Drinfeld module and Weil pairing over Dedekind domain of class number two

IF 1.2 3区 数学 Q1 MATHEMATICS
Chuangqiang Hu , Xiao-Min Huang
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引用次数: 0

Abstract

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by A. This domain corresponds to the projective line associated with an infinite place of degree two. To achieve the goals, we construct a pair of standard rank one Drinfeld modules whose coefficients are in the Hilbert class field of A. We demonstrate that the period lattice of the exponential functions corresponding to both modules behaves similarly to the period lattice of the Carlitz module, the standard rank one Drinfeld module defined over rational function fields. Moreover, we employ Anderson's t-motive to obtain the complete family of rank two Drinfeld modules. This family is parameterized by the invariant J=λq2+1 which effectively serves as the counterpart of the j-invariant for elliptic curves. Building upon the concepts introduced by van der Heiden, particularly with regard to rank two Drinfeld modules, we are able to reformulate the Weil pairing of Drinfeld modules of any rank using a specialized polynomial in multiple variables known as the Weil operator. As an illustrative example, we provide a detailed examination of a more explicit formula for the Weil pairing and the Weil operator of rank two Drinfeld modules over the domain A.
二类 Dedekind 域上的 Drinfeld 模块和 Weil 配对
本文的主要目的是推导出一个特定域(用 A 表示)上的秩一和秩二 Drinfeld 模块的明确公式。为了实现这些目标,我们构建了一对系数在 A 的希尔伯特类域中的标准一级德林菲尔德模块。我们证明,这两个模块对应的指数函数的周期网格与卡利茨模块(定义在有理函数域上的标准一级德林菲尔德模块)的周期网格表现类似。此外,我们利用安德森 t 动力得到了完整的二阶德林菲尔德模块族。这个族的参数是不变式 J=λq2+1,它实际上是椭圆曲线 j 不变式的对应变量。基于范德尔海登提出的概念,特别是关于秩二的 Drinfeld 模块的概念,我们能够使用称为 Weil 算子的专门多变量多项式来重新表述任意秩的 Drinfeld 模块的 Weil 配对。举例说明,我们将详细研究域 A 上的二阶 Drinfeld 模块的 Weil 配对和 Weil 算子的更明确公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.00
自引率
20.00%
发文量
133
审稿时长
6-12 weeks
期刊介绍: Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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