**Colloquium Publications**

Volume: 64;
2017;
391 pp;
Hardcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-1944-8
Product Code: COLL/64**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

**Electronic ISBN: 978-1-4704-4313-9
Product Code: COLL/64.E**

List Price: $104.00

AMS Member Price: $0.00

MAA Member Price: $93.60

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#### Supplemental Materials

# Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

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*Kathrin Bringmann; Amanda Folsom; Ken Ono; Larry Rolen*

2018 PROSE Mathematics Award Winner!

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Modular forms and Jacobi forms play a central
role in many areas of mathematics. Over the last 10–15 years, this
theory has been extended to certain non-holomorphic functions, the
so-called “harmonic Maass forms”. The first glimpses of this theory
appeared in Ramanujan's enigmatic last letter to G. H. Hardy written
from his deathbed. Ramanujan discovered functions he called “mock
theta functions” which over eighty years later were recognized as
pieces of harmonic Maass forms. This book contains the essential
features of the theory of harmonic Maass forms and mock modular forms,
together with a wide variety of applications to algebraic number
theory, combinatorics, elliptic curves, mathematical physics, quantum
modular forms, and representation theory.

#### Readership

Graduate students and researchers interested in modular forms and their use in number theory.

#### Reviews & Endorsements

This wonderful book is very exciting to me as it presents some very exotic and beautiful stuff that I, for one, had no notion of...I'm happy to be enlightened, if only in a preliminary way. But this book is tailor-made to lead wayward sons like me back home and to light the right kind of fire under us...Excellent.

-- Michael Berg, MAA Reviews

#### Table of Contents

# Table of Contents

## Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

- Cover Cover11
- Title page iii4
- Contents v6
- Preface xi12
- Acknowledgments xv16
- Part 1 . Background 118
- Chapter 1. Elliptic Functions 320
- Chapter 2. Theta Functions and Holomorphic Jacobi Forms 1330
- 2.1. Jacobi theta functions 1330
- 2.2. Basic facts on Jacobi forms 1633
- 2.3. Examples of Jacobi forms 2037
- 2.4. A structure theorem for 𝐽_{𝑘,𝑚} 2643
- 2.5. Relationship with half-integral weight modular forms 2744
- 2.6. Hecke theory for 𝐽_{𝑘,𝑚} and the Jacobi-Petersson inner product 3047
- 2.7. Taylor expansions 3653
- 2.8. Related topics 4360

- Chapter 3. Classical Maass Forms 4966
- 3.1. Definitions 4966
- 3.2. Fourier expansions 5067
- 3.3. General discussion 5168
- 3.4. Eisenstein series 5269
- 3.5. 𝐿-functions of Maass cusp forms 5370
- 3.6. Maass cusp forms arising from real quadratic fields 5572
- 3.7. Hecke theory on Maass cusp forms 5673
- 3.8. Period functions of Maass cusp forms 5673

- Part 2 . Harmonic Maass Forms and Mock Modular Forms 5976
- Chapter 4. The Basics 6178
- Chapter 5. Differential Operators and Mock Modular Forms 6784
- Chapter 6. Examples of Harmonic Maass Forms 83100
- Chapter 7. Hecke Theory 113130
- Chapter 8. Zwegers’ Thesis 133150
- Chapter 9. Ramanujan’s Mock Theta Functions 159176
- 9.1. Ramanujan’s last letter to Hardy 159176
- 9.2. Work of Watson and Andrews 161178
- 9.3. Third order mock theta functions revisited 163180
- 9.4. Mock theta functions as indefinite theta series 165182
- 9.5. Universal mock theta functions 167184
- 9.6. The Mock Theta Conjectures 170187
- 9.7. The Andrews-Dragonette Conjecture 171188
- 9.8. Ramanujan’s original claim revisited 173190

- Chapter 10. Holomorphic Projection 177194
- Chapter 11. Meromorphic Jacobi Forms 183200
- Chapter 12. Mock Modular Eichler-Shimura Theory 193210
- Chapter 13. Related Automorphic Forms 207224

- Part 3 . Applications 221238
- Chapter 14. Partitions and Unimodal Sequences 223240
- Chapter 15. Asymptotics for Coefficients of Modular-type Functions 245262
- 15.1. Prologue 245262
- 15.2. Asymptotic methods 246263
- 15.3. Classical holomorphic modular forms 247264
- 15.4. Weakly holomorphic modular forms and mock modular forms 251268
- 15.5. Coefficients of meromorphic modular forms 253270
- 15.6. Mixed mock modular forms 256273
- 15.7. The Wright Circle Method 258275

- Chapter 16. Harmonic Maass Forms as Arithmetic and Geometric Generating Functions 263280
- Chapter 17. Shifted Convolution 𝐿-functions 283300
- Chapter 18. Generalized Borcherds Products 291308
- Chapter 19. Elliptic Curves over \Q 307324
- Chapter 20. Representation Theory and Mock Modular Forms 323340
- Chapter 21. Quantum Modular Forms 339356
- Appendix A. Representations of Mock Theta Functions 353370
- Bibliography 367384
- Index 387404

- Back Cover Back Cover1409