We develop a dimension theory for coadmissible $\widehat {\mathcal {D}}$-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic $\mathcal {D}$-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $\underline {H}^{i}_Z(\mathcal {M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal {M}$ on $X$.

Let 𝒱 be a mixed characteristic complete discrete valuation ring with perfect residue field k. We solve Berthelot’s conjectures on the stability of the holonomicity over smooth projective formal 𝒱-schemes. Then we build a category of F-complexes of arithmetic 𝒟-modules over quasi-projective k-varieties with bounded and holonomic cohomology. We obtain its stability under Grothendieck’s six operations.

Let 𝒱 be a complete discrete valuation ring of unequal characteristic with perfect residue field. Let be a separated smooth formal 𝒱-scheme, 𝒵 be a normal crossing divisor of , be the induced formal log-scheme over 𝒱 and be the canonical morphism. Let X and Z be the special fibers of and 𝒵, T be a divisor of X and ℰ be a log-isocrystal on overconvergent along T, that is, a coherent left -module, locally projective of finite type over . We check the relative duality isomorphism: . We prove the isomorphism , which implies their holonomicity as -modules. We obtain the canonical morphism ρℰ : uT,+(ℰ)→ℰ(†Z). When ℰ is moreover an isocrystal on overconvergent along T, we prove that ρℰ is an isomorphism.

We prove that holonomic arithmetical ${\mathcal{D}}$-modules over curves have finite fibers. We also define L-functions associated with arithmetical ${\mathcal{D}}$-modules and, when the scheme is a curve, we show a cohomological formula. Furthermore, we prove that F-isocrystals over curves are holonomic.