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 $$\Psi= \Psi_p(T) \in T + T^{p-1}{\mathbb Z}[T^{p-1}]\;, $$ 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 $$\pi =\pi(\varpi) := \sum_{i\geq 0} \varpi^{(i)} pⁱ + \sum_{i<0} (\varpi^{(0)})^{p{-i}} pⁱ \in K\;. $$ 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 $\Psi_p$, 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}

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