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Topological Modular Forms

Edited by: Christopher L. Douglas Oxford University, Oxford, United Kingdom
John Francis Northwestern University, Evanston, IL
André G. Henriques Utrecht University, Utrecht, Netherlands
Michael A. Hill University of Virginia, Charlottesville, VA
Available Formats:
Hardcover ISBN: 978-1-4704-1884-7
Product Code: SURV/201
List Price: $106.00 MAA Member Price:$95.40
AMS Member Price: $84.80 Electronic ISBN: 978-1-4704-2002-4 Product Code: SURV/201.E List Price:$100.00
MAA Member Price: $90.00 AMS Member Price:$80.00
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List Price: $159.00 MAA Member Price:$143.10
AMS Member Price: $127.20 Click above image for expanded view Topological Modular Forms Edited by: Christopher L. Douglas Oxford University, Oxford, United Kingdom John Francis Northwestern University, Evanston, IL André G. Henriques Utrecht University, Utrecht, Netherlands Michael A. Hill University of Virginia, Charlottesville, VA Available Formats:  Hardcover ISBN: 978-1-4704-1884-7 Product Code: SURV/201  List Price:$106.00 MAA Member Price: $95.40 AMS Member Price:$84.80
 Electronic ISBN: 978-1-4704-2002-4 Product Code: SURV/201.E
 List Price: $100.00 MAA Member Price:$90.00 AMS Member Price: $80.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$159.00 MAA Member Price: $143.10 AMS Member Price:$127.20
• Book Details

Mathematical Surveys and Monographs
Volume: 2012014; 318 pp
MSC: Primary 55; 57; 18; 14; 13;

The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory.

This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss–Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Graduate students and research mathematicians interested in algebraic topology and the arithmetic of modular forms.

• Part I
• Corbett Redden - Chapter 1. Elliptic genera and elliptic cohomology
• Carl Mautner - Chapter 2. Ellliptic curves and modular forms
• André G. Henriques - Chapter 3. The moduli stack of elliptic curves
• Henning Hohnhold - Chapter 4. The Landweber exact functor theorem
• Christopher L. Douglas - Chapter 5. Sheaves in homotopy theory
• Tilman Bauer - Chapter 6. Bousfield localization and the Hasse square
• Jacob Lurie - Chapter 7. The local structure of the moduli stack of formal groups
• Vigleik Angeltveit - Chapter 8. Goerss–Hopkins obstruction theory
• Michael J. Hopkins - Chapter 9. From spectra to stacks
• Michael J. Hopkins - Chapter 10. The string orientation
• Michael J. Hopkins - Chapter 11. The sheaf of $E_\infty$-ring spectra
• Mark Behrens - Chapter 12. The construction of $\mathit {tmf}$
• André G. Henriques - Chapter 13. The homotopy groups of $\mathit {tmf}$ and of its localizations
• Part II
• Michael J. Hopkins and Haynes R. Miller - Ellitpic curves and stable homotopy I
• Michael J. Hopkins and Mark Mahowald - From elliptic curves to homotopy theory
• Michael J. Hopkins - $K(1)$-local $E_\infty$-ring spectra

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Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2012014; 318 pp
MSC: Primary 55; 57; 18; 14; 13;

The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory.

This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss–Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Graduate students and research mathematicians interested in algebraic topology and the arithmetic of modular forms.

• Part I
• Corbett Redden - Chapter 1. Elliptic genera and elliptic cohomology
• Carl Mautner - Chapter 2. Ellliptic curves and modular forms
• André G. Henriques - Chapter 3. The moduli stack of elliptic curves
• Henning Hohnhold - Chapter 4. The Landweber exact functor theorem
• Christopher L. Douglas - Chapter 5. Sheaves in homotopy theory
• Tilman Bauer - Chapter 6. Bousfield localization and the Hasse square
• Jacob Lurie - Chapter 7. The local structure of the moduli stack of formal groups
• Vigleik Angeltveit - Chapter 8. Goerss–Hopkins obstruction theory
• Michael J. Hopkins - Chapter 9. From spectra to stacks
• Michael J. Hopkins - Chapter 10. The string orientation
• Michael J. Hopkins - Chapter 11. The sheaf of $E_\infty$-ring spectra
• Mark Behrens - Chapter 12. The construction of $\mathit {tmf}$
• André G. Henriques - Chapter 13. The homotopy groups of $\mathit {tmf}$ and of its localizations
• Part II
• Michael J. Hopkins and Haynes R. Miller - Ellitpic curves and stable homotopy I
• Michael J. Hopkins and Mark Mahowald - From elliptic curves to homotopy theory
• Michael J. Hopkins - $K(1)$-local $E_\infty$-ring spectra
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