**abstract:**
\begin{abstract}
We give an essentially self-contained proof of the fact that a certain
$p$-adic power series
$$\Psi= \Psi_{p}(T) \in T + T^{{p}-1}{\mathbb Z}[T^{{p}-1}]\;,
$$
which trivializes the addition law of the formal group of Witt
$p$-covectors is $p$-adically entire and assumes values in ${\mathbb Z}_{p$} all
over ${\mathbb Q}_{p$.} We also carefully examine its valuation and Newton
polygons. For any perfectoid field extension $(K,

\,

)$ of
$(\Q_{p,\,}_{p)$} contained in $({\mathbb C}_{p,\,}_{p)$,} and any pseudo-uniformizer
$\varpi = (\varpi^{{}(i)})_{{i} \geq 0}$ of $K^{\flat$,} we consider the element
$$\pi =\pi(\varpi) := \sum_{{i\geq} 0} \varpi^{{}(i)} p^{i} + \sum_{{i<0}}
(\varpi^{{}(0)})^{{p}^{{}-i}} p^{i} \in K\;.
$$
We use the isomorphism between the Witt and the Cartier
(hyperexponential) group over ${\mathbb Z}_{{}(p)}$, which we extend to their
$p$-divisible closures, and the properties of $\Psi_{p$,} to show that
the map $x \mapsto \exp \pi x$, a priori only defined for $v_{p}(x) >
\frac{1}{p-1} - v_{p}(\pi) $, extends to a continuous additive character
$$\Psi_{{\varpi}:} {\mathbb Q}_{p} \to 1+K^{{\circ} \circ} \;.$$
A similar character for the cyclotomic $p$-extension of ${\mathbb Q}_{p$} appears
in Colmez' work.
I will also give the numerical computation of the first coefficients of
$\Psi_{p$,} for small $p$.
\end{abstract}

Mon 21 Sep, 9:30 - 10:20, Aula Dini

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