AROUND THE NEARBY CYCLE FUNCTOR FOR ARITHMETIC $\mathscr{D}$ -MODULES.

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$ -module theory following Beilinson's philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$ -modules whose category is equivalent to the category of overconvergent isocrystals. [ABSTRACT FROM AUTHOR]