arXiv 1402.04052014-02-03math.NT

模形式的分次代数的生成元与关系

Generators and relations of the graded algebra of modular forms

Nadim Rustom

发表信息：: Ramanujan J. 39 (2016), no. 2, 315--338
DOI：: 10.1007/s11139-015-9674-z

摘要：

本文给出了同余子群$Γ_0(N)$上系数为$\mathbb{Q}$的分次模形式代数的关系理想的生成元的度的界，其中$N$满足某些同余条件，而$Γ_1(N)$满足某些同余条件.本文给出了$Γ_1(N)$上的分次模形式的$\mathbb{Z}[\frac{1}{N}]$ -代数的相似上界，其系数在$\mathbb{Z}[\frac{1}{N}]$中.对于一个总理$p \geq 5$，我们给出了一个下界的最高权重出现在一个最小的列表中的生成器$Γ_0(p)$，我们确定了一组生成器的分次代数$M(Γ_0(p),\mathbb{Z})$的模形式在$Γ_0(p)$与系数在$\mathbb{Z}$，表明这个重量是无界的。我们推广了Serre关于$Γ_0(p)$和$SL_2(\mathbb{Z})$上的模形式之间的同余关系的结果，并使用它来确定$M(Γ_0(p),\mathbb{Z})$的一组生成元，并且我们陈述了两个猜想，进一步详细说明了该代数的结构。最后，我们提供了计算有关的数量的生成元和关系，为这些代数，以及计算证据，这些代数。

英文摘要：

We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $Γ_0(N)$ for $N$ satisfying some congruence conditions and for $Γ_1(N)$. We give similar bounds for the graded $\mathbb{Z}[\frac{1}{N}]$-algebra of modular forms on $Γ_1(N)$ with coefficients in $\mathbb{Z}[\frac{1}{N}]$. For a prime $p \geq 5$, we give a lower bound on the highest weight appearing in a minimal list of generators for $Γ_0(p)$, and we identify a set of generators for the graded algebra $M(Γ_0(p),\mathbb{Z})$ of modular forms over $Γ_0(p)$ with coefficients in $\mathbb{Z}$, showing that this weight is unbounded. We generalize a result of Serre concerning congruences between modular forms over $Γ_0(p)$ and $SL_2(\mathbb{Z})$, and use it to identify a set of generators for $M(Γ_0(p),\mathbb{Z})$, 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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英文摘要：

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