$p$ adic Galois group and $p$ adic logarithm

algebraic-number-theoryarithmetic-geometrygalois-representationsp-adic-number-theory

In the book "Theory of p-adic Galois Representations" of Fontaine, I have two questions in Prop $3.45$ on page $60$.

Let $K\subset L$ be a finte extension of local fields of characteristic $0$. And $H_L:=\chi_L$ and $H_K:=\chi_K$ are corresponding kernel of cyclotomic character, if you don't know about this, please see page $4$ of the same book.

Then we have the following exact sequence $1\rightarrow H_K/H_L\rightarrow G_K/H_L\rightarrow G_L/H_L\rightarrow1$, then $H_K/H_L$ is finite and $G_L/H_L=\mathbb{Z}_p\times \text{finite}$, so how these information imply the center of $G_K/H_L$ is open? I know the center of a Hausdorff topological group is closed, so we only need to prove the center is of finite index?

For any element $g\in G_K$, we define $n(g)=v_plog(\chi_K(g))$, then why does there exist an integer $N$ such that if $n(g)=N$, then $\bar{g}\in$ the center of $G_K/H_L$ ? Fontaine said that this can be deduced from the fact that the center of $G_K/H_L$ is open, but I don't know why.

Thanks!

Best Answer

Maybe I'm missing something but when giving a concrete meaning to your symbols I obtain something obvious

  • $K$ is a finite extension of $\Bbb{Q}_p$. The cyclotomic character of $G_K=Gal(\overline{K}/K)$ is for $\sigma\in G_K$ the $\chi(\sigma)\in \Bbb{Z}_p^\times$ such that $\sigma(\zeta_{p^r}) = \zeta_{p^r}^{\chi(\sigma) \bmod p^r}$, its kernel in $G_K$ is $H_K=G_{K(\zeta_{p^\infty})}$

  • You mean $1\rightarrow H_K/H_L\rightarrow G_K/H_L\rightarrow G_K/H_K\rightarrow1$

  • $G_K/H_L =Gal(L(\zeta_{p^\infty})/K)$

    Let $N$ be the normal closure of $L/K$ and $F= \prod_{g\in Gal(L(\zeta_{p^\infty})/K)} g(L)\subset N$.

  • If $\sigma\in Gal(L(\zeta_{p^\infty})/F),g\in Gal(L(\zeta_{p^\infty})/K)$ then $\forall a\in L,g\sigma(a)=\sigma g(a)$ and $\forall r,g\sigma(\zeta_{p^r})=\sigma g(\zeta_{p^r})$ implies $g\sigma=\sigma g$. Thus the center of $Gal(L(\zeta_{p^\infty})/K)$ contains $Gal(L(\zeta_{p^\infty})/F)$ which has finite index.

  • $Gal(L(\zeta_{p^\infty})/F)$ is closed and has finite index in $Gal(L(\zeta_{p^\infty})/L)\subset \Bbb{Z}_p^\times$ thus $Gal(L(\zeta_{p^\infty})/F)$ contains $1+p^N \Bbb{Z}_p$ and if $g\in G_L$ and $v_plog(\chi(g))\ge N$ then $g$ is in the center

  • About the same problem when $g\in G_K$ :

    Try with $K = \Bbb{Q}_3,L=K(\zeta_5,3^{1/5})$, if $g(\zeta_5)=\zeta_5,g(3^{1/5})=\zeta_5 3^{1/5},g(\zeta_{3^r})=\zeta_{3^r}$ then looking at $\chi(g)=1$ isn't enough to see that $g$ isn't in the center

Related Question