Equidistribution of Signs for Modular Eigenforms of Half Integral Weight

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.

8 pages; typos corrected, final version, accepted for publication in Archiv der Mathematik

References in corpus (1)

• Selmer Groups in Twist Families of Elliptic Curves

Cited by in corpus (9)

• On conjectures of Sato-Tate and Bruinier-Kohnen
• Sign of Fourier coefficients of modular forms of half integral weight
• A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halász' Theorem
• Equidistribution of signs for Hilbert modular forms of half-integral weight
• Sign Changes of Coefficients and Sums of Coefficients of L-Functions
• Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms
• Oscillatory behavior and equidistribution of signs of Fourier coefficients of cusp forms
• Distribution of toric periods of modular forms on definite quaternion algebras
• A note on the extended Bruinier-Kohnen conjecture