A $p$-adically entire function with integral values on ${\mathbb Q}p$ and additive characters of perfectoid fields.
speaker: Francesco Baldassarri (Università di Padova)abstract:
\begin{abstract}
We give an essentially self-contained proof of the fact that a certain
$p$-adic power series
$$\Psi= \Psip(T) \in T + T{p-1}{\mathbb Z}[T{p-1}]\;,
$$
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
$(\Qp,
\,
p)$ contained in $({\mathbb C}p,
\,
p)$, and any pseudo-uniformizer
$\varpi = (\varpi{(i)}){i \geq 0}$ of $K\flat$, we consider the element
$$\pi =\pi(\varpi) := \sum{i\geq 0} \varpi{(i)} pi + \sum{i<0}
(\varpi{(0)}){p{-i}} pi \in K\;.
$$
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 $\Psip$, to show that
the map $x \mapsto \exp \pi x$, a priori only defined for $vp(x) >
\frac{1}{p-1} - vp(\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
$\Psip$, for small $p$.
\end{abstract}
timetable:
Mon 21 Sep, 9:30 - 10:20, Aula Dini
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