Continued fractions, modular symbols, and noncommutative geometry
Continued fractions, modular symbols, and noncommutative geometry
复制标题
连分数、模符号和非交换几何
DOI：
10.1007/s00029-002-8113-3
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
M. Marcolli
中科院分区：
文献类型：
--
作者：
Y. Manin;M. Marcolli
抽象的。利用D. Mayer，我们证明了经典的Gauss-Kuzmin定理关于连分数的分布的一个扩展，特别是允许考虑连续收敛的一些同余性质。这一结果可以应用于广义相对论中的混合宇宙模型。然后，我们研究了一些涉及模符号的平均值，并表明可以通过对用连分数定义的真实的轴上的某些函数进行积分来获得与权重为2的模形式相关的狄利克雷级数。证明了商PGL（2，Z）\ P1（R）是非交换模曲线，并证明了模复形可以看作是相关叉积C*-代数的K 0-群的序列.
Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2, Z) \ P1(R) should be considered as noncommutative modular curve, and show that the modular complex can be seen as a sequence of K0-groups of the related crossed-product C*-algebras.