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The Ramanujan Journal

, Volume 48, Issue 3, pp 623–638 | Cite as

Supnorm of an eigenfunction of finitely many Hecke operators

  • Subhajit JanaEmail author
Article

Abstract

Let \(\phi \) be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaternion algebra. Assuming that \(\phi \) is an eigenfunction of Hecke operators at a fixed finite collection of primes, we prove an \(L^\infty \)-norm bound for \(\phi \) that improves upon the trivial estimate by a power of the logarithm of the eigenvalue. We have constructed an amplifier whose length depends on the support of the amplifier on Hecke trees. We have used a method of Bérard (Math Z 155: 249–276, 1977) to improve the Archimedean amplification.

Keywords

Automorphic form Hecke operator Amplifier 

Mathematics Subject Classification

11F12 11F72 

Notes

Acknowledgements

The author thanks Lior Silberman whose supervision during the author’s masters thesis helped to start this project and also for encouragement. The author thanks Paul Nelson for teaching the amplification method he used in [15] and for the numerous helpful suggestions on an earlier draft of this paper. The author also thanks Djordje Milićević, Gergely Harcos, Kevin Nowland, Felix Dräxler and the anonymous referee for several helpful feedbacks, comments and suggestions on various aspects of the work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ETH ZürichZurichSwitzerland

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