Values at s=-1 of L-functions for multiquadratic extensions of number fields, and annililation of the tame kernel

Birch-Tate公式在实二次数域的Tame核中的应用

An application of Birch-Tate formula to tame kernels of real quadratic number fields

本文利用非本原Dirichlet L值的归纳论证得到实二次数域中$L(χ_F,-1)$的2-可整除性结果，并借助Birch-Tate公式确定$K_2\mathcal{O}_F$的2-准素部分。

p-adic 伪测度的亏格理论——Tame 核与阿贝尔 p-分歧

Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

本文利用 p-adic 伪测度的亏格理论，比较实阿贝尔域中 tame 核与阿贝尔 p-分歧挠群，证明 Deng--Li 猜想并给出 p-秩下界。

A note on 4-rank densities

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

On the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\t…

Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadra…

Densities of 4-ranks of $K_2(\mathcal{O})$

Conner and Hurrelbrink established a method of determining the structure of the 2-Sylow subgroup of the tame kernel $K_2(\mathcal{O})$ for certain quadratic number fields. Specifically, the 4-rank for these fields was characterized in terms of positive definite binary quadratic forms. Numerical calculations led to questions concerning possible density results of the 4-rank of tame kernels. In this paper, we succeed in giving affirmative answers to these questions.

Kernel Groups and nontrivial Galois module structure of imaginary quadratic fields

Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $Λ$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(Λ)$ denote the locally free class group of $Λ$ and $D(Λ)$ the kernel group, the subgroup of $Cl(Λ)$ consisting of classes that become trivial upon extension of scalars to the maximal order. If $p$ is unramified in $K$, then $D(Λ) = T(Λ)$, where $T(Λ)$ is the Swan subgroup of $Cl(Λ).$ This yields upper and lower bounds for $D(Λ)$. Let $R(Λ)$ denote the subgroup of $Cl(Λ)$ consisting of those classes real…

On K_2 of certain families of curves

We construct families of smooth, proper, algebraic curves in characteristic 0, of arbitrary genus g, together with g elements in the kernel of the tame symbol. We show that those elements are in general independent by a limit calculation of the regulator. Working over a number field, we show that in some of those families the elements are integral. We determine when those curves are hyperelliptic, finding, in particular, that over any number field we have non-hyperelliptic curves of all composite genera g with g independent integral elements in the kernel of the tame symbol.

The Bloch groups and special values of Dedekind zeta functions

In this paper, we compare the two definitions of Bloch group, and survey the elements in Bloch group. We confirm the Lichtenbaum conjecture on the field $Q(ζ_p)$ under the assumption the truth of the base of the Bloch group of $Q(ζ_p)$ and the relations of $K_2$ group. We also study the Lichtenbaum conjecture on non-Galois fields. By PARI, we get some equations of the zeta functions on special values and the structure of tame kernel of these fields.

Quantitative inverse Galois problem for semicommutative finite group schemes

A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme $G$, we give a lower bound on the number of connected $G$-torsors of bounded height (such as discriminant).

偶K群的Iwasawa不变量：实二次数域中Z_2扩张的环整数

Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields

本文研究了实二次数域Z_2扩张中偶K群的Iwasawa不变量，推导了渐进行式并确定了λ、μ和ν不变量的结构。

阿贝尔概形的有限扩张与交换群层的对偶性

Finite extensions of abelian schemes and duality for commutative group stacks

该研究在离散赋值环及任意基上，证明交换群概形与群层的结构定理，推导其对偶的性质，应用Fourier-Mukai论证得到导出范畴的等价，证明了Brochard对偶猜想。