Skip to main content
SpringerLink
Log in

Weak Harmonic Maass Wave Forms

  • Chapter
  • First Online:
On the Theory of Maass Wave Forms

Part of the book series: Universitext ((UTX))

  • 891 Accesses

Abstract

In this chapter, we introduce the newly emerging concept of weakly harmonic Maass wave forms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. K. Bringmann, N. Diamantis, M. Raum, Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions. Adv. Math. 233, 115–134 (2013). https://doi.org/10.1016/j.aim.2012.09.025

    Article  MathSciNet  Google Scholar 

  2. K. Bringmann, P. Guerzhoy, Z. Kent, K. Ono, Eichler-Shimura theory for mock modular forms. Math. Ann. 355(3), 1085–1121 (2013). https://doi.org/10.1007/s00208-012-0816-y

    Article  MathSciNet  Google Scholar 

  3. K. Bringmann, K.-H. Fricke, Z. Kent, Special L-values and periods of weakly holomorphic modular forms. Proc. Am. Math. Soc. 142(10), 3425–3439 (2014). MR3238419. http://www.ams.org/mathscinet-getitem?mr=3238419; https://doi.org/10.1090/S0002-9939-2014-12092-2

  4. K. Bringmann, P. Guerzhoy, B. Kane, On cycle integrals of weakly holomorphic modular forms. Math. Proc. Cambridge Philos. Soc. 158(3), 439–449 (2015). MR3335420. http://www.ams.org/mathscinet-getitem?mr=3335420; https://doi.org/10.1017/S0305004115000055

  5. K. Bringmann, A. Folsom, K. Ono, L. Rolen, Harmonic Maass Forms and Mock Modular Forms: Theory and Applications. Colloquium Publications, vol. 64 (American Mathematical Society, Providence, RI, 2017). ISBN 978-1-4704-1944-8, ISBN 978-1-4704-4313-9

    Google Scholar 

  6. J.H. Bruinier, J. Funke, On two geometric theta lifts. Duke Math J. 125(1), 45–90 (2004)

    Article  MathSciNet  Google Scholar 

  7. J.H. Bruinier, K. Ono, R. Rhoades, Differential operators and harmonic weak Maass forms. Math. Ann. 342, 673–693 (2008). https://doi.org/10.1007/s00208-008-0252-1. Errata: Math. Ann. 345, 31 (2009). https://doi.org/10.1007/s00208-009-0338-4

  8. A. Dabholkar, S. Murthy, D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms (2012). Preprint. arXiv:1208.4074v2[hep-th]. https://arxiv.org/abs/1208.4074

  9. W. Duke, Almost a century of answering the question: what is a mock theta function? Not. Am. Math. Soc. 61(11), 1314–1320 (2014). https://doi.org/10.1090/noti1185

    MathSciNet  MATH  Google Scholar 

  10. W. Duke, Ö. Imamog~lu, Á. Tóth, Rational period functions and cycle integrals. Abh. Math. Semin. Hambg. 80, 255–264 (2010)

    Google Scholar 

  11. J.F.R. Duncan, M.J. Griffin, K. Ono, Proof of the umbral moonshine conjecture. Res. Math. Sci. 2–26 (2015). https://doi.org/10.1186/s40687-015-0044-7

  12. J.D. Fay, Fourier Coefficients of the Resolvent for a Fuchsian Group. J. Reine Angew. Math. 293/294, 143–203 (1977). https://doi.org/10.1515/crll.1977.293-294.143

  13. A. Folsom, Mock and quantum modular forms. Plenary Lecture at the Connecticut Summer School in Number Theory 2016. https://ctnt-summer.math.uconn.edu/schedules-and-abstracts/. Video: https://www.youtube.com/playlist?list=PLJUSzeW191Qzs7obSgtg7mK5Zn1qNVvno. PDF: http://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNT2016talk-Amanda-Folsom.pdf

  14. J. Gimenez, T. Mühlenbruch, W. Raji, Construction of vector-valued modular integrals and vector-valued mock modular forms. Ramanujan J. 1–64 (2014). https://doi.org/10.1007/s11139-014-9606-3

  15. S.Y. Husseini, M. Knopp, Eichler cohomology and automorphic forms. Ill. J. Math. 15, 565–577 (1971). http://projecteuclid.org/euclid.ijm/1256052512

    Article  MathSciNet  Google Scholar 

  16. S. Lang, Introduction to Modular Forms, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 222 (Springer, Berlin, 2001). ISBN 978-3-540-07833-3. https://doi.org/10.1007/978-3-642-51447-0

  17. R. Lawrence, D. Zagier, Modular forms and quantum invariants of 3-manifolds, Sir Michael Atiyah: a great mathematician of the twentieth century, Asian J. Math. 3, 93–107 (1999)

    Article  MathSciNet  Google Scholar 

  18. J. Lewis, D. Zagier, Period functions for Maass wave forms. I. Ann. Math. 153, 191–258 (2001)

    Article  MathSciNet  Google Scholar 

  19. NIST Digital Library of Mathematical Functions. Version 1.0.8. Release date 25 April 2014. https://dlmf.nist.gov/

  20. K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory. Curr. Dev. Math. 2008, 347–454 (2009). https://projecteuclid.org/euclid.cdm/1254748659

    Article  MathSciNet  Google Scholar 

  21. S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Springer, New York, 1988). ISBN 978-3-540-18726-4. MR947735. http://www.ams.org/mathscinet-getitem?mr=947735

  22. S. Ramanujan, Collected Papers of Srinivasa Ramanujan (AMS Chelsea Publishing, Providence, RI, 2000). ISBN 978-0-8218-2076-6. MR2280843. http://www.ams.org/mathscinet-getitem?mr=2280843

  23. A.M. Semikhatov, A. Taormina, I.Yu. Tipunin, Higher-level Appell functions, modular transformations, and characters. Commun. Math. Phys. 255, 469–512 (2005). https://doi.org/10.1007/s00220-004-1280-7

    Article  MathSciNet  Google Scholar 

  24. The On-Line Encyclopedia of Integer Sequences (2016). Published electronically at http://oeis.org

  25. J. Troost, The Non-Compact Elliptic Genus: Mock or Modular. J. High Energy Phys. 2010(6), 104 (2010). https://doi.org/10.1007/JHEP06(2010)104

    Google Scholar 

  26. D. Zagier, Nombres de classes et formes modulaires de poids 3∕2. C. R. Acad. Sc. Paris 281, 883–886 (1975).

    MathSciNet  MATH  Google Scholar 

  27. S.P. Zwegers, Mock theta functions. PhD thesis, Utrecht University, 2002. ISBN 90-393-3155-3. http://dspace.library.uu.nl/handle/1874/878

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, FernUniversität in Hagen, Hagen, Germany

    Tobias Mühlenbruch

  2. Department of Mathematics, American University of Beirut, Beirut, Lebanon

    Wissam Raji

Authors
  1. Tobias Mühlenbruch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wissam Raji
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mühlenbruch, T., Raji, W. (2020). Weak Harmonic Maass Wave Forms. In: On the Theory of Maass Wave Forms. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-40475-8_7

Download citation

Publish with us

Policies and ethics

Search

Navigation