Higher composition laws and applications
Higher composition laws and applications
复制标题
高等成分定律及应用
DOI：
10.4171/022-2/13
复制
发表时间：
2006
影响因子：
0.9
通讯作者：
M. Bhargava
中科院分区：
文献类型：
--
作者：
M. Bhargava
高斯在1801年提出了一个著名的二元二次积分合成定律
forms.这一发现，被称为高斯组成，不仅有深远的影响，
初等数论也奠定了理想理论和现代代数的基础
数论即使在今天，高斯合成仍然是理解
二次域的理想类群
问题是在其他空间是否也存在类似的合成定律
这些形式可以揭示其他代数数环和域的结构。在这
文章中，我们提出了几个这样的高类似物的高斯组成，我们描述了如何
这些合成定律中的每一个都可以用适当的环中的理想类来解释，
代数整数我们还讨论了这些组成定律的几个应用，包括
科恩·伦斯特拉·马丁内特启发式方法的关键案例的解决方案，以及长期存在的问题的解决方案
有界判别式的四次域和五次域的个数的计数问题。
此外，我们描述了这些不同的组成定律之间的神秘关系
和特殊李群最后，我们讨论了未来工作的前景，并以
几个悬而未决的问题
In 1801 Gauss laid down a remarkable lawof composition on integral binary quadratic
forms. This discovery, known as Gauss composition, not only had a profound influence on
elementary number theory but also laid the foundations for ideal theory and modern algebraic
number theory. Even today, Gauss composition remains one of the best ways of understanding
ideal class groups of quadratic fields.
The question arises as to whether there might exist similar laws of composition on other spaces
of forms that could shed light on the structure of other algebraic number rings and fields. In this
article we present several such higher analogues of Gauss composition, and we describe how
each of these composition laws can be interpreted in terms of ideal classes in appropriate rings of
algebraic integers. We also discuss several applications of these composition laws, including the
resolution of a critical case of the Cohen�Lenstra�Martinet heuristics, and a solution of the longstanding
problem of counting the number of quartic and quintic fields of bounded discriminant.
In addition, we describe the mysterious relationship between these various composition laws
and the exceptional Lie groups. Finally, we discuss prospects for future work and conclude with
several open questions.