Large values of modular forms

TL;DR

This paper demonstrates that certain holomorphic modular forms can attain large values proportional to the level, disproving the conjecture that their maximum size grows very slowly with the level.

Contribution

It provides a counterexample to the folklore conjecture by constructing forms with sup-norms significantly larger than previously expected.

Findings

Existence of modular forms with sup-norms growing as large as N^{1/4}

Disproof of the conjecture that sup-norms are as small as N^{o(1)}

Shows large values occur at some points for forms of arbitrary large level.

Abstract

We show that there are primitive holomorphic modular forms f of weight two and arbitrary large level N such that $|f(z)| \gg N^{1/4}$ for some point z. Thereby we disprove a folklore conjecture that the sup-norm of such forms would be as small as $N^{o(1)}$.