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.
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Last modified: April 11 2016 - 18:14:43