Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akiyama S., Tanigawa Y.: of Values of L- Functions Associated to Elliptic Curves. Mathematics of Computation 68, 1201–1231 (1999)
S. Arias-de-Reyna, I. Inam, and G. Wiese, On Conjectures of Sato–Tate and Bruinier–Kohnen, preprint (2013), arXiv:1305.5443.
Barnet-Lamb T. et al.: A Family of Calabi-Yau varities and Potential Automorphy II. Pub. Res. Inst. Math. Sci. 47, 29–98 (2011)
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.
Dummigan N.: Congruences of Modular Forms and Selmer Groups. Math. Res. Lett. 8, 479–494 (2001)
Inam I.: Selmer Groups in Twist Families of Elliptic Curves. Quaestiones Mathematicae 35, 471–487 (2012)
Kohnen W.: A Short Note on Fourier Coefficients of Half-Integral Weight Modular Forms. Int. J. of Number Theory 06, 1255–1259 (2010)
Kohnen W, Lau Y.-K., Wu J.: Fourier Coefficients of Cusp Forms of Half-Integral Weight. Math. Zeitschrift 273, 29–41 (2013)
Kumar N: On sign changes of q-exponents of generalized modular functions. J. Number Theory 133, 3589–3597 (2013)
Kohnen W., Zagier D.: Values of L−series of Modular Forms at the Center of the Critical Strip. Invent. Math. 64, 175–198 (1981)
Mazur B.: Finding Meaning in Error Terms, Bulletin of the Amer. Math. Soc. 45, 185–228 (2008)
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third Edition, Springer, 708 pp., (1990).
Niwa S.: Modular Forms of Half Integral Weight and the Integral of Certain Theta-Functions. Nagoya Math. J. 56, 147–161 (1974)
Rosser B.: Explicit Bounds for Some Functions of Prime Numbers. Amer. Journal of Mathematics 63, 211–232 (1941)
Shimura G.: On Modular Forms of Half-Integral Weight. Annals of Math. 97, 440–481 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
I.I. was supported by The Scientific and Technological Research Council of Turkey (TUBITAK). I.I would also like to thank the University of Luxembourg for having hosted him as a postdoctoral fellow of TUBITAK. G.W. acknowledges partial support by the priority program 1489 of the Deutsche Forschungsgemeinschaft (DFG) and by the Fonds National de la Recherche Luxembourg (INTER/DFG/12/10).
The authors would like to thank Winfried Kohnen and Jan Bruinier for having suggested the problem and encouraged them to write this article.
Rights and permissions
About this article
Cite this article
Inam, I., Wiese, G. Equidistribution of signs for modular eigenforms of half integral weight. Arch. Math. 101, 331–339 (2013). https://doi.org/10.1007/s00013-013-0566-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-013-0566-4