Special Number Theory Seminar

Let \(N \geq 3\). In this talk, I will sketch a proof that the ring generated by Eisenstein series of weight \(1\) on the principal congruence subgroup \(\Gamma(N)\) contains all modular forms in weights \(2\) and above. This means that the only forms that are not seen by polynomials in these Eisenstein series are cusp forms of weight \(1\). This result gives rise to a systematic way to produce equations for the modular curve \(X(N)\).