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arXiv 0910.3976math.NT
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对数向量值模形式

Logarithmic vector-valued modular forms

Marvin Knopp, Geoffrey Mason

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22 pages
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摘要：

我们考虑对数向量和矩阵值的模形式的积分重量$k$与$p$维表示$ρ: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$的模块组，仅受条件是$ρ(T)$的特征值的绝对值为1。主要结果是构造了与$ρ$相关联的亚纯矩阵值Poincaré级数.分量函数是对数$q$系列，即，$q$ -级数乘积的有限和与$\log q$的幂。我们得到几个结果，特别是我们表明，所有全纯对数向量值模形式的空间$\mathcal{H}(ρ)=\oplus_k \mathcal{H}(k, ρ)$与$ρ$是一个自由模的秩$p$上的环上的经典全纯模形式的$SL_2(\mathbb{Z})$。

英文摘要：

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $ρ: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $ρ(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to $ρ$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(ρ)=\oplus_k \mathcal{H}(k, ρ)$ of all holomorphic logarithmic vector-valued modular forms associated to $ρ$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

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