Computational methods for abelian varieties over number fields with complex multiplication

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，而且正在迅速增长；它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。这个项目是关于复数乘数域上的阿贝尔变差运算的显式算法问题。最近有大量关于具有复乘法的一维阿贝尔变种的数值结果，这也是出于在密码学中的应用。我们希望从算法和理论上解决具有复乘法的低维阿贝尔变种的各种问题。这包括以下主题：有限域上具有CM的交换簇的扭点、小不变量和显式类域理论、基于配对的密码学的曲线构造、Igusa不变量的研究、高效算法和新方法的显式实现。

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.