Hybrid sup-norm bounds for Maass newforms of powerful level. (English) Zbl 1432.11044

Saha, Abhishek

Algebra Number Theory 11, No. 5, 1009-1045 (2017).

Summary: Let \(f\) be an \(L^2\)-normalized Hecke-Maass cuspidal newform of level \(N\), character \(\chi\) and Laplace eigenvalue \(\lambda\). Let \(N_1\) denote the smallest integer such that \(N\mid N_1^2\) and \(N_0\) denote the largest integer such that \(N_0^2\mid N\). Let \(M\) denote the conductor of \(\chi\) and define \(M_1= M/\gcd(M,N_1)\). We prove the bound
\[ \| f\|_\infty\ll_{\varepsilon}N_0^{1/6+\varepsilon}N_1^{1/3+\varepsilon}M_1^{1/2}\lambda^{5/24+\varepsilon}, \]
which generalizes and strengthens previously known upper bounds for \(\| f\|_\infty\).
This is the first time a hybrid bound (i.e., involving both \(N\) and \(\lambda\)) has been established for \(\| f\|_\infty\) in the case of nonsquarefree \(N\). The only previously known bound in the nonsquarefree case was in the \(N\)-aspect; it had been shown by the author that
\[\| f\|_\infty\ll_{\lambda,\varepsilon}N^{5/12+\varepsilon}\]
provided \(M=1\). The present result significantly improves the exponent of \(N\) in the above case. If \(N\) is a squarefree integer, our bound reduces to
\[ \| f\|_\infty\ll_\varepsilon N^{1/3+\varepsilon}\lambda^{5/24+\varepsilon}, \]
which was previously proved by Templier.
The key new feature of the present work is a systematic use of \(p\)-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for \(\mathrm{GL}_2(F)\) where \(F\) is a local field.

MSC:

11F12	Automorphic forms, one variable
11F60	Hecke-Petersson operators, differential operators (several variables)
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
11F85	\(p\)-adic theory, local fields

Keywords:

Maass form; sup-norm; automorphic form; newform; amplification

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