Critical points of modular forms

TL;DR

This paper derives a formula for counting the critical points of modular forms within a fundamental domain, linking it to boundary zeros, transformations, and weights, and extends some insights to quasimodular forms.

Contribution

It provides a new explicit formula for the number of critical points of modular forms in a fundamental domain, relating it to boundary zeros and transformation parameters.

Findings

Derived a closed-form formula for critical points of modular forms.

Connected critical point count to boundary zeros and transformation data.

Extended discussion to zeros of quasimodular forms.

Abstract

We count the number of critical points of a modular form with real Fourier coefficients in a $\gamma$-translate of the standard fundamental domain $\mathcal{F}$ (with $\gamma\in \mathrm{SL}_2(\mathbb{Z})$). Whereas by the valence formula the (weighted) number of zeros of this modular form in $\gamma\mathcal{F}$ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of $\mathcal{F},$ the value of $\gamma^{-1}(\infty)$ and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form.