Harmonic Maass Forms and Mock Modular Forms

Theory and Applications

AvKathrin Bringmann,Amanda Folsom

Häftad, Engelska, 2017

Del i serien Colloquium Publications

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Beskrivning

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

Utgivningsdatum:2017-12-30
Format:Häftad
Språk:Engelska
Serie:Colloquium Publications
Antal sidor:391
Förlag:American Mathematical Society
ISBN:9781470479848

Utforska kategorier

Talteori inom Naturvetenskap och teknik

Mer om författaren

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

Recensioner i media

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

Innehållsförteckning

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