Proceedings of the Japan Academy, Series A, Mathematical Sciences

A note on non-Robba $p$-adic differential equations

Said Manjra

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Let $\mathcal{M}$ be a differential module, whose coefficients are analytic elements on an open annulus $I$ ($\subset \mathbf{R}_{>0}$) in a valued field, complete and algebraically closed of inequal characteristic, and let $R(\mathcal{M}, r)$ be the radius of convergence of its solutions in the neighborhood of the generic point $t_{r}$ of absolute value $r$, with $r\in I$. Assume that $R(\mathcal{M}, r)<r$ on $I$ and, in the logarithmic coordinates, the function $r\longrightarrow R(\mathcal{M}, r)$ has only one slope on $I$. In this paper, we prove that for any $r\in I$, all the solutions of $\mathcal{M}$ in the neighborhood of $t_{r}$ are analytic and bounded in the disk $D(t_{r},R(\mathcal{M},r)^{-})$.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 3 (2011), 40-43.

First available in Project Euclid: 3 March 2011

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Zentralblatt MATH identifier

Primary: 12H25: $p$-adic differential equations [See also 11S80, 14G20]

$p$-adic differential equations Frobenius antecedent theorem


Manjra, Said. A note on non-Robba $p$-adic differential equations. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 3, 40--43. doi:10.3792/pjaa.87.40.

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