On the arithmetic of Z_p-extensions

Abstract

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.