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Transactions of the American Mathematical Society

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Effective sup-norm bounds on average for cusp forms of even weight


Authors: J. S. Friedman, J. Jorgenson and J. Kramer
Journal: Trans. Amer. Math. Soc. 372 (2019), 7735-7766
MSC (2010): Primary 11F11, 11F72, 30F10; Secondary 47A10
DOI: https://doi.org/10.1090/tran/7933
Published electronically: September 9, 2019
MathSciNet review: 4029679
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Abstract: Let $\Gamma \subset \mathrm {PSL}_{2}(\mathbb {R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb {H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal {S}_{2k}^{\Gamma }$ of weight $2k$ for $\Gamma$, and let $\{f_{1},\ldots ,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal {S}_{2k}^{\Gamma }$ with respect to the Petersson inner product. In this paper, we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^{\Gamma }(z):=\sum _{j=1}^{d_{2k}}\vert f_{j}(z)\vert ^{2} \mathrm {Im}(z)^{2k}$ as $z$ ranges through $\mathbb {H}$.


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Additional Information

J. S. Friedman
Affiliation: Department of Mathematics and Science, United States Merchant Marine Academy, 300 Steamboat Road, Kings Point, New York 11024
MR Author ID: 772419
Email: FriedmanJ@usmma.edu

J. Jorgenson
Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031
MR Author ID: 292611
Email: jjorgenson@mindspring.com

J. Kramer
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
MR Author ID: 227725
Email: kramer@math.hu-berlin.de

Received by editor(s): January 22, 2018
Received by editor(s) in revised form: August 14, 2018, and January 14, 2019
Published electronically: September 9, 2019
Additional Notes: The views expressed in this article are the first 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.
The second author acknowledges support from numerous PSC-CUNY grants.
The third author acknowledges support from the DFG Graduate School Berlin Mathematical School.
Article copyright: © Copyright 2019 American Mathematical Society