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 $\mathfrak{X}\mathrm{\_}K$ be its generic fiber. We consider respectively over $\mathfrak{X}$ and $\text{X}\mathrm{\_}K$ the sheaves of differential operators $\mathcal{D}\mathrm{\_}\mathfrak{X},\mathrm{\infty }$ and $\text{wideparen}\text{D}\mathrm{\_}\mathfrak{X}\mathrm{\_}K$ with a rapid convergence condition. In this article, we define a characteristic variety as a subset of the cotangent space $T^{\ast }\mathfrak{X}\mathrm{\_}K$ together with a characteristic cycle for coadmissible $\text{wideparen}\text{D}\mathrm{\_}\mathfrak{X}\mathrm{\_}K$-modules. We deduce a notion of ‘‘sub-holonomicity’’ for coadmissible $\text{wideparen}\text{D}\mathrm{\_}\mathfrak{X}\mathrm{\_}K$-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic $\text{wideparen}\text{D}\mathrm{\_}\mathfrak{X}\mathrm{\_}K$ which are weakly-holonomic. Moreover, we prove that a coadmissible $\text{wideparen}\text{D}\mathrm{\_}\mathfrak{X}\mathrm{\_}K$-modules is sub-holonomic if and only if the corresponding coadmissible $\text{Di}$-module is.