Sup-norm bounds for Eisenstein series and Maass cusp forms on compact sets

Establish sup-norm bounds on fixed compact sets Ω⊂H of the form sup_{z∈Ω}|E(z,1/2+it)|≪_ε t^ε for the Eisenstein series and sup_{z∈Ω}|φ(z)|≪_ε t_φ^ε for Hecke–Maass cusp forms on Γ=SL_2(Z), uniformly as t or t_φ→∞.

Background

The authors’ approach to sign changes along geodesics would be simplified by pointwise sup-norm bounds consistent with the sup-norm conjecture. While such pointwise bounds are not currently proven, they show that analogous bounds hold on average over the spectral parameter, which suffices for full-density results.

Proving these sup-norm bounds would directly imply the L^p bounds along geodesic segments used in conditional parts of the paper and would remove remaining conditionalities in similar problems.