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.
Recenzijas
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
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
Kathrin Bringmann, University of Cologne, Germany.
Amanda Folsom, Amherst College, MA.
Ken Ono, Emory University, Atlanta, GA.
Larry Rolen, Trinity College, Dublin, Ireland.
