一类二型Fuchsian 群的局部化性质

胡小虎，陈 波

(重庆交通大学理学院，重庆 400074)

一类二型Fuchsian 群的局部化性质

胡小虎，陈 波

(重庆交通大学理学院，重庆 400074)

为了讨论整数的整二次型表示，引进整二次型的θ-级数，并发现θ-级数与模群的自守形式有紧密的联系。Fuchsian群及其自守形式是模群与模形式的重要延伸。对于一类二型Fuchsian群H()，这是 G()的一个子群。众所周知，G()中的元素要落H()中需要有诸多限制。简化这些限制条件是很有意义的。利用p-局部化方法，首先给出H()的局部化的性质。然后，给出它与G()关于模pn的一个关系。

Fuchsian群;p-进数;同余子群

1 介绍

Fuchsian群为一类重要的离散群。一方面它对刻画黎曼曲面的覆盖群具有重要意义;另一方面，由于对于二型的Fuchsian群保持庞加莱上半平面不变并且它的基本域有无限体积，于是它的自守形式为一型Fuchsian群的自守形式理论的重要补充(见文献［1-5］)。令q为大于4的正整数，本文考虑两类二型的Fuchsian群H()和G ()。

2 定理的证明

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Local Properties of Some Class of Second Fuchsian Group

HU Xiao-hu，CHEN Bo
(School of Sciences，Chongqing Jiaotong University，Chongqing 400074，China)

In order to discuss the representations of an integer by some integral quadric form，theta series of its quadric form was introduced and the close connections between them were found.Fuchsian groups and their automorphic forms were important extensions of modular group and modular forms.For the second Fuchsian group H()，it was a subgroups of G().It was well known that the elements of G()which could be in H()must satisfy many conditions.So it was significant to simplify these conditions.A local property of H)was given Byp-adic localization and relation between G()and H()by modulo pnwas established.

Fuchsian group;p-adic number;congruence subgroup.

O156.5

A

1674-0696(2011)03-0511-03

2010-11-15;

2111-03-10

胡小虎(1975-)，男，重庆铜梁人，讲师，硕士，主要从事微分几何与代数方面的研究。E-mail:huxiaohu2008@sina.com。