Supnorm of Modular Forms of half-integral Weight in the Weight Aspect

TL;DR

This paper establishes bounds on the maximum size of half-integral weight Hecke eigenforms in the Kohnen plus space as their weight increases, using Fourier analysis and amplification techniques.

Contribution

It introduces a novel combination of Fourier expansion bounds and the amplification method with a Bergman kernel to bound the supnorm in the weight aspect.

Findings

Bound on supnorm grows at a specific rate with weight

Effective bounds achieved for half-integral weight forms

Method can be adapted to other automorphic forms

Abstract

We bound the supnorm of half-integral weight Hecke eigenforms in the Kohnen plus space of level $4$ in the weight aspect, by combining bounds obtained from the Fourier expansion with the amplification method using a Bergman kernel.