Generators and relations of the graded algebra of modular forms

We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $\Gamma_0(N)$ for $N$ satisfying some congruence conditions and for $\Gamma_1(N)$. We give similar bounds for the graded $\mathbb{Z}[\frac{1}{N}]$-algebra of modular forms on $\Gamma_1(N)$ with coefficients in $\mathbb{Z}[\frac{1}{N}]$. For a prime $p \geq 5$, we give a lower bound on the highest weight appearing in a minimal list of generators for $\Gamma_0(p)$, and we identify a set of generators for the graded algebra $M(\Gamma_0(p),\mathbb{Z})$ of modular forms over $\Gamma_0(p)$ with coefficients in $\mathbb{Z}$, showing that this weight is unbounded. We generalize a result of Serre concerning congruences between modular forms over $\Gamma_0(p)$ and $SL_2(\mathbb{Z})$, and use it to identify a set of generators for $M(\Gamma_0(p),\mathbb{Z})$, and we state two conjectures detailing further the structure of this algebra. Finally we provide computations concerning the number of generators and relations for each of these algebras, as well as computational evidence for these conjectures.