Harmonic Maass Forms and Mock Modular Forms

Theory and Applications

Häftad, Engelska, 2017

Av Kathrin Bringmann, Amanda Folsom, Ken Ono, Larry Rolen

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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 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.

Produktinformation

Utgivningsdatum2017-12-30
FormatHäftad
SpråkEngelska
SerieColloquium Publications
Antal sidor391
FörlagAmerican Mathematical Society
ISBN9781470479848

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BackgroundElliptic functionsTheta functions and holomorphic Jacobi formsClassical Maass formsHarmonic Maass forms and mock modular formsThe basicsDifferential operators and mock modular formsExamples of harmonic Maass formsHecke theoryZwegers' thesisRamanujan's mock theta functionsHolomorphic projectionMeromorhic Jacobi formsMock modular Eichler-shimura theoryRelated automorphic formsApplicationsPartitions and unimodal sequencesAsymptotics for coefficients of modular-type functionsHarmonic Maass forms as arithmetic and geometric generating functionsShifted convolution $L$-functionsGeneralized Borcherds productsElliptic curves over $\mathbb {Q}$Representation theory and mock modular formsQuantum modular formsRepresentations of mock theta functions