Small Connections are cyclic

TL;DR

This paper investigates the existence of cyclic vectors in p-adic differential modules within Banach algebras, extending Katz's explicit method and establishing conditions based on the smallness of the derivation matrix.

Contribution

It provides new criteria for the existence of cyclic vectors in p-adic settings, generalizing known results from the complex case.

Findings

Existence of cyclic vectors under norm constraints

Extension of Katz's method to p-adic Banach algebras

Conditions for cyclic vector existence based on derivation matrix size

Abstract

The main local invariants of a (one variable) differential module over the complex numbers are given by means of a cyclic basis. In the $p$-adic setting the existence of a cyclic vector is often unknown. We investigate the existence of such a cyclic vector in a Banach algebra. We follow the explicit method of Katz, and we prove the existence of such a cyclic vector under the assumption that the matrix of the derivation is small enough in norm.