# Extending meromorphic connections to coadmissible D-cap-modules

## TL;DR

This paper explores conditions under which meromorphic connections on smooth rigid analytic varieties extend to coadmissible D-cap-modules, highlighting positive roots of b-functions as a key factor, and provides a counterexample illustrating limitations.

## Contribution

It establishes that meromorphic connections with positive roots of b-functions always extend to coadmissible modules, and presents an example where this extension fails.

## Key findings

- Connections with positive b-function roots extend to coadmissible modules
- Counterexample of a connection whose pushforward is not coadmissible
- Conditions for extending meromorphic connections to D-cap-modules

## Abstract

We investigate when a meromorphic connection on a smooth rigid analytic variety $X$ gives rise to a coadmissible $\mathcal{D}_X$-cap-module, and show that this is always the case when the roots of the corresponding $b$-functions are all of positive type. On the other hand, we also give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.