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.

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

“…The following theorem said that the set of primes of Theorem 3 is regular if the Chebotarev-Sato-Tate theorem satisfies certain error term. The proof is closely similar to that of [5,Theorem 4.2].…”

“…The following proposition said that the set of primes S is regular if it has a natural density that satisfies certain error term (see [5,Proposition 2.2]).…”

“…This suggestion was formulated later explicitly as a conjecture in [7]. Assuming some error term for the convergence of the Sato-Tate distribution for integral weight modular forms in [5], Inam and Wise showed when F t has no CM that half of the coefficients a(tn 2 ) are positive. They formulated this result in terms of Dedekind-Dirichlet density.…”

Oscillatory behavior and equidistribution of signs of Fourier coefficients of cusp forms

“…Before we proceed to prove the theorem, we first recall some basic facts concerning newforms with complex multiplication, following the exposition given in [7, Section 2, pp. [8][9].…”

A short note on the Bruiner–Kohnen sign equidistribution conjecture and Halász’ theorem

“…By using the celebrated Sato-Tate theorem for integral weight modular eigenforms and the Shimura lift, in [4] and in [1] (together with Sara Arias-de-Reyna), we prove results related to the Bruinier-Kohnen sign equidistribution conjecture for modular eigenforms of half integral weight. In this note we improve one of our main results to a formulation in terms of natural density.…”