Period Functions

  • Tobias Mühlenbruch
  • Wissam Raji
Part of the Universitext book series (UTX)


We introduce period functions and give several relations to (families of) Maass cusp forms. We present some remarks about Zagier’s cohomology theorem that generalizes the Eichler cohomology theorem. We conclude this chapter with a discussion of Hecke operators acting on period functions.


  1. 35.
    R.W. Bruggeman, J. Lewis, D. Zagier, Function theory related to the group \(\text{PSL}_2(\mathbb {R})\), in From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol. 28 (Springer, New York, 2013), pp. 107–201. ISBN 978-1-4614-4074-1.
  2. 36.
    R.W. Bruggeman, J. Lewis, D. Zagier, Period functions for Maass wave forms and cohomology. Mem. Am. Math. Soc. 237(1118) (2015). ISBN 978-1-4704-1407-8; 978-1-4704-2503-6.
  3. 37.
    R.W. Bruggeman, Y. Choie, N. Diamantis, Holomorphic automorphic forms and cohomology. Mem. Am. Math. Soc. 253, (1212) (2018). ISBN 978-1-4704-2855-6, ISBN 978-1-4704-4419-8.
  4. 57.
    Y.J. Choie, D. Zagier, Rational period functions for \(\mathrm {PSL}(2,\mathbb {Z})\), in A Tribute to Emil Grosswald: Number Theory and related Analysis, ed. by M. Knopp, M. Sheingorn. Contemporary Mathematics, vol. 143 (American Mathematical Society, Providence, RI, 1993), pp. 89–108Google Scholar
  5. 58.
    D. Choi, S. Lim, T. Mühlenbruch, W. Raji, Series expansion of the period function and representations of Hecke operators. J. Number Theory 171, 301–340 (2017). MathSciNetCrossRefGoogle Scholar
  6. 96.
    J. Hilgert, D. Mayer, H. Movasati, Transfer operators for Γ0(n) and the Hecke operators for period functions of \(\mathrm {PSL}(2,\mathbb {Z})\). Math. Proc. Camb. Philos. Soc. 139, 81–116 (2005).
  7. 97.
    A. Hurwitz, Ueber die angenäherte Darstellung der Zahlen durch rational Brüche. Math. Ann. 44, 417–436 (1894). MathSciNetCrossRefGoogle Scholar
  8. 116.
    S. Lang, Introduction to Modular Forms, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 222 (Springer, Berlin, 2001). ISBN 978-3-540-07833-3.
  9. 121.
    J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms. Invent. Math. 127, 271–306 (1997). MathSciNetCrossRefGoogle Scholar
  10. 122.
    J. Lewis, D. Zagier, Period functions for Maass wave forms. I. Ann. Math. 153, 191–258 (2001)MathSciNetCrossRefGoogle Scholar
  11. 141.
    L. Merel, Universal Fourier expansions of modular forms. in On Artkins Conjecture for Odd 2-Dimensional Representations, ed. by G. Frey. Lecture Notes in Mathematics, vol. 1585 (Springer, Berlin, 1994)Google Scholar
  12. 147.
    T. Mühlenbruch, Systems of automorphic forms and period functions. Ph.D. Thesis, Utrecht University, 2003Google Scholar
  13. 148.
    T. Mühlenbruch, Hecke operators on period functions for the full modular group. Int. Math. Res. Not. 77, 4127–4145 (2004). MathSciNetCrossRefGoogle Scholar
  14. 149.
    T. Mühlenbruch, Hecke operators on period functions for Γ0(n). J. Number Theory 118, 208–235 (2006). MathSciNetCrossRefGoogle Scholar
  15. 151.
    T. Mühlenbruch, W. Raji, Eichler integrals for Maass cusp forms of half-integral weight. Ill. J. Math. 57(2), 445–475 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Tobias Mühlenbruch
    • 1
  • Wissam Raji
    • 2
  1. 1.Department of Mathematics and Computer ScienceFernUniversität in HagenHagenGermany
  2. 2.Department of MathematicsAmerican University of BeirutBeirutLebanon

Personalised recommendations