Differential equations satisfied by integers and the field with one element

One can view the Fermat quotient operations on (algebraic) integers as arithmetic analogues of partial
differential operators acting on functions. This leads to a ``differential calculus over the field with one element".
When the machinery  is applied to arithmetic curves one can  obtain diophantine applications  (e.g. to Heegner points).
When the machinery is applied to rings of Witt vectors one obtains an arithmetic analogue $\Gamma$ for the integers

of the Lie groupoid G of the real line; this $\Gamma$ can be viewed as an arithmetic differential analogue of the
Galois group of the rationals over the field with one element in the same way in which the Lie groupoid
G is a differential analogue of the diffeomeorphism group of the real line. Some of the results are joint work
with B. Poonen and J. Borger.