Arithmetic differential equations

Arithmetic differential equations are analogues of (usual) ordinary differential equations in which functions are replaced by integer numbers and the derivative operator (with respect to "time") is replaced by a Fermat quotient operator with respect to a given prime. It is well known that usual differential equations can be applied to diophantine problems over function fields. In a similar way arithmetic differential equations can be sometimes applied to diophantine problems over number fields. In particular we explain an application of arithmetic differential equations to the study of linear dependence relations among special points on elliptic curves arising from modular curves. The latter is recent joint work with B. Poonen. Other applications of arithmetic differential equations include the construction of quotients of certain algebraic curves by actions of certain correspondences; these quotients do not exist in usual algebraic geometry but, rather, in a geometry obtained from algebraic geometry by replacing algebraic equations with arithmetic differential equations.