Estimates of Fourier coefficients of integral and half-integral weight cusp forms associated to cofinite Fuchsian subgroups

Abstract

Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} where the implied constant depends on the cusp form $f$, and the Fuchsian subgroup $\G$. The proof of the above estimate remains valid, for half-integral weight cusp forms of weight-$\kappa$ with $\kappa\in\big\lbrace \frac{1}{2}+m\big|\,m\in\Z_{\geq 4} \big\rbrace$, associated to the arithmetic subgroups $\G_{0}(4N)$, $\G_{1}(4N)$, and $\G(4N)$ (for $N\in\Z_{\geq 1}$).