**Mathematical Surveys and Monographs**

Volume: 201;
2014;
318 pp;
Hardcover

MSC: Primary 55; 57; 18; 14; 13;

Print ISBN: 978-1-4704-1884-7

Product Code: SURV/201

List Price: $100.00

Individual Member Price: $80.00

**Electronic ISBN: 978-1-4704-2002-4
Product Code: SURV/201.E**

List Price: $100.00

Individual Member Price: $80.00

#### Supplemental Materials

# Topological Modular Forms

Share this page *Edited by *
*Christopher L. Douglas; John Francis; André G. Henriques; Michael A. Hill*

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.

#### Readership

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

#### Table of Contents

# Table of Contents

## Topological Modular Forms

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface and Acknowledgements vii8 free
- Introduction xi12 free
- Chapter 1. Elliptic genera and elliptic cohomology 336 free
- Chapter 2. Ellliptic curves and modular forms 1750
- Chapter 3. The moduli stack of elliptic curves 2558
- Chapter 4. The Landweber exact functor theorem 3568
- Chapter 5. Sheaves in homotopy theory 4780
- Chapter 6. Bousfield localization and the Hasse square 79112
- Chapter 7. The local structure of the moduli stack of formal groups 89122
- Chapter 8. Goerss–Hopkins obstruction theory 93126
- Chapter 9. From spectra to stacks 99132
- Chapter 10. The string orientation 109142
- Chapter 11. The sheaf of 𝐸_{∞}-ring spectra 125158
- Chapter 12. The construction of tmf 131164
- Chapter 13. The homotopy groups of tmf and of its localizations 189222
- Ellitpic curves and stable homotopy I 209242
- From elliptic curves to homotopy theory 261294
- 𝐾(1)-local 𝐸_{∞}-ring spectra 287320
- Glossary 303336
- Back Cover Back Cover1353