Zeta Functions and the Distribution of Field Discriminants

PI将应用与非齐次向量空间相关的Zeta函数理论来研究场判别式的分布。关于数域判别式的研究可以追溯到1857年Hermite的工作，并且是Bhargava及其合作者最近取得突破的主题。Bhargava的工作本质上是几何性质的，PI将开发一种替代方法，使用Shintani Zeta函数和解析数论。虽然Zeta函数方法并不新鲜，但PI和他的合作者Takashi Taniguchi已经开发出一种方法，用这种方法绕过了技术困难。这导致了关于立体场的一个著名猜想的解决，以及其他结果。国际数学联合会将进一步发展这一方法来研究相关问题，包括四次和五次域的分布。高斯说：“数学是科学的女王，数论是数学的女王。”数论启发和刺激了许多数学领域的发展，也看到了实际应用，例如在密码学中。数域是代数数论的一个基本研究对象，这解释了人们对其判别式研究的浓厚兴趣。相比之下，Shintani的Zeta函数理论似乎没有得到足够的关注，特别是在日本以外的地方。PI的工作将进一步发展Shintani的理论，着眼于解决更广泛的当前感兴趣的公开问题，它还将有助于使日本数学的一个活跃领域引起美国研究人员的注意。

The PI will apply the theory of zeta functions associated to prehomogeneous vector spaces to study the distribution of field discriminants. Research on counting field discriminants dates back to 1857 work of Hermite, and is the subject of recent breakthroughs by Bhargava and his collaborators. Bhargava's work is essentially geometric in nature, and the PI will develop an alternative approach, using Shintani zeta functions and analytic number theory. Although the zeta function approach is not new, the PI and his collaborator Takashi Taniguchi have developed a method which circumvents a technical difficulty with this approach. This led to a resolution of a well-known conjecture on cubic fields, among other results. The PI will further develop this method to study related questions, including the distribution of quartic and quintic fields.Gauss said that "mathematics is the queen of the sciences and number theory is the queen of mathematics." Number theory has inspired and spurred on the development of many areas of mathematics, and has also seen practical applications, for example in cryptography. Number fields are a foundational object of study in algebraic number theory, which explains the strong interest in studying their discriminants. In contrast, Shintani's theory of zeta functions seems to have received inadequate attention, especially outside Japan. The PI's work will further develop Shintani's theory, with an eye towards solving open problems of broader current interest, and it will also help to bring an active area of Japanese mathematics to the attention of researchers in the United States.