Differential equations on an algebraic function field. (Équations différentielles sur un corps de fonctions algébriques.) (French) Zbl 0954.12004

Let \(\mathbb E\) denote the completion of a rational function field \(\mathbb C_p[t]\) in one variable over the completion \(\mathbb C_p\) of the algebraic closure of the \(p\)-adic field. Let \(\mathbb K\) be a finite algebraic extension of \(\mathbb E\). Both \(\mathbb E\) and \(\mathbb K\) are valued fields; let \(E\) and \(K\) denote the corresponding residual fields.
The derivation \(\frac{d}{dt}\) of \(\mathbb C_p[t]\) extends to a unique continuous derivation of \(\mathbb E\) and this latter extends uniquely to a derivation of \(\mathbb K\); we denote this derivation of \(\mathbb K\) by \(\frac{d}{dt}\) as well. A Frobenius endomorphism of a valued field like \(\mathbb E\) and \(\mathbb K\) is a field endomorphism \(\phi\) satisfying \(|u|\leq 1 \Rightarrow |u^p-\phi(u)|< 1\) for all \(u\) in the field.
The author is interested in two questions: first, whether \( |\frac{1}{n!}\frac{d^nu}{dt^n}|\leq |u|\) for all \(u \in \mathbb K\) and all \(n\); and second, whether every Frobenius endomorphism of \(\mathbb E\) can be extended to a Frobenius endomorphism of \(\mathbb K\).
The author proves that both questions have a positive answer when the residual extension \(K \supset E\) is separable.


12H25 \(p\)-adic differential equations
12J25 Non-Archimedean valued fields
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