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| Hardcover ISBN: | 978-1-4704-1944-8 |
| Product Code: | COLL/64 |
| List Price: | $99.00 |
| MAA Member Price: | $89.10 |
| AMS Member Price: | $79.20 |
| eBook ISBN: | 978-1-4704-4313-9 |
| Product Code: | COLL/64.E |
| List Price: | $89.00 |
| MAA Member Price: | $80.10 |
| AMS Member Price: | $0.00 |
| Hardcover ISBN: | 978-1-4704-1944-8 |
| eBook ISBN: | 978-1-4704-4313-9 |
| Product Code: | COLL/64.B |
| List Price: | $188.00 $143.50 |
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Book DetailsColloquium PublicationsVolume: 64; 2017; 391 ppMSC: Primary 112018 PROSE Mathematics Award Winner!
Free eBook for AMS Members: To receive this eBook for free, simply add it to your cart and check out as usual. Once you log in as a member, the price will update to $0. Your purchased eBook may be downloaded from your Bookshelf page immediately.
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.
ReadershipGraduate students and researchers interested in modular forms and their use in number theory.
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Table of Contents
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Background
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Elliptic functions
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Theta functions and holomorphic Jacobi forms
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Classical Maass forms
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Harmonic Maass forms and mock modular forms
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The basics
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Differential operators and mock modular forms
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Examples of harmonic Maass forms
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Hecke theory
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Zwegers’ thesis
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Ramanujan’s mock theta functions
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Holomorphic projection
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Meromorhic Jacobi forms
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Mock modular Eichler-shimura theory
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Related automorphic forms
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Applications
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Partitions and unimodal sequences
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Asymptotics for coefficients of modular-type functions
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Harmonic Maass forms as arithmetic and geometric generating functions
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Shifted convolution $L$-functions
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Generalized Borcherds products
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Elliptic curves over $\mathbb {Q}$
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Representation theory and mock modular forms
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Quantum modular forms
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Representations of mock theta functions
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Additional Material
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Reviews
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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
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Free eBook for AMS Members: To receive this eBook for free, simply add it to your cart and check out as usual. Once you log in as a member, the price will update to $0. Your purchased eBook may be downloaded from your Bookshelf page immediately.
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.
Graduate students and researchers interested in modular forms and their use in number theory.
-
Background
-
Elliptic functions
-
Theta functions and holomorphic Jacobi forms
-
Classical Maass forms
-
Harmonic Maass forms and mock modular forms
-
The basics
-
Differential operators and mock modular forms
-
Examples of harmonic Maass forms
-
Hecke theory
-
Zwegers’ thesis
-
Ramanujan’s mock theta functions
-
Holomorphic projection
-
Meromorhic Jacobi forms
-
Mock modular Eichler-shimura theory
-
Related automorphic forms
-
Applications
-
Partitions and unimodal sequences
-
Asymptotics for coefficients of modular-type functions
-
Harmonic Maass forms as arithmetic and geometric generating functions
-
Shifted convolution $L$-functions
-
Generalized Borcherds products
-
Elliptic curves over $\mathbb {Q}$
-
Representation theory and mock modular forms
-
Quantum modular forms
-
Representations of mock theta functions
-
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
