Generators of graded rings of modular forms

Rustom, Nadim

doi:10.1016/j.jnt.2013.12.008

Cited by 12 publications

(24 citation statements)

References 14 publications

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“…In fact, Lemma 1.1 allows us to show, using the same proof as in [Rus12], that the bound in part(2) of Corollary 1.2 can actually be taken to be 4.…”

Section: In [Rus12]mentioning

confidence: 93%

“…We argue as in the proof of Theorem 1 in [Rus12]. Let R 1 , · · · , R m be the relations that generate ker Φ up to degree 6.…”

Section: In [Rus12]mentioning

confidence: 99%

“…We recall the defintion of a T -form, first defined in [Sch79], which played a key role in our work appearing in [Rus12].…”

Section: Notation Definitions and Conventionsmentioning

confidence: 99%

“…In [Rus12], we studied the graded algebras of modular forms of various levels and over subrings A of C. Our main theorem in [Rus12] was that for N ≥ 5, the algebra M (Γ 1 (N ), Z[…”

Section: Introductionmentioning

confidence: 99%

“…We recall here that the T -form played a fundamental role in the algorithm we developed in [Rus12] based on the work of Scholl in [Sch79]. In particular, for f = n≥0 a n q n , we define the p-adic valuation of f as:…”

Section: Introductionmentioning

confidence: 99%

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Generators and relations of the graded algebra of modular forms

Rustom

2015

Ramanujan J

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We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in Q over congruence subgroups Γ0(N ) for N satisfying some congruence conditions and for Γ1(N ). We give similar bounds for the graded Z[ ]. For a prime p ≥ 5, we give a lower bound on the highest weight appearing in a minimal list of generators for Γ0(p), and we identify a set of generators for the graded algebra M (Γ0(p), Z) of modular forms over Γ0(p) with coefficients in Z, showing that, in contrast to the cases studied in [Rus12], this weight is unbounded. We generalize a result of Serre concerning congruences between modular forms over Γ0(p) and SL2(Z), and use it to identify a set of generators for M (Γ0(p), 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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“…In fact, Lemma 1.1 allows us to show, using the same proof as in [Rus12], that the bound in part(2) of Corollary 1.2 can actually be taken to be 4.…”

Section: In [Rus12]mentioning

confidence: 93%

“…We argue as in the proof of Theorem 1 in [Rus12]. Let R 1 , · · · , R m be the relations that generate ker Φ up to degree 6.…”

Section: In [Rus12]mentioning

confidence: 99%

“…We recall the defintion of a T -form, first defined in [Sch79], which played a key role in our work appearing in [Rus12].…”

Section: Notation Definitions and Conventionsmentioning

confidence: 99%

“…In [Rus12], we studied the graded algebras of modular forms of various levels and over subrings A of C. Our main theorem in [Rus12] was that for N ≥ 5, the algebra M (Γ 1 (N ), Z[…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generators and relations of the graded algebra of modular forms

Rustom

2015

Ramanujan J

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Rings of modular forms and a splitting of $${{\,\mathrm{TMF}\,}}_0(7)$$

Meier

Ozornova

2020

Sel. Math. New Ser.

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Among topological modular forms with level structure, TMF 0 (7) at the prime 3 is the first example that had not been understood yet. We provide a splitting of TMF 0 (7) at the prime 3 as TMF-module into two shifted copies of TMF and two shifted copies of TMF 1 (2). This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level n and introduce cubical versions of moduli stacks of elliptic curves with level structure.While our computation of mf(Γ 1 (7); Z) is not difficult to obtain, the reader should compare the simplicity of its expression with the presentation Rustom [50, Section 3.1] is forced to give by ignoring the elements of weight 1.After these explicit computations, we also prove more structural results about rings of modular forms. Theorem 1.3. For every n ≥ 2, the ring mf(Γ 1 (n); Z[ 1 n ]) is normal. Using results from [41], we give moreover a criterion for mf(Γ 1 (n); Z[ 1 n ]) being Cohen-Macaulay, which is satisfied for all 2 ≤ n ≤ 28.These commutative algebra results help us to develop cubical analogues of the stacks M 1 (n). To this purpose recall that M 1 (n) is usually defined as the normalization of the compactified moduli stack of elliptic curves M ell in M 1 (n). The stack M ell embeds into the larger stack M cub of all curves that can locally be described by a cubic Weierstraß equation. We can define M 1 (n) cub as the normalization of M cub in M 1 (n). We show:is Cohen-Macaulay. Our reason to consider these cubical stacks is that vector bundles seem to be easier to study on M cub than on M ell , a view inspired by work of Mathew [36]. This line of thought is already implicit though in the classification of vector bundles over weighted projective lines in [40, Proposition 3.4], where the idea was to extend them to vector bundles on the non-separated stack A 2 /G m . For M cub this idea takes the form that there is a nice smooth cover Spec A → M cub with A = Z[a 1 , a 2 , a 3 , a 4 , a 6 ] given by Weierstraß curves and thus quasi-coherent sheaves on M cub become equivalent to comodules over a certain explicit Hopf algebroid (A, Γ). For explicit calculations, this outweighs the disadvantage that M cub is neither Deligne-Mumford nor separated.Our wish for such explicit calculations was motivated by the results from [41].

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