Arithmetic Laplacians

TL;DR

This paper introduces an arithmetic analogue of elliptic PDEs where primes replace space coordinates and Fermat quotient operators act as derivatives, leading to new equations on algebraic groups.

Contribution

It develops a novel framework for arithmetic PDEs using primes and Fermat quotients, classifies these equations on one-dimensional groups, and studies their solutions.

Findings

Classified all arithmetic PDEs on one-dimensional groups.

Analyzed the solution spaces of these equations.

Established a new analogy between analysis and arithmetic geometry.

Abstract

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.