Generators of graded rings of modular forms

Nadim Rustom

doi:10.1016/j.jnt.2013.12.008

Generators of graded rings of modular forms

Nadim Rustom

Department of Mathematical Sciences

5 Citations (Scopus)

Abstract

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).

Original language	English
Journal	Journal of Number Theory
Volume	138
Pages (from-to)	97-118
ISSN	0022-314X
DOIs	https://doi.org/10.1016/j.jnt.2013.12.008
Publication status	Published - May 2014

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abstract = "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).",

author = "Nadim Rustom",

year = "2014",

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language = "English",

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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 -

Generators of graded rings of modular forms

Abstract

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