# The adelic zeta function associated to the space of binary cubic forms. II: Local theory.

@article{Datsovsky1986TheAZ, title={The adelic zeta function associated to the space of binary cubic forms. II: Local theory.}, author={Boris Datsovsky and David J. Wright}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={1986}, volume={1986}, pages={27 - 75} }

This series of papers aspires to fully develop the connections between the arithmetic of cubic and quadratic extensions of global fields and the properties of a zeta function associated to the natural representation of G/2 in the space of binary cubic forms. This zeta function was first studied by Shintani. In Part I of this series, we introduced an adelic Version of Shintani's theory of Dirichlet series associated with the space of binary cubic forms. In addition, this theory was situated over…

## 69 Citations

### The adelic zeta function associated with the space of binary cubic forms with coefficients in a function field

- Mathematics
- 1987

In this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms, defined over a function field. We establish its rationality, find its poles and…

### Density of discriminants of cubic extensions.

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In this paper, we shall generalize the results of Davenport and Heilbronn in [11] on densities of discriminants of cubic fields and the mean 3-class-number of quadratic fields to the case when the…

### A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms

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- 1997

About twenty-five years ago T. Shintani defined and studied four Dirichlet series whose coefficients are class numbers of integral binary cubic forms by using the theory of prehomogeneous vector…

### Fourier coefficients of Eisenstein series of the exceptional group of type G2

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- 1997

K(s) F (s) of the two Dedekind zeta functions is an entire function in the complex variable s. From the point of view of the trace formula, the above basic question is expected to be equivalent to a…

### Binary forms and orders of algebraic number fields

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- 1989

0.0. The class numbers of binary forms of degree greater than three has been scarcely studied. It seems that the finiteness of class numbers proved by Birch and Merr iman is the only general result.…

### A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field

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- 2002

Let k be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space of binary quadratic…

### Completion of local zeta functions associated with a certain class of homogeneous cones

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- 2020

It is well known that the Riemann zeta function can be completed to the Riemann xi function $\xi(s)$ in the sense that its functional equation has a higher symmetric form $\xi(1-s)=\xi(s)$. In the…

### THE SECONDARY TERM IN THE COUNTING FUNCTION FOR CUBIC FIELDS

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- 2010

Work in progress, September 21, 2010. We prove asymptotic formulas for the number of cubic fields of positive or negative discriminant less than X. These formulas involve main terms of X and X…

### Secondary terms in counting functions for cubic fields

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- 2011

We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a…

### Summary of Research

- Mathematics
- 2012

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 [Sat70, SS74] may be used to study the…

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