roots of unity, wild ramification and units of norm one in local fields

Let $\mathbf{K}$ be a finite Galois extension of $\mathbf{Q}_p$ containing the $p$-th roots of unity and let $\mathfrak{p}$ denote the unique prime of $\mathbf{K}$ lying above $p$. For each $n \geq 1$ we consider the extension $\mathbf{K}_n$=$\mathbf{K}(\zeta_{p^{n+1}})$, where $\zeta_{p^{n}}$ denotes the $p^n$-th root of unity and the following objects attached to $\mathbf{K}_n$:

• $\mathfrak{p}_n$ is the unique prime in $\mathbf{K}_n$ lying above $p$;
• $U_1(\mathbf{K}_n)$ is the group of units in $\mathbf{K}_n$ which are 1 modulo $\mathfrak{p}_n$;
• $U_n'$ is the subgroup of $U_1(\mathbf{K}{_n})$ consisting of those units whose norm to $\mathbf{Q}_p$ is equal to 1.

I am trying to establish whether the norm map $N_{\mathbf{K}_n/\mathbf{K}_m}:U_n' \to U_m'$ is surjective for all $n \geq m$. In order to solve this using class field theory, it seems to me that it is relevant to establish first whether the extensions $\mathbf{K}_n/\mathbf{K}$ are wildly ramified or not. So another (elementary) question is whether $\mathbf{K}_n/\mathbf{K}$ is indeed wildly ramified or not. This post (https://mathoverflow.net/questions/136052/cyclotomic-extension-of-p-adic-fields) shows that more caution than in the case of $\mathbf{Q}_p(\zeta_{p^{n+1}})/\mathbf{Q}_p(\zeta_p)$ is required, but one of the comments also says that $\mathbf{K}_n/\mathbf{K}$ is wildly ramified for all $n$ large enough. Why is this case?

Solution 1:

Let me try to answer successively to your two questions :

1) About "wild ramification" in the cyclotomic extension $K_n/K$ (your notations). It will be convenient to introduce the cyclotomic $\mathbf Z_p$-extension $K_{cyc} = \cup K_n$, which is Galois over $K$ with group isomorphic to $(\mathbf Z_p, +)$. The inertia subgroup $I$ is necessarily of the form $p^a\mathbf Z_p$, and $a=0$ iff $K_{cyc} /K$ is totally ramified. But, in @Lubin's terms, "some unramified behavior [could be] creeping in", more precisely the intersection $K_I = K_{cyc} \cap K_{nr}$ could be larger than $K$, where $K_{nr}$ denotes the unramified $\mathbf Z_p$-extension of $K$ . In the kummerian situation here (i.e. $K$ contains $\zeta_p$) it is not hard to determine $K_I$ inductively. Notations : let $e'_K =\frac 1 {(p-1)}ord_k (1-\zeta_p)$ (this is an integer); for $x \in K^*$, define $def_K (x) =$ max $ord_K (y-1)$ for all $y \in$ the class of $x$ mod $(K^*)^p$ ("def" is for "defect"). It is easily shown that a cyclic extension of degree $p$, $L=K(\sqrt [p]x)$, is unramified iff $def_K (x) = pe'_K$ (1). If $L$ is unramified, repeat the process on replacing $K$ by $L$, and so on until you reach $K_I$.

2) Given $n$, you want to show that the norm $U'_m \to U'_n$ (your notations) is surjective for all $m \ge n$. To ease the notations, take first $K_n$ to be the base field $K$. Then the intersection $\mathcal N_K :=\cap N_{K_n/K}(K_n^*)$ is called (for obvious reasons) the subgroup of "universal (cyclotomic) norms" of $K_{cyc}/K$. Suppose for simplification that $p$ is odd (otherwise, replace $p$ by $4$).By Kummer and local CFT (2), it is known that $x\in \mathcal N_K$ iff $N_{K/\mathbf Q_p}(x) \in p^{\mathbf Z}$. Restricting to the principal units, you get that $\mathcal N_K \cap U_1 (K) = U'_K$ . This conclusion is of course valid at any level $K_n$. Finally, one can show by direct arguments of local compacity the surjectivity of the norm $\mathcal N_{K_m} \to \mathcal N_{K_n}$ for all $m\ge n$ (2). So you are done.

References. (I give the oldest ones that I know of)

(1) B. Wyman: Wildly ramified Gamma extensions, Amer. J. of Math., 91, 1 (1969). See thm.11

(2) F. Bertrandias & J-F. Payan: Gamma extensions et invariants cyclotomiques, Ann. Sci. ENS, 4, 5 (1972). See thm.B + proof of propos.2.1, and example following prop.1.3