Zeta Functions and Number Fields

My main research has been a long exploration into how the properties of a zeta function defined by Mikio Sato and greatly developed by Takuro Shintani [#!SatoM1970!#,#!SatoShintani1974!#] may be used to study the discriminants and class numbers of algebraic number fields of low degree. These zeta functions are associated to certain representations of algebraic groups which were called prehomogeneous by Sato. My thesis work and first several papers concerned the prehomogeneous space of binary cubic forms which was first thoroughly explored by Shintani [#!Shintani1972!#] over the base field $\mathbb{Q}$. I used Weil's framework for zeta functions and Eisenstein series presented in [#!Weil1964!#,#!Weil1965!#] to generalize Shintani's work to the case where the base field is an arbitrary number field $k$ and then streamlined the proof based on the zeta function of the density of discriminants of cubic extensions of an arbitrary number field and the mean-value of the two-class-number of quadratic extensions of a number field. My friend and younger graduate student Boris Datskovsky joined me on this work towards the end in the papers [#!Wright1982!#,#!Wright1985!#,#!DatskovskyWright1986!#,#!DatskovskyWright1988!#,#!Wright1987!#].

There are several aspects of this work that remain active today. The original theorem of Davenport and Heilbronn which Datskovsky and I reproved and generalized says that the number $N_3(X)$ of cubic fields $k/\mathbb{Q}$ with discriminant not greater than $X$ in absolute value behaves like

$$N_3(X) = \frac{1}{\zeta(3)} X + o(X).$$

This comes out of the zeta function theory in much the same way that the Prime Number Theorem emerged from the Riemann zeta function. However, Shintani's zeta function theory offered much more than this. The zeta function is a function of a complex variable which has a simple pole at $s=1$ and this ultimately leads to the Davenport-Heilbronn theorem. However, there is a second pole at $s=5/6$ which suggests a precise error term

$$N_3(X) = \frac{1}{\zeta(3)} X + c_2 X^{5/6} + O(X^{(1/2)}).$$

for an explicit but complicated constant $c_2$. Datskovsky and I gave the formulas that can be used to calculate $c_2$; however, we left the value of $c_2$ unstated in our paper [#!DatskovskyWright1988!#]. When large tables of cubic discriminants became available, I checked the error term and found incredible accuracy to four decimal places up to discriminant $10^7$, and lectured about this in 1995 at the Institute for Advanced Study. This was ultimately published by Roberts in [#!Roberts2001!#], but there is a great deal more to say about this second term. In particular, the Gauss sum calculated in [#!Wright1987!#] is a crucial part of the second term story. The proof of the error term conjectured above would be a major accomplishment, and although I have no good idea on how to do this there are gifted young minds at work on it. Very recently, I was drawn back into these distribution questions by Kable, who observed that my early work in this area can answer questions on the distribution of the Steinitz ideal class for number fields of low degree. This is presented in the recent paper [#!KableWright2006!#].

Another small aspect of my early work on cubic fields that developed through the 90's lies in the distinction between fields of positive and negative discriminant. While there is a very simple relation between quadratic fields of negative and positive discriminant, namely, $\mathbb{Q}(\sqrt{D})$ corresponds to $\mathbb{Q}(\sqrt{-D})$ for all squarefree positive integers $D$, there was no known relationship between the cubic fields of positive and negative discriminants. However, Datskovsky and I pointed out in [#!DatskovskyWright1986!#] that the two corresponding zeta functions had a matrix functional equation that had an extremely simple diagonalization. After a brilliant numerical calculation correcting ancient tables of cubic fields, Ohno [#!Ohno1997!#] found that the diagonalized zeta functions conjecturally satisfy an identity that gives a correspondence between cubic fields of negative discriminants and those of positive discriminants. This identity was ultimately proved in a technical masterpiece by Nakagawa [#!Nakagawa1998!#]. Nakagawa's theorem deserves much greater study.

In 1985, I was instructed by Lusztig on the basic properties of other exotic prehomogeneous representations, including the space of pairs of ternary quadratic forms and the space of 4-tuples of alternating forms of degree 5. These, as well as the space of binary cubic forms, belong to the prehomogeneous representations of parabolic type, defined by Vinberg [#!Vinberg1975!#] in a general way from a simple Lie group and a chosen maximal parabolic subgroup. It took some time, but eventually Yukie and I fashioned in [#!WrightYukie1992!#] the basic ingredients that showed that these other representations would eventually lead to the density of discriminants of algebraic number fields of degree 4 and 5. To follow the path of the zeta function seemed to require a massive effort as the algebraic structure of these representations is much more involved than that of the space of binary cubic forms. Yukie launched on a huge personal effort to create the analytic theory of the zeta function of the space of pairs of ternary quadratic forms which was published in the book [#!Yukie1993!#]. Even with that, the technical aspects of deriving the density of quartic discriminants were daunting. At that time, it was clear to me that leadership in that area was passing to Yukie and possibly to the brilliant young student Anthony Kable, who had come to Stillwater from Oxford. In 1994 I resolved to finish the project I had worked on with Mumford on Kleinian groups.

A good problem attracts good people, though, and the world did not want to wait for the generalization of Davenport-Heilbronn to quartic fields. Manjul Bhargava took up the problem and discovered many new algebraic structures and ultimately gave the density of discriminants of totally real quartic fields in his doctoral thesis, again as an explicit constant times the bound $X$ on the size of the discriminant. Yukie has also given quartic discriminant density results, and it is likely the complete theorem about the main term may soon be known. The structure of the zeta function suggests there is a large error term of the order of $X^{5/6}$ as well. Once again, I seem to be drawn back into this subject as Lenstra and Edixhoven have organized a conference around the work of Bhargava in June, 2006, and have invited me to come and lecture as well.

Along the way of these works, my papers [#!Wright1989b!#,#!Wright1989!#] resolve certain general questions on discriminant densities of certain classes of algebraic number fields with abelian Galois group. Since that time, there has been some exciting work of Malle and Klüners [#!Malle2002!#,#!Malle2004!#] proposing conjectural discriminant density formulas for arbitrary Galois groups, and proving them in the case of nilpotent groups [#!KluenersMalle2004!#]. It is still a great mystery, though, how the general results for fields of degree less than six may be generalized to higher degree number fields.