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

Abstract

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 $\X\_K$ the sheaves of differential operators $\mathcal{D}\_{\mathfrak{X}, \infty}$ and $\wideparen{\D}\_{\mathfrak{X}\_K}$ 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 $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic $\wideparen{\D}\_{\mathfrak{X}\_K}$ which are weakly-holonomic. Moreover, we prove that a coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules is sub-holonomic if and only if the corresponding coadmissible $\Di$-module is.