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

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
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References:

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