The critical values of certain zeta functions associated with modular forms of half-integral weight
The critical values of certain zeta functions associated with modular forms of half-integral weight
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与半积分权重模形式相关的某些 zeta 函数的临界值
DOI：
10.2969/jmsj/03340649
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发表时间：
1981
影响因子：
0.7
通讯作者：
G. Shimura
中科院分区：
文献类型：
--
作者：
G. Shimura
另一个是$f$和$g$的内积$\langle f，g\rangle$，当它们具有相同的权重时。在我们以前的文章[12]，[13]和[14]中，对于整权的形式f$和g$，已经讨论过这些问题。因此，目前的调查涉及的情况下，其中一个或两个$f$和$g$有半整数的重量。为了描述问题和结果的性质，我们让$m^{\prime}$和$m$分别表示f$和g$的权重，它们是2 ^{-1}Z$的正元素。如果$m=m^{\prime}$，则$\langle f，g\rangle$和$D（s，f，g）$在$s=m$处的剩余之间存在一个众所周知的关系，因此第二个对象具有与第一个对象相似的特征。撇开这个残数不谈，出于一些自然原因，我们将$D（s，f，g）$的值的研究限制为$m<m^{\prime}$的情况。如果我们排除$m$和$m^{\prime}$都是整数的情况，有三种情况：
and the other is the inner product $\langle f, g\rangle$ of $f$ and $g$ , when they have the same weight. These have been treated in our previous papers [12], [13], and [14] for the forms $f$ and $g$ of integral weight. Therefore the present investigation concerns the cases in which either or both of $f$ and $g$ have half-integral weight. To describe the nature of the problems as well as of the results, we let $m^{\prime}$ and $m$ denote the weights of $f$ and $g$ , respectively, which are positive elements of $2^{-1}Z$. If $m=m^{\prime}$ , there is a well known relation between $\langle f, g\rangle$ and the residue of $D(s, f, g)$ at $s=m$ , and therefore the second object has a character similar to the first one. Setting this residue aside, we restrict, for some natural reasons, the study of the values of $D(s, f, g)$ to the case $m<m^{\prime}$ . If we exclude the case in which both $m$ and $m^{\prime}$ are integers, there are three cases: