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 Γ _0(N) ∖ ℍ with N squarefree or, more generally, on any hyperbolic surface Γ ∖ ℍ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over ℚ. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate ‖ Im(·)^((k)/(2)) f ‖_(∞) ≪_(ε) (k V)^((1)/(4)+ε) with absolute implied constant. For a cuspidal Maaß newform φ of eigenvalue λ on such a surface, we prove that ‖φ ‖_(∞) ≪_(λ,ε) V^((1)/(4)+ε). We establish analogous estimates in the setting of definite quaternion algebras.