Uniformizations of modular curves
Uniformizations of modular curves
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模曲线的均匀化
DOI：
10.4310/cag.1996.v4.n2.a2
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发表时间：
1996
影响因子：
0.7
通讯作者：
I. Kra
中科院分区：
文献类型：
--
作者：
Hershel M. Parkas;Y. Kopeliovich;I. Kra
在最近的一篇论文[7]中，本文的两位作者利用具有有理特征的θ常数理论研究了模群T的素数级主同余子群T(k)的自同构形式和函数。该研究的副产物是一个新的三次常数恒等式，以及k = 3,5情况下黎曼曲面EP/I^fc)在穿孔球上的显式映射。在b[5]中，这个理论从素数推广到任意正整数。使用的主要工具之一是将theta特征与群尖相关联。这再次导致新的恒等式和k = 4和6的覆盖图的明确构造。在第二篇论文中，大部分理论被推广到另外两个群族：G(k)和ro（A;）。[7]中的三次恒等式导致了[3]论文，其中k是任意奇正整数时的一般k幂恒等式。[7]中使用的技术也产生了Ramanujan型[4]的身份。[4]的主要定理是构造四次三项常数恒等式的一个工具，我们将在本文中解释它。关于模群的同余子群及其所定义的黎曼曲面，有许多事实是已知的。例如，我们从群表示理论中得知，对于每一个奇素数k bbbb3（见b[11]），有限群r/T(k)作为GL（^^，C）的子群有一个忠实表示；但再低的级别也不行。我们（几乎）用函数的具体希尔伯特空间来实现这些表示。我们只得到p （r(k)）在PGL（^i，C）中的表示；而图像组p（r(A;)）
In a recent paper [7], two of the authors of this paper used the theory of theta constants with rational characteristics to study automorphic forms and functions for the prime level principal congruence subgroups T(k) of the modular group T. A byproduct of this investigation was a new cubic theta constant identity, and explicit mappings of the Riemann surfaces EP/I^fc) onto punctured spheres for the cases k = 3, 5. In [5] this theory is extended from primes to arbitrary positive integers. One of the main tools used is the association of theta characteristics to the cusps of the group. This again leads to new theta constant identities and explicit construction of covering maps for the cases k = 4 and 6. In this second paper, much of the theory is extended to two other families of groups: G(k) and ro(A;). The cubic identity in [7] led to a paper [3] where a general k power identity was derived with k any odd positive integer. The techniques used in [7] also gave rise to identities of Ramanujan type [4]. The main theorem of [4] is a tool for the construction of quartic three term theta constant identities which we will, among other things, interpret in this paper. Many facts are known about the congruence subgroups of the modular group and the Riemann surfaces they define. For example, we learn from the theory of group representations that for every odd prime k > 3 (see [11]) the finite group r/T(k) admits a faithful representations as a subgroup of GL(^^,C); but no lower rank will do. We (almost) realize these representations using concrete Hilbert spaces of functions. We only get a representation p of r(k) into PGL(^i,C); however, the image group, p(r(A;))