Uniform bounds on sup-norms of holomorphic forms of real weight

TL;DR

This paper derives uniform bounds for the maximum size of holomorphic modular forms of any real weight, using Bergman kernels and Fourier analysis, applicable to broad classes of forms without eigenfunction assumptions.

Contribution

It extends sup-norm bounds to modular forms of arbitrary real weight and provides new techniques involving Bergman kernels and Poincaré series analysis.

Findings

Established uniform bounds for sup-norms of real weight modular forms.

Derived bounds over compact sets and the entire upper half-plane.

Proved the correct order of magnitude for the sum over an orthonormal basis.

Abstract

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis $\sum_j y^k|f_j(z)|^2$ and analysing this sum by means of a Bergman kernel and the Fourier coefficients of Poincar\'e series. As such our results are valid without any assumption that the forms are Hecke eigenfunctions. Under some weak assumptions we further prove the right order of magnitude of $\sup_{z \in \mathbb{H}} \sum_j y^k|f_j(z)|^2 $.