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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. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. This work is subject to copyright. All commercial rights are reserved by the author(s), whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Regarding these commercial rights a non-exclusive license has been granted to the publisher. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi- cation does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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 , 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).

There are also more specialized options, beginning with the introductory sec- tions in Part I and continuing as follows:

4. For quaternion algebras and analytic number theory, continue with the introductory sections in Part II (just Chapters 9–17), and then cover Part III (at least through Chapter 29). This course could follow a first-semester course in analytic number theory, enriching students' understanding of zeta functions and L-functions (roughly speaking, beginning the move from GL1 to GL2). The additional prerequisite of real analysis (measure theory) is recommended. Optionally, this course could break after Chapter 26 to avoid adeles, and perhaps resume in an advanced topics course with the remaining chapters. 5. For quaternionic applications to geometry (specifically, hyperbolic geometry and low-dimensional topology), continue with the introductory sections in Part II (through Chapter 14), and then cover Part IV (optionally skipping Chapter 32). 6. For an advanced course on quaternion algebras and arithmetic geometry, continue with Part II, the introductory sections in Parts IV and V. Chapter 41 could be read immediately after Part II. This path is probably most appro- priate for an advanced course for students with some familiarity with mod- ular forms and some hyperbolic geometry, and Chapter 42 is probably only meaningful for students with a background in elliptic curves (though the relevant concepts are reviewed at the start). 7. Finally, for the reader who is studying quaternion algebras with an eye to applications with supersingular elliptic curves, the reader may follow Chapters 2–4, 9–10, 13–14, 16–17, 23, then the main event in Chapter 42. For further reading on quaternion orders and ternary quadratic forms, I suggest Chapters 5, 22, and 24.

Sections of the text that are more advanced (requiring more background) or those may be omitted are labeled with Ã. The final chapter (Chapter 43) is neces- sarily more advanced, and additional prerequisites in algebraic and arithmetic geometry are indicated. It is a unique feature of quaternion algebras that topics overlap and fold together like this, and so I hope the reader will forgive the length of the book. The reader viii Preface may find the symbol definition list at the end to help in identifying unfamiliar notation. Finally, to ease in location, I have chosen to number all objects (theorem-like environments, equations, and figures) consecutively.

Companion Reading

Several general texts can serve as companion reading for this monograph: • The lecture notes of Vignéras [Vig80a] have been an essential reference for the arithmetic of quaternion algebras since their publication. • The seminal text by Reiner [Rei2003] on maximal orders treats many intro- ductory topics that overlap this text. • The book of Maclachlan–Reid [MR2003] gives an introduction to quaternion algebras with application to the geometry of 3-manifolds. • The book by Deuring [Deu68] (in German) develops the theory of algebras over fields, culminating in the treatment of zeta functions of division algebras over the rationals, and may be of historical interest as well. • Finally, Pizer [Piz76a] and Alsina–Bayer [AB2004] present arithmetic and algorithmic aspects of quaternion algebras over Q.

Hanover, NH, USA John Voight 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, Andrew 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 Experimental Science organized by Henri Darmon, Eyal Goren, Andrew Granville, and Mike Rubinstein at the Centre de Recherche Mathématiques (CRM) in Montréal, Québec, 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. I am very grateful for their support. After a long hiatus, the writing continued while the author was on sabbatical at the University of California, Berkeley. Several students attended these lectures and gave helpful feedback: Watson Ladd, Andrew Niles, Shelly Manber, Eugenia Ro su, Emmanuel Tsukerman, Victoria Wood, and Alex Youcis. Thanks to Ken Ribet for sponsoring my visit and helping to organize these lectures. 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. During this sabbatical, 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 Cécile Armana and Christophe Delaunay at the Laboratoire de Mathématiques de Besançon in Salins-les-Bains, France. Thanks also go to the patient participants in my Math 125 Quaternion algebras 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) and my Math 125 Geometry of discrete groups in Summer 2015 (Angelica Babei, Ben Breen, Sara Chari, Melanie Dennis, James Drain, Tim Dwyer, Jon Epstein, David Freund, Sam Kater, Michael Musty, Nicolas

