Arithmetic PDEs

We develop an arithmetic analogue of linear partial differential equations in 2 independent  "space-time" variables.
The spatial derivative is a Fermat quotient operator while the time derivative is a usual derivation. This allows
one to "flow" points in algebraic groups with coordinates in rings with arithmetic flavor. In particular we show
that elliptic curves  possess certain canonical ``arithmetic flows" which are analogous to the convection, heat,
and wave equations. Canonical convection and heat (but no wave) equations also exist on  modular curves;
the latter can be viewed as "unifying" Fourier and Serre-Tate expansions of modular forms.