On generalized Fuchs theorem over $p$-adic polyannuli

TL;DR

This paper extends Kedlaya's generalized p-adic Fuchs theorem from one-dimensional cases to higher-dimensional p-adic polyannuli, providing a broader understanding of differential modules satisfying the Robba condition.

Contribution

It proves Kedlaya's generalized p-adic Fuchs theorem in higher dimensions, expanding the scope of the theorem beyond one-dimensional cases.

Findings

Established the higher-dimensional p-adic Fuchs theorem.

Extended the decomposition theorem to p-adic polyannuli.

Confirmed the robustness of Kedlaya's generalization in multiple dimensions.

Abstract

In this paper, we study finite projective differential modules on $p$-adic polyannuli satisfying the Robba condition. Christol and Mebkhout proved the decomposition theorem (the $p$-adic Fuchs theorem) of such differential modules on one dimensional $p$-adic annuli under certain non-Liouvilleness assumption and Gachet generalized it to higher dimensional cases. On the other hand, Kedlaya proved a generalization of the $p$-adic Fuchs theorem in one dimensional case. We prove Kedlaya's generalized version of $p$-adic Fuchs theorem in higher dimensional cases.