论文标题
在$ p $ p $ extensions的$ p $ lank上
On the $p$-rank of class groups of $p$-extensions
论文作者
论文摘要
我们证明了局部全球原则，用于将全球领域的嵌入问题受到限制的影响。按照本地全球原则，对于全球字段$ k $，我们仅使用本地信息来介绍$ k $的最大$ p $ galois集团，并在满足某些GALOIS的同胞条件时，并有受限的分支。对于Galois $ p $ - extension $ k/k $，我们使用$ k $的演示结果来研究$ k $ $ k $的pro-p $ p $ galois组。然后，对于$ k = \ mathbb {q} $和$ k = \ mathbb {f} _q（t）$，带有$ p \ nmid q $，我们给出了$ k $类的$ p $ torsion组的上限和下限，这些等级是$ k $的$ p $ torsion组，这些界限仅取决于$ k $ $ k $ $ k $ $ k $ $ k/nertia sub k/k/k/k/k/k/kk/k/k的结构。最后，我们研究了$ \ mathbb {q} $的Cyclic $ p $ p $ p $ extensions的$ p $ - 级别，以及$ \ mathbb {q} $的多Quipadratic扩展名的$ 2 $ lank，用于固定的固定分支类型。
We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k=\mathbb{Q}$ and $k=\mathbb{F}_q(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-torsion group of the class group of $K$, and these bounds depend only on the structure of the Galois group and the inertia subgroups of $K/k$. Finally, we study the $p$-rank of class groups of cyclic $p$-extensions of $\mathbb{Q}$ and the $2$-rank of class groups of multiquadratic extensions of $\mathbb{Q}$, for a fixed ramification type.
