On Short Exponential Sums Involving Fourier Coefficients of Holomorphic Cusp Forms

On certain exponential sums related to GL(3) cusp forms

“…A similar question has been earlier tackled in the GL(2) setting by Karppinen and the author [1] together with estimates for short exponential sums. Namely, we proved that …”

“…is a smooth function compactly supported on [X, 2 X] and that Ψ 0 (x) is defined as in (1). Then for any fixed integer K 1 and x X 1, we have…”

“…where U 2 (Z) is the group of 2 × 2 upper triangular matrices with integer entries and ones on the diagonal, W J (z, ν, ψ 1,1 ) is the Jacquet-Whittaker function and M = diag(m 1 |m 2 |, m 1 , 1). Assume that the form is normalized in such a way that the first Fourier coefficient is one.…”

Exponential sums related to Maass forms

“…Also, let M ∈ [1, ∞[, ∆ ∈ [1, M ], and let α ∈ R. If ∆ ≪ M 2/3 , then M n M+∆ a(n) e(nα) ≪ ∆ 1/6 M 1/3+ε , where the implicit constant depends only on the underlying cusp forms and ε. Similarly, if M 2/3 ≪ ∆, then M n M+∆ a(n) e(nα) ≪ ∆ M −2/9+ε .The proof of Theorem 1 depends on an estimate for short non-linear sums, analogous to Theorem 4.1 in [12]. Fortunately, the proof in [12] works almost verbatim for Maass forms and we shall indicate the differences later.…”

“…Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums.When ∆ = M 2/3 this gives the upper bound ≪ M ϑ/3+4/9+ε , and so splitting a longer sum into sums of this length and estimating the subsums separately gives the following bound for longer sums.Actually, Theorem 1 is valid for a slightly larger range of ∆ than Theorem 5.5 in [12] is. In fact, with a minor modification [11], the proof of Theorem 5.5 of [12] can be easily modified to give the analogous result for holomorphic cusp forms:Theorem 3. Let us consider a fixed holomorphic cusp form of weight κ ∈ Z + for the full modular group with the Fourier expansion ∞ n=1 a(n) n (κ−1)/2 e(nz) for z ∈ C with ℑz > 0.…”