Number of Polynomials Vanishing on a Basis of $S_m(\Gamma_0(N))$

TL;DR

This paper determines the count of homogeneous polynomials that vanish on a basis of cuspidal modular forms for certain modular curves, linking algebraic geometry and modular forms.

Contribution

It introduces a method to compute the number of such polynomials and finds the Hilbert polynomial of the associated projective curve.

Findings

Number of vanishing polynomials computed for given parameters

Hilbert polynomial of the embedded modular curve determined

Connection established between modular forms and algebraic geometry

Abstract

In this paper we find the number of homogeneous polynomials of degree d such that they vanish on cuspidal modular forms of even weight $m\geq 2$ that form a basis for $S_m(\Gamma_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to projective space and we find the Hilbert polynomial of the graded ideal of the projective curve that is the image of this embedding.