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Full Description
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.
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 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
Bibliography
Index
