AI 中文总结
研究虚二次域k在特定条件下的Z_p - 扩张，通过Chevalley - Herbrand公式推广及p - 进调节器性质，得出不同e值下λ(K/k)、μ(K/k)的结论，给出有效可计算结果并提供相关计算程序。
AI 中文摘要
设k为虚二次域，p > 2为素数，在k中分裂为PP'。假设k的p - 类群平凡。设δ≥0为k的基本P - 单位x的P - 费马商的P' - 赋值。设K/k为任意Z_p - 扩张，p^e为P的惯性域的次数，P'完全分歧。证明了若e≥δ，则λ(K/k)=1，μ(K/k)=0；若e < δ，可从S^P - 岩泽不变量得到一个刻画。这些结果仅使用Chevalley - Herbrand公式的推广以及不完全p - 分歧中p - 进调节器R^p_δ的非零性，提供了补充岩泽理论某些方面的有效且可计算的结果。还给出了一个pari/gp程序计算p = 3，e = 1时的δ和R^p_δ。
英文摘要
Let $k$ be an imaginary quadratic field and let $p \ne 2$ be a prime number, split in $k$ into ${\mathfrak p}{\overline {\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $δ\geq 0$ be the ${\mathfrak p}$-valuation of the ${\overline {\mathfrak p}}$-Fermat quotient of the fundamental ${\mathfrak p}$-unit $x$ of $k$. Let $K/k$ be any bi-ramified ${\mathbb{Z}}_p$-extension and let $p^e$ be the degree of the inertia field of ${\mathfrak p}$, $\overline {\mathfrak p}$ being totally ramified. We prove that if $e \geq δ$, then $λ(K/k) = 1$, $μ(K/k) = 0$; if $e<δ$ a characterization is obtained from Iwasawa invariants of the $S^{\mathfrak p}$-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a $p$-adic regulator ${\mathcal R}^{\mathfrak p}_δ$ in incomplete $p$-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ${\mathbb{Z}}_p$-extension is ''exceptional''; justifications are given. A pari/gp program computes $δ$ and ${\mathcal R}^{\mathfrak p}_δ$, for $p=3$, $e=1$.
CommentsVarious minor corrections and rewriting of certain passages; addition of an appendix on the study of the p-class group of the compositum of the Zp-extensions of k. Addition of several references
