Illinois Journal of Mathematics

Eichler integrals for Maass cusp forms of half-integral weight

T. Mühlenbruch and W. Raji

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight $0$ on the full modular group introduced by Lewis and Zagier.

Article information

Source
Illinois J. Math., Volume 57, Number 2 (2013), 445-475.

Dates
First available in Project Euclid: 19 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1408453590

Mathematical Reviews number (MathSciNet)
MR3263041

Zentralblatt MATH identifier
1312.11034

Subjects
Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms
Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11F72: Spectral theory; Selberg trace formula

Citation

Mühlenbruch, T.; Raji, W. Eichler integrals for Maass cusp forms of half-integral weight. Illinois J. Math. 57 (2013), no. 2, 445--475. https://projecteuclid.org/euclid.ijm/1408453590


Export citation

References

  • K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono, Eichler–Shimura theory for mock modular forms, Math. Ann. 355 (2013), 1085–1121.
  • R. W. Bruggeman, Families of automorphic forms, Monographs in Mathematics, vol. 88, Birkhäuser, Basel, 1994.
  • R. W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms and cohomology, to appear in Memoirs of the AMS; preprint available at http://www.staff.science.uu.nl/~brugg103/algemeen/prpr.html.
  • A. Deitmar, Lewis–Zagier correspondence for higher-order forms, Pacific J. Math. 249 (2011), 11–21.
  • A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups, Forum Math. 19 (2007), 1075–1099.
  • M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298.
  • M. Knopp and G. Mason, Generalized modular forms, J. Number Theory 99 (2003), 11–28.
  • M. Knopp and W. Raji, Eichler cohomology and generalized modular forms II, Int. J. Number Theory 6 (2010), 1083–1090.
  • W. Kohnen and D. Zagier, Modular forms with rational periods, Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl. Statist. Oper. Res., Horwood, Chichester, 1984, pp. 197–249.
  • S. Lang, Introduction to modular forms, 3rd ed., Springer, Berlin, 2001.
  • J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), 191–258.
  • T. Mühlenbruch, Systems of automorphic forms and period functions, Ph.D. thesis, Utrecht University, 2003.
  • T. Mühlenbruch and W. Raji, Generalized Maass wave forms, Proc. Amer. Math. Soc. 141 (2013), 1143–1158.
  • D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), 449–465.
  • D. Zagier, Introduction to modular forms, From number theory to physics (M. Waldschmidt \betal, eds.), Springer, Heidelberg, 1992, pp. 238–291.