论文标题
Hermitian模块化组的乘数系统
Multiplier systems for Hermitian modular groups
论文作者
论文摘要
让$γ_{f，n} $为HEL BRAUN的HELMINIAN模块化$ n> 1 $相对于虚构的二次field $ f $。令$ r $为自然数字。如果并且仅当$ r = 1 $或$ 2 $时，就存在着一个重量$ 1/r $的乘数$ 1/r $（等效地等效的权重$ k+1/r $，$ k $积分）。这是由Deligne [de]与Hill [Hi]，Prasad [P]和Prasad-Rapinchuk [PR]的结果结合得多的一般结构。据我们所知，重量$ 1/2 $的系统尚未明确描述。王（WA）非常明显地给出了一个模块化形式的一半积分重量的例子。实际上，他为一组$ o（2,4）$构建了重量$ 23/2 $的Borcherds产品。该组对包含二级学位的Hermitian模块化组的$ U（2,2）$是不同意的。在本文中，我们想研究此类乘数系统。如果将它们限制在单模型组中，则获得了通常的特征。我们的主要结果指出，该特征的内核是一个非综合子组。对于足够小的$γ$，它与久保田在$ n = 2 $的情况下所描述的小组和贝斯·米尔诺·塞雷（Bass Milnor Serre）在$ n> 2 $的情况下相吻合。
Let $Γ_{F,n}$ be the Hermitian modular group of degree $n>1$ in sense of Hel Braun with respect to an imaginary quadratic field $F$. Let $r$ be a natural number. There exists a multiplier system of weight $1/r$ (equivalently a Hermitian modular form of weight $k+1/r$, $k$ integral) on some congruence group if and only if $r=1$ or $2$. This follows from a much more general construction of Deligne [De] combining it with results of Hill [Hi], Prasad [P] and Prasad-Rapinchuk [PR]. As far as we know, the systems of weight $1/2$ have not yet been described explicitly. Remarkably Haowu Wang [Wa] gave an example of a modular form of half integral weight. Actually he constructs a Borcherds product of weight $23/2$ for a group of type $O(2,4)$. This group is isogenous to the group $U(2,2)$ that contains the Hermitian modular groups of degree two. In this paper we want to study such multiplier systems. If one restricts them to the unimodular group one obtains a usual character. Our main result states that the kernel of this character is a non-congruence subgroup. For sufficiently small $Γ$ it coincides with the group described by Kubota in the case $n=2$ and by Bass Milnor Serre in the case $n>2$.
