论文标题
代数标准和p级基团在分支循环p-Extensions中的p级基团
Algebraic norm and capitulation of p-class groups in ramified cyclic p-extensions
论文作者
论文摘要
我们研究了$ p $ cluss Group $ h_k $ a a $ k $ k $ in Attally Ramified Cyclic p-Extensions $ l/k $ of度量$ p^n $的现象。使用代数规范$ν_{l/k} $的基本属性，我们表明投降的内核与通过其指数$ p^e（l）$的$ h_l $结构的“复杂性”有关$ h_l $ as $ z_p [gal（l/k）] $ - 模块。我们证明，如果$ e（l）\ in [1，n-s（l）] $ in [p^s（l），p^s（l），p^（s（l）+1）-1] $ for $ s（l）\ in [0，n-1] $（n-1] $（theorem 1.1）;这改善了“稳定性”的情况，$ \＃h_l = \＃h_k $（即$ m（l）= 1 $，$ s（l）= 0 $，$ e（l）= e（k）$）（定理1.2）。给出了数字示例（带有Pari程序），显示出$ H_K $在$ L $中的换倒数，以最简单的Abelian $ P $ - extensions $ l <k（μ__\ ell $），Primes $ \ ell = 1 $（mod $ 2p^n $）与$ P = 2 $ p = 2 $ and PREAL P = 2 $ and uce ucudratic fields和$ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3 $ p = 3。提出了一些关于这种$ \ ell $的非零密度存在的猜想（猜想1.4，2.4）。检查了其他算术不变性的投降属性。
We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $ν_{L/K}$, we show that the kernel of capitulation is in relation with the "complexity" of the structure of $H_L$ measured via its exponent $p^e(L)$ and the length $m(L)$ of the usual filtration $\{H_L^i\}_{i \ge 0}$ associated to $H_L$ as $Z_p[Gal(L/K)]$-module. We prove that a sufficient condition of capitulation is given by $e(L) \in [1, N-s(L)]$ if $m(L) \in [p^s(L), p^(s(L)+1)-1]$ for $s(L) \in [0, N-1]$ (Theorem 1.1); this improves the case of "stability" $\#H_L = \#H_K$ (i.e., $m(L) = 1$, $s(L)=0$, $e(L) = e(K)$) (Theorem 1.2). Numerical examples (with PARI programs) showing most often capitulation of $H_K$ in $L$, are given, taking the simplest abelian $p$-extensions $L < K(μ_\ell$), with primes $\ell=1$ (mod $2p^N$) over cubic fields with $p=2$ and real quadratic fields with $p=3$. Some conjectures on the existence of non-zero densities of such $\ell$'s are proposed (Conjectures 1.4, 2.4). Capitulation property of other arithmetic invariants is examined.
