The modular symbol and continued fractions in higher dimensions
The modular symbol and continued fractions in higher dimensions
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高维中的模符号和连分数
DOI：
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发表时间：
1979
期刊：
影响因子：
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通讯作者：
L. Rudolph
中科院分区：
文献类型：
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作者：
A. Ash;L. Rudolph
模符号已被证明是处理上半平面上的全纯自守形式的一个非常有用的工具。例如，它用于计算Hecke算子在自守形式上的作用，给出某些椭圆曲线的扭曲L-函数的值，并证明在模曲线的尖点上支持的零度因子在该曲线的雅可比矩阵中具有有限阶。对于这些问题，见[1]，章。第四和第五，以及那里引用的参考文献。这里是模符号的定义。设H是上半平面，即所有虚部为正的复数的集合。H的尖点被定义为任意有理数（被认为是真实的直线上的一点）或α。尖点属于H在扩展复平面上的边界。设SL（2，1 R）以通常的方式作用于H。对于SL（2，9）中任意有限指标子群F，我们可以构造商H/F。存在一个闭的、紧的黎曼曲面X（F），使得H/F双全纯于某个有限点集x 1的补。. . .，x~ in X（F）.这些点xi称为X（F）的尖点。让~？(F)表示X（F）上的全纯微分l-形式。圈上形式的积分得到从H~（X（F）;IR）到~（F）的复对偶空间的一个标准N-线性映射。黎曼曲面理论告诉我们，这个映射是同构的。现在在H的边界上选择任意两个有理尖点a和B。设l是任意合理的弧，从a开始，到B结束，在此期间停留在H中，例如，l可以是H中从a到B的测地线。然后，人们很容易地表明，积分的形象l在X（F）给出了一个线性映射？(F)我12这个线性映射以上面描述的规范方式给出了HI（X（F）; IR）的元素。我们用[a，hi]表示这个同调类。我们称[ ]为模符号。总之，“经典”模符号是从H的尖点对集到H~（X（F）;IR）的映射。该符号具有以下特性：
The modular symbol has proved a very useful tool for dealing with holomorphic automorphic forms on the upper half-plane. It serves, for instance, to compute the action of Hecke operators on automorphic forms, to give the value of twisted L-functions of certain elliptic curves in terms of a finite sum of modular symbols, and to prove that divisors of degree zero supported on the cusps of a modular curve have finite order in the Jacobian of that curve. For these matters, see [1], Chaps. IV and V, and the references cited there. Here is a definition of the modular symbol. Let H be the upper half-plane, that is, the set of all complex numbers with positive imaginary part. A cusp of H is defined to be any rational number (considered as a point of the real line) or oc. The cusps belong to the boundary of H in the extended complex plane. Let SL(2,1R) act on H in the usual way. For any subgroup F of finite index in SL(2, 9 ) we can form the quotient H/F. There is a closed, compact Riemann surface X(F) such that H/F is biholomorphic to the complement of some finite set of points x 1 . . . . ,x~ in X(F). These points x i are called the cusps of X(F). Let ~?(F) denote the holomorphic differential l-forms on X(F). Integration of forms over cycles yields a canonical N-linear map from H~(X(F);IR) to the complex dual space of ~(F). The theory of Riemann surfaces tells us that this map is an isomorphism. Now choose any two rational cusps a and b on the boundary of H. Let l be any reasonable arc beginning at a, ending at b, and staying in H in the interim for instance, l could be the geodesic in H from a to b. Then one easily shows that integration over the image of l in X(F) gives a linear map from ~?(F) to I12. This linear map gives an element of HI(X(F); IR) in the canonical way described above. We denote this homology class by [a, hi. We call [ ] the modular symbol. In summary, the "classical" modular symbol is a map from the set of pairs of cusps of H into H~(X(F);IR). This symbol enjoys properties which are the