Weyl-type bounds for twisted GL(2) short character sums

Abstract

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for $SL(2,{\mathbb {Z}})$ with normalized Fourier coefficients $\lambda _{f}(n)$. Let $\chi$ be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form

$$\begin{aligned} S_{f,\chi }(N):= \mathop \sum _{n \in {\mathbb {Z}}}\lambda _{f}(n)\chi (n)W\Big (\frac{n}{N}\Big ). \end{aligned}$$

We show that, for $0<\theta < \frac{1}{10}$,

$$\begin{aligned} S_{f,\chi }(N) \ll _{f,\epsilon }N^{3/4 + \theta /2}p^{1/6}(pN)^{\epsilon } + N^{1-\theta }(pN)^{\epsilon }, \end{aligned}$$

which is non-trivial if $N \ge p^{2/3 + \alpha + \epsilon }$ where$\alpha = = \frac{4\theta }{1-6\theta }$. Previously, such a bound was known for $N \ge p^{3/4 + \epsilon }.$