On vector differential forms attached to automorphic forms
On vector differential forms attached to automorphic forms
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论附加于自守形式的向量微分形式
DOI：
10.2969/jmsj/01230258
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发表时间：
1960
影响因子：
0.7
通讯作者：
G. Shimura
中科院分区：
文献类型：
--
作者：
M. Kuga;G. Shimura
对于群$G$的每个元素$\sigma$，其中$M（\sigma）$是$G$的张量表示。本文的目的是确定满足关系式（1）的所有全纯形式。$M$的次数为$2m-1$，我们可以给每一个$\leqq 2 m $的尖点形式附加一个具有表示llf的全纯形式$\omega$（定理1）。相反，任何满足（1）的全纯形式都可以表示为从次数为$\leqq 2 m $的尖点形式得到的形式之和;这个表达式给出了这种全纯形式的向量空间$\mathfrak{F}$的直接分解（定理2）。因此向量空间$\mathfrak{F}$的维数很容易得到，如果我们知道每个度的尖点形式的线性空间的维数。我们注意到，在[3]中所描述的意义下，附加于次数<2 m的尖点形式的形式的积分具有与0 $同调的周期。这一事实使附加于200万次尖点形式的形式区别于这些形式，这些尖点形式是[3]中研究的对象。
for every element $\sigma$ of the group $G$ , where $M(\sigma)$ is a tensor representation of $G$. The object of the present paper is to determine all holomorphic forms satisfying this relation (1). $M$ being of degree $2m-1$ , we can attach to every cusp form of degree $\leqq 2m$ a holomorphic form $\omega$ with the representation llf (Theorem 1). Conversely, any holomorphic form satisfying (1) is expressed as a sum of the forms thus obtained from cusp forms of degree $\leqq 2m$ ; and this expression gives a direct decomposition of the vector space $\mathfrak{F}$ of such holomorphic forms (Theorem 2). Hence the dimension of the vector space $\mathfrak{F}$ is easily obtained if we know the dimension of the linear space of cusp forms for each degree. We note that the integral of the form attached to a cusp form of degree $<2m$ has a period cohomologous to $0$ , in the sense described in [3]. This fact distinguishes among such forms the forms attached to cusp forms of degree $2m$ , which were the object of the investigation in [3].