ix x Acknowledgements

Petit, Sam Schiavone, Scott Smedinghoff, and Everett Sullivan). I am also grateful for the feedback given during my course at the Institute for Computational and Experimental Research in Mathematics (ICERM) at in Fall 2015, as part of the Semester Program on Computational Aspects of the Langlands Program: in particular, thanks to Henri Cohen, Edgar Costa, David Farmer, Winnie Li, David Roberts, and Drew Sutherland for their comments and questions. During the preparation of this monograph, I was supported by the NSF Division of Mathematical Sciences Awards for Quaternion algebras, Shimura curves, and modular forms: Algorithms and arithmetic (DMS-0901971, July 2009–August 2011) and CAREER: Explicit methods in arithmetic geometry (DMS-1151047, July 2012–July 2018), and Simons Collaborations in Mathematics and the Physical Sciences Award, Arithmetic geometry, number theory, and computation (from September 2017). 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. It is somehow fitting that I would find myself writing this text while a faculty member at Dartmouth College: the story of the quaternions is interwoven with history of mathematics at Dartmouth. The only mathematical output by a Dartmouth professor in the nineteenth 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 Dartmouth 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]. I can only hope that this book will receive better reviews! On a more personal note, I have benefited from the insight of incredible teachers and collaborators throughout my mathematical life, and their positive impact can hopefully be seen in this manuscript: among many, I would like to give special thanks to Pete L. Clark, Bas Edixhoven, , Hendrik Lenstra, and for their inspiration and guidance over the years. Many thanks go to the others who offered helpful comments, corrections, answered my questions, and provided feedback: Konstantin Ardakov, Sarah Arpin, Asher Auel, Andreas Bächle, John Boxall, Ben Breen, Juliusz Brzezinski, Billy Chan, Pierre Clare, France Dacar, Lassina Dembélé, Juanita Duque Rosero, Kirsten Eisentraeger, John Enns, Zak Evans, Steven Galbraith, Darij Grinberg, Joseph Gunther, Jonathan Hanke, Thorsten Herrig, Aleksander Horawa, Will Jagy, Ariyan Javanpeykar, BoGwang Jeon, Chan-Ho Kim, Robert Kucharczyk, Joel Laity, Dion Leijnse, Patrick Lenning, Chao Li, Wen-Wei Li, Benjamin Linowitz, Travis Morrison, Ariel Pacetti, Lorenz Panny, Jennifer Park, Carl Pomerance, Jose Pujol, Matthieu Rambaud, John Riccardi, Sam Roven, Nithi Rungtanapirom, Ciaran Schembri, sibilant, Jana Sotakova, Tom Shemanske, Daniel Smertnig, Jim Stankewicz, Kate Stange, Drew Sutherland, Nicole Sutherland, Jacob Swenberg, Anne van Weerden, Annalena Wernz, Mckenzie West, Dana Williams, Jiangwei Acknowledgements xi

Xue, and Vitthal Yellambalse. I am indebted to C. Emily I. Redelmeier for fabulous Technical assistance with figures. Special thanks to the anonymous referees for their constructive feedback and to Sara Chari, Peter Landweber, Joe Quinn, and Grant Molnar for a careful reading. The errors and omissions that remain are, of course, my own: please contact me at [email protected] if you find mistakes or have other suggestions (e.g., additional references, alternate proofs, further exer- cises or applications). Collected errata and a corrected version will be posted on the webpage for the text (http://quatalg.org). I am profoundly grateful to those who lent encouragement at various times during the writing of this book, when the going was tough: Srinath Baba, Chantal David, Matthew Greenberg and Kristina Loeschner, Laurie Johnson at the National Center for Faculty Development and Diversity (NCFDD), David Michaels, Tom Shemanske, and my mother Connie Voight. Thank you all! I am also grateful to Marc Strauss and Loretta Bartolini, editors at Springer, for their professional guidance. Finally, I would like to offer my deepest gratitude to my partner Brian Kennedy: this book would not have been possible without his patience, equanimity, and wit. As they say, happiness is a journey, not a destination; the same I think is true for happy mathematics, and I am blessed to have had both personal and professional companionship on this one. Contents

