L^\infty norms of holomorphic modular forms in the case of compact quotient

TL;DR

This paper establishes a sub-convex bound for the maximum value of holomorphic modular forms of large weight on a compact quotient, improving understanding of their growth behavior.

Contribution

It provides the first sub-convex estimate for the sup-norm of holomorphic modular forms on compact quotients related to quaternion division algebras.

Findings

Sup-norm of eigenfunctions is bounded by k^{1/2 - 12/131 + ε}

Result applies to L^2-normalized holomorphic modular forms

Enhances bounds in the context of compact quaternionic quotients

Abstract

We prove a sub-convex estimate for the sup-norm of $L^2$-normalized holomorphic modular forms of weight $k$ on the upper half plane, with respect to the unit group of a quaternion division algebra over $\mf Q$. More precisely we show that when the $L^2$ norm of an eigenfunction $f$ is one, | f |_\infty \ll k^{1/2 - 12/131 + \varepsilon}, for any $\varepsilon>0$ and for all $k$ sufficiently large.