Ytukyg

The theory of $(varphi,Gamma)$-modules and a reciprocity law due to Cherbonnier and Colmez involve the ring $A_K$, that is described below.

The ring $A_K$:
Set $E$ to be the set of sequences $(x^{(0)},x^{(1)}, . . . )$ of elements of $C_p$ satisfying $(x^{(n+1)})^p=x^{(n)}$, with addition given by $(x + y)^{(n)} = lim_m (x^{(n+m)} + y^{(n+m)})^{p^m}$ and $(xy)^{(n)} = x^{(n)}y^{(n)}$. Then $E$ is a complete, algebraically closed field of characteristic $p$.

After clarifying terminology in the comments, I think this is what's going on here:

Let $F$ be a local field with ring of integers $O_F$ whose maximal ideal is $(p)$, and residue field $k := O_F/(p)$. Let $A$ be the set of Laurent series

$displaystyle sum_{iin Bbb Z} a_i pi^i: a_i in O_F$ such that $lim_{ito -infty} a_i=0$

(i.e. the series can be infinite on both sides, but to the left, the coefficients go to $0$). Then one can show that $A$ is a complete DVR with maximal ideal $(p)$, whose residue field $A/p$ (that's $E_K$ in your question) is of the form $k((t))$ (usual Laurent series, with finite negative part, over the field $k$). Notice that $k((t))$ is itself a complete field with a discrete valuation, whose ring of integers is $k[[t]]$ (power series over $k$). The uniformisers of this are precisely the elements of the form $bar a_1 t + sum_{ige 2}bar a_i t^i$, where $bar a_i in k$ and $bar a_1 neq 0$.

Next, let $varpi in A$ be any lift of any such uniformiser. It is of the form

$varpi = displaystyle sum_{iin Bbb Z} a_i pi^i$ with $a_i in (p)$ for $ile 0$, and $a_1in O_F^*$

(a special case being $varpi = pi$ itself).

Claim: Every element of $A$ can also be written uniquely in the form $$displaystyle sum_{iin Bbb Z} b_i varpi^i: b_i in O_F text{ such that } lim_{ito -infty} b_i=0.$$

Now in the article they define functions $l_nu: A rightarrow Bbb Z$ by

$l_nu(sum_{iin Bbb Z} a_i pi^i) := minlbrace i: p^nu text{does not divide } a_irbrace$.

but one could also define, for each $varpi$, the variant

$l_{nu, varpi}(sum_{iin Bbb Z} b_i varpi^i) :=minlbrace i: p^nu text{does not divide } b_irbrace$

(so that $l_nu$ is the choice $l_{nu, pi}$).

Now:

• For $nu ge 2$, in the examples $varpi = p^{nu-1} + pi$ we have $l_nu(varpi) = 0$ but $l_{nu, varpi}(varpi) = 1$.




• For $nu=1$ however, a little consideration shows that both $l_{nu}$ and $l_{nu, varpi}$ can actually be described as

$l_nu(a)= l_{nu, varpi}(a) = text{ the } t-text{adic valuation of } bar a in A/(p) = k((t))$

which does not depend on the choice of $varpi$.