Theta functions, fourth moments of eigenforms and the sup-norm problem II

Abstract

Let f be an $L^2$ -normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$ . Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate $$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

Keywords

• 11F12
• 11F27
• 11F70
• 11F72
• 11D45
• 11N75
• 14G35