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

TL;DR

This paper extends the $p$-adic Fuchs theorem to relative $p$-adic polyannuli, providing criteria for module decomposition and analyzing exponents, thus advancing the understanding of $p$-adic differential equations in a relative setting.

Contribution

It generalizes the $p$-adic Fuchs theorem to relative $p$-adic polyannuli and establishes criteria for module decomposition with new conditions on exponents.

Findings

Proved $p$-adic Fuchs theorem for absolute logarithmic $ abla$-modules with non-Liouville exponents.

Established a generalized $p$-adic Fuchs theorem for relative $ abla$-modules semi-constant on fibers.

Confirmed the coincidence of two definitions of exponents and their unique weak equivalence class.

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.