Hardcover ISBN:  9781470418847 
Product Code:  SURV/201 
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Electronic ISBN:  9781470420024 
Product Code:  SURV/201.E 
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Book DetailsMathematical Surveys and MonographsVolume: 201; 2014; 318 ppMSC: 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 algebrogeometric 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.ReadershipGraduate students and research mathematicians interested in algebraic topology and the arithmetic of modular forms.

Table of Contents

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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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 algebrogeometric 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