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11-XX Number theory
11Dxx Diophantine equations
11D45 Counting solutions of Diophantine equations

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Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

Let $f$ be a Hecke–Maass cusp form of eigenvalue $λ$ and square-free level $N$ . Normalize the hyperbolic measure such that $vol (Y_{0} (N)) = 1$ and the form $f$ such that ${∥ f ∥}_{2} = 1$ . It is shown that ${∥ f ∥}_{\infty} ≪_{ϵ} λ^{\frac{5}{24} + ϵ} N^{\frac{1}{3} + ϵ}$ for all $ϵ > 0$ . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

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