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Derivations are a basic tool in algebraic geometry and in the l~itt-Kolchin differential algebra. More recently they have been used to approach some basic conjectures in diophan-tine geometry over function fields. The purpose of the lectures is to introduce an arithmetic analogue of this circle of ideas. The substitute for a derivation on the ring of integers will be the map that sends an integer n into the integer (n-nP)/p where p is a fixed prime. One may then consider "algebraic differential equations" with respect to this operator, and study such equations on curves and abelian varieties. One ends up with arithmetic applications such as

Abstracts of Closing Sessions of the Special Year in Computational Differential Algebra and Algebraic Geometry