Archiv der Mathematik

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

Equidistribution of signs for modular eigenforms of half integral weight



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 


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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

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