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Is there a good notion of holonomic $D$-modules on rigid analytic spaces?

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Yes. Although it is only beginning to be developed.

You probably want to start with Berthelot: D-modules arithmétiques I : Opérateurs différentiels de niveau fini and Introduction à la théorie arithmétique des D-modules, and other papers that can be found at Section 5 of the second paper I mentioned is perhaps most relevant.

There is also a recent paper of Caro which I cannot find online called 'Holonomie sans structure de Frobenius et criteres d'Holonomie' which removes the necessity of the Frobenius action from Berhelot's work. I suppose he would send you a copy of upon request.

Finally, in a piece of shameless self-advertising, Konstantin Ardakov and I recently put a preprint on the arXiv part of which seeks to find a framework to further develop the theory.

Update: Caro's paper mentioned above now seems to be available here: although you need a subscription to access it.

Further update (10th Feb 2015): Apologies for the further self-advertising but Konstantin Ardakov and I now have two further preprints on the topic of D-modules on rigid analytic spaces and There is no mention of holonomicity in either of these but there seems to be a natural definition of the notion in the framework outlined in these. Whether this definition behaves as one might hope is likely to be discussed in future work.

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Incidentally, a more specific question might yield a more specific answer. – Simon Wadsley Mar 1 '11 at 11:10

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