Third Italian Number Theory Meeting

# 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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