Computational methods for abelian varieties over number fields with complex multiplication

复数乘法数域上阿贝尔簇的计算方法

• 批准号: 239459353
• 负责人: Professor Dr. Andreas Stein
• 金额: --
• 依托单位: Institut für Mathematik (IFM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/239459353?language=en
• 关键词: Computational; methods; abelian; varieties; over

Computational arithmetic geometry and its related areas generalize considerably both importance and techniques of classical computational number theory. The introduction of algebraic curves to cryptography emphasized that arithmetic geometry possesses not only an extremely interesting theoretical value that is rapidly growing; it also provides us with an exciting computational side. Many questions are still unsolved and need more investigation. This project is concerned with explicit algorithmic problems in the arithmetic of abelian varieties over number fields with complex multiplication. There are a multitude of recent numerical results for dimension one abelian varieties with complex multiplication, also motivated from applications to cryptography. We wish to solve various problems for low dimensional abelian varieties with complex multiplication, both algorithmically and theoretically. This includes the following topics: Torsion points of abelian varieties with CM over finite fields, small invariants and explicit class field theory, construction of curves for pairing-based cryptography, investigation of the Igusa-invariants, efficient algorithms and explicit implementation of new methods.

计算算术几何及其相关领域在很大程度上概括了经典计算数论的重要性和技术。将代数曲线引入密码学强调了算术几何不仅具有非常有趣的理论价值，而且还为我们提供了令人兴奋的计算方面。许多问题仍然没有解决，需要更多的调查。本课题研究数域上交换簇的复数乘法运算中的显式算法问题。最近有大量的数值结果，一维阿贝尔簇与复数乘法，也从应用到密码学的动机。我们希望解决各种问题的低维阿贝尔品种与复杂的乘法，算法和理论。这包括以下主题：扭点的阿贝尔品种与CM在有限领域，小不变量和明确的类域理论，建设曲线对为基础的密码学，调查的Igusa不变量，有效的算法和明确实施的新方法。