Arithmetic partial differential equations, I

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 carry certain canonical “arithmetic flows” that are arithmetic analogues of the convection, heat, and wave equations, respectively. The same is true for the additive and the multiplicative group.