Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier co... more 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.
The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we s... more The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T N,χ ℓ,k for all primes ℓ, all weights k ≥ 2 and all characters χ taking values in {±1} splits completely modulo p has density 0, unconditionally for p = 2 and under the Cohen-Lenstra heuristics for p ≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.
Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier co... more 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.
The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we s... more The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T N,χ ℓ,k for all primes ℓ, all weights k ≥ 2 and all characters χ taking values in {±1} splits completely modulo p has density 0, unconditionally for p = 2 and under the Cohen-Lenstra heuristics for p ≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.
Uploads
Papers by G. Wiese