Harmonic Maass Forms and Mock Modular Forms

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.

subjects: Forms (mathematics),  Number theory,  Modular Forms,  Discontinuous groups and automorphic forms,  Modular and automorphic functions,  Holomorphic modular forms of integral weight,  Theta series; Weil representation; theta correspondences,  Fourier coefficients of automorphic forms,  Forms of half-integer weight; nonholomorphic modular forms,  Jacobi forms