Notes on Weierstrass points of modular curves $X_0(N)$

TL;DR

This paper investigates the conditions under which fixed points of Atkin-Lehner involutions on modular curves $X_0(N)$ are Weierstrass points, extending previous results and providing a complete classification.

Contribution

It extends Lehner and Newman's work by identifying when fixed points of partial and full Atkin-Lehner involutions are Weierstrass points on $X_0(N)$.

Findings

Conditions for fixed points of partial involutions to be Weierstrass points

Complete classification of fixed points of full involutions as Weierstrass points or not

Extension of previous results by Lehner and Newman

Abstract

We give conditions when the fixed points by the partial Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points as an extension of the result by Lehner and Newman \cite{LN}. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points or not.