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.

arXiv comment49 pages

References in corpus (18)

• Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets
• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• An additive problem in the Fourier coefficients of cusp forms
• Bounding sup-norms of cusp forms of large level
• Rankin Triple Products and Quantum Chaos
• Bounds for eigenforms on arithmetic hyperbolic 3-manifolds
• The sup-norm problem for GL(2) over number fields
• Hybrid bounds for automorphic forms on ellipsoids over number fields
• Local bounds for $L^p$ norms of Maass forms in the level aspect
• Bounds for twisted symmetric square $L$-functions via half-integral weight periods
• Theta functions, fourth moments of eigenforms, and the sup-norm problem I
• Quantum variance on quaternion algebras, II
• Beyond the spherical sup-norm problem
• Quantum variance on quaternion algebras, I
• Sup norms of newforms on $G{L}_2$ with highly ramified central character
• Hybrid bounds for the sup-norm of automorphic forms in higher rank
• Small diameters and generators for arithmetic lattices in ${\mathrm{SL}}_2(\mathbb{R})$ and certain Ramanujan graphs
• A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on $G{L}_2({\mathbb{F}}_q(T))$ with squarefree level

Cited by in corpus (1)

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