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

Abstract

In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a nontrivial nebentypus as well as Fourier coefficients of eigenforms of half-integral weight reachable by the Shimura correspondence.

A note on the extended Bruinier-Kohnen conjecture

“…Moreover, they conjectured that the signs of the sequence of Fourier coefficients of a cusp form which lies in the Kohnen's plus space are equidistributed. Through this problem in [8,9,4] Arias, Inam and Wiese proved the Bruinier-Kohnen conjecture in the special case when the Fourier coefficients of a Hecke eigenforms of half-integral weight are indexed by tn 2 with t a fixed square-free number and n ∈ N. In [1], the results of [8,4] were generalized to Hecke eigenforms with not necessarily real Fourier coefficients, based on this and empirical evidence the first author generalized the Bruinier-Kohnen conjecture to cusp forms in Kohnen's plus space with not necessarily real Fourier coefficients a(n). More precisely, he conjectured that…”

On the distribution of coefficients of half-integral weight modular forms and the Bruinier–Kohnen conjecture

“…Combining the Shimura lift with the (proved) celebrated Sato-Tate Conjecture for integral weight Hecke eigenforms, it is not very difficult to prove equidistribution of signs for the coefficients indexed by squares, see [6], [22], [23]. The sign equidistribution problem has still received much attention and is widely studied (for instance [7]), and the technique from [22] has been extended to more general automorphic forms like Hilbert modular forms in [26]. Note that this is only a partial result and the full proof of the conjecture is still an open problem and, for the moment, it is likely that there is no theoretical tool to attack this problem.…”