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# 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

#### 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