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 $\widehat{ \mathcal{ D}}^{ (0)}_{ \mathfrak{ X}, k , \mathbb{ Q}}$ with a congruence level $k \in \mathbb{ N}$ and their projective limit $\mathcal{ D}_{ \mathfrak{ X}, \infty} = \varprojlim_k \widehat{ \mathcal{ D}}^{ (0)}_{ \mathfrak{ X}, k , \mathbb{ Q}}$. In this article, we define a characteristic variety for coadmissible $\mathcal{ D}_{ \mathfrak{ X}, \infty}$-modules as a closed subset of the cotangent space $T^*\mathfrak{ X}$. For this purpose, we introduce a microlocalization sheaf of $\mathcal{ D}_{ \mathfrak{ X}, \infty}$ in which the derivation is locally invertible. We deduce a notion of "sub-holonomicity" for coadmissible $\mathcal{ D}_{ \mathfrak{ X}, \infty}$-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.
Categories: math.AG