Estimates of cusp forms for certain co-compact arithmetic subgroups

Abstract:

In this article, we compute estimates of Hecke eigen cusp forms, associated to a certain co-compact Fuchsian subgroup. Let $\Gamma \subset \mathrm {PSL}_{2}(\mathbb {R})$ be a co-compact Fuchsian subgroup associated to a division quaternion algebra $\mathcal {A}$ defined over $\mathbb {Q}$. Let $X\colonequals \Gamma \backslash \mathbb {H}$ denote the quotient space, which admits the structure of a compact hyperbolic Riemann surface. Let $\mathcal {S}^{k}(\Gamma )$ denote the complex vector space of cusp forms of weight-$k$ with respect to $\Gamma$, and let $|\cdot |_{\mathrm {pet}}$ denote then point-wise Petersson norm on $\mathcal {S}^{k}(\Gamma )$. Then, for $k\gg 1$, and any $f\in \mathcal {S}^{k}(\Gamma )$, a Hecke eigen cusp form, which is normalized with respect to the Petersson inner-product on $\mathcal {S}^{k}(\Gamma )$, and for any $\epsilon >0$, we derive the following estimate \begin{equation*} \sup _{z\in X}\big |f(z)\big |_{\mathrm {pet}}=O_{\mathcal {A},\epsilon }\big (k^{\frac {1}{2}-\frac {1}{12}+\epsilon }\big ), \end{equation*} where the implied constant depends on the quaternion algebra $\mathcal {A}$, and on the choice of $\epsilon >0$.