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

Abstract

For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate \[ \| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\epsilon} (k V)^{\frac{1}{4}+\epsilon} \] with absolute implied constant. For a cuspidal Maa{\ss} newform $\varphi$ of eigenvalue $\lambda$ on such a surface, we prove that \[ \|\varphi \|_{\infty} \ll_{\lambda,\epsilon} V^{\frac{1}{4}+\epsilon}. \] We establish analogous estimates in the setting of definite quaternion algebras.