On generalized Fuchs theorem over relative $p$-adic polyannuli

Abstract

In this paper, we study coherent locally free (logarithmic-)$\nabla$-modules on relative $p$-adic polyannuli satisfying the Robba condition and prove several criteria for decomposition of such (logarithmic-)$\nabla$-modules. Firstly we prove the $p$-adic Fuchs theorem for absolute logarithmic $\nabla$-modules where the exponents have non-Liouville differences, which generalizes a result of Shiho. Secondly, we prove a generalized $p$-adic Fuchs theorem for relative $\nabla$-modules which are semi-constant on fibers. We also prove a generalized $p$-adic Fuchs theorem for absolute $\nabla$-modules, when the derivation on the base has some specific form. In the appendix, we prove the coincidence of two definitions of exponents due to Christol-Mebkhout and Dwork and prove that the set of exponents forms exactly one weak equivalence class.