On generalized Fuchs theorem over relative $p$-adic polyannuli
Peiduo Wang
TL;DR
The paper develops a comprehensive framework to generalize the $p$-adic Fuchs theorem for coherent locally free $\nabla$-modules on relative $p$-adic polyannuli, including absolute and relative settings under the Robba condition. It introduces relative Quillen–Suslin freeness results, defines and analyzes $p$-adic exponents with Liouville partitions, and establishes invariance of Robba condition and exponents under pushforward by finite étale morphisms, complemented by a Galois-descent argument to handle non-algebraically closed bases. It then proves two generalized $p$-adic Fuchs theorems for Sigma-semi-constant and xi-constant modules, connecting to log-$\nabla$-modules and $\Sigma$-unipotence and enabling canonical decompositions in families over relative bases. These results extend canonical decompositions of $p$-adic differential modules to higher dimensions and families, with base-change stability and descent properties that enhance applicability to $p$-adic differential equations in the relative setting.
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.