Cycle caracéristique pour les D-modules coadmissibles sur une courbe formelle

Let $\mathfrak{X}$ be a formal smooth quasi-compact curve over a complete discrete valuation ring of mixed characteristic. We consider over $\mathfrak{X}$ the sheaves of differential operators ${\widehat{\mathcal{D}}}_{\mathfrak{X},k,\mathbb{Q}}^{(0)}$ with a congruence level $k\in \mathbb{N}$ and their projective limit ${\mathcal{D}}_{\mathfrak{X},\mathrm{\infty }}={\mathrm{}}_k{\widehat{\mathcal{D}}}_{\mathfrak{X},k,\mathbb{Q}}^{(0)}$. In this article, we define a characteristic variety for coadmissible ${\mathcal{D}}_{\mathfrak{X},\mathrm{\infty }}$-modules as a closed subset of the cotangent space $T^{\ast }\mathfrak{X}$. For this purpose, we introduce a microlocalization sheaf of ${\mathcal{D}}_{\mathfrak{X},\mathrm{\infty }}$ in which the derivation is locally invertible. We deduce a notion of “sub-holonomicity” for coadmissible ${\mathcal{D}}_{\mathfrak{X},\mathrm{\infty }}$-modules which is equivalent to being generically an integrable connection. Finally, we associate characteristic cycles to sub-holonomic modules proving that the latter are of finite length.