Contents...... xiii

1 Introduction ...... 1 1.1 Hamilton’s quaternions ...... 1 1.2 Algebra after the quaternions ...... 6 1.3 Quadratic forms and arithmetic ...... 10 1.4 Modular forms and geometry ...... 11 1.5 Conclusion ...... 15 Exercises ...... 16

I Algebra

2 Beginnings...... 21 2.1 . Conventions ...... 21 2.2 . Quaternion algebras ...... 22 2.3 . Matrix representations ...... 24 2.4 . Rotations ...... 26 Exercises ...... 30

3 Involutions ...... 35 3.1 . Conjugation ...... 35 3.2 Involutions ...... 36 3.3 Reduced trace and reduced norm ...... 38 3.4 Uniqueness and degree ...... 39 3.5 Quaternion algebras ...... 40 Exercises ...... 43

4 Quadratic forms ...... 47 4.1 . Reduced norm as quadratic form ...... 47 4.2 Basic definitions ...... 48 4.3 Discriminants, nondegeneracy ...... 52

xiii xiv CONTENTS

4.4 Nondegenerate standard involutions ...... 54 4.5 Special orthogonal groups ...... 55 Exercises ...... 59

5 Ternary quadratic forms and quaternion algebras ...... 65 5.1 . Reduced norm as quadratic form ...... 65 5.2 Isomorphism classes of quaternion algebras ...... 66 5.3 Clifford algebras ...... 69 5.4 Splitting ...... 73 5.5 Conics, embeddings ...... 76 5.6 Orientations ...... 77 Exercises ...... 81

6 Characteristic 2 ...... 85 6.1 Separability ...... 85 6.2 Quaternion algebras ...... 86 6.3 Ã Quadratic forms ...... 87 6.4 Ã Characterizing quaternion algebras ...... 89 Exercises ...... 93

7 Simple algebras ...... 95 7.1 . Motivation and summary ...... 95 7.2 Simple modules ...... 97 7.3 Semisimple modules and the Wedderburn–Artin theorem .... 101 7.4 Jacobson radical ...... 103 7.5 Central simple algebras ...... 105 7.6 Quaternion algebras ...... 107 7.7 The Skolem–Noether theorem ...... 108 7.8 Reduced trace and norm, universality ...... 112 7.9 Separable algebras ...... 115 Exercises ...... 117

8 Simple algebras and involutions ...... 123 8.1 . The Brauer group and involutions ...... 123 8.2 Biquaternion algebras ...... 124 8.3 Brauer group ...... 126 8.4 Positive involutions ...... 128 8.5 Ã Endomorphism algebras of abelian varieties ...... 131 Exercises ...... 133 CONTENTS xv

II Arithmetic

9 Lattices and integral quadratic forms ...... 139 9.1 . Integral structures ...... 139 9.2 Bits of commutative algebra ...... 140 9.3 Lattices...... 141 9.4 Localizations ...... 143 9.5 Completions ...... 145 9.6 Index ...... 147 9.7 Quadratic forms ...... 148 9.8 Normalized form ...... 150 Exercises ...... 152

10 Orders ...... 155 10.1 . Lattices with multiplication ...... 155 10.2 Orders ...... 156 10.3 Integrality ...... 158 10.4 Maximal orders ...... 159 10.5 Orders in a matrix ring ...... 160 Exercises ...... 161

11 The Hurwitz order ...... 165 11.1 . The Hurwitz order ...... 165 11.2 . Hurwitz units ...... 166 11.3 . Euclidean algorithm ...... 169 11.4 . Unique factorization ...... 170 11.5 Finite quaternionic unit groups ...... 172 Exercises ...... 176

12 Ternary quadratic forms over local fields ...... 181 12.1 . The p-adic numbers and local quaternion algebras ...... 181 12.2 Local fields ...... 184 12.3 Classification via quadratic forms ...... 189 12.4 Hilbert symbol ...... 193 Exercises ...... 196

13 Quaternion algebras over local fields ...... 201 13.1 Extending the valuation ...... 201 13.2 Valuations ...... 202 13.3 Classification via extensions of valuations ...... 205 13.4 Consequences ...... 208 13.5 Some topology ...... 210 Exercises ...... 212 xvi CONTENTS

