Eta-quotients and Embeddings of $X_0(N)$ in the Projective Plane

TL;DR

This paper constructs explicit projective plane models of modular curves $X_0(N)$ using eta-quotients of weight 12, focusing on those with maximal zero order at the cusp at infinity.

Contribution

It introduces a method to embed $X_0(N)$ into the projective plane via eta-quotients of weight 12, identifying those with maximal cusp zero order.

Findings

Explicit projective models for $X_0(N)$ constructed

Identification of eta-quotients with maximal cusp zero order

New embeddings of modular curves into the projective plane

Abstract

In this paper we find projective plane models of $X_0(N)$ by constructing maps from $X_0(N)$ to the projective plane using modular forms. We use eta-quotients of weight 12. We find those eta-quotients of weight 12 which have maximal order of zero at the cusp $\infty$.