arXiv 1607.005232016-07-02math.NTmath.AG

关于Z_p-扩张的算法

On the arithmetic of Z_p-extensions

Michiel Kosters, Daqing Wan

备注：: 30 pages, comments welcome

摘要：

本文共分三个部分。在第一部分中，我们给出了一个全面的综述的理论Artin-Schreier-Witt扩张：该理论允许一个理解的$\mathbf{Z}/p^n\mathbf{Z}$ -扩张的任何领域$K$的特征$p$通过$p$ -典型的Witt向量。令$W_n(K)$是$K$的长度为$n$的$p$ -典型Witt向量的环，令$\wp = F-\mathrm{id}: W_n(K)\longrightarrow W_n(K)$，其中$F$是Frobenius映射，$\mathrm{id}$是恒等映射。Artin-Schreier-Witt理论告诉我们，交换群$W_n(K)/\wp W_n(K)$表示$K$的$\mathbf{Z}/p^n\mathbf{Z}$ -扩张的集合。由于这一理论在文献中很难找到，我们在本文中包括了一个完整的处理。本文第二部分研究了特征$p>0$的局部域$K=k((T))$的$\mathbf{Z}_p$ -扩张，其中$k$是有限域。局部类域理论和Artin-Schreier-Witt理论给出了Schmid-Witt符号$$[\ ,\ ): W(K)/\wp W(K) \times \widehat{K^*} \to W(\mathbf{F}_p)=\mathbf{Z}_p,$$，它包含了$K$的$\mathbf{Z}_p$ -扩张的分歧信息.我们提出了一个新的简化公式的$[\ ,\ )$。这个公式允许一个计算分歧群，导体和判别式在一个简单的方法。第三部分研究了有限域上整体函数域的$\mathbf{Z}_p$ -扩张。首先，我们给出了一个公式计算亏格在这样的塔。我们表明，以前得到的下限的属增长的$\mathbf{Z}_p$ -扩展是不正确的，我们给一个尖锐的下限。我们还研究了当属的行为在一个“稳定”的方式。最后，我们找到唯一的代表$\mathbf{Z}_p$ -有理函数域$k(X)$的扩展，并计算在这样的塔的亏格。

英文摘要：

This paper contains three parts. In the first part, we give a thorough overview of the theory of Artin-Schreier-Witt extensions: this theory allows one to understand the $\mathbf{Z}/p^n\mathbf{Z}$-extensions of any field $K$ of characteristic $p$ via $p$-typical Witt vectors. Let $W_n(K)$ be the ring of $p$-typical Witt vectors of $K$ of length $n$ and let $\wp = F-\mathrm{id}: W_n(K)\longrightarrow W_n(K)$, where $F$ is the Frobenius map and $\mathrm{id}$ is the identity map. Artin-Schreier-Witt theory tells us that the abelian group $W_n(K)/\wp W_n(K)$ represents the set of $\mathbf{Z}/p^n\mathbf{Z}$-extensions of $K$. Since this theory is hard to find in literature, we have included a complete treatment in the paper. In the second part of the paper, we study $\mathbf{Z}_p$-extensions of a local field $K=k((T))$ of characteristic $p>0$ where $k$ is a finite field. Local class field theory and Artin-Schreier-Witt theory give us the Schmid-Witt symbol $$[\ ,\ ): W(K)/\wp W(K) \times \widehat{K^*} \to W(\mathbf{F}_p)=\mathbf{Z}_p,$$ which contains the ramification information of $\mathbf{Z}_p$-extensions of $K$. We present a new simplified formula for $[\ ,\ )$. This formula allows one to compute ramification groups, conductors and discriminants in an easy way. In the third part, we study $\mathbf{Z}_p$-extensions of global function fields over a finite field. First, we give a formula for computing the genus in such a tower. We show that a previously obtained lower bound for the genus growth in a $\mathbf{Z}_p$-extension is incorrect and we give a sharp lower bound. We also study when the genus behaves in a `stable' way. Finally, we find unique representatives of $\mathbf{Z}_p$-extensions of the rational function field $k(X)$, and compute the genus in such a tower.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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