Archiv der Mathematik

, Volume 101, Issue 4, pp 331–339 | Cite as

Equidistribution of signs for modular eigenforms of half integral weight

  • Ilker Inam
  • Gabor Wiese


Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp 2)} p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn 2)} n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.

Mathematics Subject Classification (2010)

Primary 11F37 Secondary 11F30 


Forms of half-integer weight Shimura lift Fourier coefficients of automorphic forms Sato-Tate equidistribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akiyama S., Tanigawa Y.: of Values of L- Functions Associated to Elliptic Curves. Mathematics of Computation 68, 1201–1231 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    S. Arias-de-Reyna, I. Inam, and G. Wiese, On Conjectures of Sato–Tate and Bruinier–Kohnen, preprint (2013), arXiv:1305.5443.Google Scholar
  3. 3.
    Barnet-Lamb T. et al.: A Family of Calabi-Yau varities and Potential Automorphy II. Pub. Res. Inst. Math. Sci. 47, 29–98 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. H. Bruinier and W. Kohnen, Sign Changes of Coefficients of Half Integral Weight Modular Forms, Modular forms on Schiermonnikoog, Eds.: B. Edixhoven, G. van der Geer, and B. Moonen, Cambridge University Press (2008), 57–66.Google Scholar
  5. 5.
    Dummigan N.: Congruences of Modular Forms and Selmer Groups. Math. Res. Lett. 8, 479–494 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Inam I.: Selmer Groups in Twist Families of Elliptic Curves. Quaestiones Mathematicae 35, 471–487 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kohnen W.: A Short Note on Fourier Coefficients of Half-Integral Weight Modular Forms. Int. J. of Number Theory 06, 1255–1259 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kohnen W, Lau Y.-K., Wu J.: Fourier Coefficients of Cusp Forms of Half-Integral Weight. Math. Zeitschrift 273, 29–41 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kumar N: On sign changes of q-exponents of generalized modular functions. J. Number Theory 133, 3589–3597 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kohnen W., Zagier D.: Values of L−series of Modular Forms at the Center of the Critical Strip. Invent. Math. 64, 175–198 (1981)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mazur B.: Finding Meaning in Error Terms, Bulletin of the Amer. Math. Soc. 45, 185–228 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third Edition, Springer, 708 pp., (1990).Google Scholar
  13. 13.
    Niwa S.: Modular Forms of Half Integral Weight and the Integral of Certain Theta-Functions. Nagoya Math. J. 56, 147–161 (1974)MathSciNetGoogle Scholar
  14. 14.
    Rosser B.: Explicit Bounds for Some Functions of Prime Numbers. Amer. Journal of Mathematics 63, 211–232 (1941)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shimura G.: On Modular Forms of Half-Integral Weight. Annals of Math. 97, 440–481 (1973)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesUludag UniversityGorukle/BursaTurkey
  2. 2.Faculté des Sciences, de la Technologie et de la CommunicationUniversité du LuxembourgLuxembourgLuxembourg

Personalised recommendations