Large values of modular forms

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

Full Text (PDF format)

Published 19 June 2014