Arithmetic Laplacians

Details Arithmetic Laplacians Arithmetic Laplacians

2008-05-02

arxiv.org/abs/0805.0256v1

Mathematics

Number Theory

Scientific paper

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.

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