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2016 | OriginalPaper | Buchkapitel

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

verfasst von : Joshua S. Friedman, Jay Jorgenson, Jürg Kramer

Erschienen in: Arbeitstagung Bonn 2013

Verlag: Springer International Publishing

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Abstract

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 {f 1, , 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.

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Fußnoten
1
The views expressed in this article are the author’s own and not those of the U.S. Merchant Marine Academy, the Maritime Administration, the Department of Transportation, or the United States government.
 
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Metadaten
Titel
Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights
verfasst von
Joshua S. Friedman
Jay Jorgenson
Jürg Kramer
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-43648-7_6