A regulator for curves via the Heisenberg group
A regulator for curves via the Heisenberg group
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海森堡集团的曲线调节器
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发表时间：
1981
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通讯作者：
Dinakar Ramakrishnan
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文献类型：
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作者：
Dinakar Ramakrishnan
0. 在本文中，我们提出了代数曲线 K2 调节器图 P. Deligne [4] 构造的变体。该映射可被视为代数数域中单位群的经典调节器映射的模拟，由 Spencer Bloch 首先针对具有复数乘法的椭圆曲线发现 [2, 3] 。 Deligne 的方法涉及将黎曼曲面 X 上的每对可逆全纯函数 f, g 与 X 上的连接的全纯线束关联起来，满足符号属性。我们构造的新方面是使用一个“通用”线束，其连接来自复数三维海森堡群 H 的某个商 M 的 C* x C*。在考虑双对数与椭圆模函数 X 的复合变换特性时，我们自然而然地得出 M。然后 Bloch 建议使用它来定义调节器。定义 M 上联系的规范方式是在与他的对话中演变而来的。对此的主要兴趣在于 Q 曲线与 zeta 函数特殊值的猜想关系。Bloch [2] 在椭圆曲线的情况下以及 A. Beilinson [1] 在模曲线的情况下沿着这些路线取得了一些进展。对这些问题的详细阐述，从本笔记的方法开始，加上一些补充和机器计算，将在未来的某个日期与布洛赫联合发表。
0. In this note we present a variation on a construction of P. Deligne [4] of the regulator map for K2 of algebraic curves. This map, which may be viewed as an analog of the classical regulator map for the group of units in an algebraic number field, was first found for elliptic curves with complex multiplication by Spencer Bloch [2, 3] . Deligne's method involves associating, to every pair of invertible holomorphic functions ƒ, g on a Riemann surface X, a holomorphic line bundle with connection on X, satisfying symbol properties. The new aspect of our construction is the use of a 'universal' line bundle with connection on C* x C* coming from a certain quotient M of the complex three-dimensional Heisenberg group H. We were led naturally to M while considering the transformation properties of the composite of dilogarithm with the elliptic modular function X. Then Bloch suggested using it to define the regulator. The canonical way of defining the connection on M evolved in conversation with him. The main interest in this is the conjectural relationship to special values of zeta functions of curves over Q. Some progress along these lines has been made by Bloch [2] in the case of elliptic curves, and by A. Beilinson [1] in the case of modular curves. A detailed exposition of these matters, starting from the method of this note and with some additions and machine calculations, will at some future date be published jointly with Bloch.