On Higher Order Weierstrass Points on $X_0(N)$

TL;DR

This paper generalizes the classical isomorphism between modular forms and holomorphic differentials to higher orders, providing a new perspective on Weierstrass points on modular curves and an algorithm for their detection.

Contribution

It introduces a new description of higher order holomorphic differentials in terms of modular forms and develops an algorithm for identifying higher order Weierstrass points on $X_0(N)$.

Findings

Generalization of classical isomorphism to higher order differentials

Description of the subspace $S_m^H( ext{Gamma})$ of modular forms

Implementation of an algorithm in SAGE for testing Weierstrass points

Abstract

Let $\Gamma$ be the Fuchsian group of the first kind. For an even integer $m\ge 4$, we describe the space $H^{m/2}\left(\mathfrak R_\Gamma\right)$ of $m/2$--holomorphic differentials in terms of a subspace $S_m^H(\Gamma)$ of the space of (holomorphic) cuspidal modular forms $S_m(\Gamma)$. This generalizes classical isomorphism $S_2(\Gamma)\simeq H^{1}\left(\mathfrak R_\Gamma\right)$. We study the properties of $S_m^H(\Gamma)$. As an application, we describe the algorithm implemented in SAGE for testing if a cusp at $\infty$ for non-hyperelliptic $X_0(N)$ is a $\frac{m}{2}$-Weierstrass point.