Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves

Let $\mathfrak{X}$ be a formal smooth curve over a complete discrete valuation ring of mixed characteristic and let $\mathfrak{X}_K$ be its generic fiber.

We consider respectively over $\mathfrak{X}$ and $\mathfrak{X}_K$ the sheaves of differential operators $\mathcal{D}_{\mathfrak{X}, \infty}$ and Dcap with a rapid convergence condition.

In this article, we define a characteristic variety as a subset of the cotangent space $T^*\mathfrak{X}_K$ together with a characteristic cycle for coadmissible Dcap-modules.

We deduce a notion of ''sub-holonomicity'' for coadmissible Dcap-modules which is equivalent to being generically an integrable connection.

When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic Dcap-modules which are weakly-holonomic.

Moreover, we prove that a coadmissible Dcap-modules is sub-holonomic if and only if the corresponding coadmissible $\mathcal{D}_{\mathfrak{X}, \infty}$-module is.