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

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.

This paper contains large parts of previous manuscript "On $m$-fold Holomorphic Differentials and Modular Forms" [arXiv:2101.00601]. The rest of the manuscript, related to cups forms constructed out of Wronskians, would be extended and published separately since does not fit here