Cambridge Journal of Mathematics

Volume 2 (2014)

Number 1

Large values of modular forms

Pages: 91 – 116

DOI: http://dx.doi.org/10.4310/CJM.2014.v2.n1.a3

Author

Nicolas Templier ( Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Abstract

We show that there are primitive holomorphic modular forms $f$ of arbitrary large level $N$ such that $\lvert f(z) \rvert \gg N^{\frac{1}{4}}$ for some $z \in \mathfrak{H}$. Thereby we disprove a folklore conjecture that the $L^\infty$-norm of such forms would be as small as $N^{o(1)}$.

Keywords

Whittaker functions, quantum chaos, automorphic forms, sup-norm, $L$-functions, mean value estimates

2010 Mathematics Subject Classification

11F41, 11F70

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