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Mathematical Surveys and Monographs Volume 201 Topological Modular Forms Christopher L. Douglas John Francis André G. Henriques Michael A. Hill Editors American Mathematical Society Topological Modular Forms http://dx.doi.org/10.1090/surv/201 Mathematical Surveys and Monographs Volume 201 Topological Modular Forms Christopher L. Douglas John Francis André G. Henriques Michael A. Hill Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 55N34, 55N35, 55N22, 57R77, 55T05, 55T15, 18G40, 14H52, 14D23, 13D03. For additional information and updates on this book, visit www.ams.org/bookpages/surv-201 Library of Congress Cataloging-in-Publication Data Topological modular forms / Christopher L. Douglas, John Francis, Andr´e G. Henriques, Michael A. Hill, editors. pages cm. — (Mathematical surveys and monographs ; volume 201) Includes bibliographical references. ISBN 978-1-4704-1884-7 (alk. paper) 1. Topological fields. 2. Algebraic fields. 3. Curves, Elliptic. I. Douglas, Christopher L., editor. II. Francis, John, 1982– editor. III. Henriques, Andr´e G. (Andr´e Gil), 1977– editor. IV. Hill, Michael A. (Michael Anthony), editor. QA611.T656 2014 514.23—dc23 2014029076 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Preface and Acknowledgments viii Introduction xi Part I 1 1. Elliptic genera and elliptic cohomology Corbett Redden 3 2. Elliptic curves and modular forms Carl Mautner 17 3. The moduli stack of elliptic curves Andre´ G. Henriques 25 4. The Landweber exact functor theorem Henning Hohnhold 35 5. Sheaves in homotopy theory Christopher L. Douglas 47 6. Bousfield localization and the Hasse square Tilman Bauer 79 7. The local structure of the moduli stack of formal groups Jacob Lurie 89 8. Goerss–Hopkins obstruction theory Vigleik Angeltveit 93 9. From spectra to stacks after Michael J. Hopkins 99 10. The string orientation after Michael J. Hopkins 109 11. The sheaf of E∞-ring spectra after Michael J. Hopkins 125 12. The construction of tmf Mark Behrens 131 13. The homotopy groups of tmf and of its localizations Andre´ G. Henriques 189 v vi CONTENTS Part II 207 Elliptic curves and stable homotopy I Michael J. Hopkins and Haynes R. Miller 209 From elliptic curves to homotopy theory Michael J. Hopkins and Mark Mahowald 261 K(1)-local E∞-ring spectra Michael J. Hopkins 287 Glossary 303 4 8 4 8 4 4 84 8 2 4 4 4 84 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 2 2 8 2 2 2 4 2 2 2 8 2 2 2 2 2 2 2 2 22 22 22 222 222 222 4 8 4 4 8 8 2 4 4 4 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 8 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 84 8 8 8 2 4 4 4 4 4 8 8 2 22 22 22 22 22 22 4 22 22 22 22 22 22 22 2 2 2 2 2 2 2 2 2 22 22 22 22 2 22 22 22 22 22 22 22 22 22 22 22 22 2 2 2 22 22 2 22 22 22 Preface This is a book about the theory of topological modular forms. It is also a record of the efforts of a group of graduate students to learn that theory at the 2007 Talbot Workshop, and so a book born of and steeped in the Talbot vision. In the fall of 2003, Mike Hopkins taught a course at MIT about tmf . Our gen- eration of Cambridge algebraic topologists, having survived and thrived in Haynes Miller’s Kan seminar, found in Mike’s class our next, and really our last common, mathematical crucible. The course hacked through the theory of algebraic modu- lar forms, formal groups, multiplicative stable homotopy theory, stacks, even more stacks, moduli stacks of elliptic curves, Bousfield localization, Morava K-andE- theory, the arithmetic and Hasse squares, Andr´e–Quillen cohomology, obstruction theory for moduli of associative and commutative ring spectra—by this point we were having dreams, or maybe nightmares, about the spiral exact sequence. In the middle of the course, we all flew over to M¨unster for a week-long workshop on tmf with lectures by Mike, Haynes, Matt Ando, Charles Rezk, and Paul Goerss. A transatlantic mix of students spent the late afternoons coaxing and cramming the knowledge in at a cafe off Steinfurter Strasse; there we devised a plan to reconvene and sketched a vision of what would become the Talbot Workshops: a gathering for graduate students, focused on a single topic of contemporary research interest, lectured by graduate students and guided by a single faculty mentor, having talks in the morning and in the evening and every afternoon free for discussion and outdoor activities, with participants sleeping and lecturing and cooking together under the same roof. We pitched it to Mike and Haynes and they agreed to back a ragtag summit. Talbot was born. Three years later, in 2007, we decided to bring Talbot home with a workshop on tmf , mentored by Mike Hopkins. Mike stopped by Staples on his way to the workshop and picked up a big red “That was easy” button. Throughout the work- shop, whenever he or anyone else completed a particularly epic spectral sequence computation or stacky decomposition, he’d hit the button and a scratchy electronic voice would remind us, “That was easy!” It became the workshop joke (for much of it was evidently not easy) and mantra (for shifting perspective, whether to mul- tiplicative stable homotopy or to stacky language or to a suitable localization, did make the intractable seem possible). This book is a record and expansion of the material covered in the Talbot 2007 workshop. Though the authors of the various chapters have brought their own expositional perspectives to bear (particularly heroically in the case of Mark Behrens), the contemporary material in this book is due to Mike Hopkins, Haynes Miller, and Paul Goerss, with contributions by Mark Mahowald, Matt Ando, and Charles Rezk. viii Acknowledgments We thank the participants of Talbot 2007 for their dedication and enthusi- asm during the workshop: Ricardo Andrade, Vigleik Angeltveit, Tilman Bauer, Mark Behrens, Thomas Bitoun, Andrew Blumberg, Ulrik Buchholtz, Scott Car- nahan, John Duncan, Matthew Gelvin, Teena Gerhardt, Veronique Godin, Owen Gwilliam, Henning Hohnhold, Valentina Joukhovitski, Jacob Lurie, Carl Mautner, Justin Noel, Corbett Redden, Nick Rozenblyum, and Samuel W¨uthrich. Special thanks to Corbett, Carl, Henning, Tilman, Jacob, and Vigleik for writing up their talks from the workshop. And super-special thanks to Mark for his enormous effort writing up his perspective on the construction of tmf , and for extensive support and assistance throughout the development of the book. Our appreciation and thanks also to Tilman, to Mark, and to Niko Naumann for substantial contributions to and help with the glossary, and to Paul Goerss, Charles Rezk, and David Gepner for suggestions and corrections. Our appreciation goes to Nora Ganter, who in 2003 ran a student seminar on tmf —it was our first sustained exposure to the subject; Nora also prepared the orig- inal literature list for the Talbot 2007 program. We also acknowledge the students who wrote up the ‘Course notes for elliptic cohomology’ based on Mike Hopkins’ 1995 course, and the students who wrote up ‘Complex oriented cohomology the- ories and the language of stacks’ based on Mike’s 1999 course—both documents, distributed through the topology underground, were helpful for us in the years leading up to and at Talbot 2007. The book would not have happened without Talbot, and the Talbot workshops would not have happened without the support of MIT and the NSF. Christopher Stark at the NSF was instrumental in us securing the first Talbot grant, DMS- 0512714, which funded the workshops from 2005 til 2008. MIT provided the facili- ties for the first workshop, at the university retreat Talbot House, and has provided continual logistical, administrative, and technical support for the workshops ever since.
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