Weyl-type bounds for twisted \(\mathrm{GL}(2)\) short character sums. (English) Zbl 1524.11112
Summary: Let \(f\) be a Hecke-Maass or holomorphic primitive cusp form of full level for \(\mathrm{SL}(2,\mathbb{Z})\) 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 \(\mathrm{GL}(2)\) short character sum. Here, we consider the problem of cancellation in short character sum of the form
\[
S_{f,\chi} (N):= \sum\limits_{n \in\mathbb{Z}} \lambda_f (n)\chi (n)W\Big( \frac{n}{N}\Big).
\]
We show that, for \(0<\theta < \frac{1}{10}\),
\[
S_{f,\chi} (N) \ll_{f,\epsilon}N^{3/4 + \theta /2}p^{1/6}(pN)^{\epsilon} + N^{1-\theta}(pN)^{\epsilon},
\]
which is non-trivial if \(N \geq p^{2/3 + \alpha + \epsilon}\) where \(\alpha = = \frac{4\theta}{1-6\theta}\). Previously, such a bound was known for \(N \geq p^{3/4 +\epsilon}\).
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 11M41 | Other Dirichlet series and zeta functions |
Keywords:
automorphic forms; short character sums; Weyl bound; Maass forms; holomorphic forms; Fourier coefficients of \(\mathrm{GL}(2)\) cusp formsReferences:
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