A Higher Weight Analogue of Ogg's Theorem on Weierstrass Points

TL;DR

This paper extends Ogg's theorem to higher even weights, analyzing Weierstrass points on modular curves and the behavior of cusp forms of weight $k \\geq 4$, revealing new structural properties.

Contribution

It generalizes Ogg's theorem from weight 2 to even weights $k \\geq 4$, providing new insights into Weierstrass points and cusp form vanishing orders on modular curves.

Findings

Proves a higher weight analogue of Ogg's theorem for Weierstrass points.

Characterizes cusp forms vanishing to orders exceeding the dimension.

Analyzes the structure of cusp form spaces for modular curves.

Abstract

For a positive integer $N$, we say that $\infty$ is a Weierstrass point on the modular curve $X_0(N)$ if there is a non-zero cusp form of weight $2$ on $\Gamma_0(N)$ which vanishes at $\infty$ to order greater than the genus of $X_0(N)$. If $p$ is a prime with $p \nmid N$, Ogg proved that $\infty $ is not a Weierstrass point on $X_0(pN)$ if the genus of $X_0(N)$ is $0$. We prove a similar result for even weights $k \geq 4$. We also study the space of weight $k$ cusp forms on $\Gamma_0(N)$ vanishing to order greater than the dimension.