Abstract
We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in \(\mathbb {Q}\) of level \(\Gamma _{0}(N)\) for \(N\) satisfying some congruence conditions, and of level \(\Gamma _{1}(N)\). We give similar bounds for the graded \(\mathbb {Z}[\frac{1}{N}]\)-algebra of modular forms of level \(\Gamma _{1}(N)\) with coefficients in \(\mathbb {Z}[\frac{1}{N}]\). For a prime \(p \ge 5\), 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 of level \(\Gamma _{0}(p)\) with coefficients in \(\mathbb {Z}\), showing that, in contrast to the cases studied in the study of Rustom (J. Number Theory 138:97–118, 2014), this weight is unbounded. We generalize a result of Serre concerning congruences between modular forms of level \(\Gamma _{0}(p)\) 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.
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Notes
While \(f\) might have some poles along certain vertical (i.e. fibral) divisors, it cannot have a pole along a horizontal divisor. This is because any horizontal divisor would meet the generic fiber, and so if \(f\) has a pole along a horizontal divisor, then \(f\) would have a pole when considered as a modular form over \(\mathbb {C}\), which contradicts the holomorphy of \(f\) as a complex function of a complex variable. The vertical components of the divisor of poles correspond to the primes appearing in the denominators of the \(q\)-expansion of \(f\). As these denominators are bounded, one can find a constant \(K \in \mathbb {Z}\), \(K \not \equiv 0 \pmod {p}\) such that \(Kf\) has no primes in the denominators of its \(q\)-expansion except possibly \(p\). Multiplying by such a constant obviously preserves the \(p\)-adic valuation of \(f\) at both cusps.
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Acknowledgments
The author wishes to thank Jim Stankewicz for helpful discussions and clarifications he offered regarding the paper of Deligne and Rapoport [4] and their theory of the moduli stacks.
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Rustom, N. Generators and relations of the graded algebra of modular forms. Ramanujan J 39, 315–338 (2016). https://doi.org/10.1007/s11139-015-9674-z
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DOI: https://doi.org/10.1007/s11139-015-9674-z