Hardcover ISBN:  9781470419448 
Product Code:  COLL/64 
List Price:  $104.00 
MAA Member Price:  $93.60 
AMS Member Price:  $83.20 
Electronic ISBN:  9781470443139 
Product Code:  COLL/64.E 
List Price:  $104.00 
MAA Member Price:  $93.60 
AMS Member Price:  $0.00 

Book DetailsColloquium PublicationsVolume: 64; 2017; 391 ppMSC: Primary 11;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 nonholomorphic functions, the socalled “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.

Table of Contents

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 Eichlershimura theory

Related automorphic forms

Applications

Partitions and unimodal sequences

Asymptotics for coefficients of modulartype 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


Additional Material

Reviews

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 tailormade 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 nonholomorphic functions, the socalled “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 Eichlershimura theory

Related automorphic forms

Applications

Partitions and unimodal sequences

Asymptotics for coefficients of modulartype 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 tailormade to lead wayward sons like me back home and to light the right kind of fire under us...Excellent.
Michael Berg, MAA Reviews