Colloquium

One can develop an arithmetic analogue of differential calculus in which functions are replaces by integer numbers and the derivative operator is replaced by a Fermat quotient operator. Then an arithmetic analogue of the Lie-Cartan geometric theory of differential equations can be developed. This arithmetic theory can then be applied to prove statements in diophantine geometry over number fields in the same way in which usual differential equations are being used to prove results in diophantine geometry over function fields.