Hybrid sup-norm bounds for Hecke-Maass cusp forms. (English) Zbl 1376.11030
Summary: Let \(f\) be a Hecke-Maass cusp form of eigenvalue \(\lambda\) and square-free level \(N\). Normalize the hyperbolic measure such that \(\mathrm {vol}(Y_0(N))=1\) and the form \(f\) such that \(\|{f}\|_2=1\). It is shown that \(\|{f}\|_\infty \ll_\varepsilon \lambda^{5/24+\varepsilon} N^{1/3+\varepsilon}\) for all \(\varepsilon>0\). This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.
MSC:
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
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