have opted for an organizational pattern that is “horizontal” rather than “vertical”: the text has many chapters, each represent- ing a different slice of the theory. Each chapter could be used in a (long) seminar afternoon or could fill a few hours of a semester course. To the extent possible, we have tried to make the chapters stand on their own (with explicit references to results used from previous chapters) so that they can be read based on the reader’s interests—hopefully the interdependence of the material will draw the reader in more deeply! The introductory section of each chapter contains motivation and a summary of the results contained therein, and we often restrict the level of generality and make simplifying hypotheses so that the main ideas are made plain. Hopefully the reader who is new to the subject will find these helpful as way to dive in. This book has three other features. First, as is becoming more common these days, paragraphs are numbered when they contain results that are referenced later on; we have opted not to put these always in a labelled environment (definition, theo- rem, proof, etc.) to facilitate the expositional flow of ideas, while at the same time we wished to remain precise about where and how results are used. Second, we have in- cluded in each chapter a section on “extensions and further reading”, where we have indicated some of the ways in which the author’s (personal) choice of presentation of the material naturally connects with the rest of the mathematical landscape. Our general rule (except the historical expository in Chapter 1) has been to cite specific results and proofs in the text where they occur, but to otherwise exercise restraint until this final section where we give tangential remarks, more general results, ad- ditional references, etc. Finally, in many chapters we have also included a section on algorithmic aspects, for those who want to pursue the computational side of the theory. And as usual, each section 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. For many of these exercises, there are hints at the end of the book; for any result that is used later, a complete argument is given. iii Acknowledgements This book began as notes from a course offered at McGill University in the Winter 2010 semester, entitled Computational Aspects of Quaternion Algebras and Shimura Curves. I would like to thank the members of my Math 727 class for their invaluable discussions and corrections: Dylan Attwell-Duval, Xander Faber, Luis Finotti, An- drew Fiori, Cameron Franc, Adam Logan, Marc Masdeu, Jungbae Nam, Aurel Page, Jim Parks, Victoria de Quehen, Rishikesh, Shahab Shahabi, and Luiz Takei. This course was part of the special thematic semester Number Theory as Applied and Ex- perimental Science organized by Henri Darmon, Eyal Goren, Andrew Granville, and Mike Rubinstein at the Centre de Recherche Mathematiques´ (CRM) in Montreal,´ Quebec,´ and the extended visit was made possible by the generosity of Dominico Grasso, dean of the College of Engineering and Mathematical Sciences, and Jim Burgmeier, chair of the Department of Mathematics and Statistics, at the University of Vermont. With gratitude, I acknowledge their support. The writing continued while the author was on sabbatical at the University of Cal- ifornia, Berkeley, sponsored by Ken Ribet. Several students attended these lectures and gave helpful feedback: Watson Ladd, Andrew Niles, Shelly Manber, Eugenia Rosu, Emmanuel Tsukerman, Victoria Wood, and Alex Youcis. My sabbatical from Dartmouth College for the Fall 2013 and Winter 2014 quarters was made possible by the efforts of Associate Dean David Kotz, and I thank him for his support. Further progress on the text was made in preparation for a minicourse on Brandt modules as part of Minicourses on Algebraic and Explicit Methods in Number Theory, organized by Cecile´ Armana and Christophe Delaunay at the Laboratoire de Mathematiques´ de Besanc¸on in Salins-les-Bains, France. It is somehow fitting that I would find myself composing this text while a faculty member at Dartmouth College, as the story of the quaternions has even woven its way into the history of mathematics at Dartmouth. The only mathematical output by a Dartmouth professor in the 19th century was by Arthur Sherburne Hardy, Ph.D., the author of an 1881 text on quaternions entitled Elements of quaternions [Har1881]. Brown describes it as “an adequate, if not inspiring text. It was something for Dart- mouth to offer a course in such an abstruse field, and the course was actually given a few times when a student and an instructor could be found simultaneously” [Bro61, p. 2]. On that note, many thanks go the participants in my Math 125 Quaternion al- gebras class at Dartmouth in Spring 2014: Daryl Deford, Tim Dwyer, Zeb Engberg, Michael Firrisa, Jeff Hein, Nathan McNew, Jacob Richey, Tom Shemanske, Scott Smedinghoff, and David Webb. (I can only hope that this book will receive better reviews!) Many thanks go to the others who offered helpful comments and corrections: France Dacar, Ariyan Javanpeykar, BoGwang Jeon, Chan-Ho Kim, Chao Li, Ben- iv jamin Linowitz, Nicole Sutherland, Joe Quinn, and Jiangwei Xue. The errors and omissions that remain are, of course, my own. I am profoundly grateful to those who offered their encouragement at various times during the writing of this book: Srinath Baba, Chantal David, Matthew Green- berg and Kristina Loeschner, David Michaels, and my mother Connie Voight. Thank you all! Finally, I would like to offer my deepest gratitude to my partner Brian Kennedy—this book would not have been possible without his patience and endur- ing support. To do [[Sections that are incomplete, or comments that need to be followed up on, are in red.]] Contents Contents v I Algebra 1 1 Introduction 3 1.1 Hamilton’s quaternions . 3 1.2 Algebra after the quaternions . 8 1.3 Modern theory . 11 1.4 Extensions and further reading . 13 Exercises . 13 2 Beginnings 15 2.1 Conventions . 15 2.2 Quaternion algebras . 15 2.3 Rotations . 18 2.4 Extensions and further reading . 23 Exercises . 23 3 Involutions 27 3.1 Conjugation . 27 3.2 Involutions . 28 3.3 Reduced trace and reduced norm . 30 3.4 Uniqueness and degree . 31 3.5 Quaternion algebras . 32 3.6 Extensions and further reading . 34 3.7 Algorithmic aspects . 35 Exercises . 37 4 Quadratic forms 41 v vi CONTENTS 4.1 Norm form . 41 4.2 Definitions . 43 4.3 Nonsingular standard involutions . 47 4.4 Isomorphism classes of quaternion algebras . 48 4.5 Splitting . 50 4.6 Conics . 53 4.7 Hilbert symbol . 54 4.8 Orthogonal groups . 54 4.9 Equivalence of categories . 56 4.10 Extensions and further reading . 56 4.11 Algorithmic aspects . 57 Exercises . 57 5 Quaternion algebras in characteristic 2 61 5.1 Separability and another symbol . 61 5.2 Characteristic 2 . 62 5.3 Quadratic forms and characteristic 2 .