© 2020

On the Theory of Maass Wave Forms


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Tobias Mühlenbruch, Wissam Raji
    Pages 1-92
  3. Tobias Mühlenbruch, Wissam Raji
    Pages 93-147
  4. Tobias Mühlenbruch, Wissam Raji
    Pages 149-214
  5. Tobias Mühlenbruch, Wissam Raji
    Pages 215-329
  6. Tobias Mühlenbruch, Wissam Raji
    Pages 331-371
  7. Tobias Mühlenbruch, Wissam Raji
    Pages 373-431
  8. Tobias Mühlenbruch, Wissam Raji
    Pages 433-465
  9. Back Matter
    Pages 467-509

About this book


This textbook provides a rigorous analytical treatment of the theory of Maass wave forms. Readers will find this unified presentation invaluable, as it treats Maass wave forms as the central area of interest. Subjects at the cutting edge of research are explored in depth, such as Maass wave forms of real weight and the cohomology attached to Maass wave forms and transfer operators. Because Maass wave forms are given a deep exploration, this book offers an indispensable resource for those entering the field.

Early chapters present a brief introduction to the theory of classical modular forms, with an emphasis on objects and results necessary to fully understand later material. Chapters 4 and 5 contain the book’s main focus: L-functions and period functions associated with families of Maass wave forms. Other topics include Maass wave forms of real weight, Maass cusp forms, and weak harmonic Maass wave forms. Engaging exercises appear throughout the book, with solutions available online.

On the Theory of Maass Wave Forms is ideal for graduate students and researchers entering the area. Readers in mathematical physics and other related disciplines will find this a useful reference as well. Knowledge of complex analysis, real analysis, and abstract algebra is required.


Modular forms Maass wave forms Eichler integrals Eichler-Shimura isomorphism L-series Period problems Period functions Maass wave forms of real weight Families of Maass cusp forms Transfer operator approach Weak harmonic Maass wave forms

Authors and affiliations

  1. 1.Department of Mathematics and Computer ScienceFernUniversität in HagenHagenGermany
  2. 2.Department of MathematicsAmerican University of BeirutBeirutLebanon

About the authors

Tobias Mühlenbruch is a private lecturer at the FernUniversität in Hagen, Germany. His main research interest include number theory with a focus on analytic modular forms and the connections to dynamical systems and mathematical physics. 

Wissam Raji is an Associate Professor of Mathematics at the American University of Beirut, Lebanon. His research interests include analytic number theory and modular forms with a focus on period polynomials. He is the recipient of numerous awards, including the Saylor Foundation’s 2013 Open Textbook Challenge.

Bibliographic information

  • Book Title On the Theory of Maass Wave Forms
  • Authors Tobias Mühlenbruch
    Wissam Raji
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-40474-1
  • Softcover ISBN 978-3-030-40477-2
  • eBook ISBN 978-3-030-40475-8
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XXV, 509
  • Number of Illustrations 11 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


“The level of the textbook is suitable for master students and so on, but some proofs can easily be understood by advanced undergraduate students.” (Ilker Inam, zbMATH 1456.11002, 2021)