Generators and relations of the graded algebra of

Authors: Nadim Rustom

Publish Date: 2015/02/20

Volume: 39, Issue: 2, Pages: 315-338

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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 0N for N satisfying some congruence conditions and of level Gamma 1N We give similar bounds for the graded mathbb Zfrac1Nalgebra of modular forms of level Gamma 1N with coefficients in mathbb Zfrac1N For a prime p ge 5 we give a lower bound on the highest weight appearing in a minimal list of generators for Gamma 0p and we identify a set of generators for the graded algebra MGamma 0pmathbb Z of modular forms of level Gamma 0p with coefficients in mathbb Z showing that in contrast to the cases studied in the study of Rustom J Number Theory 13897–118 2014 this weight is unbounded We generalize a result of Serre concerning congruences between modular forms of level Gamma 0p and SL 2mathbb Z and use it to identify a set of generators for MGamma 0pmathbb 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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