Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights

  • Joshua S. Friedman
  • Jay Jorgenson
  • Jürg KramerEmail author
Part of the Progress in Mathematics book series (PM, volume 319)


Let \(\Gamma \subset \mathrm{ PSL}_{2}(\mathbb{R})\) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane \(\mathbb{H}\). Consider the d-dimensional space of cusp forms \(\mathcal{S}_{2k}^{\Gamma }\) of weight 2k for \(\Gamma\), and let {f1, , f d } be an orthonormal basis of \(\mathcal{S}_{2k}^{\Gamma }\) with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity \(S_{2k}^{\Gamma }(z):=\sum _{ j=1}^{d}\vert f_{j}(z)\vert ^{2}\,\mathrm{Im}(z)^{2k}\) is bounded as \(O_{\Gamma }(k)\) in the cocompact setting, and as \(O_{\Gamma }(k^{3/2})\) in the cofinite case, where the implied constants depend solely on \(\Gamma\). We also show that the implied constants are uniform if \(\Gamma\) is replaced by a subgroup of finite index.



Jorgenson acknowledges support from numerous NSF and PSC-CUNY grants. Kramer acknowledges support from the DFG Graduate School Berlin Mathematical School and from the DFG International Research Training Group Moduli and Automorphic Forms.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joshua S. Friedman
    • 1
  • Jay Jorgenson
    • 2
  • Jürg Kramer
    • 3
    Email author
  1. 1.Department of Mathematics and ScienceUnited States Merchant Marine AcademyKings PointUSA
  2. 2.Department of MathematicsThe City College of New YorkNew YorkUSA
  3. 3.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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