驯服核和进一步的4秩密度

数域多重二次扩张的L函数在s=-1时的值及Tame核的拟合理想

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

固定全实数域的伽罗瓦扩张E/F，使得伽罗瓦群G的指数为2。设S是F的一个有限集，包含E中的无穷素数和所有分歧素数，S_E表示E中位于S中素数之上的素数，O_E^S表示E的S_E-整数环.然后我们比较了K_2（O_E^S）作为Z[G]-模的Fitting理想与更高的Stickelberger理想。这两个在Q[G]的极大序模上，并因此在Z[1/2][G]上，扩展到同一理想。在Birch-Tate猜想的假设下得到了Z[G]中的结果，特别是对于双二次扩张，我们计算了高Stickelberger理想的指数.在F的双二次扩张E包含F的分圆Z_2-扩张的第一层的情形下，我们找到了Fitting理想包含高Stickelberger理想的一个充分条件，并刻画了满足这个条件的F=Q的一类双二次扩张.

$K_2(\mathcal{O})$的4阶密度

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

Conner和Hurrelbrink建立了一种方法，确定了某些二次数域的驯服核$K_2(\mathcal{O})$的2-Sylow子群的结构。具体地说，这些领域的4-秩的特点是正定二元二次型。数值计算的问题，可能的密度结果的4秩驯服内核。本文成功地对这些问题给予了肯定的回答。

On the tame kernels of imaginary cyclic quartic fields with class number one

Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to dec…

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.

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。

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…

虚二次域的核群与非平凡Galois模结构

Kernel Groups and nontrivial Galois module structure of imaginary quadratic fields

设$K$是具有整数环$\Cal{O}_{K}$的代数数域，$p>2$是有理数素数，$G$是$p $阶循环群.令$Λ$表示阶$\Cal{O}_{K}[G].$令$Cl(Λ)$表示$Λ$的局部自由类群，$D(Λ)$表示核群，$Cl(Λ)$的子群由标量扩展到最大阶时变得平凡的类组成。如果$p$在$K$中是非分歧的，则$D(Λ) = T(Λ)$，其中$T(Λ)$是$Cl(Λ).$的Swan子群这产生了$D(Λ)$的上界和下界。令$R(Λ)$表示$Cl(Λ)$的子群组成的那些类可实现为整数环，$\Cal{O}_{L},$其中$L/K$是一个驯服的伽罗瓦扩展与伽罗瓦组$Gal(L/K) \cong G.$我们显示在上述假设下，$T(Λ)^{(p-1)/2} \subseteq R(Λ) \cap D(Λ) \subseteq T(Λ)$，这产生的条件时，$T(Λ)=R(Λ) \cap D(Λ)$和界上$R(Λ) \cap D(Λ)$。我们进行计算$K=\Bbb{Q}(\sqrt{-d}), d>0, d \neq 1$或$3.$在这种方式中，我们展示了素数$p$，这些领域有驯服伽罗瓦域扩展度$p$与非平凡伽罗瓦模结构。

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。