VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS
VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS
复制标题
向量值模形式和线性微分算子
DOI：
10.1142/s1793042107000973
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发表时间：
2007
影响因子：
0.7
通讯作者：
G. Mason
中科院分区：
文献类型：
--
作者：
G. Mason
考虑全模群Γ = SL（2，n）上整权为k的全纯向量值模形式F对应于任意有限维p的Γ的表示.假设F的分量函数线性无关，证明了不等式k ≥ 1 - p总是成立，且等式仅在p = 1和k = 0的平凡情形下成立.对于任意p ≥ 2，我们给出了如何构造k = 2 -p的大量的Γ的表示，其关键思想是考虑系数为模形式的线性微分方程的解空间上的Γ的表示.
We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p ≥ 2, we show how to construct large numbers of representations of Γ for which k = 2 - p. The key idea is to consider representations of Γ on spaces of solutions of certain linear differential equations whose coefficients are modular forms.