Israel Journal of Mathematics

, Volume 229, Issue 1, pp 357–379 | Cite as

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

  • Valentin Blomer
  • Gergely Harcos
  • Péter MagaEmail author


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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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutGeorg-August Universität GöttingenGöttingenGermany
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Central European UniversityBudapestHungary

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