  We develop an arithmetic analogue of elliptic partial differential equations.
The role of the space coordinates is played by a family of primes, and that of
the space derivatives along the various primes are played by corresponding
Fermat quotient operators subjected to certain commutation relations. This
leads to arithmetic linear partial differential equations on algebraic groups
that are analogues of certain operators in analysis constructed from
Laplacians. We classify all such equations on one dimensional groups, and
analyze their spaces of solutions.