14 Quaternion algebras over global fields ...... 217 14.1 . Ramification ...... 217 14.2 . Hilbert reciprocity over the rationals ...... 219 14.3 . Hasse–Minkowski theorem over the rationals ...... 223 14.4 Global fields ...... 227 14.5 Ramification and discriminant ...... 230 14.6 Quaternion algebras over global fields ...... 231 14.7 Theorems on norms...... 233 Exercises ...... 235

15 Discriminants ...... 241 15.1 . Discriminantal notions ...... 241 15.2 Discriminant ...... 242 15.3 Quadratic forms ...... 245 15.4 Reduced discriminant ...... 246 15.5 Maximal orders and discriminants ...... 248 15.6 Duality ...... 249 Exercises ...... 253

16 Quaternion ideals and invertibility ...... 257 16.1 . Quaternion ideals ...... 257 16.2 Locally principal, compatible lattices ...... 259 16.3 Reduced norms ...... 261 16.4 Algebra and absolute norm ...... 263 16.5 Invertible lattices ...... 265 16.6 Invertibility with a standard involution ...... 268 16.7 One-sided invertibility ...... 271 16.8 Invertibility and the codifferent ...... 273 Exercises ...... 274

17 Classes of quaternion ideals ...... 277 17.1 . Ideal classes ...... 277 17.2 Matrix ring ...... 279 17.3 Classes of lattices ...... 280 17.4 Types of orders ...... 281 17.5 . Finiteness of the class set: over the integers ...... 284 17.6 . Example ...... 285 17.7 Finiteness of the class set: over number rings ...... 287 17.8 Eichler’s theorem ...... 291 Exercises ...... 293 CONTENTS xvii

18 Two-sided ideals and the Picard group ...... 297 18.1 . Noncommutative Dedekind domains ...... 297 18.2 Prime ideals ...... 299 18.3 Invertibility ...... 300 18.4 Picard group ...... 303 18.5 Classes of two-sided ideals ...... 305 Exercises ...... 307

19 Brandt groupoids ...... 311 19.1 . Composition laws and ideal multiplication ...... 311 19.2 Example ...... 314 19.3 Groupoid structure ...... 315 19.4 Brandt groupoid ...... 318 19.5 Brandt class groupoid ...... 319 19.6 Quadratic forms ...... 321 Exercises ...... 322

20 Integral representation theory ...... 325 20.1 . Projectivity, invertibility, and representation theory ...... 325 20.2 Projective modules ...... 327 20.3 Projective modules and invertible lattices ...... 328 20.4 Jacobson radical ...... 330 20.5 Local Jacobson radical ...... 332 20.6 Integral representation theory ...... 333 20.7 Ã Stable class group and cancellation ...... 335 Exercises ...... 340

21 Hereditary and extremal orders ...... 343 21.1 . Hereditary and extremal orders ...... 343 21.2 Extremal orders ...... 344 21.3 Ã Explicit description of extremal orders ...... 346 21.4 Hereditary orders ...... 348 21.5 Ã Classification of local hereditary orders ...... 350 Exercises ...... 351

22 Quaternion orders and ternary quadratic forms ...... 353 22.1 . Quaternion orders and ternary quadratic forms ...... 353 22.2 Even Clifford algebras ...... 356 22.3 Even Clifford algebra of a ternary quadratic module ...... 359 22.4 Over a PID ...... 364 22.5 Twisting and final bijection ...... 368 Exercises ...... 371 xviii CONTENTS

23 Quaternion orders ...... 375 23.1 . Highlights of quaternion orders ...... 375 23.2 Maximal orders ...... 377 23.3 Hereditary orders ...... 378 23.4 Eichler orders ...... 382 23.5 Bruhat–Tits tree ...... 386 Exercises ...... 389

24 Quaternion orders: second meeting ...... 393 24.1 . Advanced quaternion orders ...... 393 24.2 Gorenstein orders ...... 394 24.3 Eichler symbol ...... 400 24.4 Chains of orders ...... 403 24.5 Bass and basic orders ...... 406 24.6 Tree of odd Bass orders ...... 409 Exercises ...... 410

