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.
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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.
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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
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DOI: https://doi.org/10.1007/s00013-013-0566-4