The densities for 3-ranks of tame kernels of cyclic cubic number fields

Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s satisfying t-1 s 2t-1,there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2O F)=s.It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula d∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.     

Tame kernels of pure cubic fields

In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.     

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).

     

On Tame Kernels and Ideal Class Groups

It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.     

A Fast algorithm to compute cubic fields

We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to and complex cubic fields down to have been computed.

     

Some families of Mordell curves associated to cubic fields

We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X³+Y³+Z³=0.     

On Tame Kernel and Class Group in Terms of Quadratic Forms

The paper is to investigate the structure of the tame kernel K₂O_F for certain quadratic number fields F, which extends the scope of Conner and Hurrelbrink (J. Number Theory88 (2001), 263-282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the class group. Our characterizations are in terms of binary quadratic forms X²+32Y,X²+64Y²,X²+2Py²,2X²+Py²,X²−2Py²,2X²−Py². The results are very useful for numerical computations.     

The Tate conjecture for powers of ordinary cubic fourfolds over finite fields

Recently, Levin proved the Tate conjecture for ordinary cubic fourfolds over finite fields. In this paper, we prove the Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is based on properties of so-called polynomials of K3-type introduced by the author about 12 years ago.     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

A mean value theorem for cubic fields

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define and Δ(x)=S(x)−αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of

     

On the p 2-Rank of Tame Kernels of Quadratic Fields

For any odd prime p, we prove some results connecting the p²-rank of the tame kernel of a quadratic field F with the p²-rank Cl(𝒪_E1 ), where E₁ is the maximal real subfield of F(ζ_p² ).     

On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the general cubic discriminants (real or imaginary) up to in time and space , or more generally in time and space for a freely chosen positive . A variant computes the -ranks of all quadratic fields of discriminant up to with the same time complexity, but using only units of storage. As an application we obtain the first real quadratic fields with , and prove that is the smallest imaginary quadratic field with -rank equal to .

     

Decomposition of cubic graphs with cyclic connectivity 5

Let G be a cyclically 5-connected cubic graph with a 5-edge-cut separating G into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically 5-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity 4 from 1988.     

A Remark on Computing the Tame Kernel of Quadratic Imaginary Fields

In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that is bijective when . Received October 25, 1999, Accepted February 5, 2001     

关于虚二次域Tame核计算的一个注记

本文给出了一种计算数域Tame核的方法.应用到虚二次域上,证明了当 Nv>8δD6时,(？)t／:K2S'F／K2SF→k*是双射。     

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that , where is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formf_M: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G60, 16G70, 16G20.     

The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K ₂ O _F the tame kernel of F. In this article, we determine all possible values of r ₄(K ₂ O _F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r ₄(K ₂ O _F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K ₂ O _F for infinitely many imaginary quadratic number fields F.     

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.