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On the global sup-norm of GL(3) cusp forms

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Abstract

Let φ be a spherical Hecke–Maaß cusp form on the non-compact space PGL3(ℤ)PGL3(ℝ). We establish various pointwise upper bounds for φ in terms of its Laplace eigenvalue λφ. These imply, for φ arithmetically normalized and tempered at the archimedean place, the bound

$$||\phi |{|_\infty } \ll_\varepsilon \lambda _\phi ^{39/40 + \varepsilon }$$

for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the GL3 Jacquet–Whittaker function.

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Authors and Affiliations

  1. Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, D-37073, Göttingen, Germany

    Valentin Blomer

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest, H-1364, Hungary

    Gergely Harcos & Péter Maga

  3. Central European University, Nador u. 9, Budapest, H-1051, Hungary

    Gergely Harcos

Authors
  1. Valentin Blomer
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  2. Gergely Harcos
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  3. Péter Maga
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Corresponding author

Correspondence to Péter Maga.

Additional information

First author partially supported by the DFG-SNF lead agency program grant BL 915/2-1. Second and third author supported by NKFIH (National Research, Development and Innovation Office) grants NK 104183, ERC HU 15 118946, K 119528. Second author also supported by ERC grant AdG-321104, and third author also supported by the Postdoctoral Fellowship of the Hungarian Academy of Sciences.

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Blomer, V., Harcos, G. & Maga, P. On the global sup-norm of GL(3) cusp forms. Isr. J. Math. 229, 357–379 (2019). https://doi.org/10.1007/s11856-018-1805-y

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  • DOI: https://doi.org/10.1007/s11856-018-1805-y

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