Zeta Functions and the Distribution of Field Discriminants

University South Carolina Research Foundation, Columbia SC

Investigators

Abstract

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.