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 \[ \| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_ε (k V)^{\frac{1}{4}+ε} \] with absolute implied constant. For a cuspidal Maaß newform $\phi$ of eigenvalue $\lambda$ on such a surface, we prove that \[ \|φ\|_{\infty} \ll_{λ,ε} V^{\frac{1}{4}+ε}. \] We establish analogous estimates in the setting of definite quaternion algebras.

49 pages

References in corpus (3)

• Rankin Triple Products and Quantum Chaos
• Quantum variance on quaternion algebras, II
• Small diameters and generators for arithmetic lattices in ${\mathrm{SL}}_2(\mathbb{R})$ and certain Ramanujan graphs