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L ∞ norms of holomorphic modular forms in the case of compact quotient

L ∞ norms of holomorphic modular forms in the case of compact quotient Abstract We prove a sub-convex estimate for the sup-norm of L 2 -normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over ℚ. More precisely we show that when the L 2 norm of an eigenfunction f is one, ∥ f ∥ ∞ ≪ ε k 1 2 - 1 33 + ε $ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε > 0 ${\varepsilon >0}$ and for all k sufficiently large. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

L ∞ norms of holomorphic modular forms in the case of compact quotient

Forum Mathematicum , Volume 27 (4) – Jul 1, 2015

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References (14)

Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2013-6004
Publisher site
See Article on Publisher Site

Abstract

Abstract We prove a sub-convex estimate for the sup-norm of L 2 -normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over ℚ. More precisely we show that when the L 2 norm of an eigenfunction f is one, ∥ f ∥ ∞ ≪ ε k 1 2 - 1 33 + ε $ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε > 0 ${\varepsilon >0}$ and for all k sufficiently large.

Journal

Forum Mathematicumde Gruyter

Published: Jul 1, 2015

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