III Analysis

25 The Eichler mass formula ...... 415 25.1 . Weighted class number formula ...... 415 25.2 . Imaginary quadratic class number formula ...... 416 25.3 . Eichler mass formula: over the rationals ...... 420 25.4 Class number one and type number one ...... 423 Exercises ...... 425

26 Classical zeta functions ...... 431 26.1 . Eichler mass formula ...... 431 26.2 Analytic class number formula ...... 433 26.3 Classical zeta functions of quaternion algebras ...... 436 26.4 Counting ideals in a maximal order ...... 438 26.5 Eichler mass formula: maximal orders ...... 441 26.6 Eichler mass formula: general case ...... 444 26.7 Class number one ...... 447 26.8 Functional equation and classification ...... 447 Exercises ...... 452

27 Adelic framework ...... 457 27.1 . The rational adele ring ...... 457 27.2 . The rational idele group ...... 460 27.3 . Rational quaternionic adeles and ideles ...... 462 27.4 Adeles and ideles ...... 463 27.5 Class field theory ...... 465 CONTENTS xix

27.6 Noncommutative adeles ...... 468 27.7 Reduced norms ...... 472 Exercises ...... 473

28 Strong approximation ...... 477 28.1 . Beginnings ...... 477 28.2 . Strong approximation for SL2ðQÞ ...... 479 28.3 Elementary matrices ...... 482 28.4 Strong approximation and the ideal class set ...... 483 28.5 Statement and first applications ...... 484 28.6 Further applications ...... 487 28.7 First proof ...... 490 28.8 Second proof ...... 493 28.9 Ã Normalizer groups ...... 495 28.10 Ã Stable class group ...... 498 Exercises ...... 499

29 Idelic zeta functions ...... 503 29.1 . Poisson summation and the Riemann zeta function ...... 503 29.2 . Idelic zeta functions, after Tate ...... 505 29.3 Measures ...... 507 29.4 Modulus and Fourier inversion ...... 509 29.5 Local measures and zeta functions: archimedean case ...... 511 29.6 Local measures: commutative nonarchimedean case ...... 514 29.7 Local zeta functions: nonarchimedean case ...... 517 29.8 Idelic zeta functions ...... 521 29.9 Convergence and residue ...... 525 29.10 Main theorem ...... 527 29.11 Tamagawa numbers ...... 535 Exercises ...... 537

30 Optimal embeddings ...... 541 30.1 . Representation numbers ...... 541 30.2 Sums of three squares ...... 543 30.3 Optimal embeddings ...... 545 30.4 Counting embeddings, idelically: the trace formula ...... 547 30.5 Local embedding numbers: maximal orders ...... 551 30.6 Ã Local embedding numbers: Eichler orders ...... 554 30.7 Global embedding numbers ...... 559 30.8 Class number formula ...... 560 30.9 Type number formula ...... 563 Exercises ...... 565 xx CONTENTS

31 Selectivity ...... 569 31.1 Selective orders ...... 569 31.2 Selectivity conditions ...... 572 31.3 Ã Selectivity setup ...... 573 31.4 Ã Outer selectivity inequalities ...... 576 31.5 Ã Middle selectivity equality ...... 577 31.6 Ã Optimal selectivity conclusion...... 579 31.7 Ã Selectivity, without optimality ...... 580 31.8 Ã Isospectral, nonisometric manifolds ...... 582 Exercises ...... 585

IV Geometry and topology

32 Unit groups ...... 591 32.1 . Quaternion unit groups ...... 591 32.2 Structure of units ...... 592 32.3 Units in definite quaternion orders ...... 594 32.4 Finite subgroups of quaternion unit groups ...... 596 32.5 Cyclic subgroups ...... 597 32.6 Ã Dihedral subgroups ...... 599 32.7 Ã Exceptional subgroups ...... 601 Exercises ...... 602

