The theory of arithmetic D-modules was primarily developped by Berthelot to better understand the functoriality properties of rigid cohomology. It gives a theory of coefficients for the cohomology of quasi-projective algebraic varieties over finite fields that are stable by the six Grothendieck operations, after Kedlaya and Caro. This allows a purely p-adic proof of Deligne’s Weil II theorem, that generalized the Riemann hypothesis over finite fields to the category of coefficients for cohomology (i.e., motivic sheaves).
Pierre Berthelot: D-modules arithmetiques I, II.
Kiran S. Kedlaya, Semistable reduction for overconvergent F-isocrystals I, II, III.