胡小虎，陳 波

(重慶交通大學(xué)理學(xué)院，重慶 400074)

一類二型Fuchsian 群的局部化性質(zhì)

胡小虎，陳 波

(重慶交通大學(xué)理學(xué)院，重慶 400074)

為了討論整數(shù)的整二次型表示，引進(jìn)整二次型的θ-級數(shù)，并發(fā)現(xiàn)θ-級數(shù)與模群的自守形式有緊密的聯(lián)系。Fuchsian群及其自守形式是模群與模形式的重要延伸。對于一類二型Fuchsian群H()，這是 G()的一個(gè)子群。眾所周知，G()中的元素要落H()中需要有諸多限制。簡化這些限制條件是很有意義的。利用p-局部化方法，首先給出H()的局部化的性質(zhì)。然后，給出它與G()關(guān)于模pn的一個(gè)關(guān)系。

Fuchsian群;p-進(jìn)數(shù);同余子群

1 介紹

Fuchsian群為一類重要的離散群。一方面它對刻畫黎曼曲面的覆蓋群具有重要意義;另一方面，由于對于二型的Fuchsian群保持龐加萊上半平面不變并且它的基本域有無限體積，于是它的自守形式為一型Fuchsian群的自守形式理論的重要補(bǔ)充(見文獻(xiàn)［1-5］)。令q為大于4的正整數(shù)，本文考慮兩類二型的Fuchsian群H()和G ()。

2 定理的證明

［1］Cangul I N，Singerman D.Normal subgroup of hecke groups and regular maps［J］.Math Proc Phil Soc，1998，123:59-74.

［2］Parson L A.Generalized kloosterman sum and the fourier coefficients of cusp forms［J］.Trans.Ams，1976，217:329-350.

［3］Katok S.Fuchsianian Groups［M］.Chicago:University of Chicago Press，1992.

［4］Knopp M I，Sheingorn M.On dirichlet series and hecke triangle groups of infinite volume［J］.Acta Arithmetia，1996.106(3):227-244.

［5］Iwaniec H.Spectral methods of automorphic forms［M］.Providence:Amer Math Soc，2002.

［6］Rosen D.A class of continued fractions associated with certain properly discontinuous group［J］.D.Math.J，1954，21(3):549-563.

［7］Ozgur N Y.Principal congruence subgroups Hecke Groups H()［J］.Acta Math Sinica，2006，22(2):383-392.

［8］Macbeath A M.Generators of the linear fractional groups［J］.Pure Math，Amer.Math.Soc，1969，12(1):14-32.

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，重慶銅梁人，講師，碩士，主要從事微分幾何與代數(shù)方面的研究。E-mail:huxiaohu2008@sina.com。