  We develop an arithmetic analogue of linear partial differential equations in
two independent ``space-time'' variables. The spatial derivative is a Fermat
quotient operator, while the time derivative is the usual derivation. This
allows us to ``flow'' integers or, more generally, points on algebraic groups
with coordinates in rings with arithmetic flavor. In particular, we show that
elliptic curves have certain canonical ``flows'' on them that are the
arithmetic analogues of the heat and wave equations. The same is true for the
additive and the multiplicative group.
