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Let $\mathfrak{X}$ be a formal smooth curve over a complete discretevaluation ring $\mathcal{V}$ of mixed characteristic $(0 , p)$. Let$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, \mathbb{Q}}$ be the sheaf ofcrystalline differential operators of level 0 (i.e., generated by thederivations). In this situation, Garnier proved that holonomic$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, \mathbb{Q}}$-modules as defined byBerthelot have finite length. In this article, we address this question for thesheaves $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ ofcongruence level $k$ defined by Christine Huyghe, Tobias Schmidt and MatthiasStrauch. Using the same strategy as Garnier, we prove that holonomic$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$-modules havefinite length. We finally give an application to coadmissible modules byproving that coadmissible modules with integrable connection over curves havefinite length.

$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \mathbb{Q}}$-modules holonomes sur une courbe formelle