Generators of graded rings of modular forms

TY - JOUR

T1 - Generators of graded rings of modular forms

AU - Rustom, Nadim

PY - 2014/5

Y1 - 2014/5

N2 - We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A of 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 C-algebras of modular forms over Γ1(N) and Γ0(N) with some conditions on N. For N ≥ 5, we prove that the Z[1/N]-algebra of modular forms over Γ1(N) with coefficients in 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 Γ0(N).

AB - We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A of 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 C-algebras of modular forms over Γ1(N) and Γ0(N) with some conditions on N. For N ≥ 5, we prove that the Z[1/N]-algebra of modular forms over Γ1(N) with coefficients in 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 Γ0(N).

U2 - 10.1016/j.jnt.2013.12.008

DO - 10.1016/j.jnt.2013.12.008

M3 - Journal article

SN - 0022-314X

VL - 138

SP - 97

EP - 118

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -