Closed geodesics, periods and arithmetic of modular forms
Closed geodesics, periods and arithmetic of modular forms
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闭测地线、周期和模形式算术
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发表时间：
1985
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通讯作者：
S. Katok
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作者：
S. Katok
是上半平面，F是SL（2，N）的一个通过分式线性变换作用的离散余紧子群.我们证明，在这种情况下，在闭测地线的周期发挥的作用有点类似于SL（2，Z）及其同余子群的模形式的傅立叶系数。更具体地，这些周期唯一地确定模形式（定理2）。这一结果对任何第一类Fuchsian群上的尖点形式都是有效的，并且与闭测地线相关的Poincar 6级数的研究密切相关。对于任意整数k > 2和任意闭测地线[7 o]，我们定义了权为2k的特殊尖点型，称为相对Poincar ~ 6级数Ok，t~ol，并证明了它们生成整个尖点型空间S2 k（F）（定理1）。在W 3中，我们给出了一个表达式的周期的一个相对庞加莱6系列在封闭的测地线在纯几何条件下，通过相应的测地线的交叉（定理3）。将定理1和定理3应用于SL（2，1 R）的算术子群，给出了S2 k（1 N）的两个自然有理结构（定理4）. Petersson [18，19]和Hejhal [5]（g>2）研究了广义F的相对Poincar 6级数。Wolpert [23]给出了g>2时S4（F）的基。对于SL（2，TZ），Zagier [25]，Kohnen [8]，Kohnen和Zagier [9]，以及克雷默[13]研究了相对Poincar 6级数。Kudla和Millson在[143]中讨论了构造与闭测地线相关的权为2的尖点形式的相关问题。关于下列问题：
is the upper half-plane and F is a discrete cocompact subgroup of SL(2,N) acting by fractional linear transformations. We demonstrate that in this case the periods over closed geodesics play a role somewhat similar to that of Fourier coefficients of modular forms on SL(2,Z) and its congruence subgroups. More specifically, those periods uniquely determine a modular form (Theorem 2). This result is valid for cusp forms on any Fuchsian group of the first kind with or without cusps and is closely related to the study of relative Poincar6 series associated to closed geodesics. For each integer k > 2 and each closed geodesic [7o] we define special cusp forms of weight 2k, called relative Poincar6 series Ok, t~ol and prove that they generate the whole space S2k(F ) of cusp forms (Theorem 1). In w 3 we give an expression for periods of a relative Poincar6 series over closed geodesics in purely geometrical terms through the intersection of the corresponding geodesics (Theorem 3). An application of Theorems 1 and 3 to arithmetic subgroups of SL(2,1R) gives two natural rational structures o n S2k(l N) (Theorem 4). The relative Poincar6 series have been studied for general F by Petersson [18, 19] and Hejhal [5] (g>2). Wolpert [23] gives a basis of S4(F ) for g>2. For SL(2, TZ) the relative Poincar6 series have been studied by Zagier [25], Kohnen [8], Kohnen and Zagier [9], and Kramer [13]. A related problem of constructing cusp forms of weight two associated to closed geodesics has been treated by Kudla and Millson in [143. In connection with the problem of