33 Hyperbolic plane ...... 605 33.1 . The beginnings of hyperbolic geometry ...... 605 33.2 Geodesic spaces ...... 606 33.3 Upper half-plane ...... 608 33.4 Classification of isometries ...... 611 33.5 Geodesics ...... 614 33.6 Hyperbolic area and the Gauss–Bonnet formula ...... 616 33.7 Unit disc and Lorentz models ...... 618 33.8 Riemannian geometry ...... 621 Exercises ...... 624

34 Discrete group actions ...... 629 34.1 Topological group actions ...... 629 34.2 . Summary of results ...... 632 34.3 Covering space and wandering actions ...... 632 34.4 Hausdorff quotients and proper group actions ...... 634 34.5 Proper actions on a locally compact space ...... 637 34.6 Symmetric space model ...... 639 CONTENTS xxi

34.7 Fuchsian groups ...... 640 34.8 Riemann uniformization and orbifolds ...... 642 Exercises ...... 645

35 Classical modular group ...... 649 35.1 . The fundamental set ...... 649 35.2 Binary quadratic forms ...... 655 35.3 Moduli of lattices ...... 657 35.4 Congruence subgroups ...... 658 Exercises ...... 660

36 Hyperbolic space ...... 663 36.1 Hyperbolic space...... 663 36.2 Isometries ...... 665 36.3 Unit ball, Lorentz, and symmetric space models ...... 669 36.4 Bianchi groups and Kleinian groups ...... 671 36.5 Hyperbolic volume ...... 672 36.6 Picard modular group ...... 676 Exercises ...... 680

37 Fundamental domains ...... 685 37.1 . Dirichlet domains for Fuchsian groups ...... 685 37.2 . Ford domains ...... 688 37.3 . Generators and relations ...... 690 37.4 Dirichlet domains ...... 695 37.5 Hyperbolic Dirichlet domains ...... 699 37.6 Poincaré’s polyhedron theorem ...... 700 37.7 Signature of a Fuchsian group ...... 703 37.8 The (6, 4, 2)-triangle group ...... 704 37.9 Unit group for discriminant 6 ...... 707 Exercises ...... 712

38 Quaternionic arithmetic groups ...... 715 38.1 . Rational quaternion groups ...... 715 38.2 Isometries from quaternionic groups ...... 717 38.3 Discreteness ...... 719 38.4 Compactness and finite generation ...... 721 38.5 Ã Arithmetic groups, more generally ...... 723 38.6 Ã Modular curves, seen idelically ...... 724 38.7 Ã Double cosets ...... 726 Exercises ...... 729 xxii CONTENTS

39 Volume formula ...... 731 39.1 . Statement...... 731 39.2 Volume setup ...... 734 39.3 Volume derivation...... 736 39.4 Genus formula ...... 737 Exercises ...... 741

V Arithmetic geometry

40 Classical modular forms ...... 745 40.1 . Functions on lattices ...... 745 40.2 . Eisenstein series as modular forms ...... 749 40.3 . Classical modular forms ...... 752 40.4 Theta series ...... 755 40.5 Hecke operators ...... 756 Exercises ...... 758

41 Brandt matrices ...... 763 41.1 . Brandt matrices, neighbors, and modular forms ...... 763 41.2 Brandt matrices ...... 767 41.3 Commutativity of Brandt matrices ...... 770 41.4 Semisimplicity ...... 773 41.5 Eichler trace formula ...... 775 Exercises ...... 779

42 Supersingular elliptic curves ...... 783 42.1 Supersingular elliptic curves ...... 783 42.2 Supersingular isogenies ...... 787 42.3 Equivalence of categories ...... 791 42.4 Supersingular endomorphism rings ...... 793 Exercises ...... 796

43 QM abelian surfaces ...... 799 43.1 . QM abelian surfaces ...... 799 43.2 . QM by discriminant 6 ...... 802 43.3 Genus 2 curves ...... 806 43.4 Complex abelian varieties ...... 809 43.5 Complex abelian surfaces ...... 814 43.6 Abelian surfaces with QM ...... 817 43.7 Real points, CM points ...... 823 43.8 Ã Canonical models ...... 824 43.9 Ã Modular forms ...... 826 Exercises ...... 828 CONTENTS xxiii

Symbol Definition List ...... 831

Bibliography ...... 839

Index ...... 873