Fourier coefficients of cusp forms of half-integral weight

Abstract

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N. As an application, we consider a Waring-type problem with sums of mixed powers.

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

“…where θ i (p) ∈ [0, π]. As the set {p ∈ P : θ i (p) = 0, π} is finite (see [10,Remark 2]). We may assume that θ i (p) ∈ (0, π).…”

On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring

“…For this, we require [12, Theorem 1] of Kohnen, Lau, and Wu. (1) Kohnen, Lau and Wu actually gave much stronger results in their paper [12] but this simplified version is strong enough for our use. (2) One can use an argument involving the sign changes to directly show that…”

“…In our context, this n 0 corresponds to the first sign change. Although there is some discussion in [12] about the size of n 0 , there are a number of inexplicit constants which would need to be worked out to determine the size of n 0 implied by their theorem, and it is not expected that their proof would yield a bound anywhere close to the conjectured O(p). The first author is trying to determine (and improve upon) an explicit bound for n 0 in his Masters thesis.…”

“…These sign changes were investigated by Bruinier and Kohnen [1] and later by Kohnen, Lau and Wu [12].…”

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

“…When χ = 1, Bruinier and Kohnen suggested in [3] that half of the coefficients a(n) are positive among all non-zero Fourier coefficients. 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.…”