Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms
Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms
复制标题
西格尔上半平面上的对合模变换及其在二次形式表示中的应用
DOI：
10.1016/0022-314x(86)90007-7
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发表时间：
1986
影响因子：
0.7
通讯作者：
Robert Sibner
中科院分区：
文献类型：
--
作者：
K. Hashimoto;Robert Sibner
建立了Sp （n， Z）中满足x2 + I= 0 （resp）的椭圆变换共轭类集之间的一一对应关系。x2 + X+ I= 0)和秩n / Z[√−1]的厄米特形式集合(resp。Z[(−1+√−3)/2])的行列式±1。作为应用，我们将经典事实推广到n阶的正对称积分矩阵S上，即m2 + 1的任意除数（p < 0.05）。m+ m+ 1)可以用二次形式F (X， Y)= x2 + y2 (resp。X 2+ XY+ Y 2)与相对素数X， Y：设n≤3 (n≤5)。如果det S用F表示，则S可以用F⊗n （n个F的副本）表示。证明是基于西格尔-布劳恩质量公式的厄米形式。
We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp (n, Z) which satisfy X 2+ I= 0 (resp. X 2+ X+ I= 0) and the set of hermitian forms of rank n over Z [√− 1](resp. Z [(− 1+√− 3)/2]) of determinant±1. As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of m 2+ 1 (resp. m+ m+ 1) can be represented by the quadratic form F (X, Y)= X 2+ Y 2 (resp. X 2+ XY+ Y 2) with relatively prime integers X, Y: Suppose n≤ 3 (resp. n≤ 5). Then S can be represented over Z by F⊗ n (n copies of F) if det S is represented by F as above. The proof is based on Siegel-Braun's Mass formula for hermitian forms.