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arXiv · 1209.3864

Generators of graded rings of modular forms

Nadim Rustom
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Abstract

We study graded rings of modular forms over congruence subgroups, with coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest weight needed to generate these rings as $A$-algebras. In particular, we determine upper bounds, independent of $N$, for the highest needed weight that generates the $\mathbb{C}$-algebras of modular forms over $\Gamma(N)$, $\Gamma_1(N)$ and $\Gamma_0(N)$ with some conditions on $N$. For $N \geq 5$, we prove that the $\mathbb{Z}[1/N]$-algebra of modular forms over $\Gamma_1(N)$ with coefficients in $\mathbb{Z}[1/N]$ is generated in weight at most 3. We give an algorithm that computes the generators, and supply some computations that allow us to state two conjectures concerning the situation over $\Gamma_0(N)$.

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Nadim Rustom. 2012-09-18. Generators of graded rings of modular forms. https://arxiv.org/abs/1209.